Developing Deep Knowledge in Middle School Mathematics: A Textbook for Teaching in the Age of Technology 3030685632, 9783030685638

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Springer Texts in Education

Sergei Abramovich Michael L. Connell

Developing Deep Knowledge in Middle School Mathematics A Textbook for Teaching in the Age of Technology

Springer Texts in Education

Springer Texts in Education delivers high-quality instructional content for graduates and advanced graduates in all areas of Education and Educational Research. The textbook series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, accompanied by pedagogical materials to serve students such as practical examples, exercises, case studies etc. Textbooks published in the Springer Texts in Education series are addressed to graduate and advanced graduate students, but also to researchers as important resources for their education, knowledge and teaching. Please contact Natalie Rieborn at textbooks. [email protected] or your regular editorial contact person for queries or to submit your book proposal.

More information about this series at http://www.springer.com/series/13812

Sergei Abramovich Michael L. Connell •

Developing Deep Knowledge in Middle School Mathematics A Textbook for Teaching in the Age of Technology

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Sergei Abramovich Department of Elementary Education State University of New York at Potsdam Potsdam, NY, USA

Michael L. Connell Department of Urban Education University of Houston Downtown Houston, TX, USA

ISSN 2366-7672 ISSN 2366-7680 (electronic) Springer Texts in Education ISBN 978-3-030-68563-8 ISBN 978-3-030-68564-5 (eBook) https://doi.org/10.1007/978-3-030-68564-5 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This is a textbook to be used by mathematical instructors of prospective teachers of middle school mathematics. Its content reflects on the authors’ experience in offering various technology-enhanced mathematics education courses to prospective teachers of the United States and Canada. In particular, the textbook can support one or more of 24 semester-hours courses recommended by the Conference Board of the Mathematical Sciences (2012) for mathematical preparation of middle grades teachers. It can be used in mathematics education doctoral programs by those studying mathematics teaching at the middle school level from a theoretical perspective. The textbook integrates grade appropriate content with the worldwide recommendations for teaching middle school mathematics to allow for international readership. Consequently, the book emphasizes inherent connections of mathematics to real life as many mathematical concepts and procedures stem from common sense, something that middle school students intuitively possess. In order to turn the common-sense intuition into formal knowledge, one needs to learn mathematics. For example, when students cut across the grass, rather than walk along the pavement, to get to the school building faster, they display intuitive knowledge of the triangle inequality (Chap. 9, Sect. 9.9.2.4). By using a pencil and graph paper, a student can easily draw a segment connecting any two points on the grid. However, to “teach” a computer graphing software such as the Graphing Calculator (Avitzur, 2011) to construct such a segment (or more creative images) requires skills in digital fabrication (Gershenfeld, 2005; Abramovich & Connell, 2015) through the use of inequalities (Chap. 9, Sect. 9.9.2.2). By the same token, when tossing a fair coin, it is intuitively clear that chances to have either head or tail should be the same. Yet, to “teach” a spreadsheet to simulate a large number of tosses of the coin in order to recognize almost equal chances of either outcome requires certain knowledge of Excel programming code (Chap. 12, Sect. 12.2). Making such connections of real-life situations to mathematics and turning intuitive (as well as counterintuitive) understanding into conceptual knowledge are among the main features of the textbook. v

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The textbook emphasizes the importance of prospective teachers of middle school mathematics deep understanding of the subject matter. The need for deep understanding of mathematics is at least twofold. First, the modern-day students have been actively seeking answers to conceptual questions about procedural knowledge they gain through the study of school mathematics. Second, as mathematics educators worldwide agree, “teachers must know the mathematical content they are responsible for teaching not only from a more advanced perspective but beyond the level they are assigned to teach” (Baumert et al., 2010, p. 137). A dichotomy of perspectives on the relationship between conceptual and procedural knowledge calls for a brief clarification of these terms used throughout the textbook and their connection to technology. In one of the first publications concerned with the interplay between the two types of knowledge, Gelman and Meck (1983) argued that basic principles of counting have to be developed first in order to use counting as a procedure, although the development of those principles does not imply that one has full conceptual understanding of ideas associated with counting at the level of abstraction (e.g., comprehending the infinity of the set of number names). Freudenthal (1983), one of the leading mathematics educators of the second part of the twentieth century, apparently favoring learners’ conceptual understanding in mathematics versus procedural competence, warned against nondiscriminatory use of procedures without conscious thought as “sources of insight can be clogged by automatisms” (p. 469). The advent of technology into the mathematics classroom has given strong impetus to the theme. In particular, Kaput (1992) made an argument for the need of disciplined inquiry into “the relation between procedural and conceptual knowledge, especially when the exercise of procedural knowledge is supplanted by (rather than supplemented by) machines” (p. 549). Although Peschek and Schneider (2001) put forth the notion of outsourcing procedural skills to new digital technology, programming details of spreadsheet-based computational environments used in this textbook clearly demonstrate that computational competence stems from conceptual knowledge of mathematics. The textbook integrates simple science and engineering concepts as a source of mathematical activities (Chap. 2, Sects. 2.3.1–2.3.3). It strongly emphasizes the use of technology, both physical (manipulatives) and digital (commonly available software tools). As an extension of students’ intuition and curiosity, the textbook addresses the issue of creativity which is considered being important aspect of education in the digital age in general (Henriksen et al., 2018), and in mathematics education, in particular (Freiman & Tassell, 2018). Indeed, one of the domains where creativity, stemming from curiosity, can be supported by the use of technology is digital fabrication. The authors also found it important to focus on historical aspects of mathematics education over a 40 centuries span. The textbook consists of twelve chapters and appendix. Each chapter includes an activity set comprised of problems to solve. The purpose of the appendix is to provide details of the programming of spreadsheet-based environments used throughout the chapters. Chapter 1 (written by SA) titled Teaching Middle School Mathematics: Standards, Recommendations and Teacher Candidates’ Perspectives

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demonstrates the relevance of the content of the book to pedagogical ideas of the standards for teaching mathematics in middle schools around the world including such countries as Australia, Canada, Chile, China, England, Japan, Korea, Singapore and the United States. Throughout the chapter, solicited comments by middle school teacher candidates about their learning to teach mathematics are shared. The chapter emphasizes the importance of mathematical and pedagogical content knowledge for teachers that includes the appreciation of conceptual connections among different ideas in middle school mathematics and the recognition that the diversity of teaching methods does benefit struggling students. Major contributors to the development of mathematical and pedagogical ideas are mentioned as appropriate. The chapter introduces the reader to several topics of other chapters of the textbook and discusses these ideas in the context of comments by future teachers of mathematics in middle grades. Different ways of learning to deal with students’ misconceptions are discussed. The chapter concludes with exploring issues specific for teaching in the digital era and the list of computer programs and on-line sources of information used throughout the book is provided. Chapter 2 (written by MLC) titled Modeling Mathematics in the Digital Era begins by considering mathematical modeling from two perspectives. The first perspective is that of modeling mathematics itself through hands on experiences with actual objects. The second perspective is that of modeling real world events using the principles of mathematics. These paired perspectives are used throughout the chapter to demonstrate teaching strategies leading to the acquisition of flexible middle school content understandings. An example drawn from concepts surrounding the area of a rectangle is developed starting from simple counting and advancing to a generalized fact-family for multiplication and division. This is followed by a section in which the importance of referent selection is addressed. A comparison of two graphic models for developing fraction concepts is provided demonstrating the instructional decisions which must be made when developing a conceptual model with middle school students. As was done with the rectangle example, this discussion starts with sketching and counting and proceeds to generalized methods of representing and operating upon fraction. The use of mathematical models to explore real world situations is then explored via two technology immune-technology enabled (Abramovich, 2014a) problems together with the mediational power creation of data-tables brings to the process. A method for modeling systems of equations is then developed using the Flagpole Factory. The chapter concludes with a discussion of problem solving methods drawing from Pólya (1957) leading to the introduction of the Problem Solving Board, a graphic organizer designed for the explicit inclusion of mathematical modeling as used throughout the book. Chapter 3 (written by MLC) titled Reasoning and Proof begins by revisiting connections between plausible reasoning and emerging formal logic strategies of middle school students. In particular, the importance of recognizing that students are capable of advanced creativity and reasoning that border on formal proof when utilizing developmentally appropriate materials to base their thinking on. A division by 9 example is then presented demonstrating how plausible reasoning based upon

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physical manipulatives and concrete actions can be extended to include topics far beyond that typically encountered by middle school students including infinite processes, generalized bases and establishing a rigorous justification upon which to build later more formalized proofs. Based upon elements of classical rhetoric, a framework supporting the classroom development of creativity, critical thinking, and plausible reasoning is then presented: Commonplaces Authority Reasoning Experience, or CARE. The CARE framework is then demonstrated using a rotating squares example leading to multiple student insights concerning the Pythagorean theorem which are eventually formalized to the point of being testable by a spreadsheet. This example is followed by a review of plausible reasoning strategies and methods to develop such thinking in middle school students. Finally, the chapter ends by presenting a review of some basic elements of propositional logic together with summary charts illustrating each of these elements. Chapter 4 (written by SA) titled Modeling Mathematics with Fractions begins with the introduction of two major real-life contextual motivations for teaching and learning fractions: the part-whole context and dividend-divisor context. It continues with comparing the arithmetic of whole numbers to the arithmetic of fractions (including unit, proper, improper, and mixed fractions). Tape diagrams are used to illustrate how two-dimensional models can explain conceptual meaning of procedural rules for multiplying and dividing fractions. Application of fractions to solving arithmetical word problems is demonstrated through the use of images of objects involved in the problems. Historical aspects of the use of images in mathematics teaching going back to the ancient Greek philosophy are considered. The meaning of the invert and multiply rule when (apparently more complicated) division is replaced by (seemingly less complicated) multiplication is explained as the change of referent unit. Special attention to unit fractions as benchmark fractions is given and representation of a unit fraction as a sum of two and three like fractions is discussed. This discussion leads to another historical aspect of mathematics—Egyptian fractions. The use of Wolfram Alpha in generating Egyptian fraction representation of a common fraction through the Greedy algorithm is demonstrated. The emergence of decimal fractions through the process of long division of denominator into numerator of a common fractions is illustrated for different denominators. Continued fractions are connected to the Euclidean algorithm. Representation of rational numbers in non-decimal bases is considered. The chapter concludes with demonstrating how fractions can be derived from quadratic equations with integer coefficients through developing iterative formulas for convergents to their roots. Chapter 5 (written by MLC) titled Decimal and Percent Representations of Rational Numbers begins by positioning decimal and percent as natural extensions of elementary base-ten numeracy. This is followed by reviewing key features of foundational numeracy essential for middle school students to establish these connections including the organizing principles of the base-ten, or decimal notational system. In this review, a counter-based model of number representation will be developed as a means to explore basic numerical fluency and base-ten concepts. The counter-based model will then be used to illustrate organizing principles

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present in Diene’s base-ten blocks, a base-ten representational system familiar to many middle school students. Following this, the transition from numerical fluency to the base-ten notational system is presented including the roles played by base, exponent, and place value. After these foundations of number representation and notation are reviewed, decimal and percent are developed regarding their relative position within the larger decimal notational system. As part of this development, connections are made between fractions and decimals including conversions among decimal, percent, and common fractions. Following this, problems commonly experienced by middle school students are presented including calculating winning percentages, part-whole relationships expressed as a percent, and price discounts. Drawing upon concepts first presented in Chap. 4, further explorations into repeating decimals and their fractional equivalents are presented. A classroom example concerning the comparison of the number 1 and a repeating decimal : 9 is described illustrating how student creativity and reasoning is enabled via development of numerical fluency for fraction and decimal. Taking this as a starting point, conversions between repeating decimals and common fractions are developed. Finally, considerations when extending the basic arithmetic operations into decimal and percent are discussed. Chapter 6 (written by MLC) titled Ratio and Proportion begins with proportional reasoning in general as leading to the concept of ratio. Ratios are presented as a comparison of two quantities or measures based upon multiplicative comparison between two quantities that covary. Examples including part-part and part-whole ratios are presented including counter-intuitive results with part-whole ratios. Building upon ratio, proportions are presented as being an equation with a ratio on each side and a general proportion is used to illustrate equivalent ratios. The language of verbal analogy is presented as a mechanism to allow students to “read” a proportion and enhance their proportional reasoning. Problems are then presented which draw upon this “reading” of proportions to aid in their solution and interpreting the results including shared work, discount, unit cost, interest, and recipe conversion. These problems are followed by revisiting through a proportional lens of the classic Gulliver’s travels and the film “Honey, I shrunk the kids”. An example from antiquity, Thales’ measurement of the great pyramid at Giza, is then presented together with modern problems requiring determination of object weight on differing planets and moons. Connections between proportional reasoning and geometric similarity are explored using development of PowerPoint, scale drawings and scaling factors. The roles of proportions in computing conversions between various systems of measurement are then discussed together with presentation of common conversion factors. The development of the Golden Ratio, Fibonacci numbers, the Silver Ratio, and the notion of metallic means follows. The chapter concludes by identifying the importance of proportional reasoning as a gate keeper to further progress in mathematics including the derivative associated with the limiting behavior of ratios.

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Chapter 7 (written by MLC) titled Geometry and Measurement begins with a brief history of geometry in the curriculum. Viewing geometry as a systematic exploration of relationships occurring among shapes and within space, it is suitable for middle school creative explorations including both formal and experiential approaches. The van Hiele levels are then described, together with the student thinking and objects of thought characterized by each level. Following this van Hiele overview, developing geometric vocabulary via a geometry scavenger hunt is described and suggested terms are provided. The Problem Solving Board, introduced in Chap. 2, is then modeled as a bridge to deductive reasoning. Following this, common geometric actions such as transformations, rotations, translations, and reflections are described together with their Cartesian coordinate versions. Tessellations are explored leading to an equation 1n þ m1 þ 1k ¼ 12 which defines all integer values n, m and k for which edge to edge tessellations may be made with regular polygons of n, m and k sides. Methods fitting within van Hiele levels suitable for middle school are then presented for development of the Pythagorean theorem, Pythagorean triples, and distance formula. The development of key vocabulary for triangles and quadrilaterals is then presented together with suggested terms. Measurement formulas for the circumference and area of a circle are developed using middle school friendly methods. A variety of creative methods to calculate p are then presented which include rotation of a circular disk, inscribed and circumscribed regular polygons, squeezing in through counting, and use of Pick’s theorem. The chapter concludes by revisiting van Hiele levels in light of middle school content. Chapter 8 (written by SA) titled Combinatorics begins with the introduction of basic ways of representing natural numbers through other types of numbers. The rules of sum and product are considered in real-life contexts and illustrated using tree diagrams. These diagrams are then used to introduce the concept of permutation. Through permuting letters in words describing the contexts of checking out library books and selecting tulips, combinations without and with repetition are introduced. The rule of sum is used as a tool of an alternative derivation of formulas involving both types of combinations. A well-known problem of counting rectangles on a checkerboard is explored. The sums of powers of integers are introduced in the context of counting the number of key codes available on a combination lock. A brief historical account of the summation of powers of integers is provided. The chapter concludes with presenting examples of various explorations of the sums of powers of integers and the use of those sums as combinatorial tools of counting. Chapter 9 (written by SA) titled Conceptual Approach to the Ideas of Middle School Algebra begins with the demonstration of how basic ideas of algebra including the use of variables and equations can be developed in the context of Kid Pix—a graphic software for creative activities of young children—to allow one, when solving a word problem, instead of moving from equations to unknowns to move from unknowns, found through trial and error, to equations. Different types of generalization in algebra are considered. Basic summation formulas and their visually supported rigorous proofs are presented. Addition and multiplication tables

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are considered as a milieu for developing algebraic skills through finding the sum of all numbers in each table and counting the number of multiples of two and three in these tables. Algebraic word problems are considered, and their solutions are developed either by solving the corresponding equations through the so-called conceptual shortcuts or using diagrammatic reasoning. The chapter includes the presentation of algebraic inequalities as tools of digital fabrication of points, segments, arcs, and triangles. A brief account of the triangle inequality necessary for the construction of a triangle is provided as well as the introduction of the arithmetic mean-geometric mean inequality and its proof in the case of two and three non-negative numbers. The chapter concludes with algebraic recreations supported by the use of a spreadsheet. Chapter 10 (written by SA) titled Patterns and Functions deals with visual and numeric patterns. Different ways of describing patters of the former type using the AB-language from which the latter type of pattern stems are considered. The chapter studies patterns with reference to Gestalt psychology for which the question of what one sees in an image is critical. It is demonstrated how visual patterns described by letters can provide contexts for the emergence of numeric milieus which can be used to formulate mathematical questions of different levels of complexity about numeric description of letters. This, in turn, makes it possible to move from numbers to functions included in the mathematics curriculum of the middle grades. Those functions can also form patterns, yet at a higher cognitive level. This three-step process of generalization—from visual to numeric to algebraic —can lead to the development of computational environments used for experimentation through which new patterns may be discovered and new questions can be formulated. The chapter provides prospective teachers of middle school mathematics with research-like experience and it extends the ideas of the previous chapter in which different types of algebraic generalization were introduced. In particular, the chapter shows how a numeric pattern can be generalized to an algebraic form in multiple ways when the former is not provided with a situational reference. Different ways of describing numeric patterns are introduced including closed and recursive definitions. In the context of patterns formed by quadratic functions, Wolfram Alpha and The On-Line Encyclopedia of Integer Sequences (OEIS®) are used extensively to support mathematical explorations. The chapter concludes with the development of a spreadsheet-based computational environment (the programming details of which are included in the appendix) for solving and posing problems associated with the development of patterns originated in a visual domain. Chapter 11 (written by MLC) titled Financial Literacy and Blockchain begins by noting that worldwide levels of financial literacy among youth is in a state of decline. The awareness of differences between a first job and a lifetime career is used as an introduction to the broader topic of earning, which includes calculations of take home pay, gross income and net income. Budgeting is then presented together with a sample spreadsheet illustrating typical budget categories and a 50-30-20 budgeting strategy. This is followed by a discussion of checking and types of checking accounts. Interest is defined as being a price paid when using

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money that is not yours with problems and formulas addressing both simple and compound interest presented. The compound interest formula is expanded to allow for additional applications such as determining interest rates, starting deposits needed for a specified future value, and time needed to pay back a loan. With compound interest in place, credit and borrowing are discussed including revolving credit, installment credit, service credit and mortgages. Moving from borrowing to savings, differences between types of savings accounts are presented with a focus upon simple and compound interest. Additional topics of financial literacy such as credit reports, planning and paying for college, insurance, and investments are then described together with common financial pitfalls. This is followed by an overview of blockchain basics and applications including cryptocurrencies, voting records, title records and medical records. The chapter concludes by stressing the importance of helping middle school students develop financial literacy early in life, so they reap the benefits for the rest of their adult lives. Chapter 12 (written by SA) titled Probability and Statistical Data Analysis begins with a brief review of international standards for teaching probability and statistics with an emphasize on the use of technology as a medium of experimentation. Towards this end, the chapter demonstrates how a spreadsheet can be employed as the major instrument of experimentation aimed at connecting theoretical and experimental results. Different representations of sample space are considered using the contexts of coin tossing and dice rolling. Addition and multiplication tables explored earlier in Chap. 9 are examined from the probabilistic perspective by calculating chances, given the size of a table, of having an entry with a specified property. Experimental results obtained here in the context of Wolfram Alpha are compared to the corresponding theoretical conclusions of Chap. 9. Bernoulli trials are introduced and the likely length of the longest run of heads within a sequence of 100 tosses of a coin (a Bernoulli trial) is explored. Some classic problems that stimulated the development of probability theory are presented. Simple explorations with spinners leading to a historically significant problem residing at the confluence of probability theory and number theory as well as the Palindrome conjecture introduced in Chap. 9 in the context of recreational algebra are discussed through probabilistic lens. The Monty Hall Dilemma concludes the probability part of the chapter. Connection of probability to statistical data analysis opens the second part of the chapter. Different ways of representing numeric data in a graphic form are illustrated using spreadsheets. Normal distribution is introduced in the context of coin tossing. The chapter concludes with the concept of bivariate analysis using an experiment of tossing a coin three times and the joint probability distribution of two measurements carried out within this experiment has been found. The programming details of many spreadsheet-based environments of the chapter are included in the Appendix. Appendix (written by SA) provides details of programing of spreadsheet environments used to support various problem-posing activities described in the textbook. Syntactic versatility of spreadsheets allows for alternative programming of

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computationally identical environments by modifying spreadsheet formulas presented here. Just as any reference to information available in a non-traditional print is accurate as of the day of its on-line retrieval, everything in the appendix is related to Microsoft® Excel for Mac 2020, version 16.36. Potsdam, NY, USA Houston, TX, USA

Sergei Abramovich Michael L. Connell

Acknowledgements The authors would like to acknowledge that their work on this textbook have been greatly supported at Springer by Natalie Rieborn—Editor Education, and Marianna Pascale —Asst. Editor Education. A special gratitude is extended to two unanimous reviewers of the textbook proposal and of the entire manuscript whose initial suggestions, comprehensive reviews and insightful comments significantly improved the quality of the manuscript. Last but not least, MLC would like to recognize the guidance and insights into Blockchain provided by John Connell.

Contents

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Teaching Middle School Mathematics: Standards, Recommendations and Teacher Candidates’ Perspectives . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Importance of Content Knowledge . . . . . . . . . . . . . . . 1.3 The Importance of Pedagogical Content Knowledge . . . . . . 1.4 The Need for Teachers with ‘Deep Understanding’ of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Learning to Address Misconceptions Through Mathematical Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Making Conceptual Connections . . . . . . . . . . . . . . . . . . . . 1.7 Using Technology in the Classroom . . . . . . . . . . . . . . . . . . 1.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Mathematics in the Digital Era . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modeling Mathematics and Mathematical Modeling . . . . 2.2.1 Modeling Mathematics: Selecting a Referent or Object to Act upon . . . . . . . . . . . . . . . . . . . 2.3 Modeling Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Modeling Equivalent Surface Area Functions— Stamping Functions . . . . . . . . . . . . . . . . . . . . 2.3.2 Modeling Correlations—Burning the Candle . . 2.3.3 Modeling Systems of Equations—The Flagpole Factory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Data-Tables Revisited . . . . . . . . . . . . . . . . . . . 2.4 The Problem Solving Board . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Reasoning and Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Elementary Reasoning at the Concrete Operational Stage . 3.3 Supporting Plausible Mathematical Reasoning and Formal Proofs with CARE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Commonplaces . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Authority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Reason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 Experience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 CARE and Rotating Squares . . . . . . . . . . . . . . . . . . . . . . 3.5 Basic Elements of Plausible Reasoning . . . . . . . . . . . . . . . 3.6 Basic Elements of Propositional Logic . . . . . . . . . . . . . . . 3.6.1 Conjunction, Disjunction and Implication . . . . . . 3.6.2 Application of the Logical Conjunction Truth Values in Spreadsheet Modeling . . . . . . . . . . . . 3.6.3 Application of the Logical Disjunction Truth Values in Spreadsheet Programming . . . . . . . . . 3.6.4 De Morgan Rules . . . . . . . . . . . . . . . . . . . . . . . 3.6.5 Converse, Inverse and Contrapositive Statements 3.6.6 Modus Ponens: [(p ) q) ^ p] ) q . . . . . . . . . . 3.6.7 Modus Tollens: [(p ) q) ^ :q] ) :p . . . . . . . 3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modeling Mathematics with Fractions . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Transition from Arithmetic of Integers to Arithmetic of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Meaning of Multiplying and Dividing Fractions . . . . . . 4.3.1 Multiplying Two Fractions . . . . . . . . . . . . . . . . . 4.3.2 Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conceptual Meaning of the Invert and Multiply Rule: From Integers to Fractions . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Case of Dividing Integers . . . . . . . . . . . . . . . 4.4.2 The Case of Dividing Fractions . . . . . . . . . . . . . . 4.4.3 The Invert and Multiply Rule as a Change of Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Unit Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Unit Fractions as Benchmark Fractions . . . . . . . . 4.5.2 Representation of a Unit Fraction as a Sum of Two Like Fractions . . . . . . . . . . . . . . . . . . . . 4.5.3 Representation of 1/2 as a Sum of Three Different Unit Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Egyptian Fractions and the Greedy Algorithm . . .

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From Long Division to Decimal Representation of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rational Numbers in Non-Decimal Bases . . . . . . . . . . . . . . 4.7.1 Conversion of Integers into Non-Decimal Bases . . 4.7.2 Conversion of Common Fractions into Non-Decimal Bases . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Visual Representation of Conversion of Common Fractions in Different Bases . . . . . . . . . . . . . . . . . Integer Sequences as Sources of Fractions . . . . . . . . . . . . . Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Continued Fraction Representation of Common Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Connecting Fractions to Quadratic Equations . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Decimal and Percent Representation of Rational Numbers . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Developing Numerical Fluency . . . . . . . . . . . . . . . . . . 5.3 Transitioning from Numerical Fluency to the Base-Ten Notational System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Decimals and Decimal Fractions . . . . . . . . . . . . . . . . . 5.5 Decimals and Common Fractions . . . . . . . . . . . . . . . . . 5.6 Decimal and Percent . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Repeating Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 From Representation to Operations . . . . . . . . . . . . . . . . 5.8.1 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Addition and Subtraction . . . . . . . . . . . . . . . . 5.8.3 Multiplication and Division . . . . . . . . . . . . . . 5.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Ratio and Proportion . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Part-Part Ratios . . . . . . . . . . . 6.2.2 Part-Whole Ratios . . . . . . . . . . 6.2.3 Reflecting Quotients as Ratios . 6.2.4 Reflecting Rates as Ratios . . . . 6.3 Proportions . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Scaling . . . . . . . . . . . . . . . . . . 6.3.2 Proportions and Conversions . . 6.3.3 The Golden Ratio . . . . . . . . . .

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Geometry and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Geometric Thinking and Van Hiele Levels . . . . 7.2.2 Developing Geometric Vocabulary . . . . . . . . . . 7.2.3 Deductive Proofs . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.7 Pythagorean Theorem, Pythagorean Triples, and the Distance Formula . . . . . . . . . . . . . . . . 7.2.8 Triangles and Introductory Trigonometry . . . . . 7.2.9 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Application of Scaling Factors in Measurement 7.3.2 Circumference, Area of a Circle, and P . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Rules of Sum and Product . . . . . . . . . . . . . . . . . . . . . 8.3 Tree Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Permutation of Letters in a Word . . . . . . . . . . . . . . . . . . . . 8.5 Combinations Without Repetition . . . . . . . . . . . . . . . . . . . . 8.5.1 Developing Counting Techniques . . . . . . . . . . . . 8.5.2 Counting Rectangles on a Checkerboard . . . . . . . 8.5.3 Why Are There 256 Representations of 9 Through a Sum of Ordered Integers? . . . . . . . . . . . . . . . . . 8.6 Combinations with Repetition . . . . . . . . . . . . . . . . . . . . . . 8.7 The Sum of Powers of Integers as a Combinatorial Tool . . . 8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.4 6.5 7

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9

6.3.4 The Silver Ratio 6.3.5 Metallic Means . Conclusion . . . . . . . . . . . Activity Set . . . . . . . . . .

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Conceptual Approach to the Ideas of Middle School Algebra . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 From Early Algebra to Graphing and Problem Posing with Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Three Problems from Different Grade Levels . . 9.2.2 Problem Reformulation . . . . . . . . . . . . . . . . . .

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9.2.3 9.2.4

9.3

9.4

9.5 9.6 9.7

9.8 9.9

9.10 9.11 9.12

Solution of Problem 9.5 Using Kid Pix . . . . . . . . . . Opening a Window to Traditional School Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Types of Generalization in School Algebra . . . . . . . . 9.3.1 Algebraic Generalization of the First Kind . . . . . . . . 9.3.2 Algebraic Generalization of the Second Kind . . . . . . 9.3.3 From Summation of Odd Numbers to Summation of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 The Sum of Cubes in the Multiplication Table . . . . . Conceptual Generalizations of the First Kind . . . . . . . . . . . . . 9.4.1 Patterns of Elimination Leading to Polygonal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Developing Subsequences of Fibonacci Numbers . . . . Parameterization as Conceptual Generalization of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Insufficiency of Generalization by Empirical Induction . . . . . . Numeric Tables as a Context for Developing Algebraic Skills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1 Finding the Sum of All Numbers in the n  n Addition Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Finding the Sum of All Numbers in the n  n Multiplication Table . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Exploring Divisibility of Numbers in Addition and Multiplication Tables . . . . . . . . . . . . . . . . . . . . Solving Algebraic Word Problems Through Conceptual Strategies/Shortcuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.1 Algebraic Inequalities as Tools of Digital Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9.2 Digitally Fabricating Points, Segments and Arcs . . . . 9.9.3 Arithmetic Mean-Geometric Mean Inequality . . . . . . Algebraic Recreations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Patterns and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 From Patterns to Functions . . . . . . . . . . . . . . . . . . . . 10.3 From Visual to Symbolic . . . . . . . . . . . . . . . . . . . . . . 10.4 Patterns Structured by Colors . . . . . . . . . . . . . . . . . . . 10.4.1 From Colors to Functions . . . . . . . . . . . . . . 10.4.2 From Empirical Conjectures to Mathematical Induction Proof . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Two-Color Pattern Guided by Consecutive Odd Numbers . . . . . . . . . . . . . . . . . . . . . . .

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10.4.4

More Patterns Defined by an Arithmetic Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Finding Patterns Formed by Functions . . . . . . . . . . . . . . . 10.5.1 Developing Formula for f 1 ðn; mÞ . . . . . . . . . . . . 10.5.2 The Cases of the Position of the First R and the First/Last B . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Alternative Verification of the Obtained Results . 10.6 The Case of Three Colors . . . . . . . . . . . . . . . . . . . . . . . . 10.7 The Case of p Colors . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Pattern Guided by Arithmetic Sequence with Difference One . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.2 Pattern Guided by Arithmetic Sequence with Difference Two . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.3 Pattern Guided by Arithmetic Sequence with Difference Three . . . . . . . . . . . . . . . . . . . . . . . . 10.7.4 Pattern Guided by Arithmetic Sequence with Difference Four . . . . . . . . . . . . . . . . . . . . . . . . 10.7.5 Pattern Guided by Arithmetic Sequence with Difference m . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 From Generaization to Computerization . . . . . . . . . . . . . . 10.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Financial Literacy and Blockchain . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Financial Literacy . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Earning . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Budgeting . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Checking . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Credit and Borrowing . . . . . . . . . . . . . . . 11.2.6 Mortgages . . . . . . . . . . . . . . . . . . . . . . . 11.2.7 Savings . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.8 Credit Reports . . . . . . . . . . . . . . . . . . . . 11.2.9 Planning and Paying for College . . . . . . . 11.2.10 Insurance . . . . . . . . . . . . . . . . . . . . . . . . 11.2.11 Investments . . . . . . . . . . . . . . . . . . . . . . 11.2.12 Common Financial Pitfalls . . . . . . . . . . . 11.3 Blockchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Blockchain Basics . . . . . . . . . . . . . . . . . . 11.3.2 Blockchain Security Concerns . . . . . . . . . 11.3.3 Consensus Mechanisms and Incentivation 11.3.4 Cryptocurrency and Other Applications . .

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12 Probability and Statistical Data Analysis . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Experiments with Equally Likely Outcomes . . . . . . . . . . . . . . 12.3 Randomness and Sample Space . . . . . . . . . . . . . . . . . . . . . . . 12.4 Different Representations of a Sample Space . . . . . . . . . . . . . 12.5 Fractions as Tools in Measuring Chances . . . . . . . . . . . . . . . . 12.6 Explorations with Addition and Multiplication Tables . . . . . . . 12.6.1 Computational Experiments with Pairs of Random Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Selecting Even Numbers in Addition Tables . . . . . . . 12.6.3 Selecting Even Numbers in Multiplication Tables . . . 12.6.4 Selecting Multiples of Three in Addition Tables . . . . 12.6.5 Selecting Multiples of Three in Multiplication Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.6 Theoretical Probabilities and Their Monotone Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Bernoulli Trials and the Law of Large Numbers . . . . . . . . . . . 12.8 Classic Problems that Motivated Theoretical Development . . . 12.9 A Modification of the Problem of De Méré . . . . . . . . . . . . . . 12.10 Experimental Probability Requires a Long Series of Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11 Probability Experiments with Spinners and Palindromes . . . . . 12.12 Monty Hall Dilemma as a Paradox in the Theory of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.13 Transition from Probability Theory to Statistical Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.14 Graphic Representations of Numeric Data . . . . . . . . . . . . . . . 12.15 Measures of Central Tendency . . . . . . . . . . . . . . . . . . . . . . . . 12.16 Measures of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.17 The Binomial Distribution and the Normal Curve . . . . . . . . . . 12.18 The z-Score and the Standard Deviation . . . . . . . . . . . . . . . . . 12.19 Bivariate Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.20 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.21 Activity Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11.4 11.5

11.3.5 Advantages of Blockchain . . . 11.3.6 Disadvantages of Blockchain . Conclusion . . . . . . . . . . . . . . . . . . . . . Activity Set . . . . . . . . . . . . . . . . . . . .

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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Chapter 1

Teaching Middle School Mathematics: Standards, Recommendations and Teacher Candidates’ Perspectives

1.1

Introduction

This chapter aims to demonstrate the need for linkages between textbook content and the pedagogical ideas underpinning current standards for middle school mathematics instruction, to share recommendations for the mathematical preparation of the middle grades teachers, and describe a number of notable perspectives on mathematics education that have been the backbone of the current reform movement of pre-college mathematics teaching worldwide. As will be demonstrated through multiple citations of the standards and other educational materials (available in English), there is a true congruency of the ideas of mathematics pedagogy recorded in Australia, Canada, Chile, China, England, Japan, Korea, Singapore and the United States. Throughout the chapter, the authors will share both solicited and unsolicited (over the years) comments by middle school teacher candidates about their learning to teach the subject matter at that level, indicating the candidates’ interest in acquiring deep understanding of grade-appropriate mathematical content and methods towards gaining much needed professional expertise. Some comments will also include reflections on the candidates’ own experience of learning mathematics as youngsters. In all, the comments will show that positive attitude towards knowledge acquisition stems from the candidates being passionate in helping their future students to be better mathematical thinkers. Unfortunately, this passion appears not being prevalent characteristic of practicing teachers because, as one of the teacher candidates (apparently reflecting on their pre-college experience) noted, “we live in a world where math is very important, but we have teachers who are not passionate about math”. More specific was another teacher candidate who recollects to be “good in math in elementary school, but it seems like as soon as I entered 6th grade that the teachers didn’t put much effort into teaching you the subject”. Being passionless and effortless in a mathematics classroom is likely due to © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_1

1

1 Teaching Middle School Mathematics: Standards, Recommendations …

2

teachers’ insufficient conceptual knowledge of the subject matter and its grade-appropriate pedagogy integrating concepts and procedures. Such unfortunate behavior by teachers might be the genesis of low level of success of mathematics education reform as it has been reported over the years (Batista, 1994; Kilpatrick, 1997; Handal & Herrington, 2003; Eacott & Holmes, 2010; Schoales, 2019). By providing teachers with carefully designed practical guidance, grade-appropriate learning tools and teaching materials, the odds of the improvement of teaching and success of reform integration in the context of mathematics can be greatly increased (Cohen & Mehta, 2017). This textbook includes the authors’ ideas about efforts one needs to put into the teaching of mathematics with the passion and even joy leading to the success of mathematics education reform.

1.2

The Importance of Content Knowledge

A great number of mathematics educators worldwide believe that “teachers must know the mathematical content they are responsible for teaching not only from a more advanced perspective but beyond the level they are assigned to teach” (Baumert et al., 2010, p. 137). The genesis of this position can be traced back to the concept of metacognition (Flavell, 1976) defined as “one’s knowledge concerning one’s own cognitive processes and products or anything related to them” (p. 232). In some of the early development of metacognition in mathematics education (Campione et al., 1989) it was made very clear that teachers cannot be expected to be metacognitive in a field in which they were not yet cognitive. With this in mind, teacher education programs around the world strive to provide teacher candidates with multiple opportunities to acquire strong knowledge and ‘deep understanding’ of mathematics through the appropriate mathematical and pedagogical preparation within mathematics content and methods courses that are mandatory for any such program. According to the Conference Board of the Mathematical Sciences (2012) —an umbrella organization of nineteen professional societies in the United States, concerned, in particular, with mathematics teacher preparation, desirable are courses that focus on mathematics and its pedagogy. In response to this, the textbook is written to support teaching of such courses through conceptually rich problem solving and to provide both perspectives, mathematical and pedagogical, on the “big ideas” (e.g., Schifter, 1998; Charles, 2005; Gadanidis & Hughes, 2011; Chalmers et al., 2017) of the curriculum of the middle grades. The authors followed a classis dictum from David Gilbert’s celebrated keynote address to the 1900 International Congress of Mathematicians that a good mathematical problem is characterized by “clearness and ease of comprehension … for what is clear and easily comprehended attracts, the complicated repels us” (Hilbert, 1902, p. 438).

1.2 The Importance of Content Knowledge

3

Whereas mathematical problem solving is far from being an uncomplicated cognitive endeavor, any solved problem imparts to a student the critical learning experience “to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears” (Pólya, 1957, p. 94). Those teaching mathematics in the middle grades are placed within a school system where there is a need not only to know students’ past mathematical knowledge but what is involved in mathematics learning in the upper grades as well (Association of Mathematics Teacher Educators, 2017). That is why mathematics content and methods courses for prospective middle school teachers should demonstrate “how the mathematical ideas of the middle grades connect with ideas and topics of elementary school and high school” (Conference Board of the Mathematical Sciences, 2012, p. 40). This textbook aims to be a source of connecting ideas, supported by concrete tools and practical suggestions, that penetrate the entire school mathematics curriculum with the center of gravity in the middle grades. As an example of one such connecting idea, consider the multiplication table, a mathematical structure that can be used as a medium of conceptual explorations through problem solving already in the elementary school. It is through such explorations that students develop residual mental power responsible for memorizing basic multiplication facts. For example, the numbers on the main diagonal (top left–bottom right) of the table are called square numbers as they are the products of two equal factors. The table can be revisited at the middle school level from a conceptual perspective by noting that each square number differs from its immediate predecessor by an odd number. Indeed, uncomplicated algebra shows that n2
ðn
1Þ2 ¼ n2
n2 þ 2n
1 ¼ 2n
1 : Also, the relationship between two consecutive squared integers can be explained geometrically (n = 4 in Fig. 9.8, Chap. 9), implying step-by-step that n2 ¼ ðn
1Þ2 þ 2n
1 ¼ ðn
2Þ2 þ ð2n
3Þ þ ð2n
1Þ ¼ ðn
3Þ2 þ ð2n
5Þ þ ð2n
3Þ þ ð2n
1Þ ¼ . . . ¼ ½n
ðn
1Þ2 þ ½2n
ð2n
3Þ þ ::: þ ð2n
5Þ þ ð2n
3Þ þ ð2n
1Þ ¼ 1 þ 3 þ ::: þ ð2n
1Þ: Furthermore, the reduction of a non-linear term, the square of n, to a sum of n linear terms (consecutive odd numbers) makes it possible to represent three times the sum of squares of the first n natural numbers as follows

4

1 Teaching Middle School Mathematics: Standards, Recommendations …

ð12 þ 22 þ 32 þ ::: þ n2 Þ þ 2½n2 þ ðn
1Þ2 þ ðn
2Þ2 þ ::: þ 12  ¼ 1 þ ð1 þ 3Þ þ ð1 þ 3 þ 5Þ þ ::: þ ½1 þ 3 þ 5 þ ::: þ ð2n
1Þ þ 2½n2 þ ðn
1Þ2 þ ðn
2Þ2 þ ::: þ 12  þ 3 fflþ ::: þffl3} þ 2ðn
1Þ2  ¼ ½1 þ 1 fflþ ::: þffl1} þ 2n2  þ ½3 |fflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflffl |fflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflffl n times

ðn
1Þ times

þ ½5 þ 5 fflþ ::: þffl5} þ 2ðn
2Þ2  þ ::: þ ð2n
1 þ 2  12 Þ |fflfflfflfflfflfflfflfflffl {zfflfflfflfflfflfflfflfflffl ðn
2Þ times

from where, as shown in Chap. 9 (Sect. 9.3.3), a closed formula for this sum can be derived. In particular, in the context of counting rectangles on a checkerboard—a problem-solving activity for high school (National Council of Teachers of Mathematics, 2000, p. 335), the formulas for the summation of the first n natural numbers and their squares make it possible to demonstrate already to students in the middle grades that percentage of squares among rectangles decreases monotonically with the increase of the size of the checkerboard and eventually has zero as its limit. This is an interesting mathematical phenomenon, the appreciation of which can make the students aware of the existence of qualitatively different paths to infinity of analytically expressed quantities. Furthermore, the multiplication table can be explored from a probabilistic perspective by calculating probabilities of randomly selecting entries of the table that possess certain properties. Such explorations are discussed in Chap. 12. cBig Idea Intrinsic connectivity of mathematical concepts is the major condition which affords mathematics students with multiple opportunities for collateral learning. The notion of collateral learning, being one of the seminal educational ideas of John Dewey, makes it, without any doubt, a big idea of mathematics and its pedagogy.b Making conceptual connections among variable (and, perhaps, randomly chosen) quantities and their geometric and algebraic interpretations brings about meaning to and understanding of the three major strands of pre-college mathematics —arithmetic, geometry, and algebra. As a teacher candidate noted, “The conceptual level of math is what grows from the deeper meaning. It is important that our children are being taught on a conceptual level and for this to happen, our teachers must comprehend the lessons on a deeper level”. Toward this end, manipulative materials (e.g., square tiles) can be used to demonstrate such integrated perspective on the numbers in the multiplication table, the basic geometric figures, and algebra as a generalized arithmetic. That is, using concrete materials to investigate patterns and concurrently, as a collateral learning1 outcome (Dewey, 1938), memorize facts

1

Collateral learning, while being a powerful educational idea in general, may be seen as one of the big didactical ideas of mathematics education because of innate connectivity of mathematical

1.2 The Importance of Content Knowledge

5

in the multiplication table at a lower level can be revisited at a higher level in the context of teaching and learning geometrically and probabilistically enhanced algebra. By appreciating such connectivity of ideas, middle school teachers can demonstrate to their students that the subject matter of mathematics, whatever the grade level, is indeed captivating, engaging, and conceptually rich (Romano & Vinčić, 2011). Specific cases will be developed in the subsequent chapters linking pedagogy of the middle grades and K-12 mathematics curriculum. Furthermore, when the final formal abstractions rely upon understanding developed using concrete materials and sketches of these materials, this connected nature of mathematics can be properly illustrated. cBig Idea Representing numbers as sums of other numbers can be seen as a big idea of mathematics. For example, an even number is a sum of two odd numbers, a square number of rank n is a sum the first n odd numbers, a proper fraction is a sum of unit fractions, and so on.b As will be emphasized in this textbook, the appreciation of mutual connections among mathematical ideas can be developed by demonstrating to teacher candidates through grade appropriate activities that one of the common threads (alternatively, one of the big ideas) permeating the entire school mathematics curriculum (and consequently, the teacher preparation mathematical course work) is the representation of numbers as sums of other numbers. For example, mathematics teachers in Chile appreciate that “concepts are inseparable from their representations … [the variety of which] must be provided for a particular concept, taking into account the relationships among representations … [something that] helps students achieve a deeper understanding of the concept and provides more tools for working with it” (Felmer et al., 2014, p. 35). Similarly to what was already shown above, integers (not all though) may be represented through the sums of consecutive natural numbers; squares of integers—through the sums of odd, triangular, and square numbers; unit fractions—through the sums of like fractions; integers— through irrational numbers; real numbers—through complex numbers; and so on. What is important to understand about representations is that they may not exist at all or are not unique. For instance, it is not possible to represent any power of an integer beyond the second power as the sum of two like powers. Put another way, it is not possible to extend the Pythagorean2 equation x2 þ y2 ¼ z2 beyond the second power; that is, already the cubic equation x3 þ y3 ¼ z3 does not have a solution in natural numbers. This is a statement of the celebrated Fermat’s Last Theorem3 the proof of which took mathematicians some 350 years to find. At the same time,

concepts due to which a student, indeed, learns more than “only the particular thing he is studying at the time” (Dewey, 1938, p. 49). 2 Pythagoras (ca. 570 B.C.—ca. 495 B.C.)—a Greek philosopher. 3 Pierre de Fermat (1601–1665)—a French mathematician and lawyer.

6

1 Teaching Middle School Mathematics: Standards, Recommendations …

whereas it is not difficult to prove that a prime number may not be represented as a sum of three or more natural numbers in arithmetic progression,4 the famous Goldbach5 conjecture stating that an even number greater than two can be represented as a sum of two prime numbers, sometimes in more than one way (e.g., 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 30 = 7 + 23 = 11 + 19 = 13 + 17), remains unproved for almost three centuries. Such diversity in representation of numbers calls for the diversity of methods of teaching the subject matter. Consequently, mathematics methods and content courses should include the variety of teaching strategies that reflect the broad mathematical content of middle school mathematics, its historical and cultural aspects, something that middle school teachers have to be aware of (Association of Mathematics Teacher Educators, 2017). Teachers need to learn how “to provide a more engaging, student-centered, and technology-enabled learning environment, and to promote greater diversity and creativity in learning” (Ministry of Education Singapore, 2012a, p. 17). Of a special noteworthiness is the stance that “teacher knowledge should be particularly important for the learning gains of weaker students” (Baumert et al., 2010, p. 147). Indeed, it is through the knowledge of mathematics that the diversity of teaching methods, to the benefit of struggling students, unfolds.

1.3

The Importance of Pedagogical Content Knowledge

Internationally, researchers found that mathematics education “programs that compromise on subject matter training … have detrimental effects on PCK [pedagogical content knowledge] and consequently negative effects on instructional quality and student progress” (ibid, p. 167). One should not underestimate the importance of subject matter training because, as Chilean mathematics educators put it, “many aspiring teachers have major gaps in [mathematical] knowledge and skills when entering teacher education programs” (Felmer et al., 2014, p. 158). In the United States, one of the authors’ students, when asked to reflect on learning mathematics conceptually, shared mathematical learning experience from her school years: “When I was growing up, the conceptual understanding was neglected. We were told that if a problem looks like this, solve it like this. If it looks like that, solve it that way. There was no explanation of why a certain procedure was to be used”. A teacher candidate from Texas, a non-native English speaker, shared his appreciation of using manipulative materials as a way of revealing linguistic complexity of word problems towards their expression in the language of mathematics, “Manipulatives was something that helped me learn mathematics in the U. S. I am originally from Guatemala where I was very good in math. But that all changed when I came to the U.S. I started getting lost and falling behind. What

4

As follows from formula (9.5), Chap. 9, when its right-hand side is a prime number then n = 2. Christian Goldbach (1690–1764)—a German mathematician.

5

1.3 The Importance of Pedagogical Content Knowledge

7

helped me was the use of manipulatives in the classroom, especially when it came to dealing with word problems. I believe that the use of manipulatives in the classroom can benefit not only ELLs [English language learners] but English native speakers as well”. It is interesting to connect this comment to a paper by Silver and Cai (1996) analyzing arithmetical problems posed by middle school students. These authors talked about various levels of syntactic complexity of a posed problem depending on whether it included relational (direct comparison of quantities) and conditional (indirect comparison of quantities) propositions or not. It appears that the use of manipulatives does reduce syntactic complexity of a problem by creating a tactile and visually rich environment for direct or indirect comparison of quantities. As the last comment indicates, this kind of cognitive milieu makes word problems accessible to students who are at risk of “getting lost and falling behind” in a manipulative-free learning environment. Notwithstanding, as a teacher candidate, placed in grade six of one of the public schools as an intern, reported, “manipulatives are not provided or even mentioned in the teacher’s manual even for a geometry lesson where they would have been ideal. Instead, students have to draw. But they couldn’t draw two rectangles or circles that were the same size and when they did draw them same size, they could not split them into equal-size sections”. This comment indicates the likely lack of skills of the teacher observed by the intern in the use of manipulative materials in the classroom. On a deeper level, the intern talks about students using imprecise drawings as geometric representations of unit fractions like one-half. While they draw something that does not look like one-half as an abstract concept, they may still call it one-half just because they cut it in two pieces. It appears that this kind of deficiency of visual thinking6 in dealing with self-drawn diagrams of fractions can be explained through the principle of Gestalt psychology “that many phenomena of experience are variations organized around Pragnanzstufen, phases of clear-cut structure” (Arnheim, 1969, p. 183). Whereas the concept of any fraction is an abstraction, learners of mathematics comprehend abstractions better when frames of reference can be provided by a classroom teacher. Manipulatives, both physical and virtual, can serve as such frames, especially in the case of unit fractions. However, as discussed in Chap. 4, when students need to compare non-unit fractions, the lack of precision in drawing fractions, whatever the reason, motivates the introduction of a two-dimensional model of comparison so that sloppy drawings can be accepted as frames of reference for abstractions. For example, the inequality 2/3 < 5/7 (see Fig. 4.2, Chap. 4) between non-unit fractions does not represent a clear-cut makeup to be comprehended at the pre-operational level in comparison with the inequality 1/3 < 1/2 between unit fractions. As will be shown in Chap. 4, this new model provides students with alternative “phenomena of experience” with conceptual structures of mathematics and allows them to overcome the limitations of

6

Mediating mathematical learning by visual thinking may be seen as another big idea of mathematics education. In the United States, it is most notably brought into being by the Common Core State Standards (2010) through the use of the so-called “tape diagram” (p. 87) and other drawings.

1 Teaching Middle School Mathematics: Standards, Recommendations …

8

imprecision in drawings fractions as frames of reference that are absent in the context of formal comparing of and operating on fractions as abstract symbols. cBig Idea Many great minds of the past emphasized the power of visual thinking with mathematics always being at the frontier of the use of images as mediational means of comprehending abstraction. Mediating learning by visual thinking is truly one of the big ideas of mathematics education.b

1.4

The Need for Teachers with ‘Deep Understanding’ of Mathematics

What does it mean that teacher candidates preparing to teach middle school mathematics need to possess ‘deep understanding’ of the subject matter? Why do they need to have such understanding? There are several reasons for prospective teachers to be thoroughly mathematically prepared in order to have positive effects on the progress of their students. First, in the modern mathematics classroom, students of all ages are expected and even encouraged to ask questions. In the United States, the national standards suggest, “Problems in the middle grades can and should respond to students’ questions and engage their interests” (National Council of Teachers of Mathematics, 2000, p. 258). Support for this suggestion can be found in the following comment by a teacher candidate preparing to teach mathematics in upstate New York: “I would love to be able to have a classroom where I can teach to the students and let their ideas and interests to guide our path of learning”. Just to the north of upstate New York, the Ontario Ministry of Education in Canada sets expectations for teachers to be able to “provide activities and assignments that encourage students to develop the conceptual foundation they need … to search for patterns and relationships and engage in logical inquiry” (Ontario Ministry of Education, 2005, p. 24). In order to develop such foundation, “teachers should know ways to use mathematical drawings, diagrams, manipulative materials, and other tools to illuminate, discuss, and explain mathematical ideas and procedures” (Conference Board of the Mathematical Sciences, 2012, p. 33). In England, there is a need for “teachers who have sound mathematical, pedagogical and subject-specific pedagogical knowledge … [capable of presenting] mathematics as a conceptual, coherent and cognitive progression of ideas” (Advisory Committee on Mathematics Education, 2011, pp. 1, 18). In South America, Chilean mathematics teachers are expected to “use representations, call on prior knowledge, put forward good questions, and stimulate an inquisitive attitude and reasoning among students” (Felmer et al., 2014, p. 37). In Japan, mathematics teachers are encouraged to provide students “with opportunities to do activities such as concrete manipulations or thought experiments … [in order to] further emphasize the importance of thinking process” (Takahashi et al., 2006, p. 9). In China, mathematics teachers, while seeing “both understanding and application of knowledge as equally important, they allocate more time to the latter, as they believe appropriate practice

1.4 The Need for Teachers with ‘Deep Understanding’ of Mathematics

9

can deepen and consolidate understanding” (Fan et al., 2015, p. 57). Mathematics teachers in Korea are expected to “emphasize not only the results of problem solving but also its strategies and processes and encourage students to formulate problems on their own” (Hwang & Han, 2013, p. 36). In Singapore, mathematics educators and curricula developers believe that “teachers need to have the big picture in mind so that they can better understand what … to do at their level, as well as to plan and advise students in their learning of mathematics” (Ministry of Education Singapore, 2012a, p. 11). In Australia, mathematics teachers “understand how mathematics is represented and communicated … [and] initiate purposeful mathematical dialogue with and among students” (Australian Association of Mathematics Teachers, 2006, pp. 2, 4). The repertoire of learning opportunities the teachers are expected to offer to their students includes continuous search for alternative approaches to solving problems as well as helping students to better learn a specific problem-solving strategy with which they have been struggling. A search for an alternative procedure in solving a mathematical problem, another big idea of mathematics education pedagogy, quite unexpectedly may become a source of a new conceptual development and insight. In addition, by utilizing the diversity of thinking among students as a means of individual intellectual growth, the entire class becomes confident in the correctness of the answer stemming from multiple solution strategies. As Freudenthal (1978) noted “it is independency of new experiments that enhances credibility” (p. 193). cBig Idea An emphasis of current teaching and learning standards on multifaceted nature of approaches to mathematical problem solving is a big idea of both mathematics and its pedagogy. A search for an alternative procedure in solving a problem may lead to a new conceptual development and insight.b Teachers must be able to answer foundational questions, such as “How can you tell?” and “What would happen if …?” in regard to the problems and content they are to teach. Having experience in asking and answering such conceptually-oriented questions associated with the natural extension of ideas, concepts and procedures, is useful not only because of their relevance to the entire school mathematics curriculum but such experience, as mathematics educators in Japan argue, is necessary for acquiring a rich toolkit of content knowledge and pedagogical skills with emphasis on “the importance of fostering logical thinking and intuition … [and] capability of using mathematics” (Takahashi et al., 2006, p. 13). By learning to ask ‘good questions’ or answering such questions, prospective teachers “can scaffold learning … correct a misconception, reinforce a point or expand on an idea” (Ministry of Education Singapore, 2012a, p. 27). In China, mathematics curriculum standards motivate classroom teachers to “encourage students to communicate ideas, and pose questions so as to improve their independent and creative thinking” (Huang et al., 2015, p. 317). National mathematics curriculum in England uses such terms as “practice with increasingly complex problems over time … [and] solve problems … with

10

1 Teaching Middle School Mathematics: Standards, Recommendations …

increasing sophistication” (Department for Education, 2013, p. 1). Towards this end, teachers have to be prepared to deal with situations when natural quest for inquiry leads students towards this sophistication and increase in complexity of mathematical ideas. The need for this kind of teacher preparation is confirmed by a teacher candidate who put it as follows: “In my classroom, I want to pose challenging math problems to my students. I will encourage the students to use any materials they wish to solve the problems. I will strive not to give the answers to the students but rather attempt to steer them in the right direction to find their own answers”. This comment is reminiscent of Hilbert’s (1902) classic advice to professional mathematicians that “a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts” (p. 438). If a mathematics teacher of middle grades takes notice of this advice (of course, from purely pedagogical perspective), his or her students are likely to develop understanding of “the merits of the mathematical thinking process …[including] simplicity, clarity and accuracy … [and will be] studying mathematics willingly” (Takahashi et al., 2006, p. 9).

1.5

Learning to Address Misconceptions Through Mathematical Connections

Often, an additive decomposition of a natural number in three like summands is considered by teacher candidates from a trial-and-error perspective, unlike the case of two summands when the corresponding decomposition reveals the presence of a system. It is possible for a trial-and-error approach to be turned into a system when one is asked to reduce the case of three summands to two summands by going through all possible cases for the first summand. More complicated is the situation with an additive decomposition even in two summands (without regard to their order) of the reciprocal of a natural number. Is there a conceptual basis for such a difference in complexity? For example, whereas there are five additive decompositions of the number 11 in two summands, its reciprocal (a unit fraction) has only two ways to be represented as a sum of two like fractions (see Chap. 4, Sect. 4.5.2). However, whereas replacing 11 by 12 increases the number of additive decompositions by one, the unit fraction 1/12 has eight representations as a sum of two like fractions. It is important for teachers of middle grades to understand that such variation is due to profound conceptual difference between linear and non-linear phenomena7 as manifested, respectively, by the behavior of the functions f ðnÞ ¼ n and gðnÞ ¼ 1n. Additive decompositions of integers and their reciprocals have various applications outside of arithmetic. For example, using a geoboard (Gattegno, 1971), one can discover that the equalities 12 = 1 + 11 and 12 = 6 + 6 are responsible for 7

See also graphs in Fig. 9.4 (Chap. 9) and Fig. 12.12 (Chap. 12) as examples of, respectively, linear and non-linear phenomena.

1.5 Learning to Address Misconceptions Through Mathematical …

11

(integer-sided) rectangles of perimeter 24 linear units with the smallest and the largest areas, respectively. At the upper middle school level, one can learn that whereas the latter decomposition yields the rectangle with the largest area regardless of the number system involved, in the context of rational numbers, the smaller one of the summands is, the smaller is the corresponding area. Therefore, because a rational number can be chosen as close to zero as one wishes (in other words, no number closest to zero can be found), no rectangle with the smallest area exists (whatever the perimeter). Likewise, through multiplicative decompositions of the number 9 in two factors one can discover that the equalities 9 ¼ 1  9 and 9 ¼ 3  3 are responsible for (integer-sided) rectangles of area 9 square units with the largest and smallest perimeters, respectively. But at the upper middle school level, one can learn that whereas the latter (multiplicative) decomposition yields the rectangle with the smallest perimeter regardless of the number system involved, in the context of rational numbers, the smaller one of the factors (side lengths) is, the larger is the corresponding perimeter. Once again, because a rational number can be chosen as close to zero as one wishes, no rectangle with the largest perimeter exists (whatever the area). Similarly, as will be shown in Chap. 7, the representation of 1/2 as a sum of three unit fractions can be applied to finding all edge-to-edge tessellations with three regular polygons and right rectangular prisms with surface area numerically equal to volume. Furthermore, additive decompositions of unit fractions in like fractions will be used in Chap. 9 as a mathematical model of the so-called work problem involving two or more workers. The above examples demonstrate why “middle grades teachers need to be well versed … in the domains pertaining to whole numbers and fractions … and to know how the topics they teach are connected to later topics so that they can introduce ideas and representations that will facilitate learning of mathematics in high school and beyond” (Conference Board of the Mathematical Sciences, 2012, p. 45). Expanding on the idea of decomposition and the concepts of area and perimeter, one can correct a possible misconception of students about the existence of rectangles with the smallest area and the largest perimeter, given their perimeter and area, respectively. In other words, “new concepts come through the shift of old concepts to new situations” (Schon, 1963, p. 53). Proceeding from additive decompositions of natural numbers and their reciprocals in like summands, one such new situation can be naturally brought to bear by students in the mathematics classroom who, in the quest “to uncover abstract mathematical concepts or results … investigate whether rectangles with the same perimeter can have different areas” (Ministry of Education Singapore, 2012a, p. 23). Consequently, because “nothing happens in this world in which some reason of maximum or minimum would not come to light” [Euler,8 cited in (Pólya, 1954)], a question about a rectangle with the smallest area or the largest perimeter can be raised and a teacher has to be prepared to address it in a mathematically accurate way. By having mathematical questions about naturally unfolding situations answered both procedurally and conceptually,

8

Leonhard Euler (1707–1783)—a Swiss mathematician, the father of all modern mathematics.

12

1 Teaching Middle School Mathematics: Standards, Recommendations …

students would appreciate “opportunities to extend their learning … [through] tasks that stretch their thinking and deepen their understanding” (Ministry of Education, Singapore, 2012a, p. 25).

1.6

Making Conceptual Connections

Nowadays, it goes without saying that teachers of mathematics are expected to teach by “offering students opportunities to learn important mathematical concepts and procedures with understanding” (National Council of Teachers of Mathematics, 2000, p. 3). Yet such teaching skills are not trivial. In order to develop skills that unite procedural and conceptual knowledge, teacher candidates need to possess conceptual understanding of a problem regardless it can be solved by applying a well-defined algorithm. This kind of understanding is critical for enabling a mathematics classroom to be a place where “children can learn, without drill, to deal empirically with situations involving numbers and develop a flexible set of procedures for handling such routine as is necessary” (Association of Teachers of Mathematics, 1967, p. 6). As an example of extracting a concept from a procedure consider the process of division of whole numbers focusing on remainders obtained in this process. Whereas, in general, using an algorithm does not necessarily lead to its conceptual understanding (e.g., correctly applying the “Invert and Multiply” rule when dividing common fractions does not mean that one possesses its conceptual meaning—see Chap. 4, Sect. 4.4.3), a special case may motivate making a connection between well-developed procedural skills and emerging conceptual knowledge (see Chap. 3, Sect. 3.2). Consider a number sequence the first two terms of which are equal to one and each term beginning from the third is the sum of the previous two terms. This is the celebrated Fibonacci9 number sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …. One can be asked to develop another sequence comprised of the remainders generated through dividing two consecutive terms of the constructed sequence, the larger by the smaller, beginning with the pair (2, 3). Through this process, once again, the sequence 1, 2, 3, 5, 8, 13, 21, … emerges. Then, one can be asked to explain why the process of division produced the same sequence (starting from the second term). By answering this question, one can realize that because of the rule through which the former sequence has been developed, dividing the larger number by the smaller number always yields the unit quotient. This, in turn, brings about conceptual understanding of the resulting relationship among the quantities used in the process of division: the dividend equals the divisor times the quotient plus the remainder. Therefore, one way to connect two types of knowledge, procedural and conceptual, could be through the recourse to a special case. This

9

Leonardo Fibonacci (1170–1250)—the most prominent Italian mathematician of his time, credited with the introduction of Hindu-Arabic number system in the Western world.

1.6 Making Conceptual Connections

13

observation connects Fibonacci numbers and the Euclidean10 algorithm (finding the greatest common divisor of two integers, see Chap. 4, Sect. 4.9.1 for details) as an expansion of the idea of the conceptual meaning of division. The instructional shift that Common Core State Standards (2010), the major educational (though controversial due to its innovative approach to the teaching of mathematics emphasizing the primordial character of understanding versus memorization) document in the United States of the second decade of the twenty-first century, called coherence, requires that teachers “know how the mathematics they teach is connected with that of prior and later grades” (Conference Board of the Mathematical Sciences, 2012, p. 1). The notion of connections in mathematics education has many facets and the rich interplay that exists among the concepts of mathematics is one such facet. Therefore, one of the main roles of a mathematics teacher in the classroom is to develop the appreciation of the notion of mathematical (conceptual) connections because they serve as a cornerstone of “the ability to justify, in a way appropriate to the student’s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from” (Common Core State Standards, 2010, p. 4, italics in the original). Is it always accurate to say that there are three chances out of five to pick up (without looking) an apple from a basket with three apples and two pears? How can one evaluate the chances when apples are much smaller than pears? And, after all, where does the rule of calculating chances (to be expressed as a number) come from? In order to have answers to these questions, future middle grades teachers “need courses that allow them to delve into the mathematics of the middle grades … taught with the understanding that the course-takers are future teachers so efforts should be made to connect the mathematics they are learning to mathematics they will teach” (Conference Board of the Mathematical Sciences, 2012, p. 46). More than half a century ago, the teachers of mathematics in England suggested that “teaching which tries to simplify learning by emphasizing the mastery of small isolated steps does not help children, but put barriers in their way” (Association of Teachers of Mathematics, 1967, p. 3). More recently, educators in England suggested that students “must be assisted in making their thinking clear to themselves as well as others, and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions” (Department for Education, 2013, p. 2). In order for students to develop basic mathematical skills, such skills “should be taught with an understanding of the underlying mathematical principles and not merely as procedures” (Ministry of Education Singapore, 2012a, p. 15). For example, the notion of a fixed mindset versus a growth mindset can be understood not only in terms of changeable beliefs about one’s abilities to do mathematics and the levels of achievement in completing tests but, not less important, in terms of automatism versus insight in mathematical problem solving as “sources of insight can be clogged by automatisms” (Freudenthal, 1983, p. 469). For example, the mindset developed through simplifying the equation 3x = 6 by

10

Euclid—the most prominent Greek mathematician of the third century B.C.

14

1 Teaching Middle School Mathematics: Standards, Recommendations …

cancelling out 3 as a common factor in its both sides might prompt the automatic usage of this successful cancellation practice for the inequality x2 \x by cancelling out x as a common factor to get the inequality x < 1 which includes zero and negative numbers as a solution.11 At the same time, an insight could be to recall that only positive numbers smaller than one are greater than their squares, a piece of knowledge developed through the study of common fractions as well as through graphical representation of the functions y ¼ x2 and y ¼ x studied in the middle grades. In order to be able to provide students with help in making connections through integrating different curricular strands, mathematical instruction should be engaging so that students, as educators in England suggest, can develop an appreciation “that there was more than one way of doing things” (Advisory Committee on Mathematics Education, 2007, p. 18) as they reflect on their problem-solving experience. In particular, the use of concrete materials provides diverse opportunities for teaching and learning mathematical concepts and procedures with understanding. As a result, as Canadian mathematics educators believe, “the strategies teachers employ will vary according to both the object of the learning and the needs of the students” (Ontario Ministry of Education, 2005, p. 24). Likewise, when encountering diverse learners of mathematics, teachers in Korea are expected to possess skills to “make questions as open as possible to allow students to solve a problem in a variety of ways … as part of the overall effort to stimulate thinking on the part of the students” (Hwang & Han, 2013, p. 36). It is through learning to think mathematically that teacher candidates develop skills of teaching mathematics for understanding. The chapters that follow include material encouraging productive mathematical thinking, appropriate procedural mastery, and deep conceptual understanding.

1.7

Using Technology in the Classroom

In mathematics courses for teachers, “instructors should model successful ways of using digital tools and provide opportunities to discuss mathematical issues that arise in their use” (Conference Board of the Mathematical Sciences, 2012, p. 50). Such discussion may focus on the fact that as the sophistication of the tools progresses, mathematical problem-solving practices with the use of computing technology, that until recently were considered under the lens of conceptual pedagogies, become less and less intellectually engaging. This reduction in complexity of doing mathematics in the digital era blurs the distinction between procedural and conceptual uses of computers. Put another way, a digital tool with highly sophisticated

11

This automatic cancellation practice can lead to unexpected outcomes when, after several misguided cancellations the numbers 2 and 1 become connected with an equal sign (see Chap. 2, Sect. 2.5).

1.7 Using Technology in the Classroom

15

symbolic computation capabilities enables mathematical problem solving to be reduced to a simple push of a button (e.g., using Wolfram Alpha in factoring the polynomial x3 þ 6x2 þ 11x þ 6 or demonstrating that the sum 2 þ 8 þ 14 þ ::: þ ð6n
4Þ is divisible by 3n
1.) At the same time, the type of technology usage or the level of computational sophistication associated with a particular mathematical activity may depend on what kind of tool is selected to support the activity. For example, in the context of the Geometer’s Sketchpad (Jackiw, 1991), the construction of, say, the point (1, 1) requires just typing the coordinates in the Plot Points dialogue box of the Graph menu of the program. The activity is purely procedural with no intellectual component. Yet, in the context of the Graphing Calculator (Avitzur, 2011), whereas such construction is possible, it requires digital fabrication of the point as discussed in detail in Chap. 9, Sect. 9.9.2. Unfortunately, as mathematics educators in England noted, classroom cultures might neglect the use of new technologies to advance mathematical knowledge despite the fact that students’ activities outside the classroom are heavily embedded into the use of technologies (Advisory Committee on Mathematics Education, 2011). As noted by Australian educational researchers (Eacott & Holmes, 2010), one of the reasons for the decline of students interest in the study of mathematics both at the pre-college and college levels is “the changing nature of the current generation of learners in an increasingly digital age” (p. 84). That is why, around the world, one can find multiple recommendations in favor of using digital technology as “in complex calculations, the effectiveness of learning [by Japanese students] can be enhanced by using computational tools” (Takahashi et al., 2006, p. 149), the appropriate use of computers by Australian students “can make previously inaccessible mathematics accessible, and enhance the potential for teachers to make mathematics interesting to more students, including the use of realistic data and examples” (National Curriculum Board, 2008, p. 9) so that Canadian students can “select an appropriate type of graph to represent a set of data, graph the data using technology, and justify the choice of graph” (Ontario Ministry of Education, 2005, p. 107). Likewise, in the United States, learning how to select “appropriate tools is a critical mathematical practice for middle level learners” (Association of Mathematics Teacher Educators, 2017, p. 100) while their future teachers of mathematics “engage in the use of a variety of technological tools … even if these tools are not the same ones they will eventually use with students” (Conference Board of the Mathematical Sciences, 2012, p. 50). The above citations suggest that the use of technology is also one of the big ideas of the modern-day didactics of mathematics. cBig Idea The use of computers is a big idea of mathematics and its pedagogy for the effectiveness of teaching methods depends on our knowledge and understanding of how the appropriate use of digital tools can support teaching and learning. Whereas, in many cases, computers do enable an easy path to mathematical knowledge, the concepts of mathematics can also be used to improve the efficiency of computations.b

1 Teaching Middle School Mathematics: Standards, Recommendations …

16

In this textbook, the authors will offer the selection of the following digital (computational and/or informational) tools: an electronic spreadsheet, computational knowledge engine Wolfram Alpha, The Geometer’s Sketchpad, The Graphing Calculator, Kid Pix, The Online Encyclopedia of Integer Sequences (OEIS®), and PowerPoint. As appropriate, both positive and negative affordances of a particular digital too will be discussed. Programming details of the spreadsheet environments used in the chapters that follow are included in the appendix.

1.8

Conclusion

This chapter, by reviewing various educational documents available in English, demonstrated a true congruency of ideas penetrating mathematics teaching and learning standards around the world. Solicited over the years written comments by middle grades teacher candidates indicated their aspiration to grasp both the conceptual and the procedural sides of mathematics they will be assigned to teach. With this in mind, the importance of both mathematical and pedagogical content knowledge was highlighted in order to show the main path toward the ‘deep understanding’ of the subject matter by prospective teachers. The chapter discussed the notion of interdependence of mathematical concepts and its foundational importance for recognizing the origins of misconceptions and ways to address them. The issue of using technology in the teaching of mathematics worldwide was discussed and digital tools to be used in the textbook were listed.

Chapter 2

Modeling Mathematics in the Digital Era

2.1

Introduction

cBig Idea A big idea developed within this chapter concerns a dual pedagogical role served by mathematical models. Models are developed which mathematically model real-world events as well as pedagogical models used to model the mathematics itself.b

This chapter addresses mathematical modeling from two perspectives. The first perspective is that of modeling mathematics itself through the use of hands-on experiences with actual objects. This approach will be further developed in Chap. 3 as a means to allow for the development of plausible reasoning and proof even at a concrete operational level. The second perspective is that of modeling real world events using the principles of mathematics. When developed jointly, these paired perspectives allow for a rich set of flexible content understandings to be explored that can be applied to a broad variety of later situations. Furthermore, when this is done insights developed from the application of one modeling perspective often easily apply to problems originating in the other.

© Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_2

17

18

2.2

2 Modeling Mathematics in the Digital Era

Modeling Mathematics and Mathematical Modeling

A simple example of the application of this paired approach toward modeling can be shown utilizing area of a rectangle concepts. Although relatively obvious, problems involving area of a rectangle can be used to model simple multiplication. For this example, consider that area is understood to be the number of unit squares (of area 1 square unit) required to totally fill a shape and the problem posed is that of finding the area of the following rectangle (Fig. 2.1). After some measurements are taken, assume that it is found that the dimensions of the rectangle are 4 units by 6 units and the rectangle is labeled to reflect this as shown in Fig. 2.2. Finally, to make it as clear as possible, Fig. 2.3 shows the result when each of these unit distances are drawn. This not only shows the distance of each dimension in units, but when completed shows the unit squares making up the area of the original rectangle. At this point the sketch is ready to use for modeling. Each of the dimensions have been determined and clearly labeled with the unit distances indicated. As this is done, the unit squares making up the area itself are constructed. All that is left is to count them. As shown in Fig. 2.4, it should be no surprise that no matter what direction or method is used to count the number of unit squares the result is 24. Additionally, during the process of counting patterns naturally emerge. Counting first from left to right and then top to bottom, for example, gives the following for the last value in each row: 6, 12, 18, 24. A different direction, top to bottom and then left to right, gives the following for the last value in each column: 4, 8, 12, 16, 20, 24. The modeling connection between this concrete activity and the abstract multiplication 6
4 ¼ 24 is easily established. The reciprocal task of constructing a rectangle matching a multiplication problem can also be easily accomplished from this background. As indicated earlier, insights from one modeling approach can be directly applicable to the other. For example, consider the understandings which can be drawn from the situation shown in Fig. 2.5 where the original rectangle is rotated by 90 . This rotated rectangle is the same as the original, therefore its measurements should likewise be identical—4 units in one direction and 6 units in the other. Since the rotation does not change the rectangle apart from orientation its area is the same. The only change is what constitutes left to right and top to bottom when counting the unit squares. This orientation is reflected in the related multiplications of either 4
6 or 6
4. Applying this to the abstracted multiplication it is easily seen that 4
6 ¼ 6
4 ¼ 24. Pushing the modeling a little further, if we have a rectangle of arbitrary integral dimensions a and b it can be drawn in the two orientations shown

2.2 Modeling Mathematics and Mathematical Modeling

19

Fig. 2.1 Initial rectangle

6

Fig. 2.2 Labeled rectangle

4

6

Fig. 2.3 Labeled rectangle showing unit distances and emergent unit squares

4

6

Fig. 2.4 Counting the unit squares

4

24

in Fig. 2.6. As before, since the rotation does not change the rectangle apart from orientation, the area remains the same with the only change in what constitutes left to right and top to bottom when counting the unit squares. These simple insights, developed from modeling multiplication via area, have now led to significant insights regarding multiplication. In particular, it follows

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2 Modeling Mathematics in the Digital Era

Fig. 2.5 Rotating the rectangle by 90

a

Fig. 2.6 Rectangles of equal area

b a

b or

from Fig. 2.6 that a
b ¼ b
a. Given the importance of the commutative property of multiplication this is clearly not a trivial development. Given that multiplication and division are inverse operations, the area model can be used to explore division as well. In this modeling, division can be interpreted as creating the number of equal parts specified and counting the number of unit squares within each equal part. For example, 24  4 could be shown as in Fig. 2.7. In this model, each row of 6 is an equal-part division of 24 by 4. Counting the number of unit squares within each row confirms the result of 6. Taking this insight and applying it to the abstracted case of a rectangle of arbitrary dimensions a and b, ða
bÞ  a could be shown as in Fig. 2.8. Each row of b is an equal part division of ab by a. If it were possible to actually count the number of unit squares the result would be b. Extending this even further, it would now possible to model the entire set of Fact Families1 for multiplication and division using the area model approach as shown in Fig. 2.9.

1

Fact Families in this context refers to the set of four related multiplication and division facts using the same three numbers.

2.2 Modeling Mathematics and Mathematical Modeling

21

6

Fig. 2.7 Division within the area model: 24  4

4

Fig. 2.8 Division within the area model: ða
bÞ  a

Fig. 2.9 Fact Families for Multiplication and Division

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2 Modeling Mathematics in the Digital Era

2.2.1

Modeling Mathematics: Selecting a Referent or Object to Act upon

One of the more important aspects of modeling mathematics is to ensure that the models utilized not only reflect the underlying mathematics but are easily understood by both student and teacher. They should also lend themselves to ready applicability and generalization without reliance upon special case considerations. As this section illustrates, the choice of referent can have long lasting impacts which can influence student understanding and conceptual development. cBig Idea Mathematical models should both reflect the underlying mathematics and be meaningful to the learner. This dual requirement is one of the big ideas of mathematics education.b An example outlining some of the considerations for referent selection when developing fraction concepts will be used to show the importance of this selection process. In this example, a part-whole interpretation of fractions will be developed via two approaches—a unit square and a unit circle. Or, as will be referred to here— the cake and the pizza models. In such part-whole interpretations of fractions the denominator indicates the number of equal parts created in the whole and the numerator shows how many of those equal parts are included in a particular fraction. As will be done in these paired “cake” and “pizza” examples, this representation is often accomplished using an area model. Although a unit square and unit circle are used in these examples, these are clearly not the only options. In thinking of a cake, for example, it is easy to imagine cakes with a rectangular shape. Such cakes could certainly be used in part-whole representation and may even offer advantages over the square cakes referred to in this section. The choice of a square cake in this section was made to make the contrast with unit circles, the pizza case, more immediately obvious. Nor are these the only options, in Chap. 4 for example, other popular representations such as a rectangular pie and a “tape” model will be explored. In the paired development illustrated here both models will draw upon the same mathematical understandings and shared experiences. The reasoning behind each model will follow the same structure as will common sets of actions. The primary difference will be the geometric object serving as a referent upon which these actions will be carried out. In one case, a unit square (cake) will be used. In the other, a unit circle (pizza) will be used.

2.2 Modeling Mathematics and Mathematical Modeling

2.2.1.1

23

Initial Modeling Considerations

Many of the difficulties students often have in the area of fractions is due to the lack of accurate, powerful, and easy to use models to illustrate the underlying concepts. Lacking such unifying representations, the student commonly falls back on poorly understood and developed procedures. Adding insult to injury these rule-based procedures are often misapplied with erratic results. For instance, 12 þ 35 ¼ 47 is an example familiar to nearly every mathematics teacher. In such cases, students are confusing two previously memorized fraction “rules”, one for use in addition and another to be used in multiplication. When such rules are memorized as procedures to follow and lack context confusion can arise when they are recalled for later applications.2 As a result of this confusion, a previously memorized procedure to multiply numerators and denominators together to get the answer, which is appropriate when multiplying, is misapplied to the case of addition. Lacking a model or referent this result is accepted without question. One of the authors [MLC] asked a student if this answer (which had been derived using this particular “method”) made sense to them. The response was immediate and discouraging. “No, but it makes as much sense as anything else I have done in math today.” To combat this, the referents used to model fractions and generate the operations of addition, subtraction, multiplication, and division should be both easily understood and generalizable.3 In particular, we must ensure that the models used with our students enable a firm grasp of the essential concepts underlying fractions. These should include: (a) how to represent fractions and what this representation means; (b) how to compare fractions in a meaningful fashion; (c) how to model the basic operations of addition, subtraction, multiplication and division; and (d) how to link between models and the procedures that are derived from these models. The first step in developing such a robust model is to identify a common set of experiences within which to ground the emerging model. In this case, early activities involving sharing provides a powerful set of experiences to use when building meaning for the denominator of a fraction. Nearly every student has had the opportunity to share with another— whether it is toys, time, money, or any of a broad variety of materials. This common set of experiences provides an important background within which to begin the development of fraction understanding. For example, most students realize that it would not be fair to share twelve pennies with three people like this:

2

It is possible that an alternative source of confusion when adding fractions could arise from an incorrect extension of the addition of integers to the addition of fractions. 3 Additional considerations will be developed in Chap. 3 using discussions of classroom considerations for the development of plausible reasoning. One of these, used in these paired examples, is identification of common experiences and understandings.

24

2 Modeling Mathematics in the Digital Era

Person 1 ! 1 Penny Person 2 ! 1 Penny Person 3 ! 10 Pennies To share fairly requires that each person in the sharing operation receive the same amount. This commonsense interpretation can be now applied to sharing activities in mathematics where it is required to share fairly. The fair sharing version of the twelve pennies with three people example becomes: Person 1 ! 4 Pennies Person 2 ! 4 Pennies Person 3 ! 4 Pennies This notion of fair sharing provides a common background upon which the paired fraction models, the cake and the pizza, can rest. As a first step, consider introducing some common-sense vocabulary based on this background, such as Share With, for the fraction bar. This vocabulary serves as a link between the later formal symbols and the well understood actions used to develop the underlying conceptual understandings. This is in marked contrast to the symbols themselves presented as an abstraction to be committed to memory devoid of meaningful action. Once this vocabulary is in place, it is now possible to read 2 as “share with two”, 5 as “share with five”, and so on (Jencks & Peck, 1987). In this reading, the action of “sharing” is developed in a manner that enables it to be applied to many different types of objects. In particular, students should be given the opportunity to “share” both discrete objects (such as counters, pennies, buttons, pennies, etc.) as well as continuous objects (clock faces, meter sticks, graph paper chunks, etc.). On a clock face, for example, the instruction 4, read as “share with four”, would result in creating four sections of fifteen minutes each, while on a meter stick the instruction 4, still read as “share with four”, would result in creating four sections of 25 cm each. It is possible to show such actions using either the actual continuous objects by marking off the sections with yarn, or by marking equal sections on pictures of these objects. These initial experiences expand the modeling process by grounding mathematical sharing (and hence of the denominator and division in general) to concrete actions that are easily demonstrated and understood. The symbol n, for example, is now not merely an abstracted symbol, but can be understood as a description of an assimilated action—take the object(s) under consideration and create n equal groups.

2.2.1.2

Cakes and Pizzas

Once these initial modeling considerations have been established, it is time to select a referent. In these paired examples a part-whole fraction interpretation will be developed. In part-whole interpretations a geometric object is routinely selected to

2.2 Modeling Mathematics and Mathematical Modeling

25

serve as a referent upon which these actions will be carried out. The importance of referent selection can be demonstrated by reflecting upon some factors to consider when selecting which geometric object to use. Consider how a part-whole interpretation of fractions might be developed via each of two referent selections—a unit square (cake) and a unit circle (pizza). These two objects, which we might choose for students to perform a “share with” action upon, are both familiar to students together with several important actions (such as cutting), which may be performed upon both cakes and pizzas. Coupling the earlier “share with” understandings (i.e., to share an object is to divide it into a specified number of equal parts) with the unit square based “cake” and unit circle based “pizza” we have both the appropriate actions and objects upon which to model fractions. It should be noted at this point that a little “let’s pretend” on the part of both teacher and student is helpful. In particular, whether a cake is drawn with great precision using a graphics program or sketched in haste on a student’s paper, it must be understood to represent the same underlying cake. Some middle school students have gone so far as to laughingly refer to these as magic cakes due to this “let’s pretend” consideration [MLC]. Consider what understandings emerge when the action 3, read as “share with three”, is performed upon both objects. In each case, a cake or pizza sketch is to be divided into three equal parts. These initial sets of actions are shown in Fig. 2.10. Similar activities will need to be repeated until students are able to accurately “share with” any instruction with a cake or pizza quickly. There are two modeling considerations that must be developed at this time. First among these is that each cake or pizza—whether drawn on the board or sketched on a students’ paper— represents the same idealized object and is exactly the same size and shape. Students need to play a bit more of “let’s pretend” at this point, but then this is what is required when considering an idealized object. One of the authors [MLC], has found that the more time the teacher spends on trying to make everything look perfect the less the students understand the underlying concepts being developed. The second consideration is that although each share may not look perfect, they really are all the same size.4 This is easily understood since no matter the amount of care taken it is impossible to draw perfection. Taken together, these two “let’s pretend” considerations serve to ensure that the students are performing equal partitioning upon equal items, essential if our later explanations are to be effective in the long term for them. These two considerations are not overtaxing to most students and quickly become a part of the background knowledge as they work with these actions and objects. These activities develop a robust understanding of the “share with”, or denominator, portion of fractions symbol. In developing the conceptual background for the “take” portion, or numerator, of the fraction symbol. This can be done now quite easily by reading the symbol 12, as “share with two and take one share”. The 4

See also Chap. 1, Sect. 1.3 for a deeper discussion concerning the use of self-drawn diagrams of fractions.

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2 Modeling Mathematics in the Digital Era

Fig. 2.10 Applying the action 3 to a cake and pizza

Fig. 2.11 Applying the action

1 2

to a cake and pizza

result of performing this action on both the cake and pizza is shown in Fig. 2.11 where the “take one share” action is indicated by shading an appropriate share. At this point in the modeling process the symbol 12 need not be viewed as an abstracted symbol, rather it is understood to be an instruction to follow in performing well-understood actions upon our cake and pizza objects. The symbol 12 instructs the students to draw either a cake or pizza, share it with two, and then “take” one of these two shares with the “taking” of shares shown by shading in the corresponding shares.5 At this point the students have the understandings necessary to represent any fraction using this model and identify the appropriate fraction symbol (e.g., 12) when presented with a representative sketch. Being able to thus represent fractions, places students in a strong conceptual position to develop models of the operations of comparison, addition, subtraction, multiplication and division. From a purely representative perspective both underlying objects, the cake and pizza, perform equally well. The actions, share and take, apply well to both and the resulting representations are easily understood. Once students move beyond representation and into fraction operations themselves, however, the importance of

5

A suggested method to use when representing two fractions with the cake model is to show the first one as being “shared” horizontally with the shares “taken” from top to bottom. The second fraction will then be “shared” vertically with shares “taken” from left to right. Although this is not required, this convention makes the subsequent operations of comparison, multiplication, and division seem much more natural to the students.

2.2 Modeling Mathematics and Mathematical Modeling

Fig. 2.12 Initial representation of

1 4

and

2 3

27

using both models

referent selection becomes obvious. This can be shown by illustrating comparison of 14 and 23 using both models. The first step in performing this comparison is to represent the two fractions to be compared. Following the actions developed thus far, this representation can easily be done using both cake and pizza as shown in Fig. 2.12. Although it is tempting at this point to make the comparison based upon simple visual inspection, this clearly is insufficient. In order for a comparison to be truly valid, it must be shown that each created section to be compared has the same area. A simple way of doing so for the current example would be to redraw each representation so that it can be shared with both 4 and 3. Figure 2.13 shows how this might be accomplished through cutting each of the fourths into three parts and each of the thirds into four. Different shading has been used for each share to make this action easier to observe. Each of these representations have now been shared with both 4 and 3 resulting in 12 equal-area pieces. A simple counting finishes the comparison with the result that “share with three and take two” is larger than “share with four and take one” regardless of whether a cake or pizza was used. This result is shown in Fig. 2.14 which also provides a suggested record for each set of actions. In this record, the number of equivalent pieces is shown in the small circle over each fraction. A template which can be used for future actions can be made by removing the work portions of the initial problem. The resulting record, Fig. 2.15, can now serve as a template to model additional problems and the remaining set of operations. The common form provided by the template creates a common unifying referent upon which future actions may be performed. The differences between the two referents are now quite clear. The cake model, due to the ease in which it can be divided both horizontally and vertically, was ideally suited to create equal divisions of 4, 3 and 4
3. While this task can also be accomplished using the pizza model, it was much more tedious and the resulting record of action confusing at best. Furthermore, when the cake model is used to model the general case of comparing of AB and DC additional insights are easily obtained. In this generalized example shown in Fig. 2.16, AB is understood to mean create B equal parts and shade in A of these B parts. Likewise, DC is understood to mean create D equal parts and shade in C of these D parts.

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2 Modeling Mathematics in the Digital Era

Fig. 2.13 Comparison of

1 4

and

2 3

using both cake and pizza models

Fig. 2.14 Comparison of

1 4

and

2 3

showing a suggested record of action

Fig. 2.15 Blank record of action template suitable for additional modeling

Completing the representation so that a generalized comparison may be made is accomplished by redrawing each representation so that it can be shared with both B and D. Figure 2.17 shows how this might be accomplished through cutting each of the B parts by C and each of the D parts by A. As shown in Fig. 2.17, the generalized cake model, due to the ease in which it can be divided both horizontally and vertically, is ideally suited to create equal divisions of B, D and B
D:6

6

See Chap. 4 for an alternative explanation of fraction comparisons.

2.2 Modeling Mathematics and Mathematical Modeling

29

Fig. 2.16 The Cake Model for generalized representation of

Fig. 2.17 The Cake Model for generalized comparison of

A B

A B

and

and

C D

C D

Looking at Fig. 2.17 more carefully, it is easy to see that the number of equivalent shares in AB is equal to the area of the rectangle of sides of length A and D. Likewise, the number of equivalent shares in of DC is equal to the area of the rectangle of sides of length B and C. The common denominator can also be seen as consisting of the area of the rectangle B
D. At this point the remaining operations can be easily shown. The original choice of unit square, or cake, lends itself to actions that are both simple to perform and powerful in their representational ability. The resulting record of actions, each done through the use of this sketch model, create deeper and easily understood meanings that serve to model not only fractions at a conceptual level, but also the subsequent

30

2 Modeling Mathematics in the Digital Era

Fig. 2.18 Modelling 1n ¼ n þ1 1 þ

1 nðn þ 1Þ

for n ¼ 2

operations of comparison, addition, subtraction and division. Once it is clear how to model a fraction, everything else falls into place. This is as it should be and serves to illustrate the importance of understanding the concepts. Once a concept is truly understood later procedures will fall into place quite easily. Consider, for example, how the cake approach toward modeling fractions can provide an entry point to developing the powerful generalizations drawn from unit fractions, see Chap. 4, Sect. 4.5.2. In particular, 1n ¼ n þ1 1 þ nðn 1þ 1Þ. If 1n is indeed the sum of n þ1 1 and nðn 1þ 1Þ then the cake representation for 1n must be able to be modeled by a cake with sides n and n þ 1 in order to reflect the common denominators required for addition to take place. When this is done, 1n can be shown to contain n þ 1 pieces, n þ1 1 can be shown to contain n pieces, and nðn 1þ 1Þ can be shown to contain 1 piece. So, 1n ¼ n þ1 1 þ nðn 1þ 1Þ. The case for n ¼ 2 is shown in Fig. 2.18 as an illustration. In closing, the importance of practicing any model until it is thoroughly understood and can be effectively applied must be stressed. One of the authors [MLC] remembers a teacher-candidate in an early mathematics methods course. Although able to use the fraction models as shown in Figs. 2.10, 2.11, 2.12, 2.13, 2.14, 2.15 and 2.17, the candidate always checked her work using the standard procedures which she had initially learned as a student. Even though she was among the first to admit that these rules were not understood, the teacher candidate stated, “I would like to believe these models, but my rules just won’t let me!”. As this comment serves to illustrate, it is important to practice with any model until not only our rules will let us believe them, but that they will become our default manner to model, present and solve problems.

2.3

Modeling Mathematics

cBig Idea The use of technology, when guided by pedagogical considerations, can greatly support mathematical modeling and exploration. The particulars of how this extension may be accomplished is one of the big ideas of this section.b

2.3 Modeling Mathematics

31

This section presents several easily explored situations within which mathematical models can be developed. Each of these would be classed as being Technology Immune-Technology Enabled (TITE) problems (Abramovich, 2014a) and provide multiple opportunities for mathematical modeling incorporating spreadsheets and other technology tools (Abramovich, 2019; Connell & Abramovich, 2017a, b). TITE problems, while requiring technology-enabled (TE) solution strategies, are still technology-immune (TI) in the sense that they do require using mathematical reasoning not replaceable by the modern-day technology despite its various symbolic computation capabilities. The more technology grows in the scope and sophistication of symbolic computations, the more challenging for mathematics educators is to preserve TI components of traditional problem-solving activities. TITE problems cannot be automatically solved by software; yet the role of technology in dealing with those problems is critical. The examples Stamping Functions and Burning the Candle will illustrate how the explicit inclusion frame-based data-tables serving to record initial hands-on experiences, has shown itself to be extremely useful in mediating between informal initial hands-on experience and the developed mathematical models required (Connell, 2019a). Furthermore, these data-tables serve as a guide to apply more abstracted technology-based tools such as spreadsheets. The Flagpole Factory will present an easily visualized model within which systems of equations can be introduced and explored (Connell, 1988; Connell & Ravlin, 1988). Of particular importance, as students engage in these problem situations, the data they collect will reflect the underlying mathematical principles being illustrated within the model. When this data is organized into a data-table (Figs. 2.20 and 2.23), it will be seen that the data-tables not only reflect underlying mathematics, but also an underlying cognitive “frame” model similar to that proposed by Davis (1979) and further delineated by West, Farmer and Wolff (1991) as either a “frame of Type I or Type II”. As presented by West and colleagues, a Type I frame is characterized as being a row and column arrangement of information within which there is a logical, algorithmic, or computational connection between either the rows or columns; a Type II frame is characterized as being a row and column arrangement of information within which both rows and columns are related by logical, algorithmic, or computational rules. In the examples Stamping Functions, see Sect. 2.3.1, and Burning the Candle, see Sect. 2.3.2, the data-tables which can be constructed along these frame-based properties are a nearly ideal mediational tool leading to later spreadsheet use (Connell, 2019b). By providing a standard mediational mechanism such data-tables, patterned after frames of Type I and Type II (West et al., 1991) possessing a logically based row and column structure, serve to mediate students’ initial experiences into abstractions sharing a common interpretive framework.

32

2.3.1

2 Modeling Mathematics in the Digital Era

Modeling Equivalent Surface Area Functions— Stamping Functions

The Stamping Functions activity is a TITE problem which requires students to use a variety of technologically supported tools to investigate changes in surface area as base-ten blocks—in this example, the hundreds flat—are stacked in differing configurations. Similar to the development of the “cake” fraction model outlined in Sect. 2.2.1.2, base-ten blocks of different types (ones, tens, hundreds, and thousands) are used as the objects to act upon, or referent, together with computers/ tablets with supporting software installed and internet access. As will be shown, this activity supports modeling of functions as well as links to concepts developed in Chap. 5. In the Stamping Functions activity, the Hundreds Flat and Ones Cube (Fig. 2.19) are used as the referents, or objects being acted upon. This problem asks one to imagine that the ones cube is a “stamp” which can be used to paint a single square of the hundreds flat. The task is then that of counting the number of times needed to “stamp” the hundreds flat using the ones cube to completely cover all exterior squares—in other words, to determine the surface area of the hundreds flat. Students typically quickly arrive at the same answer, 240, but are often rather surprised when they realize there are multiple differing ways of arriving at this answer. For example, it is possible to arrive at the answer by literally counting the square top of the flat (100 stamps), the square bottom of the flat (100 stamps), and each of the four sides of the flat (10), (10), (10) and (10). This could be recorded as: 100 þ 100 þ 10 þ 10 þ 10 þ 10 ¼ 240. Another approach arrives at the answer by first computing the area of the square top of the flat ð10
10Þ and multiplying this by two because the top and bottom squares are the same size and shape. This can then be added to four times the length of one side 4
10 to account for each of the four “edges”. This approach can be recorded as 2
ð10
10Þ þ 4
10 ¼ 240. Yet another approach, such as 20 þ ð10
22Þ, can be obtained by considering the two “sides” of 10 stamps each and the 10 “bands” of 22 stamps each making up the hundreds flat, can be used to obtain the same answer of 240.

Fig. 2.19 The hundreds flat and ones cube

2.3 Modeling Mathematics

33

Fig. 2.20 Stacking hundreds flats

In a simple form, the transitive property states that things equal to the same thing are equal to each other. So, if any of these approaches arrive at the same result as another it must be that these approaches resulted in procedures that are equivalent to each other. From this background, based upon physical explorations using base ten blocks, students can quickly come to the realization that the methods they used to generate their answers were equivalent. So, for the example illustrated thus far, since 100 þ 100 þ 10 þ 10 þ 10 þ 10 ¼ 240, 2
ð10
10Þ þ 4
10 ¼ 240, and 20 þ ð10
22Þ ¼ 240, then 2
ð10
10Þ þ 4
10 ¼ 100 þ 100 þ 10 þ 10 þ 10 þ 10 ¼ 20 þ ð10
22Þ. At this level, this provides an exercise in arithmetic. When the general case is presented, however, the problem quickly leads to the demonstration of the equivalence of algebraic expressions. To see how this emerges, consider Fig. 2.20 in which N is used to represent the number of hundreds flats and shows the first five hundreds flats leading to the general case of N hundred flats stacked atop one another. At this point, the data table, shown in Fig. 2.21, was developed via discussion and reference to the original referents as earlier shown in Fig. 2.20. The “How this works” descriptions, shown in Fig. 2.22, were in turn derived from data tables, see Fig. 2.21, used to record the counting method utilized. For example, each approach

Fig. 2.21 Sample data-table for Stamping Functions

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2 Modeling Mathematics in the Digital Era

Fig. 2.22 Generalizing to a formula using a spreadsheet

Fig. 2.23 Alternate stackings of hundreds flats

in Fig. 2.22 shares a realization that the number of stamps needed for one side of the stack is equal to the number of stacks, N, times the length of the hundreds flat, 10. These instructions were entered into the spreadsheet using a “point and select” method (Connell, 2019a). So, cell D3, which represents the stamps needed for one side of the stack, was entered by first selecting D3, pressing “ = ”, and then using the arrow keys to select the cell containing the value to be first used—C3, pressing the appropriate operation symbol “*”, then using the arrow keys to select the next value to be used—B3, and finally pressing “Enter” to complete the instruction entry. The overall formula for the Stamps needed, C3 * C3 * 2 + D3 * 4, was entered in cell F3 in a similar fashion. Each of the resulting formula reflected in the function bar was

2.3 Modeling Mathematics

35

thus entered by following a quite literal interpretation of concrete actions such as “take the length of the side and multiply it by the number of hundreds flats in the stacks”. Once the descriptions are entered and their results verified by checking with the original data table the formula was copied and pasted into each of the following rows. As this is done it is easily seen that many of the columns have values that are constant, such as the length of the hundreds flat, others are variable such as N—the number of hundreds flat. This gives rise to the constants and variables reflected in the formula bar and in the notation explaining how the number of “Stamps needed” for the nth row is to be calculated. In Fig. 2.22 two different student spreadsheet approaches toward addressing this problem are illustrated. Since each of these approaches result in the same number of stamps needed it follows that the instructions reflected in the formula bar, even if they appear differently as a result of the differing instructions which were used to generate them, represent equal functions. For the cases shown in Fig. 2.21 this means that the left-side spreadsheet function results in equal values to the right-side spreadsheet function. So, 10
10
2 þ 10N
4 ¼ 100 þ 100 þ 10N þ 10N þ 10N þ 10N, or in standard algebraic form 40N þ 200 ¼ 40N þ 200. Since there are many different approaches which might have been taken in addressing this problem the “How this works” area for each approach would have differing instructions for each of the methods adopted. Each of these instructions, when entered into the spreadsheet using the “point and select” method in turn gives rise to an equivalent function. In these two examples their algebraic equivalence is easily shown, however not all approaches are so easily demonstrated. Furthermore, this is obviously only one of many possible ways of stacking hundreds flats. Figure 2.23 shows several alternate physical arrangements requiring varying levels of sophistication to address. In each of these subsequent scenarios, students are tasked with coming up with a method of instructing the computer spreadsheet how to calculate the number of stamps necessary to completely cover N of the hundred flats as described. Finally, since this data is dependent upon the number of hundreds flats there will always a relationship between the rows comprising the data-table, a sign that this is a frame of Type I (West et al., 1991). This activity has been used in multiple forms by one of the authors [MLC] since the mid 1990s. As such, it has never failed to create great interest and multiple “aha!” moments on the part of the students. One student described their experience, “The stamping project gives us an idea of how to implement technology, in this example, excel, to see in person how the problem can be transferred into an image. … it allows us to record our work in a more organized way. For this problem we had to record all the ways we can add up the units that could be stamped in each picture. With this system we were able to have everything laid out in one document in an organized way.” As this discussion illustrates, this process highlights multiple connections between hands-on action, emerging records of action, metacognitive considerations, and use of technology to build a technology enhanced record suitable for further exploration.

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Modeling Correlations—Burning the Candle

Burning the Candle uses a variety of technologically supported tools to investigate some of the various factors surrounding the burn rates of candles. Students are provided with unused birthday candles of various lengths and diameters together with lighters, timers of various sorts, computers/tablets with supporting software installed and Internet access. As students gather data and plan their analysis, multiple student-posed questions are typically developed. Among these: “Is the burn rate constant over the length of the candle?”, “How long would it take to totally consume a candle of x length?”, and “What causes the change in burn rate that I observed?”. As definitions are developed and explored (burn rate, for example), students should be guided to identify potential knowns and unknowns in the problem situations. This will aid the students as they set up potential relationships between knowns and unknowns and make their exploration plans. Although it is helpful for some students to standardize the length of time between measurements, this is not essential. As long as the final data will allow for a ratio such as CandleLength UnitOfTime to be developed, the particular units are of less importance. It is quite common for different groups of students to utilize different tools in this exploration. For example, one student might select to use a ruler and stopwatch while another might choose to use a smart phone to create a digital recording of a candle being burned in front of a graduated background which can be later played back and analyzed on a laptop computer. Regardless of the choices of method and tools, however, similarly structured data-tables as shown in Fig. 2.21 can be generated as records of the initial action. A typical student data-table for this activity is shown in Fig. 2.24. Since the variables corresponding to the length of the candle and the time which the candle has burned are correlated, there is a necessary correlation between the rows and the columns. This correlational structure of an underlying link between the eventual rows and columns indicates a Type II frame (West et al., 1991). Together with this, key vocabulary such as precision, accuracy, average (over a set of measurements), outliers, and so on, can now be developed in context. Student groups of no more than four and no less than three students are recommended as the ideal group size to explore this problem within. As these groups of middle schoolers share their findings, the variables under consideration will quickly became standardized through sharing of intermediate results. Once this occurs, the group data tables become nearly identical—leading to a relatively common spreadsheet format independently generated by nearly every group. It is not uncommon for many middle school students for this to be their first experience with consequential measurement. For such students, the measurements required in data collection are important skill sets to acquire. One member of a student working group of four put it like this, “When we first started … we were very confused on whether we needed to measure the wick of the candle or just the waxy part of the candle. We started by measuring the wick but quickly noticed that

2.3 Modeling Mathematics

37

Fig. 2.24 Sample of data-table for Burning the Candle

it was not changing. We started over and did not measure the wick and we then started to see a change in the length of the candle. We also noticed a pattern that for about 3 fifteen second trials the length would stay the same and this happened twice during our 15 attempts. We also realized that the angle that we lit the candle made a huge difference in our data. We felt we got a more accurate number when R lit the candle from the top rather than at an angle from the side.” As this comment serves to illustrate, the importance of the data collection phase cannot be underestimated. Many students have had multiple experiences analyzing data sets provided to them from a text. Yet, students rarely have the essential experiences of identifying what data to collect, how to organize it, and how to create a data table for follow up analysis. These skills are assumed but rarely present in many mathematics methods courses.

2.3.3

Modeling Systems of Equations—The Flagpole Factory

By providing a plausible narrative together with an easy to apply graphic, problems supported within the Flagpole Factory model allows pre-algebra middle school students the opportunity to explore introductory systems of equations. In the flagpole world, the students need to imagine a factory with two different machines used to manufacture flagpole sections. These machines each can construct a flagpole of any specified length with the only constraint being that once they are set for a day each flagpole constructed on that day by that machine will be of the same length. Although it would be possible for each machine to be set to the same length,

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this is not done to save money. Rather than running both machines to make the same length, it is more economical to simply shut one of the machines down for the day. In Fig. 2.25 each of the machines has been set to construct a flagpole section of differing lengths and a sample flagpole section from each machine is shown. Since the machines were set for different lengths, one of the sections will be longer than the other. To make this easier to picture, their daily length is labeled with length L for longer and S for shorter, respectively. In addition, for unusual orders it is possible to draw upon a supply of older left-over flagpoles from prior days which can be combined with the current days run to meet various customers’ orders. In the flagpole story these long and short flagpole sections correspond to the variables which would occur in more formally presented algebraic equations. Working within this model, students would initially be presented with a graphic representation showing the lengths of two distinct flagpoles formed from integer combinations of long and short sections with the possible addition of a left-over section of some known length, a feature that will play an important role in the concluding discussion, see Sect. 2.4. Each pair of initial flagpoles thus corresponds to a consistent system of two equations, each of which has two unknowns. For the student, their final goal is to determine the lengths of the long and short sections produced that day when presented with a set of two differing sample flagpoles. In working towards the solution, various operations are available to the student for manipulating the flagpoles themselves. For instance, flagpoles may be made longer by integral coefficients, corresponding to the elementary row operation of multiplying a single equation by a constant; flagpoles can be compared, and the difference computed, corresponding to the elementary row operation of subtracting one equation from another; and so on. For example, imagine the situation in which the student uses a multiple of one flagpole which when subtracted from another flagpole results in the removal of all sections of a certain type. In a formal sense, this accomplishes the elimination of one of the variables. Once the length of the remaining section type is known, this newly discovered value can then be used to determine the length of the other section. At each stage of this process the newly created flagpoles can be shown graphically together with their associated values, if known, and this information is now available for further use by the student. Comparing strategically constructed flagpoles leads to derivation of the lengths of the component sections, equivalent to solving the system for each unknown. Consider the following sample problem: On Monday there were two different lengths of flagpole sections constructed by the two machines. The length of three of the shorter sections and one of the longer sections is forty-five feet. The length of two of the longer sections and one of the shorter sections is sixty-five feet. What are the lengths of each of the flagpole sections? In using the Flagpole Factory model, the first step would be to draw the flagpoles to represent the problem facts as shown in Fig. 2.26. It should be noted that the individual flagpole sections are not necessarily drawn to scale, thus the first flagpole of 45 feet actually appears longer than the second flagpole of 65 feet.

2.3 Modeling Mathematics

39

L

S

Fig. 2.25 The flagpole factory

Fig. 2.26 Modeling L þ 3S ¼ 45 and 2L þ S ¼ 65

Once each flagpole has been represented, various actions can be performed with the ultimate goal of identifying one of the two unknown lengths. For example, in Fig. 2.27 the student has decided to double the length of the first flagpole to get a common number of long sections. This corresponds to the elementary row operation of multiplying an equation by a non-zero constant. Once this is done, it is possible for the students to remove the length of the second flagpole from the first as shown in Fig. 2.28. This corresponds to subtracting one equation from the other. It is now an easy matter to show that S ¼ 25 5 ¼ 5 as shown in Fig. 2.29. By taking this newly found value and entering it into the second flagpole representation not only is progress being made, but back substitution is being modeled. This is shown in Fig. 2.30. When the length of this newly found results is removed, the effect of the known variable is removed as well. This is shown in Fig. 2.31 and the final solution, found by dividing the second flagpole by 2, is shown in Fig. 2.32. All of this work is done within a graphical representation. Otherwise formal operations are made visible through the comparison of developed sketches and the performance of potentially interesting manipulations of these sketch objects. As students work through this problem, both successful solution strategies and false starts can be analyzed by students enhancing the development and application of

40

Fig. 2.27 Doubling the length of a flagpole

Fig. 2.28 Removing one flagpole length from another

Fig. 2.29 Solving for S

2 Modeling Mathematics in the Digital Era

2.3 Modeling Mathematics

41

Fig. 2.30 Back substitution of S

Fig. 2.31 Eliminating the effect of S

metacognitive strategies. Furthermore, when these intuitive graphical operations are viewed from a formal perspective, we see that the student is building a flexible referent suitable for later more formal concepts (Connell, 1988; Connell & Ravlin, 1988).

2.3.4

Data-Tables Revisited

When students address hands-on problems, it is quite common for them to utilize an extremely diverse set of methods. It is not uncommon for differing groups of

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Fig. 2.32 Solving for L

students to choose different tools, different units, different recording schemas, etc. As a result, hands-on problem solving could be viewed as a questionable activity involving occasional chaos, lack of communicability between participants, and weak closure that in some cases only makes sense to the individual participant. Fortunately, if the activities themselves provide data suitable for mathematical analysis there will always be an underlying structure which can be used to unify the student experiences. As students record their measurements, their data gathered will come share this underlying frame regardless of the units or methods used in its creation. In some cases, this structure might only be along a single axis—the number of hundred flats, for example, as in the Stamping Functions activity. In other cases, there might be a correlation between the variables at play—time versus length, as in the Burning the Candle activity. Regardless of the individual case, however, there is an underlying cognitive “frame” which naturally emerges as students use data tables to organize their collected data. The data-tables thus generated will, of necessity, share this underlying frame. For the Stamping Functions case the frame is of Type I, and for Burning the Candle the frame is of Type II. Regardless of frame type, however, the use of data-tables proves to be equally effective in mediating students hands-on experience to an abstracted form that could be entered into the spreadsheet as shown in Fig. 2.33. Once this is done, the spreadsheet serves as a well understood record of their actions. After the creation of this record of action, the power of the technology can be utilized. In particular, the initial record of action can now be the object of further actions on the part of the student. The nature of precisely what these actions are was often dictated by the form the data was recorded in and the tools available within the spreadsheet itself. Figure 2.34 shows one worksheet that illustrates a group’s choice of actions.

2.3 Modeling Mathematics

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Fig. 2.33 A data-table mediated into spreadsheet form as a record of action

Fig. 2.34 Spreadsheet record showing basic Graph and Table views

Depending upon students’ experience and sophistication, more advanced actions are made available when they start acting upon the developed spreadsheet as an object. These might include using the actions enabled by the spreadsheet to create a regression equation, examining correlations between recorded variables, and looking for statistically significant outliers among the classroom records. By broadening the choice of available actions, the mathematics which students can experience can be greatly expanded (Connell, 2019b). Figure 2.35 illustrates some of these more advanced analyses. By creating a standardized mediational framework, frame-based data-tables allow the teacher candidates to organize their work in

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Fig. 2.35 Spreadsheet record showing advanced analysis made possible by technology

powerful structures, and to create formal records of action that may be used in later problem solving.

2.4

The Problem Solving Board

The utility of having a well understood model for mathematical operations would be hard to overstate. One of the major benefits being that such models serve to externalize and make visible otherwise hidden inner mental processes. By providing agreed upon sets of representations, mathematical models allow for students to discuss the data they are using, the relationships between these data, the goals of the problem, and potential solution paths. Furthermore, by recognizing features of the model, students are provided with a set of potential actions that might not otherwise occur to them. When students have a broad variety of models to select from in representing the basic operations of comparison, addition/subtraction, and multiplication/division these models can be used to provide flexibility and encourage creativity in their problem solving. As such, it is helpful to revisit mathematical problem solving when students possess multiple models to represent their thinking. Many of the problem-solving approaches adopted in the mathematics classrooms can be traced back to some commonsense questions first raised by Pólya in the mid 1950s (Pólya, 1957).

2.4 The Problem Solving Board

45

In approaches thus derived from Pólya, problem solving in mathematics is characterized as consisting of four main phases: understanding the problem, devising a plan, carrying out the plan, and looking back. Such characterization is easy to state and unfortunately easy to memorize. A far too common classroom result being that these steps come to be taught as a set of instructions to be memorized and not as a set of guides to direct actual problem solving. When taught in this fashion, if a student can recite the steps in order they are assumed to have “mastered” problem solving. Such “mastery” is illusory at best, the student whose example is shown in Fig. 2.37 had such “mastery” but was initially unable to apply this mastery in actual problem solving. A more productive approach lies in presenting the phases as specific actions which may be performed, documented, and discussed by students in the context of actual problem-solving situations. When this is done, understanding the problem is not the first step to recite, but includes student decision making and activity surrounding the labeling and identifying unknowns, condition(s), and data, and determining the solubility of the problem together with some sense of the operations to be performed. In a like fashion, devising a plan entails identification of an appropriate model or representation, drawing on past experience and knowledge, and even restating the problem if necessary. When this is done the plan can be carried out and checked. This final checking, Pólya’s notion of looking back, includes checking the student’s computational accuracy and adding the current problem to those that have been solved and understood. When understood by the students as guidelines and suggested actions, students can begin to use these approaches with great benefit. Indeed, an important characteristic of Pólya’s framework lies in its generality. Thanks to this generality, such Pólya-based approaches are applicable to a broad variety of problem types. When these frameworks are applied often, and in many different situations, students can gain experience and confidence in problem solving. As Pólya (1957) put it, “If the same question is repeatedly helpful, the student will scarcely fail to notice it and he will be induced to ask the question by himself in a similar situation. Asking the question repeatedly, he may succeed once in eliciting the right idea. By such a success, he discovers the right way of using the question, and then he has really assimilated it” (p. 4). There is a bit of a “chicken and egg” difficulty which arises for many students, however. While this technique has a great deal of power for students who already have had some success in solving problems, it is often very difficult for beginners who lack such experience to grasp and apply. In particular, beginning students often either get lost in the steps or fail to consider the interactions of the phases. cBig Idea The Problem Solving Board is itself a big idea. In application, it serves as a cognitive scaffold for emerging modeling, metacognition, and problem solving.b

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Fig. 2.36 The Problem Solving Board

The Problem Solving Board, see Fig. 2.36, addresses these difficulties by serving as a graphic organizer, a record for solved problems, and a tool to use in solving new problems. This tool is an expansion on the cognitive graphic organizer described in (Campione et al., 1988) and was designed to allow for explicit inclusion of mathematical modeling as described in this chapter. Since its initial use in data collection (Connell, 1995), the Problem Solving Board has proven to be an effective tool to help all students, regardless of prior experience in problem solving, use Pólya’s framework. Today, the Problem Solving Board has been widely adopted by school districts throughout the Houston area and is used yearly in methods courses, workshops, and classroom instruction. During use, each of the five areas of the Problem Solving Board prompts the student with a series of questions corresponding to one of the previously listed phases from Pólya. In addition to providing a consistent prompt in the form of an easily understood question, the Problem Solving Board supports student work by providing a framework to record this work for reflection and shared commentary. The question, “What do we know?”, together with, “What does it look like?”, serve to remind the students of the importance of understanding the problem. As mentioned earlier, this understanding includes identifying and labeling unknowns, condition(s), and data, determining the problem is soluble, and some sense of the operations to be performed. In the larger, “What does it look like?”, area the students attempt to represent the problem situation using any of the mathematical models they have developed in their repertoire. “What should we do?” requires the creation of a plan based upon the representational model and understanding generated in the first two prompts. With this plan in place, students “Solve it!” and “Check it!”. To see how this works in actual practice, consider a student example as shown in Fig. 2.37. This student was part of a remedial summer program designed to reteach the skills necessary for them to move on to seventh grade. As such, the student had already “mastered” problem solving as measured by memorizing the prescribed steps but was totally unable to apply these phases in a meaningful way. This problem being addressed in this example was, “There were 156 fifth graders and

2.4 The Problem Solving Board

47

Fig. 2.37 A student sample showing use of The Problem Solving Board

148 sixth graders at the track meet. There were 18 teachers helping time the events. How many students participated in the track meet?”. The first step in understanding the problem was to identify the relevant facts (knowns), the goal of the problem (finding the total), data (students from each grade level) and solubility (yes). The number of teachers timing the events was appropriately deemed as not relevant to the problem and correctly not included. After carefully reading the problem, 5th graders and 6th graders were both recognized as being students and included as data. Finding the total (total?) is shown as a first thought as to a solution strategy, and shown in the “What do we know?” section, see Fig. 2.37, by the student’s entry: +add. The student then drew this problem situation using an understood mathematical model, addition on a number line, and labeled this model with the appropriate data and goal statements listed in “What do we know?”. It is important to recognize that the “What does it look like?” area can be filled in with any representational model the student is comfortable with. This includes fraction sketches, data-tables, sketches, manipulatives, spreadsheet screenshots, graphs, tables, etc. Obviously, the more models the student understands and can draw upon the greater the representational flexibility which can be applied. For this problem, however, the representational model utilizing the number line for additions was quite simple. The student plan is recorded in the “What do we do?” area using a complete sentence. In this case, “We add 156 5th graders and 148 6th graders to get the sum of

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how many students went to the track meet.” The use of complete sentences in generating the plan is of utmost importance. When a thought can be expressed as a complete sentence there is a full description of what is being done (verb) and the object receiving the action (noun). As will be discussed in Chap. 3, understanding is increased by including both of these features. When “plans” cannot be expressed in such a complete sentence, it generally indicates an incomplete understanding of either the process to be followed (verb) or the data (noun). In this case, the student has correctly included all of the relevant information from the planning phase—“What do we know?” and “What does it look like?”. This inclusion is essential. Occasionally, students will reach this planning phase and make their plans based on ease of solution and not whether the problem itself is actually being addressed (Connell et al., 2018). With this work behind them, the student addresses “Solve It!” and includes the units and labels identified in the prior phases. It is at this point that the recognition that 5th graders and 6th graders are both examples of students pays off. The checking issue is most obviously shown by use of the inverse operation in “How do we know?”. Although sufficient to verify computational accuracy, this only tells part of the story. The student needed to describe how each of the phases led to this answer. This includes verifying that the same data was applied, an appropriate model was selected, the plan reflected the data and the model, the plan was followed, and a correct answer was achieved. The student was successful in this more detailed checking as evidenced by receiving 100% for this problem. The following “prescription” has shown to be highly effective when working with students. First, each day the instructor models a representative problem using the board. This ensures that students will see expert level performance on a daily basis using a consistent record of action. This consistency of presentation helps the students in their selection of questions, mathematical models, planning, and identifying the metacognitive strategies used by the instructor. Second, at least once a week the students should solve a problem of the same type as those being modeled within their working group of four. A single Problem Solving Board should then be created by this working group to serve as a record of their work for review and to share with the total class. In addition, this step allows for discussion to occur within the individual working groups, helps errors to be identified and corrected, and provides a work sample to serve as reference for later problems. Finally, each student has as weekend homework the task of completing a Problem Solving Board showing their individual work. This “prescription” ensures that students will see consistent expert performance, have the opportunity of working with others in an environment where inner mental representations are made visible, and showing individual accountability. At each stage in this process, a record of action is created which can then be used for future reference. As Pólya (1957) suggested, “The teacher should encourage the students to imagine cases in which they could utilize again the procedure used, or apply the result obtained” (pp. 15–16).

2.5 Conclusion

2.5

49

Conclusion

This chapter began by a discussion surrounding the importance of selecting an appropriate representation to use in mathematical modeling. This was presented to emphasize the importance of initial assumptions and why they matter. The importance of starting from a sensible position cannot be overstated. Without such a sensible starting point, nonsense is the only possible outcome. Helpful additional reading, together with multiple modelling examples, may be found in Vision in elementary mathematics (Sawyer, 2003). Consider the equation: Y þ 3 ¼ Y. A former student went about “solving” this without thinking about what initial assumptions have to be present in order for this statement to even be an equation and proceeded along these lines, “There is a 3 in there so, let’s multiply both side by 3.” When asked why, the student stated that it “makes as much sense as anything else I have been taught.” The rest of the solution went like this… 3ð Y þ 3Þ ¼ 3ð Y Þ which is equal to 3Y þ 9 ¼ 3Y so 9 ¼ 3Y  3Y or 9¼0 At this point, even the student recognized something had gone terribly wrong. Oddly enough, however, the student was unable to see exactly what. Lacking an appropriate model to ground the operations within, it seemed to the student that each individual step had correctly applied an appropriate rule adequate for the ultimate solution of the equation. What the student had missed was that, the assumptions underlying the initial equation were not sensible, i.e., Y cannot simultaneously equal both Y and Y þ 3. Even the most basic modeling effort on the part of the student would have flagged this difficulty immediately. In the flagpole world, for example, if Y is the length of a flagpole it is self-evident that Y 6¼ Y þ 3, see Fig. 2.38.

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Fig. 2.38 Y 6¼ Y þ 3

Y+3

Y

3

Y

Y

However, even if we have a reasonable starting point it does not always guarantee that a sensible conclusion will be reached. For example, consider the following argument: A¼B A2 ¼ AB A2  B2 ¼ AB  B2 ðA þ BÞðA  BÞ ¼ BðA  BÞ ðA þ BÞ ¼ B BþB ¼ B 2B ¼ B 2¼1 As in the Y þ 3 ¼ Y example, each individual step seems to have been correctly applied. Lacking an understanding of what assumptions must be met for a step to be taken, however, it does appear that 2 ¼ 1. It is left as an exercise for the reader to identify where the hopefully obvious error occurred. For readers who want to deepen their knowledge and understanding of mathematical modeling the following sources can be recommended (Boyer, 2012; Clawson, 2004; Davis, 1990; Hogben, 1983; Lakoff & Núñez, 2000; Maddy, 2003; Van de Walle et al., 2019).

2.6 Activity Set

2.6

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Activity Set

1. Model the comparison of the fractions 35 and 67 using both the cake and pizza models. 2. Which representation, cake or pizza, would be better suited for instruction? Why? 3. Which fraction, 35 and 67, was larger? By how much? How can you tell? 4. Which operations were modeled in the process of addressing Activity 2? 5. What would the standard forms for these operations look like? 6. Model 35 þ 67 using your choice of cake or pizza models. 7. What would the standard form for this operation look like? 8. How do you deal with mixed numbers in the cake and pizza models? 9. Model the equation 1n ¼ n þ1 1 þ nðn 1þ 1Þ for n ¼ 5. 10. Generate data tables showing two different counting procedures for each of the alternate stackings of hundreds flats shown in Fig. 2.23. 11. Create a spreadsheet similar to Fig. 2.22 showing the equivalences from the counting procedures for each of these alternative stackings of hundreds flats created in Activity 10. 12. What difference would be expected in the Burning the Candle activity if a used birthday candle were to have been used rather than a new birthday candle.7 13. What differences in a Candle Length versus Burn Time graph would you expect when the measurements are made in inches and seconds as opposed to centimeters and seconds? 14. Assume that the average rate of burn recorded in Fig. 2.24 continues. How long will it take for 12 of the candle to be burned? 15. Perform the Burning the Candle activity and construct data tables and spreadsheets corresponding to the first ten 15 s burns. 16. Assume that the average rate of burn from Activity 15 continues. Predict how long it will take for 12 of the candle length to burn. 17. Predict how long it will take to burn the candle completely until the flame goes out. Is this greater or less than your predicted time? Why? 18. Use the flagpole model to attempt to solve each of the following systems of equations: a. 2S ¼ L and 6S ¼ 3L b. 3S ¼ 2L and 3S ¼ 2L þ 1 c. S þ L ¼ 6 and L ¼ 3S þ 2 19. What difficulties did you find when addressing Activity 18? How could you address these difficulties when using the flagpole model in instruction?

7

NOTE: A new birthday candle will have sa conical tip where the wick emerges, a used candle will have a flat surface where the wick emerges.

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20. Use the Problem Solving Board to address the following problem: Shannon works in the local warehouse at 1523 North Washington Drive. At 11:00 a.m., the supervisor requested that Shannon move seven boxes. Three boxes contained TV’s and weighed 34.5 lb each. Four boxes contained computers and weighed 5.25 lb each. How much total weight did Shannon move in responding to this request.

21. Use the Problem Solving Board to address a problem from the grade level you are most interested in teaching.

Chapter 3

Reasoning and Proof

3.1

Introduction

It has occasionally been argued that from a strictly developmental perspective young children are unable to utilize strategies common to plausible reasoning and formal logic. This is then taken as justification by policy makers, curriculum developers and classroom instructors to not focus explicit instruction on reasoning and proof until late adolescence. However, this argument often rests upon inaccurate assumptions drawn from Piagetian theory (Lourenço, 2016). In particular, if an overly information-processing approach is followed, the accepted forms of representation become very narrowly prescribed and overly formal. These representational objects then come to define sets of potential activities which do not lend themselves for adequate explorations using the type of objects and activities used by beginning students. Given the emphasis placed in this book upon the objects utilized in mathematics instruction and the actions performed upon them this difference is profound and significant. A careful reading of Piaget reveals that from his perspective to know an object is to act upon it, and construct the transformation systems by acting on the object or participating in it rather than to form an abstracted formal image of such an object. According to Inhelder and Piaget (1969), to children a “… cube is perceived as a thing which can be handled and turned over and turned (p. 13)”. Compare this with more formal representational definitions such as, “a solid with six congruent square faces. A regular hexahedron”, or “…a region of space formed by six identical square faces joined along their edges. Three edges join at each corner to form a vertex” (Cube, 2011). Starting from this more robust interpretation of object, Piagetian considerations lead to a constructivist view of development and knowledge and not one focused upon a formal interpretation of representation. This difference should be kept in mind when exploring the mechanism underlying the more © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_3

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formal development of plausible reasoning and proof. This is backed up by extensive classroom experience which has repeatedly shown that younger students are capable of advanced reasoning at the concrete operation stage— and even proof—when utilizing developmentally appropriate materials to base their thinking upon (Connell, 2001, 2009; Connell et al., 2018; Fadiana et al., 2019). Indeed, these paired abilities of plausible reasoning and informal proof have come to be at the center of mathematics curriculum beginning at primary and continuing through all levels of mathematics including middle school. For the middle school mathematics educators, however, it is often important to recognize what can constitute a proof when it is presented in a non-traditional format such as might be offered by a learner coming to know at the concrete operational level. This chapter will begin by providing such an example of elementary reasoning at the concrete operational level where the reasoning is arguably very well developed, even if the elements of the proof are developed and justified using non-traditional means and materials. In reading this introductory example, one has to bear in mind the advice of Pólya (1968), who in his discussions of plausible reasoning, stated that “…the advanced reader who skips parts that appear to him too elementary may miss more than the less advanced reader who skips parts that appear to him too complex” (p. vii.). Following this introductory example of elementary reasoning, guidelines drawn from the study of formal argumentation will be used to describe reasoning strategies applicable to the classroom environments which, when adopted, create an environment within which plausible mathematical reasoning, and later more formal proofs, may emerge and flourish. Traditional elements of propositional logic will then be presented together with examples showing how these basic elements of logic can be implemented via programming within a spreadsheet environment. The chapter will end with a problem set with examples from each of these reasoning and proof methods.

3.2

Elementary Reasoning at the Concrete Operational Stage

cBig Idea When student use models they are familiar with it is possible for elementary reasoning to occur at the concrete operational stage. At the middle school level, this is a big idea as it extends instructional use of reasoning strategies beyond that typically utilized.b

3.2 Elementary Reasoning at the Concrete Operational Stage

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The division by 9 example discussed in this section is drawn from a series of classroom observations from a class of fifth grade students and their teacher who had been modeling integer division problems using base-ten blocks. These problems had been presented within a narrative where the base-ten blocks represented candies which were packaged according to increasing powers of ten. The candy factory supervisor got to eat all left-overs at the end of a division problem, thus providing an incentive to ensure the problems were done correctly. After working through multiple problems involving division by 9, one student came up with a method for determining the value of the remainder.1 In reading through this example, note how terms “left-over” and “remainder” are used interchangeably within the student’s argument. Such interchangeable vocabulary use is typical of students emerging understandings. As this example shows, the student knows what to do, can do it with full justification of actions, but does not yet know the “text-book” terminology. In essence we are seeing an emerging concept which, although clear in terms of actions and meaning, has yet to be associated with formal vocabulary. Unfortunately, such emerging concepts with their concurrent lack of formalized vocabulary are often mistaken by teachers as evidence of incorrect or weak reasoning. The student in this example initially stated her method for determining the value of the remainder for division problems (in base-ten) when dividing by 9 (note that the divisor is 10 − 1) like this: “You see all you have to do is just keep adding and adding and adding ‘till you can’t add anymore. This final add-answer is the left-overs.” Although laying out a clear and replicable procedure, albeit in non-standard terms, there is little in this description to show any understanding beyond recognition of a “trick” which happened to work. Such tricks, as helpful as they might be, do not yet rise to the level of reasoning or proof. The student’s next comments demonstrated that this trick, or procedure, was well understood in its implementation and she was able to use it effectively. She followed up by providing an example demonstrating her method in action, “If I am dividing 63,548 by 9 I would add 6 þ 3 þ 5 þ 4 þ 8 ¼ 26, but I can still add so 2 þ 6 ¼ 8. I cannot add further so my remainder is 8.” Again, this is a nice confirmation that the method worked—at least for this case. Still, however, nothing rising to a generalized conclusion or informal proof. At this point, the teacher prompted the student to provide another example verifying that the “trick” worked. This is a natural enough questioning sequence; however, it does not lead to generalizable reasoned conclusions. Whether viewed from a formal or informal standpoint, there is little here to point to any substantial depth of thinking.

1

The classroom example illustrated here is drawn from Connell (2001). This example is extended here to include alternate bases other than 10 and the final extended generalization to base-n when divided by n  1.

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3 Reasoning and Proof TEACHER: Let’s see if you get this one right… try 627 divided by 9. STUDENT: Watch! 6 and 2 is 8 and 7 is 15, but I can still add some more... 1 and 5 is 6. So the remainder is 6.

It is worth noting at this time that this question, although perfectly reasonable and in line with what developmentally appropriate pedagogy might suggest, prescribed the nature of the student’s response. By asking for another worked example, however, this question merely developed this example and not a reason as to the underlying reasons or causes. This type of questioning, although natural to pursue, is not conducive to the development of plausible reasoning or proof. As will be later shown when basic elements of plausible reasoning are discussed, this line of questioning would lead to justifications of the type: A implies B; B is true; A is slightly more credible. Credible is a worthy goal, but it falls short of a more generalizable set of reasoned justifications and proof. Fortunately for this example, this was not the end of the exchange as it might easily have been. For at this point in their interaction, the teacher commented that “you can’t just add up ones and tens and hundreds and expect to get any kind of reasonable answer”. Having said this, however, the teacher tempered her initial negative response and asked a very wise question, “How can you tell?”, and then went on to work with other students. In discussion with her later, the teacher mentioned that she had quite expected this to have ended the discussion. Fortunately, the student did not view the problem as over and continued trying to answer the powerful “How can you tell?” question posed to her. After having additional time to think this over, the student continued her discussion the following day. STUDENT: I think I can explain it to you now. My method will always work and I am not really adding ones to hundreds or anything like that. I’m just adding up the left-overs from the hundreds or the tens – or whatever. TEACHER: Ok, why don’t you show me what you’ve figured out? STUDENT: We’ll start with the one’s. Whenever I look at a bunch of ones it is easy to see the number of remainders – it’s just the number we’ve got. See, because if we’ve got 9 then it divides by nine and if we’ve got anything less than that then that is the remainder. TEACHER: Ok, but what about the tens and hundreds and all the rest? STUDENT: Look at the tens. I’ve drawn what is would look like here when I divide any ten by 9. See, I’ll always have one one left… (Fig. 3.1).

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Fig. 3.1 Dividing a “ten” by 9

Fig. 3.2 Dividing a “hundred” by 9

The students’ understanding shown here is clearly ahead of her vocabulary as evidenced by the interchangeable use of the terms “leftover” and “remainder”. Despite this, there are portions of this response that are almost poetic, “I’ve drawn what is would look like here”, for example. Although the statement “one one” might be confusing at first, a careful examination of the diagram makes the student’s intent clear. “One one” refers to the single left-over, or remainder, resulting from the division of the ten rod into nine equal integer parts. Once this is understood, the remainder of the argument now to be presented will be much clearer. STUDENT: … and it does not stop there. Look, for each hundred I’m gonna have one one because I first can divide up the hundred into nine tens with one ten left. I’ve already figured out what happens with the tens, but look – see. The ten can be divided up by nine and leaves one one (Fig. 3.2).

At this point, the teacher wisely continued her questioning of the student to determine how far the student’s thinking had gone. The result of this questioning shows that the developed reasoning was remarkably complete and indicated a degree of causality and sophistication that would be the envy of many adults. This is evidenced by noting that the student provides an answer that not only addresses the “thousands”, as directly asked, but also the “bigger numbers than these out there” qualifier. TEACHER: Ok, but what about the thousands? After all there are bigger numbers than these out there! STUDENT: Good try, but you can’t fool me! The thousands is made up of ten hundreds, each hundred will have one left over which leaves one one each. So I can add up the ten ones leftover

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3 Reasoning and Proof from each hundred, make a ten out of it, and I already know for a ten that I will have one one left. This will work all the way up the line! I will always have one one as a left-over no matter how big the number is.

The true nature of the “all the way up the line” shows an emerging sense of infinite process, which is not typically associated with concrete operational thinking. When the student applied this understanding, there was no question in her mind that each successive place value in base-ten when divided by 9 will result in “one one” for each digit within that place value. So, for the sake of simplicity, if the number under consideration is 3, 3  101 , 3  102 , or in general 3  10n , that the remainder for that division would be equal to the same number 3. Such richly developed justifications serve as strong evidence that reasoning and beginning proof can be quite rigorous even at a concrete operational level—provided the objects utilized within the proof are familiar and well understood by the student. Part of what made this easy for the student in this example in describing the generalization of her method to the numbers “all the way up the line” is the repeating pattern of shapes which make up the Diene’s base-ten blocks. This system parallels the arrangement of whole numbers in groups of three, called cycles, which are separated using a comma. Each of these cycles, whether starting from ones, thousands, millions, etc., has a “ones”, “tens”, and “hundreds” position. As shown in Fig. 3.3, in this modeling system each individual one, whether it be a unit 1, a 1-thousand, a 1-million, etc., will always looks like a cube albeit of increasingly greater size. Likewise, an individual ten will be created by placing ten of the “ones” for each triple end to end, while an individual hundred is created by placing ten “tens” to create a square. Stacking the ten “hundreds” thus created gives rise to a cube once more and the cycle is repeated. This easily visualized pattern played an important role in developing the series of justifications used by the student in her argument. It should be noted that an identical “proof” can be provided for any base—if it is modeled according to a similar organizational pattern. Consider base four. A base-four block set could be constructed with the corresponding place value of 1’s ð40 Þ, 4’s ð41 Þ, 16’s ð42 Þ, 64’s ð43 Þ, etc., as follows (Fig. 3.4): The use of the unit cube from the more common base-ten model ensures that creation of the same cycle of shapes—cube to linear group to square and back to cube—and guarantees that the same reasoning used in the student example will apply to base-four as well provided it is modeled in this fashion (Fig. 3.5). The parallel problem to the earlier classroom example when expressed in base-four terms would be to compute the “leftovers” when dividing by 3. This is because 3 in base-four is like 9 in base-ten, i.e., one less than the base. Due to this, the physical actions would be identical with those used in the divide by 9 problem for base-ten (Fig. 3.6). …and the result is the same… the student’s method would still work.

3.2 Elementary Reasoning at the Concrete Operational Stage

Fig. 3.3 Repeating shapes for first three periods in base-ten

Fig. 3.4 A base-four Diene’s like model

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Fig. 3.5 Repeating shapes for first three periods in base-four

Fig. 3.6 Division by 3 in base-four

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Nor does it stop at this point. Any arbitrary base can be modeled using the same cycle of shapes—cube to linear group to square and back to cube—resulting in the same reasoning used in the student example still being applicable to that base. When modeled in this fashion the parallel problem expressed in base-n would be to compute the leftovers when dividing by n  1. The actions would be identical with those used in the divide by 9 problem for base-ten and once more the result is the same—the method still works. The extensibility of this “proof”—based upon physical manipulatives and concrete actions—is far beyond what one would normally expect from a fifth grader. Consider a formal proof for the proposition that any base-ten number whose sum of digits is divisible by 9 is also divisible by 9. Such a proof might proceed in the following fashion. First consider the proposition that if a number N has no digits other than 9, then it is divisible both by 3 and 9. A proof could proceed as follows: Let N ¼ 99. . .99 |fflfflfflffl{zfflfflffl ffl} . Then, representing N as the sum of products of its face values knines

(digits) and the corresponding place values (powers  of 10) yields so N ¼ 9 10k þ 10k1 þ . . . þ N ¼ 10k  9 þ 10k1  9 þ . . . þ 10  9 þ 9, 10 þ 1Þ which is divisible by 9. Now, consider the proposition that a number is divisible by 9 if the sum of its digits is divisible by 9. Consider a k-digit number N ¼ a1 10k1 þ a2 10k2 þ . . . þ ak1 10 þ ak and its equivalent representation     N ¼ a1 10k1  1 þ a2 10k2  1 þ . . . þ ak1 ð10  1Þ þ a1 þ a2 þ . . . þ ak1 þ ak 2 3 5 ¼ 4a1  99. . .99 . .99 |fflfflfflffl{zfflfflffl ffl} þ a2  99. |fflfflfflffl{zfflfflffl ffl} þ . . . þ ak1  9 þ ½a1 þ a2 þ . . . þ ak1 þ ak  ðk1Þnines

ðk2Þnines

So, if the sum of the digits is divisible by 9 then the number is divisible by 9. In order to recognize such arguments as plausible and laying the foundations for later formal proofs of the type shown here, however, the teacher must be able to look past pre-existing expectations of formality and be able to see the mathematics that lives beneath the rules (Jencks & Peck, 1988). Examples such as this point to fallacies commonly emerging from an overly strict developmental perspective. They provide strong cases that even fifth grade students are fully able to utilize many of the strategies common to plausible reasoning even at the concrete operational level. This should be taken as a call to policy makers, curriculum developers and classroom instructors to encourage explicit instruction in reasoning and proof much earlier than is often implemented.

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3.3

Supporting Plausible Mathematical Reasoning and Formal Proofs with CARE

cBig Idea Student reasoning and proof require a supportive classroom environment. A big idea in developing such an environment is how to shift the role of mathematical discussions away from “winning” and toward “understanding”.b Clearly student examples of reasoning and proof such as illustrated in the division by 9 example do not arise spontaneously, but must be carefully cultivated within a supporting environment (Cuoco et al., 1996). As such, prior to any discussion of plausible reasoning strategies and proof methods it is important to describe some of the sociocultural factors leading to implausible reasoning which needs to be overcome. Human inference is, in actual practice, plagued with serious contradictory tendencies that may seem at first paradoxical. As Nisbett and Ross (1980) put it, “The same organism that routinely solves inferential problems too subtle and complex for the mightiest computers often makes errors in the simplest of judgements about everyday events. The errors, moreover, often seem traceable to violations of the same inferential rules that underlie people’s most important successes” (p. xi.). In some cases, this can be traced back to the many of abuses of language, both explicit and implicit, which are commonly used in public discourse to “win” debates. Wearing the trappings of logic in terms of their structure, these verbal tricks can convince even the most dedicated cynic in the absence of careful consideration (Gula, 2007). The use of such techniques, many of which have been used since antiquity as part of formal argumentation, have today become such a ubiquitous part of public discourse. Bennett (2012) identifies and describes over 300 logical fallacies together with methods to avoid falling prey to their use. Given that language development precedes the development of formal reasoning it is not surprising that students often fail to recognize the frameworks within which plausible reasoning leading to powerful generalizations and proof reside. cBig Idea A big idea to use in creating effective classroom environments is encapsulated in the acronym CARE—Commonplaces, Authority, Reasoning, Experience. When a classroom is crafted with CARE, effective mathematical discussion can occur.b To counter this, it is helpful to reconstruct the mathematics classroom into an environment that requires a more formal treatment of language, reasoning and even what counts as a meaningful argument. One such approach is encompassed in the acronym CARE where C refers to Commonplaces, A to Authority, R to Reasoning,

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and E to Experience (Connell, 2016). When taken together, applying these considerations create mathematics classroom environments supporting student critical thinking and reasoning. As do many of the logical fallacies documented by Bennett (2012), the principles of CARE derive from classic argumentation and will no doubt be familiar to the reader.

3.3.1

Commonplaces

The notion of commonplace has its origin in the oral histories passed down from pre-historic societies. These oral histories contain literary aspects, characters, or settings that appear again and again in stories from ancient civilizations, religious texts, and even more modern stories. Each of these serve to pass on important aspects of a culture which are so integral that it is tacitly assumed that each member of the society would be aware of them. As such, they are taken as being an aspect of culture, or universally accepted understanding, that can be used as a warrant to a claim. In classical rhetoric, a commonplace is a statement or bit of knowledge that is commonly shared by members of an audience or a community (Nordquist, 2019). This definition is expanded in common usage to include the set of cultural understandings and background knowledge that one can expect any member of the society to understand. To be effective in the classroom, an element of a commonplace should be so pervasive that it would not require instruction and could be used in instruction without extensive review as a universally understood piece of background knowledge. In geometry, for example, one could expect the informal definitions of “cube” presented by Inhelder and Piaget (1969) in the introduction to this chapter. As such, a teacher could reasonably expect their students to be able to differentiate between a cube and a sphere. Elements of the commonplace such as these often derive from earlier experiences with concrete objects such as dice and baseballs (Connell et al., 2018). To be effective in the mathematics classroom, however, a mathematics-oriented commonplace language must be developed. For example, supporting and precise vocabulary and language must be developed to expand upon the earlier imprecise notions upon which they rest. Since the entering commonplace for students include logical fallacies, difficulties in social inference, and willingness to accept vague warrants over careful reason this is not a trivial task. One of the first tasks facing the instructor, however, must be to expand the terminology and language used within the classroom commonplace into one supporting mathematical reasoning. To expand on this notion, when a teacher uses the word sphere in a geometry lesson, most students will picture a rounded ball-like shape in their mind. This memory is very “noun-like” in that it may be described, and its properties can be expanded. For example, a student could initially have thought about a baseball as indicated earlier. This evokes a different set of properties than if they had thought of

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a tennis ball, a golf ball, or a table tennis ball. If we think of a “baseball” as being a noun, then “seamed” is an adjective which may be used to describe it. Therefore, we see that a baseball has raised seams, a tennis ball is fuzzy, a golf ball is dimpled, and a table tennis ball is smooth. They also have different sizes, bounce differently, have different colors, and so on. These are all elements of the students’ commonplace, although they lack the specificity and precision necessary for rigorous discussion. Yet in a broader perspective, there is a “sphere-like” property common to each that can be recognized by all students. As a more formal concept of “sphere” is developed, this core mathematical idea plays a noun-like role.2 The properties which a sphere can have, such as radius and spatial location within a coordinate system, play the role of adjectives. A sphere can have a radius of 5 cm in the same sense that a baseball can have seams. The seams are a property of the baseball and the radius is a property of the sphere. Clearly, there are many “noun-like” portions of mathematics which play an important role in developing new ideas. This type of mathematical understanding gives us things to think about and forms the basis for later, more formalized concepts. The importance of developing supporting vocabulary is not a new one. In the Meno dilemma, Plato3 tells us that before Socrates4 agreed to begin any type of instruction with the boy, he recognized the need for a shared commonplace and precise language. The Holbo and Waring translation (2002) has Socrates stating, “Is he a Greek? Does he speak Greek?”. Bringing this question into the mathematics classroom it could easily be restated as, “What are the students’ backgrounds? Do the students speak math?” Max Beberman, a pioneer in modern mathematics education reform (e.g., Beberman, 1964), would have agreed that without a proper language in which to ask and answer questions, mathematics can degenerate into training students to give rote answers to trivial exam questions (Raimi, 2004).

3.3.2

Authority

Although an essential element in any classroom, authority can be a tricky element to negotiate in terms of developing reasoning and proof. This element relies heavily upon the role of the teacher, the role of the students, and what sources of information are deemed authoritative enough to serve as the basis of argument within the classroom. Careful consideration of the emerging classroom Commonplace is also

The notion of mathematics having a “noun-like” and “verb-like” character was perhaps first best articulated by Davis (1990). 3 Plato (ca. 428–348 B.C.)—a major figure in the history of Western philosophy. 4 Socrates (ca.470–399 B.C.)—a classical Greek philosopher, the teacher of Plato. 2

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necessary. For example, which of the many potential items of background knowledge are suitable for use in discussion? This question requires the teacher to be aware of the students’ background, educational history, and culture together with a richly developed content knowledge and potential instructional trajectories relevant to their classroom (Abramovich & Connell, 2016). From this awareness, the teacher can then powerfully serve as an authority through their selection of relevant items to include, or needed areas to reteach. Another factor contributing to the difficulty in use of authority is that much like the case for plausible reasoning, which will be developed later in this chapter, an argument from authority can only strongly suggest what is true—never prove it. Indeed, the most common use of authority—to assert a fact—often blocks student construction of personally meaningful understanding. Furthermore, in today’s Wiki and Google society it is far too easy to accept facts on an authority or source that is not valid. It has even been suggested that much of what is currently accepted within the mathematics community as having been proved might be wrong (Rorvig, 2019). In his invited address to the conference of Interactive Theorem Proving, Buzzard (2019) noted that a modern proof can easily cite tens of earlier works each of which cites tens of earlier works, etc., and if a respected mathematician cites one of these modern proofs, it, together with its citations, becomes a part of the accepted Commonplace. In this case, this is done on the basis of the Authority’s acceptance without each of the supporting details and citations being independently verified. In a very real sense, this makes it increasingly difficult for an Authority to be truly authoritative. From this perspective, it is far too easy to fall prey to a logically fallacious argument based upon an incorrect statement from recognized authority. Furthermore, many arguments ground claims in the beliefs of a source that is not authoritative. In rhetorical terms, this is often referred to as an appeal to authority. Such appeals are not only ineffective in establishing proof, but are particularly offensive when the “authority” is shown to be in error. In a 2016 survey of 78 entering first year undergraduates taking a formal reasoning course such teacher appeals to authority to establish facts were viewed as the single worst offenders present in classrooms in terms of logically fallacious arguments (Connell, 2016). Clearly, care must be taken that teachers do not overstep the significant authority they have within the classroom and make pronouncement which prove to be either inaccurate or based on faulty reasoning. It is far better for teachers to use their authority as classroom leaders to model problem solving than to claim to have all the answers already. This parallels much thinking in the sciences (Sagan, 1996) which suggests that in the sciences there are not “authorities” in the classic rhetorical sense, only experts. By adopting this mindset, mathematics teachers truly acknowledge that mathematics is a dynamic enterprise requiring creativity, a responsibility to understand what is already known relative to a problem and that as joint learners we all must to some degree become experts within our content area.

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3.3.3

3 Reasoning and Proof

Reason

Reasons are statements that support a given claim, making a claim more than a mere assertion. Reasons are statements in an argument that pass two tests. First, reasons provide answers to the hypothetical challenges: “How can you tell?” or “What would happen if…?” They also tend to follow the premise through the use of the word “because”, as in the student’s division by 9 comments when she stated, “… and it does not stop there. Look, for each hundred I’m gonna have one one because I first can divide up the hundred into nine tens with one ten left…” The premise in this case is “for each hundred I’m gonna have one one.” The reasons which follow the word because are that “I first can divide the hundred up into nine tens with one ten left…” and “The ten can be divided up by nine and leaves one one.” As was the case in this example, often the language might not yet be formally developed but the reasoning can still be effective. For example, imagine students working on the following word problem, “A man has decided to sell his horse. He bought the horse for $50 and then sold it for $60. He bought it back for $70 and now will sell it for $80. How much will he make, or lose, on this transaction?” In addition to being a recent Internet challenge (Russell, 2019), this problem has been used by one of the authors [MLC] in mathematics education classes for over 35 years. Looking back on this history, this problem came to be known as the “infamous horse problem” due to the discussions it has engendered (Fig. 3.7). Each semester this problem has never failed to engage, frustrate, and challenge both the struggling and the most mathematically adept of teacher candidates. This is reflected in the following comment from a teacher candidate, “I definitely learned that mathematics could be a lot of critical thinking which is really good for students to make logical deductions and make them want to learn something thus leading to them truly understanding the material. This is what I got from this simple yet frustrating problem. It came to my mind that even though I did not know what the answer was or if I’d ever know the answer, but I was proud of myself for coming up with a range of answers in multiple ways. My brain started thinking about the multiple ways I could go about answering this problem and I think I did a good job at looking at it differently each time. I think this is important to use when teaching students math, because math can be challenging to students. I struggled with math all the way through high school and I didn’t understand why until I started college. The way I process math and understand these problems and formulas is different from my peers. I think this will make me become a better math teacher because I know what it feels like to process it slower and not feel confident in my math skills. I feel that critical thinking has made me a better student and I hope to use it become a better teacher.5”

5

All quotes from teacher candidates Sect. 3.3 are as submitted. They have not been edited for spelling or grammar.

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Fig. 3.7 The infamous horse problem

Despite the simplicity of the operations—basic addition and subtraction—the reasoning used in solving the problem is subject to interpretation in terms of the beginning and end points of the transaction. This is how a recent teacher candidate described the experience, “During this activity I learned that it is important to not take short-cuts when figuring out a problem. In the beginning I was right to write − $50 for the purchase of the horse but instead of adding $60 for selling the horse to make his total $10, I just saw the difference of $10 so I added the $10 to the −$50 to get −$40. I continued this for the whole problem. Taking your time to properly work out a problem is important. I also learned that listening to people in your group can give you different perspectives on how to overcome a problem. It is important to listen and communicate with other people to see and hear their perspective on the activity. Without this communication I would not have realized my mistake.” Depending upon which views of the problem are believed in, and used, by the candidates a variety of strongly held “answers” are commonly generated including losing $30, breaking even, gaining $10, gaining $20, gaining $30 and even losing $70. These are often held as correct even after the correct answer is developed, modeled and explained. Recognizing the source of potential student errors is an instructor skill to be developed as recognized by a teacher candidate, “Through this Infamous Horse Problem it was seen that everybody has a unique perspective and way of thinking/solving problems. There was only one correct answer, but multiple people got an answer that was not the correct one. As a teacher if a student gets an

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answer wrong it is our job to try and see how the student got that answer, where they went wrong, and put them back on the right track to more correct answers; to change their way of thinking about the problem.”

3.3.4

Experience

The persistence of belief in the “alternate correct answers” to the horse problem serve to emphasize the primacy of personal experience in the reasoning process. This is unfortunate, however, for as a general rule personal truths which arise from personal experiences are weaker than empirical truths, for no matter how many people claim to have the same personal truths, if it’s not verifiable, it lacks the ability to be tested or repeated. Truth is not, after all, a popularity contest. It is possible for an individual to hold a variety of beliefs which are part of the broader commonplace to which they belong, but which are incorrect, incomplete, or both. That the vast majority of the commonplace holds the same beliefs might lend credibility to the belief, but ultimately has no bearing on the ultimate belief’s truth-value. There are currently over 200 members of the Flat Earth Society who claim to believe the Earth is flat (Wolchover, 2017). Within the society this is the accepted commonplace, yet it has little to offer toward the vast preponderance of reason, science and proof which argues that we live on a spherical planet. Experience is also easily swayed by emotion or other influences that bear little weight to the truth of the belief. Since the brain interprets all we see, feel and react to, if the brain is thinking in a certain way during an experience, the way we perceive the experience may be altered. However, the events which led to the experience remain unaltered. As is easily seen when considering hallucinations, no matter how real an experience might seem to you, if it never happened, it remains never having happened (Goldberg, 2019). Granting this, however, experience is an essential part of an effective classroom. As suggested earlier, one of the first actions in teaching should be to establish an effective classroom commonplace. Imagine a classroom commonplace having been previously established which included the vocabulary to be used in the activity, an ability to make use of sketching as a basis for deciding areas, and a willingness to engage in group reasoning under the authority of the teacher. The teacher has limited the use of authority to selecting the object or sketches to be used as starting points for the problem, the questions themselves, and the format to be used in recording student work.

3.4

CARE and Rotating Squares

With this groundwork in place, consider the following activity which took place in one of the authors’ [MLC] recent mathematics methods classes. You are given some square graph paper chunks of various sizes. How many squares of different

3.4 CARE and Rotating Squares

1 2 3 4

Area 1 2 3 4

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1 2 3 4

Area 1 2 3 4

1 2 3 4

Area 1 2 3 4

Fig. 3.8 Rotating squares

areas can you make inside whose corners rest on the points along the graph paper having integer coordinates of the graph paper? … and what is the area for each of these squares. Graph paper squares of various sizes was then provided with an attached template to record student work. Figure 3.8 shows a partially worked out example for a square graph paper chunk 4 on a side. Other graph paper square sizes included in this problem set included those of 3, 7, 8 and 11 on a side. The teacher’s Authority was used in asking questions leading toward generalization and extension through Reason and Experience such as, “What sizes of square graph paper would be needed to sketch inner squares with area 65 square units? Of area 85 square units? Of area 145 square units?” Finally, more general questions may be asked: “What size of square graph paper would be needed to sketch inner squares of area n square units? Given n, is such paper size unique? Does such paper size exist for any integer value of n? Why or why not?”.6 As these problems were explored a “beneath rules” approach was followed to expand the Commonplace through grounding personal Experience via Reasoning. In this approach, a remembered rule can be applied only when the reason for its’ application can be shown clearly. For example, the rule for area of a triangle is easily shown (Figs. 3.9 and 3.10) by noticing that each rectangle containing a triangle is cut into two equal parts. Since the dimensions of these rectangles make up the base and height of the triangle and the area of each triangle is equal to one half of the area of the rectangle it follows that Areatriangle ¼ 12 Basetriangle  Heighttriangle . Once this rule was established, the area of each of these triangles was established within the Commonplace to be 12  1  3 ¼ 1 12. The final step in reasoning out the area of the inner square now rested on accepted Commonplace understandings. One of these understandings being that the “inner squares” are indeed square. This would require that all four sides are of the same length and all four interior angles are equal to 90 . Figure 3.11 provides the outline of a proof which could be used to verify these requirements. For the cases shown in Figs. 3.8 and 3.11, the total area of the inner square was found by subtracting each of the triangular areas from the 4  4 area of the original square. So, for Fig. 3.8 the area of the inner square would equal ð4  4Þ    4  1 12 ¼ 16  6 ¼ 10 (Figs. 3.9 and 3.10). 6

Application of the Pythagorean theorem in this situation requires that n be equal to the sum of two squares. For example, when n ¼ 65 we have 65 ¼ 12 þ 82 ¼ 42 þ 72 .

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Fig. 3.9 Establishing that the “inner squares” are indeed squares

Fig. 3.10 An expanded view of the original problem

Height

Base

Fig. 3.11 Reasoning the area of the triangle

These experiences, being carefully grounded within an effective classroom commonplace as described, gave rise to a set of experiences which could be shared and discussed. Furthermore, by providing a template to help organize and mediate the records of the teacher candidates’ action, a commonplace was established wherein the mathematics embedded within the activity could be further developed.

3.4 CARE and Rotating Squares

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Since the referents being used were Common, as was the record of action, it was easy for the Authority of the teacher to guide the Reasoning and ensure that the Experience is effective and plausible. Once the basics of the task were understood, technology in the form of a spreadsheet, completed the development of the Commonplace by providing a tool to both record student work and to aid in further exploration. The data sheets created during initial problem exploration played a mediating role between the initial experience and aided in the creation of a Common eventual abstracted framework required by the spreadsheet (Connell, 2019a, b). Figure 3.12 shows a spreadsheet developed in this activity.7 In generating the spreadsheet, it was noticed that for each square chunk the height of each of four triangles is equal to ChunkSize  Base. The remaining columns parallel the discussion developed in Fig. 3.7 and the calculations of the selected cell can be seen in the formula bar. The Authority of the teacher is further enhanced by the computational accuracy of the abstracted framework, which in turn serves to guide the Reasoning of the students. The overall Experience is typical of that experienced in Technology Immune/Technology Enabled (TITE) problems within an Action on Objects framework (AoO) (Connell & Abramovich, 2017). As such, this newly created spreadsheet object allows for further exploration as shown in Fig. 3.13 showing the case for starting chunk sizes of 5 and 6. Finally, at this point several of the teacher candidates remembered the Pythagorean theorem and made the connection to Pythagorean Triples as they related to the area of the inner square. As the hypotenuse of the triangle, it should be equal to the sum of the squares of the base and height. In general, the area of such inner squares may be any integer which is the sum of two squares (Abramovich, 1999). This conjecture was easily verified using the spreadsheet as shown in Fig. 3.14. The use of the Pythagorean theorem can be seen by examining the formula bar. As this last example serves to illustrate, the elements of instruction contained within the CARE acronym, when applied to a technology enhanced mathematics classroom, enable a milieu within which the “How can you tell?” and “What would happen if…?” questions naturally emerge leading to plausible reasoning and later proof (Abramovich & Connell, 2016).

3.5

Basic Elements of Plausible Reasoning

In applying the principles described in the development of CARE both plausible reasoning and formal proof play important roles. In particular, plausible reasoning becomes the expected Commonplace within which further instruction takes place

7

Aunt Sarah and the Farm provides an extended example of the type of spreadsheet modeling shown in this abbreviated example (Connell & Abramovich, 2018).

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Fig. 3.12 The area of the inner square for a 4  4 chunk

Fig. 3.13 The area of the inner squares for 5  5 and 6  6 chunks

3 Reasoning and Proof

3.4 CARE and Rotating Squares

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Fig. 3.14 Verification of the Pythagorean conjecture

and is reinforced by the Authority of the instructor as backed by resource materials. As students gain additional Experience, more formal logic structures take a more important Reasoning role within the emerging Commonplace. This section includes basic examples illustrating types of plausible reasoning supporting later proofs and how these basic elements of plausible reasoning offer a foundation linking emerging reasoning to the more advanced mathematics which follows. In doing so, it is helpful to be guided by Sagan’s guide to “baloney detection” where he argues for the power of skeptical thinking which “is the means to construct, and to understand, a reasoned argument and—especially important—to recognize a fallacious or fraudulent argument. The question is not whether we like the conclusion that emerges out of a train of reasoning, but whether the conclusion follows from the premise or starting point and whether that premise is true.” (Sagan, 1996, p. 197). As was shown in the division by 9 example, prior to formal proof, much progress may be made via the development of plausible reasoning strategies. Lacking many of the universally accepted mechanisms of formal logic such as modus tollens or the De Morgan Rules which will be described later in this chapter, reasoning in the middle school mathematics classroom often tends to be a “one of” exercise. While often effective in addressing the problem at hand without careful planning, however, such strategies easily become highly idiosyncratic and stymie efforts at generalization from problem to problem. To address this difficulty, it is helpful to note that even if they lack the power of proof contained within a modus tollens argument, there are many highly powerful guidelines which can be utilized to develop plausible Reasoning within the classroom Commonplace. The following are a few of the patterns of plausible inference developed by Pólya (1968) which have proven to be highly effective in this effort. In these examples, A and B refer to statements which may be either true or false.

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Fig. 3.15 Sum of two odd numbers will always be even

The simplest case is of the form A implies B and it so happens that B is true. While not a proof, this fact certainly makes A more of a credible proposition. If, however, B is highly probably and is true then the case for A is only slightly advanced. For example, consider A to be the statement that sum of any two odd numbers will always be an even number. If B is a single example, 5 þ 7 ¼ 12 and 12 is even then the case for A is only slightly advanced. If B is the single example shown in Fig. 3.15, however, the case for A is not only significantly advanced but borders on a proof. The examples shown in Chap. 9, Sect. 9.2.1 provide other examples of generalizing from a single iconic piece of evidence. Compare this with the situation where B is considered highly improbable and yet is still true, in this situation the case for A is rendered very much more credible. Imagine a situation in which A can be shown by a variety of situations B, each of which appear markedly different to the student such that Bn þ 1 is very different from the formerly verified consequences B1 ; B2 ; . . .; Bn of A. In this situation, the case for A is significantly advanced. This case is shown in Sect. 3.4 where the areas of the inner squares were determined from a “beneath rules” discussion, a graphical interpretation, a logical proof, a spreadsheet, and comparison or formulas using the spreadsheet, and application of the Pythagorean theorem. Each of these approaches appear to be markedly different and serve to significantly advance the case for A. It only requires a few related, but seemingly unconnected examples, for A to be viewed as much more credible. This is in marked contrast to the situation in which A can be shown by a variety of situations B, each of which appear similar to the student such that Bn+1 is nearly identical to the formerly verified consequences B1 ; B2 ; . . .; Bn of A. For each successive iteration of example B, the case for A is only marginally increased. This is the situation which nearly took place in the earlier division by 9 example. Had the teacher followed her original question, “Let’s see if you get this one right… try 627 divided by 9”, with a similar set of questions only requiring a replication of the process the end result would not have led to the insights required to address the more powerful question, “How can you tell?”. Equally important, this line of questioning can easily burn out the students. Although this is an extreme example, suppose that the teacher in the division by 9 example had used these follow up questions, “Let’s see if you get this one right… try 628 divided by 9”; “… try 629 divided by 9”; “… try 630 divided by 9”; etc. Instead of thinking “how can I tell?” her student might quickly begin to think “why should I care?”. In addition, due to the role played by personal experience, generating a series of parallel examples can occasionally have a negative outcome. For the example at hand, let A be the statement, “The sum of the interior angles of a triangle is equal to

3.4 CARE and Rotating Squares

Triangle 1 2 3 4 5

Measure of A 101.83

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SUM

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36.50 178.67

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40.83 172.00

8.00

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Fig. 3.16 Mean actual measurements of 48 teacher candidates

180 ”, and B1 ; B2 ; . . .; Bn of A be a series of measurements taken by students of the interior angles of various triangles. Even in a best-case scenario with perfect measurements it must be noted that at the end of the exercise, no matter how many iterations are undergone, A will not be proven but only rendered a little more credible with each successive iteration. Furthermore, the more examples that are undertaken the greater the likelihood of student burnout. In actual practice, student measurements are prone to errors. For example, a group of teacher candidates had been given fifteen angles to measure with a protractor as part of a measurement activity. Although the candidates were unaware of this, these angles had been created from the interior angles of five triangles. One would expect that the sum of the angles making up the interior angles for each triangle would equal 180 . A table showing the average measurements of 48 candidates work is reflected in Fig. 3.16. Note that in this example, these measurements not only did NOT lead to generalization, but they served to make the original premise much less credible due to the increased power of personal experience. It can be expected that due to the cumulative effects of measurement errors it would not be very likely that 180 will emerge with any great exactness and this was exactly what occurred. Triangle 4 came to be viewed as a special case do to the “extreme pointiness” of the angles. The average sum of angle measurements, may approach 180 as they approach infinity, however, this does little to convince a finite number of students who are actually carrying out the measurement. Furthermore, as mentioned earlier, no matter how many iterations are undergone, A will not be proven but only rendered a little more credible with each set of measurements. Other guides to plausible reasoning include: (1) A is analogous to B. B is true. The case for A is made more credible; (2) A is analogous to B. B is more credible. The case for A is made slightly more credible; (3) A is implied by B. B is false. The case for A is made less credible; and (4) A is incompatible with B. B is false. The case for A is made more credible. Although this list is certainly incomplete, it goes a long way to helping establish the type of “baloney detection” mentioned earlier. Although not holding the same status as the formal proofs of propositional logic which will follow, each of the guidelines for plausible reasoning described in this section can play an important role when discussing mathematics activities within an effective classroom environment.

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3.6

Basic Elements of Propositional Logic

cBig Idea Propositional Logic, a big idea from mathematics and logic, can also play a role in the middle school classroom. As shown in this chapter, however, this requires a classroom constructed with CARE and utilizing student understandable models upon which to base the logical arguments.b In reasoning and proof logical statements play an important role. Just as the symbolic language of algebra is used to generalize from instances expressed numerically, the symbolism of propositional logic is used to generalize from particular statements which are not subjective in the sense that they may be either true or false but not both. For example, the statements “the number 11 is a prime number” and “the number 21 is an even number” are, respectively, true and false but the statement “prime numbers are more important than even numbers” is neither true nor false. In this section the following notation will be used. For AND (^)—conjunction; for OR (_)—disjunction; for NOT (:)—negation; for IF ())—conditional.

3.6.1

Conjunction, Disjunction and Implication

Logical conjunction. Let p be the statement “I will read a textbook before the exam” and q be the statement “I will pass the exam”. Then logical conjunction of p and q, p ^ q, is the statement “I will read a textbook before the exam and will pass the exam”. The p ^ q statement is only true when both p and q are true. The relationships between four different truth values involving the statements p and q are described in the table of Fig. 3.17.

3.6.2

Application of the Logical Conjunction Truth Values in Spreadsheet Modeling

Let p be the statement: the number is a multiple of two and q be the statement the number is a multiple of three. The logical conjunction p ^ q implies that the number is a multiple of six when both p and q are true statements. In order to identify multiples of six among natural numbers, one can generate the first 100 natural numbers in column A in the range [A1:A100], define in cell B1 the formula ¼ IFðANDðMODðA1; 2Þ ¼ 0; MODðA1; 3Þ ¼ 0Þ; A1; “ ”Þ and replicate it to cell B100. One can see that the negation of the logical conjunction p ^ q leaves the cell in column B blank.

3.6 Basic Elements of Propositional Logic Fig. 3.17 Logical conjunction truth values

Fig. 3.18 Logical disjunction truth values

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p

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Logical disjunction. Let p be the statement “I will read a textbook before the exam” and q be the statement “I will pass the exam”. Then the logical disjunction of p and q, p _ q, is the statement “I will read a textbook before the exam or will pass the exam”. The p _ q statement is only false when both p and q are false. The relationships between four different truth values involving the statements p and q are described in the table of Fig. 3.18.

3.6.3

Application of the Logical Disjunction Truth Values in Spreadsheet Programming

Let p be the statement: a cell in row 1 is filled with an even number and q be the statement: a cell in column A is filled with an even number. Then the logical disjunction p _ q implies the statement: the product of the corresponding entries from row 1 and column A is an even number when either p or q is true. Consequently, when row 1 and column A are filled with natural numbers, the spreadsheet formula

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Fig. 3.19 Logical implication truth values

p

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¼ IFðORðMODð$A2; 2Þ ¼ 0; MODðB$1; 2Þ ¼ 0Þ; $A2  B$1; “ ”Þ; defined in cell B2 and replicated across columns and down rows, generates in the multiplication table even entries only. As shown in Fig. 3.18, only when both p and q are false statements, the entry of the multiplication table is not an even number. Logical implication. Let p be the statement “I will read a textbook before the exam” and q be the statement “I will pass the exam”. Then the logical implication “If p then q”, p ) q, is the conditional statement “If I read a textbook before the exam then I will pass the exam”. The p ) q conditional statement is only false when q is false. The relationships between four different truth values involving the statements p and q are described in the table of Fig. 3.19.

3.6.4

De Morgan Rules8

According to these rules, :(p ^ q) = (:p) _ (:q) and :(p _ q) = (:p) ^ (:q). That is, the negation of the logical conjunction is the disjunction of negations and the negation of the logical disjunction is the conjunction of negations. For example, if one deals only with the first six natural numbers and p and q are, respectively, the statements: when three dice are rolled the numbers on the three faces form the set {1, 2, 3} and when two dice are rolled the numbers on their two faces form the set {3, 4}. Then :p = {4, 5, 6}, :q = {1, 2, 5, 6}, p ^ q = {3}, :(p ^ q) = {1, 2, 4, 5, 6}, (:p) _ (:q) = {1, 2, 4, 5, 6}. Thus :(p ^ q) = (:p) _ (:q). Likewise, :(p _ q) = {5, 6} and :(p) ^ (:q) = {5, 6}. Thus :(p _ q) = (:p) ^ (:q). The negation of a conjunction and the disjunction of negations are presented in the table of Fig. 3.20. The negation of the logical disjunction the conjunction of negations is shown in the table of Fig. 3.21.

8

Augustus De Morgan (1806–1871) was a British mathematician and logician.

3.6 Basic Elements of Propositional Logic

p

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( ¬ p)

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Fig. 3.20 The negation of a conjunction

(p ∨ q)

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Fig. 3.21 The negation of a disjunction

As an application of a Morgan rule to spreadsheet programming, consider another two statements: p: the number of apples in the box is an even number; q: the number of apples in the box is a prime number. Then the negation of p is: the number of apples is an odd number and the negation of q is the number of apples is a composite number. The conjunction of p and q is the statement that the number of apples in the box is equal to the number 2 and the negation of the conjunction of p and q is the statement that the number of apples in the box is any number but two. Now, suppose the task is to test the hypotheses p and q. Instead of verifying whether a number is a prime in finding the negation of the conjunction of p and q, one can replace this verification by using the first Morgan rule and find the disjunction of the negations of p and q. To this end, one can generate odd numbers (the negation of p) and augment them by even numbers greater than the number 2. The disjunction of the two sets of numbers yields all natural numbers but the number 2. The spreadsheet formulas are as follows: In column A consecutive natural number are displayed. In column B: = IF(mod (A1, 2) > 0, A!, “ ”). In column C: = IF(AND(A1 > 2, mod(A1, 2) = 0, A1, “ ”). In column D: IF (AND(B1, “ ”, C! + “ ”),“ ”, A1). As a result, natural numbers excluding the number 2 are displayed in column D.

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3.6.5

3 Reasoning and Proof

Converse, Inverse and Contrapositive Statements

The converse of p ) q is q ) p. Let p be the statement “The sum of digits of an integer is divisible by three” and q be the statement “The integer is divisible by three”. Then p ) q represents the following conditional statement “If the sum of digits of an integer is divisible by three, then the integer is divisible by three”. In that case, the converse q ) p is also true as if the integer is divisible by three then its sum of digits is divisible by three. The inverse of p ) q is (:p) ) (:q). In the case of divisibility by three inverse statement is also true as if the integer is not divisible by three then its sum of digits is not divisible by three. However, if p is the statement “This quadrilateral is a square” and q is the statement “This quadrilateral is a rectangle”, then p ) q is the true conditional statement “If a quadrilateral is a square, then the quadrilateral is a rectangle” the inverse if which is “If a quadrilateral is not a square, then the quadrilateral is not a rectangle”. The last logical implication is false because a quadrilateral may be a rectangle but not a square. Finally, the contrapositive of p ) q is (:q) ) (:p). The contrapositive of the true conditional statement “If a quadrilateral is a square, then the quadrilateral is a rectangle” is the conditional statement “If quadrilateral is not a rectangle then quadrilateral is not a square” which is also true.

3.6.6

Modus Ponens: [(p ) q) ^ p] ) q

In the above logical proposition, p ) q is a conditional statement, p is its hypothesis, q is its conclusion. If both the conditional statement and the hypothesis are true, then the conclusion is also true. For example, let p ) q be the statement: If an integer has the sum of digits divisible by three, then the integer is also divisible by three; its hypothesis p is the statement: the sum of digits of an integer is divisible by three; and its conclusion q is the statement: the integer is divisible by three. One can see that from the first two statements the third statement follows. The table of Fig. 3.22 shows all possible truth values of the modus ponens.

3.6.7

Modus Tollens: [(p ) q) ^ :q] ) :p

In the above logical proposition, p ) q is a conditional statement, :q is the negation of its conclusion, :p is the negation of its hypothesis. If the conditional statement and the negation of its conclusion are true, then the negation of its hypothesis is also true. For example, let p ) q be the statement: If an integer is a perfect square, then the integer has an odd number of divisors; the negation of its conclusion q is the statement: an integer does not have an odd number of divisors;

3.6 Basic Elements of Propositional Logic

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Fig. 3.22 Truth values of modus ponens

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Fig. 3.23 Truth values of modus tollens

and the negation of its hypothesis p is the statement: the integer is not a perfect square. The table of Fig. 3.23 shows all possible truth values of the modus tollens.

3.7

Conclusion

Reasoning and proof should be part of the commonplace of every mathematics classroom. When reasoning is expanded to include plausible reasoning and proof expanded to include non-standard proofs, this can take place even at the earliest of middle school grades. By using elements of plausible reasoning which parallel the later more formal logic structures, it is possible for this to occur early in the curriculum and need not be delayed. Early inclusion or plausible reasoning from meaningful experiences allows students to realize that mathematics can, and should, make sense. Even prior to being able to understand a formal proof showing A is universally true, plausible reasoning

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strategies will allow students to feel increasingly comfortable with proposition A as a result of their personally understood plausible justifications. Regardless of whether a concept was developed formally or informally, however, it must in the final analysis be sensible to the students themselves. In working toward this goal, the elements of instruction contained within the CARE acronym, when applied to the mathematics classroom, enable a milieu within which the “How can you tell?” and “What would happen if…?” questions will naturally emerge. By laying the foundation for plausible reasoning and later proof it makes later formal approaches to mathematics more understandable to the student. Within such classrooms it is possible for structured student activities to occur providing either conclusive conviction, slightly more credible belief, or willingness to acknowledge that the piece of mathematics under consideration is not only understandable but likely to be correct. This piece of personal experience goes far in overcoming the many elements of the public commonplace which are either logically fallacious or not amenable to the plausible reasoning strategies outlined in this chapter. Proofs can take many forms. For example, there is a long history in mathematics for what is often termed “proofs without words”.9 Although some might argue that these are merely examples, a more careful viewing shows that they contain the proof and in some cases are much more understandable to a beginning student than a more formal treatment. Proofs such as this should not be shunned in the course of instruction as they provide insight into the thinking underlying a subsequent more formal and propositional proof. For readers who want to deepen their knowledge and understanding of reasoning and proof the following sources can be recommended (Butterworth, 1999; Dummett, 2002; Posamentier & Spreitzer, 2018; Charles & Silver, 1988; Smullyan, 2013; Körner, 2009).

3.8

Activity Set

1. Using the dialogue from Sect. 3.2 as a guide, construct a plausible student argument for determining the remainders when dividing by 3. By 5. By 2. 2. Construct a proof showing that any base-ten number whose sum of digits is divisible by 3 is also divisible by 3. 3. Another student, when shown the argument illustrated in Fig. 3.2, said, “J clearly doesn’t get it. There is no way that the remainder is equal to 11”. (By

9

An excellent set of technology enabled examples may be found at https://www.maa.org/press/ periodicals/convergence/proofs-without-words-and-beyond-proofs-without-words-20.

3.8 Activity Set

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

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this, he meant the “one one” reported in the dialogue). What could be done to help this student? Describe how Diene’s blocks can be used to address the “all the way up the line” observation of the student described in Sect. 3.2. What is the remainder of ð34;125Þ10 ð9Þ10 ? What is the remainder of ð34;124Þ5 ð4Þ5 ? What shape would one trillion resemble? Draw it. What shape would one hundred septillions resemble? Draw it. What shape would ten billion resemble? Draw it. What size of square graph paper would be needed to sketch inner squares with an area of 100 square units? Restate the proof offered in Fig. 3.11 using a “beneath rules” approach. Using Fig. 3.15 as a guide, construct a diagram showing the result of adding any two even numbers. Using Fig. 3.15 as a guide, construct a diagram showing the result of adding one odd and one even number. What impact would increasing the accuracy and precision of measurement have upon the table from Fig. 3.16? Let p be the statement “I will increase my exercise” and q be the statement “I will lose weight” a. What is the logical conjunction p ^ q? b. What is the logical disjunction p _ q?

16. Which type of logical argument is expressed in the following statements: If today is Saturday, then I will not go to work. Today is Saturday, therefore I will go to work. 17. Which type of logical argument is expressed in the following statements: If there is smoke, there is fire. There is not fire, so there is no smoke.

Chapter 4

Modeling Mathematics with Fractions

4.1

Introduction

This chapter begins with the discussion of two major real-life contexts leading to the concept of fraction, as an extension of integer arithmetic, when the division of two integers does not results in an integer: the part-whole context and the dividend-divisor context. The division of integers is typically introduced through two modeling contexts: measurement and partition. One can say that the part-whole and the dividend-divisor contexts for fractions extend, respectively, the measurement and the partition contexts for integers. Consider the relation 6
2 ¼ 3. It can be interpreted as measuring, say, six pies by two pies as a way of creating three two-pie gift boxes. Alternatively, this division of six by two can be interpreted as partitioning six pies between two people evenly, so that each person gets three pies. Whereas replacing two by four does not result in an integer number of pies for either the measurement or the partition contexts, fractions are typically introduced when dividing a smaller number by a larger number. Consider the case of dividing 3 by 4. As an extension of integer arithmetic, one can interpret this operation as measuring 3 pies by 4 pies resulting in the number 3/4 which, in the form of abstraction, shows how many times the number 3 includes the number 4. Alternative interpretation of dividing 3 by 4 is partitioning 3 pies among 4 people evenly. This action can demonstrate physically that each person would get 3/4 of a pie. Indeed, each of the three pies can be divided in four pieces and the resulting twelve pieces are then partitioned among four people so that

© Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_4

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each person gets three pieces of the one-fourth of a pie. The former context of division is called the part-whole context and the latter context of division is called the dividend-divisor context. As noted in a number of mathematics education publications, the latter context, while being less abstract than the former one, is not commonly known by the teachers of middle school mathematics (e.g., Schifter, 1998; Conference Board of the Mathematical Sciences, 2012). This is also consistent with the authors’ observations when working with future teachers in different mathematics education courses. It appears that the dividend-divisor context often might be inadvertently omitted from teaching fractions in the schools. A reason for such omission could be due to the fact that the part-whole context is already introduced through dividing an object (e.g., a pie) into several equal parts; for example, one pie divided in two equal parts. In fact, as described by Schifter (1998), when a teacher asked a six-grader about the meaning of the fraction 1/2, expecting to hear that the fraction represents one out of two equal parts of the whole (the part-whole context), the answer was different—it is one thing divided in two (equal) parts. The teacher was even more puzzled when most of the class agreed with this interpretation of 1/2. The student, without realizing contextual meaning of this response, provided the dividend-divisor context with the numerator, 1, being the dividend and the denominator, 2, being the divisor. Often, when talking about three (identical) pies being equally divided among four people (with three being the dividend and four being the divisor), instead of recognizing an opportunity for the dividend-divisor context to be introduced, a teacher might say in passing that each person would get 3/4 of a pie without paying attention to the significance of this context as perhaps the most natural way of extending integer arithmetic to that of fractions. Of course, what is considered the most natural way depends on whether one thing is divided into four equal parts or three things are divided into four equal parts. One of the authors (SA) asked a class of about twenty students in an elementary teacher education master’s program whether they are familiar with the dividend-divisor context and nobody answered in the affirmative. The dividend-divisor context makes it possible to explain why a fraction has multiple representations through equivalent fractions and an integer has only one representation through itself. For example, 3/4 = 6/8 = 9/12 = 12/16 = …, and 4 has only one self-representation, 4 = 4. But if one recognizes that in the fraction notation the operation of division is hidden (that is, it is only identified, by presenting the dividend and the divisor, yet not completed), then at least two things become clear: there are many ways to represent four through an operation (e.g., 4 ¼ 3 þ 1 ¼ 5  1 ¼ 2  2 ¼ 12
3), and dividing three things in four equal parts gives the same result when six (identical) things are divided into eight equal parts, nine—into twelve, and so on.

4.1 Introduction

87

Fig. 4.1 Dividing 4 into 3

One can introduce the fraction 3/4 by extending the partition model for division to non-integer arithmetic; for example, by dividing three identical objects such as pies among four people. As shown in Fig. 4.1, each of the three identical rectangles (representing pies1) is divided into four equal parts. Through this process, twelve same size pieces result. Dividing 12 pieces among four people through the partition model for division yields three pieces of pie for each person, a piece being 1/4 of a pie. Thus, we have the fraction 3/4 representing the operation 3
4, the outcome of which has only been identified through the notation 3/4, yet not completed. Connection between a fraction and division is mentioned by the Conference Board of the Mathematical Sciences (2012) as one of the most fundamental ideas in arithmetic that serves as a numeric characteristic of the ratio concept in middle school mathematics when the dividend-divisor comparison of the elements of two sets yields the same fraction. Notwithstanding, the part-whole context serves as a foundation for teaching probability both from the theoretical and experimental perspectives. The teaching of fractions conceptually can be enhanced by the use of the so-called tape diagrams (Common Core State Standards, 2010) aimed at explaining formal operations and their meaning. Just as the teaching of writing was recommended to “be arranged by shifting the child’s activity from drawing things to drawing speech” (Vygotsky, 1978, p. 115), the teaching of arithmetic of rational numbers can be arranged as a transition from drawing a physical meaning of addition, subtraction, multiplication and division to describing the visual and the physical through culturally accepted mathematical notation. With this in mind, the “We Write What We See” (W4S) principle (Abramovich, 2014b) can be used in the teaching of fractions. The diagram of Fig. 4.1 is a simple example of explaining the meaning of the fraction 3/4 by drawing an image of dividing three objects in four equal parts each. We see differences and similarities when dealing with geometric 1

The focus on a pie is twofold. First, unlike a marble, pie is an object physically divisible in equal parts. Second, unlike a circular pizza, a rectangular pie may serve as a two-dimensional model for fractions (see Figs. 4.2, 4.3 and 4.4). Modeling properties related to “pies” and “cakes” were discussed in Chap. 2.

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figures or their images and appreciate different terminology to describe them; we see a relationship (known as the triangle inequality) among the lengths of three straws when trying to construct a triangle that does (or does not) allow for such a construction (e.g., Chap. 9, Fig. 9.36); we see that the sum of two consecutive triangular numbers is the square of the rank of the larger number (e.g., Chap. 8, Fig. 8.1). There are plenty examples of that kind in school mathematics and beyond.

cBig Idea One of the big ideas of mathematics pedagogy is that teaching can be orchestrated as a transition from seeing and acting on concrete objects to describing the visual and the physical through culturally accepted mathematical notation. With this in mind, the “we write what we see” (W4S) principle can be used to guide the teaching of mathematics. Yet this principle requires conceptual understanding of images to be described mathematically.b

Yet, the W4S principle works not without reservations. Although the avowal “I see” often confirms understanding, mathematical visualization, as Tall (1991) put it, “has served us both well and badly” (p. 105). Therefore, while the focus on visualization is a commonly accepted practice of mathematics teaching and learning, especially in the digital era, there is a long and sometimes challenging path from seeing things to understanding correctly their mathematical meaning or the absence thereof. Potential paths in the effort were explored in Chaps. 2 and 3.

4.2

Transition from Arithmetic of Integers to Arithmetic of Fractions

How do we add integers? For example, the meaning of the operation 2 + 3 stems from the context of adding two of something to three of something and the result is 5 (of something). But what if we add 2 apples and 3 oranges? The resulting number 5 represents neither apples nor oranges because the addends belong to different denominations. Is there a denomination to which both apples and oranges belong? The word fruit may be an answer to this question: adding two apples and three oranges yields five fruits. In the operation 2 + 3 both addends are abstractions

4.2 Transition from Arithmetic of Integers to Arithmetic of Fractions

89

de-contextualized from denominations they represent. But in this abstraction, there is one concrete thing—both 2 and 3 are comprised of the same unit of measurement. So, two units plus three units yield five units. Similar situation is with fractions. Both 1/2 and 2/3 are fractions of the same unit, or, as Bobos and Sierpinska (2017), who promoted the idea “of using the theory of fractions of quantities to go from the idea of fraction of something to a theory of abstract fraction” (p. 205), would have put it, are fractions of something. However, the situation with fractions is more difficult than with integers because this something is both a concrete thing and a unit of measurement through which both fractions can be measured. So, talking about 1/2 and 2/3 of a pie (which is the unit), we have the two fractions of the pie to measure by another, fractional unit. In doing so, 1/2 and 2/3 of the pie are measured by 1/6 of the pie to have the total 7/6 of the pie (more than the unit). In abstract form, we have the equality 1/2 + 2/3 = 7/6. The same reasoning can be applied to the operation 2/3—1/2. We will return to the discussion of the idea of a fraction as the fraction of something in Remark 4.6.

4.3

The Meaning of Multiplying and Dividing Fractions

In this section, multiplication and division of fractions will be introduced by using area model for fractions, one of the three models for teaching and learning fractions: area, measurement, and set models. Conceptual meaning of the reduction of a fraction to the simplest form (something that was not observed in the context of whole number arithmetic) and the “Invert and Multiply” rule commonly utilized for the division of fractions will be discussed. A number of word problems that provide different real-life situations for modeling mathematics with fractions will be presented. Connections of fractions to other topics such as probability, geometry, and measurement will be provided in other chapters of the book. Modeling with fractions in various contexts will be supported by an electronic spreadsheet and Wolfram Alpha.

4.3.1

Multiplying Two Fractions

Four different types of fractions can be identified: a unit fraction (e.g., 12), a proper fraction (e.g., 25), an improper fraction (e.g., 74), and a mixed fraction (e.g., 2 13 but not 2 53). How can one turn a mixed fraction into an improper fraction? For example, a symbol 2 13 is a notation used to represent the sum of the integer part of a (rational) number and its fractional part, 2 13 ¼ 2 þ 13 (thus 2 53 is not a notation used to represent a mixed fraction as 53 itself has a non-zero integer part). In order to add an integer and a fraction, one has to find a common measure for the two numbers.

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To this end, one has to turn an integer into a fraction and add two fractions (with the 1 23 þ 1 ¼ 73. This explains same denominators). In our case, 2 13 ¼ 2 þ 13 ¼ 23 3 þ 3¼ 3 conceptually why 2 13 ¼ 233þ 1 ¼ 73.

4.3.1.1

Multiplying Two Proper Fractions

Where does the rule of multiplying two fractions come from? How, for example, 10 can one justify, conceptually, that 23  57 ¼ 25 37 ¼ 21; that is, why the operation of multiplication compels multiplying numerators and denominators of two fractional factors? One can begin explanation from the multiplication of integers where 2  5 is understood as counting five things twice, or, in terms of an action, taking two groups of five objects. In the case of multiplying two fractions, repeating a fraction fractional number of times does not make sense unless this abstraction is put in context and explained through a two-dimensional diagram as shown in Fig. 4.2. Here, the large rectangular grid represents one whole or “referent unit” (Conference Board of the Mathematical Sciences, 2012, p. 28), 5/7 of which are marked (vertically) with x’s and 2/3 of which are marked (horizontally) with 0s. Which part of the whole does the product represent? Could the product be represented by a region larger than the whole? The answers to these questions are in the meaning of multiplication that does not change as the number system changes; that is, a physical meaning of multiplication is the same for objects described by integers as by fractions. How can one contextualize the product 2  5? This product represents the number of objects included in two groups of five objects. In other words, the multiplication sign is characterized contextually by the preposition ‘of’; that is, multiplying two quantities means taking a certain quantity of another quantity. Likewise, 23  57 means taking 23 of 57; that is, taking the quantity 2/3 of the quantity 5/7. In order to make this operation more concrete; in other words, in order to demonstrate the skill of contextualization of the operation, one can first take 57 of the whole (e.g., a pie as a referent unit) and then take 23 of the piece taken. This

Fig. 4.2 The product 23  57 ¼ 25 37

4.3 The Meaning of Multiplying and Dividing Fractions

91

action results in a new fraction of the whole characterized by the product 23  57. This is a region where both marks, 0s and x’s, overlap after 5/7 are marked with x’s and 2/3 are marked with 0s. Now one can see that the region representing the product is part of the whole. In other words, the product of two proper fractions, being a part of the referent unit, is smaller than the number 1. Yet the question remains: how can one describe the overlapping region through a single fraction? The overlap of x’s and 0s can be seen as 2 groups of 5 objects, that is, 2  5 ¼ 10 cells belong to the overlap. The total number of cells in the whole can be seen as 3 groups of 7 objects, that is, 3  7 ¼ 21 cells comprise the whole grid. Therefore, by moving from visual to symbolic, that is, through de-contextualization, the overlap as a fraction of the grid (the whole) can be 2 5 25 10 expressed numerically as 10 21. In other words, 3  7 ¼ 37 ¼ 21. That is, the rule (algorithm) of multiplying fractions has been developed conceptually by assisting one in seeing “where a mathematical rule comes from” (Common Core State Standards, 2010, p. 4).

4.3.1.2

Multiplying Two Improper Fractions

In order to find out when the product of two fractions is represented by a region larger than the whole, consider the case of multiplying two improper fractions using a grid. How can one construct an image of the product 32  75 proceeding from an image representing one whole? As in the case of multiplying proper fractions, we start with drawing a rectangle (the borders of which are bold lines) to represent the whole (referent unit) as shown in Fig. 4.3a. The next step, shown in Fig. 4.3b, is to divide the whole into five equal parts, each of which is 1/5 (of the whole) and then extend it to the right by another two-fifths to get an image of 7/5. Then, as shown in Fig. 4.3c, the whole, represented as five-fifths, is divided horizontally in two equal parts and then extended down by another one-half to get the image of 3/2 (of the whole) in the form of 15 cells marked with x’s. Now, one modifies the image of 7/5 shown in Fig. 4.3b by dividing it horizontally in two equal parts yielding 14 cells marked with 0s as shown in Fig. 4.3d. Finally, overlaying the (b) and the (d) parts of Fig. 4.3, in other words, taking 3/2 of 7/5 yields an image of the product 32  75 shown in Fig. 4.3e. Note that the cells marked with only x’s and with only 0s represent, respectively, 1/2 and 2/5 (of the whole). The cells with no marks represent the product 12  25, which (by using 2 cells as a new unit) can be reduced to the simplest form 1/5. Put another way, the diagram of Fig. 4.3e reflects the distributive property of multiplication over addition: 3 7 1 2 2 1 1 2 2 1 1  ¼ ð1 þ Þð1 þ Þ ¼ 1 þ þ þ  ¼ 1 þ þ þ : 2 5 2 5 5 2 2 5 5 2 5 One can see that the overlap of the images of the fractions 3/2 and 7/5, that is, the part of the diagram of Fig. 4.3e with both marks coincides with the whole.

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Unlike the case of the product of two proper fractions (Fig. 4.2) when the overlap of both marks is located within the whole, the overlap of the marks representing, respectively, two improper fractions, as shown in Fig. 4.3e, is the whole itself. Furthermore, the equality 32  75 ¼ 21 10 means that the product consists of 21 cells each of which has the name 1/10 (of the whole). In particular, the fact that the image of the product of two improper fractions includes the image of the whole implies that the product of two improper fractions is greater than the number 1.

4.3.1.3

Multiplying Proper and Improper Fractions

Likewise, the product 32  57, in which only the first factor, 3/2, is an improper fraction, can be shown being represented by a region that is larger than one whole. Of course, not every product of a proper and an improper fraction is greater than the number 1. For example, the product 32  47 is smaller than the number 1. Whereas the product of two proper or two improper fractions is, respectively, smaller or greater than the whole, the product of two fractions of different kind may be either greater or smaller than the whole. As it was done when creating Fig. 4.3, let us begin by drawing a rectangle (the borders of which are bold lines) representing the whole (referent unit) as shown in Fig. 4.4a. The next step, shown in Fig. 4.4b, is to divide the whole in two equal parts and extend it down by another one-half to get the image of 3/2 (of the whole). Now, as shown in Fig. 4.4c, the whole, already divided horizontally in two equal parts, is divided vertically in seven equal parts yielding 14 cells total, 10 of which, representing 5/7 (of the whole), are marked with x’s. The next step, shown in Fig. 4.4d, is to modify the image of 3/2 presented in Fig. 4.4b by dividing it

Fig. 4.3 From the whole to the product 32  75 as the sum of four non-overlapping regions

4.3 The Meaning of Multiplying and Dividing Fractions

Fig. 4.4 The product 32  57 as the sum of two non-overlapping regions:

93

5 7

and

1 2

of

5 7

vertically in seven equal parts. Through this process, one creates a 21-cell grid each cell of which is marked with a zero. Finally, overlaying the (c) and the (d) parts of Fig. 4.4 creates the image of the product 32  57. One can see that the whole is included in the former image as its part. In other words, the product 32  57 is greater than the number 1. Numerically, this product is equal to the fraction 15/14 as it is represented by 15 cells (10 cells marked with both x’s and 0s and 5 cells below them marked with 0s only), each of which has the name, 1/14, stemming from the whole shown in Fig. 4.4c. That is, while the product is represented by an image greater than the whole, this image, unlike the case of Fig. 4.3, is part of the whole grid shown in Fig. 4.4e. One can also see that unlike the grids shown in Figs. 4.2 and 4.3, the grid of Fig. 4.4 has all cells marked. This observation will be explained conceptually in the next section.

4.3.1.4

Reflection on Three Cases Discussed in Sects. 4.3.1.1–4.3.1.3

Remark 4.1 When all three cases of multiplying two fractions—both proper fractions, both improper fractions, and a combination of proper and improper fractions—are presented through the diagrams of Figs. 4.2, 4.3e and 4.4e, one can recognize that only in the third case, Fig. 4.4e, the entire grid is covered by the marks; in the first two cases there are cells with no marks. How can this observation be explained? In the case of multiplying two proper fractions, 2/3 and 5/7, only a part of the whole gets marks for 2/3 and another part of the whole gets marks for 5/ 7. Thus, a part of the whole that does not belong to either 2/3 or 5/7 does not get any marks. In the case of multiplying two improper fractions, 3/2 and 7/5, the first fraction, assigned the mark 0, is represented by an extension of the whole and the second fraction, assigned the mark x, is also represented by an extension of the whole. Thus, an area, which complements the second extension of the whole to get the extension by 1/2 of the first extension, gets no marks. Finally, in the case of multiplying fractions of different type, 3/2 and 5/7, the 0s, representing 3/2, are assigned to the entire extension of the whole by 1/2 and no more extensions of the whole exist.

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Remark 4.2 Note that it is never possible to cover the entire multiplication grid (be it the whole or its extension) by both marks when the area model is used to represent the product of two fractions. This is only possible when each factor is one whole. The reader may check to see that when multiplying two fractions, like 2/3 and 4/5, each of which is a unit fraction short of the whole, only one cell of the multiplication grid that represents the product of those unit fractions remains without any mark. Remark 4.3 Note that when we multiply two proper fractions, like 5/7 and 2/3, taking 5/7 of the whole yields 5/7 of 2/3. However, when we multiply two fractions of different kind, like 5/7 and 3/2, taking 5/7 of the whole does not yield 5/7 of 3/2. In that case, one first has to take 3/2 of the whole and then take 5/7 of 3/2 which would yield the correct result. Likewise, when we multiply two improper fractions, like 7/5 and 3/2, taking first 3/2 of the whole and then taking 7/5 of the whole (or vice versa, fist taking 7/5 of the whole and then taking 3/2 of the whole) does not yield either 7/5 of 3/2 or 3/2 of 7/5. Instead, taking first either 7/5 of the whole and then 3/5 of the so extended whole or vice versa yields the correct result. Remark 4.4 One can use another way of multiplying two improper fractions on a picture. Figure 4.5 shows how the fractions 5/7 and 2/3 can be written as mixed fractions and then multiplied similar to how the product of two-digit integers (not multiples of ten) can be presented as area of four rectangles. In Fig. 4.5, one such rectangle is a unit square split in 10 equal pieces, another rectangle shaded at the bottom-left represents 1/2 (split in five one-tenths), the top-right shaded rectangle (split in four one-tenths) represents 2/5, and the bottom-right shaded rectangle (split in two one-tenths) represents the product of 1/2 and 2/5. Numerically, the diagram of Fig. 4.5 shows that 3 7  ¼ 2 5

4.3.2




1 1þ 2


 2 1 2 1 2 21 1þ ¼ 1þ þ þ  ¼ 5 2 5 2 5 10

Dividing Fractions

Division is a more complicated operation than multiplication. This was already observed in the domain of integers: whereas the product of two integers is an integer, the quotient (produced when the larger integer is the dividend) may bring with it a non-zero remainder. However, the set of rational numbers is closed for all four arithmetical operations, except the case of a zero divisor. This section will demonstrate how division of fractions can be carried out conceptually using area model for fractions, explain a conceptual meaning of the “Invert and Multiply” rule, and show how the context of word problems can be used as a practice of dividing fractions both in iconic and symbolic forms.

4.3 The Meaning of Multiplying and Dividing Fractions

95

Fig. 4.5 An alternative way of multiplying two improper fractions

4.3.2.1

Dividing Whole Numbers as a Window into Fractions

As before, let us begin with dividing whole numbers by using area model. What does it mean to divide 4 into 3? Because 4 > 3, the smaller number cannot be measured by the larger number as one of the interpretations of division suggests. That is, in terms of the measurement model for division, the inclusion of 4 into 3 is an abstraction, as something larger than 3 may not be physically included in its part. Nonetheless, this abstraction can be represented in the form of a (positive) fraction smaller than one. Contextualization may help to comprehend abstraction. In our case, the partition model for division means dividing three things among four people. Obviously, one should be able to cut things into equal pieces; that is, the things to be cut should be conducive to be partitioned in smaller (equal) parts. For example, it is not possible to divide three marbles in four equal parts. Yet, it is possible to divide (fairly) three pies among four people. To do that, one has to divide each pie into four equal parts, giving each person 1/4 of a pie. With three (identical) pies, each person would get 3/4 of a pie. Put another way, in order to find how many times 4 is included into 3 one has to divide 3 by 4. How can one carry out this division? Let 3
4 ¼ x. Then x is a missing factor in the equation 4x ¼ 3. The top part of Fig. 4.6 shows the left-hand side of the last relation, that is, 4x. At the same time, the bottom part of Fig. 4.6 represents the right-hand part of the equation, that is, 3. Consequently, the (referent) unit is shaded dark and it is one-third of the top part of Fig. 4.6. The two-dimensional area model shows that because the unit consists of four cells, each of which represents the fraction 14, previously unknown x becomes known as it consists of three such cells. That is, x ¼ 34 or 3
4 ¼ 34.

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Fig. 4.6 Using two-dimensional method when dividing 4 into 3

4.3.2.2

Dividing Two Fractions as Finding a Missing Factor

Now, using the same method, let us divide two proper fractions. For example, let us find the value of 34
57. The result of this division is a number x, five-seventh of which is equal to three-fourth, that is, x is the missing factor in the equation 57 x ¼ 34. Figure 4.7 comprises three diagrams, the far-left one representing x. The middle diagram shows that x consists of seven equal sections with an unknown numerical value. In order to make it known, one has to represent the unit. Because 57 x (shown in the middle diagram) is equal to 34 (of the unit), it has to be extended by 14 in order to show the unit. This extension is shown in the far-right diagram of Fig. 4.7 using dotted lines. Through this method, the unit turned out having been divided into 5  4 ¼ 20 equal sections each of which is 1/20 of the unit. At the same time, we see that x comprises 7  3 ¼ 21 such sections. That is, x ¼ 21 20. Put another way, in order to divide 34 by 57 one has to multiply 34 by the reciprocal of 57; that is, 34
57 ¼ 34  75 ¼ 21 20. In Sect. 4.4 of this chapter, the meaning of this rule, commonly referred to as Invert and Multiply, will be explained. Remark 4.5 In the case of multiplying proper fractions using the two-dimensional area model, picture multiplication begins with drawing the (referent) unit. In the case of dividing proper fractions, the unit is unknown, and it has to be found through solving an equation with a missing factor. The same is true for improper fractions. Alternatively, as a practice in dividing fractions using the two-dimensional model, one can measure 3/7 by 4/5; in other words, one has to divide 3/7 by 4/5, that is, to find x from the equation 45 x ¼ 37. The process of solving the last equation is shown in Fig. 4.8 where the top diagram represents x, the middle diagram shows the left-hand side of the equation, and the bottom diagram shows that the equality between the two sides of the equation implies the following: the unit consists of 15 sections and therefore x consists of 10 sections each of which is equal to 1=28. As a result, we have x ¼ 15 28.

4.3 The Meaning of Multiplying and Dividing Fractions

Fig. 4.7 Using two-dimensional method when dividing

3 4

97

by

5 7

Fig. 4.8 Dividing 3/7 by 4/5 yields 15/28

Likewise (Fig. 4.9), one can divide a proper fraction by an improper fraction (alternatively, measure a proper fraction by an improper fraction, for example, 3/7 by 5/4; that is, to find x from the equation 54 x ¼ 37.

4.3.2.3

Solving Word Problems Using Fractions

When solving word problems involving fractions, once again, one has to critically rely on the W4S principle. Problem 4.1 Anna has five bottles of apple juice. If the serving glass is 3/4 of a bottle, how many servings can she make out of the bottles? Solution As shown in Fig. 4.10, in order to solve the problem, one has to measure five bottles by a glass; that is, to use the measurement model for division when dividing 3/4 into 5. This division yields the remainder. An uncritical use of the W4S principle may describe the remainder (two shaded sections within the fifth bottle) as

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4 Modeling Mathematics with Fractions

Fig. 4.9 Dividing 3/7 by 5/4 yields 12/35

Fig. 4.10 Six and two-thirds servings

1/2. However, one cannot describe servings both in terms of glass and bottle. As shown in the far-left part of Fig. 4.10, a glass (serving) is the unit (1 s). As shown in the far-right part of Fig. 4.10, extending two shaded sections by one section yields a full serving, implying that the two sections represent 2/3 of a serving. That is, from five bottles of apple juice Anna can make 6 23 servings. Note that a formal, purely arithmetical solution to Problem 4.1 does not result in any confusion. Indeed, 2 4
35 ¼ 4  53 ¼ 20 3 ¼ 6 3. Problem 4.2 Mary has 8 13 pounds of flour. It takes 3 12 pounds to make a cake. How many cakes can she make? Solution As shown in Fig. 4.11 through the use of the measurement model for division, Mary can make two full cakes. After that, 1 13 pounds of flour remain. Once again, the question is how to measure this amount by the full cake, that is, how to

4.3 The Meaning of Multiplying and Dividing Fractions

99

Fig. 4.11 What fraction of the cake can be made out of the shaded flour?

Fig. 4.12 Dividing 1 13 by 3 12

measure a fraction by a fraction. Simply put, what is the result of dividing 3 12 into 1 13? To answer this question, one has to solve the equation 3 12  x ¼ 1 13 or 72  x ¼ 43. The process of solving the last equation, which is equivalent to finding x ¼ 43
72, is shown in Fig. 4.12. The top diagram at the right shows the value of 72  x, with x shown at the top left. The bottom diagram shows that the equality 72  x ¼ 43 implies that the unit consists of 21 sections and, therefore, x consists of 8 sections each of 8 8 which is 1=21 of the unit. Therefore, we have x ¼ 21 ; that is, Mary can make 2 21 cakes. The same result follows from formal division of the two given fractions. 2 50 8 Indeed, 8 13
3 12 ¼ 25 3  7 ¼ 21 ¼ 2 21. Problem 4.3 A turtle has crept the distance of 3 13 km. If it crept 23 km per day, how many days did it take the turtle to creep the whole distance? Solve this problem as measurement and as a missing factor equation by using area model. Solution Figure 4.13 shows how one can measure 3 13 km by 23 km. The relation 3 13
23 ¼ 5 is the result of this measurement. Figure 4.14 shows a different approach—solving Problem 4.3 as a problem with a missing factor by reducing the relation 3 13
23 ¼ x to 23 x ¼ 3 13. The far-left rectangle in Fig. 4.14 represents an

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4 Modeling Mathematics with Fractions

Fig. 4.13 Measurement model for division in solving Problem 4.3

Fig. 4.14 Solving Problem 4.3 as a problem with a missing factor

3 unknown x. The grid in the middle represents 23 x which is also 10 3 . Then, separating 3 1 from 10 3 shows the unit which consists of 6 cells. Therefore, one cell represents 6. Finally, the far-right diagram shows x represented through a 30-cell grid by adding to 23 x another 13 x to get x. Because each cell is 16 (of the unit), we have x ¼ 30 6 ¼ 5.

Remark 4.6 Measurement model for division when used in the context of fractions should take into account the unit of measurement. Similarly, measurement approach to fractions proposed by Bobos and Sierpinska (2017) is based on the idea that genesis of a fraction as an abstract concept belongs to the process of taking a fraction of something, be it a distance (measured in linear units), a piece of land (measured in square units), or a space within a basement (measured in cubic units). To this end, one has to keep in mind that it is possible to go astray by formally using arithmetic of fractions in solving word problems without paying attention to the quantitative meaning of computations (Thompson and Saldanha, 2003). For example, in Problem 4.3, one measures distance in kilometers. And therefore, one measures the fraction 3 13 by the fraction 23 when dividing the former by the latter. Consider another problem about paving two parking lots, A and B, shaped as squares of sides 80 23 m and 100 18 m, respectively. It is known that 5/11 of lot A and 1/3 of lot B need new pavement. The question to be answered is which parking lot needs more pavement. If one formally takes 5/11 of 80 23 (m) and 1/3 of 100 18(m),

4.3 The Meaning of Multiplying and Dividing Fractions

101

5 the following fractions follow, respectively, 11  80 23 ¼ 36 23 and 13  100 18 ¼ 33 38. From these calculations, one might conclude that because 36 23 [ 33 38, parking lot A needs more pavement. However, in the spirit of Bobos and Sierpinska (2017), one has to recognize that in the parking lot problem this something is a square rather than its side and, therefore, one has to take fractions of squares (measured by square units) and not fractions of their sides (measured by linear units). With this in mind, one has to refine the above computations by taking fractions of areas of the 2 5 parking lots as follows2: 11  ð80 23Þ2 ¼ 2957 79 (m2) and 13  ð100 18Þ2 ¼ 3341 43 64 (m ), respectively. Now, one can see that lot B needs mote pavement than lot A, although originally, without thinking of a fraction as “fraction of something” (ibid., p. 205), the opposite conclusion could have been drawn.

In this regard, one of the Socratic dialogues, in fact the first one (Gulley, 1962, p. 11), written by Plato is worth noting (this dialogue is also referenced in Chap. 3, Sect. 3.3.1). Socrates asks his interlocutor how the twofold increase of area of a square affects the change of its side length and gets the response that it also increases twofold because “a double square comes from a double line” (cited in Arnheim, 1969, p. 221). Just as in the case of the parking lot problem, the interlocutor transfers the quantity measured in one dimension to a two-dimensional model. This is also an example of the so-called Einstellung effect (Luchins, 1942)— a psychological phenomenon evinced by an individual through a tendency to use previously learned problem-solving strategy in situations that either can be resolved more efficiently or to which the strategy is not applicable at all.3

4.4

Conceptual Meaning of the Invert and Multiply Rule: From Integers to Fractions

Typically, the Invert and Multiply rule is used when dividing two fractions and replacing division by multiplication as an operation which turns two integers into an integer. For example, in the case 25
56 it might be tempting, by analogy with multiplying two fractions when their numerators and denominators are multiplied, to proceed dividing numerators and denominators to have 2
5 5
6. This, however, brings us back to the division of the original two fractions by using the dividend-divisor context for division in the case of fractions. The Invert and Multiply rule allows one not to deal with the fraction 2
5 5
6, but rather to compute the in which the resulting numerator and demominator are integers. fraction 26 55

2

Note that the fractional computations can be outsourced to Wolfram Alpha. The significance of this dialogue is both mathematical and psychological as it requires knowledge pffiffiffi of 2 which did not have an exact numeric value at that time and, therefore, the change of the side length of the original square could not be expressed exactly in arithmetical terms.

3

102

4.4.1

4 Modeling Mathematics with Fractions

The Case of Dividing Integers

In order to understand conceptual meaning of the rule, one has to explain it in the context of the division of two whole numbers. For example, the division operation 6
2 ¼ 3 can be interpreted as finding the number of 2-tile towers built out of 6 tiles. Here, the numbers 2 and 6 point at a tile as the unit and measuring 6 tiles by 2 tiles in the process of building towers yields 3 towers. If division is replaced by multiplication and 2 is replaced by 1/2 (the inversion of 2), we have 6  12 ¼ 3. This time, we have six one-halves; that is, a tile is not the unit anymore, but it is 1/2 of a tower, which became a new unit through using the Invert and Multiply rule. Repeating half of a tower six times yields three towers. That is, the Invert and Multiply rule makes sense in the case of the division of two whole numbers when the measurement model for division is used. However, if one has to build two identical towers out of six tiles, we also have 6
2 ¼ 3, but this time, we use partition (or partitive) model for division. Replacing 6
2 by 6  12 does not result in the change of unit as 6 and 2 are comprised of different units—tiles and towers. However, due to commutative property of multiplication, 6  12 ¼ 12  6 and, therefore, 6
2 is replaced by taking half of six. So, in the case of partition model for division the change of unit does not take place through the Invert and Multiply rule. Instead, the meaning of the Invert and Multiply rule in the case of partition model for division can be understood in terms of the dividend-divisor context.

4.4.2

The Case of Dividing Fractions

Now, in order to explain the Invert and Multiply rule in the context of fractions, consider the division operation 3 13
56. The operation can be interpreted as measuring the distance of 3 13 km by the segments of length 56 km when a runner covers any segment of this length in a minute (that is, assuming the uniform movement). This kind of measurement would result in the number of minutes the runner spends to cover this distance. Here, 1 km is the unit, and the runner makes 56 of the unit per minute. Replacing division by multiplication and 56 by 65 yield 3 13  65 ¼ 106 35 ¼ 4. This time, the unit is the segment ran by the runner in a minute; this segment is smaller than 1 km which becomes 6/5 of the segment. Having 3 13 km means that 65 of this segment (that is, 1 km) when repeated 3 13 ð¼ 10 3 Þ times yields 4 min. Note that in the described case, the measurement model for division was used. The situation is more difficult in the case of division used as an operation needed to find a missing factor when another factor and the product are given. For example, when twice the distance (d) from A to B is 6 km, we have the equation 2d ¼ 6 whence d ¼ 6
2 ¼ 3. This time, just as in the case of building two-tile towers out of six tiles, the meaning of the division operation used to find d is distributing 6 km

4.4 Conceptual Meaning of the Invert and Multiply Rule …

103

within 2 distances (this distribution is the partition model for division). The result is 3 km for the distance which is the unit. Now, replacing division by multiplication, inverting 2, and using commutative property of multiplication yield 6  12 ¼ 12  6 ¼ 3. That is, the distance d became 12 of 6 km. Note that taking 12 of 6 is the same as dividing 6 by 2. That is, once again, in the case of partition model for division the Invert and Multiply rule can be understood in terms of the dividend-divisor context. Consider now a situation when 5/6 of unknown distance d is equal to 10/3 km. 10 5 Finding d from the equation 56 d ¼ 10 3 , leads to the division operation d ¼ 3
6. 10 5 This time, the process of distributing 3 km within 6 distances is difficult to imagine 5 for it is an abstraction; yet the distance d is still the unit when the operation 10 3
6 is 10 5 10 6 6 10 considered. Replacing 3
6 by 3  5 and the latter by 5  3 (due to commutative property of multiplication), means the change of unit; that is, the distance d be6 10 6 10 10 comes 65 of 10 3 km. The product 5  3 means taking 5 of 3 km, where 3 (km) is the 6 10 new unit. In that way, d ¼ 5  3 ¼ 4 (km). The meaning of operations in the case of fractions with a missing factor can only be explained by referring to a similar case that involves whole numbers; otherwise, the partition model, unlike the measurement model, is a pure abstraction.

4.4.3

The Invert and Multiply Rule as a Change of Unit

Where can one see in the Invert and Multiply rule the change of unit when one makes a transition from 23
12 to 23  2? Which one is the original unit (in the case of division) and which one is the new unit (in the case of multiplication)? To answer these questions, let 23
12 ¼ x. Then 12 x ¼ 23. The first fraction in the last relation is 12. One may assume that x, a missing factor, is the original unit. This unit and its half are shown in Fig. 4.15, parts (a) and (b), respectively. Therefore, the new unit is the region for which 12 x is equal to 23. This new unit is shown in Fig. 4.15c, one section of which is 13. At the same time, this 13 is also 14 of x. Therefore, the new unit is 34 of x or 3x/4. One can also see in Fig. 4.15d that 23  2 ¼ 2  23 ¼ x. That is, indeed, 2 1 2 3
2 ¼ 3  2. 5 10 6 Consider now the case of 10 3
6 ¼ 3  5. If we begin with the (missing factor) equation 56 x ¼ 10 3 , then x may be considered as the original unit shown in Fig. 4.16a. 5 Consequently, 6 x is shown in Fig. 4.16b. At the same time, 56 x is also 10 3 as shown in Fig. 4.16c, where the new unit, 33, consists of 15 cells. Therefore, x is 65 of 10 3 as 6 10 shown in Fig. 4.16d. One can see that x ¼ 5  3 consists of 60 cells, each of which is 1/15 of the new unit shown in Fig. 4.16c. Therefore, 10 5 10 6 6 10 60
¼  ¼  ¼ ¼ 4. When the divisor 5/6 came from the unit 6/6 and 3 6 3 5 5 3 15 then in the grid, showing 10/3, the new unit, 3/3, was located, this allowed for 10/3 to be replaced by 50/15 so that 6/5 of 50/15 became 60/15 = 4.

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4 Modeling Mathematics with Fractions

Fig. 4.15 From x being a unit to 3x/4 being a unit

Fig. 4.16 From x being a unit to

3 10

of 56 x being a unit

4.5 Unit Fractions

4.5

105

Unit Fractions

A unit fraction is the first type of fractions students become familiar with as these entities of arithmetic are introduced already in the early grades. Although a unit fraction is still a common fraction with division not being completed, a formal rule for comparing unit fractions is quite simple: the larger the denominator, the smaller the fraction. For example, 1/3 is smaller than 1/2 because 3 is greater than 2. Once again, just as 3 > 2 because both integers are comprised of the same unit, the tacit assumption behind the inequality 1/2 > 1/3 is that the two fractions are parts of the same unit. That is, a contextualization of the last inequality means that one-half of a pie is greater than 1/3 of the same pie. To a certain extent, in the context of fractions, unit fractions are like the multiples of ten in the context of integers. Indeed, one can say that 70 > 60 because 7 > 6. At the same time, in order to conclude that 74 > 65, one has to show that the difference 74–65 is a positive number; that is, to carry out an operation like in the case of comparing 4/7 and 5/6.

4.5.1

Unit Fractions as Benchmark Fractions

In integer arithmetic, the benchmark integers are multiples of ten. A problem may be to position an integer between two consecutive multiples of ten; e.g., 20 < 27 < 30. In much the same way, unit fractions are called benchmark fractions and a similar problem is to position a fraction, say, 5/12, between two consecutive unit fractions. One can see that 5/12 > 4/12 = 1/3 and 5/12 < 6/12 = 1/2. Therefore, 1/3 < 5/12 < 1/2 (Fig. 4.17). A slightly more difficult task is to position the fraction 26/69 between two consecutive unit fractions. To do that, one may write 26/69 > 26/78 = 1/3 and 26/ 69 < 26/52 = 1/2 (placing 69 between two closest multiples of 26, 2  26\69\3  26, and using the notion of equivalent fractions—another big idea of mathematics unmasking a nontrivial numeric congruity among differently written quantities). Therefore, 1/3 < 26/69 < 1/2. Sometimes, it is important to calculate the difference between a fraction and the largest benchmark fraction smaller that it. In our case, 26/69–1/3 = (26–23)/69 = 3/69 = 1/23 and therefore, 26/69 = 1/3 + 1/23. The significance of the last equality will be explained in Sect. 4.5.4. cBig Idea A possibility of numeric congruency of differently written mathematical symbols, like fractions, is a big idea of mathematics. What is hidden behind this idea is that a fraction is both a number and an operation which is only identified but has not been completed. The equivalence of differently written fractions can be revealed by teachers and understood by students through focusing on different units/wholes as the frames of reference.b

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4 Modeling Mathematics with Fractions

Fig. 4.17 Placing 5/12 between two benchmark fractions

The context of word problems such as the work and the distance problems can be used to discuss the concept of representing a unit fraction as a sum of two like fractions. A two-dimensional (area) model for fractions will be used to mediate the discussion. The general case of partitioning a unit fraction into a sum of two like fractions can be considered both from conceptual and computational perspectives and mathematical machinery associated with counting the number of such representations depending on the denominator of the partitioned fraction will be connected to the context of prime factorization, another topic studied in the middle school. A representation of the fraction 1/2 as a sum of three unit fractions will be considered in Sect. 4.5.3 and connected to the topic of edge-to-edge tessellation in Chap. 7. Egyptian fraction representations of fractions as a sum of unit fractions will be considered and the Greedy algorithm which produces such representations will be introduced.

4.5.2

Representation of a Unit Fraction as a Sum of Two Like Fractions

Consider the following problem: How many integer-sided rectangles with area being n times as much (numerically) as semi-perimeter are there? If x and y are the side lengths of such a rectangle, x  y, then xy = n(x + y). Dividing both sides of this equation by nxy yields the equation 1 1 1 þ ¼ : x y n

ð4:1Þ

In that way, the problem of finding rectangles is reduced to the problem of finding all ways to represent a unit fraction 1n as a sum of two like fractions. Many other problems can use Eq. (4.1) as a mathematical model. For example, if two workers when working together can complete a certain job in n days, one may want to know the number of days it would take them to complete this job when working alone (see Problem 9.9, Sect. 9.8, Chap. 9). Another problem can be formulated as

4.5 Unit Fractions

107

follows: If two vehicles start moving towards each other with different but constant speed (a uniform movement) and meet in n hours, find the number of hours it would take each of them to cover the whole distance. As mentioned in the Common Core State Standards (2010), “Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling” (p. 62). That is, Eq. (4.1) can be considered as a mathematical model of various concrete situations including geometry, work and distance problems. nx Equation (4.1) can be written in the form 1y ¼ 1n  1x whence y ¼ xn ¼ nðxnÞ þ n2 xn

2

n ¼ n þ xn . That is, noting that x > n (as it follows from the distance problem), Eq. (4.1) is equivalent to the equation

y ¼ nþ

n2 : xn

ð4:2Þ

Because x and y, x  y; are whole number side lengths of a rectangle, the area of which is n times as much as its semi-perimeter, the number of solutions of Eq. (4.1) coincides with the number of divisors of n2 among the differences x – n in Eq. (4.2), where 2n\x\nðn þ 1Þ: The last inequalities follow from the following observation: the values of the side lengths x ¼ y ¼ 2n and x ¼ nðn þ 1Þ; y ¼ n þ 1 satisfy Eq. (4.1). Indeed, the unit fraction 1n can always be represented both as a sum of two 1 1 equal unit fractions, 1n ¼ 2n þ 2n , and as a sum of two unit fractions with the largest possible difference between their denominators, 1n ¼ n þ1 1 þ nðn 1þ 1Þ, an identity4 in which the fraction n þ1 1 is the largest unit fraction smaller than 1n, and, therefore, the difference 1n  n þ1 1 ¼ nðn 1þ 1Þ yields the smallest possible fraction in the representation of 1n as a sum of two like fractions. Furthermore, the inequality x  y implies that the ranges for the unknowns x and y are as follows: 2n  x  nðn þ 1Þ;

n þ 1  y  2n:

ð4:3Þ

Inequalities (4.3) can be interpreted geometrically: the case x ¼ y ¼ 2n gives a square with the smallest perimeter and the case x ¼ nðn þ 1Þ; y ¼ n þ 1 gives a rectangle with the largest perimeter among all rectangles in question. The first pair of inequalities (4.3) imply the inequalities n  x  n  n2 . As n and n2 are trivial divisors of n2 , one has to find all other divisors of n2 in the interval (n, n2). Note that 2 if m, such that n < m < n2, divides n2 , then nm divides n2 as well. The inequalities n\m\n2 imply opposite inequalities among the reciprocals, n12 \ m1 \ 1n, whence, 2 multiplying each reciprocal by n2 , yields 1\ nm \n. That is, every non-trivial divisor of n2 in the range (n, n2) has the counterpart in the range (1, n). If one counts n as a (trivial) divisor of n2 twice, then the number of all divisors of n2 in the range [n, n2] is equal to half the sum of one and the total number of divisors of n2. Put 4

According to Hoffman (1998), this identity is ascribed to Fibonacci.

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4 Modeling Mathematics with Fractions

Fig. 4.18 Wolfram Alpha shows that an integer square has an odd number of divisors

another way, if D(n2) is the total number of divisors of n2, then the number of such divisors in the range [n, n2] is equal to (D(n2) + 1)/2. Finally, the last expression represents the number of ways the unit fraction 1/n can be represented as a sum of two like fractions. Now, one has to find D(n2). Note that an integer square always has an odd number of divisors. Indeed, if n ¼ pr11 pr22 . . .prkk (the prime factorization of n) then 2rk 1 2r2 n2 ¼ p2r 1 p2 . . .pk and thereby, by the Rule of Product (Chap. 8, Sect. 8.2), there are ð2r1 þ 1Þð2r2 þ 1Þ . . . ð2rk þ 1Þ divisors of n2 as each prime factor pi can be selected in 2ri þ 1 ways with the exponents ranging from zero to 2ri . This reasoning can be supported by Wolfram Alpha which, can demonstrate (in the context of its free on-line version) that square of any integer has an odd number of divisors (e.g., in Fig. 4.18 all divisors of the first 25 squared integers are displayed in brackets; one can check to see that each bracket includes an odd number of divisors of the corresponding square). Therefore, there are ð2r1 þ 1Þð2r2 þ 21Þ...ð2rk þ 1Þ þ 1 ways to partition the unit fraction 1n into a sum of two like fractions. When n = 3 we have

n2 ¼ 32 , that is r1 ¼ 1; ri ¼ 0; i [ 1 and ð2r1 þ21Þ þ 1 ¼ 3 þ2 1 ¼ 2—the number of ways the unit fraction 1/3 can be partitioned into a sum of two like fractions. 1 are the only partitions of 1/3 into a sum of two Indeed, 13 ¼ 16 þ 16 and 13 ¼ 14 þ 12 unit fractions where 6 ¼ 2  3 and 4 ¼ 3 þ 1 so that 12 ¼ 3  4 (see also solution of Eq. (9.12), Sect. 9.8, Chap. 9).

4.5 Unit Fractions

4.5.3

109

Representation of 1/2 as a Sum of Three Different Unit Fractions

Formula (4.2) can be used to find positive integer solutions to Eq. (4.1) with the goal of finding all additive decompositions of 1/2 into three unit fractions. For example it follows from Eq. (4.2) that when n = 2 we have 12 ¼ 13 þ 16 and 1 1 1 1 1 1 1 1 1 2 ¼ 4 þ 4; when n = 3 we have 3 ¼ 4 þ 12 and 3 ¼ 6 þ 6; when n = 6 we have 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 ¼ 7 þ 42,6 ¼ 8 þ 24,6 ¼ 9 þ 18, 6 ¼ 10 þ 15, and 6 ¼ 12 þ 12. Proceedings from the equality 12 ¼ 13 þ 16 and neglecting (for the brevity) the cases of equal unit fractions, the following additive decompositions of 1/2 can be developed 1 1 1 1 ¼ þ þ ; 2 3 7 42 1 1 1 1 ¼ þ þ ; 2 4 5 20

1 1 1 1 ¼ þ þ ; 2 3 8 24 1 1 1 1 ¼ þ þ : 2 4 6 12

1 1 1 1 ¼ þ þ ; 2 3 9 18

1 1 1 1 ¼ þ þ ; 2 3 10 15

The results of this section will be used in Chap. 7. Remark 4.7 Wolfram Alpha (in the context of its free on-line version) provides the above six partitions of the fraction 1/2 into a sum of three different unit fractions after the command “solve over integers 1/x + 1/y + 1/z = 1/2, x > y > z > 1” is entered into the program’s input box. Furthermore, Wolfram Alpha Pro (available by subscription only) is capable to provide all partitions of 1/2 into a sum of four unit fractions. According to Wolfram Alpha Pro, the number of such partitions is 72.

4.5.4

Egyptian Fractions and the Greedy Algorithm

An Egyptian fraction is a sum of the finite number of distinct unit fractions, for example, 13 þ 14 þ 15 þ 16 is such a fraction. Egyptian fractions were developed in ancient Egypt around 2050 B.C., replacing a hieroglyphic notation. A table, representing fractions of the form 2/n as a sum of distinct unit fractions appeared in the famous Egyptian papyrus roll (ca. 1650 B.C.) found in 1858 by Henry Rhind, a Scottish scholar and collector of antiques (Chace, Manning, Archibald, 1927). Although the genesis of using Egyptian fractions is not known, several insightful suggestions of how the representations for that table were found are offered at (“Rhind Mathematical Papyrus 2/n table”, 2020). Egyptian fractions as a special notation continued to be used into the Middle Ages. As mentioned by Graham (2013), when André Weil5 was asked about a reason Egyptians used this notation,

5

André Weil (1906–1998)—a French mathematician, known for his fundamental contributions to number theory.

110

4 Modeling Mathematics with Fractions

“he thought for a moment and then said, “It is easy to explain. They took a wrong turn!”” (p. 290, italics in the original). A unit fraction itself, like 12, is an Egyptian fraction, although it can be repre1 sented as a sum of other unit fractions, like 12 ¼ 13 þ 17 þ 42 . Sometimes, an Egyptian fraction is understood as a special representation of a fraction (alternatively, a positive rational number) through a finite sum of distinct unit fractions, so that the sum 14 þ 14 þ 15 þ 16 which is equal to 13 15 is not considered an Egyptian fraction 13 representation of 15. If one asks Wolfram Alpha to represent 13 15 as an Egyptian 1 fraction, the answer is 12 þ 13 þ 30 . However, there is another sum with three distinct unit fractions, 12 þ 15 þ 16, which is equal to 13 15 as well. Below, whereas the algorithm apparently used by Wolfram Alpha will be explained, the term an Egyptian fraction will be used for any finite sum of distinct unit fractions. Often, Egyptian fractions can be utilised to solve simple division word problems more effectively in comparison with the use of the dividend-divisor context (Sect. 4.1), such as dividing three (identical) pies among four people (see Fig. 4.1). The equality 34 ¼ 12 þ 14, the right-hand side of which is an Egyptian fraction, can be used to decide the division in a different way (Fig. 4.19): three pies are divided into eight pieces (rather than in 12 as in Fig. 4.1) and, consequently, each person would 1 can be used to get half of pie plus one-fourth of it. Likewise, the equality 35 ¼ 12 þ 10 decide the division of three (identical) pies among five people: the pies are divided into 10 pieces (rather than into 15 using the dividend-divisor context for fractions) so that each person would get half of a pie plus one-tenth of it. As an aside, note that such “creative” way of dividing pies into a smaller number of pieces through Egyptian fractions associates creativity with usefullness, thereby indeed indicating that the latter is one of the two major characteristics of the former (Plucker et al., 2004). It should be noted that an Egyptian fraction does not always provide a smaller number of pieces in comparison with the dividend-divisor context. A simple 1 example is 25 ¼ 13 þ 15 . Using this representation, two pies can be divided into three equal pieces each and one of the six such pieces is then divided into five equal pieces. As a result, we have five 1/3 pices of a pie and five 1/15 pieces of a pie, ten pieces in all. At the same time, dividing each pie into five equal pieces yields ten pieces as well. A more complicated example is provided by the Egyptian fraction 3 1 1 1 7 ¼ 3 þ 11 þ 231. The dividend-divisor context yields 21 equal pieces by dividing each pie into seven equal pieces. The Egyptian fraction yields 7 pieces, each of which is 1/3 of a pie; then 7 pieces, each of which is 1/11 of a pie; and another 7 pieces each of which is 1/231 of a pie. Once again, both divisions yield 21 pieces. 1 1 7 1 21 þ 1 Note that 7  ð13 þ 11 þ 231 Þ ¼ 73 þ 11 þ 33 ¼ 77 þ33 ¼ 99 33 ¼ 3; that is, putting all the 21 pieces together results in 3 pies. One can see that in some cases both an Egyptian fraction and the dividend-divisor context can deliver the same number of pieces. And in the specific case of dividing three pies among seven people, creativity would be to recognize the uselessness of resorting to the Egyptian fraction.

4.5 Unit Fractions

111

Fig. 4.19 An Egyptian fraction: 34 ¼ 12 þ 14

By using the identity1n ¼ n þ1 1 þ nðn 1þ 1Þ (already mentioned in Sect. 4.5.2), an Egyptian fraction can comprise any number of unit fractions.6 Among several algorithms of representing a proper fraction through an Egyptian fraction is the so-called Greedy algorithm which, just as the last identity, is also ascribed to Fibonacci (Hoffman, 1998). This algorithm (apparently utilized by Wolfram Alpha in the context of its free on-line version) can be used in developing an Egyptian fraction from a proper fraction as follows. If ab is a proper fraction (b [ a [ 1) reduced to the simplest form, then one has to find the smallest integer x1 such that x11 \ ab. One can note that x11 is the largest

benchmark unit fraction (Sect. 4.5.1) smaller than ab. The difference ab  x11 ¼ axbx1 b is 1 another proper fraction for which one has to find the smallest integer x2 such that ax1 b 1 x2  bx1 . This process might stop if the last relation turns into an equality, otherwise the process continues. For example, for the fraction 47 the value of x1 ¼ 2 1 4 1 1 so that 47  12 ¼ 427 72 ¼ 14 and, therefore, 7 ¼ 2 þ 14 is the representation sought. At the same time, selecting 13 as the first unit fraction in the representation of 47 through 5 , a sum of unit fractions yields the following chain of equalities: 47 ¼ 13 þ 21 5 1 4 4 1 1 4 1 1 1 1 21 ¼ 5 þ 105, 105 ¼ 27 þ 945. Therefore, 7 ¼ 3 þ 5 þ 27 þ 945 : Note, that in repre5 through a sum of unit fractions, the fraction 15 was the largest possible senting 21 (alternatively, 5 was the smallest possible denominator); that is, 15 was selected 1 through the Greedy algorithm. Likewise, the fraction 27 was the largest possible in 4 representing 105 . However, selecting 16 instead of 15 would yield a smaller number of 5 5 1 5 1 1 as 21 ¼ 16 þ 14 , yet 21 ¼ 15 þ 27 þ 945 . That is, an unit fractions representing 21 Egyptian fraction representation of a positive rational number is not always comprised of the smallest number of unit fractions. This demonstrates the complexity of mathematics associated with Egyptian fractions.

6

This method was developed in Chap. 2.

112

4.6

4 Modeling Mathematics with Fractions

From Long Division to Decimal Representation of Fractions

The algorithm of long division is not an outdated item of school mathematics curriculum. It can be used to demonstrate a number of concepts, including the decimal representation of a common fraction which emerges from the transition from uncompleted division (hidden in the common fraction) to completed division. For example, the common fraction 3/8 can be transformed into a decimal fraction by dividing 8 into 3 through the long division (Fig. 4.20). The result is the number 0.375; that is, 3/8 = 0.375. The right-hand side of Fig. 4.20 also shows the face values 3, 7, and 5 as quotients of three divisions until the zero remainder is reached and thus, if division continues, the remaining quotients would also be zeros. This means that the decimal representation of 3/8 is a terminating decimal. At the same time, 2/5 = 0.4 and, therefore, zero remainder is reached after the first division (of 20 by 5). The first observation that one can make from decimal representations is that numeric comparison of 0.375 and 0.4 is easier than comparison of 3/8 and 2/5. This is precisely because, while common fractions can be seen representing the dividend-divisor context which extends the division of integers to the domain of fractions when the dividend is not a multiple of the divisor, their decimal equivalents represent notations with completed division. The right-hand side of Fig. 4.20 shows an alternative representation of the long division through the sequence of relations among dividend, divisor, quotient, and remainder (DDQR), in which the quotients are the face values in the decimal representation of the common fraction 3/8. cBig Idea Decimal representation of proper fractions serves several goals including simple comparison of fractions, completing hidden division, representing fractions as a sum (finite or not) of fractions denominators of which are powers of ten, and carrying out arithmetical operations with a required precision. Achieving such goals using long division is one of the big ideas of mathematics. This idea allows for representing numbers as polynomials in the powers (both positive and negative) of ten, something that is important in digital computations.b The process of long division can demonstrate that zero remainder may never be reached. In that case, long division never terminates; however, the behavior of the face values it produces is periodic. As an example, consider the process of obtaining the decimal representation of the fraction 1/7 through long division. As shown in Fig. 4.21, the long division goes through the sequence of six DDQR relations until such relations begin repeating. This repetition is due to a simple yet conceptually profound fact that when a positive integer smaller than 7 (e.g., 1) is divided by 7, the corresponding remainder may not be greater than or equal to 7. Indeed, only six DDQR relations are possible: 1 ¼ 0  7 þ 1, 2 ¼ 0  7 þ 2, 3 ¼ 0  7 þ 3, 4 ¼ 0  7 þ 4, 5 ¼ 0  7 þ 5, 6 ¼ 0  7 þ 6.

4.6 From Long Division to Decimal Representation of Fractions

113

Fig. 4.20 Long division by 8 with 3 different remainders: 6, 4, and 0

Fig. 4.21 Long division by 7 with 6 different remainders, 3, 2, 6, 4, 5, 1

In general, when an integer smaller than n is divided by n, the largest number of different remainders is equal to n – 1. The corresponding quotients become face values and they repeat each other by forming cycles the length of which may not be greater than n – 1. In the case when the prime factorization of n consists of the powers of two and five only, the common fraction with the denominator n is represented by a terminating decimal; otherwise, the fraction is represented by a non-terminating decimal the face values of which form cycles of length not greater than n – 1. One can use Wolfram Alpha (in the context of its free on-line version) to see that the lengths of such cycles in the decimal representations of the fractions, say, 1/23, 1/31, 1/37, and 1/41 are, respectively, 22, 15, 3, and 5. One can hardly see any pattern here. This demonstrates the complexity of mathematics behind the issue of relating the length of a cycle in the decimal representation of 1/n to n; something that is beyond the scope of this textbook.

114

4.7 4.7.1

4 Modeling Mathematics with Fractions

Rational Numbers in Non-Decimal Bases Conversion of Integers into Non-Decimal Bases

Conceptual knowledge of operations in base-ten system is of the critical importance for students to understand “why division procedures work … [and] finalize fluency with multidigit addition, subtraction, multiplication, and division” (Common Core State Standards, 2010, p. 33). Consequently, learning how the rules of arithmetic work in bases other than ten can help teacher candidates to develop deep conceptual understanding necessary for teaching operations in base ten. Whereas school mathematics curriculum does not include operations in non-decimal bases, the inclusion of this topic in a mathematics course for middle school teachers can be recommended. This recommendation is consistent with a position by Vygotsky (1962) who argued, “As long as the child operates with the decimal system without having become conscious of it as such, he has not mastered the system but is, on the contrary, bound by it. When he becomes able to view it as a particular instance of the wider concept of a scale of notation, he can operate deliberately with this or any other numerical system” (p. 115). In this section, the conversion of integers into an arbitrary base system rooted in understanding of “the place value structure of our number system, which implicitly expresses numbers as polynomials in powers of 10” (Conference Board of the Mathematical Sciences, 2001, p. 5) will be considered in detail. Through extending this (implicit) expression of numbers to polynomials in powers other than ten, the corresponding conversion formulas will be developed. How can one convert a base-ten integer into a different base? Consider an example: convert a base-ten number 235 into a base-five number. First, the representation of 235 in base ten is the sum of powers of ten, the largest power being ten to the power two, as 102 \235\103 . Therefore, 235 ¼ 2  102 þ 3 101 þ 5  100 . Likewise, the representation of 235 in base five is the sum of powers of five, the largest power being five to the power three as 53 < 235 < 54. Therefore, 235 ¼ a1  53 þ a2  52 þ a3  51 þ a4  50 ;

ð4:4Þ

where ai —the digits of 235 in base five—have to be found. Note that 0  ai \5. Indeed, if one of the ai is greater than or equal to 5, then the product of ai and the corresponding power of 5 yields a power of 5 with an increased exponent. The same can be said about a polynomial-type representation of 235 (or any other integer) in any other base meaning that, just as in base ten there is no digit 10 (and 9 is the largest digit), in base B there is no digit B (and B – 1 is the largest digit). The digit a1 is relatively easy to find by dividing both sides of equality (4.4) by 53 and representing its left-hand side as the sum of the integer and the non-integer parts:

4.7 Rational Numbers in Non-decimal Bases

235 a2 a3 a4 þ 2þ 3 ¼ a1 þ 53 5 5 5

or

115

1þ

110 a2 a3 a4 þ 2 þ 3: ¼ a1 þ 53 5 5 5

a3 a2 a4 Therefore, a1 ¼ 1 and 110 53 ¼ 5 þ 52 þ 53 . Multiplying both sides of the last equality by 5 and representing its left had side as the sum of the integer and the non-integer parts yields

110 a3 a4 ¼ a2 þ 1 þ 2 2 5 5 5

or

4þ

10 a3 a4 ¼ a2 þ 1 þ 2 : 2 5 5 5

a3 a4 Therefore, a2 ¼ 4 and 10 52 ¼ 51 þ 52 . Multiplying both sides of the last equality by a4 5 yields 2 þ 0 ¼ a3 þ 51 , whence a3 ¼ 2 and a4 ¼ 0. Therefore, 23510 ¼ 14205 . In general, if N and B are base-ten positive integers, one can always find another positive integer n such that Bn1  N\Bn . Then the equality

N ¼ a1 Bn1 þ a2 Bn2 þ a3 Bn3 þ . . . þ an B0

ð4:5Þ

is the polynomial-type representation of N in base B, where the coefficients ai , which are to be found, represent the digits of N in base B, 0  ai \B; 1  i  n. The first step is to divide both sides of equality (4.5) by Bn1 to get N a2 a3 an þ 2 þ . . . þ n1 : ¼ a1 þ Bn1 B B B 1 ; 0  r1 \Bn1 . Let BNn1 ¼ N1 þ Brn1 r1 n . Then a1 ¼ N1 ; Bn1 ¼ aB2 þ Ba32 þ . . . þ Ban1 Multiplying both sides of the last equality by B yields

r1 a3 an þ . . . þ n2 : ¼ a2 þ Bn2 B B 1 2 ¼ N2 þ Brn2 ; 0  r2 \Bn2 . Let Brn2 r2 n Then a2 ¼ N2 ; Bn2 ¼ aB3 þ . . . þ Ban2 . Multiplying both sides of the last equality by B yields

r2 a4 an þ . . . þ n3 : ¼ a3 þ Bn3 B B 3 2 ¼ N3 þ Brn3 , 0  r3 \Bn3 . Let Brn3 Then a3 ¼ N3 and the process continues until all digits ai of the number N in base B are found. For example, when N = 33 and B = 2 we have 25 \33\26 , so that the equality 33 ¼ 1 þ 215 implies 3310 ¼ 1  25 þ 0  24 þ 0  23 þ 0  22 þ 0  21 þ 1  20 ¼ 25 1000012 . Likewise, when N = 55 and B = 16 we have 161 \55\162 , so that the 55 7 1 0 equality 16 1 ¼ 3 þ 161 implies 5510 ¼ 3  16 þ 7  16 ¼ 3716 . As far as technology

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4 Modeling Mathematics with Fractions

is concerned, note that whereas Wolfram Alpha Pro (available by subscription only) provides step-by-step conversion of numbers from base to base, its free on-line version provides the final result of conversion only. Remark 4.8 Setting B = 10 in (4.5) yields the polynomial-type base-ten representation of an n-digit number N. N ¼ a1 10n1 þ a2 10n2 þ a3 10n3 þ . . . þ an

ð4:6Þ

Formula (4.6) will be used in programming a spreadsheet to support topics dealing with recreational mathematics in Chap. 9, Sect. 9.10 and in Chap. 12, Sect. 12.11.

4.7.2

Conversion of Common Fractions into Non-Decimal Bases

Consider the fraction 13/15 as a base-ten number. In order to convert 13/15 into base five, the following steps have to be followed. 13 1 1. Multiplying 13 15 by 5 yields 3 ¼ 4 þ 3. This means that the first digit after the radix point in the base-five representation of the fraction 13 15 is 4. 1 5 2 2. Multiplying 3 by 5 yields 3 ¼ 1 þ 3. This means that the second digit after the radix point in the base-five representation of the fraction 13 15 is 1. 1 3. Multiplying 23 by 5 yields 10 ¼ 3 þ . This means that the third digit after the 3 3 13 radix point in the base-five representation of the fraction 15 is 3. 4. As we have arrived to 13 again, its conversion to base five was completed through steps 2 and 3. This means that the digits 1 and 3 will be repeated infinitely and thus in base 5 the fraction 13 15 is a repeating fraction, namely, 13 ¼ 0:4131313 . . . ¼ 0:413. Note that this conversion can be carried out by 15 5 Wolfram Alpha (in the context of its free on-line version) by typing “13/15 in base 5” in the input box of the program. The same outcome (i.e., without step-by-step solution) results from the use of Wolfram Alpha Pro.

In general, let N be a base-ten common fraction, i.e., 0 < N < 1. Then the equality N ¼ a1 B1 þ a2 B2 þ a3 B3 þ . . . þ Bn an þ . . .

ð4:7Þ

is the representation of N in base B. Multiplying both sides of (4.7) by B yields NB ¼ a1 þ a2 B1 þ a3 B2 þ . . . þ Bðn1Þ an þ . . .:

4.7 Rational Numbers in Non-decimal Bases

117

This means that the integer part of NB is equal to a1 —the first digit after the radix point in the base-B representation of N. Consider now the fraction N1 ¼ a2 B1 þ a3 B2 þ . . . þ Bðn1Þ an þ . . .. Multiplying both sides of the last equality by B yields N1 B ¼ a2 þ a3 B2 þ . . . þ Bðn2Þ an þ . . ., meaning that the integer part of N1B is equal to a2 —the second digit after the radix point in the base-B representation of N. The process either continues producing a repeating fraction or stops producing a terminating fraction. Note that a terminating base-ten fraction 1/4 has a non-terminating periodic representation in base-five. Indeed, multiplying 1/4 by 5 yields 5/4 = 1 + 1/4 implying that the integer part of the fraction 5/4 is the first digit after the radix point in the base-five representation of 1/4. Because the fractional part of 5/4 is also 1/4, the digit 1 will be repeated infinitely. That is (1/4)10 = (0.111…)5. At the same time, because 6/4 = 1 + 2/4 = 1 + 1/2, the first digit after the radix point in the base-six representation of 1/4 is 1. The next (and the last) step is to multiply 1/2 by 6 to get 3—an integer. Therefore 3 is the second (and the last) digit after the radix point in the base-six representation of 1/4 and the process of conversion stops. That is, (1/4)10 = (0.13)6.

4.7.3

Visual Representation of Conversion of Common Fractions in Different Bases

How can one visually represent a unit fraction 1/4 as a decimal fraction, the numeric representation of which follows from applying long division to the unit fraction? Consistent with the set model for fractions when more than one object may be considered as one whole, allowing for a proper fraction to be represented through a collection of objects (e.g., if 10 is the whole, then 6 is three-fifth), a 100-cell square grid may be considered as one whole. Likewise, in the base-ten integer arithmetic, while the place value of the first digit of a three-digit number is 100, its face value is the value of the digit itself. To answer the above question, note that in the dividend-divisor context for fractions (Sect. 4.1), the meaning of 1/4 is dividing one whole into four equal pieces. In base ten, one whole can be represented by a 10  10 grid, 1/4 of which is comprised of 25 cells (Fig. 4.22). Because each cell is 1/100 of the grid, we have 1/4 = 0. 25. The same result can be obtained if we use long division to convert a common fraction 1/4 (in which division of 1 by 4 is hidden) into a decimal fraction. The process is shown in Fig. 4.23. Now, let us represent the fraction 1/4 in base six. Regardless of base, the meaning of 1/4 is the same—dividing one whole in four equal pieces. But in base six the whole is a 6  6 grid, 1/4 of which consists of nine cells (Fig. 4.24). By analogy with the 10  10 grid, these nine cells represent 0.13 of the 6  6 grid. Therefore, 1/4 = (0.13)6. The same result can be found through carrying out long division of 1 by 4 in base six as shown in Fig. 4.25.

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4 Modeling Mathematics with Fractions

Fig. 4.22 One-fourth on a 10 by 10 grid

Fig. 4.23 Dividing 1 by 4 in base 10

Fig. 4.24 One-fourth on a 6 by 6 grid

Similarly, in order to represent the fraction 1/3 in base six, the first step is to show 1/3 of a 6  6 grid (Fig. 4.26) where one can already see that numeric representation would be simpler than that of 1/4 in base six. Indeed, noting that the fraction 1/3 (just as any common fraction) represents an uncompleted division (regardless of a base), the second step is to divide 3 into 1 having zero as an integer and then, in order to get the first number after the radix point, add zero to one. In

4.7 Rational Numbers in Non-decimal Bases

119

Fig. 4.25 Dividing 1 by 4 in base 6

Fig. 4.26 One-third of a 6  6-grid

Fig. 4.27 Dividing 1 by 3 in base 6

base six, the symbol 10 represents 6 units, two of which represent 1/3. The described process of division is shown in Fig. 4.27.

4.8

Integer Sequences as Sources of Fractions

The mathematics of iteration is another important topic that middle school teachers have to be familiar with (Conference Board of the Mathematical Sciences, 2012). This topic can be discussed in the context of fractions. With this in mind, as a way of connecting common fractions to their decimal equivalents and other topics related to middle school mathematics curriculum, this section will demonstrate how fractions can be utilized in approximating real numbers. A classic example is the approximation of the Golden Ratio by the ratios of two consecutive Fibonacci numbers. Indeed, given Fibonacci number sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …, the sequence of the ratios (fractions) 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, … pffiffi approaches (through an oscillating process) the number 1 þ2 5, known as the Golden

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4 Modeling Mathematics with Fractions

Ratio. The same is true for the ratios of any Fibonacci-like sequence; for example, given the sequence of Lucas7 numbers 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, …, the sequence of the ratios (fractions) 4/3, 7/4, 11/7, 18/11, 29/18, 47/29, 76/47, … pffiffi approaches the number 1 þ2 5 through an oscillating process as well. This interesting phenomenon, when a real number can be approached by an infinite number of sequences of rational numbers, demonstrates a generally unknown face of fractions as convergents to real numbers.

4.9

Continued Fractions

So far, three distinct representations of rational numbers were discussed: the common fraction representation, the decimal representation, and the Egyptian fraction representation. In this section, another representation of a rational number, called continued fraction, will be introduced. Just as the decimal representation works both for rational and irrational numbers, the two types of numbers have distinct continued fraction representations, finite and infinite, respectively. As any number can be represented as a sum of its integer and non-integer parts, this representation is essential for the development of continued fraction representation having been related to the Euclidean algorithm of finding the greatest common divisor of two integers. In what follows, this algorithm, recommended for the inclusion in mathematics teacher preparation programs for middle school teachers in the United States (Association of Mathematics Teacher Educators, 2017), will be discussed.

4.9.1

Euclidean Algorithm

Among topics recommended by the Association of Mathematics Teacher Educators (2017, p. 111) for sound programs at the middle school level is the Euclidean algorithm. This algorithm of finding the greatest common divisor of two integers is a recurrent procedure of successive divisions based on one of the most basic relations in arithmetic: dividend (D) is equal to the product of divisor (d) and quotient (Q) plus remainder (R), that is, D = dQ + R. Put another way, R = D – dQ. Let GCD(a, b) stand for the greatest common divisor of the integers a and b, a > b. Setting a and b as, respectively, the dividend and the divisor, one can find their remainder which becomes a new divisor with the first divisor becoming a new dividend. This process of finding remainders continues until zero remainder is

Édouard Lucas (1842–1891)—a French mathematician, known as the inventor of the Tower of Hanoi puzzle.

7

4.9 Continued Fractions

121

reached, thus making it impossible to be used as a divisor. According to Euclid, the last non-zero remainder in this process is the GCD(a, b). To justify the last statement, let a ¼ bq1 þ r1 :

ð4:8Þ

From relation (4.8) it follows that if a and b have a common divisor greater than one, then the latter should divide r1 as well. If this divisor is the GCD(a, b), it is also the GCD(a, b, r1 ). Indeed, if b and r1 have a common divisor greater than the GCD(a, b), then it should also divide a. This implication contradicts to the fact that no integer greater than the GCD(a, b) divides a and b. In that way, through the sequence of iterative relations a ¼ bq1 þ r1 b ¼ r 1 q2 þ r 2 r1 ¼ r2 q3 þ r3 . . .. . .. . .. . .. . .. . . rn2 ¼ rn1 qn þ rn rn1 ¼ rn qn þ 1 that the Euclidean algorithm of finding the GCD(a, b) entails, one can conclude that GCD(a, b) is the greatest common divisor for rn1 and rn which means that rn ¼ GCDða; bÞ. Otherwise, the relations rn1 ¼ x  GCDða; bÞ and rn ¼ y  GCDða; bÞ, where x and y are relatively prime numbers (i.e., GCD(x, y) = 1) would imply that x ¼ y  qn þ 1 and unless y = 1 the relation qn þ 1 ¼ xy contradicts to the fact that qn þ 1 is an integer. As an example, let us find the GCD(169, 140) using the Euclidean algorithm. Step 1. Divide 140 into 169 and find the remainder: 169 ¼ 140  1 þ 29. The first remainder is 29. Step 2. Divide 29 into 140 and find the remainder: 140 ¼ 29  4 þ 24. The second remainder is 24. Step 3. Divide 24 into 29 and find the remainder: 29 ¼ 24  1 þ 5. The third remainder is 5. Step 4. Divide 5 into 24 and find the remainder: 24 ¼ 5  4 þ 4. The fourth remainder is 4. Step 5. Divide 4 into 5 and find the remainder: 5 ¼ 4  1 þ 1. The fifth remainder is 1. Step 6. Divide 1 into 4 and find the remainder: 4 ¼ 1  4 þ 0. The fourth remainder is 0. The algorithm terminates as further division by zero is impossible. The last non-zero remainder was 1. Conclusion: GCD(169, 140) = 1. This means that 169 and 140 are relatively prime numbers.

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4 Modeling Mathematics with Fractions

4.9.2

Continued Fraction Representation of Common Fractions

Consider the fraction 169 140. As the numerator and the denominator of this fraction are relatively prime numbers, it is not reducible and, therefore, is represented in the 29 29 simplest form. It can be written as a mixed fraction, 1 140 , or as the sum 1 þ 140 . Using the corresponding relationships from the last section, one can continue modifying the last representation of 169 140 as follows. 169 29 1 1 1 ¼ 1þ ¼ 1 þ 140 ¼ 1 þ ¼ 1þ 24 140 140 4 þ 4 þ 29 29 ¼ 1þ

1 4þ

1 1 1 þ 24

¼ 1þ

1 4þ

1 1þ

1 4þ4 5

5

The representation

169 140

¼ 1þ

4þ

¼ 1þ

1 1þ

1 1 4þ 1 1þ1 4

1

¼ 1þ

29 24

1 4þ

1 1þ

1 4þ

¼ 1þ

1 4þ1 5 4

1 5 1 þ 24

1 4þ

1þ

:

1 1 4þ 1 1þ1 4

is called the continued fraction repre-

sentation of the fraction 169 140. Alternatively, the linear continued fraction representation is 169 ¼ ½ 1; 4; 1; 4; 1; 4 in which all the numbers are the ratios obtained 140 through the Euclidean algorithm of finding the GCD(169, 140). Note that the 140 reciprocal of the fraction 169 140, that is, 169 has the following linear continued fraction 140 representation 169 ¼ ½0; 1; 4; 1; 4; 1; 4 as the integer part of 140 169 is zero so that 140 1 169 ¼ 0 þ and the process continues with as shown above. Note that whereas 169 140 169 140

linear continued fraction representations of the reciprocals are similar, their decimal representations are quite different. For example, whereas 169 140 ¼ 1:20714265—a mixed recurring decimal with period six, its reciprocal 140 169 ¼ 0:8284. . .9940—a purely recurring decimal with period 78. One can use Wolfram Alpha (in the context of its free on-line version) to see all the digits that form a 78-digit period by simply typing the fraction in the program’s input box. It is interesting to compare Egyptian fraction and continued fraction representations of a proper fractions in terms of the number of steps it requires to develop each representation. As an example, consider the fraction 13 49. The use of the Euclidean algorithm in finding the GCD(49,13) yields the following successive divisions: 49 ¼ 13  3 þ 10;

13 ¼ 10  1 þ 3;

10 ¼ 3  3 þ 1;

3 ¼ 1  3 þ 0;

where the quotients 3, 1, 3, 3 are the entries of the fractional part of the linear form 13 of the continued fraction representation of 13 49. That is, 49 ¼ ½0; 3; 1; 3; 3. At the same time, the Greedy algorithm in writing 13 49 as an Egyptian fraction requires the 13 1 3 following steps. As 14 is the largest unit fraction smaller than 13 49 and 49  4 ¼ 196, we

4.9 Continued Fractions

123

1 3 1 3 have 13 49 ¼ 4 þ 196. Next, as 66 is the largest unit fraction smaller than 196 and 3 1 1 3 1 1 13 1 1 1 196  66 ¼ 6468, we have 196 ¼ 66 þ 6468. Therefore, 49 ¼ 4 þ 66 þ 6468. One can see 13 that developing continued fraction representation for 49 required more steps than 9 developing its Egyptian fraction representation. At the same time, for the fraction 19 9 9 1 1 1 1 we have 19 ¼ ½0; 2; 9 and 19 ¼ 3 þ 8 þ 66 þ 5016; that is, developing the contin9 ued fraction representation for the fraction 19 requires fewer steps than for its Egyptian fraction representation (which can be outsoursed to Wolfram Alpha).

4.10

Connecting Fractions to Quadratic Equations

The study of fractions can be connected to quadratic equations, another topic of middle school mathematics. To explain, let us consider the sequence 2, 5, 14, 41, 122, 365, 1094, 3281, …, the ratios of two consecutive terms of which become closer and closer to the number 3. One can check to see that 5/2 = 2.5, 14/5 = 2.8, 41/14 = 2.93 …, 122/41 = 2.92 …. One may wonder where such sequences may be coming from. To this end, one can consider a quadratic equation one root of which is 3, for example, x2  4x þ 3 ¼ 0. The modification of this equation to the form x ¼ 4  3x leads to the iterative sequence xn þ 1 ¼ 4  x3n . Setting x1 ¼ 2 yields the sequence of fractions 2/1, 5/2, 14/5, 41/14, 365/122, 1094/365, 3281/1094, … already considered above as comprised of the (monotonically increasing) convergents to the number 3, as shown in Figs. 4.28 and 4.29 (the screen shots from Wolfram Alpha used in the context of its free on-line version). A new iterative sequence of convergents to the number 3 can be developed by considering a different quadratic equation with a root 3, for example, 2x2  7x þ 3 ¼ 0. Modifying the last equation to the form x ¼ 72  2x3 leads to the iterative sequence xn þ 1 ¼ 72  2x3n from which, starting from x1 ¼ 1, the following sequence of convergents results: 1, 2, 11/4, 65/22, 389/130, 2333/778, 13,997/ 4666, …. This sequence represents another (monotonically increasing) sequence of fractions that approximates the number 3. Figures 4.30 and 4.31 are the screen shots from computations carried out by Wolfram Alpha. Comparing the behavior of the convergents to the number 3 with that of to the Golden Ratio may become a motivating factor in exploring the behavior of fractions in the context of approximating real numbers. Not every quadratic equation with a root 3 would be a source for such iterations. One can check to see that the number 1/2, the second root of the quadratic equation 2x2  7x þ 3 ¼ 0, cannot be approximated by using the iterative sequence xn þ 1 ¼ 72  2x3n , no matter which initial value is chosen to begin iterations. So, only if the number 3 is the largest in absolute value root of a quadratic equation, the equation can be used to demonstrate how a sequence of fractions forms convergents to this root.

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4 Modeling Mathematics with Fractions

Fig. 4.28 Convergents to the number 3 in the fractional (exact) form

Fig. 4.29 Convergents to the number 3 in the decimal (approximate) form

Fig. 4.30 Another set of convergents to the number 3 in the fractional (exact) form

4.10

Connecting Fractions to Quadratic Equations

125

Fig. 4.31 Another set of convergents to the number 3 in the decimal (approximate) form

Fig. 4.32 Iterating convergents within a spreadsheet

To conclude this chapter, note that Wolfram Alpha has more computational power in comparison with a spreadsheet (which also is well suited to generate iterations) as the former tool is capable of computing convergents both in the fractional and the decimal forms. The fractional form makes it possible to study integer sequences that comprise the convergents; the decimal form makes it possible to recognize the behavior of convergents. At the same time, the use of a spreadsheet in generating convergents in the decimal form requires almost no specialized knowledge of operating software. Such a spreadsheet, shown in Fig. 4.32, generates convergents to the number 3 through the iterative formula

126

4 Modeling Mathematics with Fractions

xn þ 1 ¼ 72  2x3n with x1 ¼ 1. One can say that when functioning without reliance on the user’s specialized knowledge, software displays its positive affordance. At the same time, in certain contexts, notational limitations of a digital tool can be seen through the lens of its negative affordance.

4.11

Conclusion

This chapter introduced two major contexts for fractions emerging from situations that cannot not be resolved using integer arithmetic. Consequently, the main focus of the chapter was on the use of two-dimensional models for demonstrating conceptual meaning of rules that the arithmetic of fractions follows. The discussion of unit fractions was motivated by geometry (to be considered in Chap. 7) and their use by ancient Egyptians was revealed and applied in the modern context of fair sharing as a creative alternative to the dividend-divisor context. Long division was used to demonstrate the genesis of decimal fractions. Finally, several advanced concepts associated with fractions such as fractions in non-decimal bases, continued fractions and fractional iterations as convergents to integer roots of quadratic equations were considered. For readers who want to deepen their knowledge and understanding of fractions the following sources can be recommended (Schifter, 1998; Thompson and Saldanha, 2003; Usiskin, 2007; Steffe and Olive, 2010; DeWolf et al., 2015; Bobos and Sierpinska, 2017).

4.12 1. 2. 3. 4. 5. 6. 7. 8. 9.

Activity Set

3 Place the fraction 17 between two benchmark (unit) fractions. 33 Place the fraction 79 between two benchmark (unit) fractions. 3 Convert 17 into an Egyptian fraction. 33 Convert 79 into an Egyptian fraction. Find the product 45  37 using two-dimensional area model for fractions. Represent the product as an Egyptian fraction. Find the product 43  59 using two-dimensional area model for fractions. Represent the product as an Egyptian fraction. Find the product 73  58 using two-dimensional area model for fractions. Divide 5 into 3 using two-dimensional area model for fractions. A college has to pave two parking lots, A and B, shaped as squares of sides 85 25 m and 98 25 m, respectively. It is known that 4/5 of lot A and 7/8 of lot B need

4.12

10.

11.

12.

13.

14.

15.

16.

17. 18. 19. 20. 21. 22. 23. 24.

25.

Activity Set

127

new pavement. Which parking lot needs more pavement? Use Wolfram Alpha for calculations. Divide three identical pizzas among five people fairly using dividend-divisor context and Egyptian fraction Greedy algorithm. Compare the number of pizza pieces obtained in each case. Divide five identical pizzas among six people fairly using dividend-divisor context and Egyptian fraction Greedy algorithm. Compare the number of pizza pieces obtained in each case. Divide four identical pizzas among seven people fairly using dividend-divisor context and Egyptian fraction Greedy algorithm. Compare the number of pizza pieces obtained in each case. Divide two identical pizzas among seven people fairly using dividend-divisor context and Egyptian fraction Greedy algorithm. Compare the number of pizza pieces obtained in each case. Divide three identical pizzas among eight people fairly using dividend-divisor context and Egyptian fraction Greedy algorithm. Compare the number of pizza pieces obtained in each case. Formulate two problems about dividing pizzas similar to problems 10–14: one problem when the dividend-divisor context yields the larger number of pieces that Egyptian fraction Greedy algorithm and another problem when the dividend-divisor context yields the same number of pieces that Egyptian fraction Greedy algorithm. A pizza is cut into three different pairs of equal pieces: (1/4, 1/4), (1/6, 1/6) and (1/ 12, 1/12). These pieces can be used separately to make the full pizza by repeating the first pair twice, the second pair three times, and the third pair six times. Find other ways to similarly cut pizza into three pairs of equal pieces out of which the whole pizza can be made. How can one cut pizza into four such pairs? Carry out division 45
37 using two-dimensional area model for fractions. Explain the division 25
47 through the Invert and Multiply rule as a change of unit. Convert the fraction 37 into base 9. Convert the fraction 56 into base 8. Convert the fraction 25 into base 7. Find the continued fraction and the linear continued fraction representations for the common fraction 133 120. Find the continued fraction and the linear continued fraction representations for the common fraction 247 115. Construct a quadratic equation with a root x = 2. Modify your equation to form an iterative sequence that generates convergents to this root. Write down the first six convergents. Construct a quadratic equation with a root x = 5. Modify your equation to form an iterative sequence that generates convergents to this root. Write down the first six convergents.

Chapter 5

Decimal and Percent Representation of Rational Numbers

5.1

Introduction

cBig Idea Decimals and percents can be viewed as natural extensions of the standard base-ten system. When this big idea is considered, students can use numerical fluency concepts to enhance their understanding of both decimal and percent.b

Many students enter middle school viewing decimal and percent as independent and unrelated concepts rather than as being natural extensions of the standard base-ten system in use today. This commonly held perception generally reflects a lack of basic numeracy on the part of the students concerning the foundations of number and number representations. This is a problem which must be directly addressed in initial instruction if advancement is to be made in developing decimal and percent. These are two new types of number from the perspective of such students, but are really natural extensions of ideas dating back elementary school activities. It is no accident that the base-ten notational system in common use is often referred to as the decimal notational system (Decimal, 2020). Therefore, in order to gain a full understanding of decimal and percent, middle school students must be able to see how percent and decimal is represented within the base-ten, or decimal, notational system. In middle school instruction, therefore, it is important to help students develop a solid foundation for representing numbers within the standard base-ten number system. The establishment of such a foundation can greatly aid students as they begin their decimal and percent explorations. Bearing this in mind, the chapter will begin by reviewing some key features of foundational numeracy in order to develop these facilities. © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_5

129

130

5 Decimal and Percent Representation of Rational Numbers

A particular focus will be placed upon teaching methods and examples enabling students to become numerically fluent in terms of the symbols, representations, and interpretations of number. In developing such fluency for decimal and percent understandings a similar trajectory will be followed as that used in developing basic whole number numeracy and number awareness. First, the organizing principles of the base-ten, or decimal notational system, will be reviewed. In particular, a basic organizing principle being that each increasing place value corresponds to an increasing power of ten and is composed of ten of the preceding place value units. Likewise, each 1 decreasing place value corresponds to a decreasing power of ten and is 10 the size as that represented by the larger place value. After these foundations of number representation and notation are reviewed, decimal and percent will be placed in their relative positions within the larger decimal notational system. From within this developed perspective, connections will then be made between fractions and decimals together with explorations into repeating decimals and their fractional equivalents, (Chap. 4, Sect. 4.6). Finally, considerations concerning the extensions of basic arithmetic operations into decimal and percent will be presented.

5.2

Developing Numerical Fluency

cBig Idea The big idea of numerical fluency builds strong decimal and percent understandings. When number itself is understood, the various representations used to represent that number are more easily shown to be equivalent. Likewise, understandings drawn from one representation more easily transfer to another.b When reviewing number representations, it is often helpful to remind students that numbers will continue to exist regardless of the system which is used to represent them. For example, the numbers we represent in base-ten using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 could also be represented using bear-counters as shown in Fig. 5.1. Although difficult to see in this figure, 0 could be initially represented by there being no bear-counters present. Taking this absence of bear-counters as a starting point, one single bear-counter at a time is placed from left to right until three bear-counters are reached. At that point, a new row is started with the bear counters once again being placed in increasing order from left to right. This process is then repeated three times—thus building a physical and visual representation for the students of the digits from 0 through 9. Since nearly any object could have been used in place of the bear-counters, this approach leads to the development of what could be referred to as a counter-based representational system.

5.2 Developing Numerical Fluency

131

Fig. 5.1 A counter-based representation of the digits 1 through 9 using bear counters

Although non-standard, such a counter-based representational system makes a great deal of sense when viewed from a psychological perspective (Chap. 10, Sect. 10.3). The choice of three counters per row is based in part upon subitizing (Kaufman et al., 1949), the cognitive ability to near instantly “see how many”. When there are fewer than four items, as will be utilized in this chapter, subitizing can be characterized as being rapid (40–200) ms/item, effortless in terms of cognitive processing, and highly accurate (Trick & Pylyshyn, 1994). When combined with chunking, the mental combination of basic familiar items that can be grouped together and stored in a person’s memory as a single unit (Tulving & Craik, 2000), each of these rows of bear-counters comes to constitute a “chunk” which can be stored and retrieved by the students as a single cognitive unit. When combined with subitizing these chunks enable students to create rapid and accurate visual recognition for the digits 0–9. Furthermore, the placement of counters in a left to right and top to bottom fashion was selected so as to parallel the directionality commonly used when reading and writing, thus enabling this representation to be reinforced by muscle and visual tracking memory. Since both subitizing and chunking are innate cognitive processes, they will be present in all students, barring developmental

132

5 Decimal and Percent Representation of Rational Numbers

disability, making this unusual representational system powerful for both initial instruction and to serve as a review of whole number concepts. Being based upon these two cognitive primitives (Rappaport, 1988), subitizing and chunking can be considered as “freebies” in terms of instructional time since they do not require additional explicit instruction. More importantly from an instructional perspective, they provide a basis for the development of later task-independent formalisms and abstractions. Comparing this counter-based representational system, based upon cognitive primitives, with the corresponding Roman numerals, shown in Fig. 5.2, which are based upon cultural dependent understandings, the counter-based system begins to seem more universally applicable. When using Roman numbers, for example, various cultural understandings come to play an important role in both use and interpretation. For example, there is not a symbol in Roman numerals used to represent zero (The Absence Of The Concept Of Zero in Roman Numerals System, 2020). On the few occasions when the Romans had need to represent zero, the Latin word for “none”, nulla, was used. To focus the next discussion, only whole numbers n, such that 0
n
9 will be considered at this point. Whichever representational system might be used, Roman numbers or counter-based, once the system is understood students should be able to meaningfully represent a number using this system. As such, they would be able to answer the question, “What does it look like?” with confidence. Now, imagine that a task is given to a student to represent the number “4” using a counter-based representational system and then to label it with the appropriate digit. This ability, to meaningfully represent and then accurately label a number in this fashion, is an important first step in developing numerical fluency as it ties together a concrete and more abstract representation. Furthermore, since a counter-based system was used in this task, the student’s understandings will be based upon a pair of cognitive primitives, subitizing and chunking, which lend additional meaning beyond the mere representational one to this result. The digit used to label this result will now likewise share in this deep level meaning and come to be less of an abstraction for the student. If the only thing that is done, however, is to represent a single number, this task does not develop much other than facility with a non-standard representational system. It could be considered at the same level as other such tasks such as identifying which Roman numeral corresponds to the Hindu-Arabic numeral of the same value. Although a worthy goal in terms of recognizing Roman numerals, in and of itself a task such as this does not necessarily build numerical fluency.

Fig. 5.2 Roman number representation of the digits 1 through 9

5.2 Developing Numerical Fluency

133

To see what might be required to advance student numerical fluency, imagine that this initial task is then followed by that of representing the number “3” using the same counter-based system and then labeling it with the appropriate digit. The results of these paired tasks are shown in Fig. 5.3 which shows the two counter-based number representations both labeled with their respective standardized numerals within the Hindu-Arabic system. For the participating student, the numerals are now linked with their counter-based representations, which in turn are based upon paired cognitive primitives. Helping the students to link the associated oral vocabulary terms such as “four”, “vier”, “cuatro”, “chetyre”, etc., and “three”, “drei”, “tres”, etc., to these representations will further develop the students’ numerical fluency even further. When this is done, numerical fluency has been achieved as the student would now be able to interchangeably use an object-based representational system (the counters), a symbolic representational system (the Hindu-Arabic numerals), and a linguistic representational system (the words “four”, “vier”, “cuatro”, “chetyre”, etc., and “three”, “drei”, “tres”, etc.) for number as shown in Fig. 5.4. This discussion had begun by only considering whole numbers, n, such that 0
n
9. This restriction will now be lifted such that the numbers under consideration will be expanded to include, n, where n is any whole number. Initially, this might make the question, “What does it look like?” seem to be an insurmountable cognitive obstacle for students to consider when a counter-based representational system is to be used given the limitations of subitizing and chunking described thus far. What, for example, would 134 counters look like? However, this difficulty is easily addressed by using a different counter. In particular, a 1-cm cube, which also serves as a foundational object representing the number one in the Diene’s base-ten blocks. Once this cube is used as the counter it becomes possible to create larger “chunks”, enabling the representation of any positive whole number such that students never have to visualize more than 9 counters per place value. Starting with a single 1-cm cube and using the same process as that shown in Fig. 5.1, it is possible to build numerical fluency using this cube for n 2 f0; 1; 2; 3; 4; 5; 6; 7; 8; 9g as shown in Fig. 5.5.

4

3

Fig. 5.3 Representing and labeling 4 and 3 using a counter-based system

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5 Decimal and Percent Representation of Rational Numbers

Fig. 5.4 Numerical fluency

Just as was done in the bear-counter example, shown in Fig. 5.1, this process will serve to link these numbers with their counter-based representations, which in turn will be based upon paired cognitive primitives. Unlike the bear-counters, however, the shape of these 1-cm cubes lends themselves to the creation of more powerful “chunks”. Taking a clue from the bear-counter discussion, the inborn student abilities of less than four subitizing and initial chunking will develop student fluency for n 2 f0; 1; 2; 3; 4; 5; 6; 7; 8; 9g. Trying to go further than this, perhaps by the creation of a fourth row or extension of the rows beyond three counters, although certainly possible, taxes the less than four version of subitizing and chunking. Furthermore, it can lead to a state of mental imbalance that Piaget referred to as disequilibration. In this condition, the students’ existing representational base no longer functions. Rather than being a block for further student cognitive development, however, this mental state provides an intrinsic motivation for mental restructuring to accommodate this new situation into the students’ understanding (Piaget, 1961). For the 1-cm cube situation this accommodation is greatly facilitated by the shape of the cubes themselves. Imagine that a tenth cube is placed into the representations of Fig. 5.6. Since the cubes can be easily placed end-to-end, however, a new type of cognitive chunk made up of ten cubes is easily obtained as shown in Fig. 5.6. This newly constructed chunk can be given the name “Ten” to reflect the number of cubes which it contains. This newly created unit referred to here as a “Ten” is another foundational unit in the Diene’s base-ten blocks. Furthermore, using a “Ten” as the new counter it is now possible to continue building numerical fluency using the “Ten” for n 2 f10; 20; 30; 40; 50; 60; 70; 80; 90g as the new counter as shown in Fig. 5.7. An additional rearrangement of “Tens”, see Fig. 5.8, quickly leads to both accommodation and the creation of a new cognitive chunk. As was done earlier, this newly constructed chunk can be given a name, “Hundred”, which reflects the total number of cubes which it contains. This “Hundred”, is yet another foundational unit in the Diene’s base-ten blocks and, as might be expected, can be used as the new counter to build numerical fluency for n 2 f100; 200; 300; 400; 500; 600; 700; 800; 900g as shown in Fig. 5.9.

5.2 Developing Numerical Fluency

135

Fig. 5.5 A counter-based representation of the digits 0 through 9 using 1 cm cubes

Fig. 5.6 Building a “Ten”

Similar to the previous discussions, student attempts to add an additional hundred would once more lead to mental disequilibration when relying solely upon the less than four version of subitizing and chunking. As before, however, a rearrangement of “Hundreds”, see Fig. 5.10, results in the creation of a new cognitive chunk which can be named “Thousand” for the total number of cubes which it contains.

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5 Decimal and Percent Representation of Rational Numbers

Fig. 5.7 A counter-based representation of the digits 10 through 90 using “Ten”s

Fig. 5.8 Building a “Hundred”

This newly created chunk, “Thousand”, is yet another foundational unit in the Diene’s base-ten blocks. This is significant from a student perspective since based upon the protocols used thus far can be used as the new counter which can be used expand numerical fluency for n 2 f1000; 2000; 3000; 4000; 5000; 6000; 7000; 8000; 9000g. More importantly, however, the shape of this new cognitive chunk is a cube, the same shape as the initial unit which began this discussion. So by following the protocol established it is possible for middle school students to easily imagine ten of these “Thousand”s being arranged to form a “Ten Thousand”, ten of these “Ten Thousands” arranged to form a “Hundred Thousand”, and ten of the “Hundred Thousands” being arranged into a new cube-shaped, chunkable, object. Once this newly formed object is given a name, “Million”, the process can be created once more to establish “Ten Million”, “Hundred Million” and the creation of the next cube-shaped, chunkable, object to be named on accordance with standard usage. This process can go on infinitely, giving the students a firm foundation not only for

5.2 Developing Numerical Fluency

137

Fig. 5.9 A counter-based representation of the digits 100 through 900 using “Hundred”s

Fig. 5.10 Building a “Thousand”

ones, tens and hundreds, each which can be made starting with cubes whose size is determined by how many predecessor cubes they possess, but also for the infinite nature of number. At this point there are several crucial observations which should be made from a psychological perspective. Perhaps one of the more important being that at no point in this process was it ever necessary for the middle school students to consider more than nine objects within any given place value assemblage. This positions each

138

5 Decimal and Percent Representation of Rational Numbers

grouping firmly within less than four subitizing and chunking’s efficacy. Furthermore, any newly created objects can now be mentally constructed by middle school students based upon well understood actions performed upon easily recognized shapes. The initial disequilibrium experience by the students have now resulted in a new accommodation and set of understandings.

5.3

Transitioning from Numerical Fluency to the Base-Ten Notational System

From a mathematical perspective, it is clear at this point that each of the successive objects in this process are made of 10 of their predecessors. At each stage once the 10th counter is reached a new object is created following a distinct pattern starting with a cube and proceeding to a linear arrangement of ten of these cubes, from there to a 10  10 square arrangement of 100 of these cubes, and finally to a cubic arrangement of 10  10  10 these cubes. It is also easy to see that a “One” will always have the same shape whether it be a single 1-cm cube, one of the “Thousand” cubes, one of the “Million” cubes or one of any infinite numbers of cubes which could be potentially constructed in this fashion. Likewise, a “Ten” of any size will have the same shape as will a ‘Hundred” of any size. Possessing these understandings, middle school students are well situated for a more advanced understanding of the base-ten notational system. As was shown in Sect. 3.2, Fig. 3.3, as these place values for successively greater whole numbers are represented, they are arranged into groups of three which are separated using a comma. Each of these groups are then referred by through their unique name such as ones, thousands, millions, billions, etc. Within each of these groups there is a “ones”, “tens”, and “hundreds” position which corresponds to the cube, linear arrangement and square arrangement described in the preceding paragraph. A possible rationale for this arrangement was demonstrated in Sect. 5.2 using the 1-cm cube counter model. It would now be possible to discuss with students how to go about representing a number with three ten-thousands, four hundreds, and 6 single cubes, or 3  104 þ 4  102 þ 6. In Fig. 5.11 this number is represented using a spreadsheet currently used in one of the author’s elementary and middle school methods classes [MLC]. This spreadsheet draws heavily upon the ideas of numerical fluency generated in this chapter and requires students to consider what the number being represented would look like, what is the standard vocabulary used when naming the number, and what it would look like when represented in the base-ten notational system. It is this level of numerical fluency that supported the student reasoning described in Chap. 3, Sect. 3.2.

5.3 Transitioning from Numerical Fluency to the Base-Ten …

139

Fig. 5.11 Representing three ten-thousands, four hundreds, and six single cubes using base ten

In more formal terms, since each newly constructed object is 10 time the number of its’ predecessor the base will be 10. The first group, the “Ten”, is made of 10  1 or 101 cubes, the second group, the “Hundred”, is made of 10  10 or 102 cubes, the third group, the “Thousand” is made of 10  10  10 or 103 cubes, and in general the nth group is made of 10  10  . . .10 or 10n . Since 10 is both the base |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} ntens

in the base-ten notational system and also the number raised to various powers it is now possible to generate the place values for each digit for whole number values of n within the notational system. As illustrated in Fig. 5.11, if one were to select any of the place values shown at random the larger adjacent successor place value will be ten times larger, while the 1 smaller adjacent predecessor place value will be 10 as large. So, for the general place n nþ1 value 10 the next larger place value will be 10 and the prior smaller will be 10n1 . At this point, the case of 1 deserves special consideration as it would be expressed in terms of a power of ten as 100 . Since a0 ¼ 1 for a 6¼ 0 by definition, 100 ¼ 1. With these understandings in place the following connections between base, exponent and place value can be established. For 10n , where n is a positive whole, number the base, 10, can be thought of as the number of preceding objects that are chunked together to make the next larger place value. In other words, 10 of the ones being chunked to make a Ten, 10 of these Tens chunked to make a Hundred, etc. The exponent, n, indicates how many times this chunking as taken place. Thus, 100 is easily imagined as the 1-cm cube prior to any chunking, 101 indicates that chunking has occurred once, 102 that chunking has occurred twice, and so on.1 The It is not uncommon for middle school students at this point to notice that the “shape” of the Hundred is a square and that 102 is often called “ten-squared”. Likewise, that the “shape” of the Thousand is a cube and that 103 is often called “ten-cubed”. This recognition serves the student well when encountering terms such as x2 or “x-squared”, and x3 or “x-cubed”. 1

140

5 Decimal and Percent Representation of Rational Numbers

result of this exponentiation provides the place value being generated, so 104 generates the place value 10  10  10  10 ¼ 10; 000. Moving forward, knowing that for a general place value of 10n requires that the 1 as large as 10n . So, 101 prior smaller place value will be 10n1 —a quantity that is 10 1 2 0 1 1 1 1 is 10 the size of 10 , 10 is 10 the size of 10 , and 10 is 10 the size of 100 . Since 100 ¼ 1, this last example sets up a new set of possibilities for middle school students by extending potential values of n to include both positive and negative integers. 1 to the base-ten repreThe immediate impact is the introduction of 101 or 10 sentational system. Once this is done and understood by the students, the representational system has moved beyond representation of whole numbers and into the realm of decimal fractions. The familiar decimal point, commonly represented by a small dot, is used to separate the whole number part from the fractional part of a number. Extending this process, it can be established that for whole number values of n, 10n ¼ 101n . This understanding now extends the base-ten notational system to include the representation of all fractional parts to the right of the decimal point. A compelling visual could be made at this point by imagining what would happen if the 1-cm cube from the Diene’s base-ten set were to be sliced into 10 equal parts. This can be easily visualized by imagining the Thousands cube as being 1 1-cm cube, but viewed under a magnifying glass. The next smaller unit would be 10 the size of this cube and shaped like the Hundreds. This would strongly suggest that 1 , have a similar shape. Unfortunately for the development of this poten0:1, or 10 tially powerful idea, the preponderance of decimal models currently in use do not start with a cube being used to represent one (Cramer et al., 2009). Rather, a 10  10 grid which strongly resembles the Hundreds in the Diene’s base-ten block is used. When these decimal models are introduced, it often causes confusion for many middle school students who do not realize the need to rethink how the “one” is being represented. Due to this, care should be taken when introducing decimal models based upon such a 10  10 grid to students to ensure that they fully recognize this shift in representational objects and the impact this substitution has. With the base-ten notational system now expanded to include student representation of fractional parts to the right of the decimal point it is now possible for a number, such as 235.14 to be meaningfully interpreted by middle school students. In this case it could be written in an expanded notational representation as 2  102 þ 3  1 1 þ 4  100 . 101 þ 5 þ 1  101 þ 4  102 or 2  100 þ 3  10 þ 5 þ 1  10 It should be note that the similarity of this expanded notational form directly parallels that of a polynomial. For example, consider the number 8,742. This number can be written as 8  103 þ 7  102 þ 4  101 þ 2. Compare this notation to that used for the polynomial 8x3 þ 7x2 þ 4x þ 2, which when x ¼ 10 results in the two expressions being equal. As will be shown, this similarity is part of what leads to later operations, such as addition and subtraction, being made simpler through the enabling of easy collection of like terms.

5.4 Decimals and Decimal Fractions

5.4

141

Decimals and Decimal Fractions

It should be easy at this point for students to recognize the connection between decimal and fractional representations. When considering a number such as 0.14, 1 1 þ 4  100 , see for example, students would now be able to represent it as 1  10 Sect. 5.3. As this example shows, a decimal can be thought of as a fraction that has a power of ten as the denominator. To make this connection more obvious to middle school students, rewriting the decimal in terms of an expanded fractional representation can be helpful. Consider writing 0.354 in expanded fractional form by rewriting the decimal as 3  101 þ 5  102 þ 4  103 . Since 10n ¼ 101n , an 3 5 4 300 50 4 354 þ 100 þ 1000 ¼ 1000 þ 1000 þ 1000 ¼ 1000 expanded form could be written as 10 clearly illustrating the power of ten present in each of the denominators. There is, of course, a much easier process which could be followed. In converting 0.354, for example, the first step would be to note the number of digits to the right of the decimal point, in this case, three digits. This gives the power of ten making up the denominator of the desired fraction. Since in this case there are 3, the denominator will be 103 ¼ 1000. The numerator of the desired fraction is the numbers contained 354 . This same in the decimal itself, so the decimal fraction equivalent will be 1000 process can be applied should there be a whole number component. For example, to convert 3.4372 the decimal fraction can be determined by noting the number of digits to the right of the decimal, 4. This give us the power of ten making up the denominator, 4, with the digits 4372 comprising the numerator. The expanded 4372 4372 34372 fractional representation would then be 3 þ 4372 104 ¼ 3 þ 10000 ¼ 3 10000, or 10000. It is important to note that when converting a decimal to a decimal fraction, reducing to lowest terms should only be done if the result is also fraction which has a power of ten as the denominator. If this cannot be done the resulting fraction, although equivalent, does not meet the definition of a decimal fraction. Imagine, for example, the conversion of the decimal 0.750 to a decimal fraction. If a student failed to recognize that 0:750 ¼ 0:75 and merely followed the procedure outlined in the preceding paragraph, they would identify the denominator as being 103 ¼ 1000 750 and the numerator as being 750. The student’s conversion would then be 1000 . Should this student now recognize that both 750 and 1000 have a common factor of 10 it would be acceptable to reduce the resulting fraction as follows: 750 75 ×10 75 × 10 75 = = = . Since the denominator is a power of ten this reduced 1000 100 ×10 100 × 10 100

form is also a decimal fraction and the reduction is appropriate. However, if the student were to now notice that 75 and 100 have a common factor of 25 and proceed with a further reducing, the results would be

75 3 × 25 3 × 25 3 = = = . In 100 4 × 25 4 × 25 4

this case since the denominator is no longer a power of ten, this reduced form is no longer a decimal fraction, and the reduction should not have been performed by the student.

142

5.5

5 Decimal and Percent Representation of Rational Numbers

Decimals and Common Fractions2

Common fractions differ from decimal fractions in that the denominator in a common fraction does not need to equal a power of ten. In converting a decimal to a common fraction, the process described in Sect. 5.4 can serve as an initial guide. For example, to convert 0.3125 to a common fraction the initial numerator would be 3125 and the denominator would be 104 ¼ 10000 resulting in the decimal 3125 fraction 10000 . Since a common fraction does not require the denominator to be a power of 10 this fraction may now be reduced to lowest terms as 3125 5 × 625 5 × 625 5 = = = . 10000 16 × 625 16 × 625 16

This connection between decimals and common fractions was further discussed in Chap. 4, Sect. 4.6 where long division was used as an entry point to the decimal representation of common fractions. In particular, when considering a common fraction such as 58 there is an equivalent decimal representation which can be determined by completing the division hidden within the common fraction. So, for this case 58 can be converted into its decimal equivalent by dividing 8 into 5 using long division as shown in Fig. 5.12. Once a pair of common fractions are converted to their decimal equivalents it becomes much easier for comparisons to be made, see also Sect. 5.9. For example, consider the task of comparing 25 and 38. Using the notion of hidden divisions, 25 can be converted into its’ decimal equivalent by dividing 5 into 2 using long division 2 = 5 2 = 0.4, and 38 can likewise be converted into its decimal equivalent 5 3 by dividing 8 into 3 resulting in = 8 3 = 0.375. The comparison between 0.4 and 8

resulting in

0.375 is now easily determined by looking at the digit in the largest place value, 101 , and literally seeing that 4 [ 3 in a directly analogous fashion as that shown in Fig. 5.3.

5.6

Decimal and Percent

When introducing or reviewing percent with middle school students it is helpful to remind them that “percent” may be literally translated as “per 100”. This will be second nature for those students who grew up with the metric system as this notion 1 shows up throughout common measurement where a centimeter is 100 of a meter, a 1 centigram is 100 of a gram, etc. For middle school students in the United States, a

2

In Chap. 4, Sect. 4.7.2 a more formal discussion concerning conversion of common fractions into bases other than base-ten is provided.

5.6 Decimal and Percent

143

Fig. 5.12 Converting 58 into its decimal representation

1 comparable comparison can be made by reminding them that 1 cent is 100 of a dollar. Remembering this, it is a simple step to establish that 23 cm is 23% of a meter, 35 centigrams is 35% of a gram, etc. Likewise, for U.S. students $0.55 would be 55% of a dollar. Understanding this notion of “per 100” will also serve to make conversions between decimal and percent fairly straightforward. If the number one is taken as 1 the unit to be divided into 100, 1% would be 100 which is represented in the base-ten 37 notational system as 0.01. So, 37% would be 37% ¼ 100 ¼ 0:37. In general, to convert from a percent to a decimal divide the percent by 100 and then remove the “%” symbol. For example, 55.0% would be 55:0% ¼ 55:0 100 ¼ 0:55. Remembering that within the base-ten notational system for the general place value 10n dividing by 100 results in a place value that is 10n−2. So, to divide a two-digit percent by 100, remember that the decimal point will immediately follow the second digit comprising the percent, then move the decimal two places to the left. Assuming that numerical fluency within base-ten has been achieved, this leads to an alternate conversion process making this type of conversion even easier for middle school students to perform. For example, taking another look at the 55.0% case, applying this method results in 55:0% ! 0:55. A slightly more involved problem middle school students might experience involves the conversion of a percent that includes a common fraction component. For example, consider the conversion of 12 35 % to a decimal. In addressing this problem, it would first be necessary to convert the common fraction component to a decimal by completing the hidden division resulting in 5 3 = .6. When this is added to the original 12% the percent to be converted becomes 12.6%. At this point, the conversion is straightforward resulting in 12:6% ! 0:126. For conversions from decimal to percent, this process would be reversed. So, to convert from a decimal to a percent the students would multiple the decimal value by 100 and then add the “%” symbol. So, 0.75 would be converted as 0:75  100 ¼ 75 to which the “%” symbol is added resulting in 75%. It would also be possible, assuming numerical fluency, to note that moving the decimal two places to the right is comparable to multiplying by 100, thus 0:75 ! 75%. Decimals with greater than three digits do not pose a particular problem as they would be handled in the same fashion, thus 0.125 could be converted to a percent by multiplying by 100 and adding

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5 Decimal and Percent Representation of Rational Numbers

5 the “%” symbol resulting in 0:125  100 ¼ 12:5%. Recognizing that 0:5 ¼ 10 , 5 1 this percentage could also be expressed as 12 10 % which can be reduced to 12 2 %. Percent appears in many more situations than those encountered in the conversion examples shown thus far such as interest (Chap. 11, Sect. 11.1.4), proportional reasoning (Chap. 6, Sect. 6.1), and part-whole ratios (Chap. 6, Sect. 6.2.2) to name just a few. In each, however, a key element in understanding and developing each case is the “per 100” interpretation. When taken together with the ability to convert among common fractions and decimals, this understanding provides a foundation for solving many such problems. For example, suppose it is known that the baseball team won 9 of their 15 preseason games. A very reasonable question to ask would be, “What was the team’s winning percentage?”. The first step in addressing this question would be to set up a common fraction showing the 9 wins to the total games played, 15 . Remembering the close link between decimal and percent, this common fraction would first be converted to a decimal by completing the division hidden within the common fraction resulting in for the decimal representation. The decimal, 0.6, could then be easily converted to a percent using the methods developed thus far such as 0:6  100 ¼ 60 to which the “%” symbol would be added given the answer as 60%. A more involved problem middle school students may encounter could be similar to the following: In Mr. Diego’s mathematics class 15 students were on the honor roll. If this represents 60% of the students in his class how many students are there in Mr. Diego’s class? One way of addressing this problem is to recognize that 60% would equal 15 divided by the total number of students. Since 60% is now easily converted to the decimal 0.6, it can be shown that the number of students in 15 Mr. Diego’s mathematics class, x, is 0:6 ¼ 15 x ! 0:6x ¼ 15 ! x ¼ 0:6 ¼ 25 students in Mr. Diego’s class. It is likely, however, that the most common situations students will experience percent is in shopping. It is nearly impossible to go to a store of any type without seeing markdowns of various types applied to the items on sale. For example, a student might be interested in purchasing a blouse originally priced at $65 which has been marked down by 20%. In solving such problems there are several alternate approaches which might be taken. The most straightforward would be to determine the amount of the discount and then subtract this amount from the original total. Recognizing that 20% ¼ 0:2, this discount could be calculated by $65  0:2 ¼ $13. Subtracting this discount from the original total would give the discounted price as being $65  $13 ¼ $52. Yet another highly reasonable approach would be to recognize that 20% is 15. This would result in the discount being 65 5 ¼ 13. Although straightforward, savvy shoppers will often develop a more efficient method. Since percent literally means “per 100”, if an item is discounted 20% this leaves the amount after discount as being 100%  20% ¼ 80% of the original total. Recognizing that 80% ¼ 0:8, this simplifies the calculation of final cost to be $65  0:8 ¼ $52, which would easily be computed using a calculator. So, for any given discount, the first step would be that of converting the percent being

5.6 Decimal and Percent

145

discounted to a decimal. Call this decimal d. The amount after discount would then be ð1  d ÞOriginalPrice. A $115 calculator offered at a 15% discount, using this approach, would be ð1  0:15Þ  $115 ¼ 0:85  $115 ¼ $97:75.

5.7

Repeating Decimals

Although a thorough treatment of repeating decimals is beyond the scope of this book, there are some areas that can be productively explored by middle school students. In the process of converting common fractions to decimals, for example, students cannot help but notice that some common fractions such as 12 lend themselves to relatively straightforward conversions, in this case 2 1 = 0.5, while others such as 23 are not nearly so well behaved and give rise to the infinitely repeating cycle shown in Fig. 5.13. For many students, conversions of this type provide their first experience with an infinite process which can be quite disconcerting when experienced for the first time (Chap. 4, Sect. 4.6, Fig. 4.21). As might be expected, such glimpses into the infinite can provide middle school students with opportunities to develop intuitions that will pay off in later mathematics. For example, a problem closely related to the division by 9 example presented in Chap. 3, Sect. 3.2, is the problem of interpreting how each potential remainder is represented on the student’s calculator. Students will often notice that when 196 divided by 9, using long division as shown in Fig. 5.14, it results in a remainder 7. Yet, this is not the way this remainder shows up when solved using many basic calculators. Using a Texas Instruments TI-83 Plus, for example, the problem appears as 196=9 ! 21:77777778 which provides a result that is in clear need of explanation in order to make sense for students.3 The first step in interpreting this initially strange result is for students to recognize what this remainder from the long division version of the problem represents. This remainder 7, if divided by 9, would be expressed as the common fraction 79. Like the case of 23, shown in Fig. 5.14, this is not as well behaved as many middle school students might desire and carrying out the hidden division present in these fractions will result in an infinite process. This process is here shown on the calculator in an infinite cycle of 7s, which can be denoted as 0:7 using standard notation. This dilemma has real import for students not yet familiar with decimal operations and desiring to use their calculators. How can they tell which calculator result goes with which long division remainder?

3

It is important that students are able to meaningfully interpret how the numbers displayed in their calculator, or through other technologies, relate to the problems they are working on. Otherwise, these various representational methods will be unconnected in the students minds causing a disconnect in their application.

146 Fig. 5.13 Converting decimal

5 Decimal and Percent Representation of Rational Numbers 2 3

to a

As discussed in Chap. 4, Sect. 4.6, when an integer smaller than n is divided by n, the largest number of different remainders is equal to n  1. So, when dividing by 9 the possible remainders would be 1, 2, 3, 4, 5, 6, 7, and 8. To see what the calculator equivalent for each of these remainders would be, it is necessary to interpreting them as a common fraction and then convert this common fraction to a decimal. When this is done, the following equivalences can be established: 19 ¼ 0:1, 2 3 4 5 6 7 8 9 ¼ 0:2, 9 ¼ 0:3, 9 ¼ 0:4, 9 ¼ 0:5, 9 ¼ 0:6, 9 ¼ 0:7, and 9 ¼ 0:8. These values can now be used by students in “translating” the remainders found when using long division into the calculator representations being seen when using a calculator. When exploring a similar problem with one of the authors [MLC], a group of middle school students did not immediately recognize that a remainder of 9 would result in the division being carried completely out with no remainder and so continued the string of equivalences in the prior paragraph to include 99 ¼ 0:9. This did not stand scrutiny for long, however, as the group quickly realized that 99 ¼ 1. This observation did not particularly bother many in the group who then asked the profound question, “Which is larger? 1 or 0:9?”. This question drew heavily upon these students experience with comparisons as a prelude to basic operation, see Sect. 5.8.1, and was quickly followed by the question, “By how much?”. In attempting to address these paired questions these students came up with many insights directly enabled by numerical fluency. Initial discussion led to a general student acceptance of the idea that 1 should indeed be larger. The question, “By how much?”, was the subject of a much longer and richer exploration. The first step taken by the students was to consider how much larger 1 would be for successive repetitions of the repeating decimal. Starting with one repetition, it

Fig. 5.14 Dividing 196 by 9 using long division

5.7 Repeating Decimals

147

Fig. 5.15 Which is larger1 or 0:9?

was shown that 1 would be larger than 0.9 by 0.1. This was followed by creating a spreadsheet to consider the effect on the size difference which results from increasing the numbers of nines being considered. The spreadsheet used was similar to that shown in Fig. 5.15. Although many students were now satisfied that 1 would indeed always be larger, albeit by an amount so small as to be impossible to notice, there was still a minority holding to a different belief. These students pointed out that in looking at the difference, 1 is larger in each case by a “1” that appears in the place value of the last 9 being considered. Arguing that since the 9s never ends, the “1” at the end of a never-ending repetition of 9s never has a chance to appear and to impact the difference. Because of this, these students held on to their belief that the original mistake of 99 ¼ 0:9 was not a mistake at all but was in actuality a true statement. Although lacking in sophistication, this discussion is reminiscent of the thinking underlying a Dedekind4 cut. In this approach, a Dedekind cut is a set partitioning resulting in the creation of two nonempty subsets S1 and S2 where all members of S1 are less than those of S2 and that S1 has no greatest member (Weisstein, 2020).

4

Richard Dedekind (1831–1916)—a German mathematician.

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5 Decimal and Percent Representation of Rational Numbers

Fig. 5.16 Subtracting x ¼ 0:9 from 10x ¼ 9:9

As might be supposed, the dilemma posed in the “Which is larger? 1 or 0:9?” question is far from a simple matter. Furthermore, it shows up regularly implicitly even when dealing with terminating decimals, such as 1 or 0.43. Although they are generally not considered as such, these numbers could also be written in a non-terminating and repeating fashion as 0:9 or 0:429. Making the choice of whether to choose the terminating or non-terminating and repeating form is a subtle one and beyond the scope of this book. However, lest the student’s belief that 99 ¼ 0:9 is true be thrown out completely, consider the following argument from algebra. If x ¼ 0:9 then 10x ¼ 9:9. So, by taking a hint from the Flagpole Factory (Chap. 2, Sect. 2.3.3), it would be possible to subtract one equation from the other as shown in Fig. 5.16, resulting in 9 ¼ 9x ! 1 ¼ x which is a seeming win for these students. Arguments such as these illustrate some of the difficulties facing even the more sophisticated middle school students when dealing with the various possible infinite decimal representations of number. One such representation is that of a repeating decimal, defined as a decimal number with a digit or group of digits that repeats forever. Examples of repeating decimals resulting from conversion of a common fraction through the use of hidden divisions includes rather familiar examples such as 13 ¼ 0:3 in which the “3” repeats, 29 ¼ 0:2 in which the “2” repeats, as well as examples which would be considered rather unusual by middle school students 37 such as 27 ¼ 0:285714 and 300 ¼ 0:123. The algebraic argument developed in the prior paragraph, however, can be expanded into a general procedure suitable for use by middle school students in converting a repeating decimal into a common fraction. To convert such repeating decimals to common fractions, begin by letting x equal the repeating decimal being converted. Let n be the number of repeating digit(s) within the repeating decimal and create a new equation by multiplying both sides of the equation by 10n . Solving the resulting system of equations for x will give the common fraction equivalent. To see how this process would work, consider the problem of converting 0:285714 to a common fraction. This repeating decimal is already known to be equal to 27, allowing for a quick check of computations. The first step results in the equation x ¼ 0:285714. There are six repeating digits, so multiply both sides of this equation by 106 . When this is done, a new equation, 1000000x ¼ 285714:285714,

5.7 Repeating Decimals

149

is generated. Subtracting the former equation from the latter one results in 227111337 2 999999x ¼ 285714, so x ¼ 285714 999999 ¼ 727111337 ¼ 7. As a final check of process, consider the problem of converting the other unusual repeating decimal example, 0:123, to a common fraction. The initial equation for this problem would be x ¼ 0:123. There is one repeating digit, so generate a second equation by multiplying both sides of the first equation by 101 which results in 10x ¼ 1:23. Since 1:23  0:123 ¼ 1:233  0:123 ¼ 1:11, subtracting the former equation from the latter equation results in 9x ¼ 1:11 ! 900x ¼ 111, so 337 37 x ¼ 111 900 ¼ 3300 ¼ 300.

5.8

From Representation to Operations

Up to this point, the discussions have centered upon representation of various types of numbers within the base-ten notational system together with conversions between common fractions, percent and decimal fractions. Some of these discussions have led to representational issues that are quite challenging for middle school students, such as working with repeating decimals, others have led to application problems commonly faced by students in their daily lives such as discount pricing using percent, while still others helped the students expand their representational fluency through examining processes of conversions between various representations of number. cBig Idea Operations upon number, a big idea from mathematics, including operations involving both decimal and percent, require students to possess meaningful representations of the number being acted upon. If such representations are not present, the students run the risk of relying on memorization of process at the expense of conceptual understanding.b This initial heavy emphasis upon representation is appropriate, for until students are able to represent number with numerical fluency, see Fig. 5.5, any further operations upon these numbers, regardless of their representation, will be poorly understood at best. Lacking a meaningful understanding of number representation, middle school students’ subsequent operations will not rest upon a conceptually rich foundation, but rather upon a collection of poorly understood ideas and poorly understood and developed procedures as evidenced by the 12 þ 35 ¼ 47 example presented in Chap. 2, Sect. 2.1.1.1. Once numerical fluency is achieved, however, later operations may be developed in a meaningful fashion for students. As such, the role of representation in developing number operations cannot be overstated. A powerful and well understood system of representations can make previously difficult or tedious tasks simpler and more understandable. As odd as it

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5 Decimal and Percent Representation of Rational Numbers

might seem today, at one time addition and multiplication of whole numbers was a skilled occupation. Today it is so basic as to be taught to young children. What made the difference in this case was the development of the base-ten notational system (Gardiner, 2003). By providing an improved notation for the representation of number, this notational system enables the four basic operations of addition, subtraction, multiplication and division to be reduced to relatively easy to follow and understandable procedures. As will be shown, the inclusion of comparison into the group of basic operations will add greatly to students’ numerical fluency, understanding of number, and number representation skills.

5.8.1

Comparison

In general, a comparison may be thought of as an examination of two or more items to establish similarities or differences (Comparison, 2020). Although not one of the basic operations of arithmetic, comparison is an important skill for middle school students to possess. In particular, it can aid middle school students in further developing numerical fluency and number sense. By helping students in their understanding of the relative sizes of numbers, comparison can play a key role in determining the reasonableness of an operation. Although trivial, it should be noted that it is impossible to perform a comparison when only a single number has been represented. This should not be a difficulty for middle school students with numerical fluency, however. If they are able to represent one number with meaning, meaningfully representing a second number will pose no difficulty. For students lacking numerical fluency, however, it is questionable as to whether they are able to meaningfully represent even a single number, making this most basic operation upon number mysterious to them. So, the first step in performing a mathematics comparison is to meaningfully represent two numbers. For example, 302.13 and 400.99. These numbers, in following the ideas developed in Sect. 5.3, can be written as 3  102 þ 2 þ 1  101 þ 3  102 and 4  102 þ 9  101 þ 9  102 . By this expanded notational representation, it is easy to see by examination that the largest term in each is 102 , so the only real question now facing the students being which is larger 3 or 4 as these are the coefficients of this largest term. Although the answer should be trivial, part of what would make it so for a middle school student rests upon notions of numerical fluency going back to counter-based representation. As discussed in Sect. 5.2, at no point in the base-ten notational system is a student ever required to visualize more than the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with these representations falling well within the efficacy of the less than four version of subitizing and chunking. The triviality of this decision then draws from these earlier experiences with the answer, to some degree, almost being a visual recognition task as shown in Fig. 5.17 where 4 is compared with 3 using a counter-based system.

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151

Fig. 5.17 Comparing 4 and 3 using a counter based system

5.8.2

Addition and Subtraction

Once numerical fluency has been developed for a new class of number such as decimal or percent, it is possible to expand prior student understandings and processes to include these new types of numbers and their representations. Key to this expansion, however, is the ability to meaningfully represent the numbers involved. In the comparison problem posed in Sect. 5.8.1, for example, the ability to meaningfully represent the two numbers 302.13 and 400.99 in terms of their expanded notational forms of 3  102 þ 2 þ 1  101 þ 3  102 and 4  102 þ 9  101 þ 9  102 made the comparison task for the students one of visual recognition between the coefficients of the highest powers of ten, 3 and 4. This comparison task found meaning in this example as being the answer to a question similar to “Which is larger?”. Likewise, the operation of subtraction can find meaning as the answer to a follow up question, “By how much?”. This question builds upon the comparison task, which identifies whether the two quantities being represented are equal and establishes “greater than” and “less than” size relations for those cases where they are not. By requesting an exact amount, subtraction expands upon comparison by requiring the students to quantify this difference. Since the students know which of the two numbers are larger thanks to the comparison just performed, the smaller can be subtracted from the larger and negative numbers can be avoided. For the numbers under consideration, 302.13 and

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5 Decimal and Percent Representation of Rational Numbers

400.99, one possible method would be to represent these numbers in an expanded notational form and collecting the like terms. When this is done, each term can be subtracted using the same procedures, including borrowing, which have been used for whole number subtraction by the students since earliest elementary school on a term by term basis. This method has the advantage of being analogous to that which can later be applied to polynomial forms with the notable exception that one does not “borrow” from a greater power of x.z 4 ×102 + 0 ×101 + 0 × 100 + 9 ×10−1 + 9 ×10−2

-

3 ×102 + 0 ×101 + 2 ×100 + 1×10−1 + 3 ×10−2 ______________________________________ 9 ×101 + 8 ×100 + 8 ×10−1 + 6 ×10−2

A more efficient method, however, is clearly possible applying the procedures learned for whole numbers to these new number types. Since they are being represented with the same base-ten notational system, the same processes may be used in operating upon them. 400.99 302.13 ___________________ 98.86

This immediate transfer of procedures, however, is contingent on the students being able to meaningfully represent the new class of numbers within the base-ten system. It is now possible to establish meaning for the operation of addition as the answer to a follow up question, “How many in all?”. As was done for the case of subtraction, a possible method would be to represent the two numbers in expanded notational form and then collecting the like terms. After this is done, each term could then be added by the students using the same procedures, including carrying, as those used for whole number addition since earliest elementary school on a term by term basis. 4 ×102 + 0 ×101 + 0 × 100 + 9 ×10−1 + 9 ×10−2

+

3 ×102 + 0 ×101 + 2 ×100 + 1×10−1 + 3 ×10−2 ______________________________________ 7 × 102 + 0 ×101 + 3 ×100 + 1×10−1 + 2 × 10−2

As was done for subtraction, a more efficient method is clearly possible by applying the procedures learned for whole numbers to these new number types. As they are represented with the same system, the same processes may be used in operating upon them.

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153

400.99 + 302.13 ___________________ 703.12

5.8.3

Multiplication and Division

Just as was done for addition and subtraction, once numbers are meaningfully represented within the same base-ten notational system, the familiar multiplication and division processes learned in elementary school may be applied by the students provided the implications of the representation are recognized. A subtle difference, however, that might catch some students by surprise occurs when attempting to multiply decimals without considering what these numbers represent. For example, consider the problem 0:5  0:3. Even students who have demonstrated numerical fluency will occasionally make the mistake of interpreting this from the whole number perspective that “multiplication makes things larger”. In doing so, they find themselves surprised when the product is smaller than either of the multiplicands. For students finding themselves in this situation, it is often helpful for them to express the two multiplicands using their decimal fraction 5 3 representations. In this case, 10  10 . Seeing the problem represented in fractional terms in this manner generally helps students better understand decimal multiplication. Following the multiplication of these two fractions, the answer will be a decimal fraction making conversion back to decimal easy for the student. So, 5 3 15 10  10 ¼ 100 ¼ 0:15. Following a similar process for division, i.e., begin by converting to decimal fractions, performing the operation using an approach more familiar to the students, and then converting back to decimal form will often aid students develop fluency and connections. The extra steps involved serves a more subtle psychological function by encouraging the students to put in the effort to understand the greater efficiency to be gained once a full understanding is gained.

5.9

Conclusion

Many middle school students still have incomplete or emerging notions concerning number and how number is represented in the base-ten notational system. Since the base-ten notational system is such an effective system of representation it will often allow such students to be able to calculate by following memorized procedures and to accurately achieve answers without

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5 Decimal and Percent Representation of Rational Numbers

really understanding what is being represented or what these answers might represent. In performing operations such as multiplication or long division they will often seem to be acting in a near machine-like manner following procedures in a step-by-step fashion. When they are pressed into trying to explain what the steps mean and why they should be following them, students often find themselves in a true predicament. Such students are able to get a correct answer but are often unable to understand the concepts or implications upon which this answer is predicated (Peck et al., 1989; Connell, 1990). When number and number representation are well understood, however, students are able to expand their abilities beyond being able to merely get the correct answer to a question posed to them. The possession of numerical fluency allows the creation of meaningful questions, such as that described in Sect. 5.6 involving the interpretation of a remainder when compared to the repeating decimal often provided by a calculator. Without this level of understanding, these two alternate representations would seem unconnected and poorly understood at best. The examples explored in this chapter show that the content areas of decimal and percent are not only closely related, but together can create multiple opportunities for teachers to strengthen student overall numerical fluency and number representation skills. By carefully positioning these new content areas of decimal and percent within the base-ten notational system taken for granted by middle school students, problems which once seem insurmountable become easy due to the power of the notation system itself. An immediate outgrowth of this exploration is that the sequence of whole numbers is infinite, yet it can be understood to a large degree by relying upon the two cognitive primitives of subitizing and chunking. Arbitrarily large numbers can now be viewed in terms of “ones”, “hundreds” and “thousands” with each having a recognizable shape and attributes describing their relative positions and place values. This increased representational ability allows for students to extend the notational system to include powers of ten with negative exponents, making meaningful decimal representation possible. This extension of the familiar whole number representations into decimals opens the door for additional exploration into percent, common fraction, and repeating decimals. Furthermore, when it is time to perform elementary arithmetic operations, being able to meaningfully interpret these various representations of number in terms of powers of a fixed base of ten reduces all these operations on these newly understood number types to those procedures already used by the students. For example, by recognizing similarities between number representation and polynomial, see Sect. 5.3, addition and subtraction within each of these new representations of number can be accomplished by simply collecting like terms. Of greater significance, however, fully understanding the base-ten notational system being used will enable students to extend their concept of number harmoniously in a completely natural way that is

5.9 Conclusion

155

consistent with their earlier experiences with whole number representation and operations. For readers who want to deepen their knowledge and understanding of decimal and percent the following sources can be recommended (Irwin, 2016; Brigham et al.; Heibert & Wearne, 1986; Lai et al., 2018).

5.10

Activity Set

1. What is required for students to achieve numerical fluency? 2. Describe the two foundational organizing principles of the base-ten notational system. 3. How many digits would be used in a base seven notational system? A base nine? A base-n? 4. What are three features of subitizing? 5. What is the benefit that chunking adds to number representation? 6. How would you respond to a student who refuses to work with base-ten blocks because they are only good for 1st and 2nd graders? How would you respond to a parent with the same concerns? 7. What are some of the student advantages to building visual representation for the digit 0 through 9? 8. Describe a geometric interpretation of 102 explaining why it is often referred to as ten-squared and a geometric interpretation of 103 explaining why it is often referred to as ten-cubed. 9. How could the number 103,253 be represented using powers of ten? 10. Describe the relative size of a randomly selected place value and the place value two places to its left. Do the same for the place value two places to its’ right. 11. Describe the relative roles of base, exponent, and place value when applied to the base-ten notational system. 12. How would 10−3 be represented as a decimal fraction? Which place value would this represent? 13. Describe the roles played by the decimal point within the base-ten notational system. 14. How would 143.73 be represented in expanded notational form? 15. What are the constraints placed upon decimal fractions? 16. Convert 0.439 to a decimal fraction. 17. What is meant by the term “hidden divisions”? Describe how such hidden divisions can be used in converting common fractions to their decimal representations.

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18. How would you describe the process of converting a percent to a decimal? A decimal to a percent? 19. Determine the discounted price for a computer originally priced at $432 when being sold at a 12 12 % discount using two different methods. Which method did you find easier to apply? Which method would feel easiest to teach to middle school students? 20. Name three terminating and three non-terminating decimals. 21. How could 1.4 be written as a repeating decimal? 22. Convert the repeating decimal 0:436 to a common fraction. Now convert the repeating decimal 0:1436 to a common fraction. 23. Based upon your current understanding, how would you answer the question of “Which is larger? 1 or 0:9?”. 24. Why is comparison an important part of developing numerical fluency?

Chapter 6

Ratio and Proportion

6.1

Introduction

cBig Idea Proportional reasoning, a big idea from mathematics, requires a shift in student thinking away additive thinking and toward multiplicative thinking. This shift is not trivial and unless carefully addressed can lead to student misunderstandings of scale, relation and proportion.b

At the simplest level, proportional reasoning, whether involving ratio or proportion, relies on the ability to use multiplicative thinking rather than additive thinking. In other words, instead of describing a relationship between two quantities as being larger by three or smaller by five, the relationship would be described in terms such as triple the size, one fifth the size, four times greater, etc. The ability to use proportional reasoning is essential in developing a broad variety of mathematical concepts including similarity, relative growth and size, dilations, scaling, p, constant rate of change, slope, speed, rates, percent, trigonometric ratios, probability, relative frequency, density, and direct and inverse variations.

6.2

Ratios

cBig Idea Proportion, a big idea from mathematics, is an equation with a ratio on each side. This requires ratio, another big mathematical idea, to be at least partially understood prior to beginning work with proportions.b © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_6

157

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6 Ratio and Proportion

A proportion can be thought of as an equation with a ratio on each side. As such, since proportions are constructed from ratios it is essential that middle school students have a well-developed understanding of the ratio concept prior to working with proportions. Developing these understandings will also aid in the development of proportional reasoning in general. A ratio can be thought of as a comparison of two quantities or measures that is based upon multiplication. Together with proportions, ratios reflect multiplicative rather than additive comparisons and result from multiplication or division and not from subtraction or addition. Reasoning with ratios require focus to be placed upon two quantities that covary. Ratios are closely related to fractions, but there are important differences which must be understood. Since ratios can be represented using the fraction bar, it is important to clarify these differences. It is often said that a ratio indicates how many times one number contains another or a ratio is a result of measuring one quantity by another quantity. For example, measuring the larger side of a Golden Rectangle (Sect. 6.3.3) by its smaller side results in approximately 1.618. When a ratio is written in fractional form, the fraction should generally be simplified as this aids in the understanding of the relationship between the quantities in the ratio. For example, the ratio 6:8 when written in fractional form would be simplified as 34. In some cases this can result in a much easier ratio to consider, for example the ratio

323 713

can be simplified to 12.

However, if writing a ratio in fractional form results in an improper fraction it should not be changed to a mixed number. This is important to bear in mind as a ratio reflects a relationship between two quantities, often being measured with different units. Because of these differing units, the ratio 8:6 would be written in fractional form in the improper form 86. If a beginning middle school student were to change this to a mixed number, based upon earlier experience with fractions, the result 1 26 does not preserve the relationship between the two quantities making up the ratio. Should the student continue by reducing to lowest terms the result, 1 13 would further distance the relationship underlying the ratio from the result obtained by the student. Since many ratios when expressed as a fraction result in improper fractional representation, care must be taken to preserve both the units making up each part of the ratio together with the nature of the relationship between these units. Given that ratios can be represented in several ways such as 37, 3 : 7, or 3
7 it is essential that both units and the nature of this relationship be understood and preserved by the students.

6.2.1

Part-Part Ratios

A ratio can relate one part of a whole to another part of the same whole. For example, imagine Francesca has some stamps in her stamp collection of which 19 are domestic and 6 are foreign. In this situation, if the ratio is represented with a

6.2 Ratios

159

fraction bar 19 6 it would indicate a “ratio of 19 domestic to 6 foreign stamps” and not 19 the fraction 6 . Without this understanding, it could easily appear reasonable for a student to “simplify” this ratio to 3 16. This is a common mistake with serious implications. Since there are two different types of items, domestic and foreign stamps, the original form 19 6 preserves the two parts making up the ratio. The fraction 3 16, however, does not reflect the quantities making up the ratio.

6.2.2

Part-Whole Ratios

Ratios can also be used to represent comparisons of a part to a whole. Taking another look at the stamp example, imagine that Francesca’s collection consists of a total of 25 stamps. The ratio of domestic stamps to the entire collection could be written using fractional form as 19 25. In this case, it would be proper to think of this as nineteen-twenty fifths of the total collection being domestic stamps. This situation serves to illustrate the overlap between ratio and fractional concepts. An example of a part-whole ratio familiar to fans of American baseball is the batting average. In Major League Baseball, the batting average is defined as the H number of hits1 divided by the number of at bats2 or, AVG ¼ AB . Since a batter cannot be credited with a hit without having an at bat, the at bat would be the whole part of the ratio and the hit would be the part. This average might not be as immediately recognizable in its fractional form, however, as batting averages are not be reported using a fractional representation. Rather, batting averages, AVG, are reported using three decimal places of accuracy and given without including the decimal point. So, the batting average for a batter with 358 hits and 1689 at bats 358 ¼ :212, which would be reported as “batting 212”. For the would be AVG ¼ 1689 hypothetical case of a batter having a successful hit for every at bat, the batting average would be AVG ¼ nn ¼ 1:000, and the batter would be said to be “batting a thousand”. This has led to a commonly used saying indicating a perfect, even if unachievable, performance. Part-whole ratios can be difficult for middle school students to understand as they often lead to counter-intuitive problem situations. Consider the following goldfish problem recently made popular on the Internet (Mishra, 2020). In an aquarium holding 200 goldfish, 99% of the goldfish are red. How many red goldfish must be removed in order to reduce the percentage of red goldfish to 98%? To start

A hit refers to batter safely reaches first base after hitting the ball into fair territory without the benefit of a fielder’s choice or an error. 2 An at bat is different than a plate appearance. To qualify for an at bat in addition to batting against a pitcher, the batter must NOT have received a base on balls, have hit a sacrifice fly or bunt, been replaced by another batter before their at bat is completed, been hit by a pitch, been awarded first base due to interference, or had the inning end prior to the completion of the at bat. 1

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with, if the current percentage of red goldfish is 99% of 200 there are ð:99Þð200Þ ¼ 198 red goldfish. The number of non-red goldfish would therefore be 200  198 ¼ 2. The key to solving this problem is to recognize that the percentage of red goldfish reflects a part-whole ratio. When expressed in a part-whole format, the problem 98 begins with 99%, or 198 200, of the goldfish being red. Since 98% can be written as 100, the number of red goldfish to be removed can be solved by setting up a proportion in which one ratio is the number of red goldfish following removal per total number of Red 98 goldfish following removal and the other is 98%, Totalgoldfish ¼ 100 . At this point, the goldfish part-whole considerations of the ratio become important. Since the total goldfish include the red goldfish, the total will be reduced each time a red goldfish is taken from the aquarium. So, letting x be the number of red goldfish to be removed, the Þ 98 final proportion becomes ðð198x 200xÞ ¼ 100. Solving this proportion, it can be shown that 100ð198  xÞ ¼ 98ð200  xÞ, so 19; 800  100x ¼ 19; 600  98x, 200 ¼ 2x, and finally 100 ¼ x. The need to remove 100 red goldfish to reduce a percentage by a single percentage point strikes most middle school students, as well as most adults, as highly surprising. If it were possible to magically change a red goldfish to a non-red, this situation would have been radically different. The whole in the part-whole ratio would remain a constant at 200. In this case, the proportion to be solved would be ð198xÞ 98 ð200Þ ¼ 100, 100ð198  xÞ ¼ 98ð200Þ, so 19; 800  100x ¼ 19; 600, 200 ¼ 100x, and finally 2 ¼ x. This seems much more reasonable to most students attempting the problem. When a number, as is the case in part-whole relationships such as the one described in the goldfish problem, must be removed from both numerator and denominator the overall effect on the fraction is quite small, particularly when the numerator and denominator are both large relative to the small amount which is subtracted from both. Understanding part-whole ratios can also play an important role in establishing information required in basic probability questions. For example, if a bag contains red, r, blue, b, and green, g, marbles the ratio of green marbles to the total number of marbles in the bag can be expressed as r þ bg þ g, which is a part-whole ratio. The ability to determine key information related to later problems in probability in this fashion is an important skill for middle school students to acquire. Shifting from marbles to cards, consider the ratio of Jacks, J, and Queens, Q, within a standard 52 card deck. In this case, the whole of the ratio is known to be 52, the part is found by adding the number of Jacks, 4, and the number of Queens, 4. This is straightforward for middle school students to understand and the ratio can þ4 8 2 be determined to be 4 52 ¼ 52 ¼ 13 . Now imagine a case using a non-standard deck of 52 cards. This non-standard 52-card deck will be one from which a certain number of cards selected at random is removed and then replaced by the same number of cards drawn at random from another deck. In this case, the whole of the ratio is still known to be 52. Likewise, the part of the ratio can still be determined by adding the number of Jacks, J, and Queens, Q each of which may vary in the

6.2 Ratios

161

þQ range from zero to eight. For this situation, the ratio can be determined to be J 52 . Such slightly more abstracted problems are helpful for beginning students as they plan how to construct the more general part-whole ratios typical of those required in later mathematics.

6.2.3

Reflecting Quotients as Ratios

Ratios can be thought of as a type of quotient. For example, if you can buy 5 stamps for $2.00, the ratio of dollars per stamps is $2.00 for 5 stamps, or 25, which is a quotient. Dividing these values results in $0.40 per stamp which is the unit cost. Unit costs are a common example of problems resulting from using a quotient as a ratio. When working with middle school students, unit costs should be contextualized as the cost per liter, gallon, kilogram, pound, can, package, etc., of what you want to buy. Although problems involving unit cost are generally straightforward, they can occasionally cause problems for students when the unit is not obvious, or when conversions between the units being priced must be made. For example, imagine a carton of chewing gum being sold for $9.00 which contains 10 packages of gum, each package of which contains 5 sticks of gum. The unit cost for this gum could either be $9.00, $0.90, or $0.18 depending upon how the unit is being defined in the context of the problem. Because of the occasional ambiguity of units, unit cost problems provide an excellent opportunity to review basic conversions between units of measure (Table 6.1). In many problems, as well as in real world situations, the units available do not match those present in a canonic formulation of a problem. For example, data might be available in terms of liters per mile while the problem requires an answer of miles per gallon. Being aware of the units involved makes the setting up and resolving of such situations much easier for the students.

6.2.4

Reflecting Rates as Ratios

Once a ratio is thought of as a quotient, a broad variety of applications arise. Problems involving miles per gallon, students per class, passengers per trolley, etc., can all be expressed as a ratio and as a rate. It is important to remember that rates involve two different units and how they relate to one another. This can be helpful when analyzing problem situations. Other rates that involve the same type of quantity, length for example, can also be expressed as a ratio. For example, inches per foot, feet per mile, pints per gallon, etc.… When used in this manner, rates can represent an infinite set of ratios. For example, if a pump can move 10 L per minute (a ratio), it can also move 20 L in 2 min, 30 L in 3 min, etc. Although not identical,

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Table 6.1 Common conversion factors Length U.S 1 foot (ft) = 12 inches (in.) 1 yard (yd) = 3 feet (ft) 1 yard (yd) = 36 inches (in.) 1 mile (mi) = 5,280 feet (ft)

Weight U.S 1 lb (lb) = 16 oz (oz) 1 ton (T) = 2000 lb (lb)

Metric

Conversion

1,000 mm (mm) = 1 m 100 cm (cm) = 1 m (m) 10 dm (dm) = 1 m (m) 1 decameter (dam) = 10 m (m) 1 hectometer (hm) = 100 m (m) 1 km (km) = 1000 m (m)

1 1 1 1 1 1

inch = 2.54 cm (cm) m (m)  3.28 feet (ft) mile (mi)  1.61 km (km) foot (ft)  0.30 m (m) yard (yd)  0.91 m (m) km (km)  0.62 miles (mi)

Metric

Conversion

1 1 1 1

1 oz (oz)  28.3 g 1 lb (lb)  0.45 kg (kg)

g = 1000 mg (mg) g = 100 centigrams (cg) kg (kg) = 1000 g metric ton (t) = 1000 kg (kg)

Area U.S 1 square foot (ft2) = 144 square inches (in.2) 1 square yard (yd2) = 9 square feet (ft2) 1 acre = 43,560 square feet (ft2) 1 square mile (mi2) = 640 acres

Volume U.S 1 cubic foot (ft3) = 1728 cubic inches (in3) 1 cubic yard (yd3) = 27 cubic feet (ft3) 1 cord = 128 cubic feet (ft3)

Metric

Conversion

1 cm2 = 100 mm2

1 square inch (in.2)  6.45 cm2 1 square meter (m2)  1.196 yd2 1 ha  2.47 acres

1 dm2 = 100 cm2 1 m2 = 100 dm2 1 are (a) = 100 m2 1 hectare (ha) = 100 a Metric

Conversion

1 cubic centimeter (cc) = 1 cm3 1 ml (mL) = 1 cm3 1 L (L) = 1000 mL 1 hectoliter (hL) = 100 L (mL 1 kiloliter (kL) = 1000 L (L)

1 in.3  16.39 mL 1 L  1.06 qt 1 gallon  3.79 L 1 m3  35.31 ft3 1 quart (qt)  0.95 L

Temperature F o ! C o C ¼ 59 ðF  32Þ C o ! F o F ¼ 95 C þ 32

both fraction and rate can be represented through the use of a fraction model, see Chaps. 2 and 4. When this representation is performed, the potential for an infinite number of equivalent ratios is easily demonstrated as shown in Fig. 6.1, where it is assumed that by local ordinance, there must be 2 black sheep for every 5 white sheep. We are

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Fig. 6.1 A single rate can represent an infinite number of ratios

told that Shannon is a local shepherd whose flock has always abided by the local ordinances. The problem posed is, “What are some potential combinations of sheep in Shannon’s flock?” Since the ratio of black to white sheep is 2 to 5, this ratio can be modeled using a fractional model for 25. By successive divisions, as shown in Fig. 6.1, the various potential combinations of sheep making up Shannon’s flock can be generated.

6.3

Proportions

In order to serve as a proportion, it is essential that for the general case of AB ¼ DC the two ratios making up the proportion are equal. For example, from the situation 8 . These paired ratios illustrated in Fig. 6.1, it can be shown that 25 is equivalent to 20 2 8 can therefore be placed into a proportion as 5 ¼ 20. When two nonidentical ratios are set to be equal to each other, as is done in the general case of AB ¼ DC, middle school students need to understand the consequences of this action. In particular, they should recognize that A will be larger or smaller than C by the same factor that B is larger or smaller than D. In addition, they should know that A will be larger or smaller than B by the same factor that C is larger or smaller than D. This means that CA ¼ DB will also be a proportion. When the four

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Fig. 6.2 Equivalent ratios between and within a proportion

numbers making up the proportion are arranged as in Fig. 6.2, this means that the within ratios AB, DC and the between ratios CA , DB create the paired proportions AB ¼ DC and A B C ¼ D. Recognizing the various covariations in these relationships is an important step in helping middle school students set up and solve proportions. When a proportion is thought of in this fashion, the language of verbal analogies can play an important role in developing proportional reasoning. Just as a proportion contains two equivalent ratios, an analogy contains two pairs of words that are logically related. For example, dog and puppy share the same logical relationship as cat and kitten. In thinking through such a relationship, a common approach is to “read” the analogy as “dog is to puppy as cat is to kitten”. Such problems are commonly used in standardized tests and job interviews where the goal is to show the relationship between two objects or concepts using logic and reasoning. When analogies are presented in standardized testing, they are set up in a standard format showing two terms that are related to each other in the same way that two other terms are related to each other. In the Miller Analogy Test, for example, analogy items are written as equations in the form A : B::C : D. This can be read as either “A is related to B in the same way that C is related to D” or as “A is related to C in the same way as B is related to D.” (Miller Analogies Test Study Guide, 2017). So, for our earlier example, the analogy as presented in the Miller Analogy Test would be abbreviated as dog:puppy:: cat:kitten. Given this similar structure, it should not come as a surprise that our generalized example of AB ¼ DC could be written as A : B::C : D or as A : B ¼ C : D. When this thinking is applied to a proportion, it now becomes possible to “read” the proportion using this standardized format of an analogy. For our example, AB ¼ DC could be “read” as the units in A are to the units in B as the units in C are to the units in D. By thinking of the proportion in this fashion it is easier for some students to better understand the covariation which underlies the proportion itself. In particular, “reading” the proportion helps the supporting language, showing how variation in one variable coincides with variation in the others, to be developed and discussed. Unlike the goldfish problem presented in Sect. 6.2.2, many of the problems middle school students face involving proportions involve solving for one of the four numbers comprising the ratios making up the proportion itself. In solving these type of proportion problems, cross products may be used as an intermediate step in finding the missing number. So, for the general case of AB ¼ DC using cross products

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we see that AD ¼ BC. Once the proportion is expressed in this form it makes AD solving for any individual term easier for students to visualize with A ¼ BC D,B ¼ C, BC C ¼ AD B , and D ¼ A . If we know, for example, that there are 30 white sheep in Shannon’s flock, see Sect. 6.2.4, it is possible to solve for the number of black x which could be addressed by recognizing that sheep using the proportion 25 ¼ 30 2 12 5 ¼ 30 resulting in x = 12 If this is not noticed an algebraic approach could be used, ð2Þð30Þ ¼ ð5Þð xÞ ! 60 ¼ 5x ! x ¼ 60 5 ! x ¼ 12. A related problem at this point might be to determine what sizes of flocks Shannon might have while still abiding by the local ordinance that there must be 2 black sheep for every 5 white sheep. The smallest possible flock would have 2 black sheep and 5 white, for a total of 7 sheep. Since a fractional sheep cannot exist, the flock size can be determined by adding the numbers of black sheep and white sheep. Looking at Fig. 6.1, it can be seen that Shannon’s flock could also consist of 4 black and 10 white sheep for a total of 14; 6 black and 15 white sheep for a total of 21; and 8 black and 20 white sheep for a total of 28. Furthermore, since the number of sheep in the flock is equal to the number of black sheep plus the number of the white sheep, flock size is easily computed. For the simplest case the number of sheep is shown to be 7. Each successive possible flock will thus be an integer multiple (remember, it is not possible to have fractional sheep!) of this starting flock. So, as long as Shannon’s flock consists of 7x, where x is a positive integer, the number of sheep in Shannon’s flock will continue to be in line with local ordinances. A more involved type of proportion problem often experienced by middle school students is what could be classified as a “shared work” problem. Consider the following typical example: When they work alone, Shannon can mow the local park in 4 h and Gabriel can mow the park in 3 h. How long would it take them to mow the park if they worked together? In this problem, it can be determined that since it takes Shannon 4 h to mow one park, her park per hour mowing rate is 14. Likewise, since it takes Gabriel 3 h to mow one park, his park per hour mowing rate is 13. So, 7 when working together they would be able to mow 14 þ 13 ¼ 12 of the park per hour ð127 Þ which can be written as the ratio 1 . Assuming that they continue to work at the same rate, it is now possible to write a ratio that shows the relationship between the number of parks, 1, and the amount of time it would take to mow the entire park, t. This ratio, 1t , can now be used with the ratio showing Shannon and Gabriel’s ð7Þ ð7Þ combined mowing rate of 121 to establish the proportion 121 ¼ 1t . When this is 7 done, the proportion could be “read” as “12 of a park is to 1 h as 1 park is to t hours”. Finally, this developed proportion can be solved providing a value for t of ð127 Þ 1 12 5 1 ¼ t ! t ¼ 7 ¼ 1 7 hours. Such shared work problems can prove challenging both for students still developing proportional reasoning and students unsure of the relationships between the units in various systems of measurement, see Sect. 6.3.2. For example,

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Shannon’s mowing rate might be presented in the problem as taking 60 min to mow 1 4 of the park, while Gabriel’s rate is reported as mowing the entire park in 3 h. When the units making up the problem do not seem to be related to the student, it creates additional difficulties for them in terms of setting up the problem. In addressing problems requiring conversion within a system of measurement, therefore, students may need assistance in identifying the units necessary to set up the proportion and making any necessary conversion. From a teaching standpoint, this conversion of units can also be addressed via a proportion as described in Sect. 6.3.2. When this is done, it can provide an additional opportunity to develop proportional reasoning. To be successful in the case of Gabriel and Shannon, the students must recognize that 60 min equals 1 h and be able to convert the units presented in the problem accordingly by either converting Shannon’s time to hours or Gabriel’s time to minutes. Proportions can also be used in a variety of financial contexts. One of the more common of these is that of per unit pricing. Consider the following problem: James purchased 12 rolls of paper towels for $15.42. Assuming that paper towels can be purchased at the same price for a single roll, how much would 20 rolls cost? The proportion in this case is quite straightforward with and utilizes the ratio of the total cost paid per the number of rolls purchased. Ratios such as this are often referred to as the unit cost, or per unit price, and can be extremely useful when projecting the cost of purchases. When the full proportion for James and the paper towels is set up x it becomes $15:42 12 ¼ 20 where x is the cost of 20 rolls of paper towels. It is clearly x possible to solve for x using this proportion, $15:42 12 ¼ 20 ! 20  $15:42 ¼ 12x ! x ¼ 20$15:42 ¼ $25:70. In addition, however, if the ratio of number of purchased 12 units per cost, or the unit cost is already known this could also be used. Since it is known that James purchased 12 rolls of paper towels for $15.42, the unit cost can be computed as $15:42 12 ¼ $1:285. For some students, it can be helpful to demonstrate the use of the unit cost ratio by verifying the earlier results, so since 20  $1:285 ¼ $25:70 the earlier result obtained by using proportions is verified. Although this may seem a redundant step it can help developing students see connections between the proportion and one the rates making up the proportion. Once this is established, additional problems using the unit cost can be asked leading to the generalization that by knowing this unit cost, James can determine the price of any number of paper towels assuming that each roll can be purchased at the single roll price. This observation provides important grounding for the more general rule that TotalCost = UnitCost  UnitsPurchased. In addition to this use in unit pricing, the amount paid in taxes are also proportional. For example, if the tax rate is 10% and you purchase a textbook for $100 you will pay $110 and for a $200 dollar textbook 220 you will pay $220 so it can be shown that 110 100 ¼ 200. A slightly more difficult problem emerges when the unit cost is not constant, but changes as a result of bulk pricing. For example, imagine that James revisits the store and realized that had he purchased in bulk the unit cost would have been $1.20 per roll provided at least 16 rolls were purchased. How much did James lose by not

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having taking advantage of this? At this point, many beginning students will attempt to set up a proportion to solve the problem based on their success with earlier problems seemingly of the same type. In addressing this problem, however, the unit costs are not equal being $1.285 and $1.20 respectively making it impossible to solve the problem via a proportion. In this case, however, the problem is easily addressed through the use of the two different unit costs. We know that James initially spent $25.70 for 20 rolls and paid a unit cost of $1.285. Had James have bought in bulk he would have qualified for the lower unit cost of $1.20. This means that he would have paid 20  $1:20 ¼ $24:00 resulting in his loss of $25:70  $24:00 ¼ $1:70 by not purchasing at the bulk unit cost rate. For a still more advanced financial example, consider the question of how much would need to be deposited in a savings account earning 4% ¼ :04 per year in order to have exactly $200 in two years’ time. A reasonable place to start would be to determine how much would be in the account one year following an initial deposit of $100, see Chap. 11, Sect. 11.1.4.1. Since both the time interval and interest rate are expressed in years, the interest can be calculated using the formula for simple interest which is I ¼ P  r  t, where I = Interest, P = Principal, r = annual interest rate expressed as a decimal, and t = Time. Applying this formula results in the interest earned = ð$100Þð:04Þð1Þ ¼ $4. So, at the end of one year there would be $100 þ $4 ¼ $104 in the account. This provides the information necessary to set up x the proportion $100 $104 ¼ $200, which can be “read” as “the initial deposit is to the one-year amount as the desired deposit is to the two-year desired amount”. Solving this shows that the desired amount to be deposited is x ¼ $192:31. A common use of proportions in the home is in converting and adjusting recipes. There are many cases where a recipe might need to be adjusted. For example, suppose that a party has been planned for 30 people and the desert recipe only yields 16 servings. If we know that the amount of milk required in the recipe is 2 2 x cups, it is possible to set up the proportion 16 ¼ 30 which can be “read” as “the amount in the recipe is to the yield of the recipe as the needed amount is to the desired yield”. Solving this shows that the needed amount of milk for required yield is x ¼ 3 34 cups. Once this is done, it is now also possible to determine the quantities for any other ingredients which might be in the recipe. The ratio of needed yield to recipe yield provides a conversion factor which can then be applied to each of the other ingredients. In this example, the conversion factor would be 30 16 ¼ 1:875. Given the large number of situations lending themselves to proportional reasoning, it should come as no surprise that proportions have long found their way into literature. In 1726 Jonathon Swift first introduced the world to Gulliver and his now famous travels (Kearney, 2010). These travels feature two which both lend themselves to the development of proportional reasoning. The first of these proportional adventures takes place during Gulliver’s first voyage when he is shipwrecked and washed ashore upon the island of Lilliput. Upon awakening, Gulliver finds himself help captive by a race of tiny people, the Lilliputians, who are described as being less than 6 inches tall. If the average

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Lilliputian appears to Gulliver to be 5 inches tall and Gulliver is 5′ 6″, how large would Gulliver appear to the average Lilliputian? Although easy to ask, this question requires some careful consideration in its solution, making it a rich problem within which students can explore both ratio and proportion. Having noted that some middle school students may have difficulty when the units under consideration are different, a first step would be to help these students recognize that there are 12 inches in 1 foot. Once this is understood, Gulliver’s height can be shown to be ð5  12Þ þ 6 ¼ 66 inches. Since both heights are now in terms of inches, the ratio in heights between Gulliver, G, and the G 66 Lilliputians, L, can be expressed as 66 5 which establishes the proportion L ¼ 5 ! G ¼ 13:2L making Gulliver appear 13.2 times larger than the average Lilliputian. Since the Lilliputians would correctly feel that their measurement system would be the appropriate one to use, they would interpret Gulliver’s height within that system. Reasoning that Gulliver is also of average height they would then convert Gulliver’s height into the Lilliputian system of measurement (Lilliput inches = Li , and Lilliput feet = Lf ) by multiplying their average height, which they interpret as 66Li times 13.2. This makes Gulliver’s height appear to the Lilliputians as being 66Li  13:2 ¼ 871:2Li ! 72 12 Lf . Small wonder that Gulliver appeared as such a wonder. Having a 72 12’ giant wash ashore would be amazing indeed! The second proportional adventure occurs when Gulliver is left behind in the land of the Brobdingnagians. In what could be yet another opportunity to explore proportional thinking, Gulliver estimates the height of a Brobdingnagian farmer by comparing the length of the farmer’s stride to Gulliver’s stride and computing a value of 72′ for the farmer’s height. Leaving this computation, however, a parallel question to that posed in Lilliput could also be asked at this point, “How large would Gulliver appear to the average Brobdingnagian?”. This question is slightly easier to begin than was the Lilliputian case as both Gulliver’s height and the Brobdingnagian’s height are reported in feet. This makes the ratio in heights between Gulliver, G, and the Brobdingnagians, B, capable of ð51Þ being expressed as 722 . This ratio can now be used to establishes the proportion ð512Þ ð512Þ G B ¼ 72 ! G ¼ 72 B ! G ffi :076B making Gulliver 0.076 times the size of a Brobdingnagian. Considering that the Brobdingnagians would likewise feel that their measurement system would be the appropriate one to use, they would interpret Gulliver’s height within their own system. As the Lilliputians had done, they could consider Gulliver to be of average height and would then convert Gulliver’s height into the Brobdingnagian system of measurement (Brobdingnagian inches = Bi , and Brobdingnagian feet = Bf ) by multiplying their average height, which they interpret as 66Bi times 0.076. This makes Gulliver’s height appear to them as being 66Bi  :076 ffi 5:04Bi . Gulliver, viewed by the 5-inch Lilliputian’s as a giant, is now seen in nearly the same proportional height by the Brobdingnagians as he viewed the Lilliputians! Although not asked in Gulliver’s travels, an interesting set of questions could additionally be asked at this point. How large would a Lilliputian appear to the

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Brobdingnagians? Since it has been shown that G ffi :076B, a Brobdingnagian is
1  G ffi ð13:16ÞG. So, since Gulliver is 13.2 times larger than a Lilliputian, B ffi :076 and a Brobdingnagian is 13.16 times the size of Gulliver, the Brobdingnagians are 13:2  13:16 ¼ 173:71 times larger than a Lilliputian! This theme of alternating sizes was also featured in the film, “Honey, I shrunk the kids” where the scales used is taken to an even more extreme level. In this comedy, an eccentric inventor manages to shrink a group of children to 14 inches in height. Following the approach described in Gulliver and the Lilliputians, the world experienced by the normal kids, KN , and the shrunk kids, KS can be compared using the ratio of normal to shrunk kids which would be KKNS . If we assume the height of the average KN to be 5′ ¼ 5  12 ¼ 60 inches. and knowing the height of the average KS is 14 inch all the information necessary to set up a proportion is present. Taking a clue from both the Lilliputian and Brobdingnagian cases, this ratio can now be used to establishes the proportion KKNS ¼ ð601 Þ ! KN ¼ 60  4KS ! KN ¼ 240KS making ð 4Þ the normal world appear 240 times larger than the shrunken world within which the children now find themselves. As was the case with Gulliver travels to the Brobdingnagians, “Honey, I shrunk the kids” was followed by a sequel, “Honey, I blew up the kid”. One of the more famous applications of proportions dating from antiquity is credited to Thales3 when determining the height of the great pyramid at Giza, see Fig. 6.3. Thales reportedly accomplished this task by planting a staff into the ground and comparing the length of its shadow to the shadow cast by the great pyramid. Since he realized that the triangles formed by the pyramid, the staff, and their respective shadows were similar, it would have been possible to set up the PyramidHeight StaffHeight proportion PyramidShadow ¼ StaffShadow . Thales certainly realized that when the staff’s shadow was equal to the staff’s height the pyramids height would equal its shadow length. However, this is likely not the method that was used as there are only four days each year where the geometries of the pyramid would allow these measurements to be made—and even they would result in only an approximate value (Redlin et al., 2000). The use of the PyramidHeight StaffHeight generalized proportionality PyramidShadow ¼ StaffShadow , although admittedly requiring more calculations, has the benefit of being applicable at least once a day throughout the year. Whichever methods was used, however, this feat helped cement Thales’ reputation as one of the Seven Sages of Greece (Maor & Jost, 2014). A more contemporary use of proportions is that of determining the weight objects would experience on differing planets and moons. Given the current plans for lunar exploration put forth by a number of countries and corporate interests, this topic is potentially of high interest for middle school students. Consider the

3

Thales (ca. 624–546 B.C.) was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Aristotle considered him as the first philosopher in the Greek tradition.

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Fig. 6.3 Thales’ pyramid measurement

problem of determining the weight of a 450-lb instrument package on Earth’s Moon. According to Universe Today, if you weighed 200 lb on Earth you would only weigh 33 lb on the Moon (Cain, 2015). This information allows us to set up 33 x ¼ 450 where x is the weight of the instrument package on the the proportion 200 Moon. This proportion can be “read” as “your weight on the Moon is to your weight on Earth as the weight of the instrument package on the Moon is to the weight of the instrument package on Earth”. Solving this shows that the weight of the instrument package on the Moon would be x ¼ 74 14 pounds. Furthermore, the ratio of an object’s weight on the Moon to the object’s weight on Earth provides a 33 ¼ :165 that can now be applied when planning other conversion factor of 200 potential lunar payloads. As has been noted, the underlying concepts of ratios and proportions are very closely related. For example, when starting to solve ratio word problems a common first step is to identify the known to the unknown ratios, to set up the corresponding proportion and then to solve the proportion for the missing value. As a check in this process, it is possible to use the cross product as a simple method to determine if two ratios form a proportion. For our generalized example, of AB ¼ DC, if AD 6¼ BC then the two ratios do not form a proportion. For some students this is easier than trying to find a common multiple of A and B which will result in C and D. Consider the ratios 47 and 12 21. For these students, through the use of cross products, it is relatively easy for them to show that 4  21 ¼ 7  12, thus showing that the two ratios are proportional. This approach should be taken with care, however, as a premature use of cross products can result in students losing the relationships between units comprising the underlying ratios. Prior to cross product use in actually solving proportions, students should thoroughly understand why they work and be encouraged to use reasoning to determine the missing value. In this case,

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however, the students would be using the cross product to verify that two ratios can be placed into a proportion and not to solve the proportion itself. Of course, ideally students should be able to recognize that 12 ¼ 3  4 that 21 ¼ 3  7 and then use these recognitions to establish that 47 ¼ 12 21 and thus is a proportion. However, for students lacking this ability, recognizing the linking step of 33  47 ¼ 12 21 presents a more difficult task. For such beginning middle school students lacking such facility with multiplication, this type of covariation between ratios is less obvious and subsequently more difficult for them to initially recognize.

6.3.1

Scaling

Proportional reasoning and geometric similarity can be shown to be closely related. As will be demonstrated in Chap. 7, Sect. 7.2.4, similar geometric figures can be created by dilation or expansion. This geometric transform refers to changes in the size of a preimage without changing its shape or orientation. As a result of this, such a scaling process is an example of a dilation/expansion transformation with the ratio preimage image serving as the scaling factor. In particular, two objects can be shown to be similar if one of the objects can be transformed into the other object through the application of an isotropic (uniform) scaling factor to one of the objects. This relationship makes it easy for middle school students to use geometric scaling as a mechanism to explore both ratio and proportion. By observing the relative effects of dilation/expansion upon easily visualized object’s student can gain insight into the effect application of conversion factors would have in general. In exploring the effects of scaling factors with middle school students, consider the use of the computer program PowerPoint. PowerPoint, as a tool, lends itself to the informal exploration of various scaling factors upon the dilation or expansion of an object. The PowerPoint program provides a rich technological environment within which to create geometric objects and to explore the effects of applying various scaling factors upon them. Starting with a new slide, simple geometric objects can be easily created within PowerPoint by selecting “Insert” from the ribbon bar and then “Shapes” and selecting the shape to be inserted into the slide. Once an object is created, it can easily be acted upon by right-clicking the object with the mouse. This right-click will then bring up a drop-down list of potential actions. To explore the effect of a scaling factor, the students should select the “Size and Position…” action. Once this action has been selected, the Scale Height, (yaxis), and Scale Width, (x-axis), scaling factors can be selected for application to the shape. The effects of the scaling upon the shape are shown immediately, thus providing instant feedback to the student. The examples shown in Fig. 6.4 were created within PowerPoint in this fashion and have been labeled to show the results of applying various scaling factors upon the original graphic object. These examples provide an easy to comprehend visual representation of the underlying transformation that were easily constructed using

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Fig. 6.4 Isotropic and anisotropic scaling

the “size and position” tool and actions within PowerPoint as was described above. By allowing exploration of both uniform (isotropic) scaling and non-uniform (anisotropic) scaling, it is possible for students to easily see examples and non-examples of scaling which would lead to similar figures. At this point, a rather interesting aspect of isotropic scaling can be explored through the investigation of similar rectangles. In Fig. 6.5, there are two similar rectangles which were formed through the application of an isotropic scaling factor to a larger preimage resulting in the formation of a smaller rectangle. Since the scaling was isotropic, these two rectangles are similar. When these similar rectangles are both positioned with a corresponding vertex at the origin, the scaling factor’s impact can be easily observed. Since these rectangles are similar, the following proportion will of necessity be true: xxsmaller ¼ yysmaller . So, when looking at the ðx; yÞ coordinates of the line passing larger larger through the origin and the two corresponding vertices of these two similar rectangles, it is possible to see that the slope m of this line will be equal to the scaling j93j 1j 6 factor. Therefore, for this example, m ¼ jjyx22 y x1 j ¼ j155j ¼ 10 ¼ :6. Furthermore, since this scaling factor can be expressed as a ratio, a case can be made that this is but one of an infinite number of similar rectangles which could be generated corresponding to the ðx; yÞ coordinates of points along this slope line. This prediction is easily verified by using the computer program PowerPoint to apply this scaling factor, 0.3, to the larger preimage using the “size and position” tool. To further develop the notion of isotropic scaling, the following activity can be effective in helping middle school students link geometry, symmetry, measurement, computation, application of an appropriate scaling factor, and generation of an accurate scale. In this project, the students imagine that all removable items in a room have been taken out. This includes all desks, tables, chairs, podiums, computers, etc. The task is then given for the students to draw and label a rough sketch of the room including all key features including the location of the door, windows, whiteboard, screens, and any unusual room features. Once this is done, careful measurements are taken of each of the room’s identified features and the rough

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Fig. 6.5 Isotropic scaling, scaling factor, and slope

sketch is labeled with these measurements. As students compare measurements with one another, the precision of each measurement is increased by repeating any measurement that is clearly off from that obtained by the bulk of other students. This process is repeated until the majority of observed errors are corrected. This corrected rough sketch is then used as the basis for creating a more accurate scale drawing of the classroom which includes all key features on ¼ inch graph paper. Since this graph paper can be used in either landscape or portrait mode, the choice of orientation is important. Factors leading to this decision provide an opportunity to explore scaling in general and of maps in particular. For example, when considering maps and scale drawings in general, the less dilation that occurs the more details that can be shown on the completed project. This is easily shown during this activity by comparing students who chose a landscape orientation versus a portrait orientation. Once students observe that when the longest lengths of the room and the paper are aligned, the accuracy of the developed scale drawing is increased. Once this realization is obtained, it is possible to explore exactly why this should be the case. When the length of the longest room side is aligned with the longest edge of the graph paper, the required scaling factor is less and more details can be shown. This observation can be helpful for students to better recognize the covariation which takes place within proportions. Finally, the computed scaling factor used in this sketch is reported as part of the final student record of action. Figure 6.6 shows a student’s work sample produced when participating in this activity. In addition to providing the opportunity for student practice with measuring instruments and factors impacting the precision measurements, this activity lends

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Fig. 6.6 Scaling project work sample

itself nicely to the creation of a data table of measured lengths to which the computed scaling factor can be applied. Once a data table of lengths is constructed it will provide for a natural entry point for spreadsheets, or other computational aids, to be employed.

6.3.2

Proportions and Conversions

The use of proportions in computing conversions between various systems of measurement represents an easy entry point for middle school students for the development of proportional reasoning. In addition, by gaining experience between units of measure students will be more able to set up proportions involving seemingly different units. Such facility is often called for, as was the case of Shannon and Gabriel’s mowing problem (Sect. 6.3). Setting up and using a conversion factor draws upon many of the same skills developed through the use of scaling factors and unit cost. Just as TotalCost ¼ UnitCost  UnitsPurchased, a conversion factor will allow students to recognize and use a conversion factor such as TotalInches ¼ 12InchesðNumberOfFeetÞ. Furthermore, once the conversation factor is understood any number of conversions can be performed without the need of establishing a proportional relationship. A listing of the more common conversion factors which should be familiar to middle school students is shown in Table 6.1.

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Of course, conversions within and between systems of measurement are not the only type of conversions middle school students can encounter. A common necessary conversion for students travelling internationally concerns conversion between monetary units. A student travelling between the European Union and the United States, for example, would have a definite interest in the conversation rate between Euros and US Dollars. Fortunately, the conversion rate can easily be established based upon a single transaction. Should the student learn through a purchase that it requires $27 to purchase a €25 item, the conversion factors of both dollars euros euros and dollars can be established by setting up and computing the appropriate ratio. 27 euros 25 Thus, dollars euros ¼ 25 ¼ 1:08 and the dollars ¼ 27 ¼ :93. Although currency converters are readily available online, some understandings about mental conversion should be a part of the toolkit of an educated individual. In particular, the ability to compute a conversion factor from a single set of transactions can be a highly valuable skill for people of all ages.

6.3.3

The Golden Ratio

The Golden Ratio, despite its name, is more easily understood when viewed as a proportion. Two quantities are said to be related via the Golden Ratio if their ratio is the same as the ratio of their sum is to the larger of the two quantities. This relationship is shown in Fig. 6.7, and results in the proportion x þx 1 ¼ 1x holding true for all x. So, since x þx 1 ¼ 1x we can generate the resulting quadratic equation: ðx þ 1Þð1Þ ¼ ð xÞð xÞ ! x þ 1 ¼ x2 ! x2  x  1 ¼ 0. The solution to this equation pffiffi pffiffi pffiffi is 12 5. Since 12 5 \0, we have 1 þ2 5 as the only viable root. This value is referred to as the Golden Ratio or / (phi) and is approximately equal to 1.618. Over the course of history, the Golden Ratio has come to play a large role in art and architecture due to the Golden Rectangle derived from it. This rectangle, a common construction being shown in Fig. 6.8, is said to be one of the more visually satisfying of geometric forms (Bergamini, 1980) with examples of its influence dating back to antiquity. The Golden Ratio, /, is a rich source of potential explorations for middle school students. For example, the Golden Ratio can be shown to be closely related to Fibonacci numbers. The first two numbers in the Fibonacci sequence are 1 and 1. With this as a starting point, each subsequent Fibonacci number is found by adding the two Fibonacci numbers immediately before it. So, the third term in the Fibonacci sequence is found by adding 1 þ 1 and is 2, the fourth term in the Fibonacci sequence is found by adding 2 þ 1 giving 3, etc. In general, if xn is the nth term of the Fibonacci sequence, then for n [ 2 the Fibonacci number is Fn ¼ Fn1 þ Fn2 . With this background, an initially surprising finding can be made in that the ratio formed by any two successive Fibonacci numbers rapidly approaches / for larger terms in the Fibonacci sequence. Figure 6.9 shows the first 10 terms of the Fibonacci sequence and the ratio between successive terms for

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x+1

x

1

Fig. 6.7 The golden ratio

Fig. 6.8 The golden rectangle

Fn 2 as calculated using a spreadsheet.4 Notice how the ratio FFn þn 1 is getting closer to / as n increases, a finding which was shown by Kepler5 (Tattersall, 2005). Although beyond the scope of this text, for the limiting case lim FFn þn 1 ¼ /, it can be n!1

observed in Fig. 6.9 that these approximations of / are alternatively higher and lower as they converge to /.

4

In this table, the ratio between two successive Fibonacci numbers (column 3) for the nth term (column 1) was calculated by the ratio FFn þn 1 . The value of the Fibonacci number itself was calculated for n [ 2 as Fn ¼ Fn1 þ Fn2 . 5 Johannes Keppler (1571–1630), a German astronomer and mathematician.

6.3 Proportions

177

Fig. 6.9 Fibonacci numbers and the Golden Ratio

Rao Term of between two the successive Fibonacci Fibonacci Fibonacci sequence number numbers 1 1 2 1 1.0000000000 3 2 2.0000000000 4 3 1.5000000000 5 5 1.6666666667 6 8 1.6000000000 7 13 1.6250000000 8 21 1.6153846154 9 34 1.6190476190 10 55 1.6176470588

Fibonacci numbers can also be calculated by applying Binet’s formula.6 For the Fn term, the corresponding Fibonacci number using Binet’s formula would be h pffiffin  pffiffin i Fn ¼ p1ffiffi ½/n  ð/Þn ¼ p1ffiffi 1 þ2 5  12 5 . For example, when using 5

5

Binet’s formula forthe 7th term, the corresponding Fibonacci number  pffiffito solve 7  pffiffi7 1 þ 5 1 5 1  2 ¼ 13. F7 ¼ pffiffi5 2

6.3.4

The Silver Ratio

Although the Golden Ratio is more widely known, there are other ratios that have assumed fame over time. One such is the Silver Ratio. Like the more famous Golden Ratio, the Silver Ratio is more easily understood when introduced through a proportion. Two quantities are said to be related via the Silver Ratio if the ratio of the sum of the smaller and twice the larger of those quantities to the larger quantity is the same as the ratio of the larger one to the smaller one. If a is defined to be the greater number and b the smaller, applying this definition for the Silver Ratio, often abbreviated as dS, would establish the proportion 2a aþ b ¼ ab. So, since 2a aþ b ¼ ab we can generate the resulting quadratic equation for the case where b ¼ 1: pffiffi 2 8 2a þ 1 a 2 2 2 a ¼ 1 ! 2a þ 1 ¼ a ! a  2a  1 ¼ 0: This equation, has roots of

6

Jacques Binet (1786–1856), a French mathematician.

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pffiffiffi pffiffiffi pffiffiffi which can be simplified to 1  2. Since 1  2\0 this makes a ¼ 2 þ 1 making the Silver Ratio, or dS, approximately equal to 2.414.

6.3.5

Metallic Means

As might be expected from the definitions used in the Golden and Silver ratios it would be possible to extend these ratios further. For example, a “bronze” ratio could be defined as “two quantities are said to be related via the bronze ratio if the ratio of the sum of the smaller and three times the larger of those quantities to the larger quantity is the same as the ratio of the larger one to the smaller. If a is defined to be the greater number and b the smaller, applying this definition for the “bronze” ratio would establish the proportion 3a aþ b ¼ ab.. So, since 3a aþ b ¼ ab we can generate the resulting quadratic equation for the case where b ¼ 1: 3a þ 1 a 2 2 ¼ ! 3a þ 1 ¼ a ! a  3a  1 ¼ 0. The solution to this equation is apffiffiffiffi 1 pffiffiffiffi pffiffiffiffi 3 13 3 13 3 þ 13 as 2 \0 we will only keep 2 making the “bronze” ratio be 2 . Again, p ffiffiffiffi 3 þ 13 defined as 2 . Taking this further, it would be possible to define a ratio such that “two quantities are related via a metallic ratio if the ratio of the sum of the smaller and four times the larger of those quantities to the larger quantity is the same as the ratio of the larger one to the smaller” which would establish the proportion 4a aþ b ¼ ab. When this is set up as a quadratic equation, solved, and following the precedents used to define the Golden, Silver, and Bronze ratios the value for this new ratio would equal pffiffiffi 2 þ 5. Continuing this process to a logical conclusion, it would be possible to extend these ratios, referred to as “metallic means” (de Spinadel, 1999), to the general case in defining a ratio such that “two quantities are related via the Metallic Mean if the ratio of the sum of the smaller and n times the larger of those quantities to the larger quantity is the same as the ratio of the larger one to the smaller”. For a [ b this definition would thus establish the proportion na aþ b ¼ ab. When set up as a quadratic equation, solved, and following the definitional precedents adopted thus far, the pffiffiffiffiffiffiffiffiffi 2 value for such generalized ratios would equal n þ 2n þ 4. In particular, when n ¼ 1 we have the Golden Ratio.

6.4

Conclusion

In many ways, proportional reasoning constitutes a set of gate-keeper concepts which must be mastered for further progress within mathematics. The development of proportional reasoning is essential for middle school students in order to think flexibly across a variety of number types including the

6.4 Conclusion

179

impact of powers of ten, decimals and percentages (Chap. 5), conversions, scaling factors, similarities, rates and ratios. Since one of the hallmarks of proportional reasoning is the ability to make use of multiplicative thinking, proportional reasoning serves as a requisite for a broad range of problems involving both multiplication and division. Furthermore, once rates are fully understood as involving two different units and how they relate to one another, it is possible to picture how one rate can represent an infinite set of ratios. By focusing student attention upon the covariation between units, rate problems establish a foundation for difference equations and the eventual understanding necessary to process operations such as dy dx. This was explicitly addressed in the Isotropic scaling, see

1j Sect. 6.2.4, where the slope m ¼ jjyx22 y x1 j was shown to generate the scaling factor used in the scaling process. With this background, the foundation is lain for development of essential concepts such as the derivative f 0 ðaÞ ¼

lim f ða þ hhÞf ðaÞ associated with the limiting behavior of ratios.

h!0

More immediately, even should the calculus not be reached, middle school students need a well-developed set of proportional reasoning strategies in order to achieve success in algebra. Many of the problems they will later face in algebra will require practice with multiplicative relationships, ratios, rates and proportions. Furthermore, without proportional reasoning it is far too easy to confuse fractions and ratios.

cBig Idea Ratio, a big idea from mathematics, is made up of two covarying properties. This requires students to have multiple experiences with meaningful covariation. When these experiences are in place, many potential points of confusion between ratio and fraction can be avoided.b

In order for middle school students to be successful in reasoning with ratios, it is essential that they recognize that a ratio is made up of two quantities that covary and appreciate the nature of this covariation. Once this is understood, the door is open to a broad set of problems involving miles per gallon, students per professors, proctors per test, etc., all of which can be expressed using ratio and rates. When the covariation of ratios is understood, the groundwork for developing proportions is in place. With this groundwork, a proportion can then be viewed as an equation with a ratio on each side.

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Since a proportion is an equation, it is essential that students realize that the two ratios making up the proportion must be equal. As a part of understanding the covariations present in properly established proportions it is helpful for students to recognize the equivalent ratios taking place within a proportion, see Fig. 6.2. When these relationships are understood, proportional thinking can enable middle school students to successfully analyze and address a broad variety of problems ranging from finance to cooking, scaling and product purchases. For readers who want to deepen their knowledge and understanding of ratio and proportion the following sources can be recommended (Huntley, 2015; Dole et al.; Weiland et al., 2020; Cohen, 2013; Smith & Litwiller, 2002; Beckmann & Izsák, 2015).

6.5

Activity Set

1. José has had 284 hits in 1,213 at bats, while Aaron has only had 257 hits in 810 at bats. Who is more likely to be chosen as a pinch hitter? Why? 2. How could you respond to someone arguing that José should be chosen since has had more hits? 3. Describe the effect on a fraction when a number, as is the case in part-whole relationship is removed from both numerator and denominator (Sect. 6.2.2). 4. What are some of the problems which can emerge when students do not see the relationships between units of measurements? 5. Suppose the local ordinance mentioned in Sect. 6.2.4 is changed to read, “there must be 4 black sheep for every 7 white sheep”. Use a fraction model to show the first four potential combinations of sheep in Shannon’s flock. 6. What are the first four flock sizes Shannon could have under the new local ordinance in problem 5. What rule could Shannon follow to make sure her flock will always be in line with the local ordinance. 7. For the general case of AB ¼ DC describe the covariations, proportions and equivalent ratios that can be formed. 8. What would be the best choice to complete the analogy, SHARD:POTTERY:: (____):WOOD. (a) acorn, (b) smoke, (c) chair, or (d) splinter? 9. Express the general case of a proportion, AB ¼ DC, in Miller Analogy format. 10. How could the proportion AB ¼ DC be “read”? 11. Johannes can paint a room in 2 12 hours and Paula can paint the same size room in 3 14 hours. Assuming that they continue to work at the same rate, how long would it take them to paint the same size room when working together?

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181

12. Johannes can paint a different room in 185 min and Paula can paint this new room in 3 34 hours. Assuming that they continue to work at the same rate, how long would it take them to paint the same sized room when working together? 13. Compare the steps you followed in solving problems 11 and 12. How would you help students address these additional challenges? 14. Solve the following problem by using both a proportion and unit cost: Jane purchased 16 packs of gum for $5.75. Assuming that the packs of gum can be purchased at the same price for a single pack, how much would 27 packs cost? 15. If the gum in problem 14 is offered in bulk for a price of $0.50, should Jane purchase the 27 packs in bulk? Why or why not? 16. What would be the amount in a savings account after one year when $135 is deposited in an account earning 3 12 % per year? 17. How much money should be deposited in a savings account earning 3 12 % per year in order to have $435 after a two-year period? 18. The following recipe will yield 4 servings. 2 tbsp. olive oil, divided 1 lb. ground beef 2 tsp. ground cumin 2 tsp. chili powder Kosher salt and pepper 1 yellow onion, finely chopped 1 clove garlic, pressed 1 lb. tomatoes, finely chopped 1 15-oz can cannellini beans, rinsed Adjust this recipe so that it will yield 15 servings. 19. How large would the children from “Honey, I blew up the kid” be in comparison with the Lilliputians and Brobdingagians as described in Sect. 6.3? 20. The height of the great pyramid of Giza is approximately 145 m. Assume that Thales was able to carry a staff between 34 and 1 12 times his height. What are the range of shadow lengths he could have measured? 21. How much would a 1550-lb instrument package weigh on Mars if a 200-lb person would weigh 76 lb on Mars? What would a Mars conversion factor be that could be applied to future calculations? 22. Establish that 35 ¼ 18 30 is a true proportion using two methods, one of which is not cross multiplication. 23. Use PowerPoint to explore dilation and expansion as described in Sect. 6.3.1. What problems did you encounter and how would you address them with a group of middle school students?

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24. What is the effect of applying a scaling factor greater than 1? What is the effect of applying a scaling factor less than 1? 25. Identify the coordinates of three similar rectangles which could be generated with vertices aligned with a line of slope 0.4 (Fig. 6.5). 26. Use Binet’s formula, see Sect. 6.3.3, to find F12.

Chapter 7

Geometry and Measurement

7.1

Introduction

Until recently, it was common for instruction in geometry to be delayed until the secondary school course bearing its name. Continuing with historical precedent, geometry was viewed from a formal axioms and proofs standpoint. Not surprisingly, when geometry was finally addressed at the secondary school level it centered around formal proofs—in a seeming effort as much to teach the student to think “like a mathematician” from a formal logical stance as to teach actual geometric content. This made sense to some degree. Since the prevailing view is that young children cannot operate in the arena of formal logic, pushing geometry back as long as possible seemed reasonable. Geometry, to the extent it was addressed in an K-9 curriculum, was often limited to a memory/recall process of the names of geometric figures. In rare cases a few properties such as parallel, perpendicular, and similar were addressed. Today, in addition to geometry being important as a subject in and of itself, a basic geometric understanding is invaluable to developing both mathematical thinking and modeling perspective, see Chap. 2. Many of the problems facing middle school learners, however, are best modeled with visual representations (sketches, pictures, screenshots, etc.). These representations in turn rest upon geometric understandings—which in a true chicken and egg fashion—were not adequately developed until high school. At the very least this is detrimental to many geometric concepts which could be developed in younger students. For example, middle school students routinely use geometry in other areas of mathematics, modeling multiplication

© Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_7

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using rectangular arrays would be an easily identifiable case (Chap. 2, Sect. 2.1). When viewing geometry as a systematic exploration of relationships occurring among shapes and within space, however, it is suitable for a broad variety of creative middle school explorations including both formal and experiential approaches. The chapter begins by describing the van Hiele levels, together with the student thinking and objects of thought characterized by each level. Following this van Hiele overview, developing geometric vocabulary via a geometry scavenger hunt is described and suggested terms are provided. The Problem Solving Board, introduced in Chap. 2, is then modeled as a bridge to deductive reasoning. Following this, common geometric actions such as transformations, rotations, translations, and reflections are described together with their cartesian coordinate versions. Tessellations are then explored leading to an equation 1k þ m1 þ 1n ¼ 12 which defines all integer values n, m and k for which edge to edge tessellations may be made with regular polygons of n, m and k sides. Methods fitting within van Hiele levels suitable for middle school are then presented for development of the Pythagorean theorem, Pythagorean triples, and distance formula. The development of key vocabulary for triangles and quadrilaterals is then presented together with suggested terms and measurement formulas for the circumference and area of a circle are developed. A variety of creative methods to calculate p are then presented which include rotation of a circular disk, inscribed and circumscribed regular polygons, squeezing in through counting, and use of Pick’s theorem. The chapter concludes by revisiting van Hiele in light of middle school content.

7.2

Geometry

Geometry can be thought of as the systematic exploration of relationships that occur between shapes and within space. This exploration can either be highly formal and deductive, as alluded to in the introduction, or informal and experiential and yet still be successful in developing geometric concepts. In either case, however, students in middle school should approach geometry with at least a passing understanding of basic terms relating to points, lines, planes and shapes. These will serve as the classroom commonplace within which geometry will be explored (Chap. 3, Sect. 3.3.1). As students explore geometric relationships, they should have the opportunity to measure, identify, classify, visualize and/or transform the geometric objects in order to connect the basic terms of the commonplace to personally meaningful experiences (Chap. 3, Sect. 3.3.4).

7.2 Geometry

7.2.1

185

Geometric Thinking and Van Hiele Levels

cBig Idea The five-level van Hiele model of geometric learning, a big idea from mathematics education, has significant impact on geometry curriculum. By providing a heirchy of development, these levels serve as an invaluable guide for geometry for middle school students.b One of the more influential approaches towards geometry instruction derives from the van Hiele model of geometric learning presented by in 1957 by the Dutch educators P. M. van Hiele and Dina van Hiele-Geldof. This model has been the subject of significant research and has influenced geometry curriculum worldwide. The van Hiele model at its simplest formulation is comprised of five levels which reflect different types of thinking concerning geometry. It should be noted that although all five levels will be described in this section, it would be rare for middle school students to be at either Level 3 or Level 4. These five van Hiele levels are not age-dependent, but, instead, are related more to the experiences students have had. Furthermore, unlike a strictly Piagetian approach, these levels are not as tightly linked to developmental issues. Rather, students can benefit from appropriate instructional experiences regardless of their current development status. The levels are, however, sequential and according to this theory students must pass through the levels in order, as their understanding increases. It is not possible, therefore, to be functioning at level four (the van Hiele levels begin their numeration with 0) without having experienced levels 0 through 3 in order. The descriptions of the levels are typically in terms of “students”, however, “learners” could be considered a better description (Van de Walle et al., 2019). The five levels comprising the van Hiele model establish a hierarchy of geometric conceptualization with each level describing a more abstracted representational set of concepts and processes. As students progress, the levels serve to describe the types of geometric ideas the students think about (referred to as “objects of thought”) and what the student is able to do (referred to as the “products of thought”). This description fits extremely well within the “noun-like” and “verb-like” characterization of mathematics developed in Chapter 3, as well as the Action on Object instructional approach (Connell et al., 2018; Davis, 1990; Connell, 2001; Connell & Abramovich, 2017).

7.2.1.1

Level 0—Visualization

Students functioning at this level are able to recognize shapes by their global, holistic appearance. Students at Level 0 think about shapes in terms of what they resemble and are able to sort shapes into groups that “seem to be alike.” For

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example, a student at this level might describe a triangle as an “arrow head.” The student, however, might not recognize the same triangle if it is rotated so that it “stands on its point.” A middle school student functioning at Level 0 would be able to name common geometric figures such as parallel lines, triangles, intersecting lines, etc. Students at this level would also be able to compare these identified figures based upon their appearances. This is independent of any deeper properties these figures might possess. At Level 0 the “objects of thought” would consist of the students’ experience with the objects via the senses. Although this is generally visual in nature, students with visual disability will experience these objects using alternate senses—touch, for example. The “products of thought” are limited to those based upon this sensory perception. Thus, sorting based upon a recognizable feature or naming based upon appearances would both be possible at Level 0. An example of this ability would be a student who sorts triangles and squares into two groups because “they look different”. Such students would also be able to determine angular measurement and distances as these can be determined using a “what does it look like” approach coupled with an appropriate measuring instrument—for these cases, a protractor or ruler. In working with students functioning at Level 0, a good question to ask would be, “What does it look like?”.

7.2.1.2

Level 1—Analysis

Students functioning at Level 1 are able to observe the component parts of figures (e.g., a parallelogram has opposite sides that are parallel) but are unable to explain the relationships between properties within a shape or among shapes. They are, however, able to analyze figures in terms of their components and relationships among these components. Having passed through Level 0, they would also be able to discover properties through hands-on folding, measuring, or reference to a diagram. For example, since middle school students functioning at Level 1 are able to discover properties based upon hands-on experience it would be possible to learn symmetric relations using paper folding, see Fig. 7.1. In this diagram an initial shape is shown to have four unique lines of symmetry by successive folds (shown by dotted lines). At Level 1 the “objects of thought” consist of classes of figures. For example, a set of squares, circles, or rectangles would be recognizable as belonging to the same class due to their observed properties. The “products of thought” include recognizing the properties held by these objects as being characteristic of the class of objects to which they belong. Students at Level 1 are capable of identifying and describing the properties of shapes. Thus, they are able to understand that all shapes in a group such as parallelograms have the same properties, and they can describe those properties. They are also able to consistently separate shapes into groups and describe the properties

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187

Fig. 7.1 Discovering lines of symmetry by paper folding

of given groups. For example, a rectangle would be recognized as having four sides with opposite sides congruent, four right angles, opposite sides being parallel, etc. A square would be recognized as belonging to a different group due to the additional property of having all four sides congruent. Two good questions to ask students functioning at Level 1 are, “What are the properties?” and “What does it look like?”. This latter question, “What does it look like?”, also occurs in the Problem Solving Board described in Chap. 2, Sect. 2.4. Its use with students functioning at Level 1 can help them link the properties of geometric objects and the mathematical models which those shapes can engender. 7.2.1.3

Level 2—Informal Deduction (Relationships)

Students functioning at Level 2 are able to deduce properties of figures and express interrelationships both within and between figures. In addition, these students are able to notice relationships between properties and understand informal deductive discussions about shapes and their properties. It is at this point that students begin to understand the relationships between shapes and their different characteristics. Because the students are able to see relationships between object properties and rules, they are able to create and understand informal deductive discussion concerning these shapes and their properties. As part of this emerging ability to create and follow informal deduction, students functioning at Level 2 are able to utilize if–then reasoning and can classify shapes based upon a minimal set of defining characteristics. For example, arguments like that illustrated in Fig. 7.2 can be followed. At Level 2 the “objects of thought” consist of logical relationships between object properties. For example, a set of quadrilaterals can be classified on the basis of a minimal set of defining characteristics. The “products of thought” at Level 2 include logical arguments about the properties together with informal intuitive “proofs”. Two good questions to ask students functioning at Level 2 would be, “Why?” and “What would happen if…?”. 7.2.1.4

Level 3—Formal Deduction

Middle school students are unlikely to be functioning at either Level 3 or Level 4. These descriptions and examples are included for completeness to provide an overview of the van Hiele levels.

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Fig. 7.2 Classification based upon informal logic

Students functioning at Level 3 are able to create formal deductive proofs at the level typically associated with secondary mathematics. In such deductive proofs, students begin with certain premises that are known or assumed to be true. From this beginning, students extend that set of facts based upon logical principles (Chap. 3, Fig. 3.8 for a typical geometric proof and Chap. 3, Sect. 3.6 for a review of the basics of propositional logic). For students functioning at Level 3, the “objects of thought” consist of relationships between properties of shapes and the relationships between axioms, definitions, theorems, corollaries, and postulates. Students functioning at Level 3 are able to work with abstract statements about geometric properties and make conclusions based more on logic than intuition. In addition, Level 3 students can create formal deductive proofs and understand axioms to solve problems. Furthermore, they are able to think about relationships between properties of shapes and distinguish between axioms, definitions, theorems, corollaries, and postulates. In doing so, Level 3 students are able to utilize geometric abstracts and make conclusions based on logic. A good question to ask a student functioning at Level 3 would be, “What can you prove?”.

7.2.1.5

Level 4—Rigor

Students functioning at Level 4 are able to rigorously compare different axiomatic systems. For example, they might recognize that for a given proof it might be advantageous to adopt an approach drawing upon analytic geometry, spherical geometry, or hyperbolic geometry. Such students at Level 4 are thus able to think about, select, and compare different deductive axiomatic systems of geometry and the impact each system has upon the objects of geometry. It is not expected that middle school students will be able to function at Level 4, as this is the level that college mathematics majors might use when thinking about geometry. A few good questions to ask a Level 4 student could be, “How did you prove it?”, “Which system of proof did you use?”, and “Why did you select this system?”.

7.2 Geometry

7.2.1.6

189

Summary

In general, most elementary school students will be functioning at van Hiele Level 0 or Level 1 while middle school students could possibly be at level 2. It is highly doubtful that a middle school student will be able to function at Level 3 or Level 4. This is an extremely important instructional consideration. According to van Hiele, many of the failures in teaching geometry result from teachers using the language of a higher level than is understood by the student (van Hiele, 1986). Although not a stage model, such as Piaget, there are phases which students pass through when advancing from level to level. These phases are viewed as taking place within each level; thus a similar instructional pattern can be adopted. The first phase is that of information, and consists of developing the commonplace vocabulary, examples, and non-examples (Chapter 3, Sect. 3.3.1). Following this, instructors can serve as a guide via guided orientation where students act upon the current “objects of thought” to produce “products of thought”. These actions follow a more Piagetian flavor and need to take place within the currently held level. A Level 1 student, for example, could be expected to understand folding (Fig. 7.1), measuring, and visual comparison. Further understandings are developed through explicitation where students try to make explicit in words their understanding of the relations they are able to experience. This is an essential phase as it develops the formal and informal language necessary to create, solve, and discuss geometric concepts. When explicitation occurs, students will have sufficient vocabulary to address the “What should we do?” prompt of the Problem Solving Board (Chap. 2, Fig. 2.3.5). Although not always recognized in instruction, students should be given the opportunity to create personally meaningful problems using their emerging understandings. This phase, referred to in the van Hiele levels as free orientation, is essential for students to connect their current understandings to more complex concepts and activities. The final phase is that of integration, where students summarize all that they currently understand and reflect on their actions with the goal of gaining a metacognitive awareness of the relationships developed thus far.

7.2.2

Developing Geometric Vocabulary

cBig Idea The development of mathematical vocabulary, a big idea from mathematics education, is crucial to effectively utilize the van Hiele levels in instruction. Students must be able to identify and use appropriate geometry terminology regardless of their van Hiele level.b A major component to geometric thinking lies in the ability to accurately identify and use appropriate terminology. This is present regardless of which van Hiele level a student might currently be functioning within. Such development is an important

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part of establishing the classroom commonplace and is explicitly mentioned as the van Hiele phase of information. Since many of the terms used in geometry require precise meanings that must be understood, developing this terminology can be a challenge to many middle school students. Since the majority of middle school students are functioning at least a van Hiele level of 1 or 2, however, activities that take advantage of shape recognition based upon appearances can be highly effective. One such exercise in this process is a geometry scavenger hunt augmented by technology (Connell et al., 2018). PowerPoint is a natural tool to share the results of a geometry scavenger hunt and allows the students to practice taking, editing, and selecting digital photographs representing these ideas. It must be noted, however, that these found representations are but poor approximations in several important instances. For example, the infinite extensibility of rays, lines and planes can only be suggested and not shown. For these cases these found visual representations should be expanded upon with supporting discussion and text. In finding examples, to help students recognize geometric shapes in a variety of contexts encourage students to find at two examples-one naturally occurring and one man-made-for each item from a list drawn from the grade level curriculum. Students then create a PowerPoint where each item comprises a single slide. Depending on their grade level, for each item they might: (1) give their definition, (2) describe their find, (3) provide a photograph(s), (4) identify why their example (s) should be considered accurate, and (5) show what a “textbook” example would look like. A student example of such a slide for the term “Hexagon” is shown in Fig. 7.3. The ability to quickly share, comment upon, and edit their finds allows for student ideas to be expanded to include a range of potential contexts. This information phase adds to both the vocabulary of the classroom commonplace as well as establishing a network of identifiable shapes and shape properties. For vocabulary developed using the scavenger hunt approach, although the final product is a static representation, the process leading to its creation is a highly interactive and meaningful activity. A suggested minimal set of terms to include in a middle school scavenger hunt is shown in Table 7.1. In addition to these terms, several key concepts can also be developed in this activity including parallel lines cut by a perpendicular transversal and parallel lines cut by a non-perpendicular transversal.

7.2.3

Deductive Proofs

The link between deductive proofs and the approach illustrated in the Problem Solving Board, Fig. 2.36 in Chap. 2, should be noted. This should not be surprising, given the Problem Solving Board’s use for helping students develop logical reasoning based upon models drawn from geometric principles. As such, the

7.2 Geometry

191

Fig. 7.3 Geometry scavenger hunt

Table 7.1 Suggested terms for a geometry scavenger hunt Point

Line

Plane

Space

Line

Line segment Obtuse angle

Ray Straight angle Midpoint Complementary angles Pythagorean Triple Trapezoid Regular polygon

Acute angle Perpendicular lines Triangle Scalene triangle Rhombus

Right angle Parallel lines

Transversal Acute triangle

Angle Intersecting lines Tangent Supplementary angles Quadrilateral

Right triangle Equilateral triangle Rectangle

Pentagon Ellipse

Hexagon Oval

Polygon Circle

Radius (circle)

Central angle

Parabola

Diagonal (circle) Cube

Pyramid

Prism

Face Fractal

Vertex Similar shapes

Cone Congruent shapes

Sphere

Isosceles Triangle Square Equilateral polygon Chord Inscribed angle Base Cylinder

Arc

Problem Solving Board serves to support Level 3 type thinking for students currently functioning at Level 2 or Level 1 and provides a powerful teaching tool for use within the middle school classroom.

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cBig Idea The Problem Solving Board, a big idea from mathematics education, can serve to scaffold emerging deductive proofs. By providing both a cognitive framework and record of action it can be used to mediate student understandings toward a more formal level.b For example, traditional geometric proofs typically begin with a figure illustrating what is to be proven. Often this figure is provided to the students, although they may, on occasion, have to construct this for themselves based upon a problem situation. Once the problem is adequately represented students identify the given statements (“knowns”) and the goal of the proof (“unknowns” or “to prove”). This step is identical to the “What do we know?” and “What does it look like?” questions of the Problem Solving Board. These questions serve to prompt students in their representation and planning. Within the Problem Solving Board, this includes identifying and labeling unknowns, condition(s), and data (“knowns”), determining if the problem is soluble, and initial planning of the actions arguments and justifications to be developed during the course of the problem or proof. Since students represent the problem, often using geometrically based mathematical models, in the larger “What does it look like?” the link to geometry is explicit provided the underlying model is geometrically based. Since these models can be based upon Level 1 and Level 2 experiences, the representation they create provide a connection to Level 3 development. For a geometric proof, the question “What should we do?” requires the creation of a plan based upon the geometric figure representational model and understanding generated in the first two prompts. With this plan in place, students would then complete the proof, equivalent to the “Solve it!” prompt. Since the proof is based upon deductive principles, the “Check it!” prompt can be completed through this process. To see how this works in actual practice, consider the student example shown in Fig. 7.4. The student is tasked with proving that \1 ¼ \8. Prior to starting, the student began with the following premises being known or assumed to be true based upon prior proofs and axioms: (a) corresponding angles in parallel lines have equivalent measures; (b) vertically opposite angles in parallel lines have equivalent measures; and (c) the transitive property (if a ¼ b and b ¼ c then a ¼ c). In addition, they were “given” that AB k CD.

7.2.4

Transformations

In a geometric transformation a preimage is transformed to create a different image. As this transformation is undertaken, there are two major categories: rigid and non-rigid. In a rigid transformation the preimage is moved but neither size or shape are altered. As non-rigid transformations are performed the size of the preimage

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Fig. 7.4 Student use of the Problem Solving Board in a geometric proof Fig. 7.5 A slide or translational transformation

will change but not the shape. There are three types of rigid transformations: translation, rotation, and reflection. The most basic of the rigid transformation, translation, is accomplished by moving the preimage through space without changing its size, shape, or orientation. This is similar to sliding a figure across a table top, see Fig. 7.5, and is occasionally referred to as a slide-transformation. When performing a translation on a more traditional geometric object, a triangle for example, it is helpful to utilize a coordinate plane. This makes the translation much easier for the students to visualize and carry out. Since in a translation the figure is being moved in the ðx; yÞ-plane it can be easily described by the mapping ðx; yÞ ! ðx  a; y  bÞ. For a triangle example, the translation can be easily performed by mapping and connecting each of the vertices of the triangle using the

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Fig. 7.6 A translation in the ðx; yÞ-plane

translational mapping. In Fig. 7.6, a triangle with original vertices of ð1; 3Þ; ð3; 2Þ, and ð1; 4Þ is translated −2 along the x-axis and +1 on the y-axis. The vertices of the translated triangle then become ð1; 3Þ ! ð1  2; 3 þ 1Þ ¼ ð3; 4Þ; ð3; 2Þ ! ð3  2; 2 þ 1Þ ! ð1; 1Þ; and ð1; 4Þ ! ð1  2; 4 þ 1Þ ¼ ð1; 5Þ. A rotational transformation occurs when the starting preimage is rotated about a fixed point without changing its size or shape. As Fig. 7.7 shows, the fixed point can be in, on, or external to the preimage. Coordinate-based rotations can be more difficult for middle school students, however there are some special cases which are easily within their grasp. To rotate a primage 90 with the center of rotation at the origin, the mapping is ðx; yÞ ! ðy; xÞ. This is shown in Fig. 7.8, using a triangle rotated 90 : Another common rotation, 180 , is shown in Fig. 7.9 using the same triangle. When this is done, it gives rise to the mapping for 180 as ðx; yÞ ! ðx; yÞ. In a like fashion, the mappings for 270 can be shown to be ðx; yÞ ! ðy; xÞ, Fig. 7.10. In Fig. 7.11 each of these rotational mappings are applied to our original triangle with vertices of ð1; 3Þ; ð3; 2Þ, and ð1; 4Þ. As was done with the translation example, this mapping can be applied to the vertices of any figure and then connecting the new vertices thus drawing the desired rotated image. The remaining rigid transformation of reflection, sometimes referred to as a flip-transformation, occurs when the preimage is reflected across a line as shown in Fig. 7.12. In a reflection the size and shape of the preimage does not change. Reflection transformations are also easily within the grasp of most middle school students. To reflect a primage around the y-axis the mapping is ðx; yÞ ! ðx; yÞ, to reflect a preimage around the x-axis use ðx; yÞ ! ðx; yÞ, see Fig. 7.13.

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Fig. 7.7 Rotational transformations Fig. 7.8 Results of a 90 rotation in the ðx; yÞ-plane

There is only one type of non-rigid transformation, that of dilation/expansion. Dilation or expansion refers to changes in the size of a preimage without changing its shape or orientation. As was shown in Chap. 6, scaling is an example of a dilation/expansion transformation with the ratio preimage image being the scaling factor. It is not necessary to know the scaling factor in order for a dilation/expansion to occur.

196 Fig. 7.9 Results of a 180 rotation in the ðx; yÞ-plane

Fig. 7.10 Results of a 270 rotation in the ðx; yÞ-plane

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Fig. 7.11 Rotational transformations in the ð x; yÞ-plane Fig. 7.12 Reflection or flip transformations

However, it is helpful for students to know that scaling factor values of preimage image [ 1 results in an expansion, while scaling factor values of preimage image \1 results in a dilation. As shown in Fig. 7.14, in a dilation the image is smaller than the preimage while in an expansion the image is larger than the preimage. Once students are accustomed to using the ðx; yÞ coordinate system, dilations and expansions of polygons can be easily performed by multiplying the ðx; yÞ coordinates of each vertex by the scaling factor and then connecting the transformed vertices. This is illustrated in Fig. 7.15 which shows the impact a scaling factor of 2 and 12 respectively has upon a quadrilateral with vertices at ð6; 6Þ; ð8; 6Þ; ð8; 4Þ, and ð4; 4Þ.

7.2.5

Tessellations

A tessellation is a tiling of the plane using one or more plane shapes in a repeating pattern with no gaps or overlaps. Being based upon a circle, if the sum of the angle

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Fig. 7.13 Reflection or flip transformations across the y-axis

Fig. 7.14 Dilation/expansion transformations

measures at each vertex is equal to 360 , the shapes will fit together with no overlaps or gaps. In Fig. 7.16, a fragment of a tessellation is shown with angles A, B, and C. Since this is a tessellation it must be that \A þ \B þ \C ¼ 360 . If only a single shape is used and only congruent convex regular polygons are considered this requires that there is only a small group which can tessellate a plane by themselves. This is referred to as a regular tessellation. Since the sum of the angle measures at each vertex of a tessellation must equal 360 , in a regular tessellation,1 each interior angle of the regular polygon must be a divisor of 360 . Since the sum of the interior angles of a polygon with n sides is 180 ðn  2Þ, this  requires that each interior angle measurement be 180 ðnn2Þ. It also requires that the

1

In the discussion which follows, a polygon with n sides will be referred to as a n-gon.

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Fig. 7.15 Dilation/expansion transformations in the ðx; yÞ-plane

Fig. 7.16 The sum of angles at each tessellation point must equal 360



number k of n-gons meeting at each vertex is such that ð180 ðnn2ÞÞ  k ¼ 360 , so
 ðn2Þ  k ¼ 2 ! 1  2n ¼ 2k ! 1 ¼ 2n þ 2k , and finally 12 ¼ 1n þ 1k (Chap. 4, n Sect. 4.5.3). Since both n and k must be positive integers and the smallest regular polygon is an equilateral triangle it can be shown that 12 ¼ 13 þ 16 meets the requirement, which can be interpreted as six equilateral triangles and is shown in the left diagram of Fig. 7.17. Likewise, 12 ¼ 16 þ 13 which can be interpreted as three

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regular hexagons and is shown in the right diagram of Fig. 7.17. Finally, 12 ¼ 14 þ 14 which can be interpreted as four squares as shown in the center diagram of Fig. 7.17. Higher values of n do not lead to a value of k meeting the requirement that 12 ¼ 1n þ 1k , so the only regular polygons capable of tessellating a plane are those shown in Fig. 7.17. If different regular polygons are used, other options are possible. For example, a semi-regular tessellation is shown in Fig. 7.18. A semi-regular tessellation is a tessellation with different regular polygons in which the arrangement of polygons at every vertex is identical. In Fig. 7.18, each vertex consists of a square, a regular hexagon, and regular dodecagon which share full sides and cover completely the space around the vertex point they have in common. Since each interior angle of a square is 90 , each interior angle of a regular hexagon is 120 , and each interior angle of a regular dodecagon is 150 , the sum of the angles at each vertex will be 90 þ 120 þ 150 ¼ 360 . This meets the requirement for the sum of the angle measures at each vertex to equal 360 and guaranteeing the shapes will fit together with no overlaps or gaps. These are not the only type of tessellations possible, Fig. 7.19, shows an edge-to-edge tessellation consisting of an equilateral triangle, a regular decagon, and a regular pentadecagon in which adjacent polygons share full sides having the same endpoint. Since each interior angle of an equilateral triangle is 60 , each interior angle of a regular decagon is 144 , and each interior angle of a regular pentadecagon is 156 , the sum of the angles at each vertex will be 60 þ 144 þ 156 ¼ 360 . This meets the requirement for the sum of the angle measures at each vertex to equal 360 and guaranteeing the shapes will fit together with no overlaps or gaps. For the remainder of this section, only edge-to-edge tessellations with regular polygons will be considered. Taking another look at Fig. 7.16, it can be interpreted as a fragment of an edge-to-edge tessellation with three regular polygons (k-gon, mgon, and n-gon) the angles of which are Ak, Am, and An, respectively. So, a reasonable question to ask would be what are the possible values of k, m, and n? As might be expected given the discussion thus far, the application of some algebra will be helpful in addressing this question. As was mentioned earlier, the sum of the interior angles of a polygon with   n sides is 180 ðn  2Þ, this means that \Ak ¼ 180 ðkk2Þ ; \Am ¼ 180 ðmm2Þ ; 

\An ¼ 180 ðnn2Þ. So, in order for a tessellation to occur the values of k, m, and n must meet the    m2 n2 requirement that 180 ðkk2Þ þ 180 ðmm2Þ þ 180 ðnn2Þ ¼ 360 whence k2 k þ m þ n ¼ 1  2 or 3  2 k þ m1 þ 1n ¼ 2 which can be simplified to the form 1k þ m1 þ 1n ¼ 12. The equation 1k þ m1 þ 1n ¼ 12, in a like fashion serves to define all triples of regular k-gons, m-gons, and n-gons which would allow for edge-to-edge tessellations to occur. The parallelism of this argument and the resultant equation to the case of regular tessellations should be apparent.

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Fig. 7.17 Regular tessellations

Fig. 7.18 A semi-regular tessellation

Fig. 7.19 Edge-to-edge tessellation

1 k

When k, m, and n are different, the following six solutions to the equation þ m1 þ 1n ¼ 12 can be established (Chap. 4, Sect. 4.5.3): 1 1 1 1 ¼ þ þ ; 2 4 5 20 1 1 1 1 ¼ þ þ ; 2 3 9 18

1 1 1 1 ¼ þ þ ; 2 4 6 12 1 1 1 1 ¼ þ þ ; 2 3 8 24

1 1 1 1 ¼ þ þ 2 3 10 15 1 1 1 1 ¼ þ þ : 2 3 7 42

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1 So, consider the first equality 12 ¼ 14 þ 15 þ 20 . In this example, 4, 5, and 20 each refer to the number of polygon sides earlier referred to as k-gon, m-gon, and n-gon. So, this tells us that a square, pentagon, and an icosagon will allow for an edge-to-edge tessellation, see Fig. 7.20. In much the same way, each of the other solution sets for k, m, and n can be shown to enable edge-to-edge tessellation in terms of an edge-to-edge tessellation with the three different regular polygons. Table 7.2 was constructed using the formula for the sum of interior angles of a polygon with n sides as being 180 ðn  2Þ. These values for the sum of the interior angles and of a single interior angle provide a helpful reference when checking as whether a set of figures will tessellate or not. 1 shown above, the following likewise In addition to the case, 12 ¼ 14 þ 15 þ 20 provide edge-to-edge tessellations being solutions to 1k þ m1 þ 1n ¼ 12: 1 ¼ 14 þ 16 þ 12 , 90 þ 120 þ 150 ¼ 360 (edge-to-edge tessellation with square, hexagon, and dodecagon, see Fig. 7.21); 1 1 1 1     2 ¼ 3 þ 10 þ 15 , 60 þ 144 þ 156 ¼ 360 (edge-to-edge tessellation with triangle, decagon, and pentadecagon, see Fig. 7.22); 1 1 1 1     2 ¼ 3 þ 9 þ 18 , 60 þ 140 þ 160 ¼ 360 (edge-to-edge tessellation with triangle, nonagon, and octadecagon, see Fig. 7.23); 1 1 1 1     2 ¼ 3 þ 8 þ 24 , 60 þ 135 þ 165 ¼ 360 (edge-to-edge tessellation with triangle, octagon, and icositetragon, see Fig. 7.24); and 1 1 1 1     2 ¼ 3 þ 7 þ 42 , 60 þ 128:57 þ 171:43 ¼ 360 (edge-to-edge tessellation with triangle, heptagon, and tetracontadigon, see Fig. 7.25). 1 2

7.2.6

Parallel Lines

As most middle school students will be functioning at van Hiele levels 2 or lower, it is highly important when exploring parallel lines and developing their related concepts to use language and activities appropriate to these levels. This is not the time to enforce mathematical deduction, but rather to focus on the many relationships which may be observed and described. These relationships will provide ample opportunity for inductive reasoning to be applied (Level 2), linking the many angles on the basis of their properties (Level 1), and deepening the student awareness of typical exemplars of parallelism and what they look like in a broad variety of contexts (Level 0). In exploring parallel lines, it is helpful to generate some working vocabulary describing the parallel lines themselves, the lines which intersect the parallel lines (transversals), and the angles which are formed as a result of the parallel lines being

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Fig. 7.20 Edge-to-edge tessellation with square, pentagon, and icosagon

Table 7.2 Sum of interior angles of selected regular polygons and size of one interior angle

Fig. 7.21 Edge-to-edge tessellation with square, hexagon, and dodecagon

n

S(n) = 180°(n−2)

S(n)/n = 180°(n−2)/n

3 4 5 6 7 8 9 10 12 15 18 20 24 42

180 360 540 720 900 1080 1260 1440 1800 2340 2880 3240 3960 7200

60 90 108 120 128:57 135 140 144 150 156 160 162 165 171:43

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Fig. 7.22 Edge-to-edge tessellation with triangle, decagon, and pentadecagon

Fig. 7.23 Edge-to-edge tessellation with triangle, nonagon, and octadecagon

cut by the transversal. A suggested set of working vocabulary, much of which can be developed using an informal scavenger hunt (Sect. 7.2.2), is shown in Table 7.3 and illustrated in Fig. 7.26. Once this working vocabulary is formed and is part of the classroom commonplace, it is possible for an understanding of angles to be developed based upon their observed properties and the application of inductive reasoning. In addition to illustrating parallel line vocabulary, Fig. 7.26 shows the angles of each type that would have equal measures. So, in addition to the scaffolded argument appropriate for students at van Hiele Level 2 presented in Sect. 7.2.3, students functioning at van Hiele Level 1 would recognize \1 ¼ \8 because they are alternate exterior angles.

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Fig. 7.24 Edge-to-edge tessellation with triangle, octagon, and icositetragon

Fig. 7.25 Edge-to-edge tessellation with triangle, heptagon, and tetracontadigon

Table 7.3 Suggested vocabulary for parallel lines Parallel lines

Transversal

Intersecting lines

Vertical angles

Corresponding angles

Alternate interior angles

Alternate exterior angles

Consecutive interior angles

7.2.7

Pythagorean Theorem, Pythagorean Triples, and the Distance Formula

Not only is the Pythagorean theorem one of the more useful relationships in geometry, it is also one of the most famous. In some respects, one of the challenging aspects when working with middle school students is to develop a powerful enough “proof” so that the power of the relation a2 þ b2 ¼ c2 can be confidently applied while still being within the non-deductive scope of students functioning at van Hiele Level 2. Such is possible, however, by expanding upon the Level 0

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Fig. 7.26 Parallel line vocabulary examples

ability to see “what it looks like” while coupling this with the ability to discover properties based upon hands on experiences from Level 1. This approach is readily applicable for use with middle school students and draws on an approach in geometry often referred to as “proofs without words” (Nelson, 1993; Maor & Jost, 2017). Such an approach builds upon Level 0 and Level 1 abilities to provide a visual framework within which a strongly plausible argument can be developed without the necessity or relying upon formal deduction. In Fig. 7.27, one such “proof” is offered. This particular approach is particularly well known and has been credited to Chinese scholars circa 200 B.C. (Nelson, 1993). A few clues to understanding this proof is to realize that the squares labeled “First” and “Second” are of the same area and the each of the eight triangles in these two squares are congruent. Graphics such as this provides an accessible starting point for plausible discussion can be held with middle school students. Although not proven, this discussion builds confidence in the theorem’s correctness far beyond just memorizing a2 þ b2 ¼ c2 . The payoff can be seen in problems such at the rotating squares activity, see Chap. 3, Sect. 3.4. Revisiting this activity, it can be seen that several students had begun the process of creating an alternative “proof without words” in their exploration, see Fig. 7.28. Closely related to the Pythagorean theorem is the concept of Pythagorean triples. A Pythagorean triple is any set of three positive integers which satisfy the Pythagorean theorem with one of the most common being 3, 4 and 5. It is relatively easy to show that the number of such triples is infinite. Consider the ð3; 4; 5Þ triangle shown in Fig. 7.20 as it undergoes a series of expansions. Each of these results in a similar right triangle whose sides must fit the Pythagorean theorem and are a multiple of the original triangle. So, if ð3; 4; 5Þ is a right triangle it follows for any arbitrary positive integer scaling factor p that ð3p; 4p; 5pÞ is a similar right triangle whose side lengths must satisfy the Pythagorean theorem. The argument in Fig. 7.29, while not a proof, leads to a plausibly reasonable discussion grounding this assertion.

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Fig. 7.27 A “proof without words” of the Pythagorean theorem

Fig. 7.28 The start of a “proof without words” from the rotating squares problem

Once the Pythagorean theorem is established, the distance formula is easily developed as shown in Fig. 7.30. Suppose you wish to find the distance between any two arbitrary points, Aðx1 ;y1 Þ and Bðx2 ;y2 Þ in the Cartesian plane. First, plot A and B and construct horizontal and vertical line segments to meet at a 90 angle. Once this is done, connect A and B. When this is done, the result will be a right triangle with sides of length jx2  x1 j and jy2  y1 j. Since this is a right triangle, the Pythagorean theorem applies, and so the distance from A to B, or the hypotenuse is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx2  x1 Þ2 þ ðy2  y1 Þ2 : Distance ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð x2  x1 Þ 2 þ ð y2  y1 Þ 2

ð7:1Þ

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Fig. 7.29 An argument for infinitude of Pythagorean triples

Fig. 7.30 Developing the distance formula

7.2.8

Triangles and Introductory Trigonometry

Middle school students’ experience with triangles is likely to be limited to categorizing and naming based upon observable characteristics. This is quite appropriate as these students are most likely to be functioning at van Hiele levels 2 or lower. So, just as was the case for parallel lines, it is important when extending triangle properties beyond this starting point to use language and activities appropriate to these van Hiele levels. Focusing on the observable relationships which may be observed can scaffold inductive reasoning (Level 2), extending earlier identified triangle categories to include congruence and similarity on the basis of their properties (Level 1), and deepening awareness exemplars of these newly observed properties (Level 0).

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As students extend their understanding of triangles, some working vocabulary should be developed. This should include a review of the basic triangle classes: right, equilateral, isosceles, scalene, acute and obtuse. This is also a good time to expand student vocabulary to include terms which will play a later role in the development of trigonometry: opposite, adjacent, leg, and hypotenuse. A suggested set of working vocabulary, much of which can be developed using an informal scavenger hunt (Sect. 7.2.2), is shown in Table 7.4 and illustrated in Fig. 7.31. Once this vocabulary has become an established part of the classroom commonplace, additional properties should be introduced and explored. Primary among these include the Pythagorean theorem as it applies to right triangles, congruence relations, and similarity requirements. At this level, students should recognize that two triangles are congruent if all corresponding sides and interior angles have equal measures. An easy way for Level 1 students to categorize this would be to consider these triangles as having the same size and appearance. Students at this level should be able to recognize that two equilateral triangles can be shown to be checked for congruence by measuring one of their sides. Likewise, similar triangles can be categorized as having the same shape but of differing sizes. At this point, it should also be possible for Level 1 students to describe similar triangles as being the product of either a dilation or expansion (Figs. 7.14 and 7.32).

7.2.9

Quadrilaterals

Classifying quadrilaterals on the basis of a minimal set of defining characteristics is an excellent middle school task. As each of the quadrilaterals differ in their definitions only slightly, this minimal classification is both developmentally appropriate and helpful for student use. For this to take place, the working vocabulary in Table 7.5 and illustrated in Fig. 7.33 should be developed. This can be done using visual recognition, possibly through constructing a scavenger hunt as suggested in earlier examples.

7.3

Measurement

cBig Idea Measurement, a big idea from mathematics, involves a comparison between an attribute of an object with either a standard or non-standard unit that shares the attribute being measured. Students should have multiple experience with both non-standard and standard units of measure to fully develop this important concept.b At a basic level, measurement involves a comparison between an attribute of an object or situation with either a standard or non-standard unit that shares this same attribute. In practice, when performing a measurement, it is first necessary to

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Table 7.4 Suggested vocabulary for triangles Triangle

Right triangle

Equilateral triangle

Isosceles triangle

Scalene triangle Congruent triangles Leg

Acute triangle Opposite

Obtuse triangle Adjacent

Similar triangles Hypotenuse

Fig. 7.31 Triangle vocabulary examples

Fig. 7.32 Congruent and similar triangles

Table 7.5 Suggested vocabulary for quadrilaterals

Quadrilateral

Rectangle

Square

Rhombus

Parallelogram

Trapezium (UK)

Trapezoid (US)

Kite

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Fig. 7.33 Quadrilateral vocabulary examples

determine the attribute to be measured. For example, volume would be compared using volume units, time with time units, length with length units, and so on. The objects used in this comparison will share the desired attribute and provide either standard unit or non-standard unit. Length, for example, could be measured in meters—a standard unit, or in shoestring-lengths—a non-standard unit. The selection of the comparison unit, or system of units, is a key part of the measurement process. Once the comparison unit is selected, it is used to compare the known measure of the unit with the unknown attribute to be measured. It is common for an object to have multiple attributes which could be measured. For example, a can of soda has measurements of height, radius, circumference, cross-sectional area, volume, mass, temperature, etc. As a result, one of the initial skills for middle school students to acquire is the determination of which of the potentially measurable features is applicable for a given problem situation and of an applicable system of comparison units. Often the problem situation will aid in this miles determination. If a problem is asking for gallon , for example, the student is provided with a clue that the units to be measures must determine the attributes of length (miles) and liquid volume (gallons). It may be necessary to convert between various standard and non-standard units, complicating the problem situation, but at least the characteristics of basic units have been determined. A major advantage of standardized systems of measurement lies in the availability of tools to make the comparisons easier. A standardized tool for measuring length, such as a meter stick, is much more readily available for student use than a non-standardized tool such as the length of a random shoestring. Rulers, scales, timers, protractors and stopwatches are all measurement tools that middle school students should be familiar with and skilled in using. However, students are often presented with the result of a measurement and never required to make the actual measurement itself. One such middle school student, when asked to actually measure an angle with a protractor [MLC], asked

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Fig. 7.34 Where is the point? A problem with protractors

“Where is the point?” When asked what was meant by this question the student responded, “You know, the point where the angle goes.” After some discussion it was made known that the student had never actually used a protractor and only had experience reading results from pictures and sketches. This question reflected an uncertainty in the principles of angle measurement lead to serious inaccuracies in measurement, as shown in Fig. 7.34, with this student being unable to determine whether the correct measure would be 40 or 45 . Despite the advantages offered by standardized systems of measurement there are several reasons for middle school students to experience non-standard systems. First, it allows for the concept to be developed without the extra baggage of the system of measurement. Area, for example, is often confusing to beginning students due to the need to learn both the concept and units of measurement simultaneously. By learning what area means in terms of “square units” first, students are better able to transfer this understanding to area in terms of “square centimeters”, “square inches”, etc. Secondly, without a standardized system of measurement students must carefully identify the attribute it is that they wish to measure. In doing so, they must find an additional object to use as a comparison that shares this attribute. This process can be extremely helpful in developing measurement conceptually. Finally, such experiences serve to help students recognize the strengths of standardized systems once they are encountered.

7.3.1

Application of Scaling Factors in Measurement

Going beyond basic measurements, problems involving scale drawings and application of scaling factors to measurements are commonly experienced by middle school students. As described in Sect. 7.2.4, a scaling factor can be thought of as a type of non-rigid transformation with the ratio preimage image being the scaling factor. Scaling factor values of

preimage image

[ 1 results in an expansion with the image being

larger than the preimage, while scaling factor values of preimage image \1 results in a dilation with the image being smaller than the preimage. Applying this to actual problems generally requires knowledge of the scaling factor. This topic was initially developed in Sects. 6.3 and 6.3.1 of Chap. 6.

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This scaling factor can be used to set up a proportion with the ratios of the scaled and unscaled values creating the scale. If 1 m in the real world is set to represent 1 cm in the scaled drawing, for example, it is helpful to know that 1 m = 100 cm. 1 cm With this background, the scaling factor would be 100 cm, or as is it is often represented, 1:100. Determining the size of an object in the real world whose scale representation in the scaled drawing is 1.65 cm would involve solving the pro1 ¼ 1:65 portion 100 x resulting in the actual length of the object being 165 cm, or 1.65 m.

7.3.2

Circumference, Area of a Circle, and P

In this section, p will be defined in a classical sense as being the ratio of a circle’s circumference C to its diameter D, or p ¼ DC. This definition, when viewed as a classification attribute by middle school students functioning at van Hiele Level 1, becomes a feature which can be understood to apply to all circles. This definition leads immediately to the formula for circumference of a circle, C ¼ pD. This provides the necessary starting point to generate the formula for the area of a circle in a manner understandable by middle school students. Begin with a circle (since D ¼ 2r, C ¼ 2pr). The first step is to split this circle into sectors of equal area. These sectors can then be arranged into a shape roughly resembling a parallelogram. When this is done, the height of the “parallelogram” is equal to the radius and the two remaining sides would each equal 12 C or 2pr 2 ¼ pr. Since the area of parallelogram is equal to (base) x (height), the area of this “parallelogram” would ¼ ðpr Þðr Þ ¼ pr 2 . This process is illustrated in Fig. 7.35. It should be noted that this level of “what does it look like” argument can be grasped by van Hiele Level 0 students, making this approach highly useful in the middle school. Determining a reasonable value for p that can likewise be understood by middle school students can be approached from a variety of approaches, including the obvious ones of rolling a disk of diameter 1 on a number line and observing the length covered in a single rotation, see Fig. 7.36. This rolling disc approach directly links the value for p to the circumference and diameter, going a long way towards cementing the formula for a circle’s circumference in the minds of middle school students. The accuracy which can be achieved in this manner, however, is obviously quite limited due to the inevitability of measurement errors. A famous approach providing greater accuracy than possible when using a rotating disc, dates back to Archimedes2 of Syracuse (287–212 B.C.). Archimedes 2

Archimedes is widely recognized as one of history’s greatest mathematicians. He is widely known for the method of determining p using inscribed and circumscribed polygons described in this section.

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Fig. 7.35 Generating the formula for the area of a circle

Fig. 7.36 Determining p by rotating a circular disc

recognized that the area of a regular n-gon inscribed within a unit circle3 will be smaller that the area of the circle, the area of the circle will in turn be smaller that the area of a regular n-gon circumscribed about the circle. Furthermore, as the number of sides comprising the polygon, n, approaches infinity, the two polygonal areas will provide a lower and upper bound to the area of the unit circle, i.e. p. This approach was applied using The Geometer’s Sketchpad computer program to generate Fig. 7.37. In this figure, the respective approximations of p achieved when using an octagon, n ¼ 8, a dodecagon, n ¼ 12, and an icositetragon, n ¼ 24 are shown. As n increases, a spreadsheet can be used to continue positioning p between inscribed and circumscribed polygons as shown in Fig. 7.38. In the spreadsheet shown in Fig. 7.38
the area of the inscribed polygon was calculated using the 1

expression 12 n  sin 4 cosn ð0Þ . Since cos1 ð0Þ ¼ p2 this expression is equal to 2p 1 2 n  sin n . An interesting phenomenon can be observed here: with the increase of n, the expression (a product) does not increase but, instead, approaches p—a relatively small number. Apparently, this is due to the behavior of another variable A unit circle is a circle with radius ¼ 1.

3

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215

Fig. 7.37 Using The Geometer’s Sketchpad program to approximate p using inscribed and circumscribed polygons

factor of the product, sin 2p n , the value of which decreases with the increase of n. Indeed, in a right triangle, the smaller one of its angles is, the smaller is the opposite leg along with the leg to hypotenuse ratio—the earliest and probably the most commonly known definition of sine. Thus, in the expression used to calculate the value of p, the increase of n is suppressed by the decrease of sine. Later, in the context of Calculus, this phenomenon will be described in the form of the relation   ¼ p. The area of the circumscribed polygon was calculated by the lim 12 n  sin 2p n n!1   ð2cos1 ð0ÞÞ spreadsheet of Fig. 7.38 using the expression n  tan , which is equal to n p n  tan n and it exhibits a similar phenomenon observed in the case for the inscribed polygons. In the context of Calculus, one can describe the phenomenon in  the form of the relation lim n  tan pn ¼ p. That is, such use of a spreadsheet, by n!1

allowing the value of n to grow as much as one wishes not only yields a wonderful approximation of p but demonstrates its possible uses in the context of Calculus. A fairly creative approach linking p directly to the circle’s area can also be undertaken. Although requiring nothing apart from careful drawing, counting, and Level 0 and Level 1 reasoning, this approach is made significantly easier with the aid of computer paint programs for the drawing and a spreadsheet to record the counting and to calculate the number of unit squares. In this method, students will have the opportunity to squeeze in on the value of p through a series of diagrams. To begin, consider a unit circle. Since the area formula has been established, see Fig. 7.27, the area of this circle is equal to ðpÞð12 Þ ¼ p square units. At this step, however, there is really nothing “countable” yet. This is shown as the 1st Pass in Fig. 7.39. Using the circles radius, a series of four-unit squares are drawn in the 2nd Pass. At this point, it can be argued from “what it

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Fig. 7.38 Using a spreadsheet to approximate p

looks like” that the area of the circle and thus the value of p must be less than 4 square units and greater than 0. Progress begins to be made in the 3rd and 4th Pass as shown in Fig. 7.40. In each successive pass, the length and width of the unit square is divided by two. So, in Pass 3 there will be 4 countable squares for each unit square, and by Pass 4 this number will increase to 16. In making these and all subsequent counts, a square must be completely inside or outside. If the circle passes through the square it is not counted. By counting the number of square inside and outside the circle minimum and maximum values for the area at that pass can be determined. When this is done, the result is that by the 4th Pass the value of p is found to be less than 3.75 square units and greater than 2. Progress continues in the 5th and 6th Pass as shown in Fig. 7.41 with the result that by the 6th Pass the value of p must be less than 3.2969 square units and greater than 2.8125. Progress continues in the 7th and 8th Pass as shown in Fig. 7.42 with the result that by the 8th Pass the value of p must be less than 3.1426 square units and greater

7.3 Measurement

217

Fig. 7.39 Squeezing in on p—part one

Fig. 7.40 Squeezing in on p—part two

than 2.9854. At this point, however, the counting and drawing becomes overwhelming even with the aid of technology. Plotting the results of each successive pass, as shown in Fig. 7.43, shows the minimum and maximum values converging to p.

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Fig. 7.41 Squeezing in on p—part three

Fig. 7.42 Squeezing in on p—part four

7 Geometry and Measurement

7.3 Measurement

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Fig. 7.43 Squeezing in on p—part four

There is yet another approach which is now possible thanks to the lattices generated during the Squeezing in on p approach. Pick’s theorem4 states that given a simple polygon constructed on a lattice of equal-distanced points, the area of the polygon can be determined by A ¼ i þ b2  1, where i is is the number of lattice points in the interior of the polygon and b is the number of lattice points on the boundary placed on the polygon’s perimeter. A ¼ iþ

b 1 2

ð7:2Þ

When Pick’s theorem is applied to the 4th Pass, see Fig. 7.32, careful counting yields values for b ¼ 16 and i ¼ 45. Since the radius of the circle in the 4th Pass is 4 units, p  42 ffi 16 2 þ 45  1 which would result in a predicted value using Pick’s

451Þ Theorem for p ffi ð8 þ16 ffi 52 16 ffi 3:25. Applying Pick’s theorem to the 5th Pass, see Fig. 7.33, counting yields values of b ¼ 21 and i ¼ 193. Once again, since the radius of the circle in the 5th Pass is 8 units, p  82 ffi 21 2 þ 193  1 which results in ð101 þ 1931Þ 20212 a predicted value using Pick’s theorem value for p ffi 2 64 ffi 64 ffi 3:16.

7.4

Conclusion

This chapter began by noticing that it was relatively common historically for instruction in geometry to be delayed until secondary school. Furthermore, even when such instruction was offered for it to be presented using a formal axiom and proofs-based curriculum and pedagogy. Given the broadly prevailing view that younger students cannot operate in the arena of formal logic, pushing geometry back as long as possible seemed reasonable at the time and

The theorem was first stated by Georg Alexander Pick, an Austrian mathematician, in 1899 (School of Mathematics and Statistics University of St Andrews, Scotland, 2013).

4

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still seems reasonable to many today. When taken from this mindset, to the extent geometry was addressed in a middle school curriculum, it was often limited to a memory/recall process revolving around mastering the names of geometric figures in which only limited properties such as parallel, perpendicular, and similar were addressed.

cBig Idea Geometry, clearly one of the big ideas of mathematics, can be viewed as a systematic exploration of relationships occurring between shapes and within space. This perspective lends itself to application of the van Hiele levels of instruction and bridges student experiences to logical reasoning and problem solving.b

If a broader view of geometry as being a systematic exploration of relationships that occur between shapes and within space is considered, then a much richer curriculum is not only possible but is essential. In particular, the nature of this systematic exploration can either be highly formal and deductive, or informal and experiential, and yet still be successful in developing essential middle school geometric concepts. Through such systematic explorations of geometric relationships, students should have the opportunity to actually measure, identify, classify, visualize and/or transform the geometric objects in order to develop personally meaningful experiences. It is at this point, that the van Hiele model of geometric learning has been shown to have some of its’ greatest impacts. The van Hiele model in its simplest formulation is comprised of five levels which reflect different types of thinking concerning geometry. These five van Hiele levels are not age-dependent and in their development relate more to the nature of the experience’s students have had. They do, however, establish a hierarchy of geometric conceptualization with each level describing a more abstracted representational set of concepts and processes than its predecessor. These levels can then serve to describe the types of geometric ideas the students think about (referred to as “objects of thought”) and what the student is able to do (referred to as the “products of thought”). Unlike a Piagetian approach, students can benefit from appropriate instructional experiences regardless of their current development status. The levels are, however, sequential and according to this theory students must pass through the levels in order as their understanding increases. It is not possible, therefore, to be functioning at level four (the van Hiele levels begin their numeration with 0) without having experienced levels 0 through 3 in order. Thus, it is important that great care be taken to structure geometric experiences

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to allow for an understandable and systematic exploration of geometric concepts to occur. The activities described in this chapter can, and should, be modified to accommodate the full range of van Hiele levels common to middle school students. For readers who want to deepen their knowledge and understanding of geometry and measurement the following sources can be recommended (Alexander & Koeberlein, 2020; Erdogan, 2020, Ma & Kuang, 2019; Pickover, 2001; Lachterman, 1989; Conway & Guy, 1996).

7.5

Activity Set

1. What geometric principles are drawn upon when multiplication is modeled using an area model? 2. How can geometric vocabulary be developed for middle school students functioning at van Hiele Level 0? How could the be modified for Level 1? Level 2? 3. Since van Hiele Level 0 deals with appearances, how could geometric terms and concepts be developed for students with visual disability? 4. In addition to symmetry, what other concepts could be developed using paper folding? 5. Construct a shape classification system for triangles based upon informal logic similar to that shown for quadrilaterals in Fig. 7.2. 6. How would you respond to a parent who believes that rigor cannot be achieved with a formal system of axioms and proofs being developed? 7. Which van Hiele level is most likely to be observed in middle school students? 8. What are some consequences of using language from a higher van Hiele level than that held by the students? How can these consequences be minimized? 9. How can you identify when explicitation is present in the minds of middle school students? How can you encourage the development of explicitation? 10. Create a Problem Solving Board proof scaffolding at least two proofs involving parallel lines cut by a transversal. 11. Describe the differences between a rigid and non-rigid transformation. 12. What are the four types of rotational transformations? 13. Rotate the quadrilateral with vertices of (3,5), (−3,2), (−4,−3), and (4,−1) by 270 around the origin. Describe each of the steps performed in this process. 14. What is the only non-rigid transformation? 15. What is the primary requirement for a tessellation to have no gaps or overlaps? 16. What are the three regular tessellations? 17. What are the characteristics of an edge-to-edge tessellation? 18. Which regular polygons form an edge-to-edge tessellation with an equilateral triangle? How can you tell?

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19. Draw and label a pair of parallel lines cut by a transversal. Which angles will have equal measure? What vocabulary is used to describe these equivalences? 20. Given a quadrilateral with vertices (3,5), (−3,2), (−4,−3), and (4,−1), use the distance formula to determine the distances each of the vertices. 21. Describe how a scaling factor operates. What is the result of applying a scaling factor >1? Of applying a scaling factor 63, the vehicles need less than four hours in order to meet. Now, one has to find what fraction of one hour is formed by 15 cells (left blank in Fig. 9.27). Because one hour consists of 16 cells, 15 cells represent 15/16 of an hour. Therefore, the vehicles 63 need 3 and 15/16 h in order to meet. Note that 3 15 16 ¼ 16, which is the reciprocal of 16 the fraction 63—the distance covered by two vehicles in one hour. This means that the time till they meet is the reciprocal of the distance they cover in one hour. In order to better understand the meaning of the last statement, consider a simpler situation, when 1/2 of a distance is covered in one hour. In the context of uniform movement, it is quite obvious that the whole distance will be covered in two hours. Note that the number 2 (time) is the reciprocal of the number 1/2 (velocity) in our rudimental understanding of uniform movement. In the case of a non-unit fraction, one can use a one-dimensional model for fractions (Fig. 9.28) to show how to move from one unit of measurement to another. In order to find what fraction of 16/63 (the measure of one hour in terms of distance) is 15/63 (the far-right segment in Fig. 9.28), one has to divide the latter fraction by the former one to get 15/16 (of an hour). Once again, the answer is 3 15 16 (hours). Alternatively, the answer can be given as 3 h, 22 min and 30 s after calculating 15/16 of 24 in terms of minutes and seconds. Also, one can see that the distance between A and B is an extraneous information. That is why, in problems of that kind one can use the words “a certain distance” instead of providing a specific distance. However, when providing a specific distance, it has to be realistic to relate to other data; for example, it is not realistic to cover a distance of 100 miles in seven hours by car. Problem 9.8 How many days do two workers need to complete a certain job when working together if when working alone they can complete this job in 10 and 15 days, respectively?

9.8 Solving Algebraic Word Problems Through Conceptual … Fig. 9.27 Solving the car and truck problem using rectangular grid

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Solution Similar to the assumptions of the previous problem about vehicles moving toward each other, one has to assume that the work to be done may be described as a unit of work and that the workers work with the same capacity (alternatively, speed) throughout the entire work. Then the faster worker will do 1/ 10 of the work in one day and the slower worker will do 1/15 of the work in one day. The entire work can be represented by a 10  15 grid which can be divided into six groups of 25 cells in each (Fig. 9.29). Therefore, when working together, the two workers will complete the entire work in six days. Problem 9.9 How many full days can each of the two workers complete a certain job if when working together they can complete the job in three days? Solution If x and y are the unknowns (the number of days sought) and the entire work is the unit of work (a conceptual shortcut), the reciprocals 1x and 1y represent their individual capacities (alternatively, average speed of their work) per day and the product 3ð1x þ 1yÞ represents the entire work so that the equation 3ð1x þ 1yÞ ¼ 1ð1x þ 1yÞ ¼ 1ð1x þ 1yÞ ¼ 1 or its equivalent form 1 1 1 þ ¼ x y 3

ð9:12Þ

is the mathematical model of the problem situation. In order to find x and y, all representations of 1/3 as a sum of two unit fractions have to be found. One possible solution is to partition 1/3 into the sum of two equal fractions, 1/3 = 1/6 + 1/6. Another solution may include 1/4—the largest unit fraction smaller than 1/3. One can find that 1/3 − 1/4 = 1/12. Therefore, 1/3 = 1/4 + 1/12. Finally, the only unit

16/63

16/63

16/63

15/63

Fig. 9.28 Using one-dimensional model to move between different units of measurement

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Fig. 9.29 Solving Problem 9.8 on a grid

fraction that has to be verified is 1/5. However, 1/3 − 1/5 = 2/15—not a unit fraction. Therefore, there are two ways to hire workers: a pair of equal capacity of doing work in six days as well as those capable of completing work in four and twelve days when working alone. Remark 9.7 Another tool that one can use in solving indeterminate Eq. (9.12) is purely visual. More specifically, in order to solve Eq. (9.12), one has to find two solutions without using algebraic transformations. Instead, one can reason with a tape diagram, a term coined in the United States by Common Core State Standards (2010). Such solution is shown in the diagram of Fig. 9.30. To clarify, note that the top three bars show how three days of work of workers on par with each other can be divided into six equal pieces of work, so that each worker does one piece of work daily. This means that each worker, working alone, would do the entire work in six days and 16 of work in one day. Thus, 13 ¼ 16 þ 16. The three bars in the middle of the diagram show how three days of work can be divided into nine equal pieces, so that one worker does two pieces of work daily and another (slower) worker does one piece of work daily. This means, however, that each worker, working alone, 1 can do the entire work in nine and four and a half days, i.e., 19 and 4:5 of work in one 1 1 1 day, respectively. Thus, 3 ¼ 9 þ 4:5. Finally, the three bars at the bottom of the diagram show how three days of work can be divided into twelve equal pieces, so that one worker does three pieces of work daily and another (slower) worker does one piece of work daily. This means that each worker, working alone, can do the 1 entire work in four and twelve days, i.e., 14 and 12 of work in one day, respectively. 1 1 1 Thus, 3 ¼ 4 þ 12.

9.9

Inequalities

Suppose that one has to find the sum 137 + 236 but can only add either multiples of ten or five. In addition, one knows the order of numbers and how to distinguish between one-digit evens and odds, thus being capable of comparing such integers using the inequality signs “>” and “ 130 and 137 < 140; 236 > 230 and 236 < 240. Using these comparisons and adding the corresponding multiples of ten result in the estimate 360 < 137 + 236 < 380. In other words, the sum to be found is any integer with 361 and 379 being, respectively, the smallest and the largest sums possible. In order to narrow down the number of candidates for the sum, the second step could be to compare 137 and 236 to the closest multiples of five. In doing so, one can write: 137 > 135 and 236 > 235. Therefore, as one knows how to add multiples of five (alternatively, understanding that the sum 135 and 235 is ten smaller than the sum 140 and 240, the following can be written: 137 + 236 > 370. That is, 370 < 137 + 236 < 380. Now, one can see that the sum in question may be any integer in the range from 371 to 379. The third step could be to immediately reject 376 and 277 as the sum may not have either 6 or 7 as the last digit. Furthermore, the last digit may only be an odd number as 6 is an even number and 7 is an odd number. Therefore, the sum in question may be equal to one of the following four numbers only: 371, 373, 375, or 379. So, by trying to find the sum 137 + 236, one made a significant progress by narrowing down the search to four numbers only. This progress was made possible (mainly) due to the use of inequalities.

9.9.1

Algebraic Inequalities as Tools of Digital Fabrication

In order to show a practical face of algebraic inequalities, this section will use the context of digital fabrication (Gershenfeld, 2005; Abramovich & Connell, 2015) to support middle school teacher candidates’ understanding of the applied role of inequalities as tools of mathematics. In the age of technology, digital fabrication is one such context. An educational paradigm of digital fabrication has grown out of ubiquitous movement of the 1980s to introduce a pedagogy of student-computer

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interaction into the schools as a way of improving teaching and learning of mathematics and science. With this in mind, it will be demonstrated that the process of digital fabrication of different geometric shapes formed by the combinations of graphs of the basic functions from the middle school mathematics curriculum requires deep conceptual understanding of two-variable inequalities that define those shapes. In particular, techniques of digital fabrication of a point, segment, arc, and triangle will be discussed.

9.9.2

Digitally Fabricating Points, Segments and Arcs

Pestalozzi10 argued that visual understanding is the foundation of conceptual thinking and he “forced the children to draw angles, rectangles, lines and arches, which he said constituted the alphabet of the shape of objects, just as letters are the elements of the words” (Arnheim, 1969, p, 299). In the age of computers, this pedagogical ideas of the early nineteenth century can be enhanced by using digital fabrication in the creation of such drawings. In general, drawing supports the development of higher mental functions (Vygotsky, 1978) which are fundamental for creativity. When creative drawing (that is, image fabrication) is supported by technology, the use of the latter encourages one “to visualize the results of varying assumptions, explore consequences, and compare predictions with data” (Beghetto et al., 2015, p. 90). In what follows, it will be demonstrated how digital fabrication encourages conceptual thinking through the use of mathematical concepts as tools in computing applications.

9.9.2.1

Fabricating a Point

A point in the coordinate plane can be represented visually through a small circle (a disk) or through a small square. Let us assume that we have to digitally fabricate a point with the coordinates (a, b). In the case of a circle as a point, the former has to be centered at the point (a, b) so that its interior represents a disk which looks like a bullet point. In the case of a square as a point, the square should have the center at the point (a, b). Obviously, a small disk has a small radius e. A circle of radius e centered at the point (a, b) has an equation ðx  aÞ2 þ ðy  bÞ2 ¼ e2 . Whereas this equation is (presumably) well known, it is also important to know its origin and how it can be derived. With this in mind, note that the equation of a circle stems from its definition as a set of points equidistant from a point called the center of a circle. Let (x, y) be a point on a circle. Then, the distance from the point (x, y) to the center (a, b) can be found using the Pythagorean theorem as shown in Fig. 9.31 (see also Chap. 7, Sect. 7.2.7): 10

Johann Heinrich Pestalozzi (1746–1827)—a Swiss educational reformer.

9.9 Inequalities

291

Fig. 9.31 Using Pythagorean theorem in deriving the distance formula

AB ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi AC 2 þ BC 2 ¼ ðx  aÞ2 þ ðy  bÞ2 :

That is, for any point (x, y) on a circle its distance to the center is the same. This distance is called the radius of the circle. From here, the equation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  aÞ2 þ ðy  bÞ2 ¼ e or, after squaring both sides, an equivalent quadratic equation ðx  aÞ2 þ ðy  bÞ2 ¼ e2

ð9:13Þ

follows describing the circle of radius e centered at the point (a, b). Consequently, for any point inside the circle (that is, within the disk) the inequality ðx  aÞ2 þ ðy  bÞ2 \e2

ð9:14Þ

runholds true. In order to digitally fabricate the disk defined by inequality (9.14), one can graph the inequality using the Graphing Calculator, a computer application capable of graphing relations from any two-variable equation or inequality. For example, when a = b = 0 and e ¼ 0:1, inequality (9.14) turns into the inequality x2 þ y2 \ 0:01 the image of which is shown in Fig. 9.32. Likewise, one can digitally fabricate a small square of side 2e centered at the point (a, b). Let (x, y) be a point of the coordinate plane where such a square has to be fabricated. Then, within this square, the inequality jx  aj  e, the left-hand side of which defines the distance between any two points on the coordinate plane with the first coordinates x and a, holds true. The square’s horizontal sides are described by the equalities y ¼ b  e and y ¼ b þ e; that is, within the square to be digitally fabricated, the inequality jy  bj  e, the left-hand side of which defines the distance between any two points on the coordinate plane with the second coordinates y and b, holds true. In order to complete digital fabrication of the square, one has to graph a system of the inequalities jx  aj  e and jy  bj  e. Such a square is shown in Fig. 9.33 where a = b = 1 and e ¼ 0:05.

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Fig. 9.32 Digital fabrication of a point as a small disk

9.9.2.2

Fabricating a Segment

To digitally fabricate a segment with the endpoints (a, b) and (c, d), one has to write an equation of the straight line passing through the two points in the form y ¼ b þ db ca ðx  aÞ and then graph the inequality jy  b 

db ðx  aÞj \e ca

ð9:15Þ

concurrently with the inequalities a\x\c; b\y\d. In Fig. 9.34, (a, b) = (2, 3), (c, d) = (1, 4), and e ¼ 0:1. The meaning of inequality (9.15) is as follows: when the distance from the point (x, y) to the straight line defined by the equation y ¼ b þ db ca ðx  aÞ is smaller than e, all such points represent an infinite strip of thickness 2e. Alternatively, the equation of a straight line passing through two given points can be constructed by introducing a linear function f ðxÞ ¼ kx þ m such that b ¼ f ðaÞ ¼ ka þ m; d ¼ f ðcÞ ¼ kc þ m whence d  b ¼ kðc  aÞ or k ¼ db ca . db db bcabad þ ab bcad Consequently, b ¼ f ðaÞ ¼ ca a þ m whence m ¼ b  ca a ¼ ¼ ca . ca bcad db db bcad db Finally, y ¼ db x þ ¼ ðx  aÞ þ a þ ¼ ðx  aÞ þ b; that ca ca ca ca ca ca db is, once again we have the equation y ¼ b þ ca ðx  aÞ. In that way, one uses mathematical concepts as tools in computing applications to digital fabrication.

9.9.2.3

Fabricating an Arc

To digitally fabricate an arc of a circle, one can use inequality (9.14) concurrently with the inequalities that limit x and y. For example, in constructing a 45° arc of a unit circle centered at the origin, x2 þ y2 ¼ 1, one has to graph the system of inequalities.

9.9 Inequalities

293

Fig. 9.33 Digital fabrication of a point as a small square

Fig. 9.34 Digitally fabricating a segment

jy 

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2 j  e; x 0; x 

pffiffiffi pffiffiffi 2 2 ; y 2 2

Such graph of an arc is shown in Fig. 9.35 for e ¼ 0:02.

9.9.2.4

Fabricating Triangles

One can also digitally fabricate a triangle, given its vertices. Each side can be fabricated as a segment using inequality (9.15) and corresponding inequalities for x and y. One can also digitally fabricate the interior of the triangle by graphing simultaneously the three inequalities y\f1 ðxÞ; y\f2 ðxÞ; y [ f3 ðxÞ. Here, when inequality signs are replaced by equal signs, we have the equations of the straight lines which contain the side lengths of the triangle. An interesting aspect of this construction is that whereas any three points that do not belong to the same straight line may serve as vertices of a triangle, not any three segments may be used as side lengths of a triangle as they have to satisfy the triangle inequality: The sum of any two side lengths of a triangle has to be greater than the third side length. Using the triangle inequality, one can do digital fabrication of a different kind: Construct all triangles with whole number sides the largest side of which measures

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Fig. 9.35 Digitally fabricating an arc of a unit circle

four linear units. How many such triangles exist? To answer this question, one has to find all pairs of integers each of which is not greater than four with the sum greater than four. Such pairs are (4, 4), (3, 4), (2, 4), (1, 4), (3, 3) and (2, 3). Consequently, the following six triples can be used as side lengths of the triangles to be constructed: (4, 4, 4), (3, 4, 4), (2, 4, 4), (1, 4, 4), (3, 3, 4) and (2, 3, 4). Such six triangles are digitally fabricated in Fig. 9.36.

9.9.3

Arithmetic Mean-Geometric Mean Inequality

Arithmetic mean—Geometric mean (AM-GM) inequality is one of the basic inequalities in mathematics. In the case of two positive numbers x and y, the pffiffiffiffiffi inequality connects their arithmetic mean, x þ2 y, and geometric mean, xy. In the case of two positive numbers, the inequality between their means stems from a simple geometric situation. As shown in Fig. 9.37, the four rectangles of equal area xy are located inside the square of the side length x + y. This means that if x 6¼ y, then 4xy\ðx þ yÞ2 . However, if x ¼ y (Fig. 9.38), then 4xy ¼ ðx þ yÞ2 . Thus pffiffiffiffiffi assuming x y [ 0 we have 2 xy  x þ y or pffiffiffiffiffi x þ y : xy  2

ð9:16Þ

Once inequality (9.16) is known from geometric considerations, it can be proved algebraically as follows pffiffiffi 2 pffiffiffipffiffiffi pffiffiffi pffiffiffi 2 pffiffiffi 2 x þ y pffiffiffiffiffi ð xÞ þ ð yÞ  2 x y ð x  yÞ ¼  xy ¼ 0; 2 2 2

9.9 Inequalities

295

Fig. 9.36 The triangle inequality and digital fabrication

Fig. 9.37 4xy\ðx þ yÞ2

with equality taking place when x = y. Figure 9.37 is another example when a general situation can be recognized from a particular instance (see Sect. 9.3.1). Corollary 9.1 For any positive number x, the inequality x þ 1x 2 holds true. qffiffiffiffiffiffiffiffi xþ1 Indeed, as follows from inequality (9.16), 2 x x  1x ¼ 1 whence x þ 1x 2. Corollary 9.2 The largest product of two positive numbers with a given sum is when the two numbers are equal. pffiffiffiffiffi Indeed, let a + b = p, then, as follows from inequality (9.16), ab  a þ2 b ¼ p2, pffiffiffiffiffi  2 that is, ab ¼ p2 or ab ¼ p2 when a = b. In particular, among all rectangles with a given perimeter, square has the largest area. A possible conceptual generalization of the AM-GM inequality is to add another positive number z and to consider the relationship between the arithmetic and the geometric means of three positive numbers. With this in mind, the following inequality

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Fig. 9.38 4xy ¼ ðx þ yÞ2 when x = y

xþyþz p ffiffiffiffiffiffiffi 3 xyz: 3

ð9:17Þ

may be conjectured and then proved as follows. pffiffiffi pffiffi pffiffiffi Setting a ¼ 3 x; b ¼ 3 y; c ¼ 3 z, one can write ða þ b þ cÞ3 ¼ ða þ bÞ3 þ c3 þ 3ða þ bÞ2 c þ 3ða þ bÞc2 ¼ a3 þ b3 þ c3 þ 3a2 b þ 3ab2 þ 3a2 c þ 3b2 c þ 6abc þ 3ac2 þ 3bc2 ¼ a3 þ b3 þ c3 þ 3abða þ b þ cÞ þ 3acða þ b þ cÞ þ 3bcða þ b þ cÞ  3abc;

whence a3 þ b3 þ c3  3abc ¼ ða þ b þ cÞ3  3abða þ b þ cÞ  3acða þ b þ cÞ  3bcða þ b þ cÞ.

Then, one can show that the difference between the left-and right-hand sides of inequality (9.17) is non-negative. Indeed, xþyþz p ffiffiffiffiffiffiffi 1  3 xyz ¼ ½ða3 þ b3 þ c3 Þ  3abc 3 3 1 ¼ ½ða þ b þ cÞ3  3abða þ b þ cÞ  3acða þ b þ cÞ  3bcða þ b þ cÞ 3 1 ¼ ða þ b þ cÞ½ða þ b þ cÞ2  3ab  3ac  3bc 3 1 ¼ ða þ b þ cÞða2 þ b2 þ c2  ab  ac  bcÞ 3 1 ¼ ða þ b þ cÞða2 þ b2  2ab þ a2 þ c2  2ac þ b2 þ c2  2bcÞ 6 1 ¼ ða þ b þ cÞ½ða  bÞ2 þ ða  cÞ2 þ ðb  cÞ2  0: 6

Finally, one can generalize from inequalities (9.16) and (9.17) and connect the arithmetic and the geometric means of n positive numbers x1 ; x2 ; . . .; xn as follows:

9.9 Inequalities

297

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 þ x2 þ    þ xn p n x1 x2 . . .xn : n

ð9:18Þ

A proof of inequality (9.18) is beyond the scope of this textbook.

9.10

Algebraic Recreations

The problem-solving focus of the twenty-first century standards for school mathematics worldwide (Australian Association of Mathematics Teachers, 2006; Takahashi et al., 2006; Common Core State Standards, 2010; Ministry of Education Singapore, 2012; Department for Education, 2013; Felmer et al., 2014; Fan et al., 2015; Association of Mathematics Teacher Educators, 2017) highlights the importance of mathematical enrichment activities. One aspect of such activities deals with recreational mathematics (e.g., Guy & Woodrow, 1994; O’Beirne, 2017). Recreational problems may include exploration of “puzzles and games that involve numerical reasoning using problem-solving strategies” (Western and Northern Canadian Protocol, 2008, p. 33). A characteristic didactical feature of recreational problems appropriate for the middle grades is the ease of their formulation, wealth of technology integration, appeal to one’s natural curiosity and motivation for students’ creativity. In many cases, a recreational problem motivates creative development and appropriate use of a computational environment to verify its statement, to provide a visual and/or numeric demonstration, and to discover new patterns. One class of problems in recreational mathematics deals with the properties of integers to serve as attractors for other integers when initiating a mathematically uncomplicated iterative process. Consider the following process involving three-digit numbers. Any three digits a, b and c (at least one of which is different from the other two) of a three-digit number can be arranged to form the largest and the smallest possible numbers: 100a + 10b + c and 100c + 10b + a, respectively (assuming a > c). In that way, their difference (100a + 10b + c) − (100c + 10 b + a) = 99(a − c) is the largest possible one between two numbers with the digits a, b and c, a > c. Because 1  a – c  9, the expression 99(a–c) represents a three-digit number when a – c > 1. In the case a − c = 1, one may consider 99 as an artificial number 099 and arrange its digits in the non-increasing order, 990, so that the difference 990 − 099 = 891. Thus, one can start with a three-digit number not all digits of which are the same, form the largest difference between two numbers with those digits and continue forming the largest difference with the digits of this and other like differences. This iterative process, as it was first discovered by Kaprekar (1949), very soon converges to the number 495. The number 495 has a unique property of attracting all three-digit numbers through the iterative procedure of creating and transforming the greatest differences out of three (not all the same; otherwise, one reaches zero after the first step) digits. This process is sometimes called “ordering-subtraction operation” (Trigg, 1974, p. 41).

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9 Conceptual Approach to the Ideas of Middle School Algebra

Fig. 9.39 It takes three iterations to reach 495 starting from 138

One can use a spreadsheet to explore this recreational problem. For example, the spreadsheet of Fig. 9.39 shows that applying this process to the number 138, the number 495 is reached after four steps (ordering-subtraction iterations). As shown in Fig. 9.39, the spreadsheet separates digits of three-digit numbers at each step in order to rearrange digits towards forming the largest difference. To this end, the formula ai ¼ INT

N 10ni



 10  INT

N

10ni þ 1

ð9:19Þ

where N is an n-digit number, i = 1, 2, …, n, defined by formula (4.6) of Chap. 4, Sect. 4.7.1, can be used. One can derive formula (9.19) as an independent practice in uncomplicated algebraic transformations. This computational environment can be extended to conclude (as shown in Fig. 9.40, rows 11 and 13, respectively) that out of 891 three digit numbers, with at least one digit different from the other two, 139 numbers reach 495 in one step (assuming, for the simplicity of programming, that 495 reaches itself in one step), 131—in two steps, 246—in three steps, 198—in four steps, 126—in five steps, and 51—in six steps. Among many interesting results available within this environment one result is as follows: whereas each term in the string of six numbers 116, 117, 118, 119, 120, 121 approaches 495 in 1, 2, 3, 4, 5, 6 steps, respectively, each term in another string of six consecutive numbers 161, 162, 163, 164, 165, 166 approaches 495 in just one step (e.g., 161 forms the largest difference 611 − 116 = 495). An interesting investigation could be to explain as to why these two strings are so different in their relation to 495. The programming details of this extended spreadsheet are included in Appendix. Another recreational problem deals with a different type of digits’ reversal with addition being an operation which (typically) iteratively leads to a palindrome. In mathematics, a palindrome is defined as a natural number that reads the same backward as forward. Palindromes have an interesting property to attract natural numbers through the following (iterative) “reverse-and-add” process. Start with any natural number, reverse its digits and add the two numbers. Repeat the process with the sum and continue it to see that it leads to a palindrome. For example, starting with 398 yields: 398 + 893 = 1291; 1291 + 1921 = 3212; 3212 + 2123 = 5335— a palindrome. The process took three steps (iterations) to reach a palindrome. After trying other starting numbers (in many cases, the process may take more than three

9.10

Algebraic Recreations

299

Fig. 9.40 Row 11 shows how many numbers reach 495 in 1 through 6 (row 13) steps

Fig. 9.41 The number 998 reaches the palindrome 133,697,796,331 after 17 iterations (cell A3)

steps), the following generalization known as the Palindrome conjecture can be formulated. Regardless of the starting natural number, one can always arrive at a palindrome by adding numbers with digits reversed. Whereas this conjecture remains unproven [Lines (1986, p. 62); “Lychrel number” (2020)], some computational experiments with this conjecture are appropriate for the middle grades. One such spreadsheet-based experiment is to find the smallest natural number attracted by a palindrome in more than ten steps. For example, 89 reaches the palindrome 8,813,200,023,188 in 24 steps and it is the smallest number requiring more than 10 steps. Likewise, the spreadsheet shown in Fig. 9.41 (its programming details are included in Appendix and are based, in part, on the use of formula (9.19) enabling the separation of digits of a natural number) indicates that it takes 17 steps (iterations) before the number 998 reaches a palindrome. Similar spreadsheet-based explorations are possible. Probabilistic computational experiments in the context of the Palindrome conjecture are discussed in Chap. 12, Sect. 12.11.

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9 Conceptual Approach to the Ideas of Middle School Algebra

9.11

Conclusion

This chapter demonstrated possible uses of software for creative activities of young children as context for early algebra instruction in which one moves from unknowns found through a game-based trial and error to equations rather than other way around in the traditional school algebra. Different types of algebraic generalization were introduced. Addition and multiplication tables were used as context for developing algebraic skills. Different ways of solving equations as models for word problems were discussed. Simple algebraic inequalities were used as tools of digital fabrication of basic geometric concepts. The triangle inequality and the arithmetic mean—geometric mean inequality were discussed. Finally, two recreational problems were explored using a spreadsheet. For readers who want to deepen their knowledge and understanding of school algebra the following sources can be recommended (Freudenthal, 1973; Mason & Pimm, 1984; Kieran & Wagner, 1989; Bednarz et al., 1996; Naftaliev & Yerushalmy, 2011; Abramovich, 2014c; Abramovich & Leonov, 2019).

9.12

Activity Set

1. Find the sum 2 + 4 + 6 + 10… + 2n. Prove your finding by the method of mathematical induction. Demonstrate visually how to find this sum when n = 4. 2. Find the sum 2 + 6 + 10 + 14 + … + 4n − 2. Prove your finding by the method of mathematical induction. Demonstrate visually how to find this sum when n = 4. 3. Find the sum 1  2 þ 2  3 þ 3  4 þ ::: þ nðn þ 1Þ. Prove your finding by the method of mathematical induction. Demonstrate visually how to find this sum when n = 4. 4. Find the sum 1  3 þ 3  5 þ 5  7 þ ::: þ ð2n  1Þð2n þ 1Þ. Prove your finding by the method of mathematical induction. Demonstrate visually how to find this sum when n = 3. 5. Find the sum 1  4 þ 3  6 þ 5  8 þ ::: þ ð2n  1Þð2n þ 2Þ. Prove your finding by the method of mathematical induction. Demonstrate visually how to find this sum when n = 3. 6. Find the sum of all numbers in the n  n addition table except those located on the main diagonal of the table. 7. Find the sum of all numbers in the n  n addition table except those located on the bottom left—top right diagonal of the table.

9.12

Activity Set

301

8. Find the sum of all numbers in the n  n multiplication table except those located on the main diagonal of the table. 9. Find the sum of all numbers in the n  n multiplication table except those located on the bottom left—top right diagonal of the table. 10. Prove by the method of mathematical induction than the trinomial 4n3 + 6n2 − n is a multiple of three for any natural number n. 11. Prove by the method of mathematical induction that the trinomial 4n3 + 9n2 − n is a multiple of three for any natural number n. 12. Prove by the method of mathematical induction that the product nðn þ 1Þð3n2  n  2Þ is a multiple of twelve for any natural number n. 13. Prove that any four consecutive triangular numbers tn ; tn þ 1 ; tn þ 2 ; tn þ 3 satisfy the relation tn þ 3 ¼ 3ð tn þ 2  tn þ 1 Þ þ tn : Demonstrate this relation visually in the case n = 3. 14. Prove that any four consecutive square numbers sn ; sn þ 1 ; sn þ 2 ; sn þ 3 satisfy the relation sn þ 3 ¼ 3ð sn þ 2  sn þ 1 Þ þ sn : Demonstrate this relation visually in the case n = 2. 15. Prove that the sum of three consecutive natural numbers is a multiple of three. 16. Prove that the product of three consecutive natural numbers is a multiple of three. 17. A store cells two types of packages of identical cookies and candies. The net weight of 8 cookies and 20 candies in the first package is 248 g. The net weight of 20 cookies and 9 candies in the second package is 292 g. Find the weight of a cookie and the weight of a candy by using a conceptual shortcut. 18. A store cells two types of packages of identical cookies and candies. The net weight of 17 cookies and 20 candies in the first package is 270 g. The net weight of 13 cookies and 25 candies in the second package is 255 g. Find the weight of a cookie and the weight of a candy by using a conceptual shortcut. Is there more than one way to use a conceptual shortcut? 19. The sum of ages of three children of different age is 18. The age of the oldest is twice the age of the youngest. What are the ages of the children? Solve the problem by using a conceptual shortcut. Develop a spreadsheet environment for posing new problems of that kind. 20. How many days can each of the two workers complete a certain job when working alone if when working together they can complete this job in 4 days? 21. How many days do two workers need to complete a certain job by working together if by working alone they can complete this job in 6 and 12 days, respectively? 22. It takes 5 h for a car and 7 h for a truck, moving non-stop and without changing speed, to cover distance between A and B. If the car and the truck started moving toward each other at the same time from A and B, respectively, in how many hours would they meet? 23. Define two-variable inequalities needed to digitally fabricate segments connecting the origin with the points (1, 1), (−1, 1), (−1, −1) and (1, −1).

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9 Conceptual Approach to the Ideas of Middle School Algebra

24. Define two-variable inequalities needed to digitally fabricate the sides of the triangle with the vertices at the points (1, 1), (−1, 2) and (−2, −2). 25. Define two-variable inequalities needed to digitally fabricate the triangle with the vertices at the points (1, 1), (−1, 2) and (−2, −2). 26. Define two-variable inequalities needed to digitally fabricate a circumference of radius four linear units and centered at the point (2, −2). 27. Define two-variable inequalities needed to digitally fabricate an arc of the circle mentioned in the previous problem and located in each of the four quadrants. 28. Define two-variable inequalities needed to digitally fabricate the letters L, H, A, N, and M. 29. Define two-variable inequalities needed to digitally fabricate your own initials.

Chapter 10

Patterns and Functions

10.1

Introduction

cBig Idea With the advent of computers in education enriched by advances in the development of software tools, computational experiment approach to the learning of mathematics has secured an important niche among big ideas of the pedagogy of the subject matter across the curriculum.b

This chapter deals with the recognition of visual and numeric patterns and their description through the language of functions which, in turn, form algebraic patterns to be recognized and generalized at a higher conceptual level. In the age of computers, this generalization is critical for developing a computational environment within which new numeric patterns can be revealed through a computational experiment.1 Once a new numeric pattern is discovered, it can be expressed in visual terms to be used as a springboard into new mathematical activities. Through this process, one comes full circle from visual at the basic level to visual at a higher cognitive level. The recognition of a visual pattern often deals with the constructive aspect of perception in which one can discern in an image more information than the image, in the absence of one’s insight, transmits. Such aspect of perception in the context of Gestalt psychology is called reification (Wertheimer, 1938). In addition to the reification of the

1

With the advent of computers in education, a computational experiment secured an important niche among big ideas of mathematics and its pedagogy, including the study of patterns and functions in the middle grades. © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_10

303

304

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Patterns and Functions

Gestaltists, this aspect of perception is of importance for developing metacognition (Flavell, 1976). When planning how to apply findings from metacognition to include mathematics instruction, such a cyclic reinterpretation of earlier experienced concepts at a higher abstracted level was recognized as being critical. The process of mentally “following” each subsequent representation, how it followed from the preceding representation and how it led to the next was viewed as a crucial step in developing metacognitive awareness of the concept and its formation (Campione et al., 1989). On the one hand, our perception is not definitive because certain things may be unintentionally hidden from view and one needs insight in order to reveal them. On the other hand, our perception may vary even without reliance on insight and the diversity of both perception and thinking may interpret visual patterns in more than one way. Therefore, prospective teachers of middle school mathematics should have experience with the diversity of perception in order to “support emerging mathematical practices of middle level learners … in a more sophisticated manner than they may have demonstrated in earlier grades” (Association of Mathematics Teacher Educators, 2017, pp. 94, 106) in the context of initial pattern recognition and its subsequent formal description in the language of mathematics. The recognition of numeric patterns follows by what in Chap. 9 was called algebraic generalization. Numeric patterns form foundation for the development offunctional relationships (Lee & Freiman, 2006; Wilkie & Clarke, 2016). Just as a visual pattern may be described symbolically in more than one way, a numeric pattern, if not connected to any context, that is, in the absence of “referents for the symbols involved” (Common Core State Standards, 2010, p. 6), can also be described in more than one way. For example, in the four-number sequence 2, 3, 5, 8 one can recognize at least two patterns. The first pattern stems from the equalities 2 + 1 = 3, 3 + 2 = 5, 5 + 3 = 8 where the second addend in the left-hand sides of the three equalities is the rank of the first addend in the given four-number sequence and, therefore, the pattern may continue as follows: 8 + 4 = 12, 12 + 5 = 17, 17 + 6 = 23, and so on. The second pattern stems from recognizing in the numbers 2, 3, 5 and 8 four consecutive Fibonacci numbers; in that case, the sequence extends to include 13, 21, 34, and so on. Therefore, when the four-number sequence 2, 3, 5, 8 is not connected to any context, both extensions of the sequence may be accepted. That is, context plays an important role in the teaching of mathematics enabling a teacher to decrease the negative affordance of students’ (perhaps natural) atomistic mathematical thinking and, instead, facilitate their relational thinking (Carpenter et al., 2005) and metacognitive development (Flavell, 1976). Indeed, when the numbers 2, 3, 5, and 8 describe, for example, all possible arrangements of one, two, three, and four two-sided counters in which any two red counters are separated by at least one yellow counter (Chap. 9, Sect. 9.6), then the latter extension is the only one possible. In other words, when a pattern

10.1

Introduction

305

is connected to a certain activity with clearly described rules, then the rules define the pattern. In turn, the rules may give birth to sequences of numbers which form patterns defining functional relationships, and this observation links patterns to functions. Over the last two decades, the importance of the study of functional relationships at the middle school level has been emphasized in various publications around the world (e.g., Ren & Wang, 2007; Shield, 2008; Common Core State Standards, 2010; Byun & Ju, 2012; Anabousy et al., 2014; Association of Mathematics Teacher Educators, 2017). A function can be defined as a rule that uniquely connects one set of objects (e.g., numbers) to another set of objects (numbers); that is, every element of one set is connected to one and only one element of another set. A rule may be presented either through a formula (in the case of numbers) or through a verbal description (both in terms of objects and numbers). Thus, the study of numeric patterns (stemming from visual ones) is an important pre-requisite to the study of functions. Furthermore, the study of visual and numeric patterns provides a context for developing and exploring functional relationships which, in turn, provide a context for a more advanced study of patterns formed by functions themselves. Finally, patterns formed by functions provide foundation for developing a computational environment for mathematical experimentation when experiments lead to new mathematical results hidden from view because of their complexity. But once being revealed experimentally, these results can motivate work on their formal justification and explanation. Many examples of numeric sequences when a limited number of terms enable different generalizations (that is, conclusions about a rule that governs a pattern observed may vary) are in abundance within the On-Line Encyclopedia of Integer Sequences (OEIS®). For example, in the case of the first seven Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, the OEIS® offers, among many other things, its continuation in the form 1, 1, 2, 3, 5, 8, 13, 39, 124, …, interpreting each term beginning from the third as the sum of the preceding two terms with the digits reversed. Indeed, one can check to see that 39 = 31 + 8 and 124 = 93 + 31. A similar extension (not included in the OEIS®) of the four-number sequence 2, 3, 5, 8 could stem from the rule that every number beginning from the third is the sum of the sums of digits of the previous two numbers. This rule generates the sequence 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, …, which forms a cycle of period 24. One can start a sequence with integers different from 2 and 3 to see whether, following the same rule, both a cycle and its 24-number length would result again and which number as the first term would never produce a cycle. Note that if one starts with two numbers, like 101 and 201, the sums of digits of which are 2 and 3, respectively, the following sequence results: 101, 201, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, …, in which the numbers 2 and 3 appear on the 25th and 26th places so that the first 24 numbers represented a transient process of approaching a 24-cycle with the numbers 2 and 3 as the first two terms. One can see how numeric sequences that

306

10

Patterns and Functions

develop through a certain rule can provide a rich setting for exploring patterns that the terms of the sequences follow. However, there are two steps in exploring patterns: the first step is to recognize a pattern and the second step is to explain the pattern. The second step is often based on quite involved mathematical thinking which goes beyond numbers and requires analytic skills and techniques. The goal of this chapter is to develop such skills in the broad context of middle school mathematics.

cBig Idea Patterns uniquely belong to big ideas of mathematics. When one is asked what mathematicians do, the answer is often quite definitive: they seek and study patterns. Once a pattern is found, a formulation of a mathematical proposition and its proof are in order. This chapter is about seeking patterns, generalizing patterns, justifying generalizations, and studying patterns through computational problem solving and posing.b

10.2

From Patterns to Functions

Consider the five-number sequence 1, 3, 5, 7, 9. Does one see a pattern in the development of this sequence? How can one describe the sequence analytically? One possible description is that every number beginning from the second is greater than the previous number by two.2 Indeed, 3 = 1 + 2, 5 = 3 + 2, 7 = 5 + 2, 9 = 7 + 2, and so on. This observation can be generalized as follows: the nth term, xn , is greater than the previous term, xn
1 , by two. Put another way, for all n = 2, 3, 4, … the relation xn ¼ xn
1 þ 2

ð10:1Þ

may be used as a possible symbolic representation of the above five-number sequence and its extension.

The OEIS® provides other descriptions used to continue the five-number sequence 1, 3, 5, 7, 9. A simple rule is to continue with odd numbers having odd digits only, thus not including 21, 23, 25, 27, 29 and like (odd) numbers. A more complicated rule stems from the fact that base-ten numbers 1, 3, 5, 7, and 9, being trivial (one-digit) palindromes (integers that read forward the same as backward; see Chap. 9, Sect. 9.10, and Chap. 12, Sect. 12.11), are also palindromes in base two. Indeed, 110 ¼ 12 ; 310 ¼ 112 ; 510 ¼ 1012 ; 710 ¼ 1112 ; 910 ¼ 10012 . One can use Wolfram Alpha (entering the command “base ten number N in base two” for specific values of N into the input box of its free on-line version) to see that the sequence 1, 3, 5, 7, 9 extends to include 33, 99, and 313, as 3310 ¼ 1000012 ; 9910 ¼ 11000112 ; 31310 ¼ 1001110012 . In that way, a function that defines the sequence 1, 3, 5, 9, 33, 99, 313, …, is presented in the form of the verbal description of the rule that guides the development of the sequence. Also, see Sect. 4.7.1 of Chap. 4. 2

10.2

From Patterns to Functions

307

How can one use formula (10.1) in order to find x100 ? In other words, how can one extract a procedure from a concept, in our case from the formula that connects two consecutive odd numbers? First of all, one needs to know x99 . This, in turn, requires knowledge of x98 ; the latter calls for x97 , and so on. That is, one has to know the value of x1 in order to use formula (10.1) as a generator of other terms of the sequence, including x100 . Thus, extending (10.1) to the form xn ¼ xn
1 þ 2; x1 ¼ 1; n ¼ 2; 3; 4; . . .

ð10:2Þ

is a general way of describing the sequence of consecutive odd numbers. Formula (10.2) is called a recursive definition of odd numbers. How can one describe the sequence of odd numbers through a rule that uniquely relates a number to its rank in the sequence? In other words, how can odd numbers be described in a functional form; that is, how can one represent any odd number as an algebraic expression that includes its rank? To answer this question, one can use formula (10.2) to write x2 ¼ x1 þ 2; x3 ¼ x2 þ 2; . . .; xn
1 ¼ xn
1 þ 2; xn ¼ xn
1 þ 2: Adding the last n − 1 equalities (using indices as counters) yields x2 þ x3 þ    þ xn
1 þ xn ¼ x1 þ x2 þ    þ xn
1 þ 2ðn
1Þ whence xn ¼ x1 þ 2ðn
1Þ or xn ¼ 1 þ 2n
2. That is, for all n ¼ 1; 2; 3; . . . xn ¼ 2n
1:

ð10:3Þ

Formula (10.3) represents xn through a simple algebraic expression depending on n and it may be seen as a functional description of the sequence of odd numbers through a closed formula, uniquely connecting xn and n. Sometimes, such relation is expressed in the form f(n) = 2n − 1 – a linear function of variable n. While the study of linear functions is an important strand of the middle school mathematics curriculum, Japanese students at that grade level “are expected to understand functions other than linear functions such as simple quadratic functions, and … to deepen their understanding of functions by classifying the differences and similarities of linear functions and functions which are not linear” (Takahashi et al., 2004, p. 33). In Canada, middle school students are expected “For the pattern 1, 3, 5, 7, 9, …, investigate and compare different ways of finding the 50th term” (Ontario Ministry of Education, 2005, p. 105). In England, students in middle grades are taught to “recognize, sketch and produce graphs of linear and quadratic functions of one variable” (Department for Education, 2013, p. 7). In the United States, middle grades teachers have to be proficient with “examining the patterns of change in proportional, linear, inversely proportional, quadratic and exponential functions” (Conference Board of the Mathematical Sciences, 2012, p. 43).

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Patterns and Functions

cBig Idea A functional relationship between two variable quantities, when change of one quantity (independent variable) uniquely defines change of another quantity (dependent variable), is a big idea of mathematics. Linear and quadratic functions are examples of functional relationships. The condition of uniqueness means that whereas any value of independent variable may not be associated with more than one value of dependent variable, any value of dependent variable may be associated with more than one value of independent variable.b In order to examine a quadratic function, consider another sequence of numbers 1, 4, 9, 16, 25, …. Does one see a pattern in the development of this sequence? Each number is a square of its rank in the sequence. Indeed, 1 ¼ 12 ; 4 ¼ 22 ; 9 ¼ 32 ; 16 ¼ 42 ; 25 ¼ 52 , and so on. This observation leads to the following obvious algebraic generalization x n ¼ n2 :

ð10:4Þ

Formula (10.4) is conceptually different from formula (10.2) and is similar to formula (10.3) as both (10.3) and (10.4) uniquely define any term of the corresponding sequence as a function of its rank—linear in the former case and quadratic in the latter case. How can one derive a recursive definition of the sequence of square numbers from formula (10.4)? To this end, one has to find the difference xn
xn
1 ¼ n2
ðn
1Þ2 ¼ ðn
n þ 1Þðn þ n
1Þ ¼ 2n
1, whence for all n = 2, 3, 4, …, the relations xn ¼ xn
1 þ 2n
1; x1 ¼ 1

ð10:5Þ

describe recursively the entire sequence of squared integers. One can see that whereas some numeric patterns (sequences) is easier to describe recursively, other patterns (sequences) is easier to describe through a closed formula which provides a functional description of a pattern. Nonetheless, it appears that as patterns become more sophisticated, their recursive description is easier to develop than their functional description when each term is a function of its rank. Once again, a classic example deals with the sequence of Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, …. Comparing formulas presented in Chap. 9, Sect. 9.4.2, one can see that the closed formula (see also Chap. 6, Sect. 6.3.3) is unexpectedly complex, and, as it was already mentioned, this complexity demonstrates one of the most profound mathematical ideas—representation of one type of numbers (e.g., natural numbers) through another type of numbers (e.g., irrational numbers). Such is the closed representation of the sequence 1, 1, 2, 3, 5, 8, 13, … (Fibonacci numbers Fn ) in the form pffiffi pffiffi Fn ¼ p1ffiffi5 ½ð1 þ2 5Þn
ð1
2 5Þn ; n ¼ 1; 2; 3; . . .. At the same time, the sequence 2, 3, 5, 8, 12, 17, 23, …, having recursive definition in the form xn ¼ xn
1 þ n
1; x1 ¼ 2, 2 can be represented as a (quadratic) function of n as follows: xn ¼ n
n2 þ 4.

10.2

From Patterns to Functions

309

Indeed, adding the following n – 1 relations x2 ¼ x1 þ 1; x3 ¼ x2 þ 2; . . .; xn ¼ xn
1 þ n
1, (using indices as counters) yields x2 þ x3 þ    þ xn ¼ x1 þ x2 þ    þ xn
1 þ ð1 þ 2 þ    þ n
1Þ whence, due to formula (9.1) (Chap. 9), xn ¼ x1 þ ðn
1Þn ¼ 2 þ n 2
n ¼ n
n2 þ 4. 2 The method of transition from a recursive formula to the corresponding closed formula by decomposing the former into its step-by-step linkages to reach x1 from xn does not work already for such seemingly simple recursive definition as xn ¼ 3xn
1 þ 1; x1 ¼ 1. Without going into details mathematically,3 a closed formula for the last sequence each term of which is one greater than three times the previous term, namely, 1, 4, 13, 40, 121, …, has the (exponential) form xn ¼ 12 ð3n
1Þ; n ¼ 1; 2; 3; . . .. Such difference in complexity of the three closed formula is due to the fact that arithmeric rules that govern the recursive definitions xn ¼ xn
1 þ xn
2 , xn ¼ xn
1 þ n
1, and xn ¼ 3xn
1 þ 1 are quite different. This is an example of why teacher candidates preparing to teach middle school mathematics not only need to have experience in examining “how different types of equations are used for different purposes” (Conference Board of the Mathematical Sciences, 2012, p. 43), but also need to understand (in addition to being familiar with available on-line digital tools) that, similarly to the change of an exponent in a polynomial equation, a slight change of a coefficient in a recursive equation may lead to a significant qualitative change of the corresponding solution. 2

10.3

2

From Visual to Symbolic

One important aspect in the study of visual patterns is the development of their numeric description which can then be generalized to an algebraic form. Recent standards for preparing middle school mathematics teachers in the United States emphasize the importance of having “strong understanding of algebraic thinking, noticing the central role of generalization and the use of variables to represent numbers” (Association of Mathematics Teacher Educators, 2017, p. 97). Whereas the so-called AB-patterns can serve for an early development of algebraic thinking (see Chap. 9), many patterns cannot be described using the “AB” language. The major didactical objective of using the “AB” language in the description of patterns is to demonstrate to young learners of mathematics that letters may represent different combinations of repeated images and the pattern , in which n repetitions of A are followed by m repetitions of B, can still be described as an AB-pattern.

3

A free on-line version of Wolfram Alpha provides solutions for various recurrences, both in symbolic and numeric forms; e.g., see Sect. 4.10 of Chap. 4. Also, the OEIS provides closed formulas for a great numbers of integer sequences, given their first few terms.

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Fig. 10.1 Wertheimer’s row of dots: What can one see?

Human’s perception of repetitions may vary in a number of ways. As it was mentioned in the introduction to this chapter, one of the key principles of Gestalt psychology concerns the constructive aspect of perception by which an individual can recognize in an image more information than the image, in the absence of their insight, transmits. That is why, the question of what one can see in an image is a critical one for Gestalt psychologists. For example, Max Wertheimer, one of the founders of Gestalt psychology, discussed the issue of discovery through observation of certain arrangements of objects. Among many examples, he presented a row of dots arranged in equidistant pairs (Fig. 10.1) and was curious of “what is actually seen” (Wertheimer, 1938, p. 72, italics in the original). What Wertheimer seems to be saying is that our perception is not definitive as many things may be hidden from view and one needs insight in order to reveal them. The Law of Proximity in Gestalt psychology states that perception of an individual usually arranges in groups those objects that are similar and close to each other. According to this law, our perception, perhaps, due to reliance on subitizing (Kaufman et al., 1949), sees in the row of dots (Fig. 10.1) the AB-pattern rather than the pattern A-BAB-ABABA-BABABAB-…. How can one develop new knowledge proceeding from what is known? How can one be helped to see a specific in the general? Productive thinking or insight includes seeing unfamiliar in familiar and vice versa, seeing familiar in unfamiliar. While it is unlikely seeing in the dots of Fig. 10.1 the pattern A-BAB-ABABABABABAB, a transition from visual to symbolic, that is, seeing and formulating a rule according to which a pattern develops, can be carried out through counting the dots, thereby, enriching a visual milieu by numbers. By counting the dots (from left to right), one attaches to the dots the labels in the form of the natural numbers 1, 2, 3, 4, 5, …. Now, the Law of Proximity does not work as natural numbers can be put in groups in many different ways. This, in turn, enables many mathematical questions that motivate problem solving. A simple question that can be asked is: If the row of dots in Fig. 10.1 continues as long as one wishes, which dot of a pair, the first or the second, would be given label 100? Counting by twos is the justification of an answer that it would be the second dot of the 50th pair. One can continue counting so that once the row of dots is replaced by the row of labels 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …, one can create and thus see a variety of arrangements. One arrangement can be described as A-BA-BAB-ABAB-ABABA-BABABA- … by putting natural numbers in the groups (1), (2, 3), (4, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14, 15), (16, 17, 18, 19, 20, 21), …, so that the cardinality of each group is one greater than that of the previous group. Another arrangement can be described as A-BAB-ABABA-BAB ABAB-ABABABABA-…by putting natural numbers in the groups (1), (2, 3, 4), (5, 6, 7, 8, 9), (10, 11, 12, 13, 14, 15, 16), (17, 18, 19, 20, 21, 22, 23, 24, 25), …, so that the cardinality of each group is equal to that of the previous group plus two. These ways of grouping natural numbers can continue to have the cardinalities of two neighboring groups differ by three, four, five, and so on.

10.3

From Visual to Symbolic

311

Knowing polygonal numbers (Chap. 9, Sect. 9.4) allows one to notice that in each pair of parentheses in the first grouping the largest number is a triangular number and the smallest number is the previous triangular number plus one. Indeed, the largest numbers are 1, 3, 6, 10, 15, 21, …, and the smallest numbers are 1 ¼ 0 þ 1; 4 ¼ 3 þ 1; 7 ¼ 6 þ 1; 11 ¼ 10 þ 1; 16 ¼ 15 þ 1; . . .. Formally, this statement can be proved by noting that if in the group of rank n the last number is the triangular number tn , then the sum tn þ ðn þ 1Þ, where n + 1 is the cardinality of the (n + 1)-st group, is equal to the last number in the latter group. That is, tn þ ðn þ 1Þ ¼ tn þ 1 . Similarly, in each pair of parentheses in the second grouping the largest number is a square number and the smallest number is the previous square number plus one. Indeed, the largest numbers are 1, 4, 9, 16, 25, …, and the smallest numbers are 1 ¼ 0 þ 1; 2 ¼ 1 þ 1; 5 ¼ 4 þ 1; 10 ¼ 9 þ 1; 17 ¼ 16 þ 1; . . .. Formally, this statement can be proved by noting that if in the group of rank n the last number is the square number sn , then the sum sn þ ð2n þ 1Þ, where 2n + 1 is the cardinality of the (n + 1)-st group, is equal to the last number in the latter group. That is, sn þ ð2n þ 1Þ ¼ sn þ 1 . As an aside, note that this recognition of the mathematical nature of the first and the last terms in each group facilitates finding the sum of numbers in the nth group of either way of grouping natural numbers. Moreover, the above-mentioned continuation of grouping would bring to light pentagonal, hexagonal, heptagonal, and, more generally, m-gonal numbers (Chap. 9, Sect. 9.4). Mathematics of polygonal numbers, nowadays known as elementary number theory, goes back to ancient Greece (the third century B.C.) when integers, while being abstractions, were considered as descriptors of concrete objects forming certain geometric patterns. Triangular numbers describe evolving triangles, square numbers—evolving squares, pentagonal numbers—evolving pentagons, and so on. In particular, connection of polygonal numbers to arithmetic sequences formed by the cardinalities of the above-mentioned groups of numbers was discovered by Diophantus4 (Heath, 1910). Once again, history of mathematics teaches us to appreciate concreteness as a motivation for abstraction. This may explain why Gestalt psychology is interested in studying of what one can see in an image in the presence (or absence) of insight. That is, visual, something that we see, is primary and its symbolic (e.g., numeric or algebraic) description is secondary. cBig Idea One of the best strategies to develop students’ interest in mathematics of the middle grades is to focus teaching on material that grows out of concrete situations with which students are familiar. The use of concreteness as motivation for abstraction, being rooted in classic psychological research, is commonly considered as a big idea of mathematics pedagogy.b

Diophantus of Alexandria—a Greek mathematician of the third century A.D., called “the father of algebra”.

4

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To illustrate the last point, another type of questions related to pattern recognition that stem from the first grouping of letters A and B can be as follows: if the pattern A-BA-BAB-ABAB-ABABA-… continues, what are the first and the last letters in the 100th group? How many letters A and B are there in the 100th group? To answer these questions, one can replace the AB-pattern by the numeric pattern (1), (2, 3), (4, 5, 6), (7, 8, 9, 10), (11, 12, 13, 14, 15), …. and then make an observation that the letter A is always at an odd place (as far as the entire pattern is concerned). This observation reduces the problem to finding the first and the last numbers in the 100th (numeric) group. Knowing that each such group ends with a triangular number the rank of which is the group’s number and begins with the previous triangular number increased by one, prompts one to compute t99 þ 1 ¼ 99100 þ 1 ¼ 4951 and t100 ¼ 100101 ¼ 5050. Therefore, the 100th (non-numeric) 2 2 group, having 100 letters, starts with A and ends with B thereby having 50 A’s and 50 B’s. Similarly, the pattern A-BAB-ABABA-BABABAB-ABABABABA-… (in which the letter A is still at an odd place within the entire pattern) can be explored by associating it with the corresponding numerical pattern.

10.4

Patterns Structured by Colors

Often, patterns are described in terms of colors. Consider the case of two colors shown in the diagram of Fig. 10.2 where the RED-BLUE (RB) combinations increase in size first by factor two to have RRBB, then by factor three to have RRRBBB, and so on; in other words, each color’s presence in the pattern increases by one with each new step.5 Numerically, the number of repetitions of each color in an “RB”-combination varies according to the sequence 1, 2, 3, …, n, …, which is an arithmetic sequence with difference d = 1. Through counting, the letters (that represent colors) in Fig. 10.2 can be put into one-to-one correspondence with numeric labels, thereby enriching the visual milieu by integers. One can see that this visual pattern provides a context which can be used to formulate several mathematical questions. That is, the rule formulated in context furnishes a question with the definitive interpretation. There are many ways such color-structured patterns can be developed. For example, the number of colors can be greater than two, or the increase in the repetition of colors at each step can vary according to another number sequence. In this context, activities related to the exploration of patterns can include two types of questions: questions seeking information and questions requesting explanation of the information offered (Isaacs, 1930). Whereas 5

This pattern was proposed by a student of one of the authors (SA) in a master’s childhood education program when the topic of patterns in the context of early algebra was discussed. Although the pattern cannot be described as an AB pattern taught already in preschool, teacher candidates taking part in this discussion were advised, in the spirit of Montessori (1917), to see the pattern through the lens of “the recognition of new phenomena, their reproduction and utilization” (p. 73) as important objectives of education.

10.4

Patterns Structured by Colors

R

B

R

R

313

B

B

R

R

R

B

B

B

Fig. 10.2 Which color, red or blue is in the 100th place?

information about patterns can be obtained through a specially designed computational environment, the same environment can be used to support addressing requests for explanation. An interesting aspect of such computerization is that ideas requiring insight in the transition from visual to numeric are more difficult to communicate to a digital tool through writing a computer program than those ideas that may come to mind first through a more traditional thinking. Such spreadsheet-based computational environment is introduced in Sect. 10.8 at the end of this chapter.

10.4.1 From Colors to Functions To begin, consider the following question (Fig. 10.2): If the pattern is extended, which color, red or blue, does the 100th place have? There are several ways to associate letters R and B (representing colors) with numeric labels. For example, one can describe numerically the last red cell in each combination of equal quantities of R’s and B’s. Such approach brings one very close to the answer describing the color of the 100th place: those red cells have labels 1, 4, 9, 16, 25, and so on. What is special about these numbers? They are square numbers. Furthermore, the numbers are squares of the ranks of “RB”combinations in the evolving patterns. So, the number 100, a square number, will be on the list. The equality 102 = 100 implies that the label 100 belongs to the “RB”-combination of rank 10. It is the use of insightful strategy that provided a quick answer to apparently a non-obvious question. Numerical evidence brings about the function f1 ðnÞ ¼ n2 which maps the rank of an “RB”-combination to the position of the last R within it. Note that instead of focusing on the numeric position of the last R in each “RB”combination, one can focus on the numeric position of the first R in such combinations. This focus yields the sequence 1, 3, 7, 13, 21, …. In order to find the function that relates the rank n of each term to its value f2 ðnÞ, one can use either the OEIS® or (free on-line version of) Wolfram Alpha which offer (provided that more than three terms are entered into a search box) the function f2 ðnÞ ¼ n2
n þ 1. Indeed, f2 ð1Þ ¼ 1; f2 ð2Þ ¼ 3; f2 ð3Þ ¼ 7, and so on.6 In this notation, n stands for the 6

Note that the above comment regarding the need to enter more than three terms of the sequence is due to relations 1 = 21 − 1, 3 = 22 − 1, and 7 = 23 − 1 which appear being more plausible than the values of the quadratic function f ðnÞ ¼ n2
n þ 1. Only the presence of the number 13 ð6¼ 24
1Þ as the fourth term prompts both OEIS and Wolfram Alpha to recognize the four terms as the values of a quadratic function (although three values do define a quadratic function). This

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number of repetitions of either letter in the nth “RB”-combination. Alternatively, one can recognize that each term of the five-number sequence 1, 3, 7, 13, 21 is one greater than a product of two consecutive whole numbers: 1 ¼ 0  1 þ 1; 3 ¼ 2  1 þ 1; 7 ¼ 2  3 þ 1, and so on. This may prompt one to conjecture that f2 ðnÞ ¼ ðn
1Þn þ 1 ¼ n2
n þ 1. Now, one can check to see that f2 ð10Þ ¼ 91 and f2 ð11Þ ¼ 111. In the first case, the 10th “RB”-combination, having the first R on the 91st position, would have another nine R’s, thus having an R on the 100th place. In the second case, the 11th “RB”-combination, having the first R on the 111th position, would be preceded by ten B’s, thus having a B on the 101st position and, consequently, an R on the 100th position. One can see that although focusing on the first R in an “RB”-combination makes it possible to associate color with the 100th position, the focus on the last R allowed for a more efficient solution. It is a relative simplicity of the function f1 ðnÞ ¼ n2 in comparison with f2 ðnÞ ¼ n2
n þ 1 that provided such problem-solving efficiency. In fact, the latter function can be derived from the former one through the following observation: in order to reach the first R from the last R in the nth “RB”-combination, one has to make n − 1 steps backward; that is, to subtract n − 1 from n2 to get n2
n þ 1. cBig Idea While generalizing from special cases afforded by numerical evidence is a big idea of mathematics, one must use this idea only after having some provisional speculation of how generalization might look like.b One can also focus on the positions of the last or the first B in an “RB”combination. In the former case, the positions of the last B’s would be described by the sequence 2, 6, 12, 20, …, where each term is the product of two consecutive natural numbers. This brings about a new function, f3 ðnÞ ¼ nðn þ 1Þ. The relation f3 ð9Þ ¼ 90 implies that the last B in the 9th “RB”-combination resides at the 90th place. Because this last B is followed by ten R’s, once again, the 100th place would have an R. The latter case (focusing on the first B), yields the sequence 2, 5, 10, 17, … each term of which is one greater than the square of its rank, as 2 ¼ 12 þ 1; 5 ¼ 22 þ 1, and so on. This brings about another function, f4 ðnÞ ¼ n2 þ 1. The relation f4 ð10Þ ¼ 101 identifies the position of the first B in the 10th “RB”-combination. Once again, the 100th place would have an R.

suggests three things: (i) when generalizing from numerical evidence, one should have some idea about what to expect from generalization and in order to guide students to these understandings, the teacher must be aware of and have experience with them; (ii) inductive generalization should be followed by mathematical induction (or any other) proof; and (iii) a wealth of information with an easy access, often requiring more than basic skills to deal with, attests to the duality of positive and negative affordances of technology.

10.4

Patterns Structured by Colors

315

The above exploration of the pattern shown in Fig. 10.2 brings to light the functions f1 ðnÞ ¼ n2 ; f2 ðnÞ ¼ n2
n þ 1; f3 ðnÞ ¼ nðn þ 1Þ; f4 ðnÞ ¼ n2 þ 1: When n = 10, the smallest value is provided by f2 ð10Þ ¼ 91 and the largest value is provided by f3 ð10Þ ¼ 110. In the “RB”-combination of rank 10, the difference between the positions of the last B and the first R increased by one, (110 − 91) + 1 = 20, is the length of this “RB”-combination.

10.4.2 From Empirical Conjectures to Mathematical Induction Proof The introduction of the four quadratic functions in the previous section was based on a limited numerical evidence; in other words, the generalization from “partial representations of functions” (Common Core State Standards, 2010, p. 52) was due to empirical induction. The four functions were conjectured to describe the dependence of the rank of an “RB”-combination on the position of either the last/ first R or B in that combination. In order to turn an inductive conjecture into a formal proposition, the conjecture has to be proved. One type of proof appropriate for verifying empirical results is the proof by mathematical induction (already used with detailed explanation in Chap. 9, Sect. 9.3.1). To this end, four propositions will be formulated and proved. Proposition 10.1 The position of the last R in the nth “RB”-combination (Fig. 10.2) is described by the function f1 ðnÞ ¼ n2 . Proof When n = 1 we have f1 ð1Þ ¼ 1—a true equality (as the R in a two-letter combination RB is both the first and the last R in that combination). Assuming that the conjecture f1 ðnÞ ¼ n2 is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that f1 ðn þ 1Þ ¼ ðn þ 1Þ2 . To this end, note that in the nth “RB”-combination the last R is followed by n B’s and then by (n + 1) R’s. That is, f1 ðn þ 1Þ ¼ f1 ðnÞ þ n þ ðn þ 1Þ ¼ n2 þ 2n þ 1 ¼ ðn þ 1Þ2 : Proposition 10.2 The position of the first R in the nth “RB”-combination (Fig. 10.2) is described by the function f2 ðnÞ ¼ n2
n þ 1. Proof When n = 1 we have f2 ð1Þ ¼ 1– a true equality. Assuming that the conjecture f2 ðnÞ ¼ n2
n þ 1 is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that f2 ðn þ 1Þ ¼ ðn þ 1Þ2
ðn þ 1Þ þ 1 ¼ n2 þ n þ 1. To this end, note that in the nth “RB”-combination the

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first R is followed by (n − 1) R’s and then by n B’s. One more step leads to the first R in the next “RB”-combination. That is, f2 ðn þ 1Þ ¼ f2 ðnÞ þ n þ ðn
1Þ þ 1 ¼ n2
n þ 1 þ 2n ¼ n2 þ n þ 1: Proposition 10.3 The position of the last B in the nth “RB”-combination (Fig. 10.2) is described by the function f3 ðnÞ ¼ n2 þ n. Proof When n = 1 we have f3 ð1Þ ¼ 2—a true equality (as the B in a two-letter combination RB is both the first and the last B in that combination). Assuming that the conjecture f3 ðnÞ ¼ n2 þ n is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that f3 ðn þ 1Þ ¼ ðn þ 1Þ2 þ n þ 1 ¼ n2 þ 3n þ 2. To this end, note that in the nth “RB”-combination the last B is followed by (n + 1) R’s and then by (n + 1) B’s in the (n + 1)st “RB”-combination. That is, f3 ðn þ 1Þ ¼ f3 ðnÞ þ ðn þ 1Þ þ ðn þ 1Þ ¼ n2 þ n þ 2n þ 2 ¼ n2 þ 3n þ 2. Proposition 10.4 The position of the first B in the nth “RB”-combination (Fig. 10.2) is described by the function f4 ðnÞ ¼ n2 þ 1. Proof When n = 1 we have f4 ð1Þ ¼ 2—a true equality. Assuming that the conjecture f4 ðnÞ ¼ n2 þ 1 is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that f4 ðn þ 1Þ ¼ ðn þ 1Þ2 þ 1 ¼ n2 þ 2n þ 2. To this end, note that in the nth “RB”-combination the first B is followed by (n − 1) B’s and by (n + 1) R’s in the next “RB”-combination. One more step leads to the first B in that “RB”-combination. That is, f4 ðn þ 1Þ ¼ f4 ðnÞ þ ðn
1Þ þ ðn þ 1Þ þ 1 ¼ n2 þ 1 þ 2n þ 1 ¼ n2 þ 2n þ 2:

cBig Idea Why do we need to be concerned with proof in the middle school mathematics? Why is proof a big mathematical idea to be introduced at that grade level? There are always two steps to follow when exploring patterns in the middle grades: recognizing a pattern and explaining the pattern. It is the second step that requires proof—a mutually accepted evidence that something is true. Seeking proof should not diminish students’ trust in the teacher; rather, the focus on proof indicates that the teacher’s authority is not the only means for students to be confident that something they attained in mathematics is true.b

10.4

Patterns Structured by Colors

317

10.4.3 Two-Color Pattern Guided by Consecutive Odd Numbers Consider a two-color pattern and associated counting labels shown in Fig. 10.3. One can see that the number of R’s and B’s in an “RB”-combination varies according to the sequence 1, 3, 5, …, 2n − 1, …, which is an arithmetic sequence with difference d = 2 (alternatively, the sequence of consecutive odd numbers). Once again, four functions can be introduced to relate the rank of the nth “RB”combination to the position of the last/first R or B within this new pattern. At the conclusion of the introduction of the four functions, the following query will be addressed: If the pattern shown in Fig. 10.3 is extended, which color, red or blue, does the 1225th place have?

10.4.3.1

The Case of the Last R

Focusing on the last R gives the following sequence of numbers: 1, 5, 13, 25. Using either Wolfram Alpha or OEIS®, the function g1 ðnÞ ¼ 2n2
2n þ 1 which describes the last R’s position in the extended pattern (Fig. 10.3) can be conjectured. Here n is the rank of the corresponding “RB”-combination with the total of R’s and B’s equal to 2(2n − 1)—twice the odd number of rank n. Proposition 10.5 The position of the last R in the nth “RB”-combination (Fig. 10.3) defined by d = 2 is described by the function g1 ðnÞ ¼ 2n2
2n þ 1. Proof When n = 1 we have g1 ð1Þ ¼ 1—a true equality. Assuming that the conjecture g1 ðnÞ ¼ 2n2
2n þ 1 is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that g1 ðn þ 1Þ ¼ 2ðn þ 1Þ2
2ðn þ 1Þ þ 1 ¼ 2n2 þ 2n þ 1. To this end, note that in the nth “RB”-combination the last R is followed by (2n – 1) B’s and then by (2n + 1) R’s in the (n + 1)st “RB”-combination. That is, g1 ðn þ 1Þ ¼ g1 ðnÞ þ 2n
1 þ 2n þ 1 ¼ 2n2
2n þ 1 þ 4n ¼ 2n2 þ 2n þ 1:

10.4.3.2

The Case of the First R

Focusing on the first R (Fig. 10.3) gives the following sequence of numbers: 1, 3, 9, 19. Using either Wolfram Alpha or OEIS®, the function g2 ðnÞ ¼ 2n2
4n þ 3 which describes the last R’s position in the extended pattern can be conjectured. As 1 R

2 B

3 R

4 R

5 R

6 B

7 B

8 B

9 10 11 12 13 14 15 16 17 18 R R R R R B B B B B

Fig. 10.3 Which color, red or blue, is in the 1225th place?

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before, here n is the rank of the corresponding “RB”-combination with the total of R’s and B’s equal to 2(2n − 1). Proposition 10.6 The position of the first R in the nth “RB”-combination (Fig. 10.3) defined by d = 2 is described by the function g2 ðnÞ ¼ 2n2
4n þ 3. Proof When n = 1 we have g2 ð1Þ ¼ 1—a true equality. Assuming that the conjecture g2 ðnÞ ¼ 2n2
4n þ 3 is true, one has to prove that it remains true when n is replaced by n + 1; that is, g2 ðn þ 1Þ ¼ 2ðn þ 1Þ2
4ðn þ 1Þ þ 3 ¼ 2n2 þ 1. To this end, note that in the nth “RB”-combination the first R is followed by (2n − 2) R’s and then by (2n − 1) B’s. Then one more step is needed to reach the first R in the (n + 1)th “RB”-combination. That is, g2 ðn þ 1Þ ¼ g2 ðnÞ þ ð2n
2Þ þ ð2n
1Þ þ 1 ¼ 2n2
4n þ 3 þ 4n
2 ¼ 2n2 þ 1:

10.4.3.3

The Case of the Last B

Focusing on the last B (Fig. 10.3) gives the following sequence of numbers: 2, 8, 18, 32. Using either Wolfram Alpha or OEIS®, the function g3 ðnÞ ¼ 2n2 which describes the last B’s position in the extended pattern can be developed. As before, here n is the rank of the corresponding “RB”-combination with the total of R’s and B’s equal to 2(2n − 1). Proposition 10.7 The position of the last B in the nth “RB”-combination (Fig. 10.3) defined by d = 2 is described by the function g3 ðnÞ ¼ 2n2 . Proof When n = 1 we have g3 ð1Þ ¼ 2—a true equality. Assuming that the conjecture g3 ðnÞ ¼ 2n2 is true, one has to prove that it remains true when n is replaced by n + 1; that is, g3 ðn þ 1Þ ¼ 2ðn þ 1Þ2 ¼ 2n2 þ 4n þ 2. To this end, note that the last B in the nth “RB”-combination is followed by (2n + 1) R’s and then by (2n + 1) B’s in the (n + 1)st “RB”-combination. That is, g3 ðn þ 1Þ ¼ g3 ðnÞ þ 2ð2n þ 1Þ ¼ 2n2 þ 4n þ 2:

10.4.3.4

The Case of the First B

Focusing on the first B (Fig. 10.3) gives the following sequence of numbers: 2, 6, 14, 26. Using either Wolfram Alpha or OEIS®, the function g4 ðnÞ ¼ 2ðn2
n þ 1Þ which describes the last R’s position in the extended pattern can be developed. As before, here n is the rank of the corresponding “RB”- combination with the total of R’s and B’s equal to 2(2n − 1).

10.4

Patterns Structured by Colors

319

Proposition 10.8 The position of the first B in the nth “RB”-combination (Fig. 10.3) defined by d = 2 is described by the function g4 ðnÞ ¼ 2ðn2
n þ 1Þ. Proof When n = 1 we have g4 ð1Þ ¼ 2—a true equality. Assuming that the conjecture g4 ðnÞ ¼ 2ðn2
n þ 1Þ is true, one has to prove that it remains true when n is replaced by n + 1; that is, g4 ðn þ 1Þ ¼ 2½ðn þ 1Þ2
ðn þ 1Þ þ 1 ¼ 2ðn2 þ n þ 1Þ. To this end, note that the first B in the nth “RB”-combination is followed by (2n − 2) B’s and then by (2n + 1) R’s in the (n + 1)st “RB”-combination. One step is needed to reach the first B in the (n + 1)st “RB”-combination. That is, g4 ðn þ 1Þ ¼ g4 ðnÞ þ ð2n
2Þ þ ð2n þ 1Þ þ 1 ¼ 2n2
2n þ 2 þ 4n ¼ 2ðn2 þ n þ 1Þ.

10.4.3.5

Adressing the Query: Which Color Does the 1225th Position Have?

Because the function g1 ðnÞ ¼ 2n2 , describing the position of the last B within the “RB”-combination (Fig. 10.3) of length 2(2n − 1), has the simplest form in comparison with other three functions developed through mathematical induction in Propositions 10.5, 10.6 and 10.8, one way to address the query is to find n such that 2n2 is the closest to 1225. Noting that 242 ¼ 576; 252 ¼ 625; 2  576 ¼ 1152, and 2  625 ¼ 1250, the last number is the closest of the form 2n2 to 1225. Indeed, 1250
1225 ¼ 25 and 1225
1152 ¼ 73. Therefore, the last B, positioned at the 1250th place, will belong to the 49-long chain of B’s (where 49 ¼ 2  25
1). Making 25 steps backward from the 1250th place leads to a B at the 1225th place. One can verify the correctness of this conclusion by considering the function g1 ðnÞ ¼ 2n2
2n þ 1 which describes the position of the last R in the “RB”combination of length 2(2n – 1). Now, one has to find the largest n such that the inequality 2n2
2n þ 1  1225 holds true. The inequality is equivalent to n2
n  612 whence n = 25 as the value g1 ð25Þ ¼ 2  252
2  25 þ 1 ¼ 1201 is the closest to 1225. Indeed, as g1 ð26Þ ¼ 1301 we have 1225
1201 ¼ 24 and 1301
1225 ¼ 76. Therefore, the last R in the “RB”-combination of length 98, where 98 ¼ 2ð2  25
1Þ, will be 24 steps away from a B located at the 1225th position within the same combination. Likewise, the verification that the 1225th position is colored blue can be carried out by focusing on either the first R or the first B.

10.4.4 More Patterns Defined by an Arithmetic Sequence Now, consider a two-color pattern and associated counting labels shown in Fig. 10.4. One can see that the number of R’s and B’s in an “RB”-combination varies according to the sequence 1, 4, 7, …, 3n − 2, …, which is an arithmetic sequence with difference d = 3.

320 1 R

10 2 B

3 R

4 R

5 R

6 R

7 B

8 B

Patterns and Functions

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 B B R R R R R R R B B B B B B B R

Fig. 10.4 Two-color pattern guided by arithmetic series with difference d = 3

Once again, four functions can be introduced to relate the rank of the nth “RB”combination to the position of the last/first R or B within the pattern (Fig. 10.4). Focusing on the last R gives the following sequence of numbers: 1, 6, 17, 34. Using either Wolfram Alpha or OEIS®, the function h1 ðnÞ ¼ 3n2
4n þ 2 which describes the last R’s position in the extended pattern can be developed. Here n is the rank of the corresponding “RB”-combination with the total of R’s and B’s equal to 2(3n − 2). Alternatively, assuming that such a function has to be a quadratic function of the form an2 þ bn þ c, one can enter in the input box of Wolfram Alpha (in the context of its free on-line version) the command “solve a + b + c = 1, 4a + 2b + c = 6, 9a + 3b + c = 17” to get the solution a = 3, b = -4, c = 2 (Fig. 10.5). Note that in the context of Wolfram Alpha Pro (available by subscription only), one can enjoy reviewing 14 steps leading to the final result. One can show that the conjecture h1 ðnÞ ¼ 3n2
4n þ 2 remains true when n is replaced by n + 1; that is, h1 ðn þ 1Þ ¼ 3ðn þ 1Þ2
4ðn þ 1Þ þ 2 ¼ 3n2 þ 2n þ 1. Indeed, the last R in the nth “RB”-combination (Fig. 10.4) is followed by (3n − 2) B’s and then by (3n + 1) R’s in the (n + 1)st “RB”-combination. That is, h1 ðn þ 1Þ ¼ h1 ðnÞ þ ð3n
2Þ þ ð3n þ 1Þ ¼ 3n2
4n þ 2 þ 6n
1 ¼ 3n2 þ 2n þ 1. Now, a different strategy of obtaining other three functions can be used. One can proceed using the function h1 ðnÞ ¼ 3n2
4n þ 2 describing the last R in the nth “RB”-combination (Fig. 10.4). One has to make 3n − 3 steps backward to reach the first R in the nth “RB”-combination; that is, to subtract 3n − 3 from 3n2
4n þ 2 to get h2 ðnÞ ¼ 3n2
7n þ 5. Likewise, one has to make 3n − 2 steps forward to reach the last B in the nth “RB”-combination; that is, to add 3n − 2 and 3n2
4n þ 2 to get h3 ðnÞ ¼ 3n2
n. Finally, to reach the first B in the nth “RB”-combination one has to make just one step forward; that is, to add 1 and 3n2
4n þ 2 to get h4 ðnÞ ¼ 3n2
4n þ 3. In that way, the following four quadratic functions Fig. 10.5 From numerical evidence to finding f(n) using Wolfram Alpha

10.4

Patterns Structured by Colors

321

h1 ðnÞ ¼ 3n2 þ 2n þ 1, h2 ðnÞ ¼ 3n2
7n þ 5, h3 ðnÞ ¼ 3n2
n, and h4 ðnÞ ¼ 3n2
4n þ 3 describing, respectively, the last R, the first R, the last B, and the first B, have been developed. Note that the strategy used above is perfectly rigorous and one does not need to use mathematical induction proof now. The next step would be to recognize a pattern among the four-function triple fi ðnÞ; gi ðnÞ; hi ðnÞ; i ¼ 1; 2; 3; 4, as the difference d of an arithmetic sequence that defines the pattern (d = 1 in Fig. 10.2, d = 2 in Fig. 10.3, d = 3 in Fig. 10.4) varies along the set of natural numbers. Remark 10.1 Note that f(n) is a quadratic function of n if D2 ðnÞ ¼ D1 ðn þ 1Þ
D1 ðnÞ, where D1 ðnÞ ¼ f ðn þ 1Þ
f ðnÞ, is a non-zero constant. For example, for f ðnÞ ¼ n2 we have D1 ðnÞ ¼ ðn þ 1Þ2
n2 ¼ 2n þ 1 and D2 ðnÞ ¼ 2ðn þ 1Þ þ 1
ð2n þ 1Þ ¼ 2 6¼ 0. In the case of two colors, R and B, the increase of which is guided by an arithmetic sequence with difference d, one can show that the positions of the last R in “RB”-combinations are described by a quadratic sequence as follows. Let an be the position of the last R in the nth “RB”-combination. Then, one can see in the diagram

that the positions of the last R in

the next three “RB”-combinations are described by the following three terms an þ 1 ¼ an þ 2n þ d; an þ 2 ¼ an þ 1 þ 2n þ 3d ¼ an þ 4n þ 4d; an þ 3 ¼ an þ 2 þ 2n þ 5d ¼ an þ 6n þ 9d: Therefore, as shown in the next diagram, for any four consecutive terms, the second difference between the terms is equal to 2d.

However, if only the third difference of the sequence an is a non-zero constant, then this sequence is a cubic sequence. Indeed, consider the following two-color, R and B, pattern

in which the length of R’s and B’s in each “RB”-combination is a square of its rank. The development of squares is not guided by an arithmetic sequence. In general, the five consecutive terms, an ; an þ 1 ; an þ 2 ; an þ 3 , and an þ 4 that describe the positions of the last R can be described through the diagram

322

10

Patterns and Functions

from where it follows that an ¼ n; an þ 1 ¼ 5n þ 1; an þ 2 ¼ 13n þ 6; an þ 3 ¼ 25n þ 19; an þ 4 ¼ 41n þ 44. Having five consecutive terms of the sequence an allows one to find its third D3 ðnÞ difference as follows:

One can see that whereas the second difference D2 ðnÞ depends on n, the third difference D3 ðnÞ ¼ 4 6¼ 0. However, if D3 ðnÞ ¼ 0 then D2 ðnÞ ¼ const and the corresponding sequence is a quadratic sequence of n. In the case of p colors, we have the following diagram

from where it follows that the second difference D2 ðnÞ ¼ pd—a non-zero constant:

Knowing how to find the second or the third differences of a sequence is important to decide whether the sequence is either quadratic or cubic one.

10.5

10.5

Finding Patterns Formed by Functions

323

Finding Patterns Formed by Functions

In this section, patterns formed by quadratic functions which resulted from algebraic generalizations of numeric patterns will be explored. The sequence of generalization activities—from visual to numeric to algebraic—was made possible by a conceptual approach to the ideas of middle school algebra with quadratic functions forming the foundation for learning mathematical ideas in high school. Once again, in what follows, four types of functions can be considered: those related to the last/ first R and B for different values of differences of arithmetic sequences. The goal is to define a function of two variables: the rank n of an “RB”-combination and the difference d = m of the arithmetic sequence according to which the number of R’s and B’s increases. The following notation will be used: f1 ðn; mÞ—for the case of the last R, f2 ðn; mÞ—for the case of the first R, f3 ðn; mÞ—for the case of the last B, f4 ðn; mÞ— for the case of the first B. In what follows, it will be assumed that with the increase of m the four functions will continue being quadratic about n. Therefore, aiming at generalization, one would have to look for patterns formed by (depending on m) the coefficients of the quadratic function aðmÞn2 þ bðmÞn þ cðmÞ. In the words of Arnheim (1969), “the thinker … approaches the task [of generalization] with a preliminary notion of what the concept might be like” (p. 187).

10.5.1 Developing Formula for f 1 ðn; mÞ As it was conjectured through analyzing visual data and proved by the method of mathematical induction, when m = 1 we have f1 ðn; 1Þ ¼ n2 ; when m = 2 we have f1 ðn; 2Þ ¼ 2n2
2n þ 1; when m = 3 we have f1 ðn; 3Þ ¼ 3n2
4n þ 2. Is there a pattern formed by the coefficients of the three functions? In Fig. 10.6, visually developed numerical evidence is used to conjecture f1 ðn; 4Þ ¼ 4n2
6n þ 3 and then generalize to f1 ðn; mÞ ¼ mn2
2ðm
1Þn þ ðm
1Þ. In what follows, the formula an ¼ a1 þ mðn
1Þ for the nth term of an arithmetic sequence with difference m will be used (see Sects. 9.4.1–9.4.2 of Chap. 9). This formula implies that the length of the nth “RB”-combination guided by an arithmetic sequence with difference m is equal to 2[m(n − 1) + 1]. v i s u a l

data

conjecture generalizaon

d

1

2

3

4

m

a (m )

1

2

3

4

m

b (m )

-2x0

-2x1

-2x2

-2x3

-2(m -1)

c (m )

0

1

2

3

(m-1)

Fig. 10.6 Generalizing from the cases of the last R in an “RB”-combination

324

10

Patterns and Functions

Proposition 10.9 The position of the last R in the “RB”-combination of rank n and length 2[m(n − 1) + 1] is given by the function f1 ðn; mÞ ¼ mn2
2ðm
1Þ n þ ðm
1Þ. Proof When n = 1 we have f1 ð1; mÞ ¼ m
2ðm
1Þ þ ðm
1Þ ¼ 1—a true equality as, regardless of m, in the first “RB”-combination the single R belongs to the first place. Assuming that this conjecture is true, one has to prove that it remains true when n is replaced by n + 1; that is, one has to prove that f1 ðn þ 1; mÞ ¼ mðn þ 1Þ2
2ðm
1Þðn þ 1Þ þ ðm
1Þ ¼ mn2 þ 2mn þ m
2mn
2m þ 2n þ 2 þ m
1 ¼ mn2 þ 2n þ 1: To this end, note that in the nth “ RR. . .R BB. . .B ”-combination the last R is |fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} mðn
1Þ þ 1 mðn
1Þ þ 1

followed by [m(n − 1) + 1] B’s and then by (mn + 1) R’s in the (n + 1)st combination. That is, f1 ðn þ 1; mÞ ¼ f1 ðn; mÞ þ mðn
1Þ þ 1 þ mn þ 1 ¼ mn2
2ðm
1Þn þ m
1 ¼ mn2
2mn þ 2n þ m þ 2mn
m þ 1 ¼ mn2 þ 2n þ 1: This completes the proof by the method of mathematical induction. The development of formulas for the functions f2 ðn; mÞ; f3 ðn; mÞ and f4 ðn; mÞ, while taking a different route, will be supported by the rigor of this proof.

10.5.2 The Cases of the Position of the First R and the First/ Last B When moving from the last R to the first R within the same “RB”-combination of length 2[m(n − 1) + 1], one has to make m(n − 1) steps backward; that is, to subtract m(n − 1) from f1 ðn; mÞ ¼ mn2
2ðm
1Þn þ ðm
1Þ. Thus, f2 ðn; mÞ ¼ f1 ðn; mÞ
mðn
1Þ ¼ mn2
2ðm
1Þn þ ðm
1Þ
mðn
1Þ ¼ mn2
2mn þ 2n þ m
1
mn þ m ¼ mn2
3mn þ 2n þ 2m
1 ¼ mn2
ð3m
2Þn þ 2m
1: When moving from the last R to the first B (within the same “RB”-combination), one has to make just one step forward; that is, to add 1 to f1 ðn; mÞ ¼ mn2
2ðm
1Þn þ ðm
1Þ to get f3 ðn; mÞ ¼ mn2
2ðm
1Þn þ m. Finally, when moving from the last R to the last B within the same “RB”combination of length 2[m(n − 1) + 1], one has to make m(n − 1) + 1 steps forward; that is, to add m(n − 1) + 1 to f1 ðn; mÞ ¼ mn2
2ðm
1Þn þ ðm
1Þ: Thus,

10.5

Finding Patterns Formed by Functions

325

f4 ðn; mÞ ¼ f1 ðn; mÞ þ mðn
1Þ þ 1 ¼ mn2
2ðm
1Þn þ ðm
1Þ þ mðn
1Þ þ 1 ¼ mn2
2mn þ 2n þ m
1 þ mn
m þ 1 ¼ mn2
ðm
2Þn:

10.5.3 Alternative Verification of the Obtained Results Alternatively, the same results can be confirmed through recognizing patterns formed by the functions as follows. 10.5.3.1

The Case of the Position of the First R

As it was conjectured through analyzing visual data and proved by the method of mathematical induction, when m = 1 we have f2 ðn; 1Þ ¼ n2
n þ 1, when m = 2 we have f2 ðn; 2Þ ¼ 2n2
4n þ 3, when m = 3 we have f2 ðn; 3Þ ¼ 3n2
7n þ 5. Is there a pattern among the three functions? In Fig. 10.7, visually developed numerical evidence is used to conjecture that f2 ðn; 4Þ ¼ 4n2
10n þ 7 and then generalize to f2 ðn; mÞ ¼ mn2
ð3m
2Þn þ ð2m
1Þ.

10.5.3.2

The Case of the Position of the Last B

Likewise, as it was conjectured through analyzing visual data and proved by the method of mathematical induction, when m = 1 we have f3 ðn; 1Þ ¼ n2 þ n, when m = 2 we have f3 ðn; 2Þ ¼ 2n2 , when m = 3 we have f3 ðn; 3Þ ¼ 3n2
n. Is there a pattern formed by the three functions? In Fig. 10.8, visually developed numerical evidence is used to conjecture that f3 ðn; 4Þ ¼ 4n2
2n and then generalize to f3 ðn; mÞ ¼ mn2
ðm
2Þn. 10.5.3.3

The Case of the Position of the First B

In much the same way, as it was conjectured through analyzing visual data and proved by the method of mathematical induction, when d = 1 we have f4 ðn; 1Þ ¼ n2 þ 1, when d = 2 we have f4 ðn; 2Þ ¼ 2n2
2n þ 2, when d = 3 we v i s u a l

data

conjecture

generalizaon

d

1

2

3

4

m

a (m )

1

2

3

4

m

b (m )

-(3x1-2)

-(3x2-2)

-(3x3-2)

-(3x4-2)

-(3m -2)

c (m )

2x1-1

2x2-1

2x3-1

2x4-1

2m -1

Fig. 10.7 Generalizing from the cases of the first R in an “RB”-combination

326

10 v i s u a l

data

Patterns and Functions

conjecture generalizaon

d

1

2

3

4

m

a (m )

1

2

3

4

m

b (m )

-(1-2)

-(2-2)

-(3-2)

-(4-2)

-(m -2)

c (m )

0

0

0

0

0

Fig. 10.8 Generalizing from the cases of the last B in an “RB”-combination

v i s u a l

data

conjecture

generalizaon

d

1

2

3

4

m

a (m )

1

2

3

4

m

b (m )

-2(1-1)

-2(2-1)

-2(3-1)

-2(4-1)

-2(m -1)

c (m )

1

2

3

4

m

Fig. 10.9 Generalizing from the cases of the first B in an “RB”-combination

have f4 ðn; 3Þ ¼ 3n2
4n þ 3. Is there a pattern formed by the three functions? In Fig. 10.9, visually developed numerical evidence is used to conjecture that f4 ðn; 4Þ ¼ 4n2
6n þ 4 and then generalize to f4 ðn; mÞ ¼ mn2
2ðm
1Þn þ m.

10.6

The Case of Three Colors

Consider the case of three colors—red (R), blue (B), and green (G), when the nth “RBG”-combination of length 3n consists of equal quantity of each of the three colors. Let f1;3 ðn; 1Þ be a function which maps n to the position of the last R within the nth “RBG”-combination of length 3n. Then f1;3 ðn; 1Þ ¼ ð3n
1Þn . Indeed, one can 2 check to see that the positions of the last R in the “RBG” pattern guided by the sequence 1, 2, 3, …, n, …, are described by the terms of the sequence 1, 5, 12, 22, (see Chap. 9, …, which is the sequence of pentagonal numbers pn ¼ ð3n
1Þn 2 Sect. 9.4). This can also be confirmed by using Wolfram Alpha (in the context of its free on-line version). Let f1;3 ðn; 2Þ be a function which maps n to the position of the last R within the nth “RBG”-combination of length 3(2n − 1). One can check to see that in this case, the positions of the last R within the “RBG” pattern are guided by the sequence 1, 6, 17, 34, …. Using Wolfram Alpha, one can find out that this sequence is

10.6

The Case of Three Colors

327

described7 by the function f1;3 ðn; 2Þ ¼ 3n2
4n þ 2. In much the same way, one 2 can develop the functions f1;3 ðn; 3Þ ¼ 92 n2
15 2 n þ 4 and f1;3 ðn; 4Þ ¼ 6n
11n þ 6 which map n to the position of the last R in the the nth “RBG”-combinations of lengths 3(3n − 2) and 3(4n − 3), respectively. When m = 1, 2, 3, 4, the lengths of the nth combinations are, respectively, 3ð1  n
0Þ; 3ð2n
1Þ; 3ð3n
2Þ; 3ð4n
3Þ. From here, one can generalize that for any value of m the length of the nth “RBG”-combination is equal to 3½mn
ðm
1Þ ¼ 3ðmn
m þ 1Þ. Now, the four functions f1;3 ðn; iÞ; 1  i  4, can be written in the form f1;3 ðn; 1Þ ¼ 32 n2
12 n, f1;3 ðn; 2Þ ¼ 62 n2
82 n þ 2, f1;3 ðn; 3Þ ¼ 92 n2
15 2 n þ 4, 12 2 22 f1;3 ðn; 4Þ ¼ 2 n
2 n þ 6, from where the following generalization can be developed. Proposition 10.10 Let f1;3 ðn; mÞ represent a function that maps n to the position of the last R within the nth “RBG”-combinations of length 3(mn – m + 1). Then. f1;3 ðn; mÞ ¼

3m 2 7m
6 n
n þ 2m
2: 2 2

7m
6 3m
7m þ 6 þ 4m
4 Proof When n = 1 we have f1;3 ð1; mÞ ¼ 3m 2
2 þ 2m
2 ¼ 2 ¼ 1 – a true equality as, regardless of m, in the first “RBG”-combination the single R belongs both to the first and the last place. Assuming that the con7m
6 2 jecture f1;3 ðn; mÞ ¼ 3m 2 n
2 n þ 2m
2 is true, one has to show that

3m 7m
6 ðn þ 1Þ2
ðn þ 1Þ þ 2m
2 2 2 3m 2 3m 7mn 7m n þ 3mn þ
þ 3n
þ 3 þ 2m
2 ¼ 2 2 2 2 3m 2 m
6 ¼ n
n þ 1: 2 2

f1;3 ðn þ 1; mÞ ¼

Indeed, noting that the lengths of the string of B’s and G’s in the nth “RBG”combination and the string of R’s in the (n + 1)st “RBG”-combination are equal to 2½ðmðn
1Þ þ 1 and mn þ 1, respectively, so that the sum 2½mðn
1Þ þ 1 þ mn þ 1 is the distance between the last R’s in two consecutive “RBG”combinations, one can write

7 Using OEIS®, one gets the sequence 3n2 þ 2n þ 1 which looks different from the sequence 3n2
4n þ 2. However, the former and the latter sequences generate the same numbers by starting to change n from 0 and from 1, respectively.

328

10

Patterns and Functions

f1;3 ðn þ 1; mÞ ¼ f1;3 ðn; mÞ þ 2½mðn
1Þ þ 1 þ mn þ 1 3m 2 7m
6 n
n þ 2m
2 þ 3mn
2m þ 3 ¼ 2 2 3m 2 7m
6
6m 3m 2 m
6 ¼ n
nþ1 ¼ n
n þ 1: 2 2 2 2

10.7

The Case of p Colors

10.7.1 Pattern Guided by Arithmetic Sequence with Difference One Consider the case of p colors C1 C2 . . .Cp and the pattern C1 C2 . . .Cp C1 C1 C2 C2 . . .Cp Cp C1 C1 C1 C2 C2 C2 . . .Cp Cp Cp . . . |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} p

2p

3p

C1 C1 . . .C1 C2 C2 . . .C2 . . .Cp Cp . . .Cp |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} np

in which the number of Ci’s in a p-color combination, p  2, varies according to the sequence 1, 2, 3, …, n, …. Unlike the case of two (or three) colors when a position of each color was explored, in this section we will focus on color C1 only, yet exploring patterns guided by various arithmetic sequences with the number 1 as their first term. To begin, by focusing on the last positions of color C1, the following sequence of numbers emerges: 1, p + 2, 3p + 3, 6p + 4, …. Let x, y and z stand for the coefficients of the quadratic function f1;p ðn; 1Þ that describes the position of the last C1 in the nth p-color combination of length np. Solving the system of equations x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 2; 9x þ 3y þ z ¼ 3p þ 3; using (free on-line version of) Wolfram Alpha, yields x ¼ p2 ; y ¼ 1
p2 ; z ¼ 0 whence p p f1;p ðn; 1Þ ¼ n2 þ ð1
Þn: 2 2

ð10:6Þ

One can check to see that when p = 2 we have f1;2 ðn; 1Þ ¼ n2 —the case of Proposition 10.1 (Sect. 10.4.1). Likewise, by focusing on the first positions of color C1, described by the quadratic function f2;p ðn; 1Þ ¼ xn2 þ yn þ z, the following sequence of numbers emerges: 1, p + 1, 3p + 1, 6p + 1, …. Solving the system of three linear equations

10.7

The Case of p Colors

329

x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 1; 9x þ 3y þ z ¼ 3p þ 1; using (free on-line version of) Wolfram Alpha, results in the function p p f2;p ðn; 1Þ ¼ n2
n þ 1 2 2

ð10:7Þ

that describes the position of the first C1 in the nth p-color combination of length np. One can check to see that when p = 2 we have f2;2 ðn; 1Þ ¼ n2
n þ 1—the case of Proposition 10.2 (Sect. 10.4.1). Remark 10.2 Alternatively, f2;p ðn; 1Þ can be found by subtracting n − 1 from f1;p ðn; 1Þ as it takes n − 1 steps backward to reach the first C1 from the last C1 within the string of C1’s of length n. Indeed, f2;p ðn; 1Þ ¼ f1;p ðn; 1Þ
ðn
1Þ ¼ p 2 p p 2 p 2 n þ ð1
2Þn
n þ 1 ¼ 2 n
2 n þ 1.

10.7.2 Pattern Guided by Arithmetic Sequence with Difference Two Now, consider a p-color pattern C1 C2 . . .Cp C1 C1 C1 C2 C2 C2 . . .Cp Cp Cp C1 C1 C1 C1 C1 C2 C2 C2 C2 C2 . . .Cp Cp Cp Cp Cp |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} p

3p

5p

. . . C1 C1 . . .C1 C2 C2 . . .C2 . . .Cp Cp . . .Cp |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ð2n
1Þp

in which the number of Ci’s in a p-color combination varies according to the sequence 1, 3, 5, …, 2n − 1, … of consecutive odd numbers. By focusing on the last positions of color C1, the following sequence of numbers emerges: 1, p + 3, 4p + 5, 9p + 7, …. Let f1;p ðn; 2Þ ¼ xn2 þ yn þ z be the function that describes the position of the last C1 in the nth p-color combination of length (2n − 1)p. Solving the system of three linear equations x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 3; 9x þ 3y þ z ¼ 4p þ 5; using (free on-line version of) Wolfram Alpha yields f1;p ðn; 2Þ ¼ pn2 þ 2ð1
pÞn þ p
1:

ð10:8Þ

One can check to see that when p = 2 we have f1;2 ðn; 2Þ ¼ 2n2
2n þ 1—the case of Proposition 10.5 (Sect. 10.4.3.1).

330

10

Patterns and Functions

Likewise, by focusing on the first positions of color C1, the following sequence of numbers emerges: 1, p + 1, 4p + 1, 9p + 1, …. Solving the system of equations x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 1; 9x þ 3y þ z ¼ 4p þ 1 about the coefficients x, y and z of the quadratic function f2;p ðn; 2Þ using (free on-line version of) Wolfram Alpha results in the function f2;p ðn; 2Þ ¼ pn2
2pn þ p þ 1

ð10:9Þ

that describes the position of the first C1 in the nth p-color combination of length (2n − 1)p. One can check to see that when p = 2 we have f2;2 ðn; 2Þ ¼ 2n2
4n þ 3—the case of Proposition 10.6 (Sect. 10.4.3.2). Remark 10.3 Alternatively, f2;p ðn; 2Þ can be found by subtracting 2n – 2 from f1;p ðn; 2Þ as it takes 2n – 2 steps backward to reach the first C1 from the last C1 within the string of C1′s of length 2n – 1. Indeed, f2;p ðn; 2Þ ¼ f1;p ðn; 2Þ
ð2n
2Þ ¼ pn2 þ 2ð1
pÞn þ p
1
2n þ 2 ¼ pn2 þ 2n
2pn þ p
1
2n þ 2 ¼ pn2
2pn þ p þ 1:

10.7.3 Pattern Guided by Arithmetic Sequence with Difference Three The next p-color pattern to consider is

in which the number of Ci’s in a p-color combination varies according to the sequence 1, 4, 7, …, 3n − 2, …, which is an arithmetic sequence with difference three. By focusing on the last positions of color C1, the following sequence of numbers emerges: 1, p + 4, 5p + 7, 12p + 10, …. As before, solving the system of three linear equations x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 4; 9x þ 3y þ z ¼ 5p þ 7; using (free on-line version of) Wolfram Alpha, results in the function

10.7

The Case of p Colors

331

f1;p ðn; 3Þ ¼

3p 2 7p n þ ð3
Þn þ 2ðp
1Þ 2 2

ð10:10Þ

that describes the position of the last C1 in the nth p-color combination of length (3n − 2)p. One can check to see that when p = 2 we have f1;2 ðn; 3Þ ¼ 3n2
4n þ 2 – the case of the function h1 ðnÞ described above in Sect. 10.4.4. Remark 10.4 Alternatively, the function f2;p ðn; 3Þ that describes the position of the first C1 in the nth p-color combination of length (3n − 2)p can be found by subtracting 3n − 3 from f1;p ðn; 3Þ as it takes 3n − 3 steps backward to reach the first C1 from the last C1 within the string of C1’s of length 3n − 2. Indeed, 3p 2 7p n þ ð3
Þn þ 2ðp
1Þ
3n þ 3 2 2 3p 2 7p 3p 7p ¼ n þ 3n
n þ 2p
2
3n þ 3 ¼ n2
n þ 2p þ 1: 2 2 2 2

f2;p ðn; 3Þ ¼ f1;p ðn; 3Þ
ð3n
3Þ ¼

10.7.4 Pattern Guided by Arithmetic Sequence with Difference Four Continuing in the same vein, consider the following p-color pattern

in which the number of Ci’s in a p-color combination varies according to the sequence 1, 5, 9, …, 4n − 3, …, which is an arithmetic sequence with difference four. By focusing on the last positions of color C1, the following sequence of numbers emerges: 1, p + 5, 6p + 9, 15p + 13, …. As before, using (free on-line version of) Wolfram Alpha to solve the system of three linear equations x þ y þ z ¼ 1; 4x þ 2y þ z ¼ p þ 5; 9x þ 3y þ z ¼ 6p þ 9 about the coefficients x, y and z of the quadratic function f1;p ðn; 4Þ that describes the position of the last C1 in the nth pcolor combination of length (4n − 3)p yields f1;p ðn; 4Þ ¼

4p 2 10p n þ ð4
Þn þ 3ðp
1Þ: 2 2

ð10:11Þ

332

10

Patterns and Functions

One can check to see that when p = 2 we have f1;2 ðn; 4Þ ¼ 4n2
6n þ 3—the case described above for m = 4 in Proposition 10.9, Sect. 10.5.1.

10.7.5 Pattern Guided by Arithmetic Sequence with Difference m Analyzing the form of the functions described by relations (10.6), (10.8), (10.10), and (10.11), one can conjecture that f1;p ðn; 5Þ ¼

5p 2 13p n þ ð5
Þn þ 4ðp
1Þ 2 2

ð10:12Þ

and then generalize from (10.6), (10.8), (10.10), (10.11), and (10.12) as follows f1;p ðn; mÞ ¼

mp 2 3m
2 n þ ðm
pÞn þ ðm
1Þðp
1Þ 2 2

ð10:13Þ

where p is the number of colors in the pattern, n is the rank of p-color combination of length p[m(n − 1) + 1]. For example, when p = 3 we have f1;3 ðn; mÞ ¼ 32 mn2 þ ½m
3ð3m
2Þ n þ 2ðm
1Þ ¼ 32 mn2
7m
6 2 n þ 2m
2—the 2 case already mentioned in Proposition 10.10 (Sect. 10.6). Consequenly, f2;p ðn; mÞ ¼ f1;p ðn; mÞ
mðn
1Þ, as one has to make mðn
1Þ steps backward to reach the first C1 from the last C1 in a string of length mðn
1Þ þ 1. That is, according to (10.13), mp 2 3m
2 n þ ðm
pÞn þ ðm
1Þðp
1Þ
mðn
1Þ 2 2 mp 2 3m
2 n
mn
pn þ mp þ 1
m
p
mn þ m ¼ 2 2 mp 2 3m
2 n
pn þ ðm
1Þp þ 1: ¼ 2 2

f2;p ðn; mÞ ¼

So, the position of the first C1 in the nth p-color combination of length p½mðn
1Þ þ 1 is described by the function f2;p ðn; mÞ ¼

mp 2 3m
2 n
pn þ ðm
1Þp þ 1: 2 2

ð10:14Þ

Remark 10.5 The use of Wolfram Alpha Pro (available by subscription only) allows one to review 14 stepts needed for solving (in terms of p) the systems of three linear equations in x, y, and z introduced above in Sects. 10.7.1—10.7.5.

10.8

From Generaization to Computerization

10.8

333

From Generaization to Computerization

An algorithm of relating a position number to a color in a p-color pattern, the nth combination of which has length p½mðn
1Þ þ 1, that is, a p-color pattern guided by an arithmetic sequence with difference m, can be developed using a spreadsheet. To this end, formula (10.14) can be used. Suppose N is a position number in the pattern the color of which has to be determined. Then, the inequalities f2;p ðn; mÞ  N  f2;p ðn þ 1; mÞ
1

ð10:15Þ

hold true for a certain value of n as any number may be placed between the positions of the first C1 and the last Cp, which is one position below the first C1 in the combination of rank n + 1. The first step is to find the largest value of n for which the left-hand side of inequalities (10.15), that is, f2;p ðn; mÞ  N, holds true. Turning the last inequality into the equality f2;p ðn; mÞ ¼ N and using formula (10.14) yield the quadratic equation mp 2 3m
2 n
pn þ ðm
1Þp þ 1
N ¼ 0 2 2 one root of which is equal to

n¼

3m
2 2 pþ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3m
2 2 pÞ
2mp½ðm
1Þp þ 1
N mp

:

When f2;p ðn; mÞ\N;

ð10:16Þ

the expression

INTf

3m
2 2 pþ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3m
2 2 pÞ
2mp½ðm
1Þp þ 1
N g mp

defines the largest n for which inequality (10.16) holds true, where the function INTðxÞ returns the largest integer smaller than or equal to x. So, given N, m and p, one can find n through the formula n ¼ INTf

3m
2 2 pþ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð3m
2 2 pÞ
2mp½ðm
1Þp þ 1
N g mp

ð10:17Þ

334

10

Patterns and Functions

and then caclulate the values of the expressions f2;p ðn; mÞ and f2;p ðn þ 1; mÞ
1 for the value of n defined by formula (10.17). Alternatively, the latter expression can be reduced to the former one through the formula f2;p ðn þ 1; mÞ
1 ¼ f2;p ðn; mÞ þ pðmn
m þ 1Þ
1:

ð10:18Þ

The next step is to find the ratio r of N
f2;p ðn; mÞ to ½f2;p ðn þ 1; mÞ
1
f2;p ðn; mÞ (or, to pðmn
m þ 1Þ
1). Finally, the number INTðrpÞ þ 1 when N 6¼ f2;p ðn þ 1; mÞ
1, otherwise the number INTðrpÞ is the color number of the position N. The spreadsheet pictured in Fig. 10.10 in which p = m = 2 and N = 1225 confirms an answer given in Sect. 10.4.3.5; namely, that the 1225th position has blue color in a two-color pattern. The programming of the spreadsheet of Fig. 10.10 is included in Appendix. To explain how either the value INTðrpÞ þ 1 or the value INTðrpÞ provides the value of color number, consider the case of three colors (p = 3). Let ½f2;3 ðn; mÞ; f2;3 ðn þ 1; mÞ
1 be a segment divided into three equal parts ½x0 ; x1 ; ½x1 ; x2  and ½x2 ; x3  , where f2;3 ðn; mÞ ¼ x0 \x1 \x2 \x3 ¼ f2;3 ðn þ 1; mÞ
1: Let N 2 ½x2 ; x3  —a segment filled with the third color. Then 3ðN
x0 Þ 3ðx2 þ N
x2
x0 Þ 2ðx1
x0 Þ þ N
x2 ¼ ¼ x3
x0 3ðx1
x0 Þ x1
x0 N
x2 ¼ 2þ ¼ 2 þ r1 ; 0  r1  1: x1
x0

3r ¼

If N 6¼ x3 then r1 \1 and INT(3r) + 1 = INT(2 + r1) + 1 = 2 + 1 = 3—a number that represents the third color. If N ¼ x3 then r1 ¼ 1 and, once again, INT (3r) = INT(2 + 1) = 3. In the general case of p colors, suppose the number N 2 ½x0 ; xp . The ratio 0 r ¼ xN
x ; 0  r  1; represents the fraction of the segment ½x0 ; xp  that spans from p
x0 x0 to N. Let us divide the segment ½x0 ; xp  into p equal parts x0 \x1 \x2 \. . .\xi
1 \xi \. . .\xp . If N 2 ½xi
1 ; xi ; 1  i  p, then

Fig. 10.10 Computational environment for relating position number to color

10.8

From Generaization to Computerization

335

pðN
x0 Þ pðxi
1 þ N
xi
1
x0 Þ ði
1Þðx1
x0 Þ þ N
xi
1 ¼ ¼ xp
x0 pðx1
x0 Þ x1
x0 N
xi
1 ¼ i
1þ ¼ i
1 þ r1 ; 0  r1  1: x1
x0

pr ¼

The case N ¼ xi implies r1 ¼ 1 as xi
xi
1 ¼ x1
x0 therefore pr = i and INT (pr) = i. The case N 6¼ xi implies 0  r1 \1 and INT(pr) = i − 1 or i = INT (pr) + 1. Remark 10.6 The computational environment described in this section can be developed without using formula (10.17) which works nicely in the case when a function mapping n to the position of color C1 in the nth combination of a p-color pattern is a quadratic function of n (i.e., the pattern is guided by an arithmetic sequence). However, an increase of colors may follow a more complicated pattern, for instance, when an arithmetic progression is replaced by a sequence of polygonal numbers (Chap. 9, Sect. 9.4) in which case we have to deal with polynomial functions of order greater than two. For example, consider the pattern C1 C2 . . .Cp C1 C1 C1 . . .Cp Cp Cp C1 . . .C1 . . . Cp . . .Cp . . . C1 C1 . . .C1 . . . Cp Cp . . .Cp |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} nðn þ 1Þ 6 nðn þ 1Þ p 3p 6 2 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} 6p

nðn þ 1Þ 2 p

in which the number of Ci’s in this p-color combination varies according to the sequence of triangular numbers 1; 3; 6; . . .; nðn 2þ 1Þ. The first few positions of the color C1 within the triangular number pattern are 1, p + 1, 4p + 1, 10p + 1, 20p + 1, where the coefficients in p are the partial sums of consecutive triangular numbers. Partial sums of the quadratic function tðnÞ ¼ nðn 2þ 1Þ are represented by a cubic function. In particular, the function fp ðnÞ ¼ p6 n3
p6 n þ 1 maps n to the position of first color C1 in the nth combination of the pattern guided by the sequence of triangular numbers (not an arithmetic sequence). One can check to see that fp ð1Þ ¼ 1; fp ð2Þ ¼ p þ 1; fp ð3Þ ¼ 4p þ 1; fp ð4Þ ¼ 10p þ 1: Finding n from the equation p6 n3
p6 n þ 1 ¼ N would require solving a cubic equation, something that is beyond middle (and even high school) level. Instead, one can generate the values of the fp ðnÞ and fp ðn þ 1Þ and then locate the value of n for which the inequalities fp ðnÞ  N\fp ðn þ 1Þ hold (see Fig. 10.11 where the 24th position (cell D3) in the two-color pattern

336

10

Patterns and Functions

Fig. 10.11 Computational environment without using formula (10.17)

C1 C2 C1 C1 C1 C2 C2 C2 C1 . . .C1 C2 . . .C2 . . . C1 . . .C1 C2 . . .C2 has the first color |ffl{zffl} |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} 2 10 10 6 6 6 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} 12

20

(cell K3)). The rest follows the algorithm described above. Figure 10.12 shows the use of the alternative computational environment in finding the color of the 1225th position in a two-color combination of length p½mðn
1Þ þ 1jm¼p¼2 ¼ 4n
2. The programming of the spreadsheet used for Figs. 10.11 and 10.12 is included in Appendix. This completes the construction of the computational environment which makes it possible to determine the color of the Nth position in a p-color pattern guided by an arithmetic sequence with difference m. A more complicated case of a p-color pattern guided by m-gonal numbers, the nth partial sums of which are represented by the third-degree polynomials in n, is discussed in (Abramovich, 2020). One can see that an important role of algebraic generalization in the digital era is the development of computational environments with the dual agency: to enable solving problems depending on parameters for specific values of the parameters and, in doing so, to allow for posing problems to be solved without technology. The latter agency is discussed in detail in (Abramovich, 2019). Whereas posing mathematical problems is a creative process in itself for it requires one to select data that makes a problem solvable (see Chap. 9, Remark 9.6 with reference to the

Fig. 10.12 Alternative confirmation of the result for N = 1225

10.8

From Generaization to Computerization

337

problem-solving spreadsheet of Fig. 9.5), the use of technology supports creative efforts of problem posers by confirming (or negating) the solvability of a problem with the data selected (Freiman & Tassell, 2018).

10.9

Conclusion

This chapter demonstrated how recognition of hidden potential in a teacher candidate’s far-reaching response to an open-ended question about a visual pattern allows for the development of a variety of numeric patterns and their representations in the language of functions which, in turn, form algebraic patterns. The use of Wolfram Alpha and OEIS® supported the symbolic complexity of explorations and their follow-up generalizations. Triangular and square numbers introduced in Chap. 9 were used to demonstrate connections between visual and symbolic representations. In the spirit of the Gestalt psychology educational recommendations, concreteness as a motivation for abstraction was mentioned in the context of the history of mathematics. Finally, a spreadsheet-based computational environment was constructed to solve and pose problems about numeric characteristics of visual patterns. For readers who want to deepen their knowledge and understanding of patterns and functions the following sources can be recommended (Noss et al., 1997; Marks, 2000; Rubenstein, 2002; Zazkis & Liljedahl, 2002; Vogel, 2005; Rivera, 2013; Yilmaz et al., 2018; Abramovich, 2020).

10.10

Activity Set

1. Given the recursive definition xn ¼ xn
1 ; x1 ¼ 1 of the sequence xn , find its closed formula without using technology. Then use Wolfram Alpha to check your result. 2. Given the recursive definition xn ¼ xn
1 þ 3; x1 ¼ 2 of the sequence xn , find its closed formula without using technology. Then use Wolfram Alpha to check your result. 3. Given the recursive definition xn ¼ 2xn
1 þ 1; x1 ¼ 1 of the sequence xn , find its closed formula without using technology. Then use Wolfram Alpha to check your result. 4. Given the recursive definition xn ¼ 2xn
1 þ 2; x1 ¼ 1 of the sequence xn , find its closed formula without using technology. Then use Wolfram Alpha to check your result.

338

10

Patterns and Functions

5. If the pattern A-BA-BAB-ABAB-ABABA-BABABA-BABABAB- …, in which letters A and B are put in groups of one, two, three, four, and so on letters continues, how many letters A are there in the 100th group and how many letters B are there in the 100th group? Determine the first and the last letters in the 200th group. 6. If the pattern A-BAB-ABABA-BABABAB-ABABABABA-BABABABAB AB-ABABABABABABA-…, in which letters A and B are put in groups of one, three, five, seven, and so on letters continues, how many letters A are there in the 100th group and how many letters B are there in the 100th group? Determine the first and the last letters in the 100th group. 7. If the pattern A-BABA-BABABAB-ABABABABAB-ABABABABABABA…, in which letters A and B are put in groups of one, four, seven, ten, and so on letters continues, how many letters A are there in the 100th group and how many letters B are there in the 100th group? Determine the first and the last letters in the 100th group. 8. Find a function that describes the position of the second R in the nth “RB”combination defined by an arithmetic sequence with difference d = 1. 9. Find a function that describes the position of the penultimate R in the nth “RB”combination defined by an arithmetic sequence with difference d = 2. 10. Find a function that describes the position of the second B in the nth “RB”combination defined by an arithmetic sequence with difference d = 1. 11. Find a function that describes the position of the penultimate B in the nth “RB”combination defined by an arithmetic sequence with difference d = 2. 12. Find a function that describes the position of the second R in the nth “RB”combination defined by an arithmetic sequence with difference d = 3. 13. Find a function that describes the position of the penultimate R in the nth “RB”combination defined by an arithmetic sequence with difference d = 3. 14. Find a function that describes the position of the second B in the nth “RB”combination defined by an arithmetic sequence with difference d = 3. 15. Find a function that describes the position of the penultimate B in the nth “RB”combination defined by an arithmetic sequence with difference d = 3. 16. Find a function that describes the position of the second R in the nth “RBG”combination defined by an arithmetic sequence with difference d = 1. 17. Find a function that describes the position of the penultimate R in the nth “RBG”-combination defined by an arithmetic sequence with difference d = 2. 18. Find a function that describes the position of the second G in the nth “RBG”combination defined by an arithmetic sequence with difference d = 1. 19. Find a function that describes the position of the penultimate G in the nth “RBG”-combination defined by an arithmetic sequence with difference d = 2. 20. Show that positions of the last R in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant.

10.10

Activity Set

339

21. Show that positions of the last B in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant. 22. Show that positions of the last G in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant. 23. Show that positions of the first R in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant. 24. Show that positions of the first B in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant. 25. Show that positions of the first G in “RBG”-combinations guided by an arithmetic sequence with difference d are described by a quadratic sequence; that is, by a sequence the second difference between its any two consecutive terms is a non-zero constant.

Chapter 11

Financial Literacy and Blockchain

11.1

Introduction

The importance of helping middle school students develop financial literacy is hard to overestimate. Even if viewed only from a purely mathematical perspective, the earlier one starts saving, the greater the benefit that is obtained due to increasing the length of time that the savings are subject to interest (Sects. 11.2.4.1 and 11.2.4.2). This simple example shows how educating students, while still young adults, with strategies to increase their savings alone could potentially reap huge benefits for them over the course of their lives. Going beyond mathematics, however, the benefits of possessing sound financial awareness can easily yield benefits that go beyond any interest calculation. The peace of mind coming from having basic needs met and the opportunity for wants to be obtained is hard to quantify, but will definitely contributes to a happy and productive life.

11.2

Financial Literacy

cBig Idea Most middle school students have minimal concepts of financial literacy, a big idea from mathematics. Lacking this background, they are unable to make early decisions of great import for the remainder of their lives. The earlier financial literacy can be acquired the greater students can benefit from its’ application.b

© Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_11

341

342

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Financial Literacy and Blockchain

Unfortunately, the level of financial literacy worldwide among youth is declining (Allgood & Walstad, 2013). Since most middle school students typically have minimal if any concepts of financial literacy and view themselves as years away from needing to manage their own finances, it is not surprising that they do not spend significant time concerning themselves with thinking about financial matters. Although the world of income and work can seem far in the distance as an early teen, the development of positive attitudes and productive habits towards money should be an important part of middle school mathematics education. To develop these habits and attitudes it is important to start with core financial topics which include the basics of wage earning and savings. With these established as a common background, middle school can be an ideal time for students to begin exploring ideas of how to pay for their college or post-secondary education and training. Students can easily recognize how decisions made while in middle school can come to play a major role in eligibility for scholarships and grants. Post-secondary education is much closer to the student’s current world than planning for retirement and can be a strong motivator for them to learn other financial topics. Although perhaps not as immediately important in the minds of students, middle school is also an excellent time to begin learning about the basic steps required when constructing a budget. Possessing the skills necessary to construct a budget early in life will aid in the students being able to follow a budget later. Learning to distinguish between needs and wants can be easier when it is a hypothetical situation in a classroom example than when you are faced with the reality of living from paycheck to paycheck. Middle school is also an excellent time to develop basic awareness of checking, including the actual physical process of writing a check and of balancing a checkbook. Despite the best plans and budgets, however, debt is a common part of life for most individuals. By learning the basics concerning different types of credit, loans, and borrowing options now, it will be possible for students to make more highly informed and intelligent decisions later. It is better to learn the perils of compound interest from a school lesson in middle school than from a life lesson as a young adult derived from uncontrolled credit card debt. Planning for a vehicle purchase while in middle school can serve as a strong motivation for students to learn the basics of simple interest loans. As students learn the benefits of paying these simpler loans off early, they gain important insights which can be transferred to the more complicated issues surrounding mortgage and business loans. Learning to identify and avoid common financial pitfalls will pay lifelong dividends of the life of the student. It is important for students to realize that financial literacy involves far more than just earning, buying and saving. Although far in the future for most students, their middle school years are a good time to begin thinking about insurance and investment planning. Developing financial literacy aids in the development of positive attitudes toward money and spending which can yield benefits lasting a lifetime.

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11.2.1 Earning For most students, there is a huge difference between their first job jobs and their lifetime career. First jobs are often part time without requiring a large degree of experience or skill. Depending upon the local economy, such first jobs can be relatively easy to get, equally easy to lose, and generally at the legal minimum wage. Regardless of the rate of pay, however, a first job can provide students with helpful experiences in negotiating the world of work. Concepts such as income, taxes and spending become more real to a student who is actually earning and spending their own money. No matter how much or how little a student might make, the time to create a first budget should be no later than the time of their first job. A career, however, is clearly a much different enterprise than a first job. Careers, especially in the professions, generally require advanced training and highly developed skill sets. Often prior experience, in the best cases drawn from a first job, is required to qualify for a career. A career is also a much longer-term proposition than a first job, a career in some cases will be lifelong. As such, it is extremely important to be thorough identifying the skill sets and requirements of various careers as early as possible. The middle school years are not too early to begin this identification process. Through the exploration of the skills required for a given career, students can begin matching their own developed personal skills and interests to these various career options. In many cases, this exploration helps middle school students identify the post-secondary education and training necessary to be successful in their chosen ideal career. Furthermore, this going through this exploration can also help as students try to balance the realistic costs of the required training or education with the potential increase in income and lifetime earnings brought about by working within their chosen career. A first job will also serve to provide important insights into earning and the generation of actual take home pay. Most first jobs are paid on an hourly basis, so the computation of gross earned income is easily addressed by
$  GrossIncomeIn$ ¼ ðHoursÞ Hour . Although not reflective of the true income available for use, Gross income is often used to determine eligibility for home and personal loans and when applying for housing rentals. This simple understanding concerning the computation of gross income can be used to teach the importance of checking the units involved in a calculation. For
$  example, when looking at the equation GrossIncomeIn$ ¼ ðHoursÞ Hour , if a student has worked 30 h at $5 per hour their desired question would be how much their gross income should be when expressed in $. When the calculation ⎛ $5 ⎞ GrossIncomeIn$ = ( 30 Hours ) ⎜ ⎟ = $150 is carried out, the hours drop out, leaving ⎝ Hour ⎠

the answer to be expressed in $ as expected. When students are looking forward to the income and have to do the work these lessons assume a much greater significance than when presented as a mere textbook example. When examining a first check many important lessons from mathematics may be drawn due to the reduction of Gross Income by taxes and other deductions.

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Assuming that taxes are the only reduction, the actual NetIncomeIn$ ¼
$  ðHoursÞ Hour  Taxes. Since Net Income is more reflective of the actual funds available for the earner’s use, this net income calculation provides a very meaningful and concrete situation for the earner! In looking at an actual paycheck, one of the first items is to note is the tax rate which is generally expressed as a percentage. Since prior to deductions the Gross Income is equal to 100% of the income earned and the tax rate is expressed as a percentage, students can see their actual Net
$ 
$   ðTaxRate%ÞðHoursÞ Hour . Income would be equal to ð100%ÞðHoursÞ Hour What might have been a difficult algebra problem now becomes a lesson in common sense with the realization that NetIncome ¼ ð100%  TaxRate%Þ
$  ðHoursÞ Hour and that the net income can be expressed as a percentage of the gross income. For example, assuming a pay per hour rate of $8.75 and 25 h worked, the
$ 
 GrossIncomeIn$ ¼ ðHoursÞ Hour , so GrossIncomeIn$ ¼ ð25HoursÞ $8:75 Hour ¼ $218:75 and it is this figure that middle school students initially expect to receive. When a tax rate of 12% is added to the situation however, these students quickly
 find that their NetIncome ¼ ð100%  12%Þð25HoursÞ $8:75 Hour ¼ $192:50 and their initial enthusiasm is curbed by reality.

11.2.2 Budgeting As suggested in Sect. 11.2.1, the best time to create the first budget should be no later than the first job. For students who have yet to have a first job, however, budgeting should still be taught. Even lacking a real-world job, the process followed in creating and planning to follow a budget can lead to the development of healthy spending habits and attitudes towards money. Even when creating a purely hypothetical budget, it is possible to bring up the importance of identifying budget categories to ensure that resources are allocated and will be available to meet all financial obligations. Prior to any budget creation exercise, students should engage in some goal setting. A budget, even when done as a classroom excursive, should be constructed with some goal in mind. These goals can help students identify the categories which impact all later budgeting choices. Identifying these goals in advance explicitly, students can review whether the proposed categories advance or detract from their progress towards these goals. Far too often later in life, budgets are not created until a financial challenge is already in place and are looked upon as a last resort to determine ways to save money and pay off debt. This is a sad situation since once such unplanned debt takes over being able to meet long term goals becomes nearly impossible. Even during sample budget creation problem overspending areas can be identified. Getting used to starting a budget with goal setting helps ensure that spending and savings are aligned with these goals (Bieber, 2018). As discussed in Sect. 11.2.1, Gross Income, is the total income before taxes or other deductions. This understanding needs to be further developed when

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developing a budget, however, as it should include income from all sources such as wages, pensions, interest on savings, etc. As could be expected, Gross Annual Income consists of the total yearly income prior to taxes or any other deductions (Kenton, 2020). Since Gross Annual Income is often used to determine eligibility for home and personal loans and when applying for housing rentals, this is an important quantity to be aware of even it is only partially reflective of true spendable income. Net Income is a bit trickier to consider for most adults since some of the deductions like insurance and retirement contribution actually have value for the wage earner. Determining net income requires the addition of such deductions back into the take home pay. As will be shown, there are many different strategies to follow when creating a budget. But regardless of which budgeting strategy is followed, it is important to use Net Income and not Gross Income in planning.

11.2.2.1

First Steps Toward a Budget

For classroom examples, it is important that a realistic set of data showing typical categories of where money is spent and on what type of items should be created for the students. Without this knowledge any developed sample budget is hypothetical at best and likely to be viewed as unrealistic. If all else fails, the creation of a sample set of expenses and categories can easily be done by altering the numbers and descriptions of the instructor’s own budget. If the instructor does not have a budget, this can be a wakeup call to create one as it will benefit both themselves and their students. Even a sample budget should illustrate how to track spending to determine current spending habits, expenditures, and the factors to include when creating budget. As might be expected, this process is greatly aided by using a spreadsheet program to allow for easy categorization of expenses. There are numerous examples online which can be utilized. One such sample budget (Budgets, 2020) is shown in Fig. 11.1. Even as a hypothetical exercise, developing a sample budget can help middle school students start to develop habits that ensure that their later, real-world, budgets will have categories that align with their goals and have a higher possibility of actually being followed. Furthermore, the identification and inclusions of expense categories via a spreadsheet makes these categories visual aiding the students as they practice differentiating between wants and needs. With a history of income and expenditures established in the sample budget, the students should examine the budget with the goal of identifying needs and wants. An easy way to address this is for the student to ask if they could live without it. If the answer is yes, then it is a want and not a need. Since needs are essential to life, they must be budgeted for no matter what. What is left over after the needs are each addressed can be used towards paying for the wants. Although wants may not be essential to life, when wants can be achieved the perceived quality of a person’s life increases. For both adults and middle school students, the separation of needs and wants can be extraordinarily difficult.

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Fig. 11.1 A sample budget spreadsheet within Microsoft Excel

11.2.2.2

A 50-30-20 Budgeting Strategy

Assuming that needs and wants have been successfully identified, there are many strategies for effectively allocating moneys within a budget. One simple strategy that does not require creating detailed budgeting categories is the 50-30-20 method. For students to use this approach, they would look at the expenses shown in the sample budget and place them into the categories of needs, wants, and savings/debt. Remembering the injunction in Sect. 11.2.2, students should start with the Net Income and divide this net income into these three categories setting aside 50% of the Net Income for needs, 30% for wants, and 20% for savings/debt. The debt category needs a little expansion. Minimum required payments on debt, such as credit cards and loans, would not be considered within the 20%. This expense is a need and must be included in the 50% needs category. Students need to learn that not making a minimum payment on debt negatively impacts credit ratings, results in interest compounding at a higher rate, and can lead to the application of higher interest rates or even not having future credit. Rather, the 20% savings/debt category would be money to be set aside to pay off debt faster than required or for

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savings and investments. As a textbook exercise, this budgeting strategy makes for an ideal to strive for, however in real life there are many additional factors which must be taken into account. For example, depending upon the individual’s income and the local costs of living, 50% might not be enough to actually cover basic needs.

11.2.2.3

Budgeting to Buy a Car

For some middle schools and high schools, many students require transportation and the lure of car ownership plays an increasingly large role in their personal finances. Even if this is admittedly a want and not a need, the allure of vehicle ownership can be used to lend reality to the budgeting process required. It is also the case that budgeting needed to buy a car can be motivating to many middle school students. Identifying needs and wants can be readdressed at this time. For example, the students may want the sports car but actually need the mini-van. By modeling how to identify a car that meets needs and budget, instructors can help prepare students for their later purchase decisions.

11.2.3 Checking One of the most important financial resources students should become familiar with is that of a checking account. This type of account differs from a savings account in that the funds in a checking account can easily be deposited and easily accessed. Checking accounts also differ from savings accounts, in that they generally allow the holder an unlimited number of withdrawals and deposits. This ease of deposit and withdrawal into the account makes a checking account a natural choice for use in covering normal day to day spending whether for needs or wants. A drawback for checking accounts, however, is that even when they offer interest, it is generally at a much lower rate than the interest offered for a savings account. As such, it is important for students to realize that checking accounts will pay lower interest rates —if interest is paid at all. Thus, a good strategy to share with students is that of using checking accounts only for ordinary purchases and monthly bills. Everything else should be placed into accounts with higher interest potential such as savings accounts and certificates of deposit (CDs). Actual check writing is nearly a lost art since modern checking accounts now allow for money to be accessed in ways that go far beyond the traditional writing of checks. Debit cards, for example, look like credit cards but the funds are withdrawn from moneys deposited in checking and not against a line of credit. By using a debit card connected to a checking account it is possible to take advantage of the ease of use credit cards offer without the need for additional debt or interest. Deposits may no longer require extensive forms when done via an Automatic Teller Machine (ATM) or smartphone app as shown. When using a smartphone app, as shown in Fig. 11.2, a picture is taken of the front and back of the check to deposit, the

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Fig. 11.2 Illustration of an app-based deposit

account receiving the deposit is identified, and the amount of the deposit is specified. When exploring checking account options students should be guided to look for accounts with no monthly maintenance fee and/or minimum balance requirements. Such checking accounts are not common, however, with many accounts charging fees for checking generally depending on the amount held in deposit. Finally, free checking is not always free even when these requisites are met. Students should be aware, for example, that overdraft charges and fines will still apply even to free checking accounts. Of particular interest to middle school students are basic checking accounts with low balance requirements and no monthly fees. These are sometimes available as student accounts and can be bundled together with their legal guardians checking account. These accounts will give the student access to basic check writing and the opportunity to practice this skill in a low to now cost setting.

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11.2.4 Interest An easy way to explain interest to students is that interest is a price that is paid when using money that is not yours. When the money is deposited into a savings account, however, the account holder will be paid interest by the bank for the right to use this deposited money. In the case of loans, whether from an individual or a bank, there will be a need to pay interest. Interest is based upon a percentage, referred to as the interest rate. This percentage is then applied to the money being used whether as a loan, or in a savings account. Most interest rates are expressed using an annual rate of interest. Because of this, when solving problems involving interest it is often necessary for students to compute the interest rate relative to the units of time referred to in the problem whether days, months, or years.

11.2.4.1

Simple Interest

Simple interest is the easiest type of interest to consider and it is often used when in vehicle or personal loans. The formula for simple interest is I ¼ P  r  t, where I = Interest, P = Principal, r = annual interest rate expressed as a decimal, and t = Time. Simple interest is relatively straightforward in application. For example, consider the task of determining the amount of interest earned in three years on $2500 invested at an interest rate of 3 12 % ¼ 0:035. In this case both the time interval and the interest rate are expressed in years, so the interest would be ð$2500Þð0:035Þð3Þ ¼ $262:50. Simple interest is calculated on a daily basis, so borrowers who make their payments early or on-time can benefit from its application. In application, payments made by the borrower on a simple interest loan go toward paying the current interest first. Only when this interest is paid off, does the remainder of the payment become applied to the principal. Students need to realize that since the interest always paid first it does not build up as happens in compound interest where interest ends up being paid on the interest. As simple interest is calculated on a daily basis, it is beneficial for the borrower to make early payments and to make additional, principal only, payments whenever possible. By reducing the time the borrowed money is subject to interest, it is possible to gain real savings when this can be done. Imagine a personal loan with a $10,000 balance and a 4% interest rate with payments due on the last day of the month. Consider what happens when an on-time payment is made at the end of the month and when it is made 10 days prior to this date. First, since this problem is expressed in days and not years it is necessary to compute the daily interest rate. For a 4% annual interest rate, the daily rate is 0:04 a 30-day month, the 365 . Assuming
 ð 30 Þ ffi $32:88. When the interest if paid on the last day would be ð$10; 000Þ 0:04 365 payment is made 10 days earlier the interest paid would be ð$10; 000Þ
0:04 Since the interest is paid first, an additional 365 ð20Þ ffi $21:92.

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$10:96ð¼ $32:88  $21:92Þ would go towards the principal by making the 10-day earlier payment. Over time these savings add up and can help pay the loan off more rapidly.

11.2.4.2

Compound Interest

cBig Idea Compound interest, a big idea within both mathematics and finances, plays a foundational role in developing financial literacy. Despite this, it remains a poorly understood concept requiring careful instruction at the middle school level.b The benefits of compound interest, which can be thought of as interest on interest, is hard to overestimate when considering savings. Although not primarily noted for his financial expertise, Albert Einstein is said to have observed that, “Compound interest is the eighth wonder of the world. He who understands it, earns it … he who doesn’t … pays it.” (Elliott, 2019). Students should realize that when compound interest is applied to saving, it can lead to quite large increases to the interest earned in the account. Conversely, when applied to debt, the second half of Einstein’s observations holds true and the amount to be paid back increases significantly over time. Despite this, however, only one-third of the United States population showed an ability to recognize and apply the concept of compound interest in daily situations (Lusardi & Tufano, 2009). This lack of understanding points to a true need for compound interest to be taught starting at an early age with middle school certainly not too early. The starting point for understanding compound interest is the compound interest formula:  r nt A ¼ P 1þ n

ð11:1Þ

where A is the Total amount accumulated after n years, including interest, P is the Principal Amount (the initial amount you borrow or deposit), r is the annual interest rate expressed as a decimal, t is the number of years the amount is deposited or borrowed for, and n is the number of times per year the interest is compounded. While the derivation of these formulas may be beyond the scope of middle school mathematics curriculum, the calculations involved in this formula and those shown within this section are easily within the reach of technology and calculators available to middle school students. To see how (11.1) operates, consider the problem of determining the interest earned in four years on $1500 invested at an interest rate of 3½ % per year compounded quarterly.

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  0:036 44 A ¼ $1500 1 þ ffi $1724:36: 4 As a problem for students, there is an additional step which will need to be discussed. Since the question is asking for the interest only and the amount includes the initial deposit the initial amount must be subtracted from this result to obtain the interest. Interest Earned ¼ Amount AccumulatedDeposit $1724:36  $1500 ¼ $224:36 11.2.4.3

Additional Applications of the Compound Interest Formula

Compound interest plays an important role in both personal and professional finance. To help develop this concept with students, it is worth exploring some additional applications which derive from (11.1). In each of the following situations, A is the Total amount accumulated after n years, including interest, P is the Principal Amount (the initial amount you borrow or deposit), r is the annual interest rate expressed as a decimal, t is the number of years the amount is deposited or borrowed for, and n is the number of times per year the interest is compounded. Interest rate To determine the average interest rate on loans, savings or investments, solve the compound interest formula (11.1) for rate. "  1 # A nt r¼n 1 P

ð11:2Þ

Starting principal To see much starting principal is required reach desired future value, solve (11.1) for principal. A P¼
ð11:3Þ nt 1 þ nr Time required for payback To compute the time necessary to pay back a loan the following equation, solve (11.1) for time.
 ln AP  t¼ 
ð11:4Þ n ln 1 þ nr

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11.2.5 Credit and Borrowing Most students will eventually need to borrow and incur debt. When this is done, they will enter into a credit agreement which includes specified requirements to repay the borrowed money and applicable fees and interest according to a payment schedule. It is important for students to realize that credit is not automatically given. The right to use credit must be earned and can easily be lost. This is why the minimum required payment on debt is considered a need when planning a budget, see Sect. 11.2.2. Should a borrower fail to pay back borrowed money as agreed to in the credit agreement, future credit can be denied. On the other hand, generating a history of paying back credit in a timely fashion results in future credit requests being more likely to be granted and the willingness for increased amounts of credit to be offered. There are four general types of credit accounts which students should be made aware of revolving, open, installment and service. Each has unique characteristics and would likewise have different terms in their respective credit agreement. Revolving credit refers to a line of credit which can be used at will up to the credit limit, or cap. The credit limit is an amount based upon credit history and determines how much can be borrowed at any given time. Once the credit is used, there will generally be monthly payments until the amount borrowed plus applicable interest in paid back. This is a very common type of credit with credit cards and home equity loans both being examples. Students should be taught to carefully monitor and manage revolving credit. When this is not done, it can create serious financial difficulties. Credit cards, for example, will allow the borrower to charge up to the cap. Since these charges are at the discretion of the card holder, they will vary each month depending upon how the card was used during that month. At the end of each month the credit card holder will be provided with a statement showing charges for that month and any unpaid balance. There is a grace period of 20 to 30 days within which the balance can be paid off in full without accruing interest (Brookshire, 2018). However, if the bill is not paid in full by the end of this grace period, any unpaid portions will be subject to interest compounding. It is at this point that card holders can easily enter into a financial pitfall. Since the card holder is given the option of paying an interest-only payment in the unpaid balance it is possible for the balance, or amount owed to the credit card company, to grow over time creating additional debt and making payback difficult and be subject to compounding. Once this debt is subject to compound interest, see Sect. 11.2.4.2, it is important to try to pay such debt off as quickly as possible. Credit card debt has a fairly rapid compounding frequency which can significantly increase both the time it takes to payback and the total payback amount. It is not uncommon for the card issuer to multiply the current daily balance by the daily interest rate to come up with a daily interest charge. This charge is then added to the balance the next day. This can create situations where it actually makes sense to take out a simple interest loan to pay off the credit card debt. The borrower will still need to pay back the

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amount, but it will be paid as simple interest rather than the compounded interest charged by most credit cards. Installment credit refers to a loan for a set amount which is paid back over a fixed time period. Installment credit is often the first experience students will have as young adults as they include some types of student loans and personal signature loans which have fixed, regularly occurring payments. Fortunately for first time borrowers, installment loans are relatively easy to plan for since the payment is the same each month until the loan plus interest is paid. Having the amount borrowed, the monthly payments and the time of the loan are all known in advance, thereby making budgeting easy. Service credit is another category often experienced by young adults as it impacts their cell phones. Service credit is credit that is offered by service providers such as utilities, gyms, cell phone companies, and internet providers. Although many are unaware that this is a credit being forwarded, many services are provided on a monthly basis with the expectation that they will be paid in full each month after they have been provided. This means that the service is offered as a credit until it is paid for.

11.2.6 Mortgages Although it is highly unlikely that middle school students will be faced with a mortgage, it is important for them to understand this type of loan. In a mortgage, property or real estate is used as the collateral for the loan. In a home loan, for example, the home itself is the collateral. Mortgages can range from five to thirty years with both variable or fixed interest rates and number of payments per period. Home equity loans are a type of mortgage where the equity in the home is used as the collateral for the loan. Like all services, mortgages are subject to supply and demand. This means that their interest rates can change over time. Students should learn that if interest rates drop it can be to their benefit to sign a new mortgage agreement at the lower interest rate. This refinancing process can result in significant savings over the life of the loan. Assuming that the students have had practice using the various compound interest formulas, see 11.2.4.3, it is possible to determine the monthly payment for a mortgage using the following formula: M¼P

r ð1 þ r Þn ð1 þ r Þn 1

ð11:5Þ

In (11.5), M is the total monthly mortgage payment, P is the principal loan amount, r is the monthly interest rate (lenders typically provide an annual rate, divide this rate by 12 to get the monthly rate), n is the number of payments over the loan’s lifetime (multiply the number of years in your loan term by 12).

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11.2.7 Savings Savings can be thought of as the opposite of debt. While a simple interest loan is to be preferred over a loan with interest compounding, for savings compound interest accounts are desirable. Imagine a single investment of $5000 without any further contributions. Figure 11.3 shows the value of this $5000 when simply saved with no interest, saved in a simple interest account after 20 years of 4% interest, and saved in an interest account which is compounded monthly after 20 years of 4% interest. This clearly shows the power of compound interest and the benefit additional time makes in the total interest earned. The benefit for the saver improves immensely when regular deposits are made to this savings. Figure 11.4 shows the value of a $5000 initial deposit followed by consistent $50 monthly deposits after 20 years when simply saved with no interest, saved in a simple interest account with 4% interest, and saved in an interest account which is compounded at the end of each month with 4% interest. It must be noted that these calculations are slightly more complicated in that they involve computing the compound interest for both the principal and the future value of the series of payments. The first of these may be computed using the compound interest formula (11.1). The second involves computing the future value of the series of deposits which may be computed as: "
DEP

1þ

 r nt 1 n
r  n

# ð11:6Þ

where A is the Total amount accumulated after n years, including interest, P is the Principal Amount (the initial deposit), DEP is the monthly deposit amount, r is the annual interest rate expressed as a decimal, t is the number of years the amount is deposited or borrowed for, and n is the number of times per year the interest is compounded. Taken together (11.1) and (11.6), the appropriate formula becomes "
nt #  1 þ nr 1 r nt
r  A ¼ P 1þ þ DEP n n

ð11:7Þ

The wisdom of Einstein’s compound interest observation is clearly obvious at this point. As with many things that are good for us, however, saving can be a difficult habit to acquire. Perhaps one of the best ways to start would be to include savings as a needs category in any sample budgets shared with students. An easier to follow suggestion is starting with small goals, $10 per month for example. As Fig. 11.3 shows, even a modest monthly amount can make a huge difference of a lifetime of savings.

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Impact of interest type on total savings $12,000.00

$10,000.00

$8,000.00

$6,000.00

$4,000.00

$2,000.00

$0.00

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Fig. 11.3 Impact of interest type on total savings for a one-time deposit

11.2.8 Credit Reports Given the importance of credit in today’s society, the importance of good credit is hard to overestimate. The credit available to an individual is dependent in large measure upon their credit report. This makes understanding the basics of a credit report an important aspect of financial literacy. In a credit report all of an individual’s credit activities and payback history are listed. Students need to recognize that part of being financially responsible is to periodically review their credit report and check for accuracy.

11.2.9 Planning and Paying for College Middle school is an excellent time to begin planning how to pay for college or post-secondary education and training. As budgets are created, students should recognize that scholarships and grants should be their first choice when searching for and selecting financial aid packages. Scholarships and grants are basically free education money that generally do not need to be repaid. As a general rule, scholarships tend to be based more upon merit while grants tend to be based upon student financial need. Need-based, grants can be a huge benefit for lower income/first time in college students. These are not the only types of grants available, however, less common grants are awarded on the basis of community service, occupational need (for example, teaching grants which require

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Initial deposit followed by consistent monthly deposits. $60,000.00

$50,000.00

$40,000.00

$30,000.00

$20,000.00

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$0.00

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No Interest

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Simple Interest

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Fig. 11.4 A one-time deposit followed by consistent monthly deposits

an agreement to teach high-need subjects in a low socioeconomic status1 area), and leadership. Similarly, despite being primarily merit-based, it is possible to find scholarships that are need-based or merit-based. For example, scholarship can be offered based upon a hobby, ability, etc., making them options even for less academically able students. Although it may not always be possible to create a plan to pay for college in advance, when this can be done it has many benefits which go beyond the economic factors. Peace of mind is certainly one of them. Others include greater options in terms of the university to attend and the lowered borrowing requirement. As might be expected, planning for college is directly linked to budgeting and savings. According to Sallie Mae Bank, 46% of families report creating a budget that includes college financing strategies such as dedicated savings, scholarships, student loans, and financial aid (The advantages of planning how to pay for college, 2020).

1

As used here, socioeconomic status refers to the social standing or class of an individual or group and is often measured as a combination of education, income and occupation.

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First generation in college students are particularly at risk in terms when it comes to paying for college as the costs and benefits may lack a realistic context for these students. Although student loans can make college possible for many students, they can also create debt that impacts students for years. Even if a low interest loan can be obtained, the continuous compounding can be devastating and should be undertaken carefully. Without a plan to address the expenses of higher education, students are left with a year to year coping effort which can fail at any time. The internet can be an excellent resource when searching for grants and scholarship opportunities.

11.2.10

Insurance

cBig Idea Insurance, a big idea in financial literacy, shares individual financial risks among a large pool of people. Students should be guided to recognize the values of insurances of various types and weigh the respective costs against the benefits each affords.b Insurance helps to share financial risks and losses risk among a large pool of people who are each paying premiums. The premium payments which are paid by all of the insured clients cover the costs for the emergencies of the few who need it. In setting these premiums, insurance companies rely upon a broad range of statistics to determine the level of premiums which are necessary to cover claims, costs and to maintain profitability.

11.2.10.1

Life Insurance

Given that everyone will eventually die, life insurance should be an essential part of any financial management plan. This is especially important for younger individuals as the impact of an untimely death is greatest for this group. As is the case for all types of insurance, there is a great deal of variability among insurance depending upon a number of factors including age, health, risk factors and amount of insurance desired. Life insurance, while necessary, has a number of factors which should be carefully considered prior to selection of a policy. In particular, there are several major types of life insurance middle school students should be aware of: term, whole life, and universal life (Busch, 2019). Term life insurance is the most basic type of the life insurance types and is typically the most affordable. As such, term life insurance only covers a specific time, or term. Should the insured die within this term the beneficiaries will receive the face value of the insurance policy. Term insurance is useful for people whose

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budget can only support a low insurance payment or those who will only need insurance for a specific period of time. Whole life is characterized by having a payment for as long as the holder lives, with a benefit paid to the beneficiaries upon death of the insurance holder. In addition, with whole life insurance the required premium will stay constant for the life of the policy. A potential benefit for the insured is that premiums can accumulate cash value, which is reinvested in the insurance companies’ portfolios. Once value has been accumulated in the policy, policy holders can borrow from this accumulated cash value or surrender the policy for its current cash value. Universal life insurance has a death benefit, but once the cash value of the policy is met the frequency and/or amount of premiums can be modified. In cases where enough cash value is acquired, it is often possible to use the invested cash value to pay the premium costs.

11.2.10.2

Health Insurance

Health insurance can literally save not only your life but also your finances. This can be confusing to students, however, as available health insurance and premiums depend upon a number of factors including age, family size, coverages, medical history and location. Given the importance of health insurance, however, students should at least be familiar with some basic health insurance terminology. Although not all of the following are applicable to all areas, they do represent some of these essential health care terms students should be aware of: Allowed maximum benefit is the maximum amount the insurance will pay within a specified time period. This amount is generally provided as a yearly amount, however, depending upon the service and type of insurance, it can also refer to a lifetime benefit. A copayment, or copay, is the amount that the insured must pay every time a health care service is used. The amount of the Copay can vary depending upon the service utilized with emergency room visits typically requiring the highest copays. Coinsurance is what must be paid out-of-pocket, this is typically a percentage of the service cost, once the insurance deductible has been met. Mitigating some of this expense, the amount paid in coinsurance will generally count towards the out-of-pocket maximum. The deductible is the amount which must paid by the insured prior to the health insurance being applied. Generally, insurances with high deductibles will have lower costs while insurances with lower deductibles will cost more for the insured. When applicable, essential benefits refer to the set of benefits which are covered regardless of whether the annual deductible is met. These include outpatient care, emergency services, hospitalization, pregnancy care, mental health, prescription drugs, etc. The out-of-pocket maximum is an upper limit of what the insured will have to pay. The out-of-pocket maximum can be either expressed as either a yearly or lifetime amount depending upon the policy.

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Out-of-pocket expenses refer to all health care cost which must be paid by the insured. Depending upon the insurance, occasionally there will be an out-of-pocket maximum which limits the amount the insured must pay. The amount which must be paid by the insured to be covered by insurance, typically paid on a monthly basis, is referred to as the insurance premium. For those fortunate enough to have health insurance offered through work, a portion of the premium will often be paid by the insured’s employer.

11.2.10.3

Long-Term Disability

Accidents can easily result in an inability to work for an extended period of time which can stretch to weeks, months, or even years. Just as health insurance helps cover hospitalization and medical costs, long-term disability insurance will help cover the loss of income while the insured is unable to work. Without some type of disability insurance, should a disability occur, long-term investments such as retirement and saving, will often suffer. Lack of disability insurance is one of the more common gaps in an individual’s financial planning despite that people are more likely to be disabled than die during their working years (Block et al., 2008). As is the case with all insurances there is a broad variety of options, each with varying costs, depending upon a number of factors including age, profession, and level of benefits required.

11.2.11

Investments

It is important to stress with middle school students that investments should not be viewed as a way to get rich quickly. Holding such a view will disappoint the students and lead to unproductive beliefs concerning saving and investment. A much healthier perspective is to view an investment as a way to grow and protect wealth accumulated over an extended period of time. One of the better things about investments is that they generally do not require a large sum of money to begin and there are many options available to select from. A cash investment includes the amount of actual cash held in a checking account, a savings account, or certificate of deposit (CD). Despite being relatively easy to set up and track, such investments typically provide a lower rate of return on investment than other investment options. As such, these investments will slowly lose spending power when compared with other investment options. A bond is basically a long-term document acknowledging a debt that is issued by a company or government. As such, bonds make sense for many young investors by providing a safe investment with a known rate of return on their investment. For example, if a company or government wishes to raise money it issues a bond. As a bond holder, you collect the interest until the bond matures at which point you get

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the principal invested back. At a broader level, bonds are particularly attractive investments when interest rates fall or stock markets decline. Mutual funds are created by pooling the investment money from many investors which is then re-invested in a variety of investment types. Mutual funds are quite popular due to their relatively low cost, being easy to invest in, and their allowance for a diversification of investment types. As might be expected there are a wide variety of mutual fund types. One of these is the indexed funds. An indexed mutual fund follows financial market index using a generated algorithm. Stocks are different from bonds. A holder of stock in a company actually has a partial ownership in that company. Companies issuing stock are basically selling part of itself for cash. As such, a stock holder shares in the gains and losses of the company. If the company does well the stock will increase over time. On the other hand, if the company does poorly the stock’s value will decrease or even go to zero. Stock investments tend to provide stronger long-term benefits, but are much riskier in the short term.

11.2.12

Common Financial Pitfalls

Norris (2019) has identified a listing of common financial mistakes that students should learn so that they might be avoided. Not surprisingly, excessive or frivolous spending tops the list. Even when the amounts, in and of themselves, might appear small, over time these expenses add up and reduce the resources available for other more crucial uses. Another common, but easy to overlook, pitfall is falling into the trap of never-ending payments for services that are not fully utilized. Examples would include cable television, streaming media services and gymnasium memberships which create a never-ending set of payments. Unfortunately, many young people fall into the trap of using credit cards to buy essential needs. By living on borrowed money, they open themselves up to the need to pay compound interest on life essentials. If a budget delineating the needs and wants, see Sect. 11.2.2, has been established this pitfall can be easily avoided. Closely related to this pitfall are the challenges to finances brought about through automobile ownership. Any vehicle purchase represents a significant purchase where the wants can easily sway the purchaser. As such, it should only be done after a careful review of needs versus wants. For example, is there really a need for an eight-cylinder three-quarter ton pickup if it will only be used to drive to and from school? Despite 74% of consumers reporting they have a budget, 79% of this number also report failing to follow it with the average yearly extra spending roughly $7400 (O’Brien, 2019). This points to the importance of periodic review of the budget to make sure it is realistic, covers the needs, and reflects the long and short-term goals of the individual. The purpose of a budget should be to align spending with the values of the individual. Such a review will also help to identify spending pitfalls which may have been entered into since the budget was first established.

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Financial Literacy

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Warning signs that an individual may have succumbed to one of these, or other, pitfalls tend to be quite obvious. For example, if a credit card balance is at the credit limit, if only minimum payments are being made, or if any of the credit terms are not being met.

11.3

Blockchain

cBig Idea Blockchain, a big idea in both computer science and technology, is not the same as cryptocurrency. Although most often thought of due to its role in cryptocurrency, blockchain is an emerging technology with multiple applications that are likely to significantly impact everyone’s financial lives.b Although it is still an emerging technology, blockchain is likely to significantly impact the financial lives of current middle school students. As such, it is important for students to develop a working background regarding what blockchain is and the manner in which it operates.

11.3.1 Blockchain Basics A blockchain can be thought of as a list of records or set of databases that are cryptographically shared across a peer-to-peer network of computers or “nodes”. Each individual record within such a database can consist of a variety of information types, such as business transactions. By design, blockchain is an open, distributed ledger that can record transactions between two parties efficiently and in a verifiable and permanent way. Data in any single block cannot be retroactively altered without the consensus of the network majority. This blockchain feature enforcing the data integrity results in a verification process which does not require trust between individual nodes within the peer-to-peer network. Blocks are added to the chain through various consensus mechanism algorithms to achieve agreement on data value or state of the network. Because these consensus mechanisms serve distributed systems, they do not rely on a central authority to agree on the validity of the transactions. When added to the existing chain through a consensus mechanism, new blocks are added to the existing chain of blocks and can store information. For example, suppose a transaction has occurred—Sariah sells a handcrafted hat to Shannon for $75. The initial record would include the details of the transaction in addition to a digital signature from both Sariah and Shannon. Following posting of this record, the computers in the network, often referred to as the “nodes” of the network, check the details of the record to verify that it represents a legitimate

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transaction. Following this verification process, the record will be added to other transactions in a block. As this is done, each block will contain a unique created code, referred to as a hash code, as an identifier. In addition to the hash code for the current block, each block will also contain the hash of the preceding block in the chain. Once verified, the new transaction(s) are then added to the existing blockchain together with its associated hash code. These hash codes then serve to connect the blocks in a specific order.

11.3.2 Blockchain Security Concerns The information contained within a blockchain is extremely difficult to change due to the embedded hash codes serving to identify each block in the chain and its position within the chain itself. This hash code is created by a mathematical function that takes digital information and generates a string of letters and numbers from it. This hash code has two important properties. First, no matter the size of the original file, the hash function always generates a code of the same length. So, whether we refer to the transaction between Sariah and Shannon or a thousand-page ledger documenting a list of transactions, applying the hash function to either will generate hash codes of equal length. Secondly, any change to the original input results in a new hash code being generated. Due to this forward and backward linking, any change in hash codes causes the blockchain to break. Since the following block in the chain still has the original hash code value, it would be necessary for a hacker to recalculate this value together with every subsequent hash code in the chain. Barring significant advances in quantum computing, the calculations necessary to recalculate these subsequent hash codes would require an unreasonably enormous amount of computing power.

11.3.3 Consensus Mechanisms and Incentivation Relative to traditional ledgers, blockchains are more robust as they are decentralized across a network of computers. There is not, for example, a single “master record”. All of the nodes making up the network can access the information and compete to be the next to add to the database. This competition is incentivized by the awarding of rewards—tokens for instance, or bitcoins. As the processing of a transaction occurs across a network of computers it can be accomplished extremely quickly without requiring external validation, offering significant speed increases when compared with traditional methods.

11.3

Blockchain

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Since there is not a master record, or centralized control, it is necessary that the nodes comprising the network are trustworthy. Trust is addressed via adoption of a network protocol and consensus mechanisms. A network protocol, in simplest terms, is the rulebook governing how each node in the network communicates with each other. Since each node has a local copy of the blockchain and the rulebook, it creates a redundancy building the fault tolerance of the system. The consensus mechanism is the process by which the nodes in the network verifies each transaction and reaches consensus as to what the current and accurate blockchain is with each new validated block. Two common consensus mechanisms are proof-of-work and proof-of-stake. In proof-of-work consensus individual nodes are incentivized to participate through solving increasingly difficult computational problems. In return for this “work”, the node establishes creditability, trustworthiness, and if first to solve the problem, will receive an award, generally in the form of token(s). As a result of this “work”, new transactions are joined/added to the existing blockchain in the form of block(s). These tasks require nodes to prove their trustworthiness. Validating transactions is referred to as mining and requires significant computational power. These tokens can also be purchased, establishing a value for them, and used by nodes to purchase membership stake into the network. Such membership stake plays an important role within proof-of-stake consensus where the creator of the next block is determined by various combinations of random selection and the node’s stake as defined by its wealth, or age.

11.3.4 Cryptocurrency and Other Applications Blockchain is currently most famously known as being the technological basis for cryptocurrencies, such as Bitcoin, Ripple, Ethereum and Tether. In addition to these experimental currencies, more traditional financial institutions are now investing in blockchains for a variety of reasons ranging from future market opportunities to simplifying their payments and record-keeping. These features make Blockchain extremely useful in supply chain verification through the creation of a rapid and secure method to check the history of a product. Although still early in development, it is easy to imagine additional applications of blockchain in areas requiring maintenance of historical record such as medical records, voting records, recording of property, title records, etc.

11.3.5 Advantages of Blockchain Blockchain has several major advantages over other methods of transaction processing. The first is the total transparency enabled by blockchain. Since each node in the network has access to both the ledger and rulebook, each node has shared

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visibility into all precedent transactions. This makes it is possible for any node to track the history of the blocks and records making up the chain. Second, blockchain is a truly democratic system with each node being granted equivalent control. As new blocks in the chain appear linearly in chronologic order, once a new block is assigned a hash code it is virtually impossible for the block to be altered. As mentioned earlier, such an amendment would require the hacker to alter not only the block being hacked, but also every preceding block making up the chain. Third, since blockchain transactions are self-validating, they do not require a middle-man and are instantaneous upon verification, posting, and addition to the chain. The lack of a middle-man potentially allows for huge reductions in transaction fees and avoidance of late penalties. Due to these features, Blockchain has created many new applications aimed at creating and enhancing social impact. These include agriculture and land rights, climate and environment, digital identities, financial inclusion, governance and democracy, and health (Stanford Graduate School of Business, 2019).

11.3.6 Disadvantages of Blockchain Blockchain has some potential difficulties, however. In order for it to work, there must be cooperation across all nodes in the network and all transactions added to the chain need to be transparent to each node for verification. This results in the data within a block being potentially open to hackers, even though the blockchain itself is secure against hacking. Since this impacts each node in the network, there is a potential for data loss and misuse.

11.4

Conclusion

cBig Idea Due in large part to the pivotal role played by compound interest, a big idea in mathematics and finances, developing financial literacy early can greatly contribute to students’ quality of life.b

Making monetary decisions can be challenging for anyone regardless of their age. By establishing a basic level of financial literacy early, however, students can develop the understandings and connections needed to face such challenges with confidence. When students learn financial literacy at a relatively young age, they become less impressionable to the attitudes of money held by the adults around them, which in many cases can help break generational

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Conclusion

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poverty brought about by lack of financial understanding (Why is Financial Literacy Important for Youth: Research, 2020). Furthermore, as students acquire financial management skills, they tend to keep them and use them throughout their life. Consider the Einstein quote from Sect. 11.2.4.2 concerning compound interest. It is far better to earn compound interest than to pay it. In a like fashion, it is far better to understand your finances and control them than to be controlled by the lifetime lack of resources poor planning can bring. For readers who want to deepen their knowledge and understanding of financial literacy and blockchain the following sources can be recommended (Jenkins, 2020; Bitcoin Wiki, 2020; Open Source P2P Money, 2020; Etherium Whitepaper, 2020; Nakamoto, 2020).

11.5

Activity Set

1. Why is it important for middle school students to learn about earning, buying and saving? 2. What are some differences between a first-time job and a career? 3. Why is it important for middle school students to begin personal interest explorations and careen planning? 4. What would Shannon’s Gross Income be if she works 40 h at $8.75 per hour? 5. What would Shannon’s pay have to be to have a Gross Income of $485 for the same 40 h week? 6. If Shannon is taxed at a rate of 10 12 % what would her Net Income be for questions 5 and 6? 7. What are some differences between wants and needs? Identify at least three examples of each. 8. Why is it important to use Net Income and not Gross Income when building a budget? 9. Explain the 50–30-20 budgeting strategy. 10. What are some considerations to consider when selecting a checking account type? 11. What is the difference between Simple and Compound interest? 12. How much Simple Interest would be earned in five years on a deposit of $4,000 invested at an interest rate of 4.25%? 13. Imagine a personal loan with a $15,750 balance and a 3.75% interest rate with payments due on the last day of the month. How much could be saved by making the payments 15 days early? 14. How much interest would be earned on a deposit of $2,250 invested at a rate of 3.25% per year compounded monthly?

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15. What are some differences between revolving credit, installment credit, service credit, and open credit? 16. Which would be preferable as the borrower for a loan: Simple Interest or Compound Interest? Why? 17. Which would be preferable as a saver for a savings account: Simple Interest or Compound Interest? Why? 18. What is the fundamental purpose of insurance? 19. What are some differences between stocks and bonds? 20. What is the role of the hash code in blockchain technology? 21. How are some of the ways that blockchain technology establishes trust among nodes? 22. What make the information within a blockchain difficult for hackers to modify?

Chapter 12

Probability and Statistical Data Analysis

12.1

Introduction

Teaching probability and statistical data analysis begins at the primary school level and continues through higher levels of mathematics education. At the middle school level, mathematics teaching standards related to the topics dealing with random phenomena and their disciplined description emphasize mathematical modeling, the appropriate use of physical and digital teaching tools, and the experimental character of situations associated with uncertain outcomes. Already in the middle school, a strong emphasis can be given to experimentally determining the likelihood of what in the elementary school was referred to as chance events by recording the number of trials a desired event occurs and dividing this number by the total number of trials within an experiment. The importance of using experimental techniques in the study of probability and statistics at the pre-college level is acknowledged by many standards worldwide. In the United States, students “use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations” (National Council of Teachers of Mathematics, 2000, p. 248) and “learn about the importance of representative samples for drawing inferences” (Common Core State Standards, 2010, p. 46). In Canada, students are expected to “pose and solve simple probability problems, and solve them by conducting probability experiments … [and, as they study statistics,] determine, through investigation, the appropriate measure of central tendency (i.e., mean, median, or mode) needed to compare sets of data” (Ontario Ministry of Education, 2005, pp. 85, 118). In Japan, among the main objectives of teaching probability and statistics is to make students “aware of regularity among a large number of observations of natural phenomena, and to learn the foundations for organizing data to represent that regularity … [and to help them] understand the meaning of simple statistical data correctly” (Takahashi et al., 2006, p. 59). In England, expectations for © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5_12

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students in the context of probability and statistics include the ability to “record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness … [and] describe simple mathematical relationships between 2 variables (bivariate data) in observational and experimental contexts” (Department for Education, 2013). In Singapore, students are expected to “discuss the experimental and theoretical values of probability using computer simulations” (Ministry of Education Singapore, 2012b, p. 43) and “construct a bar graph/line graph using a spreadsheet” (Ministry of Education Singapore, 2012a, p. 73). In the modern-day classroom, experiments in the context of probability and statistics can be facilitated by the use of technology. To this end, in this chapter, several means of computational experimentation will be suggested and discussed towards developing conceptual approaches to the study of probability and statistical data analysis. One of the digital tools used in this chapter is a spreadsheet that can easily generate random numbers in large quantities to support experimentation as genesis of conceptual understanding in mathematics, in general, and in the theory of probability, in particular.

12.2

Experiments with Equally Likely Outcomes

When developing conceptual approach to the theory of probability, experiments with equally likely outcomes are of a special importance. For example, tossing a fair coin, rolling an unbiased die, and drawing a card from a well-shuffled deck can be set as experiments with equally likely outcomes. It is equally likely to have either head or tail when tossing a coin, either one or six when rolling a die, and either club or heart when drawing a card. Cardano1 is credited with the definition of the concept of probability as the ratio of the number of favorable outcomes to the total number of equally likely outcomes within a certain experimental situation. This classic definition is commonly accepted nowadays in school mathematics. When solving probability problems with equally likely random outcomes, this definition gives an applied flavor to fractions in the range [0, 1] as numerical characteristics of what is considered probable and makes it possible to compare chances by using proper fractions (or their decimal equivalents). But chances (alternatively, probabilities), in order to be compared, have to be computed (measured) first. Mathematical actions of that kind should not be considered through the simplistic lens, like measuring

1

Gerolamo Cardano (1501–1576)—an Italian polymath, one of the most influential mathematicians of the Renaissance, recognized for his contributions to probability and algebra.

12.2

Experiments with Equally Likely Outcomes

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perimeters and areas of basic geometric shapes through selecting appropriate measurement tool of the modern-day geometry software. For instance, one tosses a coin and, knowing with certainty that it would not fly, cannot predict how exactly it would fall. But through tossing a coin many times and recording the results, one can recognize how randomness turns into regularity. Indeed, one can check to see2 that out of 100 tosses a coin would turn head or tail somewhere in the range [40, 60]; that is, an experimental probability of head or tail would be a number in the range [0.4, 0.6]. A more sophisticated experimentalist can attempt to measure the results of possible outcomes when a coin is tossed, say, four times, in a large series of experiments. A surprising result, which first can be established experimentally by using a spreadsheet, is that having four heads (HHHH) has the same probability as having first head, then tail, and then again head and tail (HTHT). The spreadsheet of Fig. 12.1 (the programming of which is included in Appendix) shows that as a result of 10,000 (simulated) tosses of a coin the incident HHHH happened about as many times as the incident HTHT (596 vs. 595); in other words, the chances for HHHH and for HTHT are measured, respectively, by two fractions (0.0596 and 0.0595, respectively) differing by 0.0001 (Fig. 12.1, cell F2). Multiple repetitions of this computational experiment (without duplicating the above numbers) would demonstrate that both incidents have almost equal chances within a large series of experiments. Blaise Pascal, being credited (among many other things) with bringing experimental evidence in probability theory (e.g., Shaughnessy, 1992), used his famous triangle (Fig. 12.2) for recoding the results of tossing a coin (Kline, 1985). For example, the fifth line 1 4 6 4 1 in Pascal’s triangle shows that among all possible outcomes of tossing a coin four times, the incidents HHHH and TTTT can happen only one time each, the incidents with either three heads or three tails may each happen four times, and the incidents with two heads and two tails may happen six times. Indeed, the incidents with three heads and one tail can be counted as the permutation of letters in the word HHHT; that is, 4! 3! ¼ 4. Likewise, the incidents with three tails and one head can be counted as the permutation of letters in the word TTTH; that is, 4! 3! ¼ 4. The incidents with two heads and two tails can be 4! ¼ 6. One can counted as the permutation of letters in the word HHTT; that is, ð2!Þð2!Þ see how the basic counting tools of combinatorics (Chap. 8) are used in the analysis of experiments with random outcomes. More complicated situations in terms of finding experimental and theoretical probabilities can also be considered. For example, in the case of an addition (multiplication) table one can find experimental probability of the event when two integer addends (factors) in the range 1 through 10 are randomly selected, their sum (product) is an even number. Then it can be shown how such problems can be 2

The history of probability theory acknowledges a number of similar experiments carried out over the years, one of which is attributed to Georges-Louis Leclerc, Conte de Buffon (1707–1788), a French naturalist and mathematician, who “tossed a coin 4040 times. Result: 2048 heads, a proportion of 2048/4040 = 0.5069 of heads” (Moore, 1990, p. 97).

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Fig. 12.1 HHHH versus HTHT

1 1 1 1 1 1 1

3 4

5 6

1 2 6

10 15

1 3

1 4

10 20

1 5

15

1 6

1

Fig. 12.2 Pascal’s triangle

solved theoretically. Finally, experimental and theoretical probabilities can be compared. Similar explorations with the addition and multiplication tables can be discussed using both experiment and theory. In that way, an addition (multiplication) table can be presented to the learners of mathematics not as something to memorize, but rather as an instrument of learning through conceptual exploration that integrates experiment and theory. In the context of data analysis, the chapter includes such topics (supported by a spreadsheet and Wolfram Alpha) as graphical representations of numeric data, measures of central tendency and dispersion, the normal probability distribution, and the basic ideas of bivariate analysis. The discussion of these ideas is carried out in the context of coin tossing and die rolling as a way of connecting concepts of probability and statistics.

12.3

12.3

Randomness and Sample Space

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Randomness and Sample Space

In this section, several basic concepts associated with the probability strand will be explained. The first one is randomness—a characterization of the result of an experiment that is not possible to predict but, nonetheless, is often possible to measure on the scale from zero to one, with zero assigned to something impossible (e.g., a coin flies when tossed in the still air) and one assigned to something certain (e.g., a coin falls when tossed in the still air). The very concept of randomness is not easy to define; it “has proved to be more than a little elusive, and it remains enigmatic, despite the fact that it has proved to be useful in many contexts” (Nickerson, 2004, p. 54). Nonetheless, the usefulness of randomness and its multi-contextual applicability allows one to recognize the study of randomness through rigorous methods as another big idea of mathematics. One can say that randomness does not lead to a credible pattern, although it is often difficult to conclude whether there is no pattern in a sequence of events. Nonetheless, thinking about randomness in a very informal way, can help one understand the “difference [between] predicting individual events and predicting patterns of events” (Conference Board of the Mathematical Sciences, 2001, p. 23). When tossing a coin, whereas one cannot predict how exactly it would fall, it is not impossible to have head and tail alternating in, say, five tosses, or even to have five heads or tails in a row. Yet, due to experience, one can predict a pattern in the sequence of 100 tosses. Whatever a prediction, the question to be answered is how to measure the likelihood (chances) of the prediction. cBig Idea A simple experiment with coin tossing, while having random outcome for a single toss, nonetheless, demonstrates how randomness turns into regularity when the number of tosses is large. The study of randomness using rigorous mathematical methods is one of the big ideas of mathematics that gave birth to the theory of probability the methods of which make it possible to measure the likelihood of a random outcome.b The need for measuring chances leads to the concept of the sample space of an experiment with a random outcome that is defined as the set of all possible incidents associated with this experiment. Within a sophisticated experiment it may be difficult to determine whether such a set is complete because the confirmation of completeness requires solving a separate mathematical problem. Such problems may differ significantly in terms of complexity. A simple example is the sample space of rolling a six-sided die (with the number of spots on the sides ranging from one to six) comprised of six outcomes: {1, 2, 3, 4, 5, 6}. Assuming that we deal with a fair (unbiased) die, all outcomes may be considered equally likely. Under this assumption, one can say that there is one chance out of six to cast any of the six numbers (spots).

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A more complicated example is the sample space of an experiment of changing a half dollar into nickels, dimes and quarters. The chart of Fig. 12.3 shows a ten-element sample space, where numbers in the top row and the far-left column show the ranges for quarters and dimes, respectively, and the remaining numbers show the corresponding number of nickels in a change. For example, the triple (3, 1, 1) stands for three nickels, one dime, and one quarter, so that 3  5 þ 1  10 þ 1  25 ¼ 50. Alternatively, the sample space can be described as the following set of ten triples of numbers {(10, 0, 0), (8, 1, 0), (6, 2, 0), (4, 3, 0), (2, 4, 0), (0, 5, 0), (5, 0, 1), (3, 1, 1), (1, 2, 1), (0, 0, 2)}, where each triple describes the number of nickels, dimes, and quarters, respectively, in a change. Note that there is no reason to assume that it is equally likely to get any combination of the coins from a change-making device and Fig. 12.3 represents a sample space with not equally likely outcomes. Therefore, given a change-making device, it is only experimentally that one can determine chances for a specific combination of coins in a change. Alternatively, for the purpose of problem solving, one can assume that it is equally likely to get any combination of the coins in a change out of total ten. Under this assumption, one can say that the probability of not having nickels in the change is equal to 2/10 or, in the simplest form, 1/5.

12.4

Different Representations of a Sample Space

A sample space of an experiment with random outcomes may have different representations. Here is an example: the sample space of tossing two coins can be represented in the form of a table and a tree diagram as shown in Figs. 12.4 and 12.5, respectively. Note, that the outcomes of the two tosses are independent; that is, the outcome of the second toss does not depend on what happened on the first toss. This independence is reflected in the very form of the tree diagram: each of the two possible outcomes of the first toss affords the same two outcomes for the second toss. This, however, is not always the case and in more complex situations, drawing a tree diagram to represent a sample space may be a difficult proposition (see Fig. 12.7). An outcome of an experiment is an element of its sample space. For example, the sample space of the experiment of tossing two coins shown in Fig. 12.4 (or Fig. 12.5) consists of four outcomes. Further, outcomes may be combined to form an event. For example, the outcomes HH and TT form an event that both coins turn up the same. Depending on specific conditions, the outcomes of an experiment may or may not be equally likely. All the outcomes of the experiments of tossing coins or rolling dice described in the above examples are considered equally likely assuming that one deals with fair coins or unbiased dice. An assumption about equally likely outcomes is a theoretical assumption—after all, when we deal with a coin (or a die), we assume that it is a fair coin (or an unbiased die). Outcomes are not always equally likely. Indeed, when having a holder with three red and two black markers, it is not equally likely to pick up either a red or a black

12.4

Different Representations of a Sample Space

373

Fig. 12.3 A sample space of changing a half dollar into quarters, dimes and nickels

Fig. 12.4 The sample space of tossing two coins in the form of a table

Fig. 12.5 The sample space of tossing two coins in the form of a tree diagram

HH

TH

HT

TT

HEAD HEAD TAIL

HEAD TAIL TAIL

marker. Likewise, events may be independent and dependent. In the case when from the first draw of a marker from the holder, the marker is returned to the holder, the outcome of picking up a red marker on the second draw does not depend on which color marker was picked up on the first draw. At the same time, if the marker

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Fig. 12.6 The sample space of rolling two dice

Probability and Statistical Data Analysis

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is not returned, the outcome of picking up a red marker on the second draw depends on what happened on the first draw. The sample space of rolling two dice and recording the total number of spots on two faces can be represented in the form of an addition table (Fig. 12.6) from where it follows that the highest chances are for having seven as the total number of spots on two dice. Here one can connect probability concepts to the concepts of integer partitions. Indeed, whereas twelve has more partitions than seven, in the context of rolling two dice partitions are limited to the summands that are not greater than six. Under such condition, the closer a number to six is, the more ordered partitions into two integer summands exist. For example, the sum 12 has only one possibility, 12 = 6 + 6, and the sum 7 has six possibilities, 7 = 1 + 6 = 6 + 1 = 2 + 5 = 5 + 2 = 3 + 4 = 4 + 3. Finally, consider a tree diagram representation of the sample space of the experiment of changing a half dollar into three smaller coins. Teacher candidates often ask whether one can find the number of ways to change a half dollar (or a quarter) into three smaller coins by using a tree diagram. Note that unlike the cases of tossing coins or rolling dice, in the case of getting change one deals with outcomes in the form of the triplets of integers being in relation of mutual dependency. Indeed, the number of quarters in a change depends on the number of dimes and both depend on the number of nickels. Whereas it should not be recommended to use a tree diagram for constructing the sample space of this problem, notwithstanding, it can be constructed using this method. As shown in Fig. 12.7, the form of the tree diagram (comprised of nine sub-trees where the presence of the symbol x in the row for quarter indicates the absence of a solution) reflects the notion of interdependency of possible outcomes and the didactic value of representing a sample space through a tree diagram is in demonstrating the logic of causality involved in the construction of the tree. One can see that the first sub-tree related to zero nickels in the change shows two solutions (cf. two zeroes in the table of Fig. 12.3 related to the absence of nickels in a change) and each of eight other sub-trees shows one solution only (cf. each of eight non-zero numbers appearing one time only in Fig. 12.3).

12.5

Fractions as Tools in Measuring Chances

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12.5

Fractions as Tools in Measuring Chances

Application of arithmetic to geometry brings about the idea of using a numeric measure of the likelihood of an event. One of the ways to introduce the concept of fraction as an applied tool is through using fractions as a measure of the likelihood (or probability) of an event. The idea of a geometric representation of a fraction enables its use as a means of finding the probability of an event (alternatively, measuring its likelihood). Consider equally likely outcomes of the experiments with tossing two coins and rolling two dice. The meaning of the words equally likely can be given a geometric interpretation through representing such outcomes as equal parts of the same whole. In the case of coins, we have four equally likely outcomes, which can be represented in the form of a rectangle divided into four equal parts (Fig. 12.4) so that each part is represented by a unit fraction 1/4. It is this fraction that can be considered as the value of the likelihood (probability) of any of the four (equally likely) outcomes. Dividing the rectangle in four equal parts as shown in Fig. 12.4 demonstrates two things: (i) a table representation of the sample space of tossing two coins; (ii) according to the area model for fractions (Chap. 4), each of the four parts represents the product ð1=2Þð1=2Þ, where each factor is the probability of head/tail for each of the two tosses. Extending the number of tosses beyond two, one can measure the probability of alternating heads and tails in, say, five tosses by the number ð1=2Þ5 . One can see that the same number also measures the probability of having five heads (or tails) in a row. In the case of rolling two dice, each of the 36 outcomes that comprise the table-type sample space (Fig. 12.6) is equally likely and, therefore, the fraction 1/36 is the measure of each outcome. At the same time, the likelihood of the event that after rolling two dice, the result is either eight or nine spots on both faces is measured by the fraction of the table filled with either eight or nine. As the two numbers appear nine times in the (addition) table, the probability of this event is

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Probability and Statistical Data Analysis

9/36 or 1/4. In that way, one can conclude that the chances of having HH after tossing two coins are the same as having the sum of either eight or nine on both faces when rolling two dice. Apparently, without using fractions as measuring tools for chances (probabilities) this conclusion would not be possible. One can also calculate the probability of the event that in four rolls of a fair die there appears no face with six spots. There are five chances out of six of not having six spots when rolling a die; that is, the probability is equal to 5/6. Therefore, because each roll of a die does not depend on the previous roll, the probability of not having a face with six spots over four rolls is equal to the product of four equal factors ð5=6Þð5=6Þð5=6Þð5=6Þ ¼ ð5=6Þ4  0:482. Likewise, one can measure the probability of a double six spots not appearing in a series of twenty-four rolls of two dice. As shown in the table of Fig. 12.6 (representing the sample space of rolling two dice), there are 35 chances out of 36 of not having a double six. That is, the probability of not having a double six in a series of twenty-four trials is equal to ð35=36Þ24 ¼ 0:509. The numbers 0.482 and 0.509 are somewhat close to each other being located by the different sides of 0.5 (which can serve as a benchmark number in ordering the other two). The numbers ð5=6Þ4 and ð35=36Þ24 will be mentioned below in the context of a classic probability problem from the seventeenth century. Note that in the above examples, fractions (or, more generally, numbers in the range [0, 1]) were used as tools in measuring theoretical probabilities of the outcomes of events. As will be shown below, proper fractions can also be used in estimating experimental probabilities (alternatively, relative frequencies) of events as the ratio of the number of outcomes in favor of an event to the total number of trials used in the corresponding experiment. It is important to distinguish between the notions of probability and relative frequency because, as Mises (1957) put it, “a probability theory which does not introduce from the very beginning a connexion between probability and relative frequency is not able to contribute anything to the study of reality” (p. 63) Thus, the first demonstration of this chapter was to use a spreadsheet (Fig. 12.1) comparing incidents HHHH and HTHT within 10,000 simulated tosses of a fair coin.

12.6

Explorations with Addition and Multiplication Tables

In this section, the following questions will be explored both experimentally and theoretically: If a number is chosen at random from a square-shaped addition/ multiplication table, (1) what is the probability that this number is an even number? (2) what is the probability that this number is an odd number? (3) what is the probability that this number is a multiple of three?

12.6

Explorations with Addition and Multiplication Tables

377

12.6.1 Computational Experiments with Pairs of Random Integers To begin, an experimental approach can be used. To this end, one can randomly generate 1000 pairs of integers in the range [1, n] where n varies. For example, one can first set n = 10 and use the spreadsheet function = RANDBERWEEN(1,10) to generate integers in two adjacent columns. Next, each pair of random integers can be added (in the case of an addition table) or multiplied (in the case of a multiplication table) and then the resulting sum or product can be verified in terms of divisibility by two (or by three). The computational experiment shows that, regardless of the size of addition table, out of 1000 random selections of two summands, the number of even sums is close to 500. The spreadsheet of Fig. 12.8 (the programming of which is included in Appendix) shows that when a pair of numbers is randomly selected from the range [1, 10], their sum is an even number in 506 cases out of 1000. Therefore, an experimental probability of having an even sum is equal to 0.506. Of course, this number would most likely be different for a new set of 1000 pairs of random integers. The spreadsheet of Fig. 12.9 (the programming of which is included in Appendix) shows that when a pair of numbers is randomly selected from the range [1, 10], their product is an even number in 746 cases out of 1000 (for this particular experiment). Therefore, experimental probability of having an even product in the 10  10 multiplication table is equal to 0.746. At the same time, when numbers in columns A and B (Fig. 12.9) are randomly selected from the range [1, 9], their product is an even number in 697 cases out of 1000 (for this particular experiment). For a different set of 1000 pairs of random integers and for a different size table these numbers most likely would be different; that is, experimental probability varies from experiment to experiment. A similar computational experiment can be carried out for exploring divisibility by three among the entries of an addition table. One can check to see that out of 1000 random selections of two integers, regardless of the size of an addition table, about one-third of their sums are multiples of three. The spreadsheet of Fig. 12.10 shows that when a pair of numbers is randomly selected from the range [1, 10], their sum is a multiple of three in 334 cases out of 1000. Therefore, an experimental probability of having a sum divisible by three in the 10  10 addition table is equal to 0.334 (for this computational experiment). Likewise, one can find an experimental probability of randomly selecting multiples of three among the entries of the 10  10 multiplication table. The spreadsheet of Fig. 12.11 shows that when a pair of numbers is randomly selected from the range [1, 10], their sum is a multiple of three in 543 cases out of 1000. Therefore, an experimental probability of having a product of two random integers, each of which belongs to the range [1, 10], divisible by three is equal to 0.543 (within this computational experiment).

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Fig. 12.8 Out of 1000 pairs of random integers their sums are even 506 times

Fig. 12.9 Out of 1000 pairs of random integers their products are even 746 times

Fig. 12.10 Out of 1000 pairs of random integers 334 of their sums are divisible by 3

12.6

Explorations with Addition and Multiplication Tables

379

Fig. 12.11 There are 543 products divisible by 3 out of 1000 pairs of random integers

12.6.2 Selecting Even Numbers in Addition Tables The above experimental probabilities can be compared with probabilities obtained through a theory. In Chap. 9, Sect. 9.7.3, the following result about divisibility by two in the n  n addition table was derived: when n = 2k, there are 2k2 even numbers in the table; when n = 2k + 1, there are 2k2 + 2k + 1 even numbers in the table. Therefore, the theoretical probability of selecting an even number from the 2 1 addition table of size 2k  2k is equal to 2k 4k 2 ¼ 2. Consequently, the probability of randomly selecting an odd number from the table is also 12. The theoretical probability of selecting an even number from the addition table 2 of size ð2k þ 1Þ  ð2k þ 1Þ is equal to 2kð2kþþ2k1Þþ2 1 : In particular, when k = 4, we have the 9  9 table and the probability is equal to 41 81 ¼ 0:51: One can see that this value of threoretical probability is consistent with the experimental probability of selecting at random from an addition table an even number (Fig. 12.8). 2 Note that 2kð2kþþ2k1Þþ2 1 [ 12 for any k > 0 as 4k 2 þ 4k þ 2 [ 4k2 þ 4k þ 1. However, as k grows larger, the last fractional inequality in the context of calculus enables the 2 relation lim 2kð2kþþ2k1Þþ2 1 ¼ 12 the meaning of which will be explained below in k!1

Sect. 12.6.6.

12.6.3 Selecting Even Numbers in Multiplication Tables In the case of the n  n multiplication table, it was found that when n = 2 k, there are 3k2 even numbers in the table and when n = 2k + 1, there are 3k2 + 2k even numbers in the table (Chap. 9, Sect. 9.7.3.2). Therefore, the theoretical probability of selecting at random an even number from the multiplication table of size 2k  2k

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is equal to

3k 2 4k 2

Probability and Statistical Data Analysis

¼ 34. The theoretical probability of selecting at random an even

3k þ 2k number from the multiplication table of size ð2k þ 1Þ  ð2k þ 1Þ is equal to ð2k : þ 1Þ2 2

In particular, when k = 4, the probability of selecting at random an even number from the 9  9 multiplication table is equal to 56 81  0:69: One can see that this value of threoretical probability is consistent with mentioned in the pervious section experimental probability, 0.697, of selecting at random from the 9  9 multiplication table an even number. 3k 2 þ 2k Note that ð2k \ 34 for any k > 0 as 12k 2 þ 8k\12k2 þ 12k þ 3 whence þ 1Þ2 4k þ 3 [ 0—a true inequality. However, as k grows larger, the last fractional 3k 2 þ 2k inequality in the context of calculus enables the relation lim ð2k ¼ 34 the þ 1Þ2 k!1

meaning of which will be explained below in Sect. 12.6.6.

12.6.4 Selecting Multiples of Three in Addition Tables Using results from Chap. 9, Sects. 9.7.3.3 and 9.7.3.4, one can also find the theoretical probability of selecting at random a multiple of three from addition and multiplication tables. It was found that in the 3k  3k addition table there are 3k 2 multiples of three; in the ð3k þ 1Þ  ð3k þ 1Þ addition table there are 3k2 þ 2k multiples of three; in the ð3k þ 2Þ  ð3k þ 2Þ addition table there are 3k 2 + 4k + 2 multiples of three. Therefore, the theoretical probability to select at random a 2 1 multiple of three in the 3k  3k addition table is equal to 3k 9k 2 ¼ 3, in the ð3k þ 1Þ  3k 2 þ 2k , and in the ð3k þ 2Þ  ð3k þ 1Þ2 3k 2 þ 4k þ 2 ð3k þ 2Þ addition table the probability is equal to ð3k þ 2Þ2 . In particular, when 56  0:33, and for k = 4, for the 13  13 addition table, the probability is equal to 169 66 the 14  14 addition table, the probability is equal to 196  0:34. One can check to

ð3k þ 1Þ addition table the probability is equal to

see that these values of threoretical probabilities are consistent with experimental probabilities of selecting at random from an addition table a multiple of three. 3k2 þ 2k Just as it was shown above, one can check to see that ð3k \ 13 and þ 1Þ2 3k2 þ 4k þ 2 ð3k þ 2Þ2

[ 13. However, as k grows larger, the last two inequalities in the context 3k2 þ 2k 2 k!1 ð3k þ 1Þ

of calculus enable the limiting relations lim

3k2 þ 4k þ 2 2 k!1 ð3k þ 2Þ

¼ lim

meaning of which will be explained below in Sect. 12.6.6.

¼ 13 the

12.6

Explorations with Addition and Multiplication Tables

381

12.6.5 Selecting Multiples of Three in Multiplication Tables Similarly, it was found that in the 3k  3k multiplication table there are 5k2 multiples of three; in the ð3k þ 1Þ  ð3k þ 1Þ multiplication table there are 5k2 þ 2k multiples of three; in the ð3k þ 2Þ  ð3k þ 2Þ addition table there are 5k2 þ 4k multiples of three (Chap. 9, Sect. 9.7.3.4). Therefore, the theoretical probability to select at random a multiple of three from the 3k  3k multiplication table is equal to 5k2 5 9k2 ¼ 9, from the ð3k þ 1Þ  ð3k þ 1Þ multiplication table the probability is equal to 5k 2 þ 2k , ð3k þ 1Þ2

and from the ð3k þ 2Þ  ð3k þ 2Þ multiplication table the probability is

5k þ 4k equal to ð3k . In particular, when k = 4, for the 13  13 multiplication table, the þ 2Þ2 2

88 probability is equal to 169  0:52, and for the 14  14 addition table, the probability 96 is equal to 196  0:49. As in the previous examples, one can check to see that these values of theoretical probabilities are consistent with experimental probabilities of selecting at random from a multiplication table a multiple of three. 5k 2 þ 4k 5k2 þ 2k As before, it can be proved that ð3k \ ð3k \ 59. However, as k grows þ 2Þ2 þ 1Þ2

larger, these inequalities in the context of calculus enable the limiting relations 5k2 þ 2k 5k 2 þ 4k lim ð3k ¼ lim ð3k ¼ 59 the meaning of which will be explained in the next þ 1Þ2 þ 2Þ2

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12.6.6 Theoretical Probabilities and Their Monotone Convergence The behavior of theoretical probabilities expressed by the fractions 2 2 3k 2 þ 2k 3k2 þ 2k , P3 ðkÞ ¼ ð3k ; P4 ðkÞ ¼ 3kð3kþþ4k2Þþ2 2 ; P5 ðkÞ ¼ P1 ðkÞ ¼ 2kð2kþþ2k1Þþ2 1 ; P2 ðkÞ ¼ ð2k þ 1Þ2 þ 1Þ2 5k 2 þ 2k , ð3k þ 1Þ2

5k þ 2k 5k þ 4k P5 ðkÞ ¼ ð3k and P6 ðkÞ ¼ ð3k and their respective relations to the þ 1Þ2 þ 2Þ2 2

2

fractions 12 ; 34 ; 13 ; 59 can be explained from the behavior of monotonic sequences standpoint. Consider the sequence P1 ðkÞ. Its numeric and graphic representations are shown in the spreadsheet of Fig. 12.12. The numeric representation of P1 ðkÞ, for different values of k, in the form of decimal fractions (in which division is completed, unlike in a common fraction representation) shows that each new value of P1 ðkÞ is slightly smaller than the previous one, yet they do not reach 0.5 but are bounded from below by 0.5 The same behavior with apparent convergence to 0.5 is shown in the graphic representation of P1 ðkÞ. Therefore, one can first get the idea that P1 ðkÞ decreases monotonically and then use Wofram Alpha (in the context of its free on-line version) to show that, indeed, P1 ðkÞ  P1 ðk þ 1Þ ¼ 4ðk þ 1Þ [ 0; k [ 0; that is, 12 [ P1 ðkÞ [ P1 ðk þ 1Þ for all k > 0. Likewise, ð2k þ 1Þ2 ð2k þ 3Þ2 one can show that P2 ðkÞ\P2 ðk þ 1Þ\ 34,

1 3

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Fig. 12.12 Monotone convergence of P1 ðkÞ to 0.5

P4 ðk þ 1Þ [ 13, P5 ðkÞ\P5 ðk þ 1Þ\ 59, and P6 ðkÞ\P6 ðk þ 1Þ\ 59. These observations, made possible by spreadsheet modeling and symbolic computations of Wolfram Alpha, open a window on the well known in real analysis monotone convergence theorem (Stewart, 2012). Each of the six sequences Pi ðkÞ; i ¼ 1; 2; :::; 6; either decreases monotonically and is bounded from below or increases monotonically and is bounded from above (by its limit). One can see how ideas associated with higher levels of mathematics curriculum (see also Chap. 7, Sect. 7.3.2) can be revealed through conceptual exploration of addition and multiplication tables. This kind of conceptual understanding of the relatioship among entries of the tables is critical for enabling a mathematics classroom to be a place where students “can learn, without drill, to deal empirically with situations involving numbers and develop a flexible set of procedures for handling such routine as is necessary” (Association of Teachers of Mathematics, 1967, p. 6). As was shown in Chap. 9, nowadays the set of procedures for handling conceptual explorations of sums and products of two integers can be developed in the context of spreadsheets.

12.7

Bernoulli Trials and the Law of Large Numbers

An experiment (alternatively, a trial) of tossing a fair coin one time has only two (equally likely) outcomes. When a coin is tossed twice, the outcome of the first toss has no influence on the outcome of the second toss. In other words, the two trials

12.7

Bernoulli Trials and the Law of Large Numbers

383

are independent, and the probability of head or tail is the same for each experiment. Another example of a trial with two outcomes is the birth of a child; however, unlike tossing a coin, here the outcomes are not equally likely. As an aside, note that the study of the human sex ratio by Arbuthnot3 (1710) nonetheless demonstrated that the excess of males over females is not due to chance; this study is considered not only as one of the earliest investigations on this topic, but as the pioneering work on using a representative sample to make a generalization about a population (referred to as inferential statistics). Despite the simplicity of the trials of that kind, they represent one of the most important concepts in the probability theory called Bernoulli4 trials. Let Sn be the number of heads observed within n tosses of a fair coin. Then Sn/n represents the fraction of heads in n tosses (trials). Bernoulli proved the following: the probability that the ratio Sn/n deviates from 1/2 by any number as small as one wants, tends to the unity as n increases. This statement can be extended to any kind of Bernoulli trials with the probabilities of success and failure equal, respectively, to p and q = 1 − p and it is known as the Law of Large Numbers in the form of Bernoulli. cBig Idea The Law of Large Numbers (in the form of Bernoulli) is a big mathematical idea allowing one to predict patterns of events in trials with two outcomes, success/ failure. The law makes it possible to establish the likelihood (probability) of an event experimentally, something that is especially important when it is not possible to find this likelihood theoretically. For example, when out of 10,000 drawings (without looking) from a basket with two fruits—a large pear and a small apple—the former fruit was drawn 8793 times, the probability of drawing the pear through a single draw is assumed to be equal to 0.8793.b Consider the following question: Given the number of tosses, what is the likely length of the longest run of heads (or tails)? There exists a general formula allowing one to predict the likely length of the longest success run within a sequence of n Bernoulli trials. In particular, according to this formula, within a real experiment one can predict that the likely length of the longest run of consecutive heads in 100 tosses is six. With this in mind, two groups of students may be asked to toss a coin 100 times, one group recording the results of a real experiment carried out by each member of the group, another group recording the results of imitated tosses (when a long list of 100 heads and tails is the result of subjective randomization by writing down the letters H and T in an arbitrary order by each member of the group). Then both groups are asked to report the longest run of consecutive heads. It appears that 3

John Arbuthnot (1667–1735)—a Scottish polymath, best known for his pioneering contributions to mathematical statistics due to his position as physician extraordinary to Queen Anne (1665– 1714)—Queen of Great Britain and Ireland. 4 Jacob Bernoulli (1654–1705)—a Swiss mathematician, a notable representative of the famous family of Swiss mathematicians.

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humans are not able to generate a random sequence by imitating randomness (Reichenbach, 1949) and as reported by Wagenaar (1972) “almost all experimenters [studying subjective randomization] found systematic deviations from randomness” (p. 70). In general, if p and q, p + q = 1, are the probabilities of success and failure within a single Bernoulli trial, then the largest length of consecutive successes within n Bernoulli trials (assuming nq is much larger than one) for which one could expect at least a single run of that length to occur is equal to the closest integer to log1=p ðnqÞ (Schilling, 2012). In particular, when tossing a fair coin, we have p ¼ q ¼ 1=2. In Fig. 12.13, the spreadsheet (which rounded log2 ð10i =2Þ to the closest integer value for i = 1, 2, …, 8) shows the likely lengths of the longest run of consecutive heads (or tails) within the number of trials being the integer powers of ten. Similarly, other Bernoulli trials can be examined. For example, when rolling a die 500 times, the likely maximum length of consecutive sixes is 3 as in this case we have p ¼ 1=6; q ¼ 5=6 and the closest integer to log6 ð500  5=6Þ is 3. An interesting computer exploration is to verify data included in the chart of Fig. 12.13 by generating within a spreadsheet a sequence of, say, 1000 ones and zeroes to see the appearance of same numbers nine times in a row at least once, something that may not be observed within a sequence of ones and zeroes of length 100. Such a spreadsheet (the programming of which is included in Appendix) is shown in Fig. 12.14. This exploration can also be used as a test of the quality of a random number generator.

12.8

Classic Problems that Motivated Theoretical Development

Concepts of the theory of probability were born from attempts to explain mathematically observations made in the context of the games of chance that were popular in the seventeenth century Europe. Two classic problems of different levels of complexity (and separated by a few decades), yet both relevant to the middle school mathematics can be introduced. The first problem was posed in the early seventeenth century by the Grand Duke of Tuscany (Cosimo II de’ Medici) to Galileo Galilei5: Why, when rolling three six-sided dice, does the number 11 appear more often than the number 12? Mathematically, this problem might have been posed because both 11 and 12 can be decomposed in three unordered positive integers not greater than six in exactly six ways as follows: 11 ¼ 1 þ 4 þ 6 ¼ 1 þ 5 þ 5 ¼ 2 þ 3 þ 6 ¼ 2 þ 4 þ 5 ¼ 3 þ 3 þ 5 ¼ 3 þ 4 þ 4

5

Galileo Galilei (1564–1642), an Italian polymath, court mathematician to Cosimo (appointed in 1610), the father of the major scientific developments of the seventeenth century.

12.8

Classic Problems that Motivated Theoretical Development

385

Fig. 12.13 Rounding to the nearest integer the values of log2 ð10i =2Þ for i = 1, 2, …, 8

Fig. 12.14 Cell E1 shows the number of 6 consecutive H’s in 100 tosses of a coin

and 12 ¼ 1 þ 5 þ 6 ¼ 2 þ 4 þ 6 ¼ 2 þ 5 þ 5 ¼ 3 þ 3 þ 6 ¼ 3 þ 4 þ 5 ¼ 4 þ 4 þ 4: Considering each triple of integers as a three-letter word and using the results of Sect. 8.4, Chap. 8, one can come to the following conclusion. In each of the triples (1, 4, 6), (2, 3, 6), (2, 4, 5), (1, 5, 6), (2, 4, 6) and (3, 4, 5), the elements can be permuted in six ways (like the letters in the word MAT); in each of the triples (1, 5, 5), (3, 3, 5), (3, 4, 4), (2, 5, 5) and (3, 3, 6), the elements can be permuted in three ways (like the letters in the word INN); and the elements of the triple (4, 4, 4) are not permutable. In other words, each triple from the first group generates six possibilities to roll three dice with a certain sum; each triple from the second group generates three possibilities to roll three dice with a certain sum; and the triple (4, 4, 4) generates one possibility only. Analyzing the above decompositions of 11 and 12 in three addends, one can see that there are 27 possibilities to roll the sum 11 and 25 possibilities to roll the sum 12 when rolling three dice. According to the Rule of Product (Chap. 8, Sect. 8.2), there are 63 = 216 outcomes for a single roll of three six-sided dice. Therefore, the probability of rolling the sum 11 is equal to 27/216 = 0.125 and the probability of rolling the sum 12 is equal to 25/2160.116. In other words, gambling observations of the Grand Duke of Tuscany were correct.

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The second problem is from the mid seventeenth century. History preserved the name of Chevalier de Méré, an experienced gambler and French intellectual, who, in the context of two commonly played games, observed the following. In the first game, when a die was rolled four times, the chances of having at least one six were slightly higher than not having it. In the second game, when two dice were rolled twenty-four times, the chances of having at least one double six were slightly lower than not having it. De Méré believed that this discrepancy presents a paradox because, as his reasoning and computations might have gone, 4/6 = 24/36, where 4 and 24 are, respectively, the number of trials in the first and second games and 6 and 36 are the number of possible outcomes in each game. According to a number of sources (Ore, 1960; Székely, 1986, p. 5), de Méré asked his compatriot (and, possibly, friend) Blaise Pascal to solve the problem, which ever since bears the gambler’s name. In brief, Pascal’s solution was based on the following reasoning. The sample space of the game with rolling one die four times consists of two events: having at least one six in a series of four rolls and not having it. Geometrically, the sample space represents one whole (the unity) comprised of two parts: one measured by (5/6)4 and another one measured by 1  ð5=6Þ4 . Indeed, the probability of having a six by rolling a die once is 1/6 and the probability of not having a six by rolling a die once is 5/6. Consequently, because an outcome of each roll of a die does not depend on the outcome of the previous roll, the probabilities of not having a six and having at least one six in a series of four rolls are (5/6)4 and 1  ð5=6Þ4 . The latter number is equal (approximately) to 0.517. Observations by de Méré of the first game were correct as 0:517 [ 0:483  ð5=6Þ4 . Likewise, the sample space of the (second) game with rolling two dice 24 times also consists of two parts: having a double six at least once in a series of 24 rolls of two dice and not having it. As shown in the addition table of Fig. 12.6 there are 35 chances out of 36 of not having a double six when rolling two dice. Consequently, the probabilities of not having and having at least one double six in 24 rolls of two dice are equal to ð35=36Þ24  0:508 and 1  ð35=36Þ24  0:491, respectively, and 0:508 [ 0:491. Once again, one can see that (long term) observations of de Méré were correct and, mathematically speaking, they did not present a paradox.

12.9

A Modification of the Problem of De Méré

Consider the following experiment consisting in calculating and comparing probabilities of having a one exactly one time when a die is rolled four times and having either a one or a six exactly two times when a die is rolled six times. The probabilities of having and not having one spot on a face of a die are 1/6 and 5/6, respectively. The probabilities of having and not having either one or six spots on a face of a die are 1/3 and 2/3, respectively. There are four (independent) outcomes in favor of a one appearing exactly one time in a series of four rolls of a die, with equal probabilities ð1=6Þ  ð5=6Þ3 . Indeed, the number of such outcomes is equal to the

12.9

A Modification of the Problem of De Méré

387

number of permutations of letters in the word SFFF (S—success, F—failure), that 4! is, C41 ¼ ð1!Þð41Þ! ¼ 4(see the tree diagram of Fig. 12.15). Therefore, the probability of having a one (success) exactly one time when a die is rolled four times is equal to 4  ð1=6Þ  ð5=6Þ3  0:39. There are fifteen (independent) outcomes in favor of either a one or a six appearing exactly two times in a series of six rolls of a die, with equal probabilities ð1=3Þ2  ð2=3Þ4 . Indeed, the number of such outcomes is equal to the number of permutations of letters in the word SSFFFF, that is, 6! C62 ¼ ð2!Þð62Þ! ¼ 15(see Chap. 8, Sect. 8.5.1). Therefore, the probability of having either a one or a six appearing exactly two times in a series of six rolls of a die is equal to 15  ð1=3Þ2  ð2=3Þ4  0:33. One can see that it is more probable (by about 6% = 0.39 − 0.33) to have a one over four rolls of a die than to have either a one or a six over six rolls of a die. In general, the probability of having exactly k successes in n Bernoulli trials with the probability of success p and the probability of failure q, p + q = 1, is equal to Cnk pk qnk . This classic formula will be used below in Sect. 12.17 when introducing the notion of binomial distribution. Similarly, one can compare chances (probabilities) of having a double six at least one time and exactly one time in a series of twenty-four rolls of two dice. Recall, that the probabilities of having and not having a double six when rolling two dice are, 1/36 and 35/36, respectively. Therefore, the probability of having a double six exactly one time when rolling two dice twenty-four times is equal to
 1 24! 1 35 23 C24 ð1=36Þ1 ð35=36Þ241 ¼ ð1! Þð241Þ!  36  0:349. Once again, in a series of 36 twenty-four rolls of two dice the chances of having a double six at least one time (0.491) are higher than that of having a double six exactly one time (0.349).

12.10

Experimental Probability Requires a Long Series of Observations

Experimental probability (relative frequency) of an event is defined as the ratio of the number of times the event occurred in a series of identical trials to the total number of trials in this series. As the number of trials grows larger, the difference between experimental and theoretical probabilities becomes smaller. That is, relative frequency can be used to replace theoretical probability when the number of trials is sufficiently large (cf. the Law of Large Numbers, Sect. 12.7). This relationship between experimental and theoretical probabilities is especially useful didactically when theoretical probability is difficult to determine, either from a pure mathematical perspective or because such determination requires reasoning tools of formal mathematics that are far beyond its grade-appropriate pedagogy. This is when experimental reasoning techniques can be employed to replace probability by relative frequency. In the case of the problems by de Méré, while his true observations were based on a large number of rolls of dice, a possible reasoning about equal chances of having and not having at least one six or at least one double six

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S

F

F

S

F

S

S

F S

F

S

F S

F

S

F

S

S

F

S

F

F S

S

F

S

F

F

S

F

Fig. 12.15 Having a six exactly one time in a series of four rolls of a die

was based on only four and twenty-four tosses, respectively. In the several millenniums span, people who played the games of chance learned how to make a fair die the first appearance of which dates back to the 3rd millennium B.C. (David, 1970; Bennett, 1998). Through playing such games, it was found that the chances of having and not having at least one six in a series of four rolls of a die were about the same. Likewise, the chances of having and not having a double six at least one time in a series of twenty-four rolls of two dice were about the same. But in a long practice of playing the games it was observed that chances deviated6 from 50%. The problems of de Méré thereby show “the empirical character of the theory of probability and its purpose of interpreting observable phenomena” (Mises, 1957, p. 64).

12.11

Probability Experiments with Spinners and Palindromes

Consider the following simple inquiry related to probability (e.g., Billstein et al., 1997, p. 421):“A spinner, divided into eight equal sectors each of which is labeled by one of the first eight natural numbers, is spun. What is the probability of landing on an even number?” Schoolchildren have no difficulty saying that there are four

6

The nature of these deviations was explained in Sect. 12.8.

12.11

Probability Experiments with Spinners and Palindromes

389

chances out of eight in favor of an even number. More mathematically proficient students may offer 1/2 as the answer by using Cardano’s definition of probability (Sect. 12.2) referred to by Laplace7 as the first and the second principles of the calculus of probabilities: “probability … is the ratio of the number of favorable cases [four even numbers] to that of all the cases possible [eight numbers] … that supposes the various cases equally likely” (Laplace, 1814/1951, p. 11). Just as in the case of rolling a six-sided die (Sect. 12.8) one assumes that the die is unbiased so that the six outcomes are equally likely, we assume that the outcomes of landing on any of the eight sectors are equally likely allowing for the use of Cardano’s/ Laplace’s definition of probability. Under the same assumption, the chances for an even number become smaller when the spinner is divided into nine sections labeled by the numbers 1 through 9. Indeed, 4/9 < 4/8 = 1/2. However, with ten sectors, the chances for the spinner to land on an even number are back to 1/2. One can ask: What is the probability of randomly selecting an even number from a set of consecutive natural numbers? The answer depends on the cardinality N of such a set. When N = 2k (an even number), the probability is 1/2. Yet, when N = 2k + 1 (an odd number), with the growth of the cardinality the probability (or, better, relative frequency) of selecting an even number becomes closer and closer to 1/2, a conclusion that can be first accepted by computationally exploring special cases and then confirmed algebraically by proving that for n = 1, 2, nþ1 3, … the inequality 2n nþ 1 \ 2n þ 3 holds true. Indeed, the difference between the right1 and left-hand sides of the inequality is equal to ð2n þ 1Þð2n þ 3Þ [ 0: This exploration can then be extended to selecting the multiples of 3 and using the fraction 1/3 as an appropriate measure of chances for such a selection. Very often, the random selection of a natural number with the specific divisibility property is assumed to be from the set of all natural numbers and, as the chances stabilize with the increase of the cardinality of the subset of the entire set, one heuristically (i.e., without sufficiently rigorous justification) says that the probability of selecting at random a multiple of 2 or 3 from the set of natural numbers is, respectively, 1/2 or 1/3. Similarly, the unit fraction 1/p represents the probability of selecting a multiple of a prime number p from the set of natural numbers. One can demonstrate the last statement computationally by using a spreadsheet. For example, Fig. 12.16 shows a spreadsheet which randomly selects numbers from the set of the first 10,000 natural numbers and counts the relative frequency of the multiples of 23 (cell B1) from this set. The result, 0.0431 (cell D1), is then compared to 1/23 showing that the difference is approximately equal to 0.0004 (cell F1). One can enter other prime numbers in cell B1 to see how cell F1 displays interactively a small number representing a difference between computationally found relative frequency and heuristically supported theoretical probability (in the sense of Cardano and Laplace) of randomly selecting a multiple of that number from the first 10,000 natural numbers. This exploration can be extended to a classic problem of finding the 7

Pierre-Simon Laplace (1749–1827), a French mathematician and scientist, one of the greatest scholars of all time.

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probability that a common fraction with randomly selected numerator and denominator is irreducible (i.e., the numerator and denominator are relatively prime). This problem, despite having a simple answer, 6/p2, has distinguished history and is associated with the names of many outstanding mathematicians including, most notably, Chebyshev,8 Dirichlet9 and Gauss.10 For more details see (Abramovich & Nikitin, 2017). Another probabilistic experiment may deal with the Palindrome conjecture discussed in Chap. 9, Sect. 9.10. One can find the probability that among the first 999 natural numbers (that is, having at most three digits) there exist those attracted by a palindrome in at least ten steps. Such a spreadsheet (the programming details of which are included in Appendix) is shown in Fig. 12.17. The spreadsheet calculates the number of steps (or, in some cases, including the famous case of the number 196 (see “Lychrel number” (2020)), its rough estimate from below) it takes the iterative process to reach a palindrome, marks with the number 1 in column C (beginning from cell C40), counts the number of 1’s in cell D39, and divides it by 999 in cell E39. For example, starting from the number 998 (cell B2) it takes 17 steps (cell A3) to reach a palindrome, 133,697,796,331 (cell AJ2). At the same time the probability of reaching a palindrome in exactly 10 steps for the first 999 natural numbers is very low as there are only two numbers smaller than 999, namely 829 and 928, that reach a palindrome, 88,555,588, in ten steps.

12.12

Monty Hall Dilemma as a Paradox in the Theory of Probability

Monty Hall was a popular host of a television game show of the 1970s in the United States. His name is associated with the following paradox in the theory of probability that was observed in the course of the following game show. Behind three closed doors a fancy car and two whimsical prizes are hidden and a contestant in the show must choose between the doors aiming, of course, at winning the car. He or she chooses a door at random (but does not open it), and, thereby, wins the car with a 1/3 probability. Then, the host, who knows where the car is, opens one of the other two doors with a whimsical prize behind it. Now, with one door open and two doors closed, the host asks the contestant to make the next choice: either to remain with the original selection of the door (thus winning the car with a 1/2 probability) or to switch to another (closed) door with a possibility to increase the chances of winning the car. This quandary is called Monty Hall Dilemma.

8

Pafnuty Lvovich Chebyshev (1821–1894)—a Russian mathematician. Peter Gustav Lejeune Dirichlet (1805–1859)—a German mathematician. 10 Carl Friedrich Gauss (1777–1855)—a German mathematician, commonly regarded as the greatest mathematician of all time. 9

12.12

Monty Hall Dilemma as a Paradox in the Theory of Probability

391

Fig. 12.16 Relative frequency (0.0431) of multiples of 23 among the first 10,000 integers

Fig. 12.17 Cell E39 shows relative frequency (0.046) of numbers with at least 11 steps to reach a palindrome among the first 999 natural numbers

In 1990, an American columnist Marilyn vos Savant, in her weekly Parade column, advised a contestant to switch to the other door because this switch would raise the odds of winning the car from a 1/3 probability to a 2/3 probability. Many readers of the column, professional mathematicians among them, strongly objected (see (vos Savant, 1996) for details) to the claim that switching the choice of a door would increase the probability of winning the car. Yet, vos Savant’s advice was correct. To explain, note that whereas the contestant’s original choice of a door was random, the host’s choice of the door was not random, informed by the knowledge of the location of the car. This added additional value to the door not selected by the host. A reasoning that leads to the answer 2/3 can be demonstrated through the use of a tree diagram of Fig. 12.18. Let D1, D2, and D3 be the three doors. The contestant can choose any of the three doors, with a 1/3 probability selecting a door with the car behind it. If the contestant chooses D1 and it conceals the car, then the host opens either D2 or D3 with a 1/2 probability. If the door D1 (chosen by the contestant) does not conceal the car, but D2 does, then the host will certainly open D3 with the probability one. Likewise, if the door D1 (chosen by the contestant) does not conceal the car, but D3 does, then the host will certainly open D2 with the probability one. Now, the probabilities for each situation can be computed. Just as in the case of tossing two coins when the probability of each of the four outcomes is the product of the probabilities of a single outcome, the probabilities of the two actors’ choices in the case of Monty Hall Dilemma have to be multiplied.

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The host’s choices P=1/2 P=1/3

D1 P=1/2

The contestant’s choices

P=1/3

D2 P=1

D2

1/6

D3

1/6

D3

1/3

P=1/3

D3

staying P(car) = 1/3

switching P(car) = 2/3

P=1

D2

1/3

Fig. 12.18 Resolving Monty Hall Dilemma through a tree diagram

Therefore, if door D1 is chosen by the contestant and it conceals the car, then, whatever the door the host opens, the probability of winning the car without switching doors is equal to 13  12 þ 13  12 ¼ 13. However, if D1 conceals the car, but the contestant first chooses D2 or D3, the decision of switching from D2 to D1 (with D3 open) or from D3 to D1 (with D2 open) yields the following probability of winning the car: 13  1 þ 13  1 ¼ 23. This shows the correctness of vos Savant’s advice and, thereby, resolves Monty Hall Dilemma. Other paradoxes in the probability theory are known under the names Bertrand’s Paradox (named after a French mathematician of the nineteenth century Joseph Bertrand), St. Petersburg Paradox, The Division Paradox. These (and similar) paradoxes can be found in (Székely, 1986).

12.13

Transition from Probability Theory to Statistical Data Analysis

Many concepts introduced in the context of the theory of probability can be used to introduce the ideas and tools of statistical data analysis. Statistical reasoning is often associated with mathematical models expressed through probabilities. Although in the middle school statistical data analysis is not related to probability, in what follows the models of coin tossing and die rolling, the two truly profound activities to introduce randomness, will be considered. The relation between statistics and probability was developed only in the nineteenth century. However, some basic statistical concepts such as the most frequent value (the mode, in today’s terminology) is known to be used in the fifth century, the concept of median was introduced in the context of navigation in the sixteenth century, the concept of arithmetic mean (average) was generalized to more than two values in the

12.13

Transition from Probability Theory to Statistical Data Analysis

393

eighteenth century, and the idea of graphical representation of numeric data in statistics was introduced in the eighteenth century. cBig Idea A big idea of mathematics (and a relatively recent one in comparison with other big ideas highlighted in the textbook) is to use tools of probability theory for a rigorous analysis of statistical data. In the middle grades, this idea may be demonstrated by collecting data from a fair coin tossing many times, analyzing the data collected in terms of the number of occurrences of one of the outcomes (a discrete variable) on the scale of the total number of tosses, and then describing the data in terms of the likelihood (probability) that such a variable assumes a certain value on that scale.b Statistics can be connected to probability through the concept of probability distribution. The probability distribution of the random variable x is given by the probabilities of events corresponding to the values of x. In the middle school, only discrete probability distribution is considered. In order to define a discrete probability distribution related to an experiment, one has to provide the probabilities of occurrence of all possible outcomes. For example, when tossing a fair coin, the two random outcomes, head and tail, of the experiment can be described by a random variable x which takes two values (e.g., x1 ¼ 0 and x2 ¼ 1) with a 1/2 probability. When a coin is tossed four times, the probability of having exactly one head is equal to C41 ð12Þ1 ð12Þ3 ¼ 14, where C41 ð¼ ð1!Þ4!ð3!Þ ¼ 4Þ is the number of permutations of letters in the word HTTT. When rolling a die, we have a random variable x which assumes six integer values in the range 1 through 6, each of the six values having a 1/6 probability. When a die is rolled four times, one can consider a random variable x associated with the events of having and not having one spot on the face of the die with the probabilities 1/6 and 5/6, respectively. As shown in Sect. 12.9 of this chapter, the probability of one spot appearing exactly one time in a series of four rolls of a die is equal to C41 ð16Þ1 ð56Þ3  0:39: One can see how probabilities are distributed among random variables related to all possible outcomes of an experiment.

12.14

Graphic Representations of Numeric Data

Suppose a die was rolled 20 times and the number of spots on a die assumed the following twenty values 1, 6, 4, 1, 5, 6, 1, 2, 3, 3, 5, 6, 6, 2, 3, 5, 6, 4, 4, 6. These values can be written in the non-decreasing order 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, and then organized in different graphical formats. One such format is a graph called a line plot (Fig. 12.19). Reading the graph from left to right, one can see that the first three dots indicate the number of occurrences of 1’s, the next two dots—the 2’s, …, and, finally, the last six dots—the number of occurrences of 6’s. Another graphic representation is called a histogram (Fig. 12.20). Reading the

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Line Plot 7 6 5 4 3 2 1 0

0

5

10

15

20

25

Fig. 12.19 A line plot constructed with Excel

Fig. 12.20 A histogram constructed with Wolfram Alpha

histogram, one can recognize that the heights of the six bars indicate the number of occurrences of each of the six spots on the face of a die. Numeric data can also be presented in a stem and leaf plot format. For the set of 20 one-digit numbers the stem may be defined as one of the numbers and the leaf may be defined as the number of occurrences of that numbers in the set. However, more common is to have one number representing the stem and having several leaves associated with this stem, something that a set of one-digit numbers does not follow. For example, in plotting the set of twenty integers 58, 35, 60, 57, 44, 53, 94, 81, 81, 24, 10, 95, 9, 45, 37, 5, 84, 89, 22, 89, the stem of the plot may be defined as the tens’ digits and leaves of the plot are the ones’ digits. The corresponding stem and leaf plot is shown in Fig. 12.21. In the presence of three-digit numbers, a stem may be the face value for the largest place value and the leaves are two-digit numbers. For example, the line 3| 0 7 45 89 may be read as the set {3, 37, 345, 389}.

12.15

Measures of Central Tendency

395

Fig. 12.21 Stem and leaf plot

12.15

0

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9

Measures of Central Tendency

Suppose that five children decided to contribute with as much money as each of them had towards a common goal of buying a new basketball. Their individual contributions were $3, $3, $5, $8, and $11 totaling $30. Also, they wanted to know how could $30 be collected from equal contributions and divided the number 30 by 5 to get 6. Put another way, 6 ¼ 3 þ 3 þ 55þ 8 þ 11. Statistically speaking, the number 6 can be used as a characteristic of their contributions; it is called the mean (or average) of the numbers 3, 3, 5, 8, and 11. Note that none of the five numbers is equal to six. Yet, the number 6 can be used to describe the five numbers in terms of fair sharing of moneys. In general, given the set of n numbers x1 ; x2 ; :::; xn , the mean value, x, is defined as x¼

x1 þ x2 þ ::: þ xn : n

One can also look at the above five contributions to see whether some children from the group contributed the same amount into the purchase of a basketball. An observation can be made that two children each contributed $3, something that may be considered the most common contribution among the five. In statistics, this number is called the mode. In general, if in the set of numbers there exist a single number or several numbers that occur most frequently, each one is called the mode. That is, the set of numbers may have no mode or may have one or several modes. Often, numbers collected for statistical evaluation have to be arranged in the increasing (or non-decreasing) order. In such case, depending on the total of numbers collected, there is either a number in the middle of the list or there are two neighboring numbers in the middle of the list equidistant from the first and the last numbers on

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the list. Either the number in the middle or the average of the two numbers in the middle of the list is called the median. For example, for the list 3, 3, 5, 8, 11 the median is 5 and for the list 2, 3, 4, 6, 7, 8 the median is 5 as well. That is, like the mean, the median may or may not be a part of the list. In statistics, the mean, median, and mode are considered the most basic characteristics through which a data set can be described, and they are called the measures of central tendency. These measures might be good characteristics to be used in one case and weak characteristics to be used in another case. Indeed, whereas the set of six numbers {2, 3, 4, 6, 7, 8} does not include their mean and median (which are the same), the number 5 does not deviate much from either the smallest or the greatest. At the same time, the set of six numbers {0, 1, 2, 2, 2, 23} has the same mean, 5, as the former set; yet the number 5 deviates significantly from its largest number. This observation implies that measures of central tendency may not be sufficiently good characteristics of a set of numbers in terms of their (the numbers’) deviation from the three characteristics. Therefore, new tools of statistical analysis have to be developed.

12.16

Measures of Dispersion

Consider a six-sided die. The number of spots on its faces varies in the range one through six. When the die is rolled, the sample space of this experiment is the set D = {1, 2, 3, 4, 5, 6}. In the case of a fair die, each of the six outcomes has the same probability to occur. Let us carry out a computational experiment using a spreadsheet which consists of rolling a die 10,000 times. Using the spreadsheet, one can generate 10,000 random numbers that vary within the set D. All outcomes can then be added up and divided by 10,000. The resulting number may be called the expected value of a random variable x the values of which, fxi g6i¼1 , belong to D. The spreadsheet of Fig. 12.22 (the programming of which is included in Appendix) shows that such value is equal to 3.499. This number is very close to the mean value of the set D which is equal to 1 þ 2 þ 3 þ6 4 þ 5 þ 6 ¼ 3:5. This is not a coincidence: in probability theory, the expected value of a random variable is defined as the mean value of the prolonged repetitions of the same experiment, like rolling a die 10,000 times. This explains the definition of the expected value of the number of spots on a face of a die as the mean value of the set D. More generally, the expected value, x, of the set of numbers X ¼ fx1 ; x2 ; :::; xn g is x¼

x1 þ x2 þ ::: þ xn : n

Let us assume that the set X ¼ fx1 ; x2 ; :::; Pxn g consists of n random variables xi which occur with the probabilities pi where ni¼1 pi ¼ 1. Then the expected value x

12.16

Measures of Dispersion

397

Fig. 12.22 Experimental approach to the concept of expected value when rolling a die

P of the random variable x is defined as x ¼ ni¼1 xi pi . In the case when all Pn P pi ¼ p; i ¼ 1; 2; :::; n, we have 1 ¼ i¼1 pi ¼ p ni¼1 1 ¼ pn; whence p ¼ 1n. In Pn P that case, x ¼ i¼1 xi pi ¼ 1n ni¼1 xi ¼ x1 þ x2 þn ::: þ xn —an expression already mentioned above. Consider another six-number set A = {3, 3, 3, 4, 4, 4} the mean value of which is 3.5 also. However, set D shows a greater variation about the mean than set A. Such observation was already made in the previous section about two data sets. This difference in using the mean value as a characteristic of a set of data motivates the introduction of other tools to measure data sets. The new tools are called measures of dispersion that show how the data spreads. For example, the range of set D is 6 − 1 = 5 and the range of set A is 4 − 3 = 1. One can see that whereas the spread of data is higher in D than in A, the two sets have the same mean. The range of a data set is one of measures of dispersion. Other measures of dispersion which give a more comprehensive characteristic of a data set are the variance and the standard deviation. These measures show how each value of a set deviates from its mean value x. The concept of variance, v, of the set X ¼ fx1 ; x2 ; :::; xn g is defined as the fraction v¼

ðx1  xÞ2 þ ðx2  xÞ2 þ ::: þ ðxn  xÞ2 : n

When the values xi are measurable by some unit of measurement (e.g., by cm), it is important to have a characteristic with the same unit of measurement as the variables xi . That is why, in statistics, a slight modification of the variance, called the standard deviation, s, is considered and defined as the square root of the variance; that is

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx1  xÞ2 þ ðx2  xÞ2 þ ::: þ ðxn  xÞ2 : s¼ n Likewise, when the variable x randomly assumes the values xi with the probP abilities pi ; i ¼ 1; 2; :::; n, their variance is defined as v ¼ ni¼1 pi ðxi  xÞ2 . When all pi ¼ p ¼ 1=n; i ¼ 1; 2; :::; n, then sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1X ðx1  xÞ2 þ ðx2  xÞ2 þ ::: þ ðxn  xÞ2 ; s¼ ðxi  xÞ2 ¼ n i¼1 n an expression already mentioned above. When X ¼ f1; 2; 3; 4; 5; 6g we have x ¼ 3:5 and, as computed by the spreadsheet of Fig. 12.23, s  1:7078251. One may wonder as to why the variance is defined through the squares of deviations of the elements of a set from its mean rather than simply through deviations. To clarify, note that if one consideres only deviations in the form of differences, than the differences would be both positive and negative to allow for large positive and large negative deviations to cancel each other. In some cases, like in the case of set A, the sum of the deviations is equal to zero, something that may not be used as a measure of variability within a data set. Of course, this issue of positive and negative deviations can be remidiated by using their absolute values; however, such remediation would create unnecessary difficulties in applying calculus to statistics as the absolute value function is not differentiable.

12.17

The Binomial Distribution and the Normal Curve

Consider an experiment of tossing a fair coin 100 times. The number of heads that may be recorded through this experiment varies in the range [0, 100] and may be considered as a random discrete variable x. This experiment is a Bernoulli trial. Using the expression Cnk pk qnk (see Sect. 12.9) as the probability of having exactly k successes in n Bernoulli trials with the probabilities of success and failure, respectively, p and q, p + q = 1, one can find the theoretical probability distribution for the random variable x having exactly k heads within 100 tosses of a fair coin, 0  k  100. Such probablility distribution is called the binomial disctribution due to the coefficient Cnk appearing in Newton’s11 binomial expansion P ða þ bÞn ¼ nk¼1 Cnk ak bnk . When a = p and b = q = 1 − p, the sum a + b = 1. P Consequently, the identity nk¼1 Cnk pk ð1  pÞnk ¼ 1 holds true.

11

Sir Isaac Newton (1642–1727)—an English physicist, astronomer and mathematician, one of the most influential scientists of all time.

12.17

The Binomial Distribution and the Normal Curve

399

Fig. 12.23 The standard deviation of {1, 2, 3, 4, 5, 6} from 3.5 is 1.7078251

The spreadsheet of Fig. 12.24 (the programming of which is included in Appendix) confirms computationally the last identity in the case of coin tossing. The tool displays in column A the integer values of x from the range [0, 100], in k column B the values of C100 , 0  k  100, in column C the value of ð12Þ100 , in k column D the values of C100 ð12Þk ð12Þ100k (which show the probability distribution P k 1 k 1 nk for the random variable x), and in cell E1 the sum 100 ¼ ð12 þ 12Þ100 k¼0 C100 ð2Þ ð2Þ of 101 probabilities. Figure 12.25 shows the graph of the probability distribution which looks like a bell-shaped curve the tails of which are asymptotic to the horizontal axis; that is, the tails have no points in common with the horizontal axis. Analysing the graph, one can see that the maximum value of the sequence k C100 ð12Þ100 is equal approximately to 0.08 when k = 50. Also, the probabilities of having the number of heads outside the range [40, 60] are pretty insignificant. In mathematics, the curves of that kind (Fig. 12.25) are called normal curves and a random variable with probability distributions forming a normal curve is referred to as a variable with a normal distribution. However, not all probability distributions form a normal curve. Empirically, it was found that about 68% of all data values of a normal curve lie within one standard deviation of the mean (in both directions), about 95%—lie within two standard deviations, and about 99.7% (almost all values of the random variable)—lie within three standard deviations. It can be demonstrated that a random variable x with the binomial probability distribution has the expected value x ¼ np and the variance v ¼ npð1  pÞ. Computations are quite challenging and in the modern-day classroom can be ousoursed to Wolfram Alpha (using its free on-line version) as shown in Figs. 12.26 and 12.27, respectively. Therefore, if a fair coin is tossed 100 times, the expected value of head is 100  12 ¼ 50, the variance is 100  12  ð1  12Þ ¼ 25, and the standard deviation is pffiffiffiffiffi 25 ¼ 5. In other words, about 68% of cases the number of heads would belong to the range [45, 55] and about 95% cases the number of heads would belong to the range [40, 60]. The spreasheet of Fig. 12.28 (the programming of which is included in Appendix) shows the results of the following computational experiment. A fair coin was tossed 100 times independently by 100 people. Then the occurrence of

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Fig. 12.24 The sum of 101 probability distributions is equal to one

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101

0

Fig. 12.25 A bell-type curve as a graph of binomial probability distributions

heads in each of the 100 individual experiments was recorded. In about 70 out of 100 cases (see the decimal 0.68 in cell A1) the number of heads was in the range [45, 55], that is, within one standard deviation from the expected value 50. Note that at each push of the spreadsheet to change random entries (just enter any number

12.17

The Binomial Distribution and the Normal Curve

401

Fig. 12.26 Wolfram Alpha calculates the expected value for the binomial distribution

Fig. 12.27 Wolfram Alpha calculates the variance for the binomial distribution

in an empty cell of a spreadsheet), the number in cell A1 changes. All such values appearing in cell A1 can then be averaged. That is how 68% was determined experimentally in other cases of the normal probability distribution.

12.18

The z-Score and the Standard Deviation

One can also express a particular result of an experiment through the units of the standard deviation s. For example, whereas 57 as the number of heads lies between one and two standard deviations from the mean 50, the former number can be expressed with more precision in terms of s = 5. To this end, consider the value of (57 − 50)/5 = 1.4 from where it follows that 57 = 50 + 1.4s. The value 1.4 is called the z-score of 57; in other words, 57 lies within 1.4 standard deviations to the right of the mean 50. Likewise, (43 − 50)/5 = −1.4; that is, 43 lies within 1.4 standard deviations to the left of the mean 50. In general, the fraction xi x s is denoted z and is referred to as the z-score (or the standard score) of, thereby, demonstrating how many standard deviations is above or below the mean.

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Fig. 12.28 In 68% cases the number of heads is within one standard deviation from the mean

12.19

Bivariate Analysis

In statistics, the bivariate analysis deals with a series of observations of experiments with random outcomes when exactly two measurements are carried out within each observation. In the United States, Common Core State Standards (2010) expect students in grade 8 to “investigate patterns of association in bivariate data” (p. 56). In order to be able to assist students in this kind of investigation, future teachers of mathematics at the middle grades should themselves have experience in “exploring bivariate relationships … built around seeing statistics as a four-step investigative process involving question development, data production, data analysis, and contextual conclusions” (Association of Mathematics Teacher Educators, 2017, p. 111). Towards the end of question (i.e., exploratory task) development, note that using x and y as notations for two measurements, the statistical properties of measurements carried out through a series of observations are characterized by the joint probability distribution of the pairs (x, y), something that has to be explored. If the sample space of an experiment is finite (like in the case of tossing a fair coin several times), then the joint probability distribution can be represented in the form of a two-dimensional table with the dimensions representing the values of the variables x and y. For example, if a fair coin is tossed three times (to ensure data production), the following eight outcomes, {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}, form the sample space. Within this experiment, one can pose a task to measure the number of tails observed on the first and the second tosses (x) and on the first and third tosses (y). Both x and y may assume three values only: 0, 1, and 2. Their joint probability distribution may be characterized by the probability of having one of the nine possible outcomes of measurement as shown in the tree diagram of Fig. 12.29. The tree diagram does not show joint probability distribution. Rather, it shows the number of possible pairs of (x, y) found through a series of observations. The table of Fig. 12.30 shows how a joint probability distribution can be characterized by probabilities of each pair (x, y) of nine possible values (Fig. 12.31). For example, when a tail was observed only one time on the first and the second tosses

12.19

Bivariate Analysis

403

0

0

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2

Fig. 12.29 Relating the number of tails observed for x to that of y

and two times on the first and the third tosses, we have the following sample spaces, respectively, {THH, HTH, THT, HTT} and {TTT, THT}. By analyzing data presented in the form of the sample spaces, one can see that the two sets of outcomes have only one element in common, namely, THT, which is one of eight possible outcomes when a coin is tossed three times. Therefore, the joint probability distribution can be described by the fraction 1/8—the probability of having a tail once and twice, respectively, on the first–second tosses and on the first-third tosses. Likewise, when a tail was observed only one time both on the first and the second tosses and on the first and the third tosses, we have the following sample spaces, respectively, {HTH, THH, HTT, THT} and {THH, TTH, HTT, HHT}. Once again, by continuing data analysis, one can see that the two sets of outcomes have two elements in common, namely, THH and HTT, each of which is independent of the other and is one of eight possible outcomes when a coin is tossed three times. Therefore, the joint probability distribution can be described by the fraction 1/4 (= 1/8 + 1/8), something that allows one to make the following contextual conclusion: the probability of having a tail once both on the first–second tosses and on the first-third tosses of a fair coin is equal to 1/4. Note that not only the context of statistics can be seen “as a four-step investigative process involving question development, data production, data analysis, and contextual conclusions” (Association of Mathematics Teacher Educators, 2017, p. 111). For example, in Chap. 2, Sect. 2.3.2, in the activity Burning the Candle several questions were formulated for students to explore, various tools and methods were used to produce data and develop data tables for subsequent analysis enabling the students to make contextual conclusions about the most important parameters involved in identifying answers to the questions posed. Also, just as almost any problem-solving scenario, Problem 9.1 (Chap. 9, Sect. 9.2.1) started with developing a question about possible combinations of cats and birds with the total of 18 legs and continued with producing data for investigation by drawing pictures. This was followed by analyzing and describing visual information in various symbolic forms and, finally, by reaching contextual conclusions through understanding what makes the problem solvable and how a similar problem can be formulated. Likewise, in the context of fair coin tossing (see Sect. 12.2 of this chapter), trying to understand, whether chances to have four heads in a row are the same as to have alternating heads and tails as a result of four tosses (question development), one can use a spreadsheet to collect data of tossing a coin four times

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Probability and Statistical Data Analysis

Tail: 1 & 3

HHH HHT HTH THH

HHH HHT HTH THH

HTT

HTT

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0 THT TTH TTT

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HHH HHT HTH THH

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HHH HHT HTH THH

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HHH HHT HTH THH

HHH HHT HTH THH

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HHH HHT HTH THH 1

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0 HTT

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HHH HHT HTH THH

HHH HHT HTH THH

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Fig. 12.30 A chart-type representation of the tree diagram

THT TTH TTT

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12.19

Bivariate Analysis

405

Fig. 12.31 Probabilities of nine joint distributions

0

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in a large number of experiments (data production), then analyze data through multiple repetitions of this computational experiment, and, finally, as a way of reaching contextual conclusions, recognize how randomness turns into regularity when tossing a coin many times. Other examples of applying the above-mentioned four-step investigative process can be provided when looking at the problem-solving context of the textbook through such conceptually-oriented lens.

12.20

Conclusion

This chapter emphasized an experimental character of probabilistic and statistical concepts the study of which can be significantly enhanced by the use of digital tools such as an electronic spreadsheet and Wolfram Alpha. Addition and multiplication tables used in Chap. 9 for the development of algebraic skills were explored through the lens of probability theory. Historical aspects of the theory of probability were discussed in the context of classic problems that motivated major theoretical developments. Probability experiments with spinners and the Palindrome conjecture were presented. The concept of probability distribution was discussed as a link between probability and statistics. Finally, with a reference to the modern-day standards for mathematics teacher preparation, the context of statistics as a special investigative process was connected to explorations presented in other parts of the textbook. For readers who want to deepen their knowledge and understanding of probability and statistics the following sources can be recommended (Ore, 1960; Kapadia & Borovcnik, 1991; Shaughnessy, 1992; Konold, 1995; Noss et al., 1999; Saldanha & Thompson, 2002; Schilling, 2012; Chernoff, 2014; Abramovich & Nikitin, 2017).

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12.21

Probability and Statistical Data Analysis

Activity Set

1. Two dice are rolled. Find the theoretical probability that the number of spots on two faces is ten. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 2. Two dice are rolled. Find the theoretical probability that the number of spots on two faces is eight. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 3. Two dice are rolled. Find the theoretical probability that the number of spots on two faces is six. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 4. Three dice are rolled. Find the theoretical probability that the number of spots on three faces is ten. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 5. Three dice are rolled. Find the theoretical probability that the number of spots on three faces is seven. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 6. Three dice are rolled. Find the theoretical probability that the number of spots on three faces is fifteen. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 7. Four dice are rolled. Find the theoretical probability that the number of spots on four faces is sixteen. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 8. Four dice are rolled. Find the theoretical probability that the number of spots on four faces is twenty. Construct a spreadsheet using the RANDBETWEEN function to find an experimental probability of this outcome. 9. A coin changing machine randomly changes a quarter into dimes, nickels, and pennies. Represent a sample space of this experiment in the form of a table and a tree diagram. 10. A coin changing machine randomly changes a dollar into half dollars, quarters, and dimes. Represent a sample space of this experiment in the form of a table and a tree diagram. 11. Three dice are rolled. Compare theoretical probabilities of having 11 and 12 on the three faces. Which of the two outcomes is more probable? Confirm your conclusion experimentally by using a spreadsheet. 12. A number is chosen at random from the n  n addition table. (i) What is the probability that the chosen number is a multiple of two? (ii) What is the probability that the chosen number is a multiple of three? (iii) What is the probability that the chosen number is a multiple of two but not a multiple of three? (iv) What is the probability that the chosen number is a multiple of three but not a multiple of two?

12.21

Activity Set

407

13. A number is chosen at random from the n  n multiplication table. (i) What is the probability that the chosen number is a multiple of two? (ii) What is the probability that the chosen number is a multiple of three? (iii) What is the probability that the chosen number is a multiple of two but not a multiple of three? (iv) What is the probability that the chosen number is a multiple of three but not a multiple of two? 14. A machine changed a half dollar coin into quarters, dimes, and nickels. Assuming that it is equally likely to get any combination of the coins, find the probability that (i) there is no nickel in the change, (ii) there is no dime in the change, (iii) there is no quarter in the change. 15. A machine changed a dollar coin into half dollars, quarters, and dimes. Assuming that it is equally likely to get any combination of the coins, find the probability that (i) there is no dime in the change, (ii) there is no quarter in the change, (iii) there is no half dollar in the change. 16. How many ways can one make a quarter out of pennies, nickels, and dimes? If a machine randomly changes quarters into pennies, nickels, and dimes, what is the probability that (i) there is no penny in a change? (ii) there is no nickel in a change? (iii) there is no dime in a change? 17. Determine what is more probable by calculating theoretical probabilities of the following two events: to have a six exactly two times in five rolls of a fair die or to have either one or six exactly four times in eight rolls of a fair die. 18. Determine what is more probable by calculating theoretical probabilities of the following two events: to have a one exactly one time in six rolls of a fair die or to have either one or six exactly one time in five rolls of a fair die. 19. Determine what is more probable by calculating theoretical probabilities of the following two events: to have a six exactly two times in five rolls of a fair die or to have a six exactly two times in six rolls of a fair die. 20. Determine what is more probable by calculating theoretical probabilities of the following two events: to have either a six or a five exactly two times in eight rolls of a fair die or to have either a six or a five exactly three times in eight rolls of a fair die. 21. Given data set {1, 3, 6, 10, 10, 13, 15, 15, 15, 16} determine the mean, median, mode, range, variance, and standard deviation.

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22. Find the z-scores for the elements 3, 10, and 15 from the data set of the previous problem. 23. For each of the sets of the first six, seven, and eight Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, find the standard deviation. Do you see a pattern? What is the pattern? How can this pattern be explained? 24. A fair coin is tossed three times. Find the probability of having a head once on the first–second tosses and having a head twice on the first-third tosses. 25. A fair coin is tossed three times. Find the probability of having a head twice on the first–second tosses and having a tail once on the second-third tosses. 26. Three different size towers are constructed out of six linking cubes. What is the probability that the median size tower is two-cube high? 27. Five different size towers are constructed out of fifteen linking cubes. What is the probability that the median size tower is three-cube high?

Appendix

The purpose of the appendix is to provide details of the programming of the spreadsheet environments used in this book. It is assumed that readers have basic familiarity with spreadsheets including the use of names (absolute references) and relative references in spreadsheet formulas. An absolute reference in a spreadsheet formula points to a cell the content of which does not change as the formula is replicated from cell to cell, both vertically and horizontally. A relative reference in a spreadsheet formula points to a series of cells which comprise either a row when the formula is replicated across the template (horizontally) or column when the formula is replicated down the template (vertically). Some formulas may include the so-called circular references, that is, references to a cell in which a formula is defined. Such references enable preservation of the display of the results of computations dependent on an absolute reference to a cell the value of which changes at any new step as computing moves from cell to cell using a formula with a circular reference. Also, syntactic resourcefulness of a spreadsheet enables the construction of both visually and computationally identical environments using slightly different formulas. Thereby, it is quite possible that the reader would find alternative programming techniques compared to those suggested in Appendix. Below, the notation (A1) ! will be used to present a formula defined in cell A1.

Chapter 9: Conceptual Approach to the Ideas of Middle School Algebra 1. Programming of the spreadsheet of Fig. 9.5. This spreadsheet is designed to pose problems of the pet store type when given two sets of drimps and grimps and the total number of legs among the creatures in each set, the number of legs for a drimp and for a grimp can be found. The programming of this problem-posing instrument includes formulas using which one can compute © Springer Nature Switzerland AG 2021 S. Abramovich and M. L. Connell, Developing Deep Knowledge in Middle School Mathematics, Springer Texts in Education, https://doi.org/10.1007/978-3-030-68564-5

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Appendix

in two cells the values of two unknowns using Cramer’s rule1 which solves a system of two linear equations (whatever the data is entered into cells B4, D4, F4, B10, D10, F10 using the scroll bars) and in another two cells displaying the results of calculations in the case of positive integers only. More specifically: (H2, with hidden calculations) ! = (F4 * D10 − F10 * D4)/(B4 * D10 − B10 * D4), (J2, with hidden calculations) ! = (B4 * F10 − B10 * F4)/(B4 * D10 − B10 * D4), (H6) ! = IF(OR(INT(H2) < H2, INT(I2) < I2, H2