Teaching to the Math Common Core State Standards: Focus on Grade 5 to Grade 8 and Algebra 1 946209960X, 9789462099609

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Teaching to the Math Common Core State Standards

Teaching to the Math Common Core State Standards Focus on Grade 5 to Grade 8 and Algebra 1

F.D. Rivera Departments of Mathematics & Statistics and Elementary Education, San Jose State University, California, USA

A C.I.P. record for this book is available from the Library of Congress.

ISBN: 978-94-6209-960-9 (paperback) ISBN: 978-94-6209-961-6 (hardback) ISBN: 978-94-6209-962-3 (e-book)

Published by: Sense Publishers, P.O. Box 21858, 3001 AW Rotterdam, The Netherlands https://www.sensepublishers.com/

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Contents

1.  Dear Preservice Middle Level Majors and Beginning Middle School Teachers: An Introduction 1.1 A Blended Multisourced Approach to Learning to Teach Middle School Mathematics 1.2  Overview of the Remaining Chapters 2.  Getting to Know the Common Core State Standards for Mathematical Practice 2.1  Content Activity 1: Exploring and Proving Parity 2.2 The Eight Common Core State Standards for Mathematical Practice 2.3 Content Activity 2: Building a Hexagon Flower Garden Design 2.4  Problem Solving Contexts in Middle School Mathematics 2.4.1 Content Activity 3: Different Types of Problems in Middle School Mathematics 2.5  Representations in Middle School Mathematics 2.5.1 Content Activity 4: From Multiplication of Whole Numbers to Direct Proportions to Simple Linear Relationships 2.6  Connections in Middle School Mathematics 2.6.1 Content Activity 5: The Pythagorean Theorem and Irrational Numbers 2.7  Reasoning and Proof in Middle School Mathematics 2.8  Communication in Middle School Mathematics 2.8.1 Content Activity 6: Solving Multiplication Problems with Tape Diagrams 2.9 Doing Mathematics with an Eye on the Content-Practice Standards of the CCSSM 3.  The Real Number System (Part I) from Grade 5 to Grade 6: Whole Numbers, Decimal Numbers, and Fractions 3.1 Place Value Structure of Whole Numbers and Decimal Numbers in Grade 5 3.2 Standard Algorithms for Whole Number Operations in Grades 5 and 6 3.2.1 A Brief Overview of the Addition and Subtraction Standard Algorithms in the Elementary Grades up to Grade 4 3.2.2  Multiplication Standard Algorithm in Grade 5 3.2.3 Division Standard Algorithm in Grade 6 v

1 4 5 9 10 13 15 18 20 20 20 24 25 27 29 31 32 35 36 40 40 42 45

Table of Contents

3.3  Operations with Fractions 3.3.1  Fractions in the Elementary Grades up to Grade 4 3.3.2 All Cases of Adding and Subtracting Fractions in Grade 5 3.3.3  Multiplying Fractions in Grade 5 3.3.4 Dividing Fractions in Grades 5 and 6 3.4  Operations with Decimal Numbers in Grades 5 and 6 3.4.1 Multiplying with Decimal Numbers in Grades 5 and 6 3.4.2  Dividing Decimal Numbers in Grades 5 and 6 3.4.3 Adding and Subtracting Decimal Numbers in Grades 5 and 6 3.5 Mapping the Content Standards with the Practice Standards 3.6 Developing a Content Standard Progression Table for the Real Number System Part I Domain 4. 

 he Real Number System (Part II) from Grade 6 to Grade 8 T and Algebra 1: Integers, Rational Numbers, and Irrational Numbers 4.1  Integers and Integer Operations in Grades 6 and 7 4.1.1 Concepts of Positive and Negative Integers in Grade 6 4.1.2 Addition of Integers in Grade 7 4.1.3 Subtraction of Integers in Grade 7 4.1.4 Multiplication of Integers in Grade 7 4.1.5  Divisions of Integers in Grade 7 4.2  Absolute Value in Grades 6 and 7 4.2.1 Concept of Absolute Value in Grade 6 4.2.2 Absolute Value and Distance in Grade 7 4.3  Rational Numbers and Operations in Grade 6 and 7 4.3.1  Rational Numbers in Grade 6 4.3.2 Rational Number Operations in Grade 7 4.3.3 Decimals that Transform into Rational Numbers in Grade 7 4.4 Irrational Numbers and Operations in Grade 8 and Algebra 1 4.5  Mapping the Content Standards with the Practice Standards 4.6 Developing a Content Standard Progression Table for the Real Number System Part II Domain

5.  Ratio and Proportional Relationships and Quantities 5.1  Ratio and Rate Relationships in Grade 6 5.2  Proportional Relationships in Grade 7 5.3  Graphs of Proportional Relationships from Grades 6 through 8 vi

49 49 54 54 58 60 60 62 64 65 66 69 70 70 73 77 80 84 84 84 85 86 87 87 89 91 93 93 95 97 99 99

Table of contents

5.4  Quantitative Relationships in Algebra 1 5.5 Mapping the Content Standards with the Practice Standards 5.6 Developing a Content Standard Progression Table for the Quantities and Ratio and Proportion Domains

100 101 101

6. Technology-Mediated Tools for Teaching and Learning Middle School Mathematics103 6.1  Getting to Know the TI-84+ Graphing Calculator 103 6.2  Getting to Know the GeoGebra Dynamic Software Tool 105 6.3  Getting to Know a Calculator-Based Ranger 107 6.4 Getting to Know Apps and Other Virtual-Based Manipulatives for Classroom Use 107 6.5  Understanding Technology Use in the Mathematics Classroom 108 7.  Expressions and Operations 7.1  Numerical and Algebraic Expressions in School Mathematics 7.2  Writing and Simplifying Numerical Expressions in Grade 5 7.3 Greatest Common Factors, Least Common Multiples, and Simple Factoring Involving Numerical Expressions in Grade 6 7.4 Simplifying and Operating with Numerical Expressions that Involve Integer and Fractional Exponents and Numbers in Scientific Notation in Grades 6 and 8 7.5  Algebraic Expressions in Grades 6 and 7 and Algebra 1 7.5.1 Constructing, Simplifying, and Factoring Algebraic Expressions in Grades 6 and 7 and Algebra 1 7.5.2 Adding, Subtracting, Multiplying, and Factoring Polynomials in Algebra 1 7.5.3 Understanding Simple Nonalgebraic (Transcendental) Exponential Expressions in Algebra 1 7.5.4 Deepening Variable Understanding in Algebra 1 7.6  Mapping the Content Standards with the Practice Standards 7.7 Developing a Content Standard Progression Table for the Expressions Domain 8.  Equations and Inequalities 8.1  Algebraic Thinking from Grades 6 to 8 and Algebra 1 8.2 Different Contexts for Using Variables from Grades 6 to 8 and Algebra 1 8.3  Solving Equations from Grades 6 to 8 and Algebra 1 8.4  Solving Inequalities from Grades 6 to 8 and Algebra 1 8.5 Word Problem Solving Applications Involving Equations and Inequalities from Grades 6 to 8 and Algebra 1

109 110 110 111 113 117 119 120 124 125 126 126 127 127 133 134 137 138 vii

Table of Contents

8.6  Mapping the Content Standards with the Practice Standards 138 8.7 Developing a Content Standard Progression Table for the Equations and Inequalities Domain 138 9.  Functions and Models 9.1  Rules, Patterns, and Relationships in Grade 5 9.2 Dependent Variables, Independent Variables, and Quantitative Relationships in Grade 6 9.3  Unit Rates and Proportional Relationships in Grade 7 9.4  Nonsymbolic Algebra and Functions in Grade 8 9.5  Symbolic Algebra and Functions in Algebra 1 9.6 Mapping the Content Standards with the Practice Standards 9.7 Developing a Content Standard Progression Table for the Domains of Functions and Models

139 140

10.  Measurement and Geometry 10.1 Quadrilaterals, Volumes of Right Rectangular Prisms with Whole-Number Edge Lengths, and the First Quadrant Coordinate Plane in Grade 5 10.2 The Four-Quadrant Coordinate Plane, Areas of Polygons, Nets and Surface Areas, and Volumes of Right Rectangular Prisms with Fractional Edge Lengths in Grade 6 10.3 Scale Drawings, Slicing, Drawing Shapes with Given Conditions, Circumference and Area of a Circle, Finding Unknown Angles in a Figure, and More Problems Involving Area, Surface Area, and Volume of 2D and 3D Objects in Grade 7 10.4 Volumes of Cylinders, Cones, and Spheres and Transformation Geometry, Congruence, Similarity, Informal Proofs, and the Pythagorean Theorem in Grade 8 10.5 Mapping the Content Standards with the Practice Standards 10.6 Developing a Content Standard Progression Table for the Measurement and Geometry Domains

155

11.  Data, Statistics, Probability, and Models 11.1  Representing Data in Grades 5, 6, and 8 11.2  Analyzing Data in Grades 5, 6, 7, and 8 11.3  Statistical Modeling in Algebra 1 11.4 Probability Models and Probabilities of Simple and Compound Events in Grade 7 11.5  Mapping the Content Standards with the Practice Standard 11.6 Developing a Content Standard Progression Table for the Data, Statistics, and Probability Domains

viii

142 143 143 145 154 154

156 158

159 160 162 163 165 165 166 167 168 169 169

Table of contents

12.  Content-Practice Assessment 171 12.1  General Conceptions on Assessment 172 12.2  Norm- and Criterion-Referenced Tests 176 12.3  Principles of Effective Classroom Assessments 178 12.4  Formative Assessments 178 12.5  Summative Assessments 179    12.5.1 What are Summative Assessments? 179    12.5.2 Smarter Balanced Assessment (SBA) 179 12.6  Projects and Portfolios as Alternative Summative Assessments 183 12.7 Math Journals and Lesson Investigations as Alternative Formative Assessments184 12.8  Exploring Assessment Systems 185 12.9  An Assessment Project 185 13.  Content-Practice Learning 187 13.1  Defining Learning 187 13.2 Changing Views of Learning and Their Effects in the Middle School Mathematics Curriculum 188 13.3 Math Wars: Debating About What and How Middle School Students Should Learn Mathematics 189 13.4 Understanding Piagetian and Vygotskian Views of Learning in Mathematics and Finding a Way Out of Extreme Views of Learning189 13.5  Learning Progressions in School Mathematics 190 13.6  Learning from Neuroscience 191 14.  Content-Practice Teaching 14.1  Describing (Good) Teaching 14.2  Teaching Models in Middle School Mathematical Settings    14.2.1 E-I-S-Driven Teaching    14.2.2 C-R-A Sequenced Teaching    14.2.3 Van Hiele Sequenced Teaching    14.2.4 Culturally-Relevant Teaching    14.2.5 SDAIE-Driven Teaching    14.2.6 Differentiated Instruction    14.2.7 Flip Teaching 14.3. Teaching with Manipulatives and Computer and Video Games and Apps 14.4  Teaching Mathematics with Guide Questions 14.5  Content-Practice Unit Planning 14.6  Content-Practice Lesson Planning 14.7  A Planning Project

193 193 195 195 196 196 197 197 198 198 199 200 201 206 209

ix

Table of Contents

15.  Orchestrating a Content-Practice Driven Math Class 15.1  Persistence and Struggle in Math Classrooms 15.2  Fostering Persistent Content-Practice Learners 15.3 Developing Effective Collaborative Content-Practice Learning Through Complex Instruction 15.4  Other Collaborative Content-Practice Learning Techniques 15.5 Developing an Optimal Content-Practice Learning Environment for all Middle School Students 15.6  Dealing with Potential Behavior Problems 15.7  Assigning Homework, Grading and Testing, and   seating Arrangement Issues 15.8  A Classroom Management Plan Project

x

213 213 214 215 218 218 220 221 223

Chapter 1

Dear Preservice Middle Level Majors and Beginning Middle School Teachers An Introduction

This methods book takes a very practical approach to learning to teach middle school school mathematics in the Age of the Common Core State Standards (CCSS). The Kindergarten through Grade 12 CCSS in Mathematics (i.e., CCSSM) was officially released on June 2, 2010 with 45 of the 50 US states in agreement to adopt it. Consequently, that action also meant implementing changes in their respective state standards and curriculum in mathematics. The CCSSM is not meant to be “the” official mathematics curriculum; it was purposefully developed primarily to provide clear learning expectations of mathematics content that are appropriate at every grade level and to help prepare all students to be ready for college and the workplace. A quick glance at the Table of Contents in this methods book indicates a serious engagement with the recommended mathematics underlying the Grade 5 through Grade 8 and (traditional pathway) Algebra 1 portions of the CCSSM first, with issues in assessment, learning, teaching, and classroom management pursued next and in that order. One implication of the CCSSM for you who are in the process of learning to teach the subject involves understanding shared explicit content and practice standards in teaching, learning, and assessing middle school mathematics. The content standards, which pertain to mathematical knowledge, skills, and applications, have been carefully crafted so that they are “teachable and learnable,” that is, smaller than typical in scope, coherent, inch-deep versus mile-wide, focused, and rigorous. According to the Gates Foundation that initially supported the development of the CCSSM, “(t)he new mathematics … standards were built to be teachable and concrete—there are fewer, and they are clearer. And the standards were built on the evidence of what is required for success beyond high school—these standards aim higher.” Further, “coherent” in the CCSSM means to say that the content standards “convey a unified vision of the big ideas and supporting concepts … and reflect a progression of learning that is meaningful and appropriate.” The practice standards, which refer to institutionally valued mathematical actions, processes, and habits, have been conceptualized in ways that will hopefully encourage all middle school students to engage with the content standards more deeply than merely acquiring mathematical knowledge by rote and imitation. An instance of this content-practice relationship involves patterns in numbers. When middle school students are asked 1

Chapter 1

to develop valid algorithms for combining whole numbers (i.e., operations), they should also be provided with an opportunity to engage in mathematical practices such as looking for structures and expressing the regularity of such structures in repeated reasoning. So, unlike “typical” state standards that basically provide (sound) outlines of mathematical content in different ways, the CCSSM puts premium on this contentpractice nexus with instruction orchestrating purposeful learning experiences that will enable all students to learn the expert habits of mathematicians and to struggle productively as they develop mathematical maturity and competence. To perceive the CCSSM merely in terms of redistributing mathematics content that works for all 45 states defeats the purpose in which it was formulated in the first place. That is, the CCSSM is not about covering either the content or practice of school mathematics; it is un/covering content-practice that can support meaningful mathematical acquisition and understanding of concepts, skills, and applications in order to encourage all middle school students to see value in the subject, succeed, progress toward high school, and be prepared for careers that require 21st century skills. One important consequence of this reconceptualized content-practice approach to learning middle school mathematics involves changing the way students are assessed for mathematical proficiency. Typical content-driven state assessments would have, say, fifth-grade students bubbling in a single correct answer to a multiple-choice item, for instance, choosing the largest sum from several choices of two unit-fraction addends. A CCSSM-driven assessment would have them justifying conditions that make such comparisons possible and valid from a mathematical point of view. Thus, in the CCSSM, proficiency in content alone is not sufficient, and so does practice without content, which is limited. Content and practice are both equally important and, thus, must come together in teaching, learning, and assessment to support growth in students’ mathematical understanding. “One hallmark of mathematical understanding,” as noted in the CCSSM,  is 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. There is a world of difference between a student who can summon a mnemonic device … and a student who can explain where the mnemonic comes from. The student who can explain the rule understands the mathematics, and may have a better chance to succeed at a less familiar task …. Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness. As a document that has been adopted for use in 45 US states, the CCSSM developed appropriate content standards - mathematical knowledge, skills, and applications – at each grade level that all students need to know regardless of their location and context. Such standards are not to be viewed merely as a list of competencies. As noted earlier,

2

Dear Preservice Middle Level Majors

which is worth reemphasizing, they have been drawn from documented research on learning paths that students tend to pursue, from the initial and informal phase of sense making and adaptive thinking to the more formal and sophisticated phase of necessary mathematical knowledge. Implications of various learning paths then became the basis for designing coherent progressions of mathematical concepts and processes from one grade level to the next. It is this particular constraint that should remind you that they you are teaching not to the common core of mathematics but to the CCSS. Certainly, there are other possible learning paths in, say, understanding integers and rational numbers, which should of course be encouraged and supported. But you must persevere to care and to see to it that the way you teach middle school mathematics supports the original intentions of the CCSSM. The best available evidence from research and practices across states has also informed the CCSSM content standards. Further, the standards reflect in sufficient terms the content requirements of the highest-performing countries in mathematics. Consequently, you will enter the teaching profession with this available knowledge base that you are expected to know really well and should be able to teach. Certainly, there is always room for interpretation, innovation, and creativity, especially in the implementation phase. However, the CCSSM is unequivocal about the consistent learning expectations of mathematical competence at every grade level, so fidelity is crucial in this early stage of implementation. Own it since you are its very first users. Test it to so that it makes a difference and produces a long-term positive impact on your students. Both the National Governors Association and the Council of Chief State School Officers that sponsored the CCSSM initiative point out that the CCSSM “standards are a common sense first step toward ensuring our children are getting the best possible education no what where they live” and “give educators shared goals and expectations for their students.” And a much greater focus on fewer topics based on coherent learning progressions should make the content-practice doable in every math class. Hence, under the CCSSM framework, you no longer pick and choose. Instead, you teach for content-practice expertise. With clear content-practice standards in the CCSSM, a common comprehensive assessment became a necessity. As the Gordon Commission has pointed out in relation to the future of education, the troika of assessment, teaching, and learning forms the backbone of a well-conceptualized pedagogy. While the three processes can take place independently, they should co-exist in a mutually determining context. Hence, for you, it means developing a mindset of alignment. In actual practice, in fact, it is impossible in school contexts to conceptualize teaching that has not been informed by any form of assessment of student learning. Further, the basic aim of assessment and teaching is improved learning, no less. Given this mindset of alignment, the Smarter Balanced Assessment (SBA) and the Partnership for Assessment of Readiness for College and Careers (PARCC) emerged, two independent consortia of states that

3

Chapter 1

agreed to develop “next-generation assessments” that are aligned with the CCSSM and ready to be implemented by the 2014-2015 school year. At the core of such high-stakes assessments involves measuring student progress and pathways toward college and career readiness. For you, this institutional expectation of assessment practice also means understanding what is at stake and what is needed to help your students succeed in such high-quality assessments. Suffice it to say, the realities of the CCSSM, SBA, and PARCC in schools will drive issues of learning, teaching, assessment, and classroom management, including your choice of professional growth and development. Any responsible and thoughtful professional can cope with and work within and around constraints, and this methods book will provide you with much needed support to help you begin to teach to the CCSSM. In past practices, a typical middle school methods course in mathematics would begin with general and domain-specific issues in learning and teaching with content knowledge emerging in the process. The approach that is used in this book assumes the opposite view by situating reflections and issues in learning and teaching middle school mathematics around constraints in contentpractice standards and assessment. Like any profession that involves some level of accountability, the manner in which you are expected to teach mathematics in today’s times has to embrace such realities, which also means needing to arm yourself with a proactive disposition that will support all middle school students to succeed in developing solid mathematical understanding necessary for high school and beyond, including the workplace. It is worth noting that the development of the CCSSM, SBA, and PARCC relied heavily on perspectives from various stakeholders such as teachers, school administrators, state leaders and policymakers, experts, educators, researchers, parents, and community groups. For you, this particular collaborative context of the CCSSM, SBA, and PARCC means understanding and appreciating both the efforts and the framing contexts of their emergence. It takes a village for these state-driven initiatives to succeed and it is now your turn to be in the front line of change. 1.1  A Blended Multisourced Approach to Learning to Teach Middle School Mathematics

Considering the vast amount of information that is readily available on the internet, including your digital-native disposition and competence toward conducting online searches, this book explores the possibility of a blended multisourced approach to methods of teaching middle school mathematics. Embedded in several sections of this book are activities that will require you to access online information. Becoming acquainted and getting used to this blended form of learning, as a matter of fact, provide good training for the actual work that comes. That is, once you start teaching, you no longer have to plan alone. Furthermore, you do not have to wait for a faceto-face professional development workshop to learn new ideas. You can gain access

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Dear Preservice Middle Level Majors

to resources with very minimal cost, and in many cases at no cost to you. What is always needed, however, which applies to any kind of profession, is some kind of training that involves knowing where and how to look for correct and appropriate information. Many sections in this book will require you to access links that provide the best information, including ways to obtain information by relying primarily on email communication. From a practice standpoint, the blended multisourced approach taken in this book symbolizes the collaborative nature of teaching, which you know and seasoned teachers would affirm depend on a repertoire of tools gained by learning from other sources. For convenience, you are strongly encouraged to access the following free site below, which contains all the links, articles, and reproducible worksheets in a document format that are referenced in various sections in this book. The site was informed and is respectful of copyright rules, so exercise care on matters involving dissemination. http://commoncoremiddleschoolmethods.wikispaces.com/ 1.2  Overview of the Remaining Chapters

Chapter 2 introduces you to the eight CCSSM practice standards. Content activities are provided to help you understand the content-practice dimension of teaching, learning, and assessing school mathematics. You will also learn in some detail important psychological and instructional issues surrounding problem solving, reasoning and proof, representations, communications, and connections. Take a peek at Figure 2.1 (p. 14) for an interpretive visual summary of the eight practice standards. Content-practice teaching, learning, and assessment involve establishing appropriate and meaningful correspondences between mathematical content and practices that support the development and emergence of (school) mathematical knowledge. Chapters 3 through 5 and 7 through 11 deal with content-practice, teaching, and learning issues relevant to the following domains below that comprise the Grade 5 through Grade 8 and (traditional pathway) Algebra 1 CCSSM. • • • • • • •

Real number system and operations (Chapters 3 and 4) Ratio and proportional relationships and quantities (Chapter 5) Expressions and operations (Chapter 7) Equations and inequalities (Chapter 8) Functions and models (Chapter 9) Measurement and geometry (Chapter 10) Data, statistics, and probability (Chapter 11)

Domains consist of clusters or organized groups of related content standards, where each standard defines in explicit terms what students need to understand and be able to do. 5

Chapter 1

Chapter 6 focuses on technology-enhanced tools for teaching and learning middle school mathematics from Grade 5 through Grade 8 and Algebra I. You will explore how to use a graphing calculator, Geogebra, a calculator-based ranger, and apps and other virtual-based manipulatives in ways that support student learning of mathematical concepts and processes. You will also learn how to teach visualization techniques that will help your students attend to what is mathematically relevant in visual-derived representations. “For a mathematician and a teacher,” Raymund Duval writes, “there is no real difference between visual representations and visualization. But for students, there is a considerable gap that most of them are not always able to overcome. They do not see what the teacher sees or believes they will see.” Hence, this chapter will provide you with information that will enable you to orchestrate meaningful and effective instruction (setting up tasks, using guide questions, etc.) with such visual tools. Chapter 12 addresses different assessment strategies that measure middle school students’ understanding of the CCSSM. It also introduces you to the basic structure and testing requirements of the SBA. You will learn about formative and summative assessments and norm- and criterion-referenced testing, including alternative forms of assessment such as journal writing and unit projects as an extended version of a SBA performance task. Sections that deal with the SBA can be replaced by or discussed together with PARCC, if you find it necessary to do so. In the closing section, you will develop content-practice assessment tasks for practice. Chapter 13 relies on information drawn from the preceding chapters. You will deal with issues that are relevant to middle school students’ learning of mathematics within the constraints of the CCSSM, SBA, and PARCC. The complex historical relationship between learning theories and the US school mathematics curriculum through the years, including the Math Wars, are also discussed in order to help you understand the negative consequences of holding extreme views on mathematical learning. Further, you will explore in some detail the theories of Piaget and Vygotsky with the intent of helping you understand Fuson’s integrated learning-path developmentally appropriate learning/teaching model, which emphasizes growth in middle school students’ understanding and fluency of mathematics. Fuson’s model offers a middle ground that opposes extreme views and misreadings of constructivist and sociocultural learning in mathematical contexts. Chapter 14 addresses practical issues relevant to teaching the CCSSM in middle school classrooms. You will learn about different teaching models and write contentpractice unit, lesson, and assessment plans. Chapter 15 focuses on issues relevant to setting up and running an effective mathematics classroom that is conducive for teaching and learning the CCSSM. You will become familiar with issues surrounding student persistence and motivation in mathematics, including ways to design learning and learning environments that foster flexible problem solving and mathematical disposition. You will also be introduced

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Dear Preservice Middle Level Majors

to Complex Instruction, an equity-driven approach to group or collaborative learning in the mathematics classroom. General classroom management concerns close the chapter, which deal with how to design and manage optimal learning for all students, eliminate or minimize disruptions, and address potential behavior problems.

7

Chapter 2 

Getting to Know the Common Core State Standards for Mathematical Practice

In this chapter you will learn about the eight Common Core State Standards for Mathematical Practice (CCSSMP; “practice standards”). Take a moment to access the information from the link below. http://www.corestandards.org/Math/Practice The practice standards explicitly articulate how middle school students should engage with the Common Core State Standards for mathematical content (or “content standards”) “as they grow in mathematical maturity and expertise throughout the [middle school] years” (NGACBP &CCSSO, 2012, p. 8). When you teach to the content standards, the practice standards should help inform and guide how you teach, how your students develop mathematical understanding, and what and how you assess for content knowledge. Also, consistent with the overall intent of the Common Core State Standards in Mathematics (CCSSM), the content and practice standards convey shared goals and expectations about the kinds of knowledge, skills, applications, and proficiencies that will help all middle school students succeed in mathematics and be ready for workplace demands. It is worth noting that the practice standards appear to be relevant to other subjects as well, making them powerful, interesting, and useful to middle school students who are learning by the day how making inferences, exercising formal thinking, implementing disciplined reasoning, and forming structures altogether contribute to knowledge structures that are systematic, valid, and exact. Such structures often convey patterns, and mathematics has been characterized as the science of structures. Inference involves going beyond available data, and mathematics is well known for its methods of generalization and abstraction. Both formal thinking and disciplined reasoning demand a conscious application of clearly articulated inferences on the basis of well-defined norms or rules of engagement, and mathematics is the subject that deals with verifying conjectures and proving them in a rigorous and exact manner. As you pay attention to content-practice relationships in this chapter, you should not lose sight of the assessment aspect of learning, which you will pursue in some detail in Chapter 12 in connection with the Smarter Balanced Assessment. While this chapter focuses on the learning and teaching of a few topics in middle school mathematics in order to demonstrate ways in which mathematical practices support content learning, that is, the unique content-practice dimension of the CCSSM that makes it different from typical state standards in mathematics, it is good to have 9

Chapter 2

an alignment mindset at all times. This perspective views effective and meaningful pedagogy to be a fundamental matter of alignment among mathematics curriculum, learning, teaching, and assessment. Consider the following mathematical problems on parity below, which will help you understand the significance of the eight practice standards. Students learn about the concept of parity – that is, oddness and evenness – of whole numbers as early as second grade. It is a fundamental idea in number concepts that should be familiar to all middle school students. In the elementary grades, students learn to recognize even and odd numbers. Content Activity 1 aims to deepen your and your students’ understanding of parity relationships from basic recognition to justification of general parity relationships. 2.1  Content Activity 1: Exploring and Proving Parity

Students in second grade determine the even parity of a set of objects visually by pairing objects and numerically by skip counting by 2s (2.OA.3). This elementary conception is expected to evolve from nonsymbolic to symbolic over time. Work with a pair and do the following tasks below. a. Use circle chips, unit cubes, or other available manipulatives to model the following even numbers: 2, 4, and 6. Fill in Activity Sheet 2.1. b. What patterns do you see in Table 2.1? Write a simple multiplicative expression for an even whole number involving variables (5.OA.2, 5.OA.3, and 6.EE.2). c. Sixth-grade Chase made the following claim about odd numbers: “When you form pairs of circle chips and always have 1 leftover chip, that number is odd.” Investigate with manipulatives whether Chase is correct. If he is correct, write an expression that conveys his conjecture in mathematical form (6.EE.2). d. Is the whole number 0 even or odd? How can you tell for sure? e. Use manipulatives to help you determine the parity in each case below. 1. Add two or more even numbers. 2. Add two or more odd numbers. 3. Subtract two even numbers. 4. Subtract two odd numbers. 5. Obtain the sum of an even number and an odd number. 6. Obtain the difference between an even number and an odd number. 7. Multiply two or more even numbers. 8. Multiply two or more odd numbers. 9. Obtain the product of an even and an odd number. f. The two situations below involve coin tricks that a sixth-grade teacher shared with her students. Determine how mathematics could be used to solve each problem. 1. Ms. Lamar showed her sixth-grade class a small purse filled with coins. 10

the Common Core State Standards for Mathematical Practice

Even Number

Visual Pairing Process

Sum of Two Equal Addends, or Additive Expression

Product of Two Factors or Multiplicative Expression

2

1+1

2 x 1 = 2·1

4

2+2

2 x 2 = 2·2

6

 

8

 

10

 

12

 

18

  Activity Sheet 2.1. Parity Table in Compressed Form

Ms. Lamar: Okay, I need a volunteer to take some coins from this purse. Jaime volunteered and took a handful of coins. Ms. Lamar: Jaime, count how many coins you took from the purse. Keep it to yourself or tell your classmates how many you have, but I should not know that information. Jamie decided to share the information with his classmates. Ms. Lamar: I will now take some coins from this purse. All I can say is that when I add my coins to yours, it will be even if you have an odd number of coins. It will be odd if you have an even number of coins. Ms. Lamar added her coins to Jamie’s coins. Ms. Lamar: Jaime, please count the coins. Jaime counted and then agreed that the number of coins changed to the opposite. The class was amazed! Ms. Lamar: I told you. I could change your total number of coins to its opposite! How is the trick possible? 2. Ms. Lamar tossed 10 coins on the teacher’s desk. She looked at them quickly and then looked away. 11

Chapter 2

Ms. Lamar: Okay, I need another volunteer to turn over any pair of coins and cover one coin from the pair with his or her hand. Jennifer volunteered. Ms. Lamar then looked at the exposed coins on the table. Ms. Lamar: The other coin is a “head.” Jennifer exposed the other coin and it was, indeed, a “head.” How is the trick possible? In the CCSSM, students learn about the number system properties as early as first grade. Check the CCSSM and map the correct content standards with each property listed in Table 2.1. Table 2.1. Number System Properties Explored in the Elementary CCSSM Number System Property

Applicable CCSSM Content Standards

Commutative Property of Addition, i.e., a+b=b+a

 

Commutative Property of Multiplication, i.e., a × b = b × a

 

Associative Property of Addition, i.e., (a + b) + c = a + (b + c)

 

Associative Property of Multiplication, i.e., (a × b) × c = a × (b × c)

 

Distributive Property of Multiplication over Addition, i.e., a × (b + c) = (a × b) + (a × c)

 

In addition to the stated properties in Table 2.2, another basic but important property is closure, which deals with the kinds of numbers you generate when you perform an operation involving at least two numbers in the same set. Closure for addition means that when you choose any two or more numbers from a given set, perform addition on those numbers, and obtain a sum that belongs to the set, then the set is closed under addition. The idea is the same in the case of closure for multiplication. Continue working with your pair to do the following tasks below. g. Which of the following sets of numbers are closed under addition? multiplication? 1. Whole numbers 2. Counting numbers 3. Proper fractions and mixed numbers in the elementary CCSSM h. Is the set of counting numbers closed under subtraction? Why or why not? Explain. i. Is the set of whole numbers closed under division? Why or why not? Explain. 12

the Common Core State Standards for Mathematical Practice

j. Is the set of fractions closed under subtraction? Why or why not? Explain. k. Is the set of fractions closed under division? Why or why not? Explain. l. Sixth-grade Martha used circle chips and inferred the following observation: “If you add two even whole numbers, the result is even. I also think that the set of even numbers is closed under addition.” When she was asked to obtain an expression for the sum of two such numbers, she wrote down the following explanation below. My first even number is 2a and my second even number is 2b. So, the sum of 2a and 2b is 2a + 2b. But I know that 2a + 2b is equivalent to 2(a + b). So, if a + b = c, then the sum of two even numbers is even, which is 2c. How was Martha inferring, thinking, and reasoning about her conjecture?  evise Martha’s written solution by supplying reasons where they are needed. R The improved solution that contains both statements and reasons is called a mathematical solution to the problem. If you think that Martha’s method above is valid, go back to tasks (e) and (f) and show that your results are correct by obtaining the appropriate expressions. Provide complete mathematical solutions as well. m. Is the set of odd numbers closed under addition? If your answer is yes, generate an expression for the sum and provide a mathematical solution. If your answer is no, provide one example to show that it cannot be so. Such an example is called a counterexample. You only need one counterexample to explain why a stated conjecture is not correct. n. Is the set of even numbers closed under division? If your answer is yes, generate an expression and provide a mathematical solution. If your answer is no, generate a counterexample. o. Is the set of even numbers closed under subtraction? Explain. p. Is the set of odd numbers closed under subtraction? Explain. q. Is the singleton set {1} closed under addition? subtraction? multiplication? division? Explain. 2.2  The Eight Common Core State Standards for Mathematical Practice

Figure 2.1 shows an interpretive visual model of the eight practice standards. They pertain to institutionally valued mathematical actions, processes, and habits, and have been conceptualized in ways that will hopefully encourage all middle students to engage with the content standards more deeply than merely acquiring mathematical knowledge by rote and imitation. Consider, for instance, the content-practice relationship involving parity in the preceding section. When middle school students are in the process of understanding different valid parity relationships depending on the cases being considered, they should also be provided with an opportunity to 13

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Figure 2.1. Standards for Mathematical Practice (NGACBP &CCSSO, 2012, pp. 6–8)

engage in mathematical practices such as looking for structures and expressing the regularity of such structures in repeated reasoning. Take some time to read each practice standard and the accompanying description. Continue working in pairs and respond to the following tasks below. a. To what extent did any of the problems in section 2.1 encourage you to: 1. make sense of the problems and persevere in solving them? 2. reason abstractly and quantitatively? 14

the Common Core State Standards for Mathematical Practice

3. construct viable arguments and critique the reasoning of others? 4. model with mathematics? 5. use appropriate tools strategically? 6. attend to precision? 7. look for and make use of structures? 8. look for and express regularity in repeated reasoning? b. Refer to p. 4 of the CCSSM and read the section involving “Understanding mathematics.” To what extent did the content activity encourage you to develop a mathematical understanding of parity and parity relationships? The practice standards have been derived from two important documents that describe processes and proficiencies that are central to the learning of school mathematics. The National Council of Teachers of Mathematics (NCTM) identifies problem solving, reasoning and proof, communication, representation, and connections to be the most significant processes that all middle school students should learn well. The National Research Council (NRC) characterizes mathematically proficient students as exhibiting mathematical competence in the strands of adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition. Tables 2.2 and 2.3 list the characteristics of the NCTM process standards and the NRC proficiency strands, respectively. Read them carefully and highlight terms and descriptions that you think will require some clarification. To better appreciate the CCSSM practice standards, explore the following mathematical task, Building a Hexagon Flower Garden Design, from the points of view of the NCTM process standards and the NRC proficiency strands. The task involves generating and analyzing an emerging pattern, which is an appropriate problem solving activity for fifth- through eighth-grade students (5.OA.3, 6.EE.9, 8.F.4, and 8.F.5). 2.3  Content Activity 2: Building a Hexagon Flower Garden Design

Figure 2.2 shows a hexagon flower garden design of size 4 that will be used to cover a flat horizontal walkway between two rooms. A design of size 1 consists of a black hexagon tile that is surrounded by six gray hexagon tiles. a. Figure 2.2 is a garden design of size 4. How many black and gray hexagon tiles are there? b. Work with a pair and complete Activity Sheet 2.2 together. What do the four equal signs (=) mean in the context of the activity? c. Many expert patterners will most likely infer that, on the basis of the assumptions stated for the problem and the data collected in Activity Sheet 2.3, the mathematical relationship appears to model a specific rule. What information relevant to the task and the limited data shown in Activity 2.2 will help them establish that inference? 15

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Table 2.2. The NCTM Process Standards (NCTM, 2000) Problem Solving (PS) 1. build new mathematical knowledge through problem solving; 2. solve problems that arise in mathematics and in other contexts; 3. apply and adapt a variety of appropriate strategies to solve problems; 4. monitor and reflect on the process of mathematical problem solving.

Reasoning and Proof (RP) 1. recognize reasoning and proof as fundamental aspects of mathematics; 2. make and investigate mathematical conjectures; 3. develop and evaluate mathematical arguments and proofs; 4. select and use various types of reasoning and methods of proof.

Communication (CM) Connections (CN) 1. organize and consolidate their 1. recognize and use connections among mathematical thinking through mathematical ideas; communication; 2. understand how mathematical ideas 2. communicate their mathematical thinking interconnect and build on one another to coherently and clearly to peers, teachers, produce a coherent whole; and others; 3. recognize and apply mathematics in 3. analyze and evaluate the mathematical contexts outside of mathematics. thinking and strategies of others; 4. use the language of mathematics to express mathematical ideas precisely. Representation (RP) 1. create and use representations to organize, record, and communicate mathematical ideas; 2. select, apply, and translate among mathematical representations to solve problems; 3. use representations to model and interpret physical, social, and mathematical phenomena.

Table 2.3. The NRC Mathematical Proficiency Strands (NRC, 2001) Conceptual Understanding (CU) comprehension of mathematical concepts, operations, and relations

Procedural Fluency (PF) skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

Strategic Competence (SC) ability to formulate, represent, and solve mathematical problems

Adaptive Reasoning (AR) capacity for logical thought, reflection, explanation, and justification

Productive Disposition (PD) habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy

16

the Common Core State Standards for Mathematical Practice

Figure 2.2. Hexagon Flower Garden Design of Size 4

Activity Sheet 2.2. Hexagon Flower Garden Design Task Data

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d. On the basis of the assumptions noted in (c), three Grade 5 students identified the following features of the pattern below relative to the number of gray hexagon tiles for a design of size 12. Ava: Keep adding 4, so 2, 2 + 4 = 6, 6 + 4 = 10, 10 + 4 = 14, …, 46 + 4 = 50. Bert: Add 12 groups of 4 brown tiles and then add 2 more. It’s like the times table for 4, 4, 8, 12, …, 40, 44, 48. Then plus 2 you get 50. Ces: Double 12 groups of 2 brown tiles. Then add 2 more. It’s like the times table for 2 plus 1. So 2, 4, 6, 8, …, 20, 22, 24, 25. Then you double it and you get 50.  hich student is correct and why? How does each student see his or her pattern W relative to the size-12 design? e. Since students in sixth grade learn to express mathematical relationships involving two quantities with variables and formulas (6.EE.9), the following three Grade 6 students below offered the following formulas below assuming the assumptions in (c). Dax: My recursive formula is S = P + 4. P is the succeeding answer after you add 4 to the preceding answer. So a size-1 tile has 6 gray tiles, that’s P for the succeeding one. So a size-2 tile should have 6 + 4 = 10 gray tiles. A size-3 should then have 10 + 4 = 14 gray tiles. And so on. Eve: My direct formula or rule is G = 2 + H x 4. So a size- tile has 2 + 1 x 4 = 6 gray tiles, a size-2 tile has 2 + 2 x 4 = 10 gray tiles, etc. Fey: My direct rule is T = 6 + (n – 1) x 4. So a size-1 tile has 6 gray tiles, a size-2 has 6 + 1 x 4 or 10 gray tiles, etc.  hich student is correct and why? How does each student see his or her pattern W relative to the size-12 design? How does a recursive rule or formula differ from a direct rule or formula? What are some advantages and disadvantages of each type of rule or formula? Should sixth-grade students know only one type of rule or formula? Explain. f. Ava gives Bert 75 brown hexagon tiles in groups of 10. As soon as Bert receives the last tile, he tells her, “It’s odd, so I really can’t use all of the tiles. Do you want 1 tile back?” Ava thinks Bert is not correct. What do you think? If Bert is correct, how does he know so quickly? Consider your mathematical experience with the Building a Hexagon Flower Garden Design task. Work in groups of 3 and discuss the following questions below. For each group, assign a facilitator, a recorder or scribe, and a reporter. The facilitator

18

the Common Core State Standards for Mathematical Practice

needs to make sure that all members of the group are able to share their experiences. The recorder or scribe should see to it that the group report reflects contributions from all members of your group. The reporter is responsible for sharing your group’s views in the follow up whole-classroom discussion. Exercise care and caution as you perform your respective roles so that you all feel respected and are able to make significant contributions in the process of working together. g. Refer to Table 2.2. To what extent did the above task encourage you to engage in: i) ii) iii) iv) v)

at least one of four problem solving actions? at least one of four reasoning and proof actions? at least one of four communication actions? at least one of three connection actions? at least one of three representational actions?

h. Refer to Table 2.3. This time keep in mind the needs of Grades 5 and 6 students. i) What mathematical concepts, operations, relations, and skills do they need in order to successfully construct and justify rules such as the ones the students expressed items (d) and (e)? ii) How does having a deep understanding of stable mathematical relationships help them in constructing their formulas? iii) What purpose and value do items (d) and (e) serve in their developing understanding of patterns and structures in mathematics? iv) How do patterning tasks help them develop productive disposition? In the next section you will explore the NCTM process standards in some detail, which will help you further deepen your understanding of the issues underlying the practice standards and the work that is required to implement them effectively in your own classrooms. 2.4  Problem Solving Contexts in Middle School Mathematics

Problem solving contexts in middle school mathematics are drawn from a variety of situations, as follows: 1. authentic and real situations, where mathematics emerges naturally in the context of students’ everyday experiences; 2. conceptually real situations, where mathematics emerges from well-described scenarios that are rather difficult or almost impossible to be modeled in real time but nonetheless contain sufficient information that will help students perform the relevant actions mentally; 3. simulated (i.e., imitated) situations, where mathematics emerges from situations that are modeled in a technological platform because they either cannot be easily

19

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performed at the physical or concrete level or are meant to conveniently illustrate an instructional objective; 4. context-free mathematical situations, where mathematics emerges from tasks that consist of symbols that students manipulate according to well defined rules or principles. 2.4.1 Content Activity 3: Different Types of Problems in Middle School Mathematics Access the following link below, which will take you to the Smarter Balanced Assessment released mathematics sample items for grades 5 to 8 students. Work with a pair and answer the questions that follow. http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm a. Solve all the sample items for grades 5 through 8 students first before moving on to the next two tasks below. b. Use your experiences to help you classify each task item according to the type of problem solving situation it models. c. Choose five of your favorite sample items. For each item, list the proficiencies that you think are appropriate for students to model and accomplish successfully. Your analysis does not have to reflect all five NRC proficiency strands. 2.5  Representations in Middle School Mathematics

When you ask middle school students to establish an external representation of a mathematical object, they need to find a way of portraying the object and capturing its essential attributes in some recognizable form. Representations range in type from personal to institutional, informal to formal, and approximate to exact depending on the context in which they arise in mathematical activity. Personal, informal, and approximate representations often reflect individual worldviews, so their forms tend to be idiosyncratic and situated (i.e., they make sense only to the individual). Institutional, formal, and exact representations involve the use of shared inscriptions, rules for combining them, and a well-defined and well-articulated system that links symbols and rules according to the requirements, practices, and traditions of institutions that support their emergence. Meaningful representations at the middle school level develop in an emergent context beginning with what students already know and how they convey them. There is no other way, in fact, and one goal of teaching involves finding ways to bridge prior and formal representations through negotiation either between you as the teacher and your students or among students in an engaging activity. Figure 2.3 visually captures the sense of emergence in terms of progressions in their representational competence. 20

the Common Core State Standards for Mathematical Practice

Figure 2.3. Progressive Formalization and Mathematization of Representations

2.5.1 Content Activity 4: From Multiplication of Whole Numbers to Direct Proportions to Simple Linear Relationships Consider the following fifth-grade multiplication task below. Answer the questions that follow. Watches sell for $175 each. How much would 35 watches cost? a. Read content standard 5.NBT.5. How are fifth-grade students expected to solve the problem in a multiplicative context? Set up a multiplication expression and obtain the product. b. The above one-step multiplication problem exemplifies a typical mathematical task that all students solve as early as third grade. Read the cluster of standards 3.OA and develop a version that is appropriate for third-grade students. c. The above one-step multiplication problem can be also be interpreted as a task that involves ratio and proportional reasoning. Carefully read content standards 6.RP.1, 6.RP.2, and 7.RP.2 to determine the meanings of the following terms: ratio; unit rate; and proportions; and constant of proportionality. Further, read content standards 6.RP.3 and 7.RP.2, which should provide you with different ways of representing the problem in terms of ratio and proportion. Demonstrate 21

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the following representations relative to the problem: (i) tables of equivalent ratios; (ii) tape diagrams; (iii) double number line diagrams; (iv) graph of several ordered pairs of values on a coordinate plane; and (v) linear equation. d. Solve the following word problem below in different ways. Use the graphic organizer shown in Activity Sheet 2.3 to help you organize your work. Mrs. Lee, a grocery store owner, bought 3 cantaloupes for $6. How many cantaloupes can Mrs. Lee buy if she has $120? Problem in Words

Table of Equivalent Ratios

Tape Diagram

Double Number Line Diagram

Graph on a Coordinate Plane

Linear Equation

Activity Sheet 2.3. Representing Mathematical Relationships in Different Ways

Personal, approximate, and informal representations involve the use of situated gestures, pictures, and verbal expressions. For example, some fifth-grade students might initially solve the multiplication problem involving the watches through repeated addition of the same addend with very little to no understanding of the conceptual relationship between multiplication of whole numbers and repeated addition. Others might draw a box, label it with the quantity $175, and then perform a series of repeated addition processing. Overall, such representations convey models that are situated (i.e., context-dependent) and either unstructured (i.e., unorganized and naïve) or structured (i.e., sophisticated) in form. Further, they evolve as presymbolic models, that is, they are inductively inferred as a result of seeing repeated actions on similar problems but nothing else beyond that. Students who manage to solve the above problem and other similar examples in a presymbolic mode oftentimes 22

the Common Core State Standards for Mathematical Practice

do not think at the level of deductive rules that apply to all problems of the same type. The presence of a real-life context also characterizes the presymbolic and, thus, nonsymbolic, dimension of such representations. Your goal as the teacher is to help them develop and manipulate symbolic representations. Symbolic representations in (school) mathematics convey decontextualized mathematical relationships that operate and make sense at the level of structures and rules alone. Classroom representations mark the beginning of progressive formalization. Students’ representations in this phase are purposeful and intentional as a result of their intense and shared interactions with you and others learners. Through meaningful discussions they learn to use tape diagrams and ratio tables, which replace actions of extensive and (un)structured listing or repeated addition of the same addend. Classroom representations achieve their meaningful progressively formal state when middle school students begin to coordinate the following two nonsymbolic aspects of their representations below. 1. Analytic Condition: Knowing how to process problems deductively by applying the appropriate rules; 2. Abstract Entity Condition: Using formal symbols to translate problems in external form. In this particular characterization of classroom representation, middle school students think in terms of rules. In the presymbolic phase, specific instances over rules dominate representational action. In the nonsymbolic phase, the reverse is evident. However, such representations remain grounded in the nonsymbolic phase because they are still strongly linked to the context of their emergence. For example, in the case of the above multiplication task involving watches, the numbers 175 and 35 are classified as arithmetical quantities, that is, numerals with units. The number sentence 35 × 175 = 6125 actually means

or,

35 watches x $175/watch = $6,125.

Another interesting characteristic of classroom representations is the condensed or contracted nature of the forms involving alphanumeric expressions that replace the oral, verbal, and pictorial descriptions that are evident in the presymbolic phase. Institutional representations convey formal and shared representations at the community level. Since school mathematics content reflect the values, traditions, and practices of the entire mathematics community, middle school students need to acquire appropriate and intentional ways of representing objects and relationships in order to participate meaningfully in the larger community. Institutional representations evolve into symbolic models as soon as students begin to decontextualize problems, that is, 23

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by shifting their attention from the contexts to the processing of the abstract entities. Think about the above multiplication tasks together. Such problems evolve in purpose over time in the middle school mathematics curriculum. In third grade, students process them by using equal groups, arrays, drawings, or simple equations with appropriate symbols for the unknown numbers to represent the problems (3.OA3). In fourth grade, they represent them in the context of multiplicative comparisons and multiples (4.OA.2 and 4.OA.3) and with larger numbers (4.NBT.5). In fifth grade, they either focus on writing simple expressions that record the relevant calculations without having to solve the problems (5.OA.2). Or, if they need to solve them, they are expected to use the standard algorithm and deal with factors involving multidigit numbers (5.NBT.5). From sixth- through eighth-grade, they learn to see the structures of such problems in terms of ratios (6.RP.1 to 6.RP.3), proportions (7.RP.1 to 7.RP.3), and simple linear function or power relation of the type y = mx (7.RP.2c, 7.RP.2d, 8.EE.5, 8.F.1 to 8.F.2, F-IF.1, F-IF.2, and F-IF.1 to F-IF.7a). Moving on to a different aspect of representation, some mathematical objects are known to generate multiple representations, that is, they can be expressed in several different ways. For example, Activity Sheets 2.2 and 2.3 illustrate different ways of representing patterns and proportional relationships, respectively. Work with a pair and accomplish the following activity below. e. Think about all the mathematical objects you know that have multiple representations. Many middle school teachers tend to teach all the relevant representations involving the same object at the same time. Do you agree? Discuss possible concerns with this particular teaching practice. 2.6  Connections in Middle School Mathematics

One important aspect of your job involves helping middle school students establish connections among mathematical ideas. If you think of connections in terms of relations, then mathematical ideas are in fact linked in at least two different ways depending on the context of activity, as follows: • Representational connections involve matters that pertain to equivalence, which means that two or more representations convey the same idea despite the superficial appearance of looking different. Consider, for example, the multiple representations for the same pattern shown in Activity Sheet 2.2. The table, ordered pairs, graph, and rule are all equivalent representations of the pattern shown in Figure 2.2. They hold the same assumptions about the pattern, which support their equivalence. Also, take note of the four equal symbols shown in Activity Sheet 2.2, which convey the sense that equal means “is the same as.” • Topical connections involve linking two or more separate topics either from the same subject (e.g., multiplication of whole numbers and ratio and proportion) or from different subjects (e.g., the Pythagorean Theorem in geometry and irrational numbers in algebra). 24

the Common Core State Standards for Mathematical Practice

Meaningful connections should facilitate transfer, which is a core process that supports thinking and learning about structures. Transfer involves applying what one has learned in one situation to another situation. But that is usually easier said than done. Transfer is a subjective experience for individual learners, meaning to say that they need to establish the connections or relationships between two or more situations, representations, or topics themselves. Certainly, the subjective process can be smoothly facilitated by providing them with tools at the appropriate time that will enable them to see those connections as similar in some way. In other words, connections among representations and topics do not transfer on their own. Learners need to construct them with appropriate resources at their disposal. Such resources can be concrete (e.g., using manipulatives) or interventional (e.g., using purposeful guide questions) to help them accomplish successful transfer. The following activity exemplifies topical connections. Work with a pair and answer the questions that follow. 2.6.1 Content Activity 5: The Pythagorean Theorem and Irrational Numbers Middle school students first learn about the Pythagorean Theorem in eighth grade. Content standard 8.G.6 expects them to provide a mathematical explanation for the correctness and validity of the theorem for any right triangle. Furthermore, content standard 8.G.7 expects them to apply the theorem in situations that involve determining an unknown side length in a right triangle in both contextual and decontextualized situations. Interestingly enough, they also learn about radicals (8. EE.1 and 8.EE.2), especially irrational numbers (8.NS.1 and 8.NS.2), for the first time in eighth grade. In this section, you will learn how the Pythagorean Theorem is topically related to the mathematical way of establishing the existence of square roots (i.e., radicals with an index of 2). Work with a group of 3 students and do the following tasks below. a. Discuss the following two prerequisite concepts before moving on to the next step: (i) Make sure that you remember how to read and write expressions involving whole-number exponents (6.EE.1). Students are exposed to exponents for the first time in the context of representing the decimal number system in terms of powers of 10 (5.NBT.2); (ii) Students also learn about the area of a square with a side length of s units in third grade (3.MD.7b). Explain why its area A is s2 square units. b. The diagrams in Figure 2.4 provide two visual ways to verify the correctness of the Pythagorean Theorem. The theorem states that any right triangle with side lengths a, b, and c units, where c is the hypotenuse or the longest side of the triangle, will obey the mathematical relationship a2 + b2 = c2.  or the diagram on the left, obtain the areas of the two squares. Then use a pair of F scissors to cut out figures 2 through 5 along the dotted segments. Next, use all five figures to form another square. How is the new square related to the length of the hypotenuse of the right triangle? Verify that it is, indeed, the case that  a2 + b2 = c2. 25

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 he diagram on the right starts with four copies of the same right triangle. Move T the triangles around so that you form the square shown. Then calculate the areas of the larger square and the individual components of the smaller squares. How are they related to each other? Verify that it is, indeed, the case that a2 + b2 = c2.

Figure 2.4. Exploring the Pythagorean Theorem in Two Ways

Find the length of the hypotenuse in each right triangle below.

c. Figure 2.5 shows a square with a side length of 1 unit. Cutting the square along a diagonal yields a right triangle. Calculate the length of the hypotenuse. Can you find a whole number that will complete the equation shown in Figure 2.5? How about a fraction? Would a decimal number work? Try to see what happens when you fill in the blank with the exponential expression (“square root of 2,” could also be written as ).

Figure 2.5. Exploring

26

or, in radical form,

the Common Core State Standards for Mathematical Practice

 is not a whole number, it is reasonable to ask if it is a rational number d. Since (i.e., a quotient of two integers). Investigate what happens when you assume that it is a rational number by following the argument below. If

is a rational number, then

, where a and b are integers that do

not share any common factor and b ≠ 0 (why?). Squaring each side of the equation, which is permitted, yields the equivalent equation

, or a2 = 2b2. Why?

Now, observe that a2 is even. Why? Consequently, a must be even and a2must have a factor of 4. Why? Consequently, too, both b2and bmust be even. Why? Since a and b are both even, this contradicts an assumption. Which one? cannot be a rational number. It is, in fact, an irrational number. Thus, How does content standard 8.NS.1 describe an irrational number? ? What does content standard 8.NS.2 say about the approximate value of How can you use the number line below and the right triangle in Figure 2.5 to visually verify the approximate value?

Figure 2.5. Number Line Diagram

e. Use your knowledge of the Pythagorean Theorem to generate five irrational numbers. Approximate their values visually and numerically. 2.7  Reasoning and Proof in Middle School Mathematics

Knowing how to reason and prove mathematically is just as equally important as knowing how to process problems deductively by applying the appropriate rules. Work through the following activity below with a partner. a. There exists an infinite number of irrational numbers. Show that finding a right triangle whose side lengths yields a hypotenuse of

exists by units. Then

use the argument structure in section 2.6.1 item (d) to develop an explanation that is an irrational number. Also, obtain an approximate value of

visually

and numerically. 27

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Reasoning and proof are fundamentally tied to beliefs. When students reason, they convey beliefs that are usually based on inferential norms that often map to their level of experience. Certainly, those beliefs evolve over time depending on how well they are able to coordinate factors relevant to perceptual and conceptual competence and representational ability. • Perceptual competence involves making inferences on the basis of what one sees at the moment. • Conceptual competence involves making analytical inferences on the basis of one’s ability to interpret valid mathematical relationships, properties, features, or attributes of objects beyond the limits of perceptual competence. • Representational ability involves employing gestures, drawing pictures and diagrams, and using abstract entities (numeric forms, variables, and combinations) that enable one to externalize concepts and percepts. Reasoning and proof activities help deepen middle school students’ developing mathematical understanding. Activities that encourage them to generate propositions or conjectures and claims or provide explanations that justify their truth values (e.g.: why they are either true or false; how they are true or false; under what conditions they are either true or false) help them establish certainty and necessity in their developing mathematical knowledge. Many of them, of course, tend to reason through examples that are intended to empirically demonstrate certainty at their level of experience. Further, the examples they use and the repeated actions that come with them convey how they sense and capture the necessity of an interpreted structure. These kinds of explanations and actions should be encouraged and supported. When you teach middle school students to reason mathematically, you should aim to help them achieve competence in three different kinds of explanations, as follows: • Abductive reasoning, which involves forming conjectures or claims; • Inductive reasoning, which involves performing repeated actions as a way of demonstrating the validity of conjectures or claims; • Reasoning that emerges from a deductive argument, which arises from abductive and inductive reasoning together. For example, consider the following steps that demonstrate the parity of the sum of any pair of even whole numbers. I. 2 + 2 = 4. The sum is even. II. 4 + 6 = 10. The sum is even. III. 2 + 8 = 10. The sum is even. IV. 6 + 6 = 12. The sum is even. V. Two even addends yield a sum that is even. VI. So 26 + 12, which equals 38, should be even. Each claim under steps I, II, III, and IV closes with an abduction, that is, a guess or conjecture. Steps I through IV together support step V, which illustrates a statement 28

the Common Core State Standards for Mathematical Practice

that emerges from inductive reasoning. Step VI involves reasoning that emerges from the preceding five steps; it exemplifies what is called a deductively-closed argument that is empirically supported by both abduction and induction. When you ask middle school students to reason about any mathematical object or relationship, you should aim for deductive closure at all times, which is a fundamental characteristic of the analytical condition of mathematical representations. Work with a pair and deal with the following two questions below. g. What happens when middle school students are not provided with sufficient opportunities to engage in abductive and inductive reasoning? h. What happens when middle school students are able to engage in abductive and inductive reasoning but unable to achieve deductive closure? 2.8  Communication in Middle School Mathematics

When you ask middle school students to communicate in the mathematics classroom, it means that you want them to pay serious attention to forming, using, reasoning with, and connecting various forms of representations. Communication involves externalizing an idea in some recognizable form for personal and public access. Personal access is a form of intra-communication that helps them monitor what and how they are thinking when they are in the process of conceptualizing or problem solving. However, meaningful communication takes place in a public domain, where individuals have access to each other’s representations. Hence, public access is a form of inter-communication that helps them monitor and learn from each other with the goal of developing shared intent, knowledge, and practices over time. Meaningful communication in public also targets the development of more refined and sophisticated forms of representation and reasoning, including the relevant connections that make them interesting. Furthermore, because personal communication of mathematical representations among middle school students is still in a developing phase, which can be inexact or situated and organized or unorganized, their experiences in public communication provide them with structured opportunities that enable them to negotiate and bridge conceptual and representational gaps between personal and formal or institutional versions. Thus, this purposeful public communication in mathematical activity supports the emergence and full development of exact and valid forms of representations. Do not be surprised if the initial forms of public communication among middle school students appear to be predominantly verbal, gestural, pictorial, or combinations of at least two such forms. Also, depending on grade level, some of them will need manipulatives and other concrete tools to help them communicate. Consider the first task in Content Activity 4, which involves determining the total cost of purchasing 35 watches that sell for $175 each. Some students who are proficient in English (or in an official language) are already capable of explaining their thinking in a mathematical way. However, there are students such as English learners and those who are not ready to share their personal thoughts in public who 29

Chapter 2

will employ gestures or draw pictures as their preferred ways of communication. Verbal, gestural, and pictorial means of public communication should be supported, of course. However, they are good tools to think but only when students are able to extract from public discourse the analytic conditions of representations across specific instances, examples, or tasks. Knowing such conditions is, in fact, the essence of academic language acquisition in a public forum. No less. Writing solutions is the most significant form of public communication in mathematics. Among middle school students, it involves drawing pictures and diagrams, setting up number expressions and number sentences (i.e., equations and inequalities), graphing mathematical relationships, constructing tables of values, etc. Unlike verbal and gesture-driven solutions as forms of communication, however, having them write their solutions involves asking them to externalize in paper their understanding of “all aspects relevant to language use, from vocabulary and syntax to the development and organization of ideas” (NGSS/CCSS, 2011, p. 19). Further, “a key purpose of writing is to communicate clearly to an external, sometimes unfamiliar audience” and “begin to adapt the form and content of their writing to accomplish a particular task and purpose” (ibid, p. 18). While these perspectives on writing have been taken from the appropriate sections of the Common Core State Standards for English Language Arts and Literacy in History/Social Studies, Science, and Technical Subjects, they also apply to writing solutions in school mathematics. When you ask middle school students to write solutions to mathematical problems, your most significant task is to help them coordinate between the following actions below, which they need to clearly manifest in order to develop a good writing structure. • Processing action, which involves drawing pictures and diagrams that show how they perceive and conceptualize relationships in a given problem. • Translating action, which involves converting the relevant processing action in correct mathematical form. Asking them to draw pictures as an initial step can help them make the transition to the formal phase of writing solutions. Further, drawn pictures can evolve into (schematic) diagrams over time with increased conceptual competence. Pictures are iconic forms of representation, meaning they remain faithful to the depicted objects. For example, a word problem that involves apples and oranges in a ratio problem might yield a written solution with drawn apples and oranges. Some drawn pictures might also contain features that are irrelevant to a problem. However, with appropriate scaffolding, middle school students can be taught to transform those pictures into diagrams. Diagrams are either indexical or symbolic forms of representation. Indexical forms are figures that emerge out of association, while symbolic forms are figures that emerge from shared rules of discourse. For example, the square root form shown in section 2.6.1 item (c) involving a radical sign is not symbolic but indexical since the form reminds students to perform a square root process and nothing else. Variables such as X, Y, and Z are symbols that the mathematics community use for 30

the Common Core State Standards for Mathematical Practice

certain purposes. Indexical and symbolic diagrams can be mere skeletal shapes of figures or placeholder figures that do not need to resemble the original source. Middle school students are naturally drawn to process mathematical problems due to the interesting contexts in which they arise or are framed. However, many of them often find it difficult to convert their processing actions in correct and acceptable mathematical form. Explore the following activity below with a pair. Pay attention to potential issues in processing and converting representations in a problem-solving activity for fifth-grade students. 2.8.1 Content Activity 6: Solving Multiplication Problems with Tape Diagrams By the end of fifth grade, students are expected to be able to solve real world problems involving multiplication of fractions either by using visual fraction models or equations (5.NF.6). Visual fraction models are instances of tape diagrams, which the CCSSM describes as drawings “that look like segments of tape, used to illustrate number relationships.” Tape diagrams are otherwise known as “strip diagrams, bar models, fraction strips, or length models.” Consider the following problem below. Two sisters sold caramel apples in a school fair. Maria’s earnings were of Nini’s earnings. Together, they earned $520. How much did Nini earn? a. Figure 2.6 provides a beginning tape diagram processing of the problem. How does the diagram help you solve the problem?

Figure 2.6. A Tape Diagram Solution

b. Convert the processing actions you performed in (a) in mathematical form. That is, write a complete mathematical solution to the given problem. c. Process the following problem below using tape diagrams. Then provide a complete mathematical solution. Three fifth-grade students, Mark, June, and Rhianna, compared their weekly savings in the following manner: Mark:    The value of my weekly savings is Rhianna:  In my case my total value is only

of June’s weekly savings. of your weekly savings.

If June’s weekly savings is $30, how much does Rhianna save in a week?

31

Chapter 2

2.9  Doing Mathematics with an Eye on the Content-Practice Standards of the CCSSM

Hopefully the preceding sections have given you sufficient and meaningful insights and experiences that encouraged you to reflect on how you might begin to structure content teaching and learning with the practice standards in mind. You will learn more about the relationships between content and practice standards in the next eight chapters that focus on the core domains in the CCSSM middle school mathematics curriculum. Domains consist of clusters – organized groups – of related standards, where each standard defines in explicit terms what every middle school student needs to understand and be able to do. The eleven middle school mathematics domains up to Algebra 1 in the CCSSM are as follows: • • • • • • • • • • •

Operations and algebraic thinking in Grade 5; Numbers and operations in base 10 in Grade 5; Numbers and operations involving fractions in Grade 5; Ratios and proportional relationships from Grade 6 to Grade 7; Number system from Grade 6 to Grade 8 and Algebra 1; Quantities in Algebra 1; Expressions and equations (and inequalities) from Grade 5 to Grade 8 and Algebra 1; Functions in Grade 8 and Algebra 1; Geometry from Grade 5 to Grade 8; Measurement and data in Grade 5; Statistics and probability from Grade 6 to Grade 8 and Algebra 1

An example of a cluster of three standards under the CCSSM Grade 5 Operations and Algebraic Thinking domain is shown in Figure 2.7. Write and interpret numerical expressions.

Figure 2.7. CCSSM Format

32

the Common Core State Standards for Mathematical Practice

Note that Figure 2.1 is an interpretive visual model of the CCSSM practice standards. While the CCSSM considers all practices as essential in supporting growth in mathematical understanding, the perspective that is assumed in this book embraces a stronger position. The solid triangle in Figure 2.1 conveys three core mathematical practices that all middle school students need to demonstrate each time they learn mathematical content. Having a positive disposition toward mathematical problems and problem solving is central. Knowing which tools to use, formal or informal, and when to use them further support and strengthen such disposition. Also, given the symbolic nature of school mathematical content, which is both formal and exact, attending to precision at all times introduces an algebraic disposition toward tool use in which case emerging and meaningful symbols convey expressions that are certain and necessary in the long term. Any of the remaining practice standards can then be coupled with the solid triangle depending on the goals of mathematical activity. Work with a pair and accomplish the following two activities below. First, access the following link which describes in detail the eight practice standards: http://www.corestandards.org/Math/Practice. For each practice standard, develop a checklist of specific actions that is appropriate for middle school students. Some statements under each practice standard may not be appropriate for middle school students, so consult with each other and decide which ones matter. You may refer to Table 2.4 for a sample of a beginning checklist structure for Practice Standard 1. Second, in this chapter you explored six content activities. Use the checklist in Table 2.5 to determine practice standards that you can use to help your students learn each content activity. When you teach with a content-practice perspective, you (and your colleagues) decide which practice standards to emphasize over others, but you should also see to it that such choices are aligned with your teaching, learning, and assessment plans. Table 2.4. Sample of a Beginning Checklist of Actions for Practice Standard Practice Standard 1: Make sense of problems and persevere in solving them. Provide middle school students with every opportunity to – 1.1

start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

1.2

analyze givens, constraints, relationships, and goals.

1.3

 

1.4

 

1.5

 

etc.

 

33

34    

   

   

Different Types of Problems in Middle School Mathematics

From Multiplication of Whole Numbers to Direct Proportions to Simple Linear Relationships

The Pythagorean Theorem and Irrational Numbers

Solving Multiplication Problems with Tape Diagrams

 

 

 

 

   

PS 2

Exploring and Proving Parity

PS 1

Building a Hexagon Flower Garden Design

Content Activity

 

 

 

 

 

 

PS 3

 

 

 

 

 

 

PS 4

 

 

 

 

 

 

PS 5

 

 

 

 

 

 

PS 6

 

 

 

 

 

 

PS 7

 

 

 

 

 

 

PS 8

Table 2.5. Practice Standards Checklist for the Six Content Activities in this Chapter (“PS” means “Practice Standard”)

Chapter 2

Chapter 3

The Real Number System (Part I) from Grade 5 to Grade 6 Whole Numbers, Decimal Numbers, and Fractions

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the following CCSSM number-related domains: Numbers and Operations in Base Ten (NBT) in Grade 5; Numbers and Operations – Fractions (NF) in Grade 5; and portions of Number System (NS) in Grade 6. In Chapter 4, you will reconceptualize them in terms of their relationships with other structures of numbers (i.e., integers, rational numbers, irrational numbers, and real numbers). Table 3.1 lists the content standards that are covered in this and the next chapter for your convenience. Table 3.1.  The Real Number System Domain in the CCSSM Grade Level Domain

Standards

Page Numbers in the CCSSM

Covered in This Chapter

5.NBT

1 to 7

35

Yes

Covered in Chapter 4  

5.NF

1 to 7

36-37

Yes

6.NS

1 to 8, except 4

42-43

Yes, 1 – 3

7.NS

1 to 3

48-49

8.NS

1 and 2

54

Yes

N-RN (Algebra 1 Traditional Pathway)

1 to 3

60

Yes

 

Yes, 5 – 8 Yes

Work in groups of 4 and accomplish the following tasks below. This beginning activity is meant to help you obtain initial impressions of the expected content progression in students’ conceptions of numbers and number systems from kindergarten to grade 6. Each remaining section in this chapter addresses particular number-related conceptions in some detail. a. Access the link below, which should take you to the CCSSM. http://www.corestandards.org/Math

35

Chapter 3

The headings in Table 3.2 refer to the domains of content standards that deal with number concepts and operations from kindergarten to Grade 6. b. Read each domain and synthesize the stipulated content knowledge that students need to acquire by grade level. For instance, the kindergarten domains K.CC, K.OA, and K.NBT expect young children’s conceptions of whole numbers to include mathematical proficiency in counting by tens and by ones from 1 to 100 and composing and/or decomposing up to 19. In terms of number operations, they are expected to add and subtract whole numbers to 10. c. Consider the situation in which parents of middle school students would like to learn how their children’s conceptions of numbers and operations are expected to progress from elementary to middle school. Provide an appropriate response. 3.1  Place Value Structure of Whole Numbers and Decimal Numbers in Grade 5

Given a set of objects, students can count the total number of objects in the set (i.e. cardinality) in different ways. Elementary students initially learn about whole numbers and especially number systems in base 10 early in kindergarten when they begin to count to 100 by ones and by tens (K.CC.1). Grouping or composing objects together into tens and ones also takes place in kindergarten. Certainly, there are other ways of grouping and counting objects. Access the link below which illustrates an Oksapmin child demonstrating his culture’s base-27 counting system. Counting systems shape quantity-related systems such as currency values, trading, and measurement which also tend to influence the conduct of everyday transactions. http://www.culturecognition.com/ Work with a pair and do the following tasks below. a. Gather a handful of blocks and demonstrate how elementary students are expected to count in base 10. When middle school students consistently group by tens, hundreds, thousands, ten thousands, and so on up to millions, which convey grouping by powers of 10, they learn to count in the context of the decimal number system. In this particular numeration structure, they also learn to distinguish between digit, place value, and value. Digit corresponds to the number of groups of a certain power of 10, place value refers to a stipulated power-of-10 grouping, and value is the product of a certain digit and its place value. For example, the whole number 3256 has four digits (i.e., 3, 2, 5, and 6), has place values of thousands, hundreds, tens, and ones, the place value of the digit “3” is 1000 and its value is 3000. There are important long-term consequences when students understand the decimal number system really well. The practical ones draw on applications in 36

           

Fraction Concept

Fraction Addition

Fraction Subtraction

Fraction Multiplication

Fraction Division

Subtract to 10

Whole Number Subtraction

Whole Number Division

Add to 10

Whole Number Addition

 

Count by 10s & 1s to 100; de/compose to 19

Whole Number Concept

Whole Number Multiplication

K.CC, K.OA, & K.NBT

Type of Number Concept and/or Operations

 

 

 

 

 

 

 

 

 

 

1.OA, 1.NBT, & 1.G

 

 

 

 

 

 

 

 

 

 

2.OA, 2.NBT, & 2.G

 

 

 

 

 

 

 

 

 

 

3.OA, 3.NBT, 3.NF, & 3.G

 

 

 

 

 

 

 

 

 

 

4.OA, 4.NBT, & 4.NF

 

 

 

 

 

 

 

 

 

 

5.OA, 5.NBT, & 5.NF

Table 3.2.  Number Concepts Progression from Kindergarten to Grade 6

 

 

 

 

 

 

 

 

 

 

6.NS.1 to 6.NS.3

Dear Preservice Middle Level Majors

37

Chapter 3

everyday life such as having the ability to engage in financial transactions that involve currency exchanges. In an academic context, five consequences are worth nothing. The first four points below pertain to multiplicative thinking and the last point addresses a generalization issue. • Considering the manner in which the decimal number system emphasizes unit relationships, it provides middle school students with an opportunity to understand the implications of implementing a consistent multiplicative structure, including the need to pay attention to units. With whole numbers, they learn to perceive that 1 group of 10 is equivalent to 10 ones, 1 group of 100 is equivalent to 10 tens, 1 group of 1000 is equivalent to 10 hundreds, etc. Thus, working with the decimal system enables them to describe relationships among quantities, where one quantity can be used to describe another quantity. • While students in the elementary grades use measurement thinking in a variety of multiplicative situations, having a solid mathematical understanding of the decimal number system should help them in high school and in the workplace when they have to deal with problems that involve quantities such as those used in the physical sciences (e.g., acceleration), modeling (e.g., batting averages), social sciences (e.g., rates and per capita income), and everyday applications of quantification (e.g., currency conversion). In fact, number and quantity form a major conceptual category in the high school CCSSM. • Students’ awareness about the need to pay attention to units in a multiplicative context, which stems from their understanding of the decimal number system, prepares them well for proportional thinking, which is a major domain in both the sixth and seventh grade CCSSM. For example, the following reasoning exemplifies equivalent multiplicative and proportional thinking in a simple number task: Since there are 10 ones in 1 ten, 4 tens equal 40 ones. Or, 10 ones to 1 ten = m ones to 4 tens, that is,

. Hence, multiplication problems

(involving decimal conversion) basically convey proportional relationships. • A consistent awareness about the need to pay attention to units in the decimal number system prepares middle school students to distinguish between similar/ like and dissimilar/unlike terms and combine similar/like terms, skills that are useful and necessary in high school Algebra 1 when they learn about the structure of algebraic expressions and operations. • The increasing power-of- 10 structure of whole numbers is extended to include the decreasing power-of-

structure of decimal fractions (5.NBT.3a). Fifth-

grade students learn to read and write decimal fractions to the thousandths place using base-10 numerals, number names, and the expanded form. For example, the decimal fraction 13.568, which should be read as “13 and 568 thousandths” and not simply as “13 point 5 6 8” in order to emphasize place value and fractional relationships, has the following equivalent (or expanded) form in base 10: 38

Dear Preservice Middle Level Majors

In fifth grade, they write the expanded form of whole numbers with powers of 10 using whole number exponents. Since negative integers are introduced in sixth grade, they can rewrite the expanded form of all decimal fractions using negative and whole-number exponents. For example, the above expanded form in sixth grade can be rewritten as , which illustrates the power-of-10 structure of real numbers involving positive, zero, and negative integer exponents. This phase of rewriting is still optional in sixth grade since they deal with negative exponents formally in eighth grade. Continue working with your pair and do the following task below. b. Read content standard 5.NBT.3a. Generate a graphic organizer that is similar to the ones shown in Activity Sheets 2.2 and 2.3. See to it that the equivalent forms of representation for decimal fractions are accounted for in your organizer, including standard and visual forms. Provide an example to test your organizer. c. Read content standard 5.NBT.1. What does it say about how fifth-grade students should perceive relationships between two “neighboring” or adjacent digits? Extend the idea to nonadjacent pairs of digits. For each number below, express in precise terms the multiplicative relationship between the digits 2 and 8. (i) 3218.723  (ii) 174.872  (iii) 24.78  (iv) 235784 d. Gather a handful of blocks and demonstrate counting in base 2. What are you digits? What are your digits if you count in base: (i) 3? (ii) 4? (iii) 5? (iv) 9? Generalize to any counting number n. Gather 51 blocks. Express the number in expanded form. What are the standard and expanded forms of 51 in base 4? What are the standard and expanded forms of 13542 in base 27? Consistent with their elementary experiences with whole numbers, middle school students are also expected to compare and order two or more decimal numbers up to the thousandths place (5.NBT.3b). Comparing and ordering decimal numbers deepen their understanding of the place value structure of numbers in the NBT domain. Furthermore, having them order decimal numbers either together with or without whole numbers reinforces useful mathematical terms such as ascending, descending, increasing, and decreasing. Continue working with your pair and do the following task below. e. Assume that fifth-grade students have a firm mathematical understanding of visual representations for whole numbers and decimal numbers. Develop a visual 39

Chapter 3

strategy for comparing two or more decimal numbers. What rules could possibly emerge from the visual activity? f. Some fifth-grade students rely heavily on drawing diagrams when they compare two or more numbers. The visual approach aids them in their processing and should not be discouraged. Your job involves using their their visual experiences to help them transition to rules in the form of algorithms (i.e., stable step-by-step procedures). How might you assist them in this translation process? Also, how might they use their algorithms to order numbers? Search online resources that could help address these issues. Students begin to round whole numbers in third grade (3.NBT.1). In fifth grade, they extend rounding activity to include decimal numbers (5.NBT.4). Access the following web resources below to learn different rounding approaches. Continue working with your pair to accomplish the tasks that follow. • Number line approach, http://www.youtube.com/watch?v=UP7YmXJc7Ik • Numerical approaches, http://www.mathsisfun.com/rounding-numbers.html • Practical approaches, http://www.mathcats.com/grownupcats/ideabankrounding.html g. Develop a poster for rounding decimal numbers in fifth grade. h. Identify situations in everyday life that support (i) rounding up and (ii) rounding down. Why should middle school students need to learn to be proficient in rounding numbers? i. When the digit to be rounded is a 5, “the” rule tells us to round up. Explore how this rule is justified and what it means in terms of the types of justification and reasoning that are accepted in mathematics. 3.2  Standard Algorithms for Whole Number Operations in Grades 5 and 6

Prior to fifth grade, students in fourth grade are expected to be fluent in adding and subtracting multidigit whole numbers using the standard algorithm (4.NBT.4). In fifth and sixth grade, their fluency in applying the standard algorithm is extended to multiplication (5.NBT.5) and division (6.NS.2). Hence, by the end of sixth grade, they need to be fluent with all four operations involving whole numbers, which should help them deal with operations involving integers in seventh grade (7.NS.1 to 7.NS.3). 3.2.1 A Brief Overview of the Addition and Subtraction Standard Algorithms in the Elementary Grades up to Grade 4 A standard algorithm is a numerical or symbolic contraction of a long version that consists of steps in a sequence of instructions or rules. The standard algorithms

40

Dear Preservice Middle Level Majors

for adding and subtracting whole numbers always proceed from the right, that is, beginning at the ones place. Figures 3.1 and 3.2 illustrate one consistent approach at the elementary level called Fuson’s math drawing model that links visual processing and symbolic conversion in a systematic manner. In Figure 3.1, visual processing involves: (1) initially representing each addend in terms of squares, sticks, and circles; (2) regrouping or recomposing from right to left as needed; and (3) finally recording the sum. In Figure 3.2, visual processing involves: (1) initially representing the minuend, the quantity from which another quantity called the subtrahend is to be subtracted, in terms of squares, sticks, and circles; (2) taking away the subtrahend one digit at a time from right to left with or without the need to ungroup and regroup; and (3) finally recording the difference. Certainly, students may choose to add and subtract whole numbers mentally and in nonstandard ways, which should also be encouraged and supported. Well-articulated and justified standard algorithms represent sophisticated, efficient, general, and formal methods that can deal with all possible cases in a convenient manner. In some cases, they emerge in the classroom through negotiation and agreement. In some other cases, they are institutionally supported processes. Regardless of the context of emergence, however, they need to be validated and grounded conceptually, as illustrated by the visual processing examples shown in Figures 3.1 and 3.2.

Figure 3.1. Adding Whole Numbers with Regrouping

41

Chapter 3

Figure 3.2. Subtracting Whole Numbers with Regrouping

3.2.2  Multiplication Standard Algorithm in Grade 5 In the elementary grades, students’ mathematical understanding of multiplication of two whole numbers involves repeated addition. That is, for whole numbers a and b,

Figure 3.3 illustrates how visual processing and numerical translating can be coordinated to facilitate fourth-grade students’ understanding of multiplication involving a 4-digit number by a single-digit number (4.NBT.4 and 4.NBT.5). Observe how the right-to-left numerical approach, which illustrates the standard algorithm for multiplication, makes sense from the point of view of convenience. That is, it is reasonable to start regrouping in base 10 from the right, where 10 ones become 1 ten, 10 tens becomes 1 hundred, 10 hundreds become 1 thousand, etc. Furthermore, notice how the product conveys a numerical recording of what remains under each column once all required grouping actions are completed. In the case of numerical-based left-to-right multiplication, students need to first calculate sums of partial products. For instance, the area model (also called array or grid model) in Figure 3.4 illustrates a visual structure for multiplying a multidigit 42

Dear Preservice Middle Level Majors

On Saturday, 1355 people visited the zoo. Three times as many people visited on Sunday than on Saturday. How many people visited the zoo on Sunday?

Figure 3.3. Multiplying 3 × 1355 in Fourth Grade

number by a single digit number. The numbers in the boxes, which correspond to the numbers below the bar in the numerical version, convey three partial products that when added together yield a sum of 702, the product of 234 and 3. The model shown in Figure 3.4 can be extended to cases involving two multidigit factors. In Figure 3.5, fifth-grade students calculate nine partial products before arriving at the final answer. The expected closure involves fluency in multiplying all cases of multidigit whole numbers using the standard algorithm, which proceeds from right to left (5.NBT.5). Work with a pair to accomplish the following tasks below. 43

Chapter 3

Figure 3.4. Area Model for Multiplication

Figure 3.5. Area Model for Multiplying Two Multidigit-Number Factors

a. The visual representation in Figure 3.4 shows 3 × 234. However, the numerical representation shows 234 × 3 in a vertical format. Why is that an acceptable practice? b. The visual representations in Figures 3.4 and 3.5 model the distributive property of multiplication over addition. Explain how so. c. Vina, a fourth-grade student makes the following claim about multiplying multidigit numbers by a single-digit number: “I think that we should simply multiply from the left in situations that do not involve regrouping.” Is she correct? Illustrate.  heska, Vina’s classmate, also makes a bold claim, as follows: “It really doesn’t C matter where you start to multiply as long as there’s no regrouping in any place.” Is she also correct? Illustrate. d. Many textbooks offer the lattice method to help fifth-grade students conveniently multiply two multidigit whole numbers. Access the following link to learn how and why it works: http://youtu.be/Yt2atjULffY.

44

Dear Preservice Middle Level Majors

Use the lattice method to solve the following multiplication compare problem below. Draw a bar diagram first to help you set up a correct mathematical relationship. A rubber band is 234 cm long. How long will the rubber band be when it is stretched to be 125 times as long? Round the numbers appropriately to help you perform the multiplication process quickly. How reasonable is your estimate in comparison with the actual value? e. Establish a pattern and state a generalization. 2 × 1 = 2       3 × 2 = 6       15 × 2 = 30 2 × 10 = 20       3 × 20 = 60      15 × 20 = 300 2 × 100 = 200      3 × 200 = 600     15 × 200 = 3000 2 × 1000 = 2000     3 × 2000 = 6000    15 × 2000 = 30000 2 × 10,000 = 20,000    3 × 20,000 = 60,000   15 × 20,000 = 300,000 20 × 10 = 200      30 × 200 = 6000    150 × 2000 = 30,000 Obtain the products below by drawing on your inferred generalization. 1. 3400 × 2 2. 4000 × 700 3. 213000 × 200 What is the significance of this patterning task on fifth-grade students’ understanding of multiplying whole numbers by powers of 10 (5.NBT.2)? f. Develop a reasonable multiplication template that will help fifth-grade students establish a relationship between the visual and numerical methods for multiplying whole numbers. g. Search the internet for free online apps and games involving multiplication of multidigit whole numbers and assess potential benefits and possible concerns. 3.2.3 Division Standard Algorithm in Grade 6 By the end of third grade, students learn that every multiplication fact is related to two division facts. For example, the multiplication fact 3 × 4 = 12 is related to 12 ÷ 4 = 3 (a measuring problem) and 12 ÷ 3 = 4 (a partitioning problem). Hence, when they solve a division problem, they know that the problem can be converted in multiplicative form, which partly justifies why they need to learn to commit the multiplication table to memory in the elementary grades because of its relative usefulness in processing division problems. In third grade, they learn that a quotient c relative to a division situation in which a certain whole number a is divided by a counting number b, that is, a ÷ b = c, corresponds to the unknown or missing factor that when multiplied to b, the known factor, yields a, the known product (3.OA.6). 45

Chapter 3

In the elementary grades, students also learn to conceptualize division from a visual perspective. Figure 3.6 reinterprets the division problem 40 ÷ 5 in terms of finding the missing dimension when both the area and one dimension of a rectangle are given. In fourth grade, students continue to divide up to four-digit whole numbers by one-digit counting numbers with and with no remainders (4.NBT.6). When a certain division yields no remainder, it means to say that a dividend is divisible by (or is a multiple of) a divisor (or factor). The situation with division becomes complicated when remainders are involved. Relying on their expertise with the multiplication table and divisibility rules in fourth grade (4.OA.4), they learn to quickly see that the division problem 41 ÷ 5 involves a remainder of 1 when 41 is divided by 5, that is, 41 is not divisible by 5. They also learn to write their answer in the correct format. They may use the division algorithm form, as illustrated by the numbr sentence below, 41 = 5 × 8 + 1= 5 • 8 + 1, or, they may use the alternative form involving fractions, as follows: or

+

.

The informal expression 8r1 is not a number even though everybody understands what it conveys. Practice Standard 6 requires students to attend to precision in the use of mathematical symbols, which explains the reference to the correct way of writing division sentences. Furthermore, fifth-grade students already understand fractions and their notation, including improper fractions or mixed numbers, meaning to say that they should be capable of writing their answers in the correct format.

Figure 3.6. Division Problem 40 ÷ 5 in Terms of Area-Dimension Relationship

46

Dear Preservice Middle Level Majors

Dividing a whole number by a counting number without regrouping and with no remainder can be performed from either left to right or right to left. Figure 3.7 illustrates how the visual and numerical methods can be coordinated in fourth grade to help students understand the relevant notations for division. In the example, dividing 336 by 3 can be interpreted as a sequence of three divisions. Proceeding from the left, 3 friends receive 1 square, 1 stick, and 2 circles each. Of course, values matter (i.e., 300 ÷ 3 = 100; 30 ÷ 3 = 10; 6 ÷ 3 = 2). Also, the context (i.e., whether the division problem is partitioning or measuring) influences the manner in which grouping action is performed visually. In fifth grade, students can draw on the same coordinated process when they deal with 4-digit dividends and two-digit divisors (5.NBT.6). Some cases of division with regrouping and no remainders are rather complicated to accomplish with right-to-left processing. Figure 3.8 shows a simple division with regrouping problem that has been recorded in a modified long division format. Figure 3.9 shows a complex division with regrouping problem that has been recorded in a vertical format and a textbook-based long division format. Fifth-grade students may still need to draw on the corresponding visual processing model to remind them about the value of each digit in the dividend. In Figure 3.8, the numbers in superscript form are the remainders. So, assuming a partitioning context and beginning from the left: 3 thousands equal 3 groups

Figure 3.7. Models for Dividing 336 by 3 in a Partitioning Context

Figure 3.8. Division Involving Simple Regrouping Using a Modified Long Division Format

47

Chapter 3

Figure 3.9. Division Involving Simple Regrouping Using Vertical and Long Division Formats

of 1 thousand with no remainder; 4 hundreds equal 3 groups of 1 hundred with a remainder of 1 ten; 16 tens (regrouped) equal 3 groups of 5 tens with a remainder of 1 ten; and 18 ones (regrouped) equal 3 groups of 6 ones with no remainder. Assuming a partitioning context and beginning from the right: 8 ones equal 3 groups of 2 ones with a remainder of 2 ones; 6 tens equal 3 groups of 2 tens and no remainder; 4 hundreds equal 3 groups of 1 hundred and a remainder of 1 hundred; and 3 thousands equal 3 groups of 1 thousand and no remainder. Since there are still remainders (at the superscript level) to deal with, perform another division from right to left, as follows: 2 ones cannot be divided into 3 equal groups; the remainder of 10 tens equals 3 groups of 3 tens and a remainder of 1 ten. Performing a third and final round of division from right to left, 12 ones equal 3 groups of 4 ones and no remainder, which completes the right-to-left division processing. Note the changes that need to be made with the affected partial quotients. In Figure 3.9, dividing from the left is easier to accomplish than dividing from the right. Assuming a partitioning context and beginning from the left: regroup the first two digits in the dividend to form 10 hundreds, which equal 3 groups of 3 hundred with a remainder of 1 hundred; regroup to form 10 tens, which equal 3 groups of 3 tens with a remainder of 1 ten; and regroup to form 15 ones, which equal 3 groups of 5 ones with no remainder. Assuming a partitioning context and beginning from the right: 5 ones equal 3 groups of 1 one with a remainder of 2 ones; regroup all the way to the thousands place by converting 1 thousand into 10 hundreds; regroup by decomposing 10 hundreds into 9 hundreds and 1 ten; regroup by converting 1 ten into 10 tens, which equal 3 groups of 3 tens with a remainder of 1 ten; 9 hundreds equal 3 groups of 3 hundred with no remainder; and the remaining 12 ones equal 3 groups of 4 ones with no remainder. 48

Dear Preservice Middle Level Majors

Work with a pair and accomplish the following three tasks below. a. Make up a partitioning word problem involving a four-digit dividend and a twodigit divisor (5.NBT.6). Use both vertical and long division methods to solve the problem. Make up a measuring word problem involving a three-digit dividend and a twodigit divisor. Use the definition of division as repeated subtraction to solve the problem. b. Carefully describe a general method for dividing multidigit whole numbers fluently using the standard algorithm (6.NS.2). c. Search online resources for alternative methods or strategies involving division of multidigit whole numbers and explain why they work. d. Develop reasonable division templates that will help fifth- and sixth-grade students establish a relationship between the visual and numerical methods for dividing whole numbers. e. Some concerned stakeholders claim that asking middle school students to learn the long division method is a “waste of time” since calculators can do the job quickly and accurately. What do you think? f. Some middle school students think that doing division problems that involve multidigit divisors (6.NS.2) is nothing else but “busy work.” How do you deal with this issue? g. Search the internet for relevant online apps and games on division for fith- and sixth-grade students. Then assess potential benefits and possible concerns. 3.3  Operations with Fractions

In this section you will deal with all four arithmetical operations involving fractions. Pay special attention on the structural relationships between whole numbers and fractions, including similarities and differences in the way the different operations are conceptualized. 3.3.1  Fractions in the Elementary Grades up to Grade 4 Students’ initial exposure to fractions in the CCSSM occurs in first grade through concrete activities that ask them to partition whole pieces into equal halves and fourths. In first grade they also describe each region and groups of equal pieces in words. In second grade equal partitioning action is extended to thirds. In third grade, they convert their verbal expressions of fractions with denominators 2, 3, 4, 6, and 8 in numerical terms. Furthermore, they learn to view proper fractions as multiplicative expressions, that is, corresponds to a copies or multiples of the unit fraction

, where a and b are whole numbers and

. While proper fractions

49

Chapter 3

continue to be represented in terms of equal smaller pieces that are derived from the same whole piece, they also begin to visualize and construct them as points on a unit segment. Similar to their structural experiences with whole numbers, they compare and order proper fractions. Furthermore, they learn about improper fractions and equivalent fractions. Work with a pair and discuss the following question below. a. Two third-grade students talked about their experiences with fractions in the following manner below. Martha:  I think fractions are like whole numbers. We use whole numbers to count objects and to add and subtract. I think fractions are the same way. If I have a group of congruent half pieces, I can count them the same way I count whole numbers: one-half, 2 halves, 3 halves, 4 halves, 5 halves, and so on. Also, 3 halves and 5 halves together equal 8 halves or 4 whole pieces. Jack:  I don’t agree. I think whole numbers are different from fractions. We use whole numbers to count. With fractions, we shade parts. What do you think? In fourth grade students deal with fractions that involve denominators 5, 10, 12, and 100. They also visually establish the important general relationship where

and

,

, n, a, and b are whole numbers. For example, Figure 3.10

illustrates

. They should see that since the same

multiplicative action is applied to all equal shares of the fraction, the value of the fraction remains unchanged. However, it is true that multiplying a fraction by , which equals 1, has the effect of increasing the number of equal shares. As a consequence of knowing the equivalence relation

, fourth-grade

students should be able to visually and numerically compare two fractions with different denominators by creating common denominators relative to the same whole

Figure 3.10. Visual Processing for

50

Dear Preservice Middle Level Majors

(4.NF.2). For example, their earlier visual experiences with unit fractions should easily convince them that

relative to the same unit whole. In numerical

terms, the relationship is also true because

or

.

In practical terms, both requirements of comparing fractions relative to the same unit whole and converting to equivalent fractional forms that have the same denominator should enable them to progressively transition to a numerical process for comparing any pair of fractions. For example, a numerical processing of fraction tasks that have the same numerator (4.NF.2) such as creating common denominators. So,

involves initially

. From the equivalent forms, than on the right. Alternatively,

since there are more sixth pieces on the left since

.

Continue working with your pair and address the following three tasks below. b. Lamar, 4th grader, makes the following claim: “I’m thinking that in the case of 5/2 and 5/3, I really don’t need to use the numerical process. I know that halves will always be bigger in size than thirds relative to the same whole, so 5 of halves should be greater than 5 of thirds.” Does Lamar’s reasoning make sense? Can you generalize his claim to all problems of the same type? Explain. c. Standard 4.NF.2 also recommends having students compare fractions relative to a benchmark fraction such as is greater than

and

. Consider, for example, the task is less than

, then

. Since

. Discuss possible

issues that students might have with this particular approach. d. Marianne, Lamar’s classmate, asked her father to help her try to make sense of comparing fractions. The next day, she shared the following strategy with her classmates: “I know a super quick way of comparing fractions. My father taught me to cross multiply. So,

because 5 × 5, which equals 25, is greater than

51

Chapter 3

2 × 6, which equals 12.” Discuss possible issues with this particular process for comparing fractions. By the end of fourth grade all students should know the following types of fractions

, where a is a whole number,

, and b is restricted to whole numbers

2, 3, 4, 5, 6, 8, 10, 12, and 100: • • • • •

unit fractions if a = 1; proper fractions if a < b; whole numbers if b = 1; 1 if a = b; improper fractions if a ≥ b. Continue working with your pair and address the following issue below.

e. What happens if b = 0? a = b = 0? Fourth-grade students should also understand that fractions of the form multiplicative expressions, as follows:

are

Because of the multiplicative structure of fractions, they need to understand the central role that common units play in such a structure. Having a common unit, in fact, enables them to decompose fractions into sums of fractions in several different ways (4.NF.3b). The graphic organizer for

shown in Figure 3.11 can help them

see connections among the different representations of the same fraction. Continue working with your pair and accomplish the following activity below. f. Name a proper fraction and make a graphic organizer similar to the one shown in Figure 3.11. In the case of an improper fraction that does not simplify to a whole number, having a firm understanding of common unit should help fourth-grade students understand why its equivalent form results into a mixed number that involves the sum of a whole number and a proper fraction. For example, the improper fraction 52

can be

Dear Preservice Middle Level Majors

Figure 3.11. Graphic Organizer Showing Different Representations of a Proper Fraction

decomposed into

, which is equivalent to the mixed number form is a shortcut for +

mixed expression

and does not mean q

. The . Continue

working with your pair and do the following tasks below. g. Three fourth-grade students offered the following responses below when they were asked to describe the visual representation shown in Figure 3.12. Martina: It’s Drew: No, it’s

. .

Roman: My common unit is a whole piece, and I see 2 whole pieces. But the second whole piece tells me it’s a half. So I think it’s

. If I were to

think like Martina, my common unit is a half, so 2 halves and a half gives me three halves,

. If I were to agree with Drew, my common unit

would be a fourth, so 1 fourth, 2 fourths, three fourths,

. 53

Chapter 3

Figure 3.12. What Do You See?

Which student is correct and why? h. Use Roman’s visual processing in wholes to help you convert

into an

improper fraction. Then use Martina’s visual processing in unit fractions as a second solution strategy. Which method makes sense, and why? i. Jake, 4th grader, makes the following claim: “To convert a mixed number into an improper fraction, my mother told me to simply multiply the denominator and the whole number, add the numerator, and copy the denominator. This MAD strategy always works.” How might you process this strategy in class? j. How does skip counting help students convert an improper fraction into its mixed form? Illustrate with an example. 3.3.2  All Cases of Adding and Subtracting Fractions in Grade 5 By the end of fourth grade, students should be able to add and subtract proper and improper fractions involving like denominators (4.NF.3c and 4.NF.3d). In fifth grade they deal with proper and improper fraction addition and subtraction with unlike denominators (5.NF.1 and 5.NF.2). They also establish the following general rule for fraction addition:

±

, where a, b, c, and d are whole

numbers and both b and d are not equal to zero. Work with a pair and do the following task below. k. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Number and Operations – Fractions page. Click on the Grade 5 link. Then open the “Show all” page and click the link “see illustrations” under A.1. Solve all seven tasks completely. 3.3.3  Multiplying Fractions in Grade 5 Since fractions are multiplicative expressions, the transition to the simple case of fraction multiplication is reasonable and occurs in fourth grade (4.NF.4). They can draw on the definition of multiplication of two whole numbers to establish multiplication involving a whole number (as a multiplier) and a fraction. Work with a pair and do the following activity below. l. How can fourth-grade students use their knowledge and understanding of multiplication of two counting numbers to help them process and obtain the products of the following multiplication problems? 54

Dear Preservice Middle Level Majors

(i)

; (ii)

In fifth grade students deal with the remaining two cases of fraction multiplication, that is, when the multiplier is a fraction and when both factors are fractions. Multiplication as repeated addition can still be made to work in cases when the multiplier is a fraction. For example, the expression

as “5 repeated

obviously does not make any sense. However, if you interpret it as “

times”

 group of 5,”

that seems to be reasonable and can in fact be illustrated in a valid visual manner, as follows. In third grade students conceptualize multiplication of two whole numbers in terms of “equal groups or sets of.” Since multiplication as scaling is introduced in fifth grade (5.NF.5), scaling can be extended to include scalar multipliers. Consequently, there should be no trouble interpreting the meaning of the operation × in terms of “equal groups of.” Figure 3.13 shows a few examples, including “a part of a partition” interpretation as an alternative model for thinking about multiplication involving a fraction multiplier (5.NF.4a). The processing and translation activity in Figure 3.13 should help them understand   of c means: (1)

the following two generalizations about the product obtaining

 groups from c whole units and reconfiguring the groups to generate

a new fraction whose value corresponds to the product

(5.NF.5b); and (2)

forming “a parts of a partition of c into b equal parts” (5.NF.4a). In the case of the product

    

, the repeated-addition conception of

multiplication fails. However, they can still use either the equal group (scaling or resizing) or the part-of-a-partition model to help them develop a reasonable rule. Consider, for example, the task of finding the product of Figure 3.14, if they take 1 part of a partition of obtain

. Alternatively, if they take a half of

. In

into 2 equal parts, then they , they also obtain

. Either way,

. In terms of scaling, Figure 3.15 visually shows how taking number line results in a product that is smaller than

of

on a unit

, the multiplicand (5.NF.5b).

Fifth-grade students can also use an area model as an alternative approach to help them understand multiplication involving two fractions. Consider once again the 55

Chapter 3

Figure 3.13. Motivating

Figure 3.14. Part of a Partition of

into 2 Equal Parts or an Equal Half of

Figure 3.15. Visual Rescaling Example Involving

56

Dear Preservice Middle Level Majors

task of obtaining the product of

. Ask them to draw a unit square (i.e., a

square with unit dimensions) and shade two parts when they partition the square into three equal (horizontal) parts (see Figure 3.16). Next ask them to obtain

of

,

which means taking 1 part when they partition into two equal (vertical) parts. So, 1 2 2 × = or . 2 3 6 A third approach involves tiling (5.NF.4b). Begin with a rectangular tile that has unit fractional dimensions

¥

(refer to Figure 3.16). Use 6 such tiles to

completely cover a unit square. Hence, a rectangle with height

1 2 2 yields an area × of= or 2 3 6

, which illustrates

1 2 2 ¥× = = 2 3 6

and width

.

The preceding three visual models should help fifth-grade students understand a c ac how  × = , where a, b, c, and d are whole numbers and both b and d are not b d bd equal to zero. Work with a pair and do the following tasks below. m. Find the product of

5 2 × in four different ways. 4 5

n. Create five story multiplication problems involving the equation a ¥ b = c with the following conditions below. Then solve them using any method of your choice. 1. Both factors are whole numbers. 2. Both factors are proper fractions.

Figure 3.16. Area Model for

57

Chapter 3

3. a is a proper fraction and b is a mixed number. 4. b is a proper fraction and a is a mixed number. 5. Both factors are mixed numbers. o. Authors Tom Parker and Scott Baldridge suggest that if you try to multiply two fractions in the same way you add two fractions, that is, a c a× c × = , b b b t hen that test rule fails to satisfy at least one arithmetic property of multiplication. Consequently, it is not a valid rule. Which of the following properties of multiplication are not satisfied: commutativity; associativity; distributive property of multiplication over addition; and multiplicative identity? p. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Number and Operations – Fractions page. Click on the Grade 5 link. Then open the “Show all” page and click the link “see illustrations” under B.5. Solve all six tasks by analyzing them from the point of view of scaling or resizing. q. Search the internet for free online apps and games that deal with multiplication of fractions and assess potential benefits and possible concerns. 3.3.4 Dividing Fractions in Grades 5 and 6 Fractions in the elementary grades are conceptualized as numbers that have a multiplicative structure, which should be consistent with their experiences with whole numbers. In fifth grade the concept of fractions is extended to convey an operation, that is, in terms of division of a numerator by a denominator,

= a ÷ b,

where a and b are whole numbers and b  ≠  0 (5.NF.3). Consequently, when they view fractions in terms of division of two numbers, the problems they pursue are extended to tasks that involve equal sharing. For example, consider the following equal sharing problem below. If 5 whole rectangular pizzas are shared equally among 6 people, how much is each person’s share? The problem translates into the following division expression: 5 ÷ 6. Figure 3.17 shows a visual processing of the problem. Intuitively, you can partition each whole rectangular piece into 6 equal shares, a strategy that many students in the lower elementary grades tend to use in the absence of formal instruction. Since each person is expected to receive a sixth share from each whole piece, then the person’s total share is 5 ¥

=

. Hence, 5 ÷ 6 = 5 ¥

verify that this answer is correct by multiplication, that is, 6 ¥ 58

=

. You can

= 5(why?). With

Dear Preservice Middle Level Majors

Figure 3.17. Partitioning Each Whole Piece Into Six Equal Shares

a sufficient number of similar everyday examples, students should be able to verify that, indeed, a ÷ b= a×

1 a = , b b

where a and b are whole numbers and b ≠ 0. It is important to emphasize academic language at this stage. That is, dividing by 6 and multiplying by

convey the same

mathematical action. Work with a pair and do the following task below. r. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Number and Opaerations – Fractions page. Click on the Grade 5 link. Then open the “Show all” page and click the link “see illustrations” under B.3. Solve the task called What is 23 ÷ 5? 1 a s. Knowing that a ÷ b = a × =, extend the values of a and b to include unit b b fractions. Standard 5.NF.7 does not include division of a fraction by a fraction, which students pursue in sixth grade (6.NS.1). Follow the procedure in (r) to access the following tasks below. Solve them completely and carefully attend to both processing and translation aspects of your work. 1. Dividing a unit fraction by a nonzero whole number: Click the link “see illustrations” under 7.a and solve the problem Painting a Room. 2. Dividing a whole number by a unit fraction: Click the link “see illustrations” under 7.b and solve the problem Origami Stars. t. Content standard 6.NS.1 suggests different ways of helping students understand quotients of two fractions. (1) Illustrate the different ways by using two proper fractions. (2) Explain why the following rule for division works in two different ways:

a c ad ÷ = , where a, b, c, and d are whole numbers and both b and d are b d bc

not equal to zero. u. Since all middle school students learn to view fractions as quotients of two whole numbers by the time they complete fifth grade, they also need to understand situations involving division by 0. Use the relationship between multiplication 59

Chapter 3

a and division to explain why: (1) , where a is a nonzero whole number, is 0 0 is indeterminate. undefined; and (2) 0 3.4  Operations with Decimal Numbers in Grades 5 and 6

In this section you will deal with all four arithmetical operations in the context of decimal fractions. Think about the underlying conceptual structures that all fifthand sixth-grade students must have in order to mathematically understand what it means to add, subtract, multiply, and divide decimal numbers. Also, reflect on ways in which the content-practice learning and teaching of whole numbers, fractions, and decimal numbers can be presented to middle school students so that they develop a coherent and unified understanding of numbers and their relationships and operations across type. 3.4.1  Multiplying with Decimal Numbers in Grades 5 and 6 Track a single-digit whole number, say, 2, each time it is multiplied by a power of 10. In Figure 3.18, observe how the number 2 as a digit moves to the left when it is multiplied by increasing powers of 10 and moves to the right when it is multiplied by decreasing powers of 1/10. The digit 2 stays in its current position when it is multiplied by 1 (= 100). Location of “2” 2 × 1 = 2 × 100 = 2 2 stays in the ones place 2 × 101= 20 2 moves to the tens place 2 2 × 10 = 2 × 100 = 200 2 moves to the hundreds place 3 2 × 10 = 2 × 1000 = 2000 2 moves to the thousands place 2×

1 2 = = 0.2 10 10

2 moves to the tenths place

2×

2 1 = = 0.002 100 100

2 moves to the hundredths place

Thinking in terms of money, since 2 dimes have a total value of 0.20¢ and 2 pennies have a total value of 0.02¢, students should see rather easily that 2 × 0.1 = 0.2 and 2 × 0.01 = 0.02. Thinking in terms of patterns with the decimal point, it helps to view the number 2 in its decimal form 2.0 and that multiplying it by 0.1 moves the decimal point in 2.0 one place to the left and multiplying it by 0.01 moves the decimal point in 2.0 two places to the left. The experience is not totally new to students. For instance, they know that 2 $1 bills have a total value of $2 (i.e., 60

Dear Preservice Middle Level Majors

Figure 3.18. Tracking Multiplication of 2 by Powers of 10

no decimal point movement in 2.0), 2 $10 bills yield a value of $20 (i.e., move the decimal point in 2.0 one place to the right or, equivalently, move the number 2 to the left of 0 once), 2 $100 bills equal $200 (i.e., move the decimal point in 2.0 two places to the right), etc. Extending the same patterning action to two decimal numbers, when you obtain the product of, say 0.5 × 0.5, one possible mental strategy is to ignore the decimal points and multiply the corresponding whole numbers first, that is, 5 × 5 = 25, which is the same as 25.0. Then drawing on the findings from the preceding paragraph, move the decimal point in 25.0 twice to the left (why?). So, 0.5 × 0.5 = 0.25. Certainly, this particular mental strategy is conceptually enhanced when fraction multiplication is involved, as follows:

Or,



0.5 × 0.5 =

1 1 1 25 25 × = × = = 0.25. 2 2 4 25 100

0.5 × 0.5 =

5 5 25 × = = 0.25 . 10 10 100

When you teach the preceding patterns in your own class, use “friendly numbers” in the beginning phase to help your students enjoy and appreciate the underlying pattern structures. For example, have them quickly multiply the following decimal numbers below mentally first and then have them verify their results with a calculator. 0.25 × 3   2.25 × 0.1   3.04 x. 02   15.2 x. 04   150.01 x. 05 Work with a pair and do the following four tasks below. a. Use labeled circles to find the product of 3 x 0.28. Then translate the visual process numerically by using a vertical format for multiplying numbers. b. Construct a template with all the necessary components that will help students perform decimal number multiplication in an organized manner. Then use your template to multiply the following decimals fractions: 345.23 × 0.12; 54.32 × 0.7. 61

Chapter 3

c. Multiplying two whole numbers involves repeated addition. How might you conceptualize multiplying two decimal numbers? Discuss situations in everyday life that justify the need to know decimal multiplication. d. Trace the content progression of multiplication involving decimal numbers from fifth to sixth grade in the CCSSM. 3.4.2  Dividing Decimal Numbers in Grades 5 and 6 The division algorithm for decimal numbers should follow the same structure that students developed when they initially learned to divide two whole numbers in a systematic manner. When the divisors in a division problem are decimal numbers, however, students need to transform them into their whole-number forms in order to perform the division process in a convenient manner. From a psychological perspective, the conversion to a whole-number divisor should help fifth-grade students link the idea of dividing two decimal numbers to their earlier experiences of partitioning and measuring in division. For example, to obtain the quotient of 10 ÷ 2.5, moving the decimal point in the divisor one place to the right also requires performing the same action in the dividend. So, 10 ÷ 2.5 = 100 ÷ 25, which equals 4. They can then verify that, indeed, 25 × 4 = 100 and 2.5 × 4 = 10. When their beginning experiences with decimal division problems are reasonable and can be quickly verified, such experiences can help them understand and appreciate the additional step in the division process involving decimal-number divisors. Furthermore, once they are fully proficient in fraction division (5.NF.7b and 6.NS.3), they can always transform the relevant decimal numbers into their equivalent decimal fraction forms and use fraction division to help them obtain the correct quotients, as illustrated below. 25 5 1 1  = 10 ÷ = 10 ÷  5 ×  = 2 ÷ = 4   10 2 2 2 25 10 100 10 ÷ 2.5 = 10 ÷ = 4, = 10 × = 10 25 25

10 ÷ 2.5 = 10 ÷ or



or

10 ÷ 2.5 = 10 ÷

25 100 25 = = = 100 ÷ 25 = 4. 10 10 10

Figure 3.19 shows both vertical and long division formats relative to the division task 2.368 ÷ 0.4. First, move each decimal point one place to the right (why?). Then perform the usual division process. Interpreting the task in a partitioning context, consider a division problem that involves determining the amount that one person is expected to receive when $23.68 is split evenly among four people. First, there is not enough 2 $10 bills that can be shared equally among the four of them. However, 23 62

Dear Preservice Middle Level Majors

Figure 3.19. Dividing  2.368 ÷ 0.4

$1 bills can do the task. After each friend receives 5 $1 bills, three $1 bills remain. Since $1 = 10 dimes, a total of 36 dimes can be shared equally among them with each one receiving exactly 9 dimes. The remaining 8 nickels can also be shared equally among them. Hence, each friend should receive 5 $1 bills, 9 dimes, and 2 nickels or, $5.92. Hence, 2.368 ÷ 0.4 = 23.68 ÷ 4 = 5.92. Continue working with your pair and do as follows: e. Provide a visual representation that goes with the division process shown in Figure 3.19. f. Divide 465.75 by 2.5. Interpret the task in the context of a measurement problem and explain each step in your division process. g. Divide 0.385 by 12.5. What possible issues might occur when fifth-grade students perform the division process? h. Construct a template with all the necessary components that will help students perform decimal number division using the standard algorithm in an organized manner (6.NS.3). Generate two sixth-grade decimal division problems and use your template to help you solve them. i. The rules for multiplication and division of decimal numbers might encourage procedural fluency over conceptual understanding. What kinds of teaching and learning situations might promote such a gap or dichotomy in students’ thinking relative to these content standards, and how might you prevent them from occurring in your own class? j. How does rounding decimal numbers support fifth-grade students’ understanding of multiplication and division of decimal numbers? Provide examples to illustrate your claims and arguments. k. Trace the content progression of division involving decimal numbers from fifth to sixth grade in the CCSSM. l. Search the internet for free online apps and games that pertain to the topics in this section and assess potential benefits and possible concerns. 63

Chapter 3

3.4.3  Adding and Subtracting Decimal Numbers in Grades 5 and 6 The addition and subtraction rules for multidigit whole numbers and decimal numbers are the same. The only issue that you and your students will need to be concerned about is the possible misalignment of the digits when adding or subtracting decimal numbers in a vertical format. For example, when they add 2.54 + 3.2, some of them will rewrite the numbers vertically in the following incorrect manner below.   2. 5 4 + 3.2 ________ One way to prevent that from taking place is to include placeholder zeroes as illustrated in the following situation shown below.   2. 5 4 + 3. 2 0 ________   5. 7 4 Coins are very effective visual representations for teaching addition and subtraction of decimal numbers in fifth grade, which might still be necessary to use in sixth grade. Labeled circles are also equally effective. Since all students learn about the relationship between fractions and decimals in fourth grade, you can reinforce decimal addition and subtraction by manipulating the relevant decimal numbers as fractions. For example, the sum of 2.54 and 3.2 can be shown in two different ways. The first way stays at the level of decimal number representation. The second way involves initially converting the two decimal numbers into decimal fractions. When they perform the indicated process, they should know how to add two fractions with unlike denominators, which they also learn in fifth grade (5.NF.1). So, for example, 2.54 + 3.2 =

254 32 10 254 320 574 + × = + = = 5.74. 100 10 10 100 100 100

Continue working with your pair and do as follows: m. Construct one addition put-together story problem that involves three decimal numbers as addends and regrouping. Use labeled circles to demonstrate the visual addition process. Also, record your process in numerical form. n. Develop one subtraction take-apart story problem that involves decimal numbers and regrouping. Use labeled circles to demonstrate the visual subtraction process. Also, record your process in numerical form. o. Consider the following problem: “Chuck has $100. He goes to a pet store and spends $63.65. How much money does he have left?” Solve this subtraction problem by employing a counting-on strategy. 64

Dear Preservice Middle Level Majors

p. Construct addition and subtraction templates with all the necessary components that will help students perform decimal number addition and subtraction using the standard algorithms in an organized manner (6.NS.3). Generate two addition and two subtraction problems for sixth-grade students and use your template to help you solve them. q. Find the sum or difference by first converting all the decimal numbers below in fractional form. Then express your final answers in decimal number form. 1. 21.315 + 77.05 + 9.02 2. 1456.125 – 57.25 3.5  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following two tasks below. 1. Use the checklist you developed in Table 2.4 to map each content standard under The Real Number System Part I domain with the appropriate practice standards. Use the structure shown in Table 3.3 to record your responses. Table 3.3. Mapping the Real Number System Part I Domain to the Practice Standards and Proficiency Strands

             

             

             

Productive disposition

             

Adaptive reasoning

             

Strategic competence

             

Procedural fluency

             

Conceptual understanding

             

Look for and express regularity in repeated reasoning

Look for and make use of structure

             

Attend to precision

             

NRC Mathematical Proficiency Strands

Use appropriate tools strategically

             

Model with mathematics

             

Construct viable arguments and critique the reasoning of others

5.NBT.1 5.NBT.2 5. NBT.3 … … … 6.NS.3

Reason abstractly and quantitatively

Make sense of problems and persevere in solving them

CCSSM Practice Standards CCSSM Real Number System Part I Content Standards

             

65

Chapter 3

2. Determine which among the five NRC proficiency strands (see Table 2.3) can be used to learn each content standard. Remember that the strands, like the CCSSM practice standards, should not be interpreted as single stand-alone proficiencies. Since they target different dimensions of mathematical learning, using several strands in mathematical activity will help strengthen students’ mathematically proficiency. 3.6  Developing a Content Standard Progression Table for the Real Number System Part I Domain

Work on your own to develop a reasonable content standard progression table for this particular domain (i.e., a domain table) over a course of nine months. A content standard progression is a gradual development of a content standard from the relatively simple case(s) to the target case(s) over time. For example, the second row in Table 3.4 shows a content progression for the two NBT clusters that all fifthgrade students are expected to learn over three months. A domain table should show the different content standard progressions and their interrelationships over time. You may use the table shown in Table 3.4 for a beginning structure. Remember that progressions and domain tables are negotiated in actual practice (i.e., either at the school or district level). This activity should prepare you for instructional planning, which you will pursue in detail in Chapter 14.

66

X (Introduce) X (Introduce)  

 

X (Addition and subtraction of decimals to hundredths)

5.NBT.3b

5.NBT.4

5.NBT.5

5.NBT.6

5.NBT.7

X (Review)

X (Review of all four operations)

X (Multiplication and division of decimals to hundredths; Review of addition and subtraction)

X (Review)

X (Divide multi-digit whole numbers by 2-digit divisors

X (Multiply multidigit whole numbers using the standard algorithm

X (Review)

X (Review)

X (Review)

X (Review)

X (Introduce) X (Introduce)

5.NBT.2

5.NBT.3a

X (Review)

X (Introduce)

November

5.NBT.1

October

September

CCSM Content Standards

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

December January

 

 

 

 

 

 

 

 

February

 

 

 

 

 

 

 

 

March

Table 3.4. A Content Standard Progression Table for the Real Number System Part I Domain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

April May

 

 

 

 

 

 

 

 

June

Dear Preservice Middle Level Majors

67

Chapter 4

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1 Integers, Rational Numbers, and Irrational Numbers

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Number System (NS) domain of the CCSSM from Grade 6 to Grade 8 and Algebra 1. In particular, this chapter will help you establish connections between whole numbers and fractions in the earlier grades and integers, rational numbers, and irrational numbers in the middle grades. Table 4.1 lists the content standards that are covered in this and the preceding chapter. Table 4.1. The Real Number System Domain in the CCSSM Grade Level Domain 5.NBT 5.NF 6.NS 7.NS 8.NS N-RN (Algebra 1 Traditional Pathway)

Standards

Page Numbers in Covered in This the CCSSM Chapter

1 to 7

35

Yes

1 to 7 1 to 8, except 4 1 to 3 1 and 2

36–37 42–43 48–49 54

Yes Yes, 1 – 3

1 to 3

60

Covered in Chapter 4   Yes, 5 – 8 Yes Yes Yes

There are two fundamental views about real numbers that all middle school students are expected to learn by the end of eighth grade and Algebra 1. First, real numbers consist of two sets of numbers, namely, rational numbers that can be expressed as quotients of two integers a and b, never be expressed in the

, where

, and irrational numbers that can

form. Second, real numbers are conceptualized as

points on a number line, which justifies in part why negative numbers are possible. Furthermore, since negative numbers are opposites of their positive counterparts on a number line, students learn to express distances between two points on the line in terms of absolute value. The CCSSM expects all eighth grade and Algebra 1 students to understand the densely populated context of the real number line, that is, such a 69

Chapter 4

line that is everywhere dense with no holes asserts the completeness of the system of real numbers. Work in groups of 4 and accomplish the following tasks below. The tasks build on the work you started in the previous chapter in relation to Table 3.2. a. Access the link below which should take you to the CCSSM. http://www.corestandards.org/Math  he column headings in Table 4.2 refer to the domains of content standards that deal T with number concepts and operations from kindergarten to Grade 8 and Algebra 1. b. The gray region in Table 4.2 corresponds to Table 3.2, so you may consider that as given information or assumed prior knowledge. Read each remaining content standard from Grades 6 to 8 and Algebra 1 that pertains to integers, rational numbers, and irrational numbers and provide a brief description of what students need to acquire at the indicated grade level. For instance, the content standards 6.NS.5 to 6.NS.8 expect sixth-grade students to understand concepts and representations involving integers, rational numbers, and absolute value of a rational number. In terms of applications, they are expected to solve everyday and mathematical tasks that involve such numbers, graph points in all four quadrants of the rectangular coordinate system, and calculate lengths of horizontal and vertical segments. c. Consider the situation in which parents of middle school students are interested in understanding how their children’s conceptions of numbers and operations are expected to progress from middle school to Algebra 1 in high school. Develop a reasonable response. 4.1  Integers and Integer Operations in Grades 6 and 7

In this section you will learn a variety of representational contexts that will help you teach integers and integer operations in a meaningful way. Reread section 2.5, representations in middle school mathematics, and think about ways that will help you link the knowledge you acquired in that section with the issues raised in the following sections below. 4.1.1 Concepts of Positive and Negative Integers in Grade 6 Integers consist of 0, the counting (or natural) numbers 1, 2, 3, 4, …, and the opposites of counting numbers –1, –2, –3, –4, …. Whole numbers consist of 0 and the counting numbers. The opposite of a counting number a is denoted by the indexical symbol –a. Some textbooks use a hyphen notation in superscript form such as –a to help students remember that there is a difference between a negative-number notation and minus notation for subtraction. The negative sign in the expression –a conveys “the opposite of a,” while the negative sign in the expression b – a refers to the operation of subtraction. Sixth-grade students need to understand the difference 70

 

           

Fraction Multiplication

Fraction Division

Integer and Rational Number (I&RN) Concepts

Integer & Rational Number Operations

Irrational Number Concept

Irrational Number Operations

   

   

Whole Number Division

Fraction Concept

Fraction Addition

 

Whole Number Multiplication

Fraction Subtraction

 

 

Whole Number Subtraction

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.OA, 1.OA, 1.NBT, & 2.NBT, & 2.G 1.G

Whole Number Concept

K.CC, K.OA, & K.NBT

Whole Number Addition

Type of Number Concept and/or Operations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.OA, 3.NBT, 3.NF, & 3.G

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.NS.1 to 5.OA, 4.OA, 4.NBT, & 5.NBT, & 6.NS.3 5.NF 4.NF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6.NS.5 to 6.NS.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.NS

Table 4.2. Number Concepts Progression from Kindergarten to Grade 8 and Algebra 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

8.NS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algebra 1 N-RN

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

71

Chapter 4

between the two uses of the hyphen notation and be ready to employ the correct academic language in situations that involve either negative numbers or subtraction. Numerical Expression –4 7–2 3 – –5

Correct Reading “negative 4” “the opposite of 4” “7 minus 2” “3 minus negative 5” “3 minus the opposite of 5”

Incorrect Reading “minus 4” “7 negative 2” “3 minus minus 5”

Work with a pair and do the following tasks below. a. The simplest way to introduce integers to sixth-grade students is to ask them to construct a number line with negative integers, 0, and positive integers (i.e., counting numbers). Draw a number line that shows negative integers, 0, and positive integers. Illustrate the following expressions on your number line: 5, –5, and –(–5) (6.NS.6c). What does it mean to say that 5 and –5 are opposites of each other on a number line (6.NS.6a)? Find an equivalent expression for –(–m), where m is any integer (6.NS.6a). b. What does 0 mean in each context below (6.NS.5)? 1. temperature above and below 0; 2. elevation above and below sea level; 3. credit/debit; 4. positive/negative electric charge. c. Gather a set of ten unit cubes or circle chips and lay them on the positive region of the mat shown in Figure 4.1. Demonstrate the meaning of –10 on the mat. If you gather an arbitrary set of m unit cubes or circle chips, what does –m mean in concrete terms?

Figure 4.1. Algeblocks-Based Binary Mat

72

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

 se the mat in Figure 4.1 to visually illustrate –4 and –(–4). Obtain an equivalent U expression for –(–m), where m is an integer. Does the equivalent expression hold for all integers? Explain. d. Sixth-grade students initially learn about the coordinate system and graph points in the first quadrant of the plane in fifth grade (5.G.1 and 5.G.2). In sixth grade, they plot points in all four quadrants of the coordinate plane (6.NS.6b and 6.NS.6C). What patterns can they generate from their point-plotting experiences (see 6.NS.6a for an example). What content-related issues are likely to occur? Also, search online resources and activities that can help you teach integer pointplotting effectively. e. Access the link http://www.illustrativemathematics.org/standards/k8. Open The Number System page. Click on the Grade 6 link. Then open the “show all” page and click the link “see illustrations” under C.5. Solve the two tasks completely. f. Plot the following integers on a number line: 8, –4, 4, –8, 7, 0, –3. Present an everyday context that uses all seven integers in increasing order (6.NS.7b). g. Search at least three online resources that talk about the history of negative numbers. How does your knowledge of the history of negative numbers help you teach them and the relevant content standards in class? h. Develop a blog entry for parents and other teachers that describes how you expect sixth grade students to develop their understanding of integers and their applications. 4.1.2 Addition of Integers in Grade 7 Seventh-grade students should be able to draw on their elementary experiences when they add two or more integers. The basic process of adding whole numbers involves combining or putting things together, which also applies in situations involving addition of integers. When students add two or more integers, they need to coordinate two different processing actions, namely: (1) determine the sign of the sum; and (2) obtain the actual sum. Work with a pair and do the following tasks below. i. The expression 3 + 2 is shown in Figure 4.2. Find the sum and its sign (i.e., whether it is positive or negative).

Figure 4.2. Adding Two Positive Integers

73

Chapter 4

Seventh-grade Chuck makes the following claim: “Adding two or more positive

integers is the same way as adding whole numbers.” Is he correct? Explain. If Chuck is correct, establish a general rule for adding two or more positive integers. j. The expression –3 + –2 is shown in Figure 4.3. Find the sum and its sign.

Figure 4.3. Adding Two Negative Integers

 eventh-grade Martha says: “Adding two or more negative integers is the same S way as adding whole numbers, but you need to prefix a negative sign to the result.” Is she correct? Explain. If Martha is correct, establish a general rule for adding two or more negative integers. k. The expression –5 + 3 is shown in Figure 4.4. In the mat you see three zero pairs. A zero pair is a pair of integers with a positive and a negative sign whose sum is 0 (7.NS.1a and 7.NS.1b).

Figure 4.4. Adding a Negative and a Positive Integer

 eventh-grade Myrna makes the following claim: “Adding a positive and a S negative integer is like subtracting whole numbers.” Is she correct? Explain. If Myrna is correct, establish a general rule for finding the sum of a positive integer and a negative integer. 74

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

l. Use the rules you developed in (i) to (k) to accomplish the following tasks: 1. Find the sum: –345 + 76; 45 + –89; –56 + –3; –13 + –45; –12 + 19 + 38 + –65 2. Without calculating, arrange the following expressions below in increasing order. –7 + 2   7 + –2   7 + 2   –7 + –2 What do you want your students to learn from this particular task? m. The opposite of an integer m is –m. If you add m and –m, the sum is 0. Thus, m + –m = –m + m = 0. Another term for the opposite of an integer is additive inverse. Find the additive inverse of: (i) 3; (ii) –4; and (iii) 0. n. An alternative way of teaching integer addition involves using a number line diagram (7.NS.1b). In this case, signed integers are viewed as vector quantities. A vector quantity describes both magnitude (numerical value or scalar) and direction. In the case of positive and negative integers, the positive sign can be interpreted as “moving forward” and the negative sign conveys its opposite, that is, “moving backward.” On a number line, the initial or “default” orientation of a moving object, say, a turtle, is towards the positive side of the number line beginning at 0. Consider the following examples below. 1. 1 + 3: The turtle moves 1 unit forward from 0 and then moves 3 units forward from 1.

Figure 4.4. Demonstrating 1 + 3 = 4 on a Number Line

2. –1 + –3: The turtle moves 1 unit backward from 0 and then moves 3 units backward from –1.

Figure 4.5. Demonstrating –1 + –3 = –4 on a Number Line

75

Chapter 4

3. –1 + 3: The turtle moves 1 unit backward from 0 and then moves 3 units forward from –1.

Figure 4.6. Demonstrating –1 + 3 = 2 on a Number Line

4. 1 + –3: The turtle moves 1 unit forward from 0 and then moves 3 units backward from 1.

Figure 4.7. Demonstrating 1 + –3 = –2 on a Number Line

 ithout actually determining the sum, arrange the following expressions below W in decreasing order. Imagine having a moving turtle on a number line to help you obtain a sense of each sum. –64 + 81   –64 + –72   64 + 81   64 + –81 What can students to learn from this particular task? o. Demonstrate on a number line why the sum of an integer and its additive inverse is 0. p. Access the 5-minute classroom video What’s Your Sign: Integer Addition from the site below. https://www.teachingchannel.org/videos/adding-integers-lesson-idea  ow did Ms. Allison Krasnow, 8th grade teacher, use the number line model to H help her students understand integer addition? What guide questions did she use to highlight essential features of the target content standard? q. Search online resources and activities that will help you teach integer addition in other ways and using other contexts. Discuss potential benefits and issues when you teach them in class. r. Write story problems involving the following expressions: (1) 12 + –7; (2) –12 + 7; and (3) –12 + –7. 76

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

4.1.3 Subtraction of Integers in Grade 7 In the elementary grades students deal with the following contexts of subtraction involving whole numbers: part-whole; take away; and comparison. Take a few minutes to learn about these contexts by analyzing the problems shown in Table 1 (p. 88) of the CCSSM. One simple and easy way to introduce seventh-grade students to subtraction of integers involves using patterns to help them intuitively observe why a – b = a + –b for any integers a and b. Figure 4.8 illustrates how a table of expressions can be used to assist them abduce and induce the idea that subtracting an integer and adding the opposite of the integer are equivalent actions.

Figure 4.8. Using Patterns to Demonstrate a – b = a + –b

When students intuitively understand the relationship a – b = a + –b for any integers a and b, they can, in fact, process all subtraction expressions in terms of their equivalent addition expressions. Continue working with your pair and address the question below. s. Seventh-grade Bert says: “Subtracting two positive or two negative integers is the same way as subtracting whole numbers, but you need to make sure that you prefix the correct sign to obtain the apppropriate result.” Is he correct? Explain. t. Find the difference: –345 – 76; 45 – –89; –56 – –3; –13 – –45; –4 – 8 – –6 u. Arrange the following expressions below in decreasing order. –7 – 2   7 – –2   7 – 2   –7 – –2 v. Fill in each blank below with the correct set of values that will complete the sentence.   (i)  –2 + ____ = a positive integer (ii)  ___ + –8 = a positive integer (iii)  15 + ____ = a negative integer (iv)  ___ + –9 = a negative integer w. Seventh-grade Erica made the following claim: “I think a – b = a + –b makes sense for all integers because if 5 – 2 = 3, then I know that 2 + 3 = 5. So if I add –2 on both sides of 2 + 3 = 5, then 2 + 3 + –2 = 5 + –2. But 2 + 3 + –2 = 2 + –2 + 3 = 0 + 3 = 3 = 5 + –2. So 5 – 2 = 3 = 5 + –2.” Is Erica correct? If she 77

Chapter 4

is correct, provide a justification for each step in her explanation. If Erica is not correct, obtain a counterexample. Seventh-grade students can also use the number line to understand how subtracting two integers a and b involves adding the additive inverse, that is, a – b = a + (–b) (7.NS.1c). Consider a simple example that involves finding the difference between 4 and 2. To obtain the answer, they simply need to either count up 2 units to 4 or count down 2 units from 4. Note that the opposite action of counting down 2 units from 4 is counting up -2 units from 4. Hence, 4 – 2 = 4 + –2 = 2. In Figure 4.9, the turtle begins at 0 and moves 4 units to the right. Then it moves 2 units backward and stays at 2.

Figure 4.9. Demonstrating 4 – 2 = 4 + –2 = 2 on a Number Line

Once again the subtraction expression a – b, which involves taking away b units from a units, is equivalent to the addition expression a + –b, which involves adding –b units from a units. Consider the following additional examples below. i) 1 – 3: The turtle moves 1 unit forward from 0. Since subtracting 3 from 1 is equivalent to counting up –3 from 1, the turtle moves 3 units backward from 1. Hence, 1 – 3 = 1 + –3 = –2.

Figure 4.10. Demonstrating 1 – 3 = –2 on a Number Line

ii) –1 – –3: The turtle moves 1 unit backward from 0. Subtracting –3 from –1 is equivalent to counting up 3 units from –1, so the turtle moves 3 units forward from –1. Hence, –1 – –3 = –1 + 3 = 2.

Figure 4.11. Demonstrating –1 – –3 = 2 on a Number Line

78

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

iii) –1 – 3: The turtle moves 1 unit backward from 0. Since subtracting 3 from –1 and adding –3 and –1 are equivalent actions, the turtle moves 3 units backward from –1. Hence, –1 – 3 = –1 + –3 = –4.

Figure 4.12. Demonstrating –1 – 3 = –4 on a Number Line

iv) 1 – –3: The turtle moves 1 unit forward from 0. Since subtracting –3 from 1 and adding 3 and 1 are equivalent actions, the turtle moves 3 units forward from 1. Hence, 1 – –3 = 1 + 3 = 4.

Figure 4.13. Demonstrating 1 – –3 = 4 on a Number Line

Continue working with your pair and address the following questions below. x. The following “rules” below have also been suggested in relation to the two minus signs that emerge in the context of learning about integer operations. If the minus sign appears as a negative notation, then interpret it to convey the action of “moving backward.” If the minus sign appears as a subtraction operation, then interpret it to convey the action of “turning around.”  heck to see whether the above “rules” apply to the different subtraction C situations shown in Figures 4.9 through 4.13. Discuss potential issues or concerns with such rules. y. Access the following Youtube links below and discuss potential concerns with the suggested approaches for teaching integer subtraction with binary chips and tiles. i) http://www.youtube.com/watch?v=zqt-fxcUPek ii) http://www.youtube.com/watch?v=yApCm8_A364 79

Chapter 4

iii) http://www.youtube.com/watch?v=8DI-Za7ifPo iv) http://www.youtube.com/watch?v=l6MKcyuqUYs z. Locate the following numerical expressions below on a number line. –11 – –2   –11   –11 + 2   –11 – 2 aa. Fill in each blank below with the correct set of values that will complete the sentence.   i)  –8 – ____ = a positive integer ii)  7 – ___ = a positive integer iii)  3 – ____ = a negative integer iv)  –12 – ____ = a negative integer   v)  ___ – 15 = a positive integer vi)  ____ – –3 = a negative integer bb. Verify that the following properties of operations below apply to integers (7.NS.1d). i) Commutative property of addition ii) Associative property of addition iii) Additive inverse property iv) Additive identity property Does a whole number have an additive inverse? Explain. Does a counting or natural number have an additive identity? Explain. Is the set of integers closed under addition? subtraction? Is the set of whole numbers closed under addition? subtraction? cc. Use the properties of additive inverse and commutativity for addition to formally explain why –(–m) = m for any integer m (6.NS.6a)? dd. Seventh-grade Jane made the following claim below: For any integers a and b, the expression a – b, which means subtracting b from a, is the same thing as saying adding –b to a. Subtracting an integer from a minuend is like obtaining the sum of the opposite of the integer and the minuend.  se the missing-addend definition of subtraction and some of the properties of U operations for integers identified in (z) to formally prove Jane’s claim. 4.1.4 Multiplication of Integers in Grade 7 Seventh-grade students may continue to use patterns of related expressions to make sense of the multiplication rules for positive and negative integers (7.NS.2a). Each table in Figure 4.14 addresses a particular multiplication rule. Figures 4.15 and 4.16 illustrate a × –b, where a and b are positive integers, using a binary mat and a number line, respectively. The visual representations shown in Figures 4.17 and 4.18 illustrate the more complicated case of –a × b, which implicitly assume the mathematical relationship –a × b = –(a × –b). When students are able to visually make sense of the rule for –a × b, that can help them deal with the case involving –a × –b. Figures 4.19 and 4.20 illustrate how the case of –a × –b can be visually performed on a binary mat and a number line, respectively. 80

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

Figure 4.14. Patterns Involving Three Cases of Integer Multiplication

Figure 4.15. Modeling 3 × -2 = –6 on a Binary Mat

Figure 4.16. Modeling 3 × –2 = –6 on a Number Line

81

Chapter 4

Figure 4.17. Modeling –(3 × 2) = –3 × 2 = –6 on a Binary Mat

Figure 4.18. Modeling –(3 × 2) = –3 × 2 = –6 on a Number Line

Figure 4.19. Modeling –(3 × –2) = -3 × –2 = 6 on a Binary Mat

82

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

Figure 4.20. Modeling –(3 × –2) = -3 × –2 = 6 on a Number Line

Continue working with your pair and address the following questions below. ee. Verify that the following properties of operations below apply to integers (7.NS.2c). i) Commutative property of multiplication ii) Associative property of multiplication iii) Distributive property of multiplication over addition property Is the set of integers closed under multiplication? ff. Seventh-grade teacher Ms. Jeanne offered the following illustration below to her class when asked to explain why “a negative times a positive is a negative.” Use the properties of operations listed in items (aa) and (ee) to provide reasons for each step in her thinking process. i) –3 × 2 = (–3 × 2) + 0 ii) –3 × 2 = (–3 × 2) + [(3 × 2) + –(3 × 2)] iii) –3 × 2 = [(–3 × 2) + (3 × 2)] + –(3 × 2) iv) –3 × 2 = [(–3 + 3) × 2] + –(3 × 2) v) –3 × 2 = 0 + –(3 × 2) vi) –3 × 2 = –(3 × 2) gg. Seventh-grade student Jeremiah made the following claim: “Since I know that 3 × –2 = –6, I can just assume that –2 × 3 is also –6.” Do you agree? hh. Access the following link below which justifies the rule “a negative times a negative is a positive.” http://learnzillion.com/lessons/1028-prove-that-a-negative-times-anegative-equals-a-positive  uppose a student raised the following issue in class: “Why do I need to know S that the multiplication rules make sense by using the properties of operations? I think it’s enough that I understand them visually and through patterns.” How might you respond to the student’s concern? ii. Access the following Youtube links below which provide different ways of explaining why “a negative times a negative is a positive.” Discuss potential issues with such explanations. 83

Chapter 4

I) http://www.youtube.com/watch?v=hLm5lRxt1rE II) http://www.youtube.com/watch?v=RAnQNQnKziM III) http://www.youtube.com/watch?v=I4SNnTmaJe8 IV) http://www.youtube.com/watch?v=S4l86En9lbs jj. Develop story problems illustrating each of the following four cases of multiplication of integers: (I) positive × positive; (II) positive × negative; (III) negative × positive; and (IV) negative × positive. Solve them using different models for multiplying integers. 4.1.5  Divisions of Integers in Grade 7 Continue working with your pair and address the following questions below. kk. Use the missing-factor definition of division involving whole numbers to establish the following rules for division involving integers: I) positive ÷ positive = positive; II) positive ÷ negative = negative; III) negative ÷ negative = positive; IV) negative ÷ positive = positive. ll. Write one story problem for each case in item (kk). mm. Ms. Martha, seventh-grade teacher, made the following claim about division of integers: Since my seventh grade students know something about the slope of a line (i.e., as a unit rate that measures the steepness of the line; 7.RP), I can use the concept of unit rate and fractions to further help them understand division of integers. In fact, the cases involving negative divisors make so much sense if the kids learn them through slopes first. What do you think? How are slopes related to division of integers? 4.2  Absolute Value in Grades 6 and 7

The concept of the absolute value of a number is very difficult for many students. The difficulty persists all the way to precalculus, in fact, and oftentimes the source of difficulty deals with a narrow view of absolute value. For many students the idea that “absolute value makes numbers positive” is good enough for them. In simple cases when an integer n is positive, |n| = n is a reasonable generalization. However, in cases when an integer n is negative, they find the general statement |n| = –n difficult to comprehend. In this section pay special attention on how the concept of absolute value stems from relations of distance between pairs of points on a number line.

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The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

4.2.1 Concept of Absolute Value in Grade 6 The notion of absolute value as conveying distance from the origin or representing the magnitude of a positive or negative quantity in a real world context is one important concept that all sixth-grade students learn about integers and their representation as points on a number line (6.NS.7c). For example, in Figure 4.21, the distance of –5 from the origin is 5. In notational terms, you record that as |–5| = 5. In the general case, you say |m| = –m for any negative integer m. Also, in Figure 4.21, the distance of 7 from the origin is 7, thus, |7| = 7. In the general case, you say |m| = m for any positive integer m.

Figure 4.21. Number Line

Continue working with your pair and address the following questions below. nn. Simplify the following expressions: |0|; |–9|; |245|; and –|–6|. oo. Consider the following conversation among three sixth-grade students: Alfinio: OK, the absolute value of a positive integer should be the same positive integer. So |x| = x. So if you want to find the absolute value of a negative integer, just drop the negative sign and you’ll be fine. Casey: So how come we need to keep the negative sign in |x| = –x when x is a negative integer? I think it’s always x no matter what the number is. Alfinio: You’re right, Casey. I agree. Zach: You’re both wrong. –x in the statement |x| = –x when x is negative is an expression of distance from the origin, so –x is a positive number when x is negative. Casey: I don’t know what you mean by –x as an expression of distance. They’re all numbers to me.  ow might you process the above conversation in class so that all H your students develop a correct understanding of the definition of the absolute value of an integer? pp. Read content standards 6.NS.7c and 6.NS.7d. Provide everyday examples that will help sixth-grade students distinguish: (1) between an integer and its absolute value; and (2) between absolute value and order concepts of integers.

85

Chapter 4

4.2.2 Absolute Value and Distance in Grade 7 When seventh-grade students understand the number line model for subtracting integers, they can easily deal with interesting applications such as those that involve calculating distances between pairs of points on a horizontal or vertical number line. Consider the following example below. Mount Kilimanjaro is the highest mountain in Africa at 5893 meters. Lake Assal is the lowest point in Africa at 155 meters below sea level. What is the difference between the two elevations? The difference between 5893 and –155 is 5893 – –155 = 5893 + 155 = 6048. Hence, the difference between the two elevations is 6048 meters. The preceding seventh-grade example provides a useful context for applying absolute value. Note that in sixth grade, students interpret the absolute value of an integer in terms of its distance from the origin. In seventh grade, they obtain the distance between any pair of points on a number line by calculating the absolute value of their difference (7.NS.1c). For example, in Figure 4.22, the distance between the points 5893 and –155 is |5893 – –155| = |6048| = 6048. They can also find the same distance by calculating |–155 – 5893| = |–6048| = 6048. Thus, in sixth grade, they initially conceptualize the absolute value of an integer in terms of its distance from the origin, that is, |m| = |m – 0| = |0 – m|. In seventh grade, they calculate the distance between any pair of points a and b on a number line, that is, |a – b|. Continue working with your pair and address the following question below. qq. Obtain the length of each segment (I) through (VI) in Figure 4.23. Why do such tasks exemplify absolute value relationships (6.NS.8 and 7.NS.1c)?

Figure 4.22. Visual Processing of the Elevation Problem

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The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

4.3  Rational Numbers and Operations in Grade 6 and 7

4.3.1 Rational Numbers in Grade 6 Sixth-grade students’ knowledge of integers and fractions provide the context for extending fractions to rational numbers. Rational numbers consist of both positive and negative fractions. Furthermore, like integers, they are points on a number line (6.NS.6). Students should also be made aware that without positive and negative fractions that do not simplify to integers, the number line with just the integers would have too many holes. Hence, both sets of numbers are needed to construct a number line in which every point on the line corresponds to a unique number. Work with a pair and do the following tasks below. a. Refer to your Tables 3.2 and 4.2. How are fractions conceptualized from grades 1 to 5? b. Read content standards 6.NS.5 through 6.NS.8. Given sixth-grade students’ experiences with integers and fractions, what problems or issues might they have in relation to understanding the content standards relevant to rational numbers?

Figure 4.23. Obtaining Lengths of Segments

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4.3.2 Rational Number Operations in Grade 7 Continue working with your pair and do the following tasks below. c. Read content standards 7.NS.1a through 7.NS.1d. Discuss potential issues that seventh-grade students might have in understanding the following operations involving rational numbers: (1) addition; and (2) subtraction. d. Find two ways to simplify the following subtraction task:

.

Discuss potential issues that seventh-grade students might have with the task. e. Create four story problems involving different cases of addition and subtraction of rational numbers. Solve them. f. Seventh-grade Akron made the following claim below regarding division of rational numbers. I think I can use the rules for multiplying integers to help me better understand division of integers in a way that uses rational numbers. Say I want to find the quotient of 4 ÷ –2, which I know is –2 since –2 × –2 = 4. I know from fifth grade fractions that dividing by a number is the same thing as multiplying by the reciprocal of the number (5.NF.3). So, 4 ÷ –2 is the same thing as

. So,

. Since a positive times a negative is a negative and 4 times a half is 2, then the answer is –2. Hence, 4 ∏ –2 = –2.  hat do you think? Does Akron’s multiplicative approach work in all cases W of division problems (7.NS.2)? Obtain the quotients of the following division problems: (1)

; (2)

; (3)

; and (4)

g. How do you explain the fact that wrong with the following statement:

.

(7.NS.2b)? What is ?

h. Create four story problems that involve products of rational numbers (7.NS.2a). Solve them. i. Create four story problems that involve quotients of rational numbers (7.NS.2b). Solve them. j. Verify that the following properties of operations below apply to rational numbers (7.NS.1d and 7.NS.2c). 1. Commutative property of addition 2. Commutative property of multiplication 3. Associative property of addition 88

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

4. Associative property of multiplication 5. Additive inverse property 6. Additive identity property 7. Multiplicative inverse property 8. Multiplicative identity property 9. Distributive property of multiplication over addition I s the set of rational numbers closed under addition? subtraction? multiplication? division? k. Access the link http://www.illustrativemathematics.org/standards/k8. Open The Number System page. Click on the Grade 7 link. Then open the “show all” page and click the link “see illustrations” under A.1 and A.3. Solve all tasks completely. 4.3.3 Decimals that Transform into Rational Numbers in Grade 7 Since rational numbers are quotients of two integers a and b, that is

, where

, they can also be expressed as either terminating or repeating decimals (7.NS.2d). For example, 0.5 and 0.32 are terminating decimals, while 2.3333… and 0.14141414… are repeating (and nonterminating) decimals. The three-dot ellipsis notation in repeating decimals indicates that the digits are expected to repeat forever. Note that the repeating decimals 2.3333… and 0.14141414… can also be written as – – 2.3 and 0.14, respectively, where the digits under each bar (or vinculum) convey the shortest repeating string (called repetend) of the decimal number. The term period refers to the length of the repetend. Continue working with your pair and do the following tasks below. l. Convert the following decimal numbers into fractions: 0.5; 0.25; 0.150 and 0.568. Make an observation about the denominators of the corresponding fractions. Seventh-grade student Leila made the following claim: “I think that all the fraction equivalents of terminating decimals have denominators that contain only the factors of 2 and 5.” Is Leila correct? Explain. m. Obtain the decimal representations of the following fractions: 

and

. Jake, Leila’s classmate, made the following claim: “When I obtained the quotient of the numerator and the denominator in each case, I noticed that the period of the repetend is less than the denominator.” Is Jake correct? Explain. Maria, Jake’s classmate, said: “I agree with Jake. Also, when you have factors other than 2 and 5, the fraction becomes repeating and nonterminating.” Is Maria correct? Explain.

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n. Find an easy way of determining which fraction below yields either a terminating or repeating decimal representation. Then convert all of them into decimals.

o. Seventh-grade students learn how to solve single-variable one-step equations in sixth grade (6.EE.5 and 6.EE.6), a skill that they can use to help them convert repeating decimal numbers into fractions. Consider the following examples below. 1. Convert

in fraction form. This is an easy problem. Students (should) know

that it is equivalent to

. A different way of verifying the equivalence other

than performing the actual division is illustrated below. Let x = 0.3333… 10x = 3.3333… (Why?) –– x = 0.3333… (Why?) 9x = 3     x= 2. Convert

(Why?)  (Why?)

in fraction form. First, we let x = 2.342424242…. 1000x = 2342.4242424242….(Why?) –– 10x = 23.4242424242…   (Why?) 990x = 2319            (Why?) x=

     (Why?)

Obtain the fraction form of each repeating decimal number below.   p. Obtain the exact difference: q. Explain why 0.99999… = 1.

and 1.03¯

; .

r. Consider a decimal number of the type period of the repetend the decimal number is

, where p stands for the

. Explain why the equivalent fraction form of , where D corresponds to the integer

.

Also, determine whether the fraction form is always in lowest terms. For example, , and the integers 1 and 3 are relatively prime. 90

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

s. Access the link http://www.illustrativemathematics.org/standards/k8. Open The Number System page. Click on the Grade 7 link. Then open the “show all” page and click the link “see illustrations” under A.2d. Solve the two tasks completely. 4.4  Irrational Numbers and Operations in Grade 8 and Algebra 1

In eighth grade students pursue nonrepeating-nonterminating decimal numbers. Famous examples of such numbers include pi, π (≈3.14159), and (≈1.41421356237). Nonrepeating-nonterminating decimal numbers do not have repetends. Also, since the digits go on forever, they do not terminate. Hence, nonrepeating-nonterminating decimals are called irrational numbers, that is, they can never be expressed as quotients of two integers (8.NS.1). However, their approximate values can be expressed in fraction form such as

and

.

Rational and irrational numbers together complete the real number line, meaning to say that the line has no holes and is everywhere dense. In practical terms, it means that you can always find a real number when you need it. The equivalent decimal fraction representations of nonrepeating-nonterminating decimals consist of an infinite number of terms. For example, can be expressed as

Work with a pair and do the following activities below. a. Express  as a sum of an infinite number of decimal fractions. b. In section 2.6.1 you learned how to construct irrational numbers using the Pythagorean Theorem. Recall how you locate their approximate locations on a number line (8.NS.2) and use that knowledge to help you locate the approximate locations of the following irrational numbers on the same line. (i)

; (ii)

; (iii)

; (iv)

; (v) 2 3 − 4 ; (vi)

Content standard 8.NS.2 also recommends approximating the values of irrational numbers by initially truncating the decimal expansion parts and then performing a series of interval divisions by tenths, hundredths, thousandths, etc. that enable students to construct the decimal expansion parts. Continue working with your pair and do the following tasks below. c. To approximate on a number line, divide the interval between 1 and 2 into tenths and obtain the approximate location of point P between 1.4 and 1.5.  ay attention on the interval between 1.4 and 1.5 and divide it into ten subintervals P of equal length. If P is one of the nine new points within the interval, then you 91

Chapter 4

are done. If not, divide the interval between 1.41 and 1.42 into ten subintervals of equal length and determine once again if P is one of the nine new points within that interval. Keep doing this division process several more times to help you obtain a good approximation of d. Use the process in (c) to approximate the location of π on the number line to the nearest ten thousandths place.  ince eighth grade students also learn that square roots and cube roots represent S solutions to and , where P is a rational number (8.EE.2), the numerical process of taking the nth root of a number provides another way of approximating the value of an irrational number, although it is not as precise as the process described in task (c) above. Continue working with your pair and do the following tasks below. e. Since 1 < 2, then 12 < 22. Also, –1 < 3 implies . Is it correct to say that a < b implies a2 < b2? If the statement is true for all real numbers a and b, explain how so. If it is false, provide a counterexample. Under what conditions is the statement always true? f. To help students obtain a general sense of the values of irrational numbers, say, , they can construct an interval that contains the number 2 and with endpoints corresponding to perfect squares. Hence, 12 = 1 < 2 < 4 = 22. Taking the square roots, they obtain the statement .  Which two integers contain each irrational number below? (i)

; (ii)

; (iii)

; (iv)

; (v)

he Algebra 1 content standards under N-RN further deepen students’ T understanding of real numbers in both aspects of notational representation (N-RN.1 and N-RN.2) and properties (N-RN.3). Continue working with your pair and do the following tasks below. g. Which properties of operations involving the rational numbers in section 4.3.2 item (j) apply to real numbers? h. Access the link http://www.illustrativemathematics.org/standards/hs. Open the Number and Quantity page. Click on the Real Number System link and open the “show all” page. Solve all the tasks specified under A.1, A.2, B, and B.3. In addition, under B.3, prove all valid conjectures in item (c) using the properties of real numbers. i. Arrange the following real numbers below in increasing order. ,

,

,

,

,

, –8,

j. Access the link below and check out the High School problem labeled Rationals and Radicals (#42906) under the Concepts and Procedures released items. Solve it completely. http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm 92

The Real Number System (Part II) from Grade 6 to Grade 8 and Algebra 1

4.5  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below.  se the checklist you developed in Table 2.4 to map each content standard U under the Real Number System Part II domain with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 4.6  Developing a Content Standard Progression Table for the Real Number System Part II Domain

Continue to work on the table that you constructed in section 3.9 to develop a ninemonth progression of content standards under the RNS Part II domain. First, add the appropriate number of rows corresponding to the RNS Part II content standards. Then carefully plan a reasonable content path or trajectory for each standard over the indicated timeline. Remember that your table at this stage should reflect a reasonably tight mapping of interrelationships between and among the different content standard progressions involving both the RNS Part I and RNS Part II domains.

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Chapter 5

Ratio and Proportional Relationships and Quantities

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Ratio and Proportion (RP) and Quantities (N-Q).domains of the CCSSM from Grades 6 to 8 and Algebra 1. Quantities are numbers with units (e.g.: centimeters, 0.5 square inches, and 8 cubic meters). They play an important role in the development of middle school students’ understanding of ratios and proportions, which describe relationships between two (or more) quantities. In the CCSSM, ratios are linked to fractions by way of unit rates and proportions are linked to one-step multiplication problems and their graphical representations in the form of points that lie on a straight line. Table 5.1 lists the content standards that are covered in this chapter. Table 5.1. Quantities and Ratio and Proportion Domains in the CCSSM Grade Level Domain

Standards

Page Numbers in the CCSSM

6.RP

1-3

42

7.RP

1-3

48

8.EE

5

54

1 to 3

60

N-Q (Algebra 1 Traditional Pathway)

To help you obtain an initial sense of the connections among the topics noted in the preceding paragraph, consider the following third-grade multiplication problem below. If a triangle has 3 sides, how many sides do 8 separate triangles have in all? Fill in Figure 5.2 which shows different ways of analyzing the same problem by grade level content expertise. Continue to read the following numbered list below to learn more about what students need to exhibit under each type of thinking. 1. Elementary Product Thinking. In third grade students convert the problem symbolically in terms of a product involving two whole numbers a × b. Set up a correct multiplication expression for this particular problem and obtain the product. What are the units associated with each factor and the answer? 2. Ratio Thinking. In sixth grade students learn about the concept of ratio m:n, which conveys a relationship between two quantities m and n. For example, the ratio 1: 3 95

Figure 5.2. Thinking Progressions in the Analysis of a Simple One-Step Multiplication Problem

Chapter 5

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Ratio and Proportional Relationships and Quantities

means “each triangle has 3 sides.” Complete the table under the ratio thinking box in Figure 5.2 by generating equivalent ratios (i.e., by multiplying each quantity in the ratio by the same numerical value). Then check to see whether the entries in the eighth column of your table match the answer you obtained in (1). 3. Proportional Thinking. In seventh grade students construct proportions, that is, equivalent ratio relationships. Refer to the proportional thinking box in Figure 5.2. Fill in each blank with the correct number. See to it that your statement of proportion makes sense in terms of the relationship between the two quantities and their units. 4. Linear Relationship Thinking. In sixth grade students plot a set of ordered pairs of points on a coordinate plane corresponding to the table of values shown under the ratio thinking box in Figure 5.2. In eighth grade, they see both representations as instances of a direct proportional relationship (i.e., x and y are directly proportional if the ratio

is constant). Under the linear relationship thinking box in Figure 5.2,

the coordinate plane consists of the horizontal axis, which corresponds to the number of triangles n, and the vertical axis, which corresponds to the total number of sides s for n triangles. Plot points in the plane corresponding to the entries in your ratio table and then connect the points to generate a graph. Carefully describe the graph. If you extend the graph, it passes through the origin. Why does that happen? Is it reasonable to extend the graph in both directions? Why or why not? Do you need to connect the points in order to obtain a correct graph? Explain. Find an equation that describes the relationship between n and s. 5.1  Ratio and Rate Relationships in Grade 6

Work with a pair and do the following tasks below. a. What is a ratio (6.RP.1)? How are ratio relationships converted in symbolic form? Access the link http://www.illustrativemathematics.org/standards/k8. Open the Ratios and Proportional Relationships page. Click on the Grade 6 link. Then open the “show all” page and click the link “see illustrations” under A.1. Solve the two tasks completely. b. How are unit rates related to ratios (6.RP.2)? Why do students need to carefully attend to the units of quantities in the way they construct unit rates? Study the two examples provided in 6.RP.2. How are students are expected to use unit rate language correctly? c. Consider the following ratio situation below. 60 men and 40 women were invited to attend a job fair for 100 people. Sixth-grade Carla said: “The ratio of men to women is 6:4.” Jeremy, Carla’s classmate, said: “No, Carla, the correct ratio is 60:40.” What do you think? 97

Chapter 5

Follow the instructions in (a) relative to the RP page and solve all five tasks completely under A.2. d. Describe at least four different ways of processing and reasoning with ratios and rates (6.RP.3). Search the Youtube channel for video resources that you can use to teach the different ways to sixth-grade students. Then solve the following problems below in different ways. Sixth-grader Mikayla sells three caramel apples for $4. A. How much does Mr. Robert, Mikayla’s teacher, need to pay her if he intends to purchase 24 apples for his class? B. Thinking of his family, Mr. Robert wants to purchase 30 apples. How much will that cost him? C. Mr. Robert has $60 to spend. How many caramel apples can he purchase? Some teachers are wary about teaching all the different ways to students. What do you think? Follow the instructions in (a) relative to the RP page and solve all four sets of tasks completely under A.3. e. Are ratios numbers? Are they fractions? Explain. Discuss possible issues when students convert ratios to fractions. f. A three-term ratio is a relationship involving three quantities. For example, if there are 2 red marbles, 3 blue marbles, and 4 green marbles in a bag, then the ratio of red marbles to blue marbles to green marbles can be expressed symbolically as 2:3:4. Describe everyday and mathematical situations that involve three-term ratios. All three problems below involve three-term ratios. How might sixth-grade students draw on their knowledge of unit rates involving two-term ratios to solve them? 1. The number of chickens, dogs, and pigs on Mr. Rico’s farm are in the ratio of 2:3:4. If there are 72 pigs, then how many chickens are there? 2. A store sells shorts in three colors: black, brown, and red. The colors are in the ratio of 3:4:5. How many shorts does the store have in all if they have brown shorts in stock? 3. Edward’s cereal mixture has wheat, corn, and rice in the ratio of 3:5:7. If a bag of mixture contains 6 pounds of wheat, how much rice does it contain? g. Sixth-grade students begin to deal with conversion problems in measurement in fourth grade. How can ratio and rate reasoning be used in contexts that involve

98

Ratio and Proportional Relationships and Quantities

metric and nonmetric units (6.RP.3d)? For example, since 12 inches = 1 foot, there is a unit rate for converting from either inches to feet or feet to inches. Another example involves the meter diagram shown in Figure 5.3, which uses the same structure shown in Figure 3.15. The meter diagram helps students understand the metric conversion process involving meters such as 1 kilometer = 1000 meters and 1 millimeter =

meter. That is, there is always a unit rate that can be used

to convert from one kind of measure to another.

h. Access the 6th-grade performance task called Field Trip from the site below. Solve it completely. http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm i. Why is ratio and rate reasoning a good example of multiplicative (versus additive) thinking? 5.2  Proportional Relationships in Grade 7

In seventh grade students formulate equations involving ratios, that is, proportions. Unit rates are renamed as constants of proportionality. Work with a pair and do the following activities below. a. Read content standards 7.RP.1 to 7.RP.3. Then access the link http://www. illustrativemathematics.org/standards/k8 and open the RP page. Click on the Grade 7 link and open the “show all” page. Solve all the tasks in this cluster. b. Access online resources that use the Singapore visual model (bar diagrams) for solving proportional problems. Learn the model and then assess which problems in (a) can be solved using bar diagrams. c. Reanalyze each problem in item 5.1(f) by providing two different solution approaches, as follows: (1) using bar diagrams; (2) setting up an equation. d. Authors Tom Parker and Scott Baldridge note that setting up an equation is easier than constructing a bar diagram in the case of the following ratio problem below. Two numbers are in a ratio of 3:5. If you subtract 11 from each number, the new ratio is 2:7. What are the two numbers? What do you think?

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Chapter 5

5.3  Graphs of Proportional Relationships from Grades 6 through 8

The following content standards address various aspects of the general graph of two quantities that are in a direct proportional relationship: 6.RP.3a, 7.RP.2a, 7.RP.2c, 7.RP.2d, 8.EE.5, and 8.EE.6. Work with a pair and do the following activities below. a. In sixth grade students generate ratio tables and graphs to help them make sense of ratio problems. Consider once again the problem in 5.1(d). Use Table 5.2 to generate equivalent ratios. The table describes a relationship showing the total cost C of buying n number of apples from sixth-grader Mikayla. Table 5.2. Ratio Table Number of Apples Bought (n)

Total Cost (C)

 3  6  9 12

Plot the ordered pairs of values in a coordinate plane. Label your axes appropriately. How does this representation of ratios help sixth-grade students reason about ratios and rates? What connections should students form between a ratio table and the corresponding graph? b. In seventh grade students use the same representations in (a). However, they focus their attention on the concept of a unit rate r as a constant of proportionality and the fact that the general graph consists of points that lie on a straight line. Further, they learn to recognize that the points (0, 0) and (1, r) are always included in the graph. One way to calculate r involves using the Singapore method of unitizing. Consider Table 5.2. Since 3 apples cost $4, that is, , then if you divide both numbers by 3, you obtain the equivalent relationship 1 apple costs $

. Check to see that your graph contains the points (0, 0) and

. How do you expect your seventh-grade students to explain the r value of ? Further, does the r value apply only to the ordered pairs

and (3, 4) or

does it apply to any pair of points on the line? Explain. c. In eighth grade students reconceptualize unit rates in terms of slopes of lines. They also use similar triangles to explain why pairs of points on a straight line 100

Ratio and Proportional Relationships and Quantities

have the same slope. Furthermore, they derive the linear equation y = mx and compare it with the linear equation y = mx + b, where b is the point where the graph crosses the y-axis (i.e., y-intercept). 1. How might eighth grade students use their knowledge of proportions and slopes to help them describe arithmetical patterns in the multiplication table, which they learned in third grade (3.OA.9), in both graphical and equation formats? 2. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 8 link. Then open the “show all” page. Solve completely all the three sets of tasks under B. 5.4  Quantitative Relationships in Algebra 1

Work with a pair and do the following tasks below. a. Read content standards N-Q.1 to N-Q.3. What purpose do these standards serve in terms of the expected level of understanding of units at the high school level? b. Compare the manner in which units are conceptualized and used in middle school and high school mathematical contexts. c. Access the following link below. http://www.illustrativemathematics.org/standards/hs Open the Number and Quantity page. Click “show all” under the Quantities link. Solve all the tasks under this conceptual domain. 5.5  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard under the Quantities and Ratio and Proportion domains with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 5.6  Developing a Content Standard Progression Table for the Quantities and Ratio and Proportion Domains

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level RP or N-Q domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving the RNS, RP, and N-Q domains are coherently developed. 101

Chapter 6

Technology-MEDIATED Tools for Teaching and Learning Middle School Mathematics

Like physical and concrete manipulatives, technology-mediated tools for teaching and learning middle school mathematical concepts and processes provide students with an opportunity to actively engage in thinking while they tinker with the relevant objects in a virtual context. Such dual processing, also called thinkering, helps students construct and infer deep mathematical relationships and structures on the basis of their experiences with the objects. Consequently, the nature of instruction and learning also changes from mere reception to modeling. In a modeling context they generate insights from empirical results, produce schemes and structures, develop and/or employ necessary reasoning, and ultimately predict outcomes. Complications in learning mathematics with technology-mediated tools are consistent with documented difficulties that some students have with physical and concrete manipulatives. Raymond Duval’s distinction between visual representations and visualization is worth noting in this regard. While physical and virtual tools provide students with visual representations that fulfill different functions, visualization requires them to attend to what is mathematically relevant (i.e., the necessary figural units) in those representations. “For a mathematician and a teacher,” Duval writes, “there is no real difference between visual representations and visualization. But for students, there is a considerable gap that most of them are not always able to overcome. They do not see what the teacher sees or believe they will see.” Hence, when you use technology-mediated tools in your own math classrooms, you need to find ways to implement instruction (setting up tasks, using guide questions, etc.) in ways that enable them to acquire all the necessary figural units in a visual representation. Think about how you might accomplish that as you learn some of the technology-mediated tools that are appropriate for middle school students in the following sections below. 6.1  Getting to Know the TI-84+ Graphing Calculator

Access the following links below which provide information about the different basic functions of a TI 84+. Skip those sections that address mathematical content beyond what you know at this stage. In this section, your basic objective is to learn how to use a TI-84+ in order to perform certain arithmetical and graphical processes that are appropriate at the middle school level.

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http://mtl.math.uiuc.edu/non-credit/basic84plus/ti84plus-tutorial/ti84-tutorial. toc.html http://web.clark.edu/math/Calculator/#general http://www.freewebs.com/ti84tutor/index.html Access the link below if you are interested in learning directly from the official guidebook. http://education.ti.com/en/us/guidebook/details/en/C4D11EB6D86B47D19C D768E54A967441/84p Work with a pair and do the following tasks below. a. The link below contains several calculator activities that require knowledge of the keys and functions in a TI-83. Accomplish the same activities using your TI-84+. http://archives.math.utk.edu/ICTCM/VOL11/C049/paper.pdf b. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Ratios and Proportional Relationships page. Click on the Grade 7 link. Then open the “show all” page and click the link “see illustrations” under A.2. Analyze the Buying Bananas task using your TI-84+ by following the instructiosn below. i)

Set up a table of values on your TI-84+ showing five different prices for different quantities of bananas. How do you describe your first column entries? second column entries? How much does 1 pound of bananas cost? What is the unit rate? ii) Graph the table of values you generated in (i). Describe the behavior of the points relative to each other. Is there a trend or a pattern? iii) Generate an equation for the problem. Describe your variables in precise terms. iv) Use the STAT key of the TI-84+ to help you determine an equation for the table of values you generated for the problem. Does your equation match the one you generated in (iii)? Explain. v) The graph appears to cross the origin. Explain the significance of the origin with respect to the problem. vi) Does your equation contain your unit rate or slope of the curve containing all your points? If so, explain what it means graphically and in the context of the problem.

c. Consider the following variations of the Buying Bananas task: I)

Carlos bought

lbs of bananas for $5.00.

II) Carlos bought

lbs of bananas for $4.70.

III) Carlos bought

lbs of bananas for $8.50.

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Teaching and Learning Middle School Mathematics

Obtain the slope, equation, and graph that describe each situation above. Use your TI-84+ to graph all three equations in the same view screen or coordinate plane. Also, graph the equation you obtained in (iii) so that the screen shows all four graphs. How do you explain the fact that all four graphs appear to pass through the origin? How might eighth-grade students make sense of the slopes and the continuous graphs? d. You may use the Transformation Graphing App of the TI-84+ to help eighth grade students investigate slopes and y-intercepts of graphs of linear equations involving two quantities. Access the following link below and explore the tasks with the aid of the graphing app. http://www.trianglehighfive.org/pdf/007_exploring_transformation.pdf e. How does the use of technology support growth in students’ ability to engage in processes of abduction and induction? 6.2  Getting to Know the GeoGebra Dynamic Software Tool

Access the following link below to install the GeoGebra software tool into your desktop or laptop. http://www.geogebra.org/cms/en/download/ Then access the link below which provides you with an introduction to GeoGebra by Gerrit Stols. http://www.geogebra.org/workshop/en/GerritStols-GeoGebra-in10Lessons.pdf Work with a pair and do the following tasks below. a. Learn the Geogebra user interface and related terms (p. 1). Explore the different menus and construction tools (pp. 2–4). b. Study Lessons 1 to 4 (pp. 5–16). In Lesson 4, explore the linear equation y = mx + b. c. Access the link below to learn how to set up a table of values. http://lokar.fmf.uni-lj.si/www/GeoGebra4/Spreadsheet/create_table/ create_table.htm In fifth grade students analyze patterns and relationships (5.OA.3). Use the patterning task in the CCSSM under this content objective (p. 35) to construct two tables of values in GeoGebra. Plot both sets of ordered pairs in the same coordinate plane using two different colors to distinguish one set from the other. For each set of points, connect the points to form a continuous straight line and then determine the equation that describes the line. How does the equation capture or record the characteristics of the line? Construct three more tables beginning with the number 0 using the rules “Add 3,” “Add 8,” and “Add 10.” Graph them 105

Chapter 6

on the same coordinate plane and determine their equations. Make an observation about their slopes. Use a different GeoGebra interface. Set up five tables of values beginning with the number 0 using the following rules: Add Add

; Add

; Add

; Add

; and

. Determine the linear equations. Make an observation about their slopes.

Establish an empirical generalization regarding slopes of lines that are: (i) greater than 1; and (ii) between 0 and 1. d. Is it possible to generate a table of values with a slope of 0? If so, construct one such table, graph the points, and determine the equation that describes the points. Are there everyday experiences that you can use to explain lines with a slope of 0? e. Use a different GeoGebra interface to obtain a graph with the following table of values below. 2

2

2

2

2

2

2

2

0

-2

2

3

-3

4

-4

5

Connect the points to form a straight line. What kind of line do you see? What is its slope? Describe everyday experiences that model this particular kind of graph. f. Consider the following ratio and proportion task below. How much would it cost eighth-grader Jeremy to purchase 80 stamps if each stamp is worth $0.46? Solve the task in two ways, as follows: i)

ii)

Use GeoGebra to set up a table of values, graph, and determine the equation that describes the relationship between number of stamps bought and cost of purchase. How can you use any of these of representations to answer the question? Set up the correct proportion in fractional form.

g. Access the following links below that teach you how to develop technologymediated dynamic worksheet activities using Geogebra. Such worksheets are meant to encourage students to model abductive and inductive reasoning that can then be a basis for deductive reasoning. When middle school students engage in a dynamic worksheet activity, they manipulate a mathematical form (e.g., graph of a straight line) in order to establish conjectures about some or all of its relevant features. Raymond Duval uses the term figural unit to refer to what is significant, relevant, and informative in a visual representation, which explains the purpose of visualization in mathematics. That is, “visualization can fulfill all the functions important to understand and use mathematics when one becomes able to discern all the figural units that are mathematically relevant in a given visual representation.”  Dynamic worksheets and other technology-mediated 106

Teaching and Learning Middle School Mathematics

activities in the general case are meant to provide intentional insight and visual support and strengthen the link between visual representations (e.g., graph of a straight line) and visualization (e.g., significance of the slope of a line). http://techtraining.brevard.k12.fl.us/BETC/Handouts/Cross_ Creating%20a%20GeoGebra%20Dynamic%20Worksheet.pdf http://mathgr3-5.pds-hrd.wikispaces.net/file/view/GeoGebra_WS_6.pdf Do the following activities below. i) ii)

Develop a dynamic worksheet that will help eighth-grade students visually understand the Pythagorean Theorem (8.G.6 through 8.G.8). Develop a dynamic worksheet that will help eighth-grade students visually establish the fact that the slope is constant between any pair of points on a straight line in a coordinate plane by using similar triangles. 6.3  Getting to Know a Calculator-Based Ranger

A calculator-based ranger (CBR) is a sonic motion detector that can be used with the TI-84+ to “collect, view, and analyze motion data without tedious measurements and manual plotting.” CBR activities enable students to explore and establish mathematical (and scientific) relationships such as the effects of increasing positive values of slopes on distance-time graphs. Work with a group of 4 students to accomplish the following tasks below. a. Access the following link below, which provides guidance and instructions for using a CBR effectively in the classroom. Work through the details and accomplish all five activities together. http://www.pa.uky.edu/~ellis/Instr_Labs/CBR%20Manual.pdf b. Discuss potential issues when you teach with the CBR in class. 6.4  Getting to Know Apps and Other Virtual-Based Manipulatives for Classroom Use

Educational apps and other virtual-based manipulatives provide powerful resources for supporting middle school students’ learning of school mathematics. Work with a pair and do the following activities below. a. Search the internet for information regarding the benefits of using educational apps and virtual manipulates to help students learn mathematics better. b. There are internet sites that list interesting apps for teachers to use in the classroom. Explore at least one app together that you can use in your mathematics classroom and be prepared to share your findings with others.

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c. Access the following link below which contains activities that use interactive tools to help students learn mathematical concepts and processes in problemsolving contexts. http://illuminations.nctm.org/ Explore one activity together and be prepared to share your learnings with others. d. Explore the National Library of Virtual Manipulatives site, http://nlvm.usu.edu/, and discuss potential learning and classroom implementation issues. 6.5  Understanding Technology Use in the Mathematics Classroom

Work in groups of 4 and accomplish the following tasks below. a. Access the following articles below that tackle issues relevant to the use of disruptive technologies and blended learning in education. Hill, P. (2000). The innovator’s dilemma. Education Week, 19(40), 52. Patterson, G. (2012). Blending education for high-octane motivation. Phi Delta Kappan, 94(2), 14–16. Richardson, J. (2011). Disrupting how and where we learn. Phi Delta Kappan, 92(4), 32–38. How might new and emerging disruptive technologies and blended learning environments change instruction and learning in mathematics? b. Access the following article below that tackles the value of using Google apps in supporting 21st century learning. Nevin, R. (2009). Supporting 21st century learning through Google apps. Teacher Librarian, 37(2), 35–38. What is cloud computing? How might it support student learning of mathematics? How might you use Google apps to support assessment? How might you implement Google apps effectively in class? c. Access the following article below which provides insights on how you might think about the role of technology in your own math classrooms. Goldenberg, E. P. (2000). Thinking (and talking) about technology in math classrooms. Issues in Mathematics Education (pp 1–8). Retrieved from http://www2.edc.org/mcc/pdf/iss_tech.pdf. What does the author say about what constitutes a “good use” of technology in the math classroom?

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Expressions and Operations

In this chapter you will deal with content-practice, teaching, and learning issues relevant to constructing and manipulating expressions from Grade 5 to Grade 8 and Algebra 1 of the CCSSM. Table 7.1 lists the specific domains and relevant content standards that are covered in this chapter. Numerical and algebraic expressions in mathematics basically consist of numbers, variables, and their arithmetical combinations. In fifth grade they learn to write and simplify simple numerical expressions under the Operations and Algebraic Thinking domain. In sixth grade they use their beginning knowledge of least common multiples and greatest common factors to engage in simple factoring of numerical expressions under the Number Sense domain. Also in sixth grade and much of seventh grade they use their understanding of numerical expressions with and without wholenumber exponents to make sense of algebraic expressions under the Expressions and Equations domain. In eighth grade they deepen their knowledge of numerical expressions by exploring properties of integer exponents, evaluating square roots of perfect squares and cube roots of perfect cubes, and using and operating with scientific notation in a variety of interesting applications. In Algebra 1 they explore the structures of algebraic expressions and simple nonalgebraic transcendental expressions under the Seeing Structure in Expressions domain. They also simplify polynomials under the Arithmetic with Polynomials and Rational Expressions domain. Table 7.1. The Expressions Domain in the CCSSM from Grade 5 to 8 and Algebra 1 Grade Level Domain 5.OAT

Standards

Page Numbers in the CCSSM

1 to 2

35

6.NS

4

42

6.EE

1 to 4 and 6

43-44

7.EE

1 to 2

49

8.EE

1 to 4

54

A-SSE (Algebra 1 Traditional Pathway)

1, 2, 3a, and 3c

64

A-APR (Algebra 1 Traditional Pathway)

1

64

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7.1  Numerical and Algebraic Expressions in School Mathematics

Work with a pair and do the following tasks below. a. Search online for various meanings of the term “expressions” in (school) mathematics. Develop a clear and precise definition on the basis of your findings. b. Content standard A-SSE.1.a provides a structure for interpreting parts of an algebraic expression that also applies to numerical expressions. Conduct an internet search and establish the meanings of the following parts: terms, factors, and coefficients. Consider the algebraic expression 5x3y – 3xy2 + 2xy + 4. Name its terms, factors, and coefficients. Now consider the numerical expression 3 × 5 + 4 ÷ −2 + 3 − 8. Name its terms and factors. How are numerical expressions similar to and different from algebraic expressions? c. When eighth-grader Josh learned that the part “4” in the algebraic expression 5x3y – 3xy2 + 2xy + 4 was called a constant because it appeared as a term without variables, he felt confused. He said, “I think it’s a coefficient for the term 4x0 y0 since I know that x0 = y0 = 1 for all nonzero real numbers x and y.:” What do you think? d. A monomial is a one-term expression. Generate examples of monomials involving numerical and algebraic expressions. A binomial is a two-term expression. Generate examples of binomials involving numerical and algebraic expressions. A trinomial is a three-term expression. Generate examples of trinomials involving numerical and algebraic expressions. A multinomial is an expression with more than three terms. Generate examples of multinomials involving numerical and algebraic expressions. Is the numerical expression (3 + 4)2 a monomial or binomial expression? Explain. 7.2  Writing and Simplifying Numerical Expressions in Grade 5

Continue working with your pair and do the following tasks below. a. Explain the order of operations in mathematics. Search online for interesting resources that you can use to teach the topic. 110

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b. Ms. Jamie’s fifth-grade class wanted to know more about the order of operations. She said, “Don’t worry about it too much. The rule is a conventional agreement and not a consequence of logical necessity.” What do you think? c. Read content standard 5.OA.1. What is the significance of using grouping symbols to evaluate numerical expressions? d. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Operations and Algebraic Thinking page. Click on the Grade 5 link. Then open the “show all” page and click the link “illustrations” under A. Solve completely all the tasks under 5.OA.A, 5.OA.A.1, and 5.OA.A.2. 7.3  Greatest Common Factors, Least Common Multiples, and Simple Factoring Involving Numerical Expressions in Grade 6

Continue working with your pair and do the following tasks below. a. Gather one set of 3 unit cubes and another set of 5 unit cubes. Form two rectangles with areas of 3 and 5 square units that share: (i) a common side (or dimension, divisor, or factor); (ii) the largest or greatest common side (or dimension, divisor, or factor; i.e., GCD or GCF; 6.NS.4). b. Follow the instructions in (a) to find the GCF of each pair of numbers below. (i) 3 and 8; (ii) 5 and 7; (iii) 11 and 13; (iv) 2 and 9 What can you conclude about the GCF of any pair of prime numbers? Explain. c. Find the GCF of each pair of numbers below. (i) 2 and 4; (ii) 2 and 8; (iii) 2 and 16; (iv) 3 and 9; (v) 4 and 16 Find the GCF of: (i) 2n and 2m, where n < m; (ii) 5a and 5b, where a > b; and (iii) 2a3b5c and 2m3n5p, where a < m, b > n, and c = p. d. Gather a horizontal (or vertical) chain consisting of 3 unit cubes (i.e., a 3-chain) and another chain of 5 unit cubes (i.e., a 5-chain). Build copies or multiples of a 3-chain on one row and copies or multiples of a 5-chain on another row and stop when they both reach the same length. This first instance of common length is called the least common multiple (or LCM) of 3 and 5 (6.NS.4). Generate two more common multiples of 3 and 5. e. Follow the instructions in (d) to find the LCM of each pair of numbers below. (i) 2 and 3; (ii) 7 and 5; (iii) 3 and 7; (iv) 2 and 11 What can you conclude about the LCM of any pair of prime numbers? Explain. f. Find the LCM of each pair of numbers below. 111

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(i) 2 and 4; (ii) 2 and 8; (iii) 2 and 16; (iv) 3 and 9; (v) 4 and 16 Find the LCM of: (i) 2n and 2m, where n < m; (ii) 5a and 5b, where a > b; and (iii) 2a3b5c and 2m3n5p, where a < m, b > n, and c = p. g. Search online resources for teaching GCF and LCM. Discuss advantages and possible concerns when they are implemented in the classroom. h. Sixth-grade teacher Mr. Nash made the following claim: “I don’t like those visual approaches for teaching GCFs and LCMs of whole numbers. They will just confuse my kids. As long as they know how to systematically list all the factors and multiples, they should be fine and able to find either GCF or LCM.” For example, Figures 7.1 and 7.2 below illustrate how to find the LCM and GCF of two numbers by a listing approach, respectively. Find the LCM of 12 and 32. Solution: Factors of 12: 1, 12, 2, 6, 3, 4 Factors of 32: 1, 32, 2, 16, 4, 8. Therefore, the GCF is 4. Figure 7.1. Finding the Common Factors and GCF of 12 and 32

Find the GCF of 12 and 32. Solution: Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108. Factors of 32: 32, 64, 96, 128, 160, 192 Therefore, the LCM is 96. Figure 7.2. Finding the Common Factors and GCF of 12 and 32

Ms. Jenna, Mr. Nash’s colleague, offered the following response to his claim: “Well, I use a visual approach to help my kids understand why GCFs and LCMs are done differently. For example, in finding the LCM of 12 and 32, their respective prime-power products, 22 × 3 × 1 and 25 × 1 should help them see why the GCF of the two numbers should be 22and the LCM 25 × 3 × 1. From their visual experience, they should be able to decide which factors to choose.” What do you think? i. Use the distributive property of multiplication over addition to express each sum of numbers below with a common factor as a multiple of a sum of two whole numbers with no common factor (6. NS.4). For example, 2 + 4 = 2(1 + 2). This content standard introduces sixth-grade students to simple factoring using the GCF.

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(i) 18 + 36; (ii) 100 + 64: (iii) 75 + 225; (iv) 14 + 35; (v) 24 + 80 j. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Number System page. Click on the Grade 6 link. Then open the “show all” page and click the link “illustrations” under B.4. Solve completely all five tasks. 7.4  Simplifying and Operating with Numerical Expressions that Involve Integer and Fractional Exponents and Numbers in Scientific Notation in Grades 6 and 8

While middle school students learn to express powers of 10 with whole-number exponents in fifth grade (5.NBT.2), they deal with exponential expressions with whole-number exponents in sixth grade (6.EE.1) and all integer and simple fractional exponents in eighth grade (8.EE.1). Fractional exponents in eighth grade, however, are limited to square roots and cube roots, and integer exponents span positive, zero, and negative integers. Work with a pair and do the following tasks below. a. The exponential expression am with base a and exponent (or power or index) m, where a and m are positive integers, captures the sense in which a appears as a factor m number of times (6.EE.1). That is,

Search the internet for interesting everyday activities that will help eighth-grade students appreciate and understand exponential expressions. b. Which product grows faster, am or a × m, where a and m are positive integers? Use Geogebra to help you develop a visual solution. c. How might you use patterning, paper folding, or everyday contexts to help your students make sense of the expression a0, where a is a positive integer. d. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 6 link. Then open the “show all” page and click the link “illustrations” under A.1. Solve all three tasks completely. e. Complete Tables 7.2 and 7.3 below.

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Table 7.2. Powers of 2 24

16

23

8

22

4

21

 

20

 

2–1

 

2

–2

 

2

–3

 

Divide by ____ Or Multiply by ___

Table 7.3. Powers of 5 54

625

5

125

52

25

1

5

 

50

 

–1

5

 

5–2

 

5

 

3

–3

Divide by ____ Or Multiply by ___

Suppose a is a natural number and m is a positive integer. What can you conclude about the simplified form of a–m (8.EE.1)?

it seems to me that

 

Eighth-grader Chuck made the following claim: “Looking at Tables 7.2 and 7.3, .” What do you think?

(i) am · an = am+n; (ii) (ab)m = am . bm ; (iii) ; and (v)

114

.

 

f. Read content standard 8.EE.1. Search the internet for activities that will help eighth-grade students learn the following different properties of integer exponents in interesting ways: Assume a  ≠  0, m, and n are whole integers. , where m ≥ n; and (iv)

Expressions and Operations

g. Use the definition of exponents provided in (a) and the relevant properties of arithmetic to explain why the exponential rules in (f) are correct. h. What happens to rule (iii) in (e) if m < n? How is the result related to (e)? i. Authors Tom Parker and Scott Baldridge provide two different ways of thinking about the expression 00. The first way is to complete Tables 7.4 and 7.5 below. Table 7.4. Powers of 0 Table 04

0

3

0

0

02

0

1

0

 

00

 

Table 7.5. Whole Numbers Raised to 0 40

1

0

3

1

20

1

0

1

 

00

 

Complete the sentence: Drawing on Tables 7.4 and 7.5, 00 = ______? The second way is to use Rule (iii) in (e). Complete the following sentence: 00 = 03–3 =

= ______.

What can you conclude about 00? j. Students can use rule (v) in (f) to simplify simple expressions that are raised to a half and a third (i.e., square root and cube root). For example, and . Why, and how so? How can they use the rule to provide them with another way of verifying that is an irrational number (8.EE.2)? k. Eighth-grader Marla made the following comment regarding expressions with fractional exponents (i.e., radicals): “Radicals fail to satisfy our definition of an exponential expression, which involves repeated multiplication. For example, it doesn’t make sense to say is 2 repeated a half time?” How might you respond? 115

Chapter 7

l. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 8 link. Then open the “show all” page and click the link “illustrations” under A.1. Solve the two tasks completely. In fifth grade students use whole-number exponents to convey power-of-10 relationships in situations involving whole numbers and decimal numbers up to the thousandths place (5.NBT.2). In eighth grade, since the domain covers all integer exponents, they learn to express and manipulate numbers in scientific notation form (8.EE.3 and 8.EE.4). Continue working with a pair and do the following activities. m. Search the internet for at least five resources that you can use to introduce scientific notation to eighth graders. Engage in a content analysis and identify the basic concepts and processes that they would need to know in the beginning phase. Pay attention on possible misconceptions and other difficulties that might occur. n. How would you respond to students who think that learning scientific notation is a waste of time? o. Develop “friendly” sense-making and other mental activities that will help eighth-grade students perform simple simplification or estimation involving products and quotients of numbers in scientific notation (8.EE.3). For example, the quotient of is 20 since 5 ÷ 2.5 = 2, 1010 ÷ 109 = 10, and 2 × 10 = 20. When you develop such activities, make sure you include tasks that will require students to mentally draw on the rules stated in (f). p. How do you use your TI – 84 + to enter numbers in scientific notation and convert between scientific notation and standard form? q. Access the link below and solve all the applied problems that involve estimating quantities. http://tulsa.curriculum.schooldesk.net/Portals/Tulsa/Curriculum/docs/Math/ Grade%208/Unit%201/Lessons/Lesson%208.pdf r. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 8 link. Then open the “show all” page and click the link “illustrations” under A.3 and A.4. Solve all the tasks completely.

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7.5  Algebraic Expressions in Grades 6 and 7 and Algebra 1

Elementary students spend several years learning the structure of whole numbers and decimal numbers. They also deal with variables informally through tasks that ask them to solve for unknown whole-number values in equations that relate three whole numbers such as 5 + ? = 13 in first grade (1.OA.8) and ? × 5 = 45 and 7 = ( ) ÷ 3 in third grade (3.OA.4). In such cases, variables are conveyed through boxes, question marks, and other symbols that act as placeholders for specific unknowns. In middle school their understanding of variables undergoes several conceptual progressions. In sixth grade they formally pursue the idea of variables as letters that represent one or more numbers (6.EE.2). In sixth and seventh grade they manipulate algebraic expressions that contain terms with variables and their equivalent forms by drawing on their earlier experiences with numerical expressions (6.EE.2, 6.EE.3, 6.EE.4, and 7.EE.2). That is, like numerical expressions, algebraic expressions consist of, in the words of Colin Maclaurin, “operations and rules that are similar to those in common arithmetic, founded upon the same principles.” In Algebra 1 they continue to explore equivalent relationships in algebraic expressions and to attend to the meanings of variables through activities that situate algebraic expressions in particular contexts. In section 7.1(b) you learned that algebraic expressions consist of terms, factors, and coefficients. Aside from knowing such parts that define the structure of algebraic expressions, there is a way to classify them. Work with a pair and do the following tasks below. a. Draw on your knowledge of numbers from Chapters 3 and 4 to help you establish a real number system map that involves the following types of numbers: (i) counting numbers; (ii) natural numbers; (iii) whole numbers; (iv) zero; (v) terminating decimals; (vi) repeating decimals; (vii) nonrepeating decimals; (viii) fractions; (ix) positive integers; (x) negative integers; (xi) rational numbers; (xii) irrational numbers; and (xiii) real numbers. To help you create such a map, first record each number type in a post-it note. Next, reassemble the labeled post-it notes in a way that makes sense to you. Make sure that you present a well-reasoned and logical map showing correct and valid relationships between number types. b. Figure 7.1 shows 16 examples of algebraic expressions. Cut along the edges and classify the expressions in a way that makes sense to you.

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Figure 7.1. Types of Algebraic Expressions

Consider the monomial unit xn. If n is a whole number, then the monomial unit is a polynomial expression. For example, x2, x5, and x are polynomials. If n is an integer, then the monomial unit is a rational expression. For example,

, where x ≠ 0,

is a rational expression. Rational expressions are quotients of two polynomial expressions assuming, of course, that the denominator is not equal to 0. If n is a real number, then the monomial unit is a radical expression. For example, and are radicals. Continue working with your pair and address the following questions below. c. Note that other abstract entities (e.g., letters in the English or Greek alphabet system) can replace the variable x in the monomial unit xn. Generate examples of polynomials, rational expressions, and radicals involving different letter variables. d. Are polynomial expressions rational expressions? Are rational expressions polynomial expressions? Are polynomial and rational expressions radical expressions? Are radical expressions polynomial and rational expressions? Explain. 118

Expressions and Operations

e. The above classification for monomial algebraic expressions can also be extended to terms that have two or more variable factors. For example, the monomial expression ambn has variables a and b as factors. Is the expression a–1b a polynomial, rational, or radical expression? Explain. f. Is the expression

the same as the expression

? Explain.

g. The above classification for the monomial unit can be extended to binomial, trinomial, and multinomial algebraic expressions. Generate examples of binomial expressions that are (i) polynomials; (ii) rational expressions; and (iii) radicals. h. Are real numbers polynomial expressions? Explain. i. Reclassify the algebraic expressions in Figure 7.1 by type of expression. 7.5.1 Constructing, Simplifying, and Factoring Algebraic Expressions in Grades 6 and 7 and Algebra 1 Continue working with your pair and do the following tasks below. j. Read content standards 6.EE.2 through 6.EE.4 and 7.EE.1 to 7.EE.2. What kinds of algebraic expressions are covered in sixth and seventh grade? k. Focus on content standard 6.EE.2b. Consider the numerical expression 3(3 + 7). i. Identify the parts of the numerical expression using the appropriate mathematical terms. ii. Sixth-grader Marco claims that the given expression is a binomial. How do you respond? iii. Sixth-grader Marcia read the given expression as follows: “Three times three plus seven.” Was she correct in the way she read the expression? Explain. l. Focus on content standards 6.EE.2a, 6.EE.2c, 6.EE.3, and 6.EE.4 that show examples of algebraic expressions. What is the nature of coefficients that matters in sixth grade? Compare the examples with the ones stipulated in seventh grade under content standards 7.EE.1 and 7.EE.2. What does “linear expressions with rational coefficients” mean? m. Focus on content standard 6.EE.2c. What does it mean to evaluate expressions? What do students learn when they evaluate expressions? n. Read content standard 7.EE.1. How is this standard related to content standards 6.EE.3 and 6.EE.4? o. Search the internet for resources that teach students how to use algebra tiles, algeblocks, or other virtual manipulatives to learn content standards 6.EE.3

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Chapter 7

and 6.EE.4. What does it mean to simplify an expression? What mathematical concepts and academic language matter? p. How does knowing content standard 7.EE.2 provide students with a different way of perceiving tasks that involve simplifying expressions? q. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Do the following sets of tasks below. i. Open the Grade 6 link. Then open the “show all” page and click the link “illustrations” under A. Solve the tasks under 6.EE.A and 6.EE.A.2 completely. ii. Open the Grade 7 link. Then open the “show all” page and click the link “illustrations” under A. Solve all four tasks completely. r. In 6.NS.4 students use the concept of GCF and the distributive property of multiplication over addition to express sums of numbers with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, 2 + 4 = 2(1 + 2). One consequence of learning content standard 6.EE.3 involves factoring simple polynomial expressions by using the GCF. Demonstrate how the same arithmetical principles can be applied to the following situations below involving polynomials with or without the aid of concrete or virtual manipulatives. (i) 12x + 9;  (ii) 9 + 6b;  (iii) 15x + 24y;  (iv) 3m + 12n + 15p. 7.5.2 Adding, Subtracting, Multiplying, and Factoring Polynomials in Algebra 1 The algebraic expression below represents the general form of a polynomial in a single variable x. anxn + an−1xn−1 + an−2xn–2 + .... + a2x2 + a1x + a0, where n is a whole number, an,an−1,an–2,....,a2,a1,a0 are real numbers, and an ≠ 0. The degree of the polynomial is n, which refers to the highest power of the variable x. If n = 0, the polynomial is a constant real number. If n = 1 the polynomial is a linear expression. If n = 2 the polynomial is a quadratic or second-degree expression. If n =3 the polynomial is a cubic or third-degree expression. If n = 4 the polynomial is a quartic or fourth-degree expression. If n = 5 the expression is a quintic or fifthdegree polynomial. For polynomials with degree greater than 5, students may simply refer to them as degree-n expressions. The concept of degree in expressions applies to polynomials only. In cases of polynomials having two or more variables that are each raised to an exponent, the degree is simply the sum of the exponents. For example, the polynomial 3x2y3 has degree 5 while the polynomial xy has degree 2 (why?). Every polynomial expression has a leading coefficient which corresponds to an; it is the coefficient of the term that contains the highest exponent of the variable x. The real number a0 is the constant term of the polynomial expression. Continue working with your pair and do the following tasks below. 120

Expressions and Operations

s. In Algebra 1 students deal with both linear and nonlinear polynomial expressions. Search the internet for resources that use algebra tiles, algeblocks, or other virtual manipulatives to help students visualize polynomial expressions. t. How might you use manipulatives to help your students understand the difference between 2x and x2, 2y and y2, and x2y and xy2? u. What mathematical concepts and academic language do Algebra 1 students need to fully understand the following processes below? i. adding polynomials; ii. subtracting polynomials; iii. multiplying polynomials. v. Identify possible difficulties that Algebra 1 students might have in learning each operation listed in (t). w. Read content standard A-APR.1. Prove that the system of polynomials is closed under the operations of addition, subtraction, and multiplication. x. Martha and James, Algebra 1 students, made the following remarks in relation to content standard A-APR.1: James: A  dding, subtracting, and multiplying polynomials seem to resemble the way we perform place-value addition, subtraction, and multiplication involving whole numbers. Martha: I agree except that we do not have to worry about regrouping and all that stuff. What do you think? y. An Algebra 1 teacher made the following comment in relation to adding and subtracting polynomials: “It seems to me that my students will need to draw on their multiplicative thinking ability quite extensively.” Do you agree? Explain. z. Refer to Figures 3.4 and 3.5 in section 3.2.2 which illustrate an area model for obtaining the products of two whole numbers. Can Algebra 1 students use the same model to help them make sense of multiplying two polynomial expressions? Illustrate. aa. Algebra 1 teacher Mr. Bilbo made the following comment in relation to multiplying polynomials: “When I teach my students to multiply polynomials, I use the FOIL approach: First; Outer; Inner; Last. For example, (3a + 1)(2a + 3) = (3a)(2a) + 3a(3) + 1(2a) + 1(3) = 6a2 + 9a + 2a + 3 = 6a2 + 11a + 3.” Discuss advantages of and concerns with this method. Compare this method with the area model in item (y). bb. In item (r) you learned to factor polynomial expressions by using the GCF. A second factoring technique involves the difference of two squares a2  −  b2 121

Chapter 7

(A-SSE.3a). To obtain the correct factors, consider the two squares shown in Figure 7.2 and answer the questions that follow.

Figure 7.2. Factoring the Difference of Two Squares

i. Replace each question mark in Figure 7.2 with the correct expression. ii. Use a pair of scissors to help you cut out the smaller (gray) square from the larger square. What expression corresponds to the area of the “new” figure? iii. Cut along the broken segment of the “new” figure to produce two trapezoids. Reconfigure the two trapezoids to form a rectangle. What expressions correspond to the dimensions of the rectangle? iv. Explain why a2 − b2= (a – b)(a + b). Then use this fact to obtain the factors of the following expressions: x2 – y2; 4m2 – n2; 25a2 – 36b2; (2x + 1)2 – 9b2; (3 – 2y)2 -49y4; x4 – y2; 16m4 – 81n4; (A-SSE.2). v. Use factoring by GCF and difference of two squares to factor the following expressions completely: 27x2 – 3y2; 16a2 – 36b2; 9x3 – 4x. vi. Can you factor a sum of two squares, a2 + b2, in such a way that you generate products of expressions with rational number coefficients? (Note: It is possible to factor polynomials into products of expressions having complex number coefficients, but this particular content expertise is pursued in Algebra II.) cc. In items (r) and (aa) you learned to factor polynomial expressions by using the GCF and the difference-of-two-squares structure. Another interesting factoring task involves obtaining factors of quadratic trinomial expressions whose basic structure takes the form ax2 + bx + c, where  and a, b, and c represent rational numbers (A-SSE.3a). Do the following tasks: i. Why is the expression ax2 + bx + c called a quadratic trinomial? ii. Search the internet for online resources that use algeblocks or algebra tiles to help students obtain factors of quadratic trinomials (A-SSE.2). iii. An Algebra 1 teacher made the following comment: “It seems to me that obtaining the factors of a quadratic trinomial is like finding the dimensions of a rectangle with a known area.” What do you think? 122

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iv. The 3 × 3 tic-tac-toe method shown in Figure 7.3 is a trial-and-error factoring strategy that can be used to obtain the two factors of any quadratic trinomial (A- SSE.2). It has been derived from students’ visual experiences with algeblocks or algebra tiles. The second column in a tic-tac-toe diagram contains the three terms of a trinomial expression in standard order. To obtain the factors of the trinomial you first generate factors for the first and last terms. Then you take the sum of the corner products along the two diagonals of the square to see if it matches the linear term of the trinomial. If the match is correct then the correct factors are shown under the first and third columns and read top to bottom (or bottom to top) as sums of binomials. If the match is not correct you need to consider other combinations of factors. Figure 7.4 shows two additional examples involving more complicated cases of factoring quadratic trinomials.

Figure 7.3. Tic-Tac-Toe Method for Factoring General Trinomials

2x2 + x – 6 = (2x – 3)(x + 2)  6x2 – 17x + 5 = (3x – 1)(2x – 5)

Figure 7.4. Applying the Tic-Tac-Toe Method on Two Quadratic Trinomials

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v. Use the tic-tac-toe method to factor the following quadratic trinomials: 5y2 – 13y + 6; 8y2 + 10y – 3; 6a2 – 13a + 5; 64x2 – 9; 9m2  + 12m + 4; and (x – 2)2 –5(x – 2) + 6. vi. Use the GCF and tic-tac-toe method to obtain the factors of the following  expressions: 4x3 + 4x2 + x; 5xy – 3y + 2yx2; and 2x2 (2 – x) + 3x (2 – x) – x + 2. vii. Algebra 1 students should know that a prime number is a whole number whose only factors are 1 and itself. Give examples of quadratic trinomials that are irreducibly prime, that is, they cannot be expressed as products of two factors with rational number coefficients. 7.5.3 Understanding Simple Nonalgebraic (Transcendental) Exponential Expressions in Algebra 1 In the previous sections you learned about the different kinds of algebraic expressions by imposing restrictions on the monomial expression xn. You also learned to generate terms and factors by forming arithmetical combinations of such monomial expressions and their multiples. Expressions of the type nx, where x is a variable and nx is a real number, are different, however. They are called (nonalgebraic) transcendental expressions. Continue working with your pair and do the following tasks below. dd. Search online for the meaning of the term transcendental expressions. How are they different from algebraic expressions? ee. Use either a TI–84+ or a Geogebra to plot the following sets of points on the same coordinate plane. Context: You are building a sequence of growing squares.

124

Square with side length of n units

Area A of square in square units

1

1

2

4

3

9

4 5

Context: A certain bacteria colony on a petri dish doubles its population every hour. Assume that you start with 1 bacterium. Hour Population or amount of bacteria (t) P at the end of each hour 0

1

16

1

2

25

2

4

3

8

4

16

5

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i. For each set of points connect the points to form a graph. What do you see? ii. For each set of points determine a generalization in the form of a direct expression that will enable you to conveniently predict or calculate an output for any given input (i.e., A in terms of n and P in terms of t). iii. How does this activity help Algebra 1 students see the difference between the algebraic expression x2 and the transcendental expression 2x? Exponential expressions in Algebra 1 generally take the form C × at or Cat, where C is a real number, a is a positive real number, and . Continue working with your pair and do the following tasks below. ff. Generate several different examples of exponential expressions. gg. What happens if a = 1 in Cat? What happens when a assumes a negative value in Cat? hh. Certain drugs or substances are subject to a half-life condition. Conduct an online search to help you understand what this condition means. Then consider the following problem below. The half-life of a certain radioactive substance is 1 hour. You begin with 50 grams of this substance. What is a direct expression for the amount of substance remaining after t hours? How might you help Algebra 1 students process and translate the given problem in mathematical form? ii. Access the link http://www.illustrativemathematics.org/standards/hs. Open the Algebra page. Open the Seeing Structure in Expressions page and access the “illustration” task under B.3c. Solve the indicated task completely. 7.5.4 Deepening Variable Understanding in Algebra 1 Work in groups of three and accomplish the following tasks below. jj. Access the link http://www.illustrativemathematics.org/standards/hs. Open the Algebra page. Open the Seeing Structure in Expressions page and access the “illustrations” tasks. Solve the following tasks completely: (i) Mixing Fertilizer; (ii) Increasing or Decreasing? Variation 1; (iii) Quadrupling Leads to Halving; (iv) Kitchen Floor Tiles; (v) Animal Populations; (vi) Delivery Trucks; (vii) Delivery Trucks; and (viii) Seeing Dots. kk. Consider the tasks you solved in (a) which represent examples of problems that students pursue relative to content standard A-SSE.1. What do you want your Algebra 1 students to learn from such tasks that target interpreting algebraic expressions in different contexts? What makes such tasks different from those noted in content standard A-APR.1?

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ll. Read content standards 6.EE.3, 6.EE.4, 7.EE.2, and A-SSE.2. What mathematical knowledge do students acquire when they work on tasks that address these standards? 7.6  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard listed in Table 7.1 with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 7.7  Developing a Content Standard Progression Table for the Expressions Domain

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving several domains are coherently developed.

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Equations and Inequalities

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the equations and inequalities domain of the CCSSM from Grades 6 to 8 and Algebra 1. Table 8.1 lists the appropriate pages for your convenience. To better appreciate the clusters of content standards that deal with equations and inequalities in school mathematics, it is helpful to begin with a clear description of algebra, at least one that is compatible with the CCSSM at the middle school level. This matter is pursued in section 8.1 through Albrecht Heeffer’s two views of algebra, one nonsymbolic and the other symbolic. The ideas that are pursued in that particular section, in fact, provide the foundation for understanding the remaining sections in this chapter and the changing contexts of algebra from middle school to Algebra 1. Table 8.1. Page Reference for the Equations and Inequalities Domain in the CCSSM Grade Level Domain

Standards

Page Numbers in the CCSSM

6.EE

5–9

44

7.EE

4

49

8.EE

7–8

54–55

N-Q

1–3

60

A-SSE (Algebra 1 Traditional Pathway)

3a, 3b

64

A-CED (Algebra 1 Traditional Pathway)

1–4

65

A-REI (Algebra 1 Traditional Pathway)

1, 3–7, 10–12

65–66

8.1  Algebraic Thinking from Grades 6 to 8 and Algebra 1

Take some time to reflect on your early experiences in mathematics. As you engage in reflective thinking, access the first three sites below, which capture attitudes that people have in relation to the subject of algebra. Gather at least ten to fifteen more examples online and try to categorize them in a way that makes sense to you. Assemble them in a mini-poster. When you are done, access the fourth link below which should take you to Andrew Hacker’s controversial

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July 2012 opinion essay regarding the problematic status of algebra in American education. • http://luannesmath.files.wordpress.com/2012/07/equation-cartoon.jpg • http://marklolson.files.wordpress.com/2009/03/sidney-harris-cartoon-a-miracleoccurs-here.gif?w=300&h=364 • http://picrust.files.wordpress.com/2011/08/algebra-cartoon1.jpg • http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary. html?pagewanted=all Share your poster with a pair and discuss categories and responses that are similar and different. Why does algebra over other mathematics subjects receive so much critical reception from the public that appears to be more negative than positive? What can you do to help change the negative reception towards the subject? Logician and philosopher of science Albrecht Heeffer’s reflections in relation to a recent reinterpretion of the early Babylonian problem texts have led him to suggest two views of algebra, which should help clarify the nature of algebraic thinking at the middle school level in the CCSSM. The paragraph below describes Heeffer’s characterization of nonsymbolic algebra and symbolic algebra. Read it slowly and draw on your early mathematical experiences to see whether the distinction between the two views of algebra makes sense to you. Let us call (nonsymbolic) algebra an analytical problem-solving method for arithmetical problems in which an unknown quantity is represented by an abstract entity. There are two crucial conditions in this definition: analytical, meaning that the problem is solved by considering some unknown magnitudes hypothetical and deductively deriving statements so that these unknowns can be expressed as a value, and an abstract entity that is used to represent the unknowns. This entity can be a symbol, a figure, or even a color …. More strictly, symbolic algebra is an analytical problem-solving method for arithmetical and geometrical problems consisting of systematic manipulation of a symbolic representation of the problem. Symbolic algebra thus starts from a symbolic representation of a problem, meaning something more than a shorthand notation. One easy way to think about the two views is to consider CCSSM practice standard 2, reasoning abstractly and quantitatively, which is another way of interpreting symbolic and nonsymbolic algebraic approaches to solving mathematical problems. Practice standard 2 emphasizes both aspects of contextualizing and decontextualizing in problem solving activity. Applied to arithmetical problem solving, contextualizing and nonsymbolic algebra map well together since entities (e.g., whole numbers and fractions) and relationships emerge in the context of arithmetical problems and that the various operations and manipulations are seen as applications of certain arithmetical processes. Decontextualizing and nonsymbolic algebra also map well together in which case students attend to the analytical methods that enable them to manipulate the representing symbols as if they have a life of their own, without 128

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necessarily attending to their referents. In the following statement below, also taken from Practice Standard 2, think of numbers and their units as representing abstract entities and the relevant calculations and properties of operations and objects that are used to manipulate the entities as the analytic method. From the description, the initial phase in quantitative reasoning is appropriately nonsymbolic algebraic in character, as follows: Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Hence, arithmetical problem solving is not just a simple matter of doing arithmetic since the underlying processes are nonsymbolic-algebraic in context. Any instruction that simply takes a nonalgebraic approach to problem solving is most likely going to produce students who can only demonstrate rote learning and problem spotting. For example, mathematical tasks that simply ask students to basically apply steps in a procedure without providing them with an opportunity to understand some underlying algebraic structure will most likely cultivate a view in which arithmetic and algebra are seen as two separate and distinct subjects. Another example is illustrated in the two contrasting responses below from two fictitious seventh-grade students named Myrna and James. The two responses capture common and daily occurrences of thinking among older students who are learning to add two signed numbers for the first time. James relied heavily on a binary mat (see Figures 4.2 to 4.4) to help him obtain sums of positive and negative integers. Unfortunately the merely arithmetical and nonalgebraic character of his thinking prevented him from seeing meaningful analytical relationships across the two tasks. Myrna’s nonsymbolic algebraic approach enabled her to reason in a more efficient and sophisticated manner. Her consistent responses reflected the use of a deductively closed argument that linked the operations of addition and subtraction involving integers. Also, while Myrna clearly understood James’s concrete process for adding signed numbers, she further established a systematic way of manipulating them without having to depend on the concrete representations. More formally, Myrna demonstrated the use of a deductively-driven analytical method for adding integers, unlike James who primarily engaged in a nonalgebraic arithmetical way of combining quantities, that is, he was simply doing arithmetic. Teacher: Show me how to find the sum of -4 and 8. James, seventh-grade student, initially constructs a binary mat. He then draws four minuses and eight pluses in the appropriate regions. James: 1, 2, 3, 4. There are 4 plus in the positive region. 1, 2, 3, 4, 5, 6, 7, 8. There are 8 minuses in the negative region. So I see one, two, three, and four zero pairs. I take them out, which leaves me 4 pluses. So, -4 + 8 equals 4. 129

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Teacher: Good job, James. Do we have other ways of finding the sum? Myrna raises her hand and offers another suggestion. Myrna: It looks like subtraction to me, like 8 – 4. I take away 4, which to me means taking away 4 zero pairs, leaving me with +4. Teacher: Good job, Myrna. How about finding the sum of 23 and -13? James: That’s a lot. So I have I, 2, 3, 4, …, 21, 22, and 23 in the plus region. Then 1, 2, 3, 4, …, 11, 12, 13 in the negative region. So, 1 and -1 form a zero pair, another 1 and -1 form another zero pair, …, so there are 13 zero pairs, which are all equal to 0, so I have 1, 2, 3, …, 10 positives left. So, 23 + -13 is 10. Myrna: Oh, it should be 10 since 23 take away 13 leaves me with 10 since it’s like taking away 13 zero pairs. Even if the problem has been changed to -23 + 13, you think the same way. So 23 – 13 is 10, but since there are more negatives than positives, the correct answer is -10. Teaching to the CCSSM means that all students need algebra to do arithmetic, a view that fits nicely with Heeffer’s point about the role of “deductively deriving statements” in solving mathematical problems. Otherwise every problem is a new problem as James and many other older students tend to demonstrate in mathematical activity. Suffice it to say, algebra is not something that is simply imposed from the outside like ornaments on a Christmas tree. Meaningful arithmetical activity needs to be nonsymbolic algebraic at the very least in order to help students avoid the gap trap,which sees arithmetic and algebraic as two different types of mathematical activity. Algebraic thinking through fifth grade in the CCSSM employs numbers as entities. Furthermore, all students should have been provided with an informal introduction to variables by the end of fifth grade. In fact, their initial exposure to variables takes place in first grade. They learn them as unknowns that have fixed values (e.g., what unknown number goes in the blank in order to make the equation true: 7 = __ - 3 (1.OA.8)) and placeholders that can represent a range of values (e.g.: the situation * + ? yields the same result as ? + * (1.OA.3)). In fourth grade, they also employ variables indirectly in the context of patterning activity (e.g., given the rule “add 3” and the starting number 1, generate the first ten terms in the resulting sequence and identify and justify patterns you observe (4.OA.5)). In sixth grade, algebraic thinking includes using letter variables formally to represent one or more numbers in various arithmetical situations (6.EE.2). When elementary students through fifth grade use numbers as placeholders in order to demonstrate an arithmetical method, such numbers are often referred to as quasi-variables that operate within the context of a general power of a method. Some scholars refer to the latter as a scheme, that is, a pattern of action, while other scholars prefer to call it structure. Both terms can be used interchangeably for the time being. 130

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Heeffer notes that among Babylonian (and Arabic) mathematicians, the absence of letter variables in the way they processed arithmetical problems did not deter them from grasping the abstract nature of the unknowns and the analytic procedures that they used to solve the problems. While their solutions to problems primarily employed numbers, they, in fact, used them as quasi-variables, conveying how they actually perceived the generality of the corresponding methods of computation that applied to all problems that belonged to the same category. In such cases, variables are intuited or tacit in the examples with numbers. Visual diagrams can also be used to introduce middle school students to variables as unknowns that represent specific values. For example, the US-adopted Singapore texts consistently employ bar diagrams in arithmetical problem solving as visual placeholders for letter variables. Consider the following sixth-grade California Department of Education Practice SBA task below, which involves setting up and solving an equation (6.EE.7). Investigate how the bars in Figure 8.1 have been used to process and solve the problem and the corresponding translation of an appropriate mathematical relationship in the form of an equation.

Figure 8.1. A Visual Approach to Solving a Simple Linear Equation in Sixth Grade

Various pattern generalization tasks such as the Building a Hexagon Flower Garden Design in section 2.2 provide another instance of nonsymbolic algebraic activity. Fifthgrade students learn to abduce – that is, look for and make use of structures (Practice Standard 7) – and induce – that is, express regularity in repeated reasoning (Practice Standard 8), two practice standards that also reflect core processes in the construction of generalization in mathematics. Such tasks are nonsymbolic because the structures and generalizations that emerge are still linked to the relevant contexts. For example, the generalizations offered by Ava, Bert, and Ces in relation to the pattern shown in 131

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Figure 2.2 have been drawn from the assumptions that the three students employed in that particular activity. In section 9.5, you will deal with pattern generalization once more in both nonsymbolic and symbolic algebraic contexts. All arithmetical problems require students to manipulate numbers, perform operations, and establish general mathematical relationships in a decontextualized activity. These requirements enable them to experience the symbolic algebraic nature of arithmetical activity early in their learning of mathematics. For example, sixthgrade students learn to use the distributive property of multiplication over addition to represent the expression 225 + 75 with a common factor in terms of a multiple of a sum of two whole numbers with no common factor, which introduces them to simple factoring using the greatest common factor (6.NS.4). Thus, symbolic algebra becomes a tool for manipulating the representing symbols as if they have a life of their own, without necessarily attending to their referents (Practice Standard 2). Hence, while it is easy to dismiss decontextualized arithmetical problems as being all about producing answers or results by applying the correct procedures, asking students to provide reasons, showing work, and discussing them in class together will help reinforce the important role of algebraic thinking in such arithmetical contexts. Suffice it to say, teaching and learning about numbers, operations, equivalent relationships, and patterns at the CCSSM middle school level involve conceptualizing the relevant abstract entities and analytic processes in an algebraic context. If you investigate the clusters of content standards in the number, algebra, operations, and geometry domains carefully from sixth through eighth grade, they address the following concepts and processes below, which characterize the big ideas in symbolic algebra in high school. • (6–8) Implementing the four fundamental operations as general methods for combining and generating integers, rational numbers, and decimal numbers; • (8) Understanding the real number system as consisting of rational and irrational numbers; • (6–8) Writing, forming, and interpreting equivalent numerical and variable-based expressions, equations, and inequalities; • (6–8) Understanding the general meaning of the equal sign; • (6–8) Seeing solutions of arithmetical problems in a general way; • (6–8) Simplifying expressions, solving single and pairs of linear equations and inequalities, and reasoning about them; • (6–8) Pattern generalizing as instantiations of nonsymbolic and symbolic algebraic structures; • (6–8) Interpreting arithmetical and patterning activity in geometric terms; • (6–8) Representing and analyzing quantitative and functional relationships between dependent and independent variables and quantities. Hence, all fundamental and important algebraic concepts and processes in Algebra 1 and beyond reflect, in the words of Colin Maclaurin, a famous mathematician, “operations and rules similar to those in common arithmetic, founded 132

 Equations and Inequalities

upon the same principles.” Teaching arithmetic to middle school students should, thus, strive to help them view and employ nonsymbolic algebra in concrete contexts. 8.2  DIFFERENT Contexts for Using Variables from Grades 6 to 8 and Algebra 1

Work with a pair and do the following tasks below. a. Read content standards 6.EE.5, 6.EE.6, 6.EE.7, 6.EE.8, 7.EE.4a, and 7.EE.4b. How are students expected to use variables in such contexts? b. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 6 link. Then open the “show all” page and click the link “illustrations” under B. Analyze all the tasks under each content standard. c. Read content standard 7.EE.4. Process the two problems identified under each substandard and then determine whether changes are needed in seventh-grade students’ use of variables. d. Read content standards 8.EE.7, 8.EE.8, A-CED.1, A-CED.3, and A-REI.2 through A-REI.7. Work out the sample tasks associated with each content standard and determine whether changes are needed in eighth-grade and Algebra 1 students’ use of variables. e. Read content standards A-CED.4 and A-REI.3. Then access the link https://www. illustrativemathematics.org/standards/hs. Open the Algebra page, the “show all” page under A-CED, and “illustrations” under standard 4. Analyze the two tasks completely. How are variables used in such contexts? f. Read content standards 6.EE.9 and A-CED.2. Then access the link http://www. illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 6 link and open the “illustrations” page under 6.EE. Analyze the task listed under 6.EE.9 (Chocolate Bar Sales). When you have completed your analysis, analyze an Algebra 1 problem task that targets the same content standard. Open the link https://www.illustrativemathematics.org/standards/hs. Click on the Algebra domain and “illustrations” under the A-CED cluster. Analyze the task Clea on an Escalator completely. How are variables used in such contexts? g. What are the different uses of variables from sixth grade to Algebra 1? 8.3  Solving Equations from Grades 6 to 8 and Algebra 1

Work with a pair and accomplish the following tasks below. a. Read content standard 6.EE.5. What does it mean to solve an equation in one variable? How does it differ from simplifying expressions? Read content standard 8.EE.8a. What does it mean to solve: (1) an equation in two variables? (2) a system of two linear equations in two variables?

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b. The following content standards below pertain to the process of solving equations from sixth grade to eight grade and Algebra 1. 6.EE.7; 7.EE.4a; 8.EE.7a & b; 8.EE.8a through c; A-CED.1 through 4; A-REI.1 through 7 (except 2); and A-REI.10 through 12.  onstruct a concept map showing one way of helping students develop their C ability to solve equations from sixth grade to eighth grade and Algebra 1. Use your concept map to respond to the following question: What does it mean to solve equations algebraically and graphically? c. What are equivalent expressions and equations? What are the different properties of equations, and why do students need to learn them? d. Algeblocks or algebra tiles can be used to help students understand the process of solving linear and quadratic equations. The document below provides instructions on how to solve different types of linear equations in one variable using algeblocks. Read pages 3-8 and use the properties of equations to provide a justification for each visual step in the solution process (A-REI.1). http://scsweb1.spotsylvania.k12.va.us/instruction/LinkClick.aspx?filetic ket=7uRCW83gcuU%3D&tabid=4609 e. Content standard A-REI.4 recommends several different ways of solving quadratic equations in one variable. Simple quadratic equations of the type x2 = a2 can be solved by inspection. Using algeblocks, one possible solution to the equation x2 = 4 is x = 2 by simply comparing their sides (Figure 8.2). However, since x is a real number, x  =  −2 is another possible solution since (−2)2 = 4. Solve the following equations: x2 = 16, 3y2 = 27, x2 − 1 = 8 and z2 = 5. Discuss potential issues that students might encounter when they solve such equations.  uadratic equations of the type ax2 + bx + c = 0 can be solved by: (1) factoring; Q (2) completing the square; and (3) using the quadratic formula. Refer to section 7.5.2 to help you recall the algeblocks-mediated process for factoring quadratic

Figure 8.3. Using algeblocks to solve x2 = 4

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trinomials. When we solve quadratic equations by factoring, we use the zeroproduct theorem, which states that if ab = 0, then either a = 0 or b = 0, where a and b are real numbers. Solve the following quadratic equations by factoring: x2 −5x + 6 = 0, 2s2 + 5s = 8, and x2 − 4 = −3x. Discuss potential issues that students might encounter when they solve quadratic equations by factoring. How might you justify the zero-product theorem in class?  lgeblocks can also be used to help students understand the process of solving A quadratic equations by completing the square. For example, consider the equation x2 +4x = 8. Figure 8.4 provides an algeblocks-based representation of the equation.

Figure 8.4. Visual Representation of x2 + 4x = 8

 otice that if we reconfigure the square and 4 linear blocks to form an incomplete N bigger square, we would need to add 22 =4 unit squares in order to complete the bigger square. Figure 8.5 shows the completed square on the left and the 12 unit cubes on the right.

Figure 8.5. Visual Representation of (x + 2)2 = 12

Note that x + 2 =  number, x + 2 = −

, or x  =  −2 + , or x = −2 −

(why?). However, since x is a real . We record this process in the following

manner below. 135

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(x+2)2 = 12

|x+2| =  x+2 = ±  x = −2 ±

(Why?) (Why?) (Why?)

olve the following quadratic equations by completing the square: S x2  −5x + 6 = 0, 2s2 +5s  = 8, x2  −4  = −3x, and x2 +bx = c. Discuss potential issues that students might encounter when they solve quadratic equations by completing the square.  he method of completing the square can be used to help students understand and T derive the quadratic formula. Start with the general quadratic equation ax2 + bx + c = 0. Use the method of completing the square to solve for the two values of x, which take the following general formula: x= 

.

Then use the general formula to solve the following quadratic equations: x2 −5x + 3 = 0, 3y2 = 2y + 2, and m2 = 0.2m + c. Discuss potential issues that b students might encounter when they solve quadratic equations by using the quadratic formula.  rom the preceding discussion, it quadratic equations in a single variable can be F solved in several different ways. When is it appropriate to use one method over another? f. Algebra 1 students can also obtain the solutions of equations by using technological tools such as graphing calculators and Geogebra. Read standard A-REI.11 and then search the internet for resources that will help you learn a graphical process based on the ideas stipulated in the standard.  hat concepts do students need to in order to successfully model a graphical W approach when they solve equations?  ome parents and teachers are concerned about encouraging Algebra 1 students S to use such tools to help them solve equations. Do you agree and why? What benefits do Algebra 1 students gain from learning the technology-based process for solving equations? What issues might arise when Algebra 1 students rely too heavily on such tools to help them solve equations?

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 Equations and Inequalities

g. What are the different types of equations and why do students need to learn them? How can you assist them to understand the different types algebraically and graphically? h. Access the link https://www.illustrativemathematics.org. Explore tasks that pertain to the content standards listed in (b) and identify possible teaching and learning issues when they are used in class. 8.4  Solving Inequalities from Grades 6 to 8 and Algebra 1

Work with a pair and accomplish the following tasks below. a. Read content standard 6.EE.5. What does it mean to solve an inequality in one variable? Read content standard A-REI.12. How are students expected to understand the solutions of a: (1) linear inequality involving two variables; (2) system of linear inequalities in two variables? Reread the responses you provided in 8.3(a) and determine whether they fit with the responses you provided in the case of solving inequalities. b. The following content standards below pertain to the process of solving inequalities from sixth grade to eight grade and Algebra 1. 6.EE.8; 7.EE.4b; A-CED.1 and 3; A-REI.3 and 12.  onstruct a concept map showing one way of helping students develop their C ability to solve inequalities from sixth grade to Algebra 1. What does it mean to solve inequalities algebraically and graphically? c. What are equivalent inequalities? What are the different properties of inequalities and why do students need to learn them? Janice, seventh-grader, makes the following claim: “Solving an inequality is like solving an equation. All the properties we use to solve an equation work for any inequality.” Is she correct? Why or why not? d. Search the internet for resources that will allow you to solve inequalities by using technological tools. What concepts do students need to in order to successfully model a graphical approach when they solve inequalities? What issues might arise when they solve inequalities graphically with the aid of such tools? e. What are the different types of inequalities and why do students need to learn them? How can you assist them to understand the different types algebraically and graphically? f. Access the link https://www.illustrativemathematics.org. Explore tasks that pertain to the content standards listed in (b) and identify possible teaching and learning issues when they are used in class. How are they similar and/or different from tasks that you explored in the preceding section?

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8.5  Word Problem Solving Applications Involving Equations and Inequalities from Grades 6 to 8 and Algebra 1

Work with a pair and accomplish the following tasks below. a. Access the link https://www.illustrativemathematics.org. Explore context-based problem solving tasks that pertain to the following content standards listed below. 6.EE.7 and 8; 7.EE.4a and 4b; and A-CED.1 through 4  ompare and contrast tasks across the grade levels. By the end of Algebra 1, C what impressions might students have about what it means to solve problems in mathematics? 8.6  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard listed in Table 8.1 with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 8.7  Developing a Content Standard Progression Table for the Equations and Inequalities Domain

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving several domains are coherently developed.  

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Chapter 9

Functions and Models

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of functions and models in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 9.1 lists the appropriate pages for your convenience. Functions in the CCSSM are formally defined in eighth grade – that is, as rules that assign to each input exactly one output. However, all students learn about the defining attributes of a function beginning in fifth grade. In fifth grade they learn to convey rules of relationships between two quantities in the context of patterns. In sixth grade they learn to represent quantitative relationships between independent and dependent variables. In seventh grade they pursue direct proportional relationships, which exemplify one particular instance of functions. In eighth grade they translate among and interpret the different representations of functions in a variety of contexts and problem situations. They also learn about linear functions and linear function modeling involving two quantities. In Algebra 1 they continue to interpret and model functional relationships in both linear and nonlinear contexts. They acquire all the formal terms and notations and build new functions from known functions in the context of their experiences with linear, quadratic, absolute value, step, exponential, and sequence functions. Across grade level, students learn that meaningful and well-defined rules, expressed in either recursive or explicit form, oftentimes convey functional relationships between two quantities in nonsymbolicand symbolic-algebraic contexts. Models and modeling play a central role in students’ mathematical understanding of functions in the CCSSM. While constructing multiple representations of functions (i.e., tables, equations, graphs, etc.) provides them with an opportunity to explore equivalent relationships, having them interpret and build models allows them to see why and how such representations emerge in the first place. Take a moment to learn more about models and modeling by reading: (1) the contexts in which all students model with mathematics, which is one of eight standards for mathematical practice (NGACBP & CCSSO, 2012, p. 7); and (2) the introduction to modeling, which is a conceptual category in high school mathematics (ibid, pp. 72-73). Work with a pair and address the following questions: i) What are models in mathematics? ii) What does mathematical modeling entail and what situations enable students to engage in modeling activity? iii) What insights can students develop from mathematical models and modeling?

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Table 9.1. Page Reference for the Functions and Model Domain in the CCSSM Grade Level Domain 5.OAT

Standards

Page Numbers in the CCSSM

3

35

6.EE

9

44

7.RP

2 (a to d)

48

8.EE

5-6

54

8.F

1 to 5

55

A-SSE (Algebra 1 Traditional Pathway)

3b

64

F-IF (Algebra 1 Traditional Pathway)

1 to 6, 7a, 7b, 7e (linear, quadratic, absolute value, step, piecewise-defined, and exponential functions), 8, and 9

69-70

F-BF (Algebra 1 Traditional Pathway)

1a, 1b, 2, 3 (linear, quadratic, absolute value, and exponential only), and 4a (linear only)

70

F-LE (Algebra 1 Traditional Pathway)

1-3, 5

70-71

iv) Explore the mathematical modeling cycle diagram presented on p. 72. Provide examples of middle school mathematical problems that can be used to illustrate the process. v) Compare descriptive modeling from analytic modeling. Provide examples from middle school mathematics that illustrate each type. 9.1  Rules, Patterns, and Relationships in Grade 5

Well-defined patterns in mathematics have explicit rules that describe relationships between two sets of terms in precise terms (5.OA.3). Work with a pair and accomplish the following activities below. a. A group of Grade 5 students divided themselves into two groups to observe two different auditoriums. Tables and chairs in the first auditorium were arranged in the following way:

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Functions and Models

Figure 9.1. Table for 4 Pattern

Tables and chairs in the second auditorium were arranged in the following way:

Figure 9.2. Table for 2 Pattern

When they regrouped to describe what they saw, the teacher initially asked them to complete the table below. Table 9.2. Table of Values for Figures 9.1 and 9.2 Number of tables

0

1

2

3

4

8 12

25

Number of chairs in the first auditorium

0

4

8

12  

Number of chairs in the second auditorium

0

2

4

6

100

 

Several students then made the following comments: Amy: To obtain the total number of chairs in the first auditorium, just keep adding 4. Bryn: To obtain the total number of chairs in the second auditorium, keep adding 2. Ces:  Looking at the table, it seems to me that each term in the third row is twice the corresponding term in the second row. Diem:  For me each term in the second row is half the corresponding term in the third row. Elna:  Adding 4 is a good rule, but I think a much better rule for the total number of chairs is “number of tables x 4.” So 25 tables will require 25 x 4 =100 chairs in all. 141

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Which student is correct and why? b. Each table below describes a numerical pattern. Fill in the missing entries. Table 9.3  Tables for Patterns A and B Pattern A Rule: Add 3 Position Number

Pattern B Rule: Add 9

Total Number of Sticks

Ordered Pair

 1

 3

(1, 3)

 1

 9

(1, 9)

 2

 6

(2, 6)

 2

18

(2, 18)

 3

 9

 

 3

27

 

 4

12

 

 4

36

 

 5

 

 

 5

 

 

 

 8

 8

 

10

 

 

Position Number

10

Total Number of Sticks

Ordered Pair

   

   

 raph each set of ordered pairs on the same coordinate plane. Make an observation G about how the terms in Pattern A relate with the terms in the Pattern B. 9.2  Dependent Variables, Independent Variables, and Quantitative Relationships in Grade 6

Read standard 6.EE.9. Work with a pair and accomplish the following tasks below. a. What are quantities? variables? independent variables? dependent variables? b. An explicit equation is an equation that expresses a dependent variable in terms of an independent variable. Give examples of explicit equations. Give nonexamples of explicit equations (i.e., implicit equations). Why do sixth-grade students need to learn to write explicit equations? c. Refer to Pattern A in Table 9.3. Let the variables P and T denote position number and total number of sticks, respectively. Three sixth-grade students constructed their explicit equations in the following manner below. Equation Succeeding T = Current T + 3 T of (P + 1) = T of P + 3 T = 3P

Type of Equation Recursive Covariational Correspondence

Justification Keep adding 3. As P increases by 1, T increases by 3. T is three times P.

Three Grade 6 teachers discussed the students’ solutions in the following manner below. 142

Functions and Models

Ms. Oliver: I think we should stop teaching our students to think recursively about patterns because the equations they construct are useless. Mr. Merck: I agree. Even covariational thinking is not significant since they don’t learn proportions and slopes in sixth grade. Ms. Jamie: You both make sense. Right now they just need to be proficient in setting up equations based on correspondences between independent and dependent variables. What do you think? d. Refer to Table 9.2. Let the variables T and C denote number of tables and number of chairs, respectively. Express the mathematical relationships that Amy and Elna articulated for the pattern in Figure 9.1 in terms of explicit equations. e. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Expressions and Equations page. Click on the Grade 6 link. Then open the “show all” page and click the link “illustrations” under C. Analyze the task called Chocolate Bar Sales. Item (e) is an example of an inverse problem. How so? Furthermore, what CCSSM standards do students learn in the process of learning to deal with inverse problems? 9.3  Unit Rates and Proportional Relationships in Grade 7

Work with a pair and accomplish the following tasks below. a. Read standard 7.RP.2 a through d. Discuss the following terms: (1) two quantities in a proportional relationship; (2) constant of proportionality or unit rate; (3) a point (x, y) on the graph of a proportional relationship; and (4) the meaning of the unit rate r relative to the origin (0, 0) and (1, r). b. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Ratios and Proportional Relationships page. Click on the Grade 7 link. Then open the “show all” page and click the link “illustrations” under standard 2.c. Analyze the task called Proportionality. c. Consider Tables 9.2 and 9.3 and their graphs. Discuss whether such representations convey correct direct proportional relationships between two quantities. If so, determine the unit rates and explain what they mean in terms of the given situations. 9.4  Nonsymbolic Algebra and Functions in Grade 8

Work with a pair and do the following tasks below. a. Read standards 8.EE.5 and 8.EE.6. How are slopes and unit rates related? Use the similar triangles test to help you better understand the graphs you generated for Table 9.3. 143

Chapter 9

b. Read standard 8.F.1. Define the following concepts in precise terms: (1) function; (2) input; (3) output; and (4) graph of a function. How are these concepts related to the ideas that you learned in sections 9.1 through 9.3? c. In mathematics, we define a relation as a set of ordered pairs. Are all functions relations? Are all relations functions? Generate examples and counterexamples to help you explain your answers. Give examples and nonexamples of functions. Search the internet for resources that will help you understand the process called the Vertical Line Test (VLT). Access the site http://www.ixl.com/math/algebra-1/ identify-functions-vertical-line-test and use the VLT to help you deal with the graphical activity. d. Read standard 8.F.3. What is a linear function? Give examples of functions that are not linear. Consider the linear function y = mx + b. What do the variables x, y, m, and b represent? What happens when: m equals 0? b = 0? both m and b are equal to 0? e. Read standard and 8.F.2. What are the different ways of representing functions? Generate graphical, numerical (i.e., tabular), and verbal representations for the linear pattern task below. Tables and chairs are arranged like this in the school auditorium:

Explain how many chairs are needed for each long table. Figure 9.3  Adjacent Tables Pattern

f. Read standard 8.F.4. What should students know about what it means to develop a mathematical model for describing linear functional relationships? When asked to generate a linear function model for the pattern in Figure 9.3 using variables n and C, where n corresponds to the number of tables and C the total number of chairs, four Grade 8 students offered the following linear functions (also called explicit formulas) below. Dale: My explicit formula is C = 2 + 2n. Earl: My explicit formula is similar to Dale’s formula. My linear function is C = n x 2 + 2.

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Functions and Models

Frae: My linear function is C = n x 4 – (n – 1) x 2. Gina: My explicit formula is C = 4n – 2(n – 1). Henry: My explicit formula is C = 4 + 2(n – 1). Which student is correct and why? If two or more explicit formulas are correct, how Should students establish their equivalence algebraically? g. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Functions page. Then open the “show all” page and click the link “illustrations” under B.5. Read standard 8.F.5 and then analyze the four tasks completely. What does it mean to provide a qualitative description of a functional relationship between two quantities? h. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Functions page and click the link “illustrations.” Each standard contains at least one illustration. Try a sample of the problem tasks in order to help you understand what Grade 8 students need to learn about functions, linear functions, and nonlinear functions. 9.5  Symbolic Algebra and Functions in Algebra 1

Work with a pair and accomplish the following tasks below. a. Read standards F-IF-1 through F-IF-3. How are these standards related to the standards noted in sections 9.1 through 9.4? b. The domain and range of a function (and all relations) are usually conveyed using interval notation. Search the internet for resources on how to express inequalities in terms of interval notations. Then complete the Interval Notation Puzzle activity in Figure 9.4 and discuss advantages and possible issues when it is implemented in class. c. Obtain the domain and range of each relation in Figure 9.5. Discuss potential incorrect answers that students might generate in each case. d. What issues might students have in relation to the function notation f(x)? What does the notation mean algebraically and graphically? e. Are patterns sequences? Explain. f. The following recursive relations below for the pattern in Figure 9.1 are equivalent. Next = Current + 4 f(n) = f(n - 1) + 4 an = an - 1 + 4 Discuss all unfamiliar symbols. Graph the relation on a coordinate plane and state its domain and range. How do you use the TI-84+ and Geogebra to help you calculate terms in a recursive relation (also called a sequence function)?

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Below are 12 rectangular cards. Cut out each card and assemble the cards into a closed trail. All the cards should be used.

Figure 9.4. Interval Notation Puzzle

Figure 9.5. Domain and Range of Graphs of Relations

146

Functions and Models

For each task below, obtain a recursive relation on the basis of your assumptions about the initial sequence of terms. 1. Begin with 2. Subtract 3 to the current term to obtain the next term. 2. -2, -6, -18, -54, -162, … 3. Assume that a pair of rabbits, one male and one female, are born on Labor Day. When they mature in 2 months, they give birth to a new pair of rabbits. Every month afterwards they give birth to another pair. Assuming that all rabbits generate pairs in the same manner and do not die, how many pairs of rabbits will there be after n months? 4. Search the internet to learn about the famous Tower of Hanoi puzzle. Generate a recursive relation and explain the meaning of each term in the relation. Figure 9.6. Patterns and Recursive Relations

g. Construct a recursive relation for each pattern in Figure 9.6. h. Read standards F-IF.4 through F-IF.6. Interpreting functional relationships involves describing components or attributes of such relationships in precise terms. Use internet resources to help you describe the following relevant terms: y-intercepts; x-intercepts; intervals where a function is increasing; intervals where a function is decreasing; intervals where a function is constant intervals where a function is positive; intervals where a function is negative; relative and absolute maximum; relative and absolute minimum; even function; odd function; end behaviors; average rate of change of a function over a specified interval; and appropriate domain (and range) of a function model. Then draw graphs of functions that illustrate the above components and attributes. i. Access the link https://www.illustrativemathematics.org/standards/hs. Open the Functions page and click the link “illustrations” under Interpreting Functions. Solve the following tasks: F-IF How is the Weather? F-IF Warming and Cooling; F-IF Influenza; F-IF Telling a Story with Graphs; F-IF Throwing Baseballs; F-IF Oakland Coliseum; F-IF The High School Gym; F-IF.6 Mathemafish Population; F-IF Temperature Change; and F-IF Laptop Battery Charge 2. Discuss potential issues that students might have in processing the tasks. j. Use a graphing calculator or Geogebra to obtain the graph of the linear function f(x) = x. Notice that you only need two points on the graph of f(x) = x to determine the unique straight line that represents the function. The domain of the function is R or (−∞,∞), the set of all real numbers. Graph the following linear functions: f(x) = 2x − 1 and f(x) = 0.5x + 3. Express the domain and range in each case in either set notation or interval notation form. Also, check the components and attributes listed in item (h) and determine which ones apply. Graph the linear function g(x) = x over the restricted domain -3 ≤ x ≤ 3. Describe the graph and its range in both set notation and interval notation forms. 147

Chapter 9

Graph the linear function f(x) = 2x  −  1 over the restricted domain 2  ≤  x  0? a < 0. Consider the general quadratic function y  = ax2 + bx + c. Use the components identified in item (h) to completely describe its graph when: a > 0? a < 0. Consider the absolute value function y = a|x|. Use the components identified in item (h) to completely describe its graph when: a > 0? a < 0. p. A piecewise-defined function consists of two or more pieces of functions (F-IF.7b). , For example, the function is made up of 2 pieces. Each piece , represents a function that is defined over a restricted domain. Use what you learned in item (j) to graph the piecewise function in the same coordinate plane. For each piecewise-defined function below, determine the: (1) values of f(−2) and f(2); (2) interval/s where the function is/are increasing; (3) interval/s where the function is/are decreasing: (4) domain and range; (5) (estimated) value of x when f(x) = 1.



i)   

, ,

       iv) 

 y = x 2 , x < 0  ii)     2  y = − x x ≥ 0  iii)    

v)     



What issues might students have when they graph piecewise-defined functions? q. Access the link https://www.illustrativemathematics.org/standards/hs. Open the Functions page and click the link “illustrations” under Interpreting Functions. Solve the following tasks: Graphs of Quadratic Functions; Which Function?, Springboard Dive, and Throwing Baseballs.

150

Functions and Models

r. Search the internet for resources involving step functions. Examine their graphs and determine concepts that Algebra 1 students need to know in order to express them algebraically. Access the link http://www.algebra-class.com/step-functions. html and analyze the two given tasks. Also, check the components and attributes listed in item (e) and determine which ones apply. What benefits do Algebra 1 students obtain from learning about step functions? What issues might students have when they graph step functions? s. A simple exponential function is a function of the form y = ax, where a ≠ 0, a ≠ 1, and a > 0. The variable x is called the exponent and a is the base of the function. Use a graphing calculator or Geogebra to graph the following exponential functions below in the same coordinate plane. y = 2x, y = 2.5x, y = 3x, y = 4.75x, and y = 10x Make an observation about the graph of y = ax when a > 0. Express its domain and range in both set notation and interval notation forms. All five graphs do not touch the x-axis. In fact, the horizontal line y = 0 is called a horizontal asymptote. What does it mean in a numerical context? Also, check the components and attributes listed in item (h) and determine which ones apply (F-IF.7e). Use a graphing calculator or Geogebra to graph the following exponential functions below in the same coordinate plane. x

x

1 19 , y=  and y = 0.95x 3 35 Make an observation about the graph of y = ax when 0  0. 152

Functions and Models



 y = x2    2  y = ( x − 2)   2  y = ( x + 1) 

y =  y =   y =

 y =| x |     y = | x | − 2  y = | x | +1   

x   1  y =     3   2x  x−2      1  2 x − 2     y =  3     x +1   2 x +1  1  y =     3   

Task C: Graph each set of functions below and determine what happens when the graph of y = f(x) is replaced by y = k · f(x), where k > 0. y y y y

= 2x = 4x = 8x = 15 x 1 y= x 2 2 y= x 3 11 y= x 12

y y y y

y = 2 x2 y = 4x2 y = 8x2 y = 15 x 2 1 y = x2 2 2 y = x2 3 11 2 y= x 12



= 2| x | = 4| x| =8| x| = 15 | x | 1 y = | x| 2 2 y = | x| 3 11 y = | x| 12

Task D: Refer to the functions you investigated in task (o). Carefully describe what happens when the graph of y = f(x) is replaced by y = k · f(x), where k < 0. Task E: Investigate graphical relationships between y = f(x) and y = f(−x) in the x  1 following cases: f(x) = 2x, f(x) = 0.5x + 1, f(x) = 2x, and f ( x) =   .  4 Carefully describe the effect on the graph of y = f(x) when f(x) is replaced by either f(kx) or f(−kx), where k > 0. Task F: Explain the effect on the graph of y = f(x) when f(x) is replaced by a · f(b(x + c)) + d for specific positive and negative values of a, b, c, and d. Task G: Develop interesting activities or puzzles such as the one shown in Figure 9.4 that you can use in your own class to teach this particular content standard. bb. Content standard F-BF.3 provides students with different ways to build new and more complicated functions from simple functions. Another popular method involves generating the inverse of a function. In the case of Algebra 1, finding 153

Chapter 9

inverses is limited to linear functions. Search the internet for resources involving inverse functions and applications. 9.6  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard listed in Table 9.1 with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 9.7  Developing a Content Standard Progression Table for the Domains of Functions and Models

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving several domains are coherently developed.  

154

Chapter 10

Measurement and Geometry

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of geometry and measurement from Grade 5 to Grade 8 of the CCSSM. Table 10.1 lists the appropriate pages for your convenience. The development of fifth-grade students’ mathematical understanding of measurement follows the same multiplicative structure that they already learned in the elementary grades (i.e., identify a (standard) unit and perform repeated iterations of the same unit). Toward the end of eighth grade they develop the understanding that all quantities drawn from measuring line segments, angles, areas of 2D objects, and volumes of 3D figures utilize the same multiplicative process. Measurement further supports growth in students’ understanding of analytic geometry which basically utilizes ordered pairs of numbers to represent points on geometric objects in a rectangular coordinate system. Students learn about the rectangular coordinate plane in fifth and sixth grade and then investigate geometrical relationships in algebraic terms from fifth through eighth grade. The classic synthetic approach to geometry enables them to construct, state, and establish precise definitions, descriptions, and explanations. From fifth through eighth grade they explore the defining attributes and properties of 2D and 3D objects and other well-known geometrical relationships (e.g.: interior angle-sum and exterior angle theorems of a triangle; Pythagorean Theorem; angle-pairs formed when parallel lines are cut by a transversal). They also develop and construct informal proofs (i.e., explanations and justifications) in most cases by drawing on various concrete, technology-based, and other empirical representations (e.g.: manipulatives; paperfolding; Geogebra). By the end of eighth grade they understand how to use transformation geometry to establish congruent and similar relationships in both synthetic and analytic contexts. Table 10.1. Page Reference for the Geometry and Measurement Domains in the CCSSM Grade Level Domain

Standards

Page Numbers in the CCSSM

5.G

1 to 4

38

5.MD

1, 3–5

37

6.G

1 to 4

44–45

7.G

1 to 6

49–50

8.G

1 to 9

55–56

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Chapter 10

10.1  Quadrilaterals, Volumes of Right Rectangular Prisms with Whole-Number Edge Lengths, and the First Quadrant Coordinate Plane in Grade 5

Standards 5.G.3 and 5.G.4 deal with figural property discernment and construction which supports deep structure or concept attainment. That is, students’ visual representations of geometric objects are accompanied by precise mathematical descriptions in terms of their defining attributes. Work with a pair and accomplish the following tasks below. a. Establishing definitions is an important structuring activity in the middle school mathematics curriculum. Access the following link below to learn about the difference between stipulated and extracted definitions. http://ir.library.oregonstate.edu/xmlui/bitstream/handle/1957/21549/ EdwardsBarbara.Mathematics.Role%20of%20Mathematical Definitions.pdf?sequence=1 Focus on pp. 224–225: What kind of definitions is acceptable in mathematics and why? Focus on p. 229: What are the pedagogical objectives of activities that focus on establishing and understanding definitions? Explain how classifying quadrilaterals involves knowing both necessary and preferred features. Triangles and quadrilaterals are special examples of polygons. Search online for resources that will help you understand what polygons mean in both necessary and sufficient terms. Examples and nonexamples are also used to help students establish and understand definitions. Examples exhibit all the stipulated attributes in a definition, while nonexamples include some but not all of them. Provide examples and nonexamples of triangles and quadrilaterals. b. Search online for resources involving the different kinds of quadrilaterals and classify them in a hierarchy based on their side and angle properties. What issues might students have in understanding inclusive or hierarchical relationships (refer to the example provided in 5.G.3). How might you help them deal with those issues? c. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page, the Grade 5 link, the “show all” page, and the “illustrations” page under B. Analyze the task labeled What is a Trapezoid? to learn more about the significance of establishing clear definitions in school mathematics. In fifth grade students begin to construct and graph points in the first quadrant of the coordinate plane (5.G.1 and 5.G.2).

156

Measurement and Geometry

d. The introduction to coordinate geometry in fifth grade represents a conceptual shift from students’ elementary experiences with geometric objects. What might have been the motivation for this new representation of geometry? Conduct an online search that will help you understand the difference between analytic and synthetic (or Euclidean) geometry. What purpose does analytic geometry serve in the upper grades? e. What should fifth grade students know about the coordinate plane (5.G.1)? What issues might they have in relation to graphing points in the first quadrant of the coordinate plane (5.G.2)? f. Search appropriate online activities that will help fifth-grade students become proficient in plotting points in a coordinate plane. g. Develop coordinate geometry activities that can be used to support fifth grade students’ understanding of quadrilaterals and types. h. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page, the Grade 5 link, the “show all” page, and the “illustrations” page under A. Analyze the two tasks. When fifth-grade students draw segments to connect pairs of points in the first quadrant of the coordinate plane, they use a ruler to measure the length of each segment. They also know that the resulting quantity depends on the choice of measurement unit. i. Read standard 5.MD.1. What prerequisite skills do fifth-grade students need to have in order to be successful on tasks that involve conversion among differentsized standard measurement units? Search appropriate online activities (e.g., games) that will help you teach measurement conversion in an interesting way. Measuring line segments, angles, areas of polygon regions, and volumes of solid figures follows a multiplicative structure. j. Read standards 5.MD.3 and 5.MD.4 and carefully describe how fifth-grade students are expected to learn volume measurement. Why does such a process exhibit a multiplicative structure? How is volume measurement affected if we count with unit cubes using: cubic cm? cubic inch? cubic foot? other improvised cubic unit? k. Standard 5.MD.5 provides recommended strategies for finding the volumes of right rectangular prisms with whole-number side or edge lengths. Try the strategies on one simple right rectangular prism and one complex solid figure consisting of two or more non-overlapping right rectangular prisms.

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l. Discuss potential issues that students might have when confronted with tasks that involve finding the volumes of solid figures. m. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Measurement and Data page and the “illustrations” link associated with Grade 5. Analyze the tasks under 5.MD.A.1, 5.MD.C, 5.MD.C.5, 5.MD.C.5.a, and 5.MD.C.5.b. 10.2  The Four-Quadrant Coordinate Plane, Areas of Polygons, Nets and Surface Areas, and Volumes of Right Rectangular Prisms with Fractional Edge Lengths in Grade 6

Work with a pair and accomplish the following tasks below. a. Read standard 6.G.3. What issues might students have about plotting points in coordinate plane? Access the following link below and model the Cartesian classroom activity with your partner. http://www.cpalms.org/Public/PreviewResource/Preview/5723 Learn how to use Geogebra to plot points in the coordinate plane and to record the corresponding ordered pairs in a spreadsheet table. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 6.G. Analyze the task called Polygons in the Coordinate Plane and determine concepts and processes that students will acquire as a result of engaging with the task. b. Read standard 6.G.1. Search the internet for resources that will help you use paperfolding and/or cutting to teach sixth-grade students visual-based strategies for establishing areas of certain polygons. Then access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 6.G. Analyze the following tasks: Same Base and Height (both variations); Finding Areas of Polygons; Base and Height; and Sierpinski’s Carpet. How does using each task strengthen students’ understanding of areas of polygons? c. Read standard 6.G.2. How is it related to standard 5.MD.5? d. Read standard 6.G.4. The following link below provides suggestions for helping lower elementary school students learn something about nets, surface areas, and 3D figures. http://lovewhatyouteach.com/2012/11/25/using-nets-to-develop-childrensunderstanding-of-3d-shapes/

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Explore the link and develop tasks that are appropriate for sixth grade students. 10.3  Scale Drawings, Slicing, Drawing Shapes with Given Conditions, Circumference and Area of a Circle, Finding Unknown Angles in a Figure, and More Problems Involving Area, Surface Area, and Volume of 2D and 3D Objects in Grade 7

Work with a pair and accomplish the following tasks below. a. Read standard 7.G.1. Access the following link below and explore the tasks and activities with your partner. http://illuminations.nctm.org/lesson.aspx?id=1675 Discuss potential issues when the entire unit is implemented in class. b. How is standard 7.G.1 related to the 7.RP cluster of standards? c. Read standard 7.G.2. Search internet resources that will help you teach the standard using: Geogebra; hands-on tools; other virtual manipulatives. d. Read standard 7.G.3. Access the link http://www.illustrativemathematics.org/ standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 7.G. Analyze the task called Cube Ninjas! and develop a similar activity using other solid figures. e. Explore the activity in Figure 10.1. How does it support seventh-grade students’ understanding of standard 7.G.4? Search the internet for a visual demonstration and derivation of the mathematical relationship between the circumference and area of a circle. What concepts do students need in order to fully understand and appreciate the relationship? Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 7.G. Analyze the six tasks listed under 7.G. B and 7.G.B.4. f. The remaining two geometry standards 7.G.5 and 7.G.6 focus on problem solving that involve basic concepts that students already know. Develop appropriate problem tasks for the concepts listed in 7.G.5 that also assess their understanding of standard 7.EE.4a. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 7.G. Analyze the task under 7.G.B.6 and discuss how it is related to standard 7.RP.

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A. Draw a big circle on a construction pad. Construct a diameter of the circle and use a piece of string to measure its length. 1. How many copies of the string do you need to measure the “perimeter” of the circle?       Fill in the blank: The circumference of a circle is slightly more than ______ times its diameter. 2. Draw two circles having different diameters and follow the same procedure in (1). What can you infer about the circumference of a circle relative to its diameter? B.  Measure at least 15 different circular objects. Carefully measure the circumference of each object with a piece of string. Then use a metric ruler to measure the length of string in millimeters. Also measure the diameter of each object in millimeters. Record your values by creating a spreadsheet consisting of three columns. Enter the circumference of each item you measured in column A. Enter the corresponding diameter in Column B. In column C, create a formula for the value of π by drawing on what you learned in (1) and (2). Below is an example of a spreadsheet table for this activity. Object Number 1 2

A 27 13

B 8 4

C 3.375 3.25

Use your spreadsheet to answer the following questions below. 3.  What does your spreadsheet tell you about the value of π ? 4. Search the internet for the correct approximate value of π rounded to the nearest hundredths. If your value of π is different from this value, what factors might have contributed to the difference in value? Figure 10.1. Exploring π 10.4  Volumes of Cylinders, Cones, and Spheres and Transformation Geometry, Congruence, Similarity, Informal Proofs, and the Pythagorean Theorem in Grade 8

Work with a pair and accomplish the following tasks below. a. Read standards 6.G.2, 7.G.4 and 8.G.9. How might eighth grade students use these standards to derive and use the formula for the volume of a cylinder?

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Search online resources that establish a mathematical relationship between the volume of a cone and the volume of a cylinder in a visual manner. What assumptions do students need to consider in the visual-derived explanation? Search the internet for at least three different ways of deriving the formula for the volume of a sphere that are appropriate for eighth grade students. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 8.G. Analyze the four tasks listed under 8.G.C.9. b. Transformation geometry (also called rigid motions and isometries) provides eighth grade students with a powerful visual introduction to the concepts of congruence and similarity. Search the internet to learn something about the meanings of transformation, rigid motions, and isometries in mathematical and geometrical contexts. Eighth-grade students initially learn basic transformation properties of rotations (turn), reflections (flip), and translations (slide) in the context of their experiences with lines, segments, angles, and parallel lines. They also learn to associate a point in a plane with some another point in the same plane. Is transformation geometry a function? What do the terms preimage and image mean? Read standard 8.G.2. When are two-dimensional figures said to be congruent? Search the internet for eighth-grade resources that will enable you to teach the basic principles involvling reflections, translations, and rotations. Reread standard 8.G.2 and explain what “a sequence of motions” precisely entails when students need to exhibit congruence between two or more 2D figures. Draw a line with three or more points on the same line (i.e., collinear points). Do reflections, translations, and rotations preserve betweeness among three points, collinearity, and distance between any pairs of points? Explore transformations involving angles, parallel lines, triangles, and quadrilaterals as well (8.G.1). When you feel comfortable with the three transformation concepts, explore them in the context of 2D figures in a coordinate plane (8.G.3). Search online resources that involve transformation geometry activities using coordinates and assess issues that eighth-grade students might have in when they apply transformations in a coordinate geometry context. Learn to use the transformation tools in Geogebra as well. Read standard 8.G.4. When are two-dimensional figures said to be similar? Are congruent figures similar? Are similar figures congruent? Why or why not? Does similarity preserve angle measure, betweeness, collinearity, and distance? Why or why not? Use Geogebra to learn how to construct two similar figures.

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An eighth-grade teacher made the following claim: “My students simply need to know that congruent figures have exactly the same size and same shape and similar figures have the same shape but not necessarily the same size.” Do you agree? Why or why not? Access the link http://www.illustrativemathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 8.G. Analyze all the tasks listed under 8.G.A and identify potential conceptual and implementation issues. c. Read standard 8.G.5. What does an informal mathematical argument mean and how does it differ from a formal mathematical argument? Use Geogebra and paperfolding which to help you establish the sum of the interior angles of any triangle. Both activities provide eighth-grade students with an informal or concrete (and empirical) way of understanding the stable relationship. Next, search the internet for a formal proof that involves the use of parallel lines. Use this particular experience to help you distinguish between an informal and formal argument in mathematics. Use Geogebra or paperfolding to informally establish facts relevant to the following famous geometrical relationships: exterior angle of a triangle; pairs of congruent angles formed when two or more parallel lines are cut by a transversal; and the angle-angle criterion that helps establish similarity between two triangles. d. Refer to section 2.6.1(b) for two visual demonstrations of the Pythagorean Theorem. What kind of argument does each model demonstrate? Search the internet for two or more (formal and/or informal) proofs of the theorem that are appropriate for eighth-grade students (8.G.6). What does the term converse mean in mathematics? Explore the converse of the Pythagorean Theorem using several specific examples and use them to help you provide a proof. Read standards 8.G.6 and 8.G.7. Access the link http://www.illustrative mathematics.org/standards/k8. Open the Geometry page and the “illustrations” link associated with Grade 8.G. Analyze all the tasks listed under 8.G.B and identify potential conceptual and implementation issues. 10.5  Mapping the Content Standards with the Practice Standards

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard listed in Table 10.1 with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses.

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10.6  Developing a Content Standard Progression Table for the Measurement and Geometry Domains

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving several domains are coherently developed.

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Data, Statistics, Probability, and Models

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of data, statistics, and probability in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 11.1 lists the appropriate pages for your convenience. Since data, statistics, and probability involve relationships between quantities, middle school students can use descriptive and analytic modeling processes to represent and analyze those relationships in a structured way. In the case of data and statistics, they learn to construct different representations of data from grades 5 through 8, beginning with line plots in fifth grade, dot plots, histograms, and box plants in sixth grade, and scatter plots in eighth grade. By the end of eighth grade, they are able to describe, summarize, and analyze (bivariate) measurement and categorical data using several descriptive statistical measures (e.g., center, spread or variation, and skewness) and construct simple linear models. In Algebra 1 they engage in statistical modeling activity, which employs all the concepts and processes that they learned in the earlier grades, and also learn to distinguish between correlational and causal relationships. Probabilities of simple and compound events and probability models are explored in seventh grade when students begin to use statistics to generate valid generalizations about populations by constructing representative samples drawn from random sampling techniques. Table 11.1. Page Reference for the Data, Statistics, and Probability Domains in the CCSSM Grade Level Domain

Standards

Page Numbers in the CCSSM

5.MD

2

37

6.SP

1 to 5

45

7.SP

1 to 8

50–51

1 to 4

56

1–9 except 4

81

8.SP S-ID (Algebra 1 Traditional Pathway)

11.1  Representing Data in Grades 5, 6, and 8

Work with a pair and do the following tasks below. a. Read standard 5.MD.2. What are line plots? Describe situations in real life that require using line plots to display unit fraction measurements. 165

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Access the link http://www.illustrativemathematics.org/standards/k8. Open the Measurement and Data page and analyze the task called Fractions on a Line Plot that appears under standard 5.MD.B.2. b. Read standard 6.SP.4. How are dot plots, histograms, and box plots used to represent numerical data? What issues might students have in representing numerical data in such formats? Access the link http://www. illustrativemathematics.org/standards/k8 and analyze the two tasks that appear under standard 6.SP.B.4. c. Read standard 8.SP.1. What is bivariate measurement data? How are scatter plots and clustering used to describe and model relationships or associations between two quantitative variables? Search the internet for examples of relationships that model positive linear association, negative linear associations, and nonlinear association. Access the link http://www.illustrativemathematics.org/standards/k8 and analyze the four tasks that appear under standard 8.SP.A.1. Learn to use either a graphing calculator or Geogebra to generate scatter plots. Read standard 8.SP.4 which deals with bivariate categorical data. Search the internet for resources that will help you teach the difference between bivariate measurement data and bivariate categorical data and the different ways they can be represented in visual form. Also, learn to use either a graphing calculator or Geogebra to generate two-way tables that summarize data involving two categorical variables. d. What do middle school students need to know about what it means to represent numerical data in a variety of visual formats? 11.2  Analyzing Data in Grades 5, 6, 7, and 8

Continue exploring the following issues below with your pair. a. Search the internet for at least five mathematical tasks that use line plots in fifth grade. What types of questions might you ask to help students infer and/or understand relationships that you want them to draw from line plots? b. Standard 6.SP.1 addresses the nature of an appropriate statistical question. Access the link http://www.illustrativemathematics.org/standards/k8. Open the Statistics and Probability page and analyze the two tasks that appear under standard 6.SP.A.1. What do sixth-grade students need to know about what it means to state a statistical question? c. Standards 6.SP.2, 6.SP.3, and 6.SP.5 are related. Provide definitions for the following concepts which are used to describe and summarize a set of data: number of observations; attributes, measurement, and units; distribution; measures 166

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of center; measures of spread or variation; and shape or pattern of distribution. Access the link http://www.illustrativemathematics.org/standards/k8 and analyze the three tasks that appear under standards 6.SP.B.4 and 6.SP.B.5.d. d. Seventh-grade students learn about the significance of using random sampling techniques to generate representative samples which enables them to draw generalizable inferences about a population and informally compare inferences about two populations (7.SP.1 through 7.SP.4). Search the internet for resources that will help you answer the following questions: What constitutes a random (or representative) sample? What is a nonrepresentative sample? What are some well known random sampling techniques? What does it mean to establish valid generalizations about a population from a sample? What factors do students need to carefully consider when they are asked to compare two numerical data distributions or populations? Access the link http://www.illustrativemathematics.org/standards/k8 and analyze all the tasks that appear under standards 7.SP.A and 7.SP.B. For each task identify potential learning and implementation issues. e. Read standards 8.SP.2 and 8.SP.3. Straight lines can be used to model relationships between two quantitative variables that suggest a linear association. Those lines are called lines of best fit and such relationships exemplify linear models. Access the link http://www.illustrativemathematics.org/standards/k8 and analyze all the tasks that appear under standards 8.SP.A.2 and 8.SP.A.3 with the aid of either a graphing calculator or Geogebra. How are such tasks helpful in deepening students’ understanding of slopes and y-intercepts? f. Standard 8.SP.4 deals with bivariate categorical data. Access the link http://www. illustrativemathematics.org/standards/k8 and analyze the two tasks that appear under standard 8.SP.A.4 with the aid of either a graphing calculator or Geogebra. 11.3  Statistical Modeling in Algebra 1

Continue exploring the following issues below with your pair. a. The traditional Algebra 1 pathway includes statistical modeling and covers standards S-ID.1 through S-ID.9 except S-ID.4. Read all eight standards and assess your level of familiarity with the stated standards. Based on the work you accomplished in sections 11.1 and 11.2, determine new statistical concepts that students need to learn in Algebra 1. Why does it seem appropriate to include statistical modeling in Algebra 1? b. Access the link https://www.illustrativemathematics.org/HSS and analyze several tasks under the statistics and probability page. Why is it important for students to know correlation and causation and be able to distinguish one from the other? 167

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c. Access the eighth-grade 14-minute classroom video Experimenting with STEM: The Barbie Bungee Jump from the link below. https://www.teachingchannel.org/videos/stem-lesson-ideas-bungee-jump Is the activity appropriate for eighth-grade and Algebra 1 students? Use the relevant content standards to assess the quality of the worksheet that the students used to carry the experiment. 11.4  Probability Models and Probabilities of Simple and Compound Events in Grade 7

Standards 7.SP.5 through 7.SP.8 focus on concepts related to probability. Work with a pair and do the following tasks below. a. Search the internet for a precise definition of sample space. How is this concept related to the concept of random samples in standard 7.SP.1? b. Read standard 7.SP.5. Search the internet for a precise definition of a chance event. Give examples of chance events with probability: 0;   1 ; 3 ; 2 ; and 1. 2  4  3  Carefully describe such events in precise terms. c. Read standard 7.SP.6. How is the concept of probability of a chance event related to the concept of relative frequency? Access the link https://www. illustrativemathematics.org/standards/k8 and analyze the three tasks under 7.SP.C.6. Discuss potential conceptual and/or linguistic issues in understanding the tasks. d. Read standard 7.SP.7. What is a probability model? Describe situations that illustrate the two different types of probability models noted in the standard. From the situations you just described, carefully define the probability of an event. Search online activities that will help you teach these concepts to seventh-grade students. e. Read standard 7.SP.8. What is a compound event? Search the internet for activities that will provide seventh-grade students with an opportunity to construct different ways of representing a sample space that they can use to establish the probability of a compound event. Access the link https://www.illustrativemathematics. org/standards/k8 and analyze all the tasks under 7.SP.C.8. Discuss potential conceptual and/or linguistic issues in understanding the tasks.

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11.5  Mapping the Content Standards with the Practice Standard

Work with a pair to accomplish the following task below. Use the checklist you developed in Table 2.4 to map each content standard listed in Table 11.1 with the appropriate practice standards and NRC proficiency strands. Make a structure similar to the one shown in Table 3.3 to organize and record your responses. 11.6  Developing a Content Standard Progression Table for the Data, Statistics, and Probability Domains

Continue working on your own domain table. Add the appropriate number of rows corresponding to your grade-level domain standards. Then map a reasonable content trajectory for each standard over the indicated timeline. Make sure that the different grade-level content standard progressions involving several domains are coherently developed.

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Content-Practice Assessment

In this chapter you will explore different content-practice assessment strategies that will help you measure middle school students’ understanding of the CCSSM. The term measure should not be conceptualized merely in terms of scaled values or proficiency labels relevant to students’ performances on a contentpractice standard being assessed. Think of measure in terms of patterns that they exhibit relative to the standard. Following Robert Mislevy, such patterns may indicate regularities that can tell you how they learn mathematics and provide you with information about various opportunities to learn and resources that are available to them before, during, and after assessment. In fact, when you engage with colleagues to study student work for a large number of students, your findings will most likely yield interesting and relevant patterns that are useful in supporting improved and high-quality content-practice teaching and learning. Consequently, viewing assessment findings in terms of patterns would mean expecting different kinds of results each year since various circumstances influence and shape student performances on tasks. In this chapter you will also learn to assess middle school students’ contentpractice proficiency in ways that are consistent with the structure of the Smarter Balanced Assessment (SBA). The SBA has been referred to as the “next-generation assessment.” At the very least, SBA tasks are meant to measure 21st century skills. They are carefully aligned with the CCSSM and are intended to measure student progress toward college and career readiness, that is, the requisite mathematical competence that all students need to possess in order to succeed in entry-level creditbearing coursework and a demanding high-skilled workforce. For middle school students the SBA should be able to measure their progress toward high school mathematics. Each grade-level SBA, which begins in third grade and culminates in eleventh grade, consists of selected-response questions (e.g., multiple choice and true or false), constructed response items (e.g., draw a diagram and provide solutions), technology-enriched (i.e., computer-mediated) response items, and performance tasks. Performance tasks, especially, are meant to engage middle school students in in-depth projects that enable them to apply their analytical and real-world problem solving skills. Toward the end of this chapter, you will deal with alternative forms of assessment. A good way to start thinking about content-practice assessment involves reflecting on your own experiences as former students in math classrooms. Work with a group and address the following tasks below. 171

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a. Reflect on your previous (student) experiences of testing in your favorite subject. What kinds of tests did you enjoy the most and why? Which ones were difficult and/or challenging, and why? b. This time consider your experiences of testing in any of your previous math classes. Think about the style and quality of math items that you had to answer. Was content prioritized over practice or vice-versa? Or was there a balance between content and practice? Remember “practice” refers to the CCSSM standards for mathematical practice. c. Draw on the responses of your group to assemble possible dos and donts about testing. Then access the reference below to read Dylan William’s view on assessment. William, D. (2006). Assessment. Journal of Staff Development, 27(1), 16–20. Compare your responses with the author’s general assessment-for-learning strategies. Which strategies on your list align with his suggested strategies? What new ideas are you eager to test? When you assess students’ mathematical proficiency in the CCSSM, you will need to use tasks that are framed around the content standards (i.e., Claim 1-assessment target items) and the eight practice standards (i.e., Claim 2-, 3-, and 4-assessment target items). For example, Figure 12.1 exemplifies an eighth-grade content-practice item under the Expressions and Equations domain (8.EE.7a) that assesses at least one content standard and one practice standard. Both the style of testing and the thinking that is needed to accomplish the task depart from older and simpler content-driven assessments such as the one shown in Figure 12.2. Continue to work with your group and deal with the following task below. d. Solve the two tasks shown in Figures 12.1 and 12.2. What content and practice standards are being assessed on each task? Carefully read content standard 8.EE.7 and William’s five assessment-for-learning strategies. How might you use the assessment task in Figure 12.1 to develop learning intentions and success criteria with all students and orchestrate effective classroom discussions and tasks around similar items? 12.1  General Conceptions on Assessment

The following tasks below will help you construct an appropriate meaning of assessment in an educational context. Work with a pair and accomplish them together. a. Search online for various meanings of assessment in an educational context. Access the link http://www.wordle.net/ to create a wordle. Paste into the wordle box at least ten different characterizations of assessment. When you are done

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Figure 12.1. Common Core Content-Practice Standard Question Involving 8.EE.7

Figure 12.2. Traditional Content-Driven Question Involving 8.EE.7

constructing your assessment wordle, generate initial impressions. For example, which terms in your wordle appear to be more prominent than others? Which terms caught your attention? b. Share your wordle with another pair of students. Work together to develop a definition of assessment. When you have formulated your group definition, consider once again your earlier individual responses to items (a) and (b) in the introduction. Assess the extent to which your personal experiences of assessment reflect various aspects of your group definition of assessment. c. If your group is part of a whole class, assemble a gallery walk of group definitions of assessment. Then formulate a shared definition of assessment. The preceding activity should provide you with the basic characteristics of assessments in an educational context. The following three valuable views below regarding assessments, which have been drawn from the Gordon Commission report on the future of education, are also worth keeping in mind. 1. The troika of assessment, teaching, and learning forms the backbone of a wellconceptualized pedagogy. While the three processes can take place independently, their mutual coordination is necessary in order to guarantee meaningful pedagogical experiences for all students. The key is alignment. It is impossible in school contexts to conceptualize teaching that is not informed by any form of assessment of student learning. Further, both assessment and teaching should aim for improved student (and teacher) learning. 2. Assessment is a process of reasoning from evidence. However, it is also important to remember that evidence is imprecise to a certain extent. All findings from any 173

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kind of assessment basically provide an estimate of what students either know or are able to accomplish. 3. Fundamental to assessment planning is a clearly articulated model of cognition and learning. You need to know the best evidence possible regarding how students learn, process, and represent a target content standard and the relevant practice standards, including effective means that can help them attain proficiency in both content and practice standards. To gain an alignment mindset, work on the following tasks below with a group. d. First, refer to your CCSSM domain tables and decide on a cluster of content standards that your group is interested in exploring together. Before you fill out Table 12.1, discuss the following concerns below. i) Identify a cluster (also called Claim 1 assessment target in SBA terms) that you are interested in assessing together. Specify a particular content standard in that cluster that is reasonable to assess in a single 50-minute classroom session. ii) Identify a practice standard that you also want to assess. Refer to your Table 2.3, which provides you with a checklist of specific actions (also called Claim 2, 3, or 4 assessment targets in SBA terms) under each practice standard. Specify one specific action that you intend to assess in relation to the content standard you identified in (ii). iii) Describe an evidence that you need to establish that will indicate students’ success in achieving the two assessment targets you intend to measure. What type of assessment items or tasks and task models are appropriate and will provide you with that evidence? iv) How might students go about learning the relevant content and practice standards that will enable them to exhibit mathematical proficiency in your planned assessment? v) How might teaching be orchestrated to support students’ meaningful and successful learning of the articulated content and practice standards? Regarding item (iii), an evidence statement that restates your content standard as a student-driven objective should be sufficient. For example, two evidence statements for the task shown in Figure 12.1, which is based on the relevant content standard 8.EE.7, are as follows: The student can determine whether a linear equation in one variable has one solution, infinitely many solutions, or no solutions by successively transforming the equation into simpler forms until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). The student can solve linear equations with rational number coefficients and whose solutions require expanding expressions using the distributive property and collecting like terms. 174

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Assessment items can be drawn from any of the following sources: • tasks that are aligned with district/benchmark and other released sample assessment items; • tasks that are aligned with current district and state standards and benchmarks. Task models provide information about important features of assessment items or tasks that enable students to exhibit the stipulated evidence. Figure 12.1 is an example of a task model that shows three nonstandard linear equations involving a single variable that students need to simplify completely on the basis of the content recommendations stated in 8.EE.7. The simplification should help them determine the correct solutions. Regarding assessment items or tasks, develop or use those that model SBA tasks such as the ones noted below. Access the following link below, which will take you to the released SBA sample tasks, to help you gain a better understanding of each task description. http://www.smarterbalanced.org/sample-items-and-performance-tasks/ • Selected-response tasks involve selecting one or more responses from a set of options (e.g., single- or multiple-response multiple choice items or true-or-false items); • Technology-enhanced tasks assess deeper understanding of content and skills that cannot be accomplished in the context of traditional tasks. For example, simulations that demonstrate mathematical phenomena provide students with an opportunity to engage in simple conceptual investigations and establish inferences within the given constraints; • Constructed-response tasks involve constructing a detailed solution; • Performance tasks involve integrating knowledge and skills across multiple content standards. They measure depth of understanding, research skills, and other forms of complex analysis that cannot be adequately assessed with selectedor constructed-response items. You may also use released items from the Partnership for Assessment of Readiness for College and Careers (PARCC). Access the link below to access sample items and prototypes by grade level. http://www.parcconline.org/samples/item-task-prototypes Regarding item (iv), focus on your domain table and briefly map a conceptual development of your target content standard from the initial to the final phase. If applicable, consider as well the approach(es) noted in the CCSSM for that particular standard. For example, content standard 8.EE.7 recommends having eighth-grade students solve linear equations that (1) have rational number coefficients and (2) contain expressions that need to be transformed and simplified by applying the distributive property and collecting like terms. Keep in mind that the content standards 175

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have been drawn from extensive research on student-driven learning progressions that show how they evolve and are meaningfully acquired by students over time. In the case of content standard 8.EE.7, students learn to deal with equivalent expressions and different types of solutions in the context of solving linear equations. These content pieces are not covered in an earlier grade-level related content standard (7. EE.4a) that requires them to fluently solve linear equations of the type px + q = r and p(x + q) = r, where p, q, and r are rational numbers. Regarding item (v), briefly suggest instructional tools and activities that can help your students learn the articulated content and practice standards. For example, content standard 8.EE.7 can be taught by initially presenting students with several different or contrasting ways of solving complicated linear equations in one variable and asking them to evaluate them for validity or correctness and approach. Figure 12.3 shows an example. Individual time and paired or group work in that order can then be utilized as a follow up activity in order to give them time to work on similar problems. 12.2  Norm- and Criterion-Referenced Tests

One important concept in assessment involves norm- and criterion-referenced testing. In norm-referenced testing (NRT) an individual student’s performance is compared with other students’ performances on the same valid and reliable test. NRT

Figure 12.3. Four Different “Solutions” Involving the Same Linear Equation

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Table 12.1. Initial Assessment-Learning-Teaching Plan Group (names):

Grade Level: Class Type:

Relevant standard for mathematical content

Relevant standard for mathematical practices (check only one)

Domain:

General: ____ Make sense of problems and persevere in solving them ____ Reason abstractly and quantitatively ____ Construct viable arguments and critique the reasoning of others ____ Model with mathematics ____ Use appropriate tools strategically ____ Attend to precision ____ Look for and make use of structure ____ Look for and express regularity in repeated reasoning

Cluster: Standard:

Specific: Evidence of attainment of content standard

Planned assessment item or task that will be used to establish evidence

How students will learn the content and practice standards (a rough sketch of steps will suffice)

How the content and practice standards can be taught effectively (a rough sketch of steps will suffice)

produces results that rank students on a curve, producing labels such as high and low achievers. Examples of NRT are standardized assessments such as the Scholastic Aptitude Test and college entrance examinations. In criterion-referenced testing (CRT) an individual student’s performance is assessed relative to some well-defined criteria or standards. CRT produces results that tell whether the student has achieved mastery of a specific competence. Examples of CRT include the SBA, PARCC, and school/district/wide benchmark assessments. In the case of many school/district/ wide assessments an individual student’s performance in mathematics is labeled either advanced (i.e., achieving beyond grade-level expectations), proficient (i.e., meeting grade-level expectations), basic (i.e., not meeting grade-level expectations), below basic, and far below basic. In the California Assessment of Student Progress and Performance (i.e., state assessment) “proficient” is the desired minimum achievement level for all students. 177

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Work with a group and do the following tasks below. a. Access the following two links below and discuss advantages and disadvantages of NRT and CRT. Can one test be used for two different purposes? Explain. http://www.fairtest.org/criterion-and-standards-referenced-tests http://www.cshe.unimelb.edu.au/assessinglearning/06/normvcrit6.html b. Access the following link below and discuss different kinds of scores that are generated by NRT and CRT and different ways in which they are interpreted. http://www.cal.org/twi/evaltoolkit/5when2usetests.htm c. Access the following link below and discuss item characteristics of NRT and CRT. http://www.edpsycinteractive.org/topics/measeval/crnmref.html 12.3  Principles of Effective Classroom Assessments

Access the following link below and answer the questions that follow. http://www.ets.org/Media/Tests/TOEFL_Institutional_Testing_Program/ ELLM2002.pdf a. How are classroom assessments generally helpful to teachers? b. When is an assessment considered to be valid, reliable, and fair? Why should such characteristics matter? c. What are some suggestions for planning, writing, and implementing good assessments? d. How do you prepare students to value classroom and other school-based assessments? What should they know before and after taking an assessment? 12.4  Formative Assessments

Obtain a copy of the following article below by Greg Conderman and Laura Hedin that discusses the nature and practice of formative assessments. Then work with a group to discuss the questions that follow. Conderman, G., & Hedin, L. (2012). Classroom assessments that inform instruction. Kappa Delta Pi Record, 48, 162–168. a. What are formative assessments (also called adaptive instruction)? How do they differ from summative assessments? b. Describe formative assessment strategies that can be used before, during, and after instruction. What findings are pertinent in every phase, and how can they be processed effectively in class? 178

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c. Identify a CCSSM domain, a cluster, and one content standard in that cluster. Identify one CCSSM practice standard that can be assessed with the content standard. Develop at least one formative assessment task before, during, and after a 55-minute classroom session. Obtain a copy of the following article below by Brent Duckor that deals with explicit formative assessment strategies that can be easily implemented in the classroom. Work with the same group to discuss the questions that follow. Duckor, B. (2014). Formative assessment in seven good moves. Educational Leadership, 71(6), 28–32. d. Describe each formative assessment move and assess its applicability in the mathematics classroom. Develop an appropriate situation that will enable you to model the stipulated qualities or characteristics. e. Discuss potential stakeholder concerns with tagging and binning as formative assessment moves. 12.5  Summative Assessments

This section provides you with an opportunity to learn the SBA system in detail. You may find the PARCC system interesting to learn as well. In case you study both summative assessment tools, determine similarities, differences, and essential components. 12.5.1 What are Summative Assessments? Access the link below to read the perspective of Ezekiel Dixon-Roman about the role of formative and summative assessments in teaching and learning. http://www.gordoncommission.org/rsc/pdfs/vol_1_no_2_18654.pdf a. What are summative assessments? How do they differ from formative assessments? What does the author mean when he claims that both formative and summative assessments have a dynamic relationship? How can summative assessments be embedded in formative assessments? b. Describe different types of summative assessment strategies and their purposes. 12.5.2 Smarter Balanced Assessment (SBA) Access the link below to download several key PowerPoint modules that will help you understand the SBA system. http://www.smarterbalanced.org/smarter-balanced-assessments/item-writingand-review/ 179

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Choose Item Writing and Review under the Smarter Balanced Assessments tab. Download the PowerPoint module labeled Introduction to Smarter Balanced Item and Performance Task Development. Provide responses to the questions below. c. What is the SBA? What is the goal of the SBA Consortium? What are the basic key features of the SBA system? d. Slide 9 describes the SBA item development process. For our purpose, focus only on steps 1 and 2. Briefly describe the evidence-centered design (i.e., assessment triangle) that guides each step in the process. e. How are content and item specifications useful in the development of SBA items and tasks? f. Learn the following SBA item ad task types below and their different purposes. 1. selected response items; 2. constructed response items; 3. extended response items; 4. performance tasks; 5. and technology-rich items (i.e., technology-enabled and technology-enhanced tasks).  g. Describe important terms and key concepts that are central to the SBA system. Download the PowerPoint module labeled Mathematics Selected Response, Constructed Response, and Technology-Enhanced Items. This module provides an in-depth discussion of three of the four SBA item and task types, including guidelines and requirements for developing and writing such items. Provide responses to the following questions below. h. What are selected response items? What are the benefits and limitations of using such items? How are stems and distractors in traditional selected response items developed? What is the purpose of a distractor analysis? How are stimuli and stems in nontraditional selected response items developed? What is the purpose of a scoring rubric? Jump to slide 15 to learn about the qualities of a rubric. What are some basic guidelines for writing nontraditional selected response items? i. What are constructed response items? How are stimuli and stems in constructed response items developed? What is the purpose of a scoring rubric? What are some basic guidelines for developing constructed response items? j. What are some general guidelines and requirements for developing exemplary selected and constructed response items? What can be learned from the flawed items? k. What are technology-enabled items? How are they constructed? l. What are technology-enhanced items? How are they constructed? What are the key components in a technology-enhanced item? How are interaction spaces and scoring rules constructed? Study the template for creating technology-enhanced items. 180

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m. Explain similarities and differences between technology-enabled, technologyenhanced, and paper-based items. Access the link below which provides information about how to develop and write good performance tasks. Provide responses to the questions that follow. http://www.oakland.k12.mi.us/LinkClick.aspx?link=Learning%2FDRAFT+P erformance+Task+Specifications_022812.pdf&tabid=1876&mid=7256 n. What are the characteristics of good performance tasks? o. How are SBA performance tasks in mathematics constructed and administered in classrooms? How are they scored? How are rubrics constructed? p. Refer to pp. 15-17 for details regarding the structure of a performance task in mathematics. Access the following link below to investigate the Grade 6 sample performance task called Taking a Field Trip. Solve the task and evaluate the task specification and scoring rubric that came with the sample. http://www.smarterbalanced.org/sample-items-and-performance-tasks/ Download the PowerPoint module labeled Mathematics Grade Level Considerations for Grades 6-8. This module provides information regarding item and task development and content specification issues that apply to grades 6 through 8. Provide responses to the following questions below. q. What are some basic suggestions regarding vocabulary, style, the use of words and numerals, use of commas in numbers, item difficulty, and contexts that are appropriate for grades 6 to 8 students? r. SBA items and tasks are categorized not only by type but also by content specification. Content specification consists of four foundational claims and their respective assessment targets. Describe each claim, the relevant assessment targets, and item and task types that can be used to collect evidence. How are the CCSSM clusters used in Claim 1? How are the CCSSM practice standards used in Claims 2 through 4? Note that a student’s Claim 1 score represents 40% of his or her total SBA math score, while the three remaining claims factor in 20% each. s. Download the PowerPoint module labeled Mathematics Grade Level Considerations for Grades 3-5. Address the same questions listed in (q) and (r) to help you learn something about task considerations that are appropriate for elementary students and how they compare with considerations at the middle school level. Download the PowerPoint module labeled Mathematics Content Specifications, Item Specifications, and Depth of Knowledge. Pay careful attention to the cognitive rigor matrix that is used to develop high-quality items and tasks that are framed around the four foundational claims and their respective assessment targets. Provide responses to the following questions below. t. What is the basis for SBA’s cognitive rigor matrix? 181

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u. How are middle school students assessed based on the four levels of depth of knowledge? v. Explore the cognitive rigor matrix shown in slide 12. How can the information provided in the table help you develop different levels of items and tasks? w. The table shown in slide 15 identifies the number of assessment targets for Claim 1 under each grade level. Verify that each count is correct by inspecting the appropriate reference in the CCSSM document. For instance, p. 41 of the CCSSM contains 10 clusters of content standards that correspond to Claim 1 assessment targets for sixth grade. x. Carefully investigate the structure of the sample Claim 1 assessment target shown in slide 16. How are headings and full description constructed? Focus on the assessment description and the notations that you see on the first line. Refer to pp. 29-39 and 79-81 of the following document below for more details regarding the different cluster emphases (i.e.: major, additional, and supporting). Note that only major Claim 1 clusters are assessed in the SBA. http://www.smarterbalanced.org/wordpress/wp-content/uploads/2011/12/ Math-Content-Specifications.pdf y. Assessment targets for Claims 2, 3, and 4 are drawn from the CCSSM practice standards and, hence, they are not grade-level specific. Notice that the targets under each claim should be similar to your Table 2.3 checklist of specific actions. Carefully investigate the structure of each set of assessment targets shown in slides 32 through 34. Which practice standards are assessed in Claim 2? 3? 4? Refer to pp. 55-78 of the document in (v) for more details. z. Carefully investigate the structure of the sample item specification table shown in slide 36. See to it that you understand every component in the structure. aa. What is an evidence statement? How was the sample evidence statement shown in slide 37 constructed? bb. What is a task model? How were the sample task models shown in slide 38 constructed? cc. Slide 39 shows a sample of a SBA item that consists of multiple claims and assessment targets. How are such items processed? Download the PowerPoint module labeled Mathematics Stimulus Considerations. This module provides information about the purpose, types, and guidelines for developing and using stimuli in nontraditional selected and constructed response items. dd. What are the different parts of a nontraditional selected or constructed response item? ee. What are some suggestions for developing appropriate stimuli material? Access the link below and refer to pp. 12 to 14 of the document which list additional suggestions for developing, writing, and implementing classroom-based 182

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performance tasks and selected response items. Provide responses to the question that follow. http://www.ets.org/Media/Tests/TOEFL_Institutional_Testing_Program/ ELLM2002.pdf ff. What should teachers do before and after administering a performance task? gg. Which among the tips for writing good selected response items appear new to you? 12.6  Projects and Portfolios as Alternative Summative Assessments

Performance tasks provide one instance of projects. In this section you will explore several other kinds of projects, including portfolios. Access the following link below and answer the questions that follow. http://www.intel.com/content/www/us/en/education/k12/project-design/ design.html a. What are projects? How are they different from worksheets in terms of purpose, scope of work, level of engagement, etc.? b. What are some characteristics of good and well-defined projects? Investigate the sample project units for Grades 3-5 and 6-8 students. What information should be included in a project unit? c. What are the benefits of project-based learning? How does it support and enrich traditional classrooms? What do you need to consider so that all students are able to experience success in accomplishing project plans and goals? Access the link Project in Action for more information about changes in instruction as a result of implementing effective project-based learning in class. d. What are some misconceptions regarding project-based learning? e. Open the file labeled Plan a Project. Learn the basic planning components and determine how they can be aligned with the structural components of the CCSSM and SBA. f. Open the file labeled Project Rubric. How are projects assessed and graded? Scoring rubrics are used to evaluate projects. They are either analytic or holistic in structure and content. Access the following links below to learn the difference between the two types in terms of purpose, advantages and disadvantages, development and design, guidelines, and scoring practices. http://www.heinemann.com/shared/onlineresources/e00278/chapter4.pdf http://www.uni.edu/chfasoa/analyticholisticrubrics.pdf http://www.teachervision.fen.com/teaching-methods-and-management/ rubrics/4524.html 183

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https://resources.oncourse.iu.edu/access/content/user/mikuleck/Filemanager_ Public_Files/EFL_Assessment/Unit_3/Metler_Designing_scoring_rubrics_ for_your_classroom.pdf Portfolios are collections of students’ work that showcase their efforts, progress, and achievements within a stipulated timeframe. Access the following links below to learn about traditional and electronic portfolios, including essential components, samples, and scoring guidelines. http://www.gallaudet.edu/clerc_center/information_and_resources/info_to_go/ transition_to_adulthood/portfolios_for_student_growth.html#MiddleRubrics http://www.educationworld.com/a_tech/tech/tech112.shtml 12.7  Math Journals and Lesson Investigations as Alternative Formative Assessments

Some teachers use math journals in a formative context. Math journal writing should encourage middle school students to reflect and communicate their ideas and dispositions. a. Access the following link below to learn about different journal writing strategies that apply in school mathematical contexts. http://www.mcrel.org/~/media/Files/McREL/Homepage/Products/01_99/ prod19_Writing_in_math.ashx Discuss advantages of and potential issues with each strategy. b. Access the following link below    which provides additional strategies for embedding writing in instruction and learning. http://www-tc.pbs.org/teacherline/courses/rdla230/docs/session_1_burns.pdf Which strategies are familiar? Which ones are new? c. Access the following link below which provides samples of math prompts about attitudes and dispositions. http://www.readwritethink.org/files/resources/lesson_images/lesson820/ MathPrompts.pdf Generate short samples of math prompts that specifically address the practice standards. Lesson investigations are short mini-projects or hands-on activities that students accomplish in one classroom session. For example, sixth-grade students who accomplish Activity Sheets 2.1 and 2.2 are provided with an opportunity to search for patterns and develop structures for understanding the relevant concepts in their own way. Effective math investigations should enable them to explore and construct 184

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ideas, collect data, and perform experiments that begin informally and/or creatively. Furthermore, there is no prescribed mathematical method since they are meant to encourage students to use sense-making or adaptive strategies and personal problemsolving techniques. Through purposeful scaffolding from others, students begin to develop formal knowledge. Eventually all results in math investigation should yield correct and valid mathematical understanding that support the development of structures and routine knowledge. They can also become the basis for more formal learning and instruction, and evaluation can be either holistic or analytic. Access the following link below for samples, guidelines, and a template. http://montclairgifteded.wikispaces.com/file/view/Math+Investigation+Hand out.doc 12.8  Exploring Assessment Systems

Access the following document below which explores different ways that you can develop a system of assessment for deeper learning. Work with a group and discuss the questions that follow. Conley, D.T., & Darling-Hammond, L. (2013). Creating systems of assessment for deeper learning. Stanford, CA: Stanford Center for Opportunity Policy in Education. a. What are 21st century skills and what role do new systems of assessment play in the development of such skills? b. What are systems of assessment and why should they matter to student learning? Study the given examples of local and international state systems of assessment and identify shared characteristics of such systems across context. c. What useful implications can be drawn from the assessment continuum shown in Figure 10 that can inform the manner in which you design and develop various performance tasks? 12.9  An Assessment Project

In this section you will develop assessment items and tasks and provide details that draw on the SBA structure. Access the link below which will direct you to the mathematics sample items and performance tasks page. http://sampleitems.smarterbalanced.org/itempreview/sbac/index.htm Click on the tab that says “View More Mathematics Sample Items.” The table shows grade bands, the four foundational claims, and performance tasks. Open the sample item labeled Integer Expressions (item 42960) which addresses a Claim 1 assessment target. Click on the tab that says “About This Item.” For the given selected response item, the following summary specification has been provided: (1) 185

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item name; (2) grade level; (3) claim; (4) target assessment; (5) CCSSM content standard; and (6) a brief description of the task in the form of an evidence statement and the relevant task model. Open the sample item labeled Sandbags (item 43026) which addresses a Claim 2 assessment target. For this particular constructed response item, the “About This Item” specification addresses the above 6 components and an analytic rubric (why?). Open the sample item labeled Field Trip. It is a performance task that seeks to evaluate six assessment target claims. The item specification addresses the following components: (1) classroom activity; (2) student task; (3) task specifications; and (4) a detailed analytic scoring rubric (why?). Work with a pair and accomplish the following tasks below. a. Identify a grade level and a CCSSM domain and develop five SBA-type items and tasks. One assessment item should be a performance task and the remaining four items should assess four different claims. Open the link below for more detailed samples of items under each claim. http://www.smarterbalanced.org/wordpress/wp-content/uploads/2011/12/ Math-Content-Specifications.pdf b. Be sure to provide an item distractor analysis in the case of selected response items. Also, develop scoring rubrics on all constructed response items and performance tasks as well. c. See to it that every assessment item that you develop comes with an “About This Item” specification. Develop task you construct should follow the structure provided in the case of the Field Trip sample. d. Make sure that all items and tasks are assessed according to the cognitive rigor matrix that you explored in section 12.5.  

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Content-Practice Learning

In this chapter you will deal with issues relevant to middle school students’ contentpractice learning of mathematics. In section 13.1 you will establish and articulate a general definition of learning that will help guide the manner in which you expect middle school students to learn the CCSSM. In section 13.2 you will explore a historical interpretation of the complex relationship between learning theories and the US school mathematics curriculum through the years, which will help you understand various contentious issues and concerns in mathematics education. In section 13.3 you will learn about relevant issues surrounding the Math War and possible consequences when you hold extreme or narrow views of learning school mathematical knowledge. In section 13.4 you will further deepen your understanding of the theories of Piaget and Vygotsky as they relate to mathematics learning and Fuson’s middle-ground perspective – learning-path developmentally appropriate learning/teaching model –that emphasizes growth in middle school students’ understanding and fluency of mathematics. To fully appreciate Fuson’s learning model you will explore the notion of a learning progression (LP) in some detail in section 13.5. LPs are useful to know for other reasons. The evidence and research that were used to support the content structure of the CCSSM drew significantly on LPs. You will also use LPs in Chapter 13 on content-practice teaching when you develop and write your unit and lesson plans. In section 13.6 which closes this chapter, you will learn about current issues and concerns involving brain-based studies and their implications to middle school students’ learning of mathematics. 13.1  Defining Learning

Access the following link below to view a 14-minute classroom video involving a class of eighth-grade students who were engaged in a modeling activity called The Barbie Bungee Jump. Answer the questions that follow. https://www.teachingchannel.org/videos/stem-lesson-ideas-bungee-jump a. Which Common Core State content and practice standards in mathematics did the students exhibit in the video? Refer to p. 4 of the CCSSM for a description of mathematical understanding. Assess the extent to which the students mathematically understood the relevant concepts and processes. How did their teacher, Mr. Patrick Roda, utilize group learning in class? Pay close attention on his reflective comments toward the end of the video regarding the positive 187

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impacts of interesting modeling activities on his students’ mathematical experiences. b. The students in Mr. Roda’s class exhibited some type of learning in class. Search the internet for at least ten definitions of learning and construct a Learning wordle. Which terms appear prominently in your wordle? Formulate a tentative definition of learning. c. Access the article below which discusses several different meanings of learning. De Houwever, J., Barnes-Holmes, D., & Moors, A. (2013). What is learning? Psychon Bull Rev, 20, 631–642.  ompare the definition of learning that you developed in (b) with the authors’ C functional definition of learning. Which components in the authors’ definition of learning are reflected in your own definition of learning? Which ones are different? Which ones are not mentioned? Which ones appear to be essential? Do you agree with their definition of learning? How is growth in mathematical understanding accounted for in their definition of learning? d. At this stage you should have developed a well-conceptualized perspective on learning. Think about the eighth-grade students in Mr. Roda’s class and assess how they learned in his class. What did they actually learn from the Barbie Bungee Jump activity? 13.2  Changing Views of Learning and Their Effects in the Middle School Mathematics Curriculum

Access the following article below which provides an interesting historical analysis of factors that contributed to cycles of change in the US school mathematics curriculum. Answer the questions that follow. Lambdin, D., & Walcott, C. (2007). Changes through the years: Connections between psychological learning theories and the school mathematics curriculum. NCTM 69th Yearbook. Reston, VA: NCTM. a. Read the narrative provided under each phase in the history of the school mathematics curriculum and then address the following questions related to learning: 1. Describe the social conditions that motivated the change in teaching practice and assessment of student learning. 2. Use De Houwever, Barnes-Holmes, and Moors’s three components of learning to assess how students were expected to learn and understand mathematics. 3. Explore how content-practice learning might have taken shape in math classrooms. b. Describe shared, different, and, more importantly, essential views of learning across the six phases. 188

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c. Consider your own recent experience in the classroom. Pay special attention on how you learned mathematics. Are there concerns, issues, or other factors relevant to your own learning experiences that have not been addressed in the article and are worth noting? 13.3  Math Wars: Debating About What and How Middle School Students Should Learn Mathematics

Access the following two articles below which provide detailed commentaries regarding the Math Wars in the history of US school mathematics education. Answer the questions that follow. Schoenfeld, A. (2004). The math wars. Educational Policy, 18(1), 253–286. Crary, A. & Wilson, S. (2013). The faulty logic of the “math wars.” The New York Times Opinionator. a. What motivated the Math Wars between traditionalists and reformists? Describe issues and concerns raised by each camp and determine whether they made sense to you. b. How might each camp analyze the manner in which the eighth-grade students in Mr. Roda’s class exhibited their learning of lines of best fit from the initial problem solving phase to the final whole-group discussion phase? What concerns might each camp have before, during, and after the implementation of the activity? c. Another unfortunate consequence that emerged from the Math Wars involves making a distinction between routine knowledge and flexible or adaptive knowledge. Learning that yields routine knowledge exhibits understanding and correct application of simple, complex, and sophisticated routines in efficient ways. Learning that yields adaptive knowledge also exhibits some level of routine competence, but there is also an acquired disposition toward inventing strategies and thinking about solutions in different ways. Assess the video for instances of routine and flexible knowledge. d. Is there a way out of the Math Wars? What can you learn from this unfortunate event in the history of US school mathematics education? 13.4  Understanding Piagetian and Vygotskian Views of Learning in Mathematics and Finding a Way Out of Extreme Views of Learning

In this section you will further deepen your understanding of Piagetian and Vygotskian views in relation to middle school students’ learning of mathematics. Access the article below and read the first six pages. Answer the questions that follow. 189

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Fuson, K. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching? Cognitive Development, 24, 343–348. a. How are middle school students expected to learn mathematics in a Piagetian context? What are some unfortunate consequences of misinterpreting mathematics learning in this context? b. How is mathematical learning expected to occur in a Vygotskian context? Compare your answers to your findings in (a). What are some unfortunate consequences of misinterpreting mathematics learning in a Vygotskian context? c. Fuson recommends a balanced learning-teaching path model as a way out of extreme misreadings of Piaget and Vygotsky. How are students expected to learn mathematics in Fuson’s context? Go back to the eighth-grade classroom video and assess the extent to which Mr. Roda modeled the different components of the balanced learning-teaching model listed under Table 2 on p. 347 of the article. 13.5  Learning Progressions in School Mathematics

You have come across the term learning progressions (LPs) several times in the preceding chapters and in various references from articles you have read. As you know the content standards in the CCSSM have been carefully informed by LPs. Consequently, constructing lesson plans and activities in the mathematics classroom involves developing appropriate learning paths that are informed by LPs. Access the following links below to learn more about LPs in mathematics and answer the questions that follow. http://www.cpre.org/sites/default/files/researchreport/1220_learningtrajecto riesinmathcciireport.pdf https://www.mheonline.com/assets/pdf/program/building_blocks_learning_ trajectories.pdf http://www.k12.wa.us/assessment/ClassroomAssessmentIntegration/pubdocs/ FASTLearningProgressions.pdf a. What are LPs? How are they similar to and different from learning trajectories? How are they similar to and different from typical scope-and-sequence planning activities? What are the essential characteristics in a LP? b. What should you be concerned about regarding LPs in school mathematics? c. How can LPs be used to inform content-practice teaching and assessment? The following site below contains links to detailed LPs involving the major content domains that comprise the Grades 5-8 and Algebra 1 CCSSM. Work

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with a group of four students and explore a specific LP together. Follow the instructions below. http://ime.math.arizona.edu/progressions d. Read the LP for the domain. Remember that middle school students are learning a significant amount of formal mathematical concepts and processes for the first time, so always put yourself in their situation and determine all new mathematical knowledge that they are they expected to learn in every phase of the LP. Assess the logic and reasonableness of the proposed progression. Discuss possible benefits to students in the long term, including potential issues and concerns. e. All middle school teachers need to prepare well for Back to School Night. A nerve-wracking part of the evening involves presenting to parents, administrators, and colleagues information about each curriculum that matters to students and instructional activities or methods that will be used to help them achieve the relevant learning goals. Develop a LP poster for your domain. Construct either a map or a path and present details of your LP in a format that is interesting and sensible. Access any of the following links below, which provide initial sources of information about making posters. http://abacus.bates.edu/~bpfohl/posters/#titles (Creating a Poster in PowerPoint) http://clt.lse.ac.uk/poster-design/ (Poster Design Tips) http://www.iris.edu/hq/files/programs/education_and_outreach/poster_ pilot/Poster_Guide_v2a.pdf (Pedagogical Power of Posters and Tips) 13.6  Learning from Neuroscience

Recent discussions on learning in schools recommend the use of brain-based interventions that depend on valid neuroscientific evidence. In this section, you will learn about the nature of such evidence and the manner in which results need to be interpreted in light of the various constraints in studies that generated them. Access the following article below to learn about neuroscientific evidence and implications to middle school students’ learning of mathematics. Answer the questions that follow. Rivera, F. (2012). Neural correlates of gender, culture, and race and implications to embodied thinking in mathematics. In H. Forgasz & F. Rivera (Eds.), Towards equity in mathematics education (pp. 515–543). New York, NY: Springer. a. Distinguish between descriptive and prescriptive information in relation to neural mechanisms that support cognitive processes in mathematical thinking and learning. What does this mean in practical terms? 191

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b. Neuroscience evidence is basically correlational and not causational in nature. Furthermore, there is a difference between neural and psychological evidence. What do such types of evidence mean in relation to the teaching and learning of mathematics? c. How is learning defined in neuroscientific terms? Discuss the different types of memory and their implications to cognitive processing. d. Describe the four lobes that divide the human cerebral cortex and their individual functions. e. What do you learn from Nieder’s research regarding the neurobiological evolution of symbolic thinking and reasoning in humans and nonhuman primates? f. Describe Dehaene’s neural network of triple coding and its implications to mathematical learning. g. What do you learn from neural studies that target mathematical, linguistic, and visuospatial processing? h. What are some implications of neural studies that focus on gender, race, and culture to mathematics teaching, learning, and assessment at the middle school level?  

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Content-Practice Teaching

In this chapter you will deal with content-practice issues relevant to teaching middle school mathematics. You will learn different teaching models and write unit plans and lesson plans. The component of teaching completes the alignment mindset that you have been asked to frequently bear in mind in your developing professional understanding of teaching to the CCSSM. That is, as indicated in Figure 14.1, issues in content-practice teaching are also about issues in content-practice learning and assessment. The pedagogical troika shapes the manner in which the CCSSM is implemented in individual classrooms.

Figure 14.1. Grounding a CCSSM-Structured Curriculum Around the Pedagogical Troika

14.1  Describing (Good) Teaching

Access the fifth-grade classroom video A Passion for Fractions from the link below. Watch the 13-minute video and answer the questions that follow. Work with a group and deal with the questions together. https://www.teachingchannel.org/videos?page=2&categories=subjects_ math,organizations_national&load=2 193

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a. Identify the Common Core State content and practice standards in mathematics that the fifth-grade teacher, Ms. Betty Pittard, wanted her students to learn relative to the day’s lesson. Pay special attention on the CCSSM-suggested mathematical strategies for processing the content. b. Use the peer classroom observation tool shown in Table 14.1 to help you gather information about key content-practice events that transpired in Ms. Pittard’s class. Content-practice events pertain to teaching/learning episodes when teachers employ particular mathematical practices in order to support students’ learning of a target content standard/s. For each key content-practice event in Ms. Pittard’s class, try to capture the questions that she used to facilitate meaningful discussion on the topic in her class. Use the cognitive rigor matrix you learned in Chapter 12 to help you analyze the type and depth of the questions she used in class. c. What do you learn from Ms. Pittard’s classroom session about what it means for teachers to teach to the CCSSM? Table 14.1. Peer Classroom Observation Tool Peer Classroom Observation for _________________ (Name of Teacher)

Grade Level and Class Type: _____________________

CCSSM Content Standard/s Domain:

CCSSM Practice Standard/s Cluster:

Cluster:

Specific Actions:

Standard/s: Sequence of Key ContentPractice Events and Relevant Materials Used

Supportive Questions Used to Facilitate the Event

Cognitive Rigor Analysis (Depth of knowledge and type of thinking exhibited by students)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Access the following article below which describes general characteristics of (good) teaching. Answer the questions that follow. Tell, Carol. (2001). Appreciating good teaching: A conversation with Lee Shulman. Educational Leadership, 6–11. d. How does Lee Shulman describe (good) teaching? e. To what extent does Ms. Pittard’s teaching style model the characteristics you identified in (d)? Access the following article below which describes three important activities that encourage students to engage in meaningful mathematical activity in the classroom. Answer the questions that follow. Fisher, D., Frey, N., & Anderson, H. (2011). Thinking and comprehending in the mathematics classroom. In M. Pitici & F. Dyson (Eds.), The best writing on mathematics (pp. 188–202). New Jersey, NJ: Princeton University Press. f. Describe the ninth-grade teacher’s daily instructional routine and identify possible consequences on content-practice assessment and learning. g. The authors identified modeling, vocabulary development, and productive group work as three effective instructional strategies for engaging students in the mathematics classroom. How does each strategy support good content-practice learning? Reevaluate Ms. Pittard’s teaching style based on these strategies. 14.2  Teaching Models in Middle School Mathematical Settings

In this section you will explore six models of teaching mathematics at the middle school level. It should be noted that while the models are discussed separately, effective teaching often involves employing combinations of two or more models. As you learn more about each model and its components, keep in mind ways in which content-practice strategies can be integrated into the components. 14.2.1 E-I-S-Driven Teaching The acronym E-I-S stands for Jerome Bruner’s three modes of representation, Enactive-Iconic-Symbolic. Whenever you introduce middle school students to new representations in school mathematics, a good entry point involves providing them with opportunities to act on objects that represent a target knowledge (enactive). Hopefully, over instructional time, they are able to think about such representations mentally (iconic) and linguistically (symbolic). Language in the symbolic phase is the domain of codes in contracted or condensed forms, generalizations, and abstractions. In this phase, especially, students overcome limitations in enactive- and iconic-based representations. Work with a group and do the following two tasks below.

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a. Search the internet for more discussions regarding the E-I-S progression. Provide detailed characteristics of each type of representation and activities that support them. Discuss advantages and disadvantages of each type. Should E-I-S be taught in that order? b. Identify a CCSSM content-practice standard and illustrate how it can be taught in an E-I-S manner. 14.2.2 C-R-A Sequenced Teaching The acronym C-R-A stands for Concrete-Representational-Abstract. C-R-A sequenced teaching emerged from intervention studies involving students with learning difficulties. Both the C-R-A and E-I-S models share common features such as using various multisensory techniques (i.e., visual, auditory, kinesthetic, and tactile strategies) that support students’ thinking in the abstract phase of learning. But the R in C-R-A involves much more than the I in E-I-S in the sense that students are encouraged to draw pictures as a way of helping them transition to the abstract phase. In school mathematical contexts, meaningful and effectively drawn pictures are diagrammatic in nature. That is, they reflect schemes or patterns that convey necessary mathematical relationships. Hence, constructed diagrams on problem solving tasks are not mere pictures or faithful copies of objects but convey structural relationships (e.g., mathematically relevant figural units) that support mathematical thinking and understanding. Certainly such diagrams may emerge from pictures of concrete or everyday objects. c. Continue working with the same group and address the same two tasks listed in 14.2.1 in the context of the C-R-A teaching model. 14.2.3 Van Hiele Sequenced Teaching A Van Hiele sequenced teaching approach engages students in five phases of geometric processing: visual; analysis; abstraction; deduction; and rigor. The model initially emerged from the research studies of Dina van Hiele-Geldof and Pierre van Hiele in relation to students’ thinking about school geometric concepts. Think about your learning experiences in Chapter 5. Notice how the different sections encourage you to progress in your geometric thinking from the visual phase (recognizing and sorting objects based on surface appearances) to the analysis phase (developing initial properties about objects) and the abstract phase (constructing definitions based on common properties of a set of objects and making sense of inclusive relationships). The deductive and rigor phases involve formal mathematical reasoning and systems thinking, skills that students learn in high school geometry and beyond. d. Keep working with the same group and address the same two tasks listed in 14.2.1 in the context of the van Hiele teaching model. 196

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14.2.4 Culturally-Relevant Teaching Culturally-relevant teaching emerged from the work of Gloria Ladson-Billings and it involves employing culturally-responsive instructional strategies and activities. That is, one way of teaching mathematics in a meaningful manner involves adapting instruction and developing activities that draw on students’ culture which exerts a significant influence in the way they think and learn. Effective culturallyresponsive contexts motivate them to participate in class and develop a productive mathematical disposition. Continue working with your group and address the following two tasks below. e. Conduct an internet search that provides reliable information about the race and ethnic profiles of students in today’s mathematics classrooms. f. Search the internet for more discussions regarding culturally-relevant teaching. What conceptions and misconceptions do you need to know about this teaching approach? g. Describe a few general instructional strategies or activities that either draw on cultural referents or model cultural relevance. Use at least one of them to help you teach the content-practice standard you identified in (b) in a culturally responsive manner. h. Drawing on students’ funds of knowledge is a related culturally-responsive perspective that, according to Luis Moll, Cathy Amanti, Deborah Neff, and Norma Gonzalez, “refers to the historically accumulated and culturally developed bodies of knowledge and skills essential for household or individual functioning and well-being.” Search the internet for discussions regarding this perspective and ways in which mathematics teachers have used them to inform their own instruction and pedagogical practices. 14.2.5 SDAIE-Driven Teaching The acronym SDAIE stands for Specially Designed Academic Instruction in English. A SDAIE-driven teaching approach targets the needs of English learners who need appropriate scaffolding or “comprehensible input” to help them obtain full and equal access to academic concepts and language. Continue working with the same group and address the two tasks below. i. Who is an English learner (EL)? How can he or she be supported to succeed in the context of the CCSSM? Access the following links below for a good overview of concepts surrounding ELs, language demands, and the CCSSM. Pay attention on key factors that can inform the way you teach mathematics to ELs. http://www.ccsso.org/Documents/2013/Toward_a_Common_Definition_ 2014.pdf?utm_source=buffer&utm_campaign=Buffer&utm_content= buffer3f9b6&utm_medium=twitter 197

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http://ell.stanford.edu/sites/default/files/pdf/academic-papers/10-Santos%20 LDH%20Teacher%20Development%20FINAL.pdf j. Address the same tasks described in (f) and (g) in 14.2.3 but with the SDAIE teaching model in mind. 14.2.6 Differentiated Instruction Differentiated instruction is a teaching model that employs a variety of instructional strategies as a way of adapting to the different needs of individual learners in a classroom. Effective teachers who exhibit this teaching model design their curricular and classroom activities in ways that enable all their students to achieve the same standards and proficiency levels despite differences in cognitive processing ability. Such differences stem from a variety of factors such as readiness level, learning profile, interests, etc k. Continue working with the same group and address tasks (f) and (g) in 14.2.3 relative to the differentiated instruction model. l. Access the following link below that shows how an Algebra 1 teacher, Ms. Marie Barchi, employs differentiated instruction to help her students learn to solve simple one- and two-step linear inequalities proficiently. https://www.teachingchannel.org/videos/student-daily-assessment?utm_source= Teaching+Channel+Newsletter&utm_campaign=66a9b38bed-Newsletter_ September_28_2013&utm_medium=email&utm_term=0_23c3feb22a66a9b38bed-291836021 How does Ms. Barchi use exit cards use in her Algebra 1 class to help her achieve effective differentiated instruction? Discuss advantages and potential issues. 14.2.7 Flip Teaching Flip teaching is a teaching model that basically reverses classroom time for traditional instruction and homework time for practice. This model begins with the assumption that all students have open access to either a computer or the internet. Teachers assign video lessons for them to study on their own (as a homework task) that often deal with introductory or basic information. Class time is then used to engage them on problem-solving tasks that require the use of higher-order skills. Since flip teaching promotes blended learning in the sense just described, lecturing is almost minimized or even eliminated and is replaced by problem-solving activity. Continue working with the same group and address the following two tasks below. m. Access the following link below to learn more about how this teaching model is used in particular schools. Generate a few initial impressions. Also, assess which 198

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among the teaching models in the preceding subsections are compatible with this particular model. http://www.boston.com/bostonglobe/editorial_opinion/oped/articles/2011/ 09/18/flipping_for_math/ n. Deal with tasks (f) and (g) in 14.2.3 with the flip teaching model in mind. 14.3.  Teaching with Manipulatives and Computer and Video Games and Apps

Manipulatives are objects that middle school teachers use to model concepts, ideas, and processes in school mathematics. For example, the algeblocks shown in Figure 7.3 represent concrete monomial expressions that support students’ mathematical understanding of polynomial operations. Manipulatives such as the algebra balance scales drawn from the National Library of Virtual Manipulatives are virtual, online, and technology-enhanced versions of their concrete forms. Access the link below for samples of virtual manipulatives. http://nlvm.usu.edu/en/nav/vlibrary.html Access the following articles below that address various aspects relevant to the nature and use of manipulatives in middle school classroom settings. Read them together first and then answer the questions that follow. Stein, M.K., & Bovalino, J. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6(6), 356–359. Stewart, M. (2003). From tangerines to algorithms. Instructor, 112(7), 20–23. Hunt, A., Nipper, K., & Nash, L. (2011). Virtual vs. concrete manipulatives in mathematics teacher education: Is one type more effective than the other? Current Issues in Middle Level Education, 16(2), 1–6. a. How do you use manipulatives effectively in the mathematics classroom? Generate a list of dos and donts. b. Discuss the role of manipulatives in each teaching model in section 14.2. All education-related computer and video games and apps provide every student with an opportunity to (visually) engage in mathematical knowledge building and development at their own pace. They involve a good amount of interactive or dynamic component, enabling students to generate abductions (i.e., formulate conjectures), perform inductions (i.e., repeatedly test conjectures), and experience deductive closure (i.e., successfully apply acquired rules to several more problems). Teaching with games and related apps should help students make progressive transitions in their understanding of a target content knowledge, skill,

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or application. Certainly, powerful and effective games and apps should enable them to also employ practice standards. c. Address the same tasks described in (f) and (g) in 14.2.3 but with the games and app Access the following article below, which provides a good synthesis of research on the efficacy of teaching mathematics with concrete manipulatives. Carbonneau, K., Marley, S., & Selig, J. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. d. Revisit the list you constructed in (a) and compare your responses with the findings noted in this particular article. Add more to your list, if necessary. e. In previous sections you explored several different computer and video games and apps for learning mathematics. To what extent do the authors’ findings and recommendations relative to concrete manipulatives apply to games and apps? 14.4  Teaching Mathematics with Guide Questions

Regardless of teaching approach, asking the right questions when middle school students are engaged in content-practice learning is just as equally important as knowing when and how to use manipulatives in mathematical activity. Employing guide questions can take place before, during, and after instruction in either individual or group contexts; it is a useful formative (i.e., ongoing) assessment tool. The use of guide questions also set the mood and tone of content-practice learning in class. They influence the direction of both the lesson and the depth or quality of students’ learning experiences and content-practice driven knowledge. In fact, an effective way to help students develop a consistent mindset toward content-practice learning is to use appropriate guide questions that will cue them to specific practice-based actions. Access Table 3 on pp. 14-16 from the link below for examples of good guide questions under each practice standard. http://www.cde.ca.gov/ci/ma/cf/documents/aug2013overview1.pdf#search= examples%20of%20questioning%20strategies%20teachers%20might%20 use%20to%20support%20mathematical%20176%20thinking%20and%20 stude&view=FitH&pagemode=none Access the following three links below which provide an introduction to the nature of good questions, how they are created, what teachers need to do when posing good questions, and samples of possible generic and key questions to ask in a variety of contexts. Work with a group and answer the questions that follow. http://www4.uwm.edu/org/mmp/PDFs/Yr5_PDFs/ThoughtProvokingQuestionsslides.pdf 200

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http://www-tc.pbs.org/teachers/_files/pdf/TL_MathCard.pdf http://www.fcps.org/cms/lib02/MD01000577/Centricity/Domain/97/The%20 art%20of%20questioning%20in%20math%20class.pdf a. Generate guide questions for each classroom situation presented on pages 6-7 of the third document. b. Access Ms. Pittard’s classroom video once again and reassess the type and quality of the guide questions that she used to help her fifth-grade students achieve the target content and practice standards relevant to the concept of multiplication involving two proper fractions. 14.5  Content-Practice Unit Planning

Unit planning is one very exciting aspect of a teacher’s job. Regardless of school or district context and resources, teachers always need to plan a unit before they can write sensible and logical lesson plans. The unit-planning process can be accomplished either individually, which rarely happens these days, or collectively with grade- or subject-level colleagues. Research on effective mathematics departments indicates that teachers who plan units, lessons, and assessments together produce much better overall results in student learning than teachers who prefer to plan alone. However, for the purpose of this section and considering the fact that you are learning to teach middle school mathematics, you will need to develop your own unit plans in order to experience firsthand what this process entails. Also, since feedback matters, you will need to share your unit plan with a group and critique each other’s work. In schools, inservice teachers often rely on feedback from other individuals (e.g.: grade-level teammates; math coach; subject-matter coordinator; principal), so you need to get used to it. Content-practice unit planning (CPUP) involves developing, managing, and assessing a sequence of related content-practice lesson plans that you plan to implement within a stipulated range of time. CPUP involves the following aspects below. • There is content-practice lesson sequence to be developed, taught, and assessed. • Content-practice lessons are coherent and focused. • Considering the preparation and time that are needed for planning, the implementation phase requires at least one work week. There is no scientific basis for the minimum and maximum number of sessions that comprise a CPUP. However, in practical terms, a really long unit plan (say, one month) can be rather onerous to achieve. While the ultimate decision rests on you (or your group), it is always important to obtain feedback at all times. Another very important factor in your decision-making process involves how well you know your grade-level content-practice standards. Other factors worth considering pertain to various psychological and institutional constraints that affect instructional 201

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delivery and time. Examples include the following: accounting for students’ needs; demographic information; learning profiles of students in your class, grade level, school, and district; grade-level planning with colleagues; district-mandated pacing guides and periodic benchmark assessment issues; and school- and district-wide extracurricular activities and affairs. Developing a coherent and well-articulated CPUP involves addressing the thirteen content-practice related tasks listed in Table 14.2. Taken together, they target left-foot unit planning. Right-foot unit planning takes into account all the intended and unintended psychological and institutional constraints that shape and complicate the daily task of teaching. Hence, a successful unit-walk orchestration, which is a perennial task for all novice and experienced teachers, involves learning to smoothly coordinate between left- and right-foot requirements. Chapter 14 addresses rightfoot planning concerns. In the remainder of this section, you will focus on left-foot CPUP issues. Table 14.2. Content-Practice Unit Planning Steps

Pre-CPUP Concern 0. Obtaining first impressions of a target unit

Content-Practice Framing Concerns 1. Labeling the unit 2. Identifying primary (and, if applicable, secondary) big idea/s 3. Identifying key concepts and processes that are relevant to understanding the big idea/s 4. Developing specific essential questions for each concept and process 5. Developing a unit content overview, abstract, or summary 6. Producing a well-organized and well-justified unit content trajectory in the form of a concept map 7. Generating rough overviews or drafts of individual content-practice lesson plans that will accomplish each essential question identified in step (4). 8. Developing a content-practice unit assessment plan 9. Identifying prerequisite knowledge, skills, and content-practice standards  Content-Practice Implementation Concerns 10. 11. 12.

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When you begin CPUP you need to decide on how to define your unit. A mathematics curriculum can be structured as units in several different ways, as follows: • • • • •

Domain-driven units; Cluster-driven units; Content standard-driven units; Chapter-driven units; Pacing guide-driven units.

The first three structures reflect the CCSSM framework while the fourth and fifth structures reflect textbook- and school- or district-mandated structures, respectively. Regardless of the type of unit structure, however, both the CCSSM content-practice standards and SBA (or PARCC) need to be embedded in all the lessons. As an example, refer to p. 41 of the CCSSM, which shows the mathematical content overview for sixth-grade students. There are at least three ways to reorganize the content overview. If you choose to organize your units by domain, the Number System (NS) as one unit will require more time to accomplish than the Geometry unit which means you may need to break down the NS domain into several smaller units. If you choose to organize your units by cluster then you will need to develop ten units since the overview contains ten clusters. To some extent, chapter-driven units are organized by cluster. If you choose to organize your units by content standard you will need to develop twenty-nine units. District- and school-developed pacing guides are often organized in several different ways, and grade-level teams usually work together to organize unit planning around such constraints. Work with a group and address the following task below. a. Search the internet for at least three samples of pacing guides and describe common, different, and essential components and/or structures. For each guide, determine a way to organize the content by units. To better understand the steps in Table 14.2, access the following link below, which shows a sample Grade 7 chapter drawn from the textbook, Connected Mathematics Project 3, published by Pearson. Consider the textbook chapter as your unit that consists of fifteen lessons. Continue working with the same group and respond to the following tasks below. Use the unit plan template shown in Table 12.3 to help you organize and record your responses. http://www.pearsonschool.com/index.cfm?locator=PS1yJe&PMDbProgramI d=110081&PMDbChildProgramId=110083&sampleId=11222 b. Step 0 involves obtaining first impressions of a target unit. Browse through the chapter, the different lessons, and the chapter exercises. Gather information about concepts and processes that all students need to learn by the end of the chapter.

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Check the CCSSM and determine the applicable domain, cluster, and content standards that map with the chapter. Assess the chapter’s examples and tasks in terms of the examples and recommendations provided in the CCSSM, including sample SBA items that are available. Identify the relevant CCSSM practice standards that can be used to support and facilitate content knowledge acquisition. c. Step 1 involves labeling the unit. Labeling enables you to convey in explicit terms the primary content knowledge (i.e., the topic) that you want students to learn. Provide a label for this particular unit. d. Step 2 involves identifying the primary and, if applicable, secondary big idea/s for the unit. Randall Charles defines a big idea in mathematics in the following manner: “A Big Idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole.” Provide a big idea statement for this particular unit. Limit to two sentences. Note that such a statement is not a complex outline of content and practice standards but simply a concise and clear expression of a synthesized mathematical idea that students will explore throughout the unit implementation phase. e. Step 3 involves identifying key mathematical concepts and/processes that matter in acquiring knowledge about the big idea. The analysis involved in this step is still content-driven and does not have to be expressed in terms of content and practice standards. Regardless of standard, in fact, this step should help you identify content and academic language that need to be emphasized throughout the unit. List down all key concepts, processes, and/or academic language that are relevant in this particular unit. f. Step 4 involves developing specific essential questions that will help you tackle your big idea/s for the unit. Following Grant Wiggins and Jay McTighe, an essential question expresses what students need to answer in order to say that they have learned, understood, and made sense of an aspect of the big idea. Once again, such questions do not need to depend on particular content-practice standards. Identify essential questions for this particular unit. Later, in step 6, you will be asked to map individual content-practice lesson plans with the relevant essential question/s. g. Step 5 involves developing a unit content overview. This step requires you to reread your earlier responses in order to help you develop a coherent narrative of the unit that you can share with interested stakeholders (e.g., parents and colleagues). When you write your narrative description, you may begin to point out relationships between the unit and the relevant CCSSM content and practice standards, which convey your target learning outcomes for the unit. You may also want to point out how lessons are connected to each other and the relevant academic language that matter. Highlight materials and resources that you intend to use as well. Develop an overview for this particular unit. h. Step 6 involves producing a unit content trajectory in the form of a concept map. Generally, a concept map is a graphical representation that shows relationships between concepts in the unit. A content trajectory can be conveyed through a concept map, but it should also convey a (multi-)path sequence that models one 204

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plausible progressive emergence of content knowledge from initial conceptions to the sophisticated conceptual form. All sequences of content-practice lesson plans are, in fact, based on information drawn from a unit content trajectory. Follow each step below which will help you construct a content trajectory in the form of a concept map for this particular unit. 1. For each lesson in the chapter, identify key concepts and processes that students need to learn. Generate a label for each concept and process and then record the label in a post-it note. 2. Reassemble the labeled post-it notes that you developed for the entire unit in the form of a concept map. Check to make sure that your concept map reflects a well-reasoned and logical content trajectory, showing transitions in content knowledge from the simple and informal or beginning phase to the complex and sophisticated forms of representation. i. Steps 7 and 8 go together. These two steps involve mapping the content-driven analysis in the preceding steps with the relevant Common Core content and practice standards. Step 7 involves generating rough drafts of lesson plans that will help you accomplish the different aspects of the unit. Table 12.4 provides a template for organizing lesson plan drafts. The primary intent in writing lesson plan drafts is to gather initial impressions of basic requirements that you will need to answer the target essential questions. The requirements for each lesson involve identifying: the relevant essential question; the applicable CCSSM content and practice standards; activities, materials, and resources that will help you teach the lesson; the appropriate formative and summative assessment tasks that will help you evaluate student learning of the lesson; and the amount of time you need to complete the lesson. Note that counting the total number of sessions that is needed to accomplish a lesson involves taking into account both instructional days and additional time for assessment and reteaching days. Step 8 involves developing a unit assessment plan. Each lesson plan should include information about possible formative and summative assessment tasks. Following the SBA style that you learned in Chapter 11, make sure that you label each task you intend to use for easy referencing in the lesson plan. Use Table 14.5 to help you assess how your tasks are aligned with the CCSSM content and practice standards. Further, use Table 14.6 to help you assess their levels of cognitive rigor (see section 11.5.2 item (t) for details regarding the cognitive rigor matrix). Develop an assessment plan for this unit. Make sure that you clearly provide a label for each assessment task that you intend to use so that it is easy to trace its location in Tables 14.5 and 14.6. j. Step 9 involves identifying the relevant prerequisite information (i.e., knowledge, skills, and content and practice standards) that students need to have in order 205

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successfully accomplish the goals of the unit. For this particular unit, identify the relevant prior information. k. Step 10 involves planning for effective time management. This pertains to the last column in Table 14.4. Do not underestimate this aspect of the CPUP. Beginning teachers usually have trouble estimating the appropriate amount of time that students will need to learn each lesson in a unit. You need to remind yourself that your students are learning formal mathematical concepts and processes for the first time, so be reasonable. You also need to allot time for everyday or ongoing formative assessments, weekly and unit summative assessments, and, if necessary, reteaching sessions. Provide a breakdown and total number of 55-minute classroom sessions that you need to achieve the goals and objectives of your unit. l. Step 11 involves identifying relevant materials and resources that can help you implement your activities for the unit. Technology resources pertain to hardware and software tools (e.g., apps). Printed resources and materials include worksheets, textbooks, and other visual tools for learning. Supplies pertain to typical schoolrelated learning aids and tools, accessories, and manipulatives. Internet resources are links that are relevant to teaching and learning a unit. For this particular unit, identify the supplies, resources, and materials that matter. m. Step 12 involves articulating accommodation-related issues and resources. This last component in CPUP encourages you to reflect on the specific needs of your class. For example, if you have English learners and/or students with mathematical difficulties, you may briefly discuss explicit strategies that you intend to use to help them succeed in learning the unit goals and lesson objectives. For this particular unit, identify accommodation strategies that you can use to help English learners and students with mathematical difficulties learn to perform operations on integers and rational numbers proficiently before, during, and after each lesson. 14.6  Content-Practice Lesson Planning

Content-practice lesson planning involves developing, managing, and assessing an essential question in a detailed manner. You can construct such plans either daily, weekly, or based on the nature and complexity of a target content-practice objective. Every exemplary content-practice lesson plan is usually clear, systematic, logical, and sufficiently thorough, which means to say that it is possible for any interested stakeholder, and especially substitutes, to imagine how the lesson is expected to unfold by simply reading the plan. Do remember that while a lesson plan operates like a script, it is not supposed to be read directly in class. It is meant to help you anticipate or hypothesize an emerging content-practice structure. Overall, a good lesson plan should help you implement mathematics instruction that is coherent and provides a conducive and optimal environment for student and classroom learning. 206

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Ultimately, James Scrivener points out, “in class, [you] teach the learners [and] not the plan.” Do the following tasks below. a. Access the following links below to learn more about effective lesson planning. https://docs.google.com/a/sjsu.edu/document/d/17H0IYY0ICX4PUtdpRGJXal_ jemRhf2PIqpVK2U9TdcU/edit http://www.crlt.umich.edu/gsis/p2_5 http://www.scholastic.com/teachers/article/new-teachers-guide-creatinglesson-plans http://www.cal.org/caela/tools/program_development/elltoolkit/Part2-29 LessonPlanning.pdf b. Use what you learned from the above links to construct a 55-minute contentpractice lesson plan based on the CPUP you developed in the preceding section. Use the template shown in Table 14.7 to help you organize your lesson plan. Consider the following additional comments and instructions below in relation to Table 14.7. Under the CCSSM Content Standard row, this is a simple cut-and-paste from the CCSSM document. Identify the appropriate domain, cluster, and content standard/s that apply to the lesson. There is no need to paraphrase. Further, under Standard, if a content standard contains two or more related smaller standards, highlight in either bold or underline form the specific line (or lines) that applies (apply) to your lesson. Under the CCSSM Practice Standard/s row, this is also a simple cut-and-paste from the CCSSM document. Refer to your Table 2.3 for specific primary practice action/s that you want your students to employ in learning the content standard you identified in the preceding section row. Do not be too eager to identify all practice standards that apply. Fill in the secondary cluster and related practice standard/s, if necessary. Under the Prerequisite CCSSM Content Standard/s row, this is also a simple cutand-paste from the CCSSM document. Identify the prerequisite content standard/s that students need to know to learn the new lesson well. You may need to highlight in either bold or underline form the appropriate line/s from the content standard/s describing the prerequisite knowledge. Under the Essential Math Question row, this is a simple cut-and-paste from your unit plan. Under the CR Objective/s row, the acronym SWBAT stands for “Students Would Be Able To.” There are two things that you need to accomplish. First, the relevant CCSSM objective may already provide you with the appropriate language in which case you simply cut and paste. Second, use the Cognitive Rigor Matrix to help you decide whether you need to construct a more specific and measurable objective or

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set of objectives than the one provided in the CCSSM document. In either case, analyze the learning objective according to the matrix and record your result by coding in the following manner: [DOK #, TOT], where DOK # refers to the level of depth of knowledge and TOT pertains to the target type of thinking. Remember that each CR Objective is a learning outcome that will be assessed in either formative or summative context. Also, be realistic about the number of objectives that you intend to target in a 55-minute lesson. Under the Target SB Summative Assessment Tasks (Claim/s & Item/s) row, while you are not expected to implement a summative assessment task at the end of a 55-minute lesson, being aware of such tasks can inform the manner in which you develop appropriate formative tasks. In fact, any summative assessment task can be used a formative context with some modification. Try to obtain released summative assessment tasks that match your content-practice standards and CR objectives. Also, identify the appropriate claim and item number that applies to each task. Refer to item (v) in section 11.5 to help you recall how this task assignment processing is accomplished. Under the Supporting Material/s and Resource/s row, draw on your unit plan and identify relevant technological tools, printed matter, supplies, and, if applicable, internet resources that you intend to use to accomplish your lesson objective. Your teaching model should also help you decide what you need. Under the Lesson Activity row, you need to address the following key events in your 55-minute lesson: beginning event; during event; and ending event. You may want to revisit the above links to gather suggestions for this very important phase in lesson planning. Accomplish the following two steps: First, determine the teaching model/s that you intend to use to teach your lesson. For illustration purposes, assume the C-R-A and SDAIE teaching models, with levels of collaborative learning actions (i.e.: work independently, work together in small groups, and work together as a class) providing additional teaching support. Second, describe under Event Sequence where each phase of the C-R-A, particular SDAIE strategies, and each level of collaborative learning will be implemented in your lesson. Also include under each phase your planned activities, academic language strategies, and labeled formative assessment tasks that will be used to support teaching and learning. Further, provide appropriate practice-driven guide questions and target response/s to help you gain a sense of how you expect your lesson activity to evolve. See to it that your guide questions are aligned with your practice standard objective/s. Budget your time wisely so that you are able to accomplish every part of your lesson activity within the allotted time and in a reasonable manner. Under the Formative Assessment Task (Claim/s & Item/s) row, provide details of all labeled formative assessment tasks that you identified in your Lesson Activity section. Include an analysis by initially assigning a claim number. Do remember that all tasks, stipulated CCSSM content and practice standards, and CR learning objectives must align well together. 208

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Under the Additional Notes row, identify important issues, concerns, reminders, and other details that you think are pertinent to know regarding your lesson. For example, if your lesson will employ a SDAIE strategy, you may identify your target academic language objective/s that your students need to achieve in addition to the CCSSM content-practice standard objectives. Another example involves the use of differentiated instruction in which case your notes can provide details that will help you identify concerns that you need to consider, which cannot be simply integrated in the lesson activity section of your plan. A third example that requires a note involves conducting collaborative learning. You simply do not ask students to work together in pairs or in groups, which means you need to provide details about what you need to accomplish and/or indicators to carry through an effective group work activity. Refer to sections 14.3 and 14.4 for details regarding how to set up, implement, and process collaborative learning in the math classroom. A fourth example that may require notes is when you intend to use particular technological tools before, during, and/or after instruction. 14.7  A Planning Project

In this section you will develop your own CPUP, lesson plans, and assessments based on certain requirements. Do the following tasks below. a. Choose a chapter in any Grade 5-8 or Algebra 1 mathematics textbook that has at least five individual lessons or sections. b. Develop a CPUP for the chapter. Use the template shown in Tables 14.3 through 14.6 to help you organize your CPUP. c. Develop a lesson plan: (1) involving one lesson in the chapter that combines C-R-A and SDAIE; (2) involving another lesson in the chapter that combines C-R-A and cultural responsiveness; (3) involving at least one lesson in the chapter for a flipped class. Use Table 14.7 for guidance. See to it that all assessment tasks are presented and analyzed following the appropriate forms. d. Develop a project as an alternative form of summative assessment for the unit. Refer to section 11.6 for details regarding essential project components and scoring rubrics and guidelines.

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Chapter 14 Table 14.3. Content-Practice Unit Plan Template Subject and Grade Level Unit Title

 

  Unit Content Big Idea(s) and Essential Questions

Big Idea Statement

 

Key Concepts and/or Processes

 

Essential Question for Each Section in the Unit

  Unit Content Overview

  Unit Content Trajectory in the Form of a Concept Map Rough Drafts of Lessons See Table 14.4 Unit Formative Summative Assessment Plan, Details, and Analysis See Tables 14.5 and 14.6 for the analysis Prerequisite Knowledge to the Unit    

Breakdown and Total Number of 55-Minute Classroom Sessions Needed to Achieve Unit Goals and Lesson Objectives Unit Materials and Resources Technology Resources

 

Printed Resources and Materials

 

Supplies

 

Internet Resources

 

Accommodation Issues and Resources

 

Table 14.4. Lesson Plan Sequence Template Draft Formative and Number of 55-Min Summative Sessions Assessment Tasks

Essential Question

Content Standard

Practice Standards

Lesson Overview

Appropriate Activities and Relevant Materials and Resources

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

210

 

CCSSM Domains

CCSSM Practices

 

Look for and express regularity in repeated reasoning

 

 

 

   

Attend to precision

   

   

Model with mathematics Use appropriate tools strategically Look for and make use of structure

 

 

 

 

 

Make sense of problems and persevere in solving them

 

 

 

 

 

Reason abstractly and quantitatively

 

Geometry

   

Construct viable arguments and critique the reasoning of others

   

Fractions and Operations Measurement and Data

 

 

   

Operations and Algebraic Thinking

 

E.g., Labeled   Task 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Formative Assessment Tasks

 

Whole Numbers and Operations in Base 10

Counting and Cardinality

Content-Practice-Assessment Analysis

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E.g., Labeled   Task 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summative Assessment Tasks

Table 14.5. Formative and Summative Content-Practice Assessment Mapping

Content-Practice Teaching

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Chapter 14 Table 14.6. Task Analysis Template By Depth and Type of Thinking

Remember

 

 

Understand

 

Level 4 Extended Thinking

Level 3 Strategic Thinking and Reasoning

Level 2 Basic Skill and Concepts

Level 1 Recall and Reproduction

Depth of Thinking + Type of Thinking

 

 

E.g., labeled formative task 1

 

 

Apply

 

 

 

 

Analyze

 

 

 

 

Evaluate

 

 

 

E.g., labeled summative task 1

Create

 

 

 

 

Table 14.7. Content-Practice Lesson Plan Template Date

 

Teacher

CCSSM Content Standard

CCSSM Practice Standard/s

Domain

 

Cluster

 

Standard

 

Primary Cluster

 

Specific Action/s

 

 

Subject & Grade Level

Secondary Cluster, if   applicable Specific Action/s

 

Prerequisite CCSSM Content Standard/s   Essential Math Question

 

CR Objective/s

At the end of the 55-minute lesson, SWBAT

Target SB Summative Assessment (Claim/s & Item/s)

 

Supporting Material/s and Resource/s

 

Lesson Activity

Allotted Time  

Formative Assessment (Claim/s & Item/s) Additional Notes

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Event Sequence

Guide Question/s

Beginning

 

Target Response/s  

 

During

 

 

 

Closing

 

 

   

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Orchestrating a Content-Practice Driven Math Class

In this chapter you will deal with various issues relevant to setting up and running an effective content-practice driven mathematics classroom. Such classrooms typically provide a climate that is conducive for teaching and learning mathematics, that is, a sociocultural condition that supports meaningful engagement with the content and practice standards. Having such a climate will enable you to teach in ways that will help your middle school students develop a positive mathematical disposition and see intellectual struggle as an integral aspect of learning mathematical concepts, skills, and applications. The first four sections in this chapter will address issues relevant to persistence and struggle, including ways that you can support your students to perform consistently in such intense moments in both individual and collaborative contexts. From a practical standpoint all stakeholders want effective math classrooms to be taking place at all times. They want you and your students to succeed together. When your math class is effective, everybody is, in fact, happy beginning with you and your students and then your colleagues in the same building and in adjacent rooms, your math coach, your principal, and your superintendent and the parents of your students. Setting up an effective math classroom will require a strong management plan that is capable of providing optimal learning for all students, eliminating or minimizing disruptions, and dealing with potential behavior problems. The last three sections in this chapter address such general issues, which will help you conceptualize meaningful and appropriate preventive and supportive strategies as students learn different aspects of classroom work requirements. However, it is wise to remember that your model colleagues (e.g.: math coach and seasoned and awardwinning veteran teachers) can also provide additional and necessary contextual information and advice that you should not hesitate to seek out especially prior to the start of classes. 15.1  Persistence and Struggle in Math Classrooms

Access the following link below which compares Eastern and Western perspectives on the nature of intelligence and struggle. Answer the questions that follow http://www.npr.org/blogs/health/2012/11/12/164793058/struggle-forsmarts-how-eastern-and-western-cultures-tackle-learning

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a. How are notions of “struggle” and “academic excellence” perceived in Eastern and Western contexts? b. How should you think about studies that compare differing cultural practices? Access the following article below which discusses ways in which learning and learning environments can be designed to effectively support cognitive growth and mathematical disposition among students. Answer the questions that follow. De Corte, E. (1995). Fostering cognitive growth: A perspective from research on mathematics learning and instruction. Educational Psychologist, 30(1), 37–46. c. Describe a few characteristics of effective learning processes. Develop possible implications for mathematics teaching and learning. d. Learn the five design principles for powerful learning environments and develop implications for your own practice. e. What is Realistic Mathematics Education (RME)? What can we learn from RME about meaningful and effective mathematics teaching, learning, and management? Access the following articles below which tackle the issue of lack of engagement in school mathematics in different contexts. As you read the articles, focus on the implications of the authors’ findings in your developing classroom management practice. Answer the questions that follow. Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the decision of some students not to engage in learning mathematics in schools is deliberate. Educational Studies in Mathematics, 62, 81–99. Singh, K., Granville, M., & Dika, S. (2002). Mathematics and science achievement: Effects of motivation, interest, and academic engagement. Journal of Educational Research, 95(6), 323–332. f. What are some possible sources of students’ (lack of) perseverance as they relate to engagement in mathematics? g. How might you set up your classroom environment in ways that will effectively eliminate students’ underparticipation and underachievement in mathematics? 15.2  Fostering Persistent Content-Practice Learners

Engaging middle school students to pursue tasks that are aligned with the CCSSM means engaging them in problem solving activity. In section 2.4 you learned about four different problem-solving contexts. Reread that section before you engage in the following tasks below. Access the following link below which discusses strategies for helping students to develop persistence in flexible problem solving contexts. Answer the questions that follow. 214

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http://mason.gmu.edu/~jsuh4/Persistent%20Flexible%20Problem%20 Solvers2008.pdf a. Discuss target characteristics that will help students to become good problem solvers in mathematics. b. Slide 26 shows a universal design for learning framework (UDL) for mathematics instruction. According the Higher Education Opportunity Act, a UDL is “a scientifically valid framework for guiding educational practice that (a) provides flexibility in the ways information is presented, in the ways students respond or demonstrate knowledge and skills, and in the ways students are engaged; and (b) reduces barriers in instruction, provides appropriate accommodations, supports, and challenges, and maintains high achievement expectations for all students, including students with disabilities and students who are limited English proficient.” How might you design content-practice mathematical tasks that promote flexible problem solving? c. What can you learn from the reflections provided by teachers Brooke and Gwen when they restructured their classrooms around problem solving? Access the following link below which talks about the different aspects of mathematical activity in a social context. http://www.tlu.ee/~kpata/haridustehnoloogiaTLU/constructivsit.pdf d. What is the emergent perspective on learning mathematics in a classroom context? e. What are social norms, and how are they distinguished from sociomathematical norms? How do both norms contribute to the emergence of classroom mathematical practices? f. Table 15.1 shows a classroom manifesto based on the authors’ emergent perspective. How might the manifesto shape or influence the manner in which instruction and learning occur in the math classroom? 15.3  Developing Effective Collaborative Content-Practice Learning Through Complex Instruction

Complex Instruction (CI) emerged from the work of Elizabeth Cohen and her colleagues who were concerned about providing meaningful and equitable collaborative learning experiences for all students in everyday classrooms. CI employs a very structured approach to collaborative learning in the classroom; there are very clear group work and group task requirements that need to be accomplished. Certainly, it is easy to plan classroom work by having all students work in groups. However, the most difficult problem is how to make group learning work effectively for all members in every team. CI assumes the following three conditions below.

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Figure 15.1. A Learning Manifesto

• There is a diverse curriculum that enables all students with different abilities to experience success in developing higher-order thinking skills through purposeful group work activities. • Instruction focuses on training students to work effectively in groups through activities that employ and develop cooperative norms and explicit team role playing. • There is value in equitable learning that deals with treat status issues proactively through activities and reflection that encourage all learners, especially the 216

Orchestrating a Content-Practice Driven Math Class

underperforming ones, to participate equally in discussions and value their contributions to group work. CI tasks should be groupworthy, multiple ability-based, and sufficiently openended so that all students are encouraged to work together and contribute to the discussion. Think about groupworthy tasks in terms of the UDL framework noted in the preceding section (i.e., item 15.2 (b)). Once you start to implement CI, your task as the classroom teacher involves observing groups, providing timely feedback, and dealing with treat status issues. Feedback can take many forms, of course, but you need to see to it that all team members are exhibiting their assigned roles in a respectful manner. Work with a team of four students and do the following tasks below. a. Access the following links below, which provide details relevant to designing classroom instruction in mathematics around CI. http://math.arizona.edu/~cemela/english/content/workingpapers/2010N CTMRossTsinnajinnieCivil04_24.pdf http://nrich.maths.org/content/id/7011/nrich%20paper.pdf http://complexinstruction5.wikispaces.com/ I dentify teaching practices that support the use of CI in classroom activity. What other issues are not addressed in CI? b. One complicated issue among middle school students involves helping them learn to deal with their emotions. Since CI capitalizes on meaningful and supportive interactions among members in a team, older children may still need some explicit instruction in terms of how to process their emotions in a group activity. Access the following articles below, which address various aspects of social emotional learning within and outside mathematics classrooms. http://www.nytimes.com/2013/09/15/magazine/can-emotional-intelli gence-be-taught.html?_r=0 Jones, J., Jones, K., & Vermette, P. (2009). Using social and emotional learning to foster academic achievement in secondary mathematics. American Secondard Education, 37(3), 4–9. Elias, M. (2006). The connection between academic and social-emotional learning. In M. Elias & H. Arnold (Eds.), The educator’s guide to emotional intelligence and academic achievement (pp. 4–14). Thousand Oaks, CA: Corwin Press.  hat is social-emotional learning (SEL)? How can teachers use SEL strategies and W implement a conducive SEL climate that can support effective CI-based learning? c. Given what you know about CI and SEL, discuss how both strategies can assist in facilitating middle school students’ learning of the CCSSM. 217

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15.4  Other Collaborative Content-Practice Learning Techniques

Access the file “classactivities” from the link below which describes nine strategies that encourage students to work cooperatively in groups. Work with a group of three students and answer the questions that follow. http://complexinstruction5.wikispaces.com/ a. Which cooperative learning strategies are doable in the math classroom? Which ones might be difficult to implement and why? b. Do you need to implement all the cooperative learning strategies that you learned in this section during your first year of teaching? Explain. c. How might you prepare yourself to teach in a collaborative context that supports content-practice learning of mathematics? d. What do you need to do before and after students work in groups and during the time they are working in groups? 15.5  Developing an Optimal Content-Practice Learning Environment for All Middle School Students

Work with a pair and do the following tasks below. a. The What Works Clearinghouse in Education link below provides lots of evidencebased information around effective classroom instruction in mathematics. http://ies.ed.gov/ncee/wwc/default.aspx Search “Back to School Tips “in the WWC for: (1) recommendations regarding how to help students deal with mathematical problem solving; and (2) evidence of effective math interventions in schools and districts. b. Access the following two links below which provide information regarding how to effectively organize instruction and study time and plan effective interventions for middle school students. http://ies.ed.gov/ncee/wwc/pdf/practice_guides/20072004.pdf http://ies.ed.gov/ncser/pubs/20133001/pdf/20133001.pdf What kinds of classroom environment and instructional practices are likely to support successful learning among older children? What educational practices can help improve the manner in which they learn mathematics? c. Access the following two links below which provide information about best instructional practices, in particular, Response-to-Intervention (RtI) strategies, for students with mathematical difficulties. http://nichcy.org/wp-content/uploads/docs/eemath.pdf

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http://ies.ed.gov/ncee/wwc/pdf/practice_guides/rti_math_pg_042109. pdf http://educationnorthwest.org/resource/1679  hat is RtI applied to mathematics, and what does it recommend insofar as W instruction in mathematics and the use of intervention materials are concerned? d. Consider a specific example involving fraction content. Access the link below which provides instructional recommendations for teaching fractions. http://www.edvanceresearch.com/images/fractions_pg_093010.pdf  raw on your responses in items (a) through (c) to assess how well the general D recommendations for effective instruction are aligned with the stipulated ones in the case of fraction learning at the middle school level. e. Access the following link below which provides instructional recommendations for effectively supporting females in middle school classrooms. http://ies.ed.gov/ncee/wwc/pdf/practice_guides/20072003.pdf  ow can instruction be used to help young females strengthen their beliefs about H their ability to do mathematics? What strategies can promote their interest toward the subject and encourage them to pursue a career path in mathematics? What activities can help them develop powerful spatial skills that are pertinent to successful mathematical learning and understanding? f. You already know something about SDAIE-driven teaching in section 13.2.4. Access the follow links below and identify additional instructional strategies for supporting English learners to obtain cognitive academic language proficiency in mathematics. http://schools.nyc.gov/NR/rdonlyres/9E62A2F2-4C5C-4534-968B5487A7BD3742/0/GeneralMathStrategiesforELLs_082811.pdf https://uteach.utexas.edu/sites/default/files/files/SixKeyStrategiesELL. pdf http://www.tsusmell.org/downloads/Conferences/2005/Moore-Harris_ 2005.pdf https://www.mheonline.com/glencoemath/pdf/ell.pdf 15.6  Dealing with Potential Behavior Problems

In this section you will learn general classroom management strategies (CMS) that provide foundational support to help you run an effective mathematics classroom. CMS should be implemented proactively so that all students understand that they are meant to help them learn mathematics in a safe environment. Two basic facts 219

Chapter 15

make CMS complicated and difficult to implement. The first reality is that your personality, confidence, and knowledge of students’ cultures influence the manner in which you implement CMS. The internet will provide you with good, professional, and legal CMS for just about any kind of behavior problem. However, your personal, social, cultural, and other environmental constraints will ultimately shape your CMS. Some teachers, though, share the view regarding the need to develop a professional mindset about CMS. The second reality is that despite the existence of available information about effective CMS, there is no any one-size-fits-all CMS that will work effectively for all students. However, that should not discourage you from learning about successful stories of CMS. Do the following tasks below. a. Access the following resources below and answer the question that follow. Englehart, J. (2012). Five half-truths about classroom management. The Clearing House, 85, 70–73. http://www.calstatela.edu/faculty/jshindl/cm/Ten%20Biggest%20 Mistakes%20Made%20by%20Teachers%209-04b.htm Identify at least ten ideas that will help guide your emerging CMS. b. Central to an effective implementation of CMS involves the necessity of frontloading. For example, some teachers employ the 3F frontloading practice, which involves having students practice routines and other classroom rules on the first of day, the first week, and the first month of the school year. CMS that are practiced in class are intended to be preventive. Access the following link below and develop a list of top ten specific actions that you want your students to be able to accomplish by the end of the: (1) first day; (2) first week; and (3) first month of the school year. https://www.msu.edu/~dunbarc/dunbar3.pdf c. Classroom CMS require a firm understanding of school and district rules and policies regarding attendance, suspension, time off, and so on. Search the internet for two to three samples of such rules and policies. Discuss similarities, differences, and essential components. d. Access the following link below which discusses three types of discipline that characterize various aspects of CMS. http://ci.columbia.edu/ci/tools/0511/  se what you know about the discipline types to assess the following views from U teachers regarding how they deal with an incident in class. 1. When an incident occurs, small or big, I quickly react to it by implementing a corrective strategy. They need to show me the appropriate behavior all the time. 2. I like to confront disruptive action with an offensive reaction. That way my students know who is the boss in my class. 220

Orchestrating a Content-Practice Driven Math Class

3. I cannot help but hold a grudge against a student who misbehaves in my class. I tend not to move on until the student knows what proper behavior means. 4. When an incident happens, I reinforce my expectations quickly rather than enforce a corrective rule mixed with some kind of punishment. 5. My students were so distracted by the manipulatives we used in class today. As a consequence, I told them that we will never use any manipulatives ever again. They are not ready. 6. Today my first-period class entered my room in a disorganized manner. Several students came in late after the bell. When they sat down, they refused to listen and simply started talking with each other. Since they were not interested in learning and being respectful, I decided to give them free time. I hope they realize how much of their own time is wasted whenever they feel like behaving like that in my class. I will not teach a class that does not know how to behave in a respectful manner. e. Access the following link below which provides research-based recommendations for reducing common behavior problems among students. Compare the CMS you generated in (a) and (b) with the recommendations noted in the document. http://ies.ed.gov/ncee/wwc/pdf/practice_guides/behavior_pg_092308. pdf 15.7  Assigning Homework, Grading and Testing, and SEATING ARRANGEMENT ISSUES

Should middle school students do homework, where homework means tasks that students accomplish during nonschool hours? What are some benefits of having students do homework problems or tasks? Are there compelling reasons that can tell us that doing homework is not helpful to children’s learning and (academic) development in the long term? These perennial issues about the benefits and disadvantages of assigning homework are still not settled because there is still no solid and converging research evidence that indicates strong correlational and causal relationships between student achievement and doing homework. But it is important to be aware of the issues and open-minded enough to consider the pros and cons of assigning homework, especially when older children are involved. Do the following tasks below with a group. a. Search the internet for at least three to five articles that discuss the benefits and disadvantages of assigning homework. Identify the claims under each perspective and then develop either your own or your group’s perspective on the matter. b. Access the following links below which provide homework tips from the National Council of Teachers of Mathematics (NCTM). http://www.nctm.org/resources/content.aspx?id=6338 http://www.nctm.org/news/content.aspx?id=13816 221

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uppose you intend to incorporate homework in your developing teaching S practice. Generate possible homework rules and strategies for processing homework in class. c. How might you assign homework tasks in such a way that they target growth in students’ competence in content-practice learning of mathematics? What potential issues do you anticipate parents or responsible caretakers might have on homework tasks that are designed around the practice standards? How might you address such issues so that parents and caretakers know that such tasks are intended to help their children and wards succeed in mathematics, progress through middle school, and prepare them for future workplace and 21st century skills? Grading is an important aspect of a teacher’s job. Unlike homework, grading these days is linked to local, state, and federal accountability structures that schools and districts carefully develop, implement, and monitor. Do the following tasks below. d. Search the internet to learn about Adequate Yearly Progress (AYP) and Academic Performance Index (API). What are AYP and API all about? How are schools and districts affected by AYP and API scores? e. Search the internet to learn about benchmark assessments (BAs) in mathematics. What are BAs and how do schools and districts develop them and use the results to inform classroom instruction? f. Access the following link below which provides a list of recommendations for using student achievement data to support instructional decision making. http://ies.ed.gov/ncee/wwc/pdf/practice_guides/dddm_pg_092909.pdf  arefully read each recommendation and assess its significance and value in C terms of how and what you teach in the classroom. g. Access the following links below, which provides grading tips from the NCTM. http://www.nctm.org/resources/content.aspx?id=6336  hich suggestions are familiar to you and which ones are new? Share other tips W that you might want to add to the list. h. Inservice middle school teachers usually discuss grading schemes either in gradelevel teams or at the school level. In some school districts, such schemes are implemented as a district policy. Search the internet for at least three grading schemes in mathematics at the middle school level. How are they similar and different? Which components appear to be essential? i. Students will earn good grades if learning, teaching, and assessments are well aligned. For many teachers, a good and responsible instructional practice involves preparing them to take tests. Access the following link below, which provides test preparation tips from the NCTM. http://www.nctm.org/resources/content.aspx?id=2147483737 222

Orchestrating a Content-Practice Driven Math Class

 hich suggestions do you share and which ones do you find troubling? Share W other tips that you might want to add to the list. j. Access the following link below which introduces you to issues surrounding high stakes testing (HST). http://www.nctm.org/uploadedFiles/Lessons_and_Resources/dialogues/ May-June_1998/1998-05.pdf  hat is HST? Why do you need to be concerned about it? Reflect on the various W responses on HST in the document and develop your own response to the question of whether HST should drive mathematics curriculum and instruction. Assigning middle school students to their seats in the classroom is one very important aspect of classroom management. Optimal learning, especially in mathematics classrooms, depends on how well you are able to arrange individual students in your class(es). Work with a team and do the following task below. k. Search the internet for effective classroom seating arrangement tips and rules. Explain how each tip or rule matters to learning. l. Search the internet for seating arrangement tips for middle school students: (1) with behavior problems; (2) with attention deficit hyperactivity disorder; and (3) that require English language assistance. 15.8  A Classroom Management Plan Project

Use the structure from the link below to help you develop a classroom management plan (CMP). See to it that you address all five components in your CMP in sufficient detail by drawing on your learnings from the preceding chapters on teaching, learning, and assessment and the sections in this chapter. http://people.umass.edu/~afeldman/beingnewteacher/classmanageplan.html The following sites below provide well-conceived samples that can inform your own CMP. http://www.calstatela.edu/faculty/jshindl/cm/Example%20CMPs.htm http://sarahsmalley.wordpress.com/244-2/ http://web.utk.edu/~rmcneele/classroom/management.html http://users.manchester.edu/Student/JLCollins/ProfWeb2/CM--Classroom% 20Management%20Plan.pdf  

http://caje33.wikispaces.com/file/view/BSM_ManagementHandout4.pdf

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