Mine the Gap for Mathematical Understanding, Grades 3-5: Common Holes and Misconceptions and What To Do About Them (Corwin Mathematics Series) 2016026612, 1506337678, 9781506337678

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What Your Colleagues Are Saying... “Wow! Mine the Gap for grades 3–5 teachers really unpacks! Not only does the book provide dozens of great mathematical tasks, but—perhaps more importantly—it gives teachers the chance to consider each task; anticipate student responses—a critical first step in formative assessment; analyze actual student responses; and then consider what they would do in the classroom. Thoughts about modifying each task provide another option for each task. At a time when teachers are encouraged to regularly provide students with tasks that promote reasoning and problem solving, this book will help teachers dig deeper as they mine for understandings.” 

—Francis (Skip) Fennell, Project Director, Elementary Mathematics Specialists and Teacher Leaders Project; Past President, National Council of Teachers of Mathematics

“Too often students (and parents and teachers) have the notion that the goal in math should be to get the right answer.  Yet insight into misconceptions and students’ thinking can tell us much more about what students know (and don’t know) beyond simply a correct or incorrect answer.  SanGiovanni offers teachers a treasure trove of rich tasks and student work on those tasks.  Examples of how to analyze student thinking and next instructional steps make this a volume that should be on every 3–5 math teacher’s desk!” 

—Linda Gojak, Past President, National Council of Teachers of Mathematics

 “This work does what other books only attempt to do. It combines instruction, assessment, and practice with open-ended and rich tasks that allow for teachers to not only immediately implement the ideas but also understand the content and pedagogy behind them. The tasks, which are immediately implementable and customizable, engage each and every learner.  They are based on cutting edge and research-based instructional frameworks and provide countless learning opportunities for students.” —Zachary Champagne, Assistant in Research, Florida Center for Research in Science, Technology, Engineering, and Mathematics at Florida State University “Mine the Gap for Mathematical Understanding is a much-needed and anticipated resource for teachers, mathematics coaches, mathematics specialists, administrators, and other stakeholders. The easy to follow, teacher-friendly format, the accompanying commentary for each student work sample, along with the thoughtful reflection questions will quickly make this resource a ‘go to’ professional development tool.” —Latrenda Knighten , Elementary Mathematics/Professional Development, East Baton Rouge Parish School System, Baton Rouge, LA  “Mine the Gap is a great tool for teachers to use to grow their own understanding of student misconceptions and incomplete understandings and how to address them. This is an indispensable resource for all involved in supporting students’ growth in mathematics.”  —Nathan Rosin, Sun Prairie Area School District “More than just a nice collection of problems, this book shares a road map for teachers looking to enhance the quality of the math tasks they use with students. Teachers will appreciate the examples of actual student work paired with tips for analysis and instruction.” —Delise Andrews, Mathematics Coordinator, Lincoln (NE) Public Schools

“John SanGiovanni continues to provide teacher-friendly, must-have books. They empower teachers by deepening their understanding of content and teaching.” —Megan Dooley, Mathematics Coach, Indianhead Elementary, Charles County, Maryland “This book helps navigate how to use student work to drive instruction with rich engaging tasks, which will help all students become better mathematicians.  SanGiovanni has done an excellent job of helping teachers to carefully look at student work to identify how students solved math problems, using this evidence to identify those students who understand the targeted skill, along with the misconceptions or misunderstandings of other students, with suggestions of how to move all students forward in their thinking.”    —Cynthia Baumann, Mathematics Supervisor, Omaha Public Schools

MINE GAP the

for Mathematical Understanding

For Oscar—a lot of your music accompanied a lot of this writing. Looking forward to buying your first album.

MINE GAP the

for Mathematical Understanding Common Holes and Misconceptions and What To Do About Them

John SanGiovanni

grades

3–5

FOR INFORMATION: Corwin A SAGE Company 2455 Teller Road Thousand Oaks, California 91320 (800) 233-9936 www.corwin.com SAGE Publications Ltd. 1 Oliver’s Yard

Copyright © 2017 by Corwin All rights reserved. When forms and sample documents are included, their use is authorized only by educators, local school sites, and/or noncommercial or nonprofit entities that have purchased the book. Except for that usage, no part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All trademarks depicted within this book, including trademarks appearing as part of a screenshot, figure, or other image, are included solely for the purpose of illustration and are the property of their respective holders. The use of the trademarks in no way indicates any relationship with, or endorsement by, the holders of said trademarks.

55 City Road London, EC1Y 1SP United Kingdom SAGE Publications India Pvt. Ltd. B 1/I 1 Mohan Cooperative Industrial Area Mathura Road, New Delhi 110 044 India SAGE Publications Asia-Pacific Pte. Ltd. 3 Church Street

Printed in the United States of America. Library of Congress Cataloging-in-Publication Data Names: SanGiovanni, John.

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Title: Mine the gap for mathematical understanding : common holes and

Singapore 049483

misconceptions and what to do about them (3-5) / John SanGiovanni, Howard County Public School System. Description: Thousand Oaks, California : Corwin, A Sage Company, [2017] | Includes bibliographical references. Identifiers: LCCN 2016026612 | ISBN 9781506337678 (pbk. : alk. paper) Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Elementary school teachers—Training of. | Mathematics teachers—Training of. Classification: LCC QA135.6 .S258 2017 | DDC 372.7—dc23 LC record available at https://lccn.loc.gov/2016026612 This book is printed on acid-free paper.

Acquisitions Editor:  Erin Null Associate Editor:  Julie Nemer Editorial Assistant:  Nicole Shade Production Editor:  Libby Larson Copy Editor:  Gillian Dickens Typesetter:  C&M Digitals (P) Ltd Proofreader:  Sarah J. Duffy Interior and cover Designer:  Scott Van Atta Marketing Manager:  Rebecca Eaton

16 17 18 19 20 10 9 8 7 6 5 4 3 2 1

THE BOOK AT-A-GLANCE A quick-reference matrix provides a snapshot of the Big Ideas in the book, along with descriptions of the associated tasks.

BIG BIG IDEAS IDEAS &&TASKS TASKS AT-A-GLANCE AT A GLANCE

Big Idea No.

Big Idea

Task No.

Description

1

Adding within 1,000

1A

Students represent addition of three-digit addends on a number line.

1

Adding within 1,000

1B

Students add three-digit numbers represented with base ten blocks.

1

Adding within 1,000

1C

Students consider if three-digit addends can be decomposed by place value and added partially.

1

Adding within 1,000

1D

Students represent three-digit addition on a modified hundred chart.

2

Reasoning about addition within 1,000

2A

Students consider if adjusting addends for friendlier computation always works.

2

Reasoning about addition within 1,000

2B

Students compare sums by reasoning about the relationship between addends in two different expressions.

2

Reasoning about addition within 1,000

2C

Students create new expressions that will have a sum equal to a given expression.

2

Reasoning about addition within 1,000

2D

Students find addends for a given three-digit sum.

3

Subtraction within 1,000

3A

Students represent three-digit subtraction with four different representations.

3

Subtraction within 1,000

3B

Students represent subtraction with three-digit numbers on number lines.

3

Subtraction within 1,000

3C

Students are asked to break apart one or both numbers to make the subtraction of large numbers friendlier.

3

Subtraction within 1,000

3D

Students represent subtraction with three-digit numbers using a modified hundred chart.

Big Ideas & Tasks at a Glance

xiii

CHAPTER

2

ADDITION AND SUBTRACTION WITHIN 1,000 THIS CHAPTER HIGHLIGHTS HIGH-QUALITY TASKS FOR THE FOLLOWING: zz

Big Idea 1: Adding Within 1,000

Chapter Overviews highlight and explain the Big Ideas covered in each chapter.

Multi-digit addition can be represented with different models, including place value models and number lines. Work with these models builds understanding and lays the foundation for flexible strategies for addition. zz

Big Idea 2: Reasoning About Addition Within 1,000 There is a relationship between addends and sums. Sums change as addends are changed. We can manipulate addends to make addition more friendly. Although addition strategies always work, the efficiency of the strategy relates to the numbers in the situation and the individual’s own number sense.

zz

Big Idea 3: Subtraction Within 1,000

Mining Hazard icons signal examples of incomplete thinking that students may encounter.

Multi-digit subtraction can also be represented with place value models and number lines. Subtraction can be thought of as taking away, breaking apart, or comparing two values. We can count back (subtract) or count up (add) to find differences. zz

Big Idea 4: Reasoning About Subtraction Within 1,000 Reasoning about subtraction situations can help us determine accurate differences. To do this, we need to understand the relationship between the minuends, subtrahends, and differences. As with addition, we can manipulate numbers to subtract more efficiently.

zz

Big Idea 5: Problem Solving With Addition and Subtraction

BIG IDEA 1

Problems can be thought of as any situation in which the solution path isn’t apparent. We use addition and subtraction to solve problems. Story problems are one small subset of the types of problems we encounter in mathematics or everyday life. Problem solving requires making sense of the problem, knowledge of strategies, reasoning, and justification.

Adding Within 1,000

BIG BIGIDEA IDEA

11

TASK 1A Use the number lines to add 358 + 453. Use the number line to show how you added. Add 371 + 361. Use the number line to show how you added.

MINING TIP It is important to

14

connect ticked number

Mine the Gap for Mathematical Understanding

Each Big Idea starts by describing one related high-quality task.

About the Task We can represent addition with larger numbers on number lines. However, the size of these numbers limits the model to an open or empty number line. These number lines do not have tick marks for each number. In some cases, these number lines do not have defined endpoints either. In this task, students add different three-digit numbers on an open number line. The openness allows them to apply flexible strategies to the computation.

lines with open number lines to support our students’ transition to open number lines. We can adjust the intervals of tick marks to support the transition. For example, we can change the intervals

Anticipating Student Responses

The highlighted task is explained in depth and potential student responses are predicted and described in detail.

Students are likely to decompose one or both numbers by their place value. For 358 + 453, these students may begin with 358 and make a jump of 400 (to 758), a jump of 50 (to 808), and then of 3 (to 811). Some students may decompose an addend and then make repeated jumps of the place value. In other words, a jump of 400 would be represented by four jumps of 100. Other students may jump by place value but begin with the ones place. They would first make a jump of 3, then 50, and then 400. Some students may make endpoints of 0 and 1,000 on their number line. Other students may assign one of the addends to the left endpoint and then jump/count on from there. Some of our students may find the sum of the numbers with an algorithm or similar procedure and then represent the addends and sum on the number line.

from 1, 2, 3 to 10, 20, 30 or 100, 200, 300.

MINING HAZARD Students who make a jump of 400 as four jumps of 100 are mathematically accurate. However, this strategy is less efficient than making one jump of a larger amount.

Pause and Reflect sections invite teachers to think about the task in relation to their practice and their own students.

PAUSE AND REFLECT zzHow

Does this Task Compare to Tasks I’ve Used?

zzWhat

Might My Students Do In This Task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 15

MINING TIP We sometimes marvel at students who offer creative, even complicated, jumps on the number line. It’s

WHAT THEY DID

Student 1 Student 1 shows that he doesn’t understand the meaning of the equations. He adds up from one addend to the other. This would be a viable strategy for finding the difference between two numbers on the number line. We can be encouraged that he makes use of friendly numbers in the first prompt.

What They Did sections analyze how the students’ work gives insight into their thinking.

important to remember that we want our students to work towards efficient and accurate strategies.

Student 2 Student 2 decomposes the addends into unique chunks. For the first prompt, he jumps by 50, then 40, and then two jumps of 5. The sequence is equivalent to 100 but slightly more complicated. We can also note that he mixes in two jumps of 5 before then adding a large jump of 300. In the second prompt, he jumps by 200 before breaking apart the remaining 100 to smaller jumps. His mathematics is accurate but inefficient.

USING EVIDENCE

What would we want to ask these students? What might we do next?

MINING TIP We may need to work with physical models and an open number line with two-digit addends before moving to three-digit numbers.

MINING HAZARD We want our students to be efficient mathematicians. The strategies that we develop in them should support their efficiency. We have to be mindful of students who apply strategies both

Student 1 Our first action with Student 1 is to ask him to describe the meaning of the expressions. It is possible that he misread the problem, thinking subtraction instead of addition. Assuming that he did read it correctly, we know that we have work to do to develop understanding of the expressions and the operation. It would be wise to put the expression into context and work with models of the quantities with base ten blocks or similar models. We can work to count up by place values, making use of expanded form. We can compare the sum of the base ten blocks with the representation and location on the number line.

Student 2

Mining Tip icons offer additional notes about mathematics content, misconceptions, or implementing the related tasks. Using Evidence sections identify questions and instructional next steps to address gaps in student understanding.

It is likely that Student 2 is quite comfortable manipulating numbers. His complicated jumping may be a “look what I can do” statement. It’s also possible that he has a notion of benchmarks and is trying to navigate them through the computation. For example, on the first number line, he jumps to 408, which is close to the 400 benchmark. His next jump of 40 lands him at 448, which is also close to the benchmark of 450. Student 2 serves as a reminder that our students can exaggerate ideas of our mathematics instruction. We should discuss with him how to add parts of the addend with fewer jumps. We may also want to shift focus to decomposition of TASK numbers to friendly chunks. For example, 453 might be thought of as 1A: 400 +Use 50 +the 3 number lines to add 358 + 453. Use the number line to show how you added. Add 371 + 361. Use the number line to show how or 450 + 3.

inappropriately and

you added.

unnecessarily. Student 2 is a good example of the latter.

Student Work 1

16

Mine the Gap for Mathematical Understanding

Mining Hazard icons also offer insight and advice as to where teachers themselves sometimes go awry in their own thinking. Student Work 2

Each task is highlighted at the top of the page, with the related student work showcased below.

Chapter 2: Addition and Subtraction Within 1,000

17

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 1B: Dennis got out some base ten blocks (235). Jackie got out some base ten blocks (137). How many blocks did Dennis and Jackie get out altogether? Use pictures, numbers, or words to explain your answer. MODIFYING THE TASK Modifying the Task: We can extend the task by asking students to create an addition problem and then represent it with base ten blocks. Students can draw/represent base ten blocks with dots (ones), lines (tens), and squares (hundreds).

Addition within 1,000 should build from conceptual understanding of numbers and the operation as it does with smaller addends. We can use similar tools to develop this understanding. This problem prompts students to add two three-digit numbers represented by base ten blocks. The blocks are not arranged in order of place value. Students will need to show that they have made sense of each number. They will also need to show how they found their sum. Some students may simply count all of the blocks for each place value to find the sum. Doing so shows that they understand the meaning of addition, but it also shows that they rely on a lower-level strategy (counting on) and physical models or drawings to combine larger numbers. It will be important to explicitly connect the representations with computation on number lines and equations.

TASK 1C: Kelly added 348 + 256 by breaking both numbers apart. She created 300 + 40 + 8 and 200 + 50 + 6. She said she can then just add the hundreds, tens, and ones to get the sum. Do you agree with Kelly? Will this always work? Create a new equation to show if it will or won’t work. Flexibly decomposing numbers enhances our ability to compute efficiently and mentally. In this task, students are asked to consider if we can decompose two addends by their place values and then add by their place values. Student understanding of decomposition, in this case expanded form, may be their greatest challenge. Students who understand this will note that you are still adding the same numbers, so it does and always will work. Others will state that there is a difference between the number (348) and the expanded form of it (300 + 40 + 8). The task notes that the numbers are then added by hundreds, then tens, and lastly ones. Students may believe that you can only add by starting with the ones place. Yet, this is only necessary when applying a traditional algorithm. For these students, we can add using expanded form of two addends starting with ones and then add again starting with hundreds, noting that the sum remains the same regardless of which place value we begin with.

TASK 1D: Use the following hundred chart (701 – 800) to add 732 + 59. Use the following hundred chart (501 – 600) to add 514 + 77. Hundred charts are quite useful for adding within 100. We can modify them to model addition within different centuries. In this task, students use a 700 chart and 500 chart to add numbers within the respective century. Some students may count on from one addend by ones. Many students are likely to add by tens and then ones or by ones and then tens. We should look for students who make a jump of a multiple of ten. Doing so shows a more refined approach to computation. We can connect the computation represented on hundred charts with it represented on number lines. This helps students transfer their understanding to new models. We can also connect it to equations to lay the groundwork for developing understanding of symbolic representations and eventually an algorithm. 20

Mine the Gap for Mathematical Understanding

Modifying The Task marginal notes provide suggestions for further adaptation and exploration.

Other Tasks sections provide three additional high-quality tasks related to each Big Idea, along with relevant explanations and analyses.

CONTENTS

Big Ideas & Tasks at a Glance xiii Acknowledgments xxiii Publisher’s Acknowledgments xxiv About the Author xxv Introduction xxvi

CHAPTER 1: RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKS 1 CHAPTER 2: ADDITION AND SUBTRACTION WITHIN 1,000 BIG IDEAS    1: Adding Within 1,000   2: Reasoning About Addition Within 1,000   3: Subtraction Within 1,000   4: Reasoning About Subtraction Within 1,000   5: Problem Solving With Addition and Subtraction

CHAPTER 3: MULTIPLICATION AND DIVISION BIG IDEAS   6: Representing Multiplication   7: Reasoning About Multiplication   8: Properties of Multiplication   9: Representing Division 10: Reasoning About Division 11: Problem Solving With Multiplication and Division 12: Connecting Multiplication and Division 13: Representing Multi-Digit Multiplication 14: Reasoning About Multi-Digit Multiplication 15: Representing Multi-Digit Division 16: Reasoning About Multi-Digit Division

14 15 21 27 33 40

48 49 57 63 69 75 79 85 89 97 104 113

CHAPTER 4: FOUNDATIONAL FRACTION CONCEPTS BIG IDEAS 17: Representing Fractions 18: Connecting Representations of Fractions 19: Fractions on a Number Line 20: Fractions Greater Than 1 on a Number Line 21: Decomposing Fractions 22: Equivalent Fractions on a Number Line 23: Comparing Fractions 24: Reasoning About Fractions 25: More Reasoning About Fractions

CHAPTER 5: DECIMAL CONCEPTS BIG IDEAS 26: Representing Decimals 27: Representing Decimals as Numbers 28: Estimating and Rounding Decimals 29: Decomposing Decimals 30: Comparing Decimals 31: Addition With Decimals 32: Subtracting With Decimals 33: Problem Solving With Decimals

CHAPTER 6: ADDITION AND SUBTRACTION WITH FRACTIONS BIG IDEAS 34: Addition With Fractions on Number Lines 35: Reasoning About Addition With Fractions 36: Subtraction With Fractions 37: Reasoning About Subtraction With Fractions 38: Problem Solving With Addition and Subtraction of Fractions 39: Addition and Subtraction With Mixed Numbers

CHAPTER 7: MULTIPLICATION AND DIVISION WITH FRACTIONS AND DECIMALS BIG IDEAS 40: Multiplication of Fractions 41: More With Multiplication of Fractions 42: Division With Fractions 43: Problem Solving With Multiplication and Division of Fractions 44: Multiplication With Decimals 45: Division With Decimals

CHAPTER 8: WHAT DO WE DO NEXT? References and Additional Resources

120 121 130 140 148 156 165 171 177 184

192 193 200 209 215 223 230 237 244

252 253 260 268 276 285 292

300 301 309 317 324 333 339

346 350

BIG IDEAS & TASKS AT A GLANCE

Big Idea No.

Big Idea

Task No.

Description

1

Adding within 1,000

1A

Students represent addition of three-digit addends on a number line.

1

Adding within 1,000

1B

Students add three-digit numbers represented with base ten blocks.

1

Adding within 1,000

1C

Students consider if three-digit addends can be decomposed by place value and added partially.

1

Adding within 1,000

1D

Students represent three-digit addition on a modified hundred chart.

2

Reasoning about addition within 1,000

2A

Students consider if adjusting addends for friendlier computation always works.

2

Reasoning about addition within 1,000

2B

Students compare sums by reasoning about the relationship between addends in two different expressions.

2

Reasoning about addition within 1,000

2C

Students create new expressions that will have a sum equal to a given expression.

2

Reasoning about addition within 1,000

2D

Students find addends for a given three-digit sum.

3

Subtraction within 1,000

3A

Students represent three-digit subtraction with four different representations.

3

Subtraction within 1,000

3B

Students represent subtraction with three-digit numbers on number lines.

3

Subtraction within 1,000

3C

Students are asked to break apart one or both numbers to make the subtraction of large numbers friendlier.

3

Subtraction within 1,000

3D

Students represent subtraction with three-digit numbers using a modified hundred chart.

Big Ideas & Tasks at a Glance

xiii

Big Idea No.

xiv

Big Idea

Task No.

Description

4

Reasoning about subtraction within 1,000

4A

Students compare differences by reasoning about the relationships between numbers and expressions.

4

Reasoning about subtraction within 1,000

4B

Students consider if adjusting subtraction expressions for friendlier computation always works.

4

Reasoning about subtraction within 1,000

4C

Students investigate how changing a subtrahend affects the difference.

4

Reasoning about subtraction within 1,000

4D

Students are provided with an equation and asked to find differences of similar equations.

5

Problem solving with addition and subtraction

5A

Students use a data table to solve addition and subtraction problems.

5

Problem solving with addition and subtraction

5B

Students use addition or subtraction to solve an unconventional problem.

5

Problem solving with addition and subtraction

5C

Students solve a change unknown problem featuring base ten blocks.

5

Problem solving with addition and subtraction

5D

Students use addition or subtraction to find missing digits in equations.

6

Representing multiplication

6A

Students are asked to create two different representations for the same multiplication fact.

6

Representing multiplication

6B

Students establish that different factors can produce the same product.

6

Representing multiplication

6C

Students recognize that some products can be made with different factors.

6

Representing multiplication

6D

Students identify multiplication situations and justify their reasoning.

7

Reasoning about multiplication

7A

Students recognize that known products or equations can be used to find unknown products.

7

Reasoning about multiplication

7B

Students note the relationship between factors in different expressions and identify how doubling a factor affects the product.

7

Reasoning about multiplication

7C

Students compare products of two expressions by reasoning about the relationship between the factors.

7

Reasoning about multiplication

7D

Students compare products without multiplying.

8

Properties of multiplication

8A

Students apply properties of multiplication to compare multiplication expressions.

8

Properties of multiplication

8B

Students make use of both sides of a number line to represent the commutative property.

8

Properties of multiplication

8C

Students identify the distributive property by examining two different expressions.

Mine the Gap for Mathematical Understanding

Big Idea No.

Big Idea

Task No.

Description

8

Properties of multiplication

8D

Students consider expressions as examples of the associative property.

9

Representing division

9A

Students represent division by making equal groups.

9

Representing division

9B

Students represent division in two different ways.

9

Representing division

9C

Students examine numbers that can be divided in different ways.

9

Representing division

9D

Students consider if different division expressions can have the same quotient.

10

Reasoning about division

10A

Students consider a misconception about the results of a division problem.

10

Reasoning about division

10B

Students use a known division equation to solve other division expressions.

10

Reasoning about division

10C

Students examine patterns in division expressions and extend the pattern to new situations.

10

Reasoning about division

10D

Students compare division expressions without finding exact quotients.

11

Problem solving with multiplication and division

11A

Students solve multiplication and division problems representing different problemsolving situations.

11

Problem solving with multiplication and division

11B

Students solve different problems representing different problem-solving situations.

11

Problem solving with multiplication and division

11C

Students solve a problem and then work with an extension of the problem.

11

Problem solving with multiplication and division

11D

Students write a word problem for a given multiplication or division equation.

12

Connecting multiplication and division

12A

Students create multiplication or division expressions by making connections between the operations.

12

Connecting multiplication and division

12B

Students find missing values in expressions by connecting multiplication and division.

12

Connecting multiplication and division

12C

Students write about the use of a multiplication fact to solve a division expression.

12

Connecting multiplication and division

12D

Students write about how the connection between multiplication and division can be useful.

13

Representing multi-digit multiplication

13A

Students represent two-digit multiplication with an area model.

13

Representing multi-digit multiplication

13B

Students represent multi-digit multiplication in two different ways.

Big Ideas & Tasks at a Glance

xv

Big Idea No.

xvi

Big Idea

Task No.

Description

13

Representing multi-digit multiplication

13C

Students find the missing factors and partial products represented with an open area model.

13

Representing multi-digit multiplication

13D

Students demonstrate their understanding of partial products.

14

Reasoning about multi-digit multiplication

14A

Students reason about factors to compare products to a known value.

14

Reasoning about multi-digit multiplication

14B

Students demonstrate understanding of multiplication by making connections between similar problems.

14

Reasoning about multi-digit multiplication

14C

Students use reasoning skills to estimate a product without using multiplication.

14

Reasoning about multi-digit multiplication

14D

Students use reasoning skills to estimate the range of a product without using multiplication.

15

Representing multi-digit division

15A

Students represent multi-digit division.

15

Representing multi-digit division

15B

Students work with an area/array model that is missing a dimension.

15

Representing multi-digit division

15C

Students explain the partial quotient strategy.

15

Representing multi-digit division

15D

Students indicate models they prefer for dividing multi-digit numbers.

16

Reasoning about multi-digit division

16A

Students estimate quotients by adjusting dividends or divisors.

16

Reasoning about multi-digit division

16B

Students justify how expressions compare to a known quantity, 50.

16

Reasoning about multi-digit division

16C

Students use a known equation to find unknown expressions making use of multiplication and division with multiples of 10.

16

Reasoning about multi-digit division

16D

Students compare division equations to a known quantity, 25.

17

Representing fractions

17A

Students consider different representations of a fraction.

17

Representing fractions

17B

Students represent

17

Representing fractions

17C

Students determine if shaded or unshaded parts describe a fraction.

17

Representing fractions

17D

Students consider shading and equal partitioning of models.

18

Connecting representations of fractions

18A

Students connect different representations of fractions.

18

Connecting representations of fractions

18B

Students determine fractions represented on number lines.

Mine the Gap for Mathematical Understanding

3 in different ways. 4

Big Idea No.

Big Idea

Task No.

Description

18

Connecting representations of fractions

18C

Students place fractions on number lines in which the location of 1 changes.

18

Connecting representations of fractions

18D

Students recognize fractions represented by area models and connect them to number line models.

19

Fractions on a number line

19A

Students find endpoints on number lines by considering the placement of a fraction on the number line.

19

Fractions on a number line

19B

Students locate 1 on a number line by making sense of a given fraction placed on the number line.

19

Fractions on a number line

19C

Students place fractions on number lines with varied endpoints.

19

Fractions on a number line

19D

on two different numbers Students place 3 lines and explain how they found the location.

20

Fractions greater than 1 on a number line

20A

Students work with mixed numbers on number lines.

20

Fractions greater than 1 on a number line

20B

Students create numbers to place on a number line with mixed-number endpoints.

20

Fractions greater than 1 on a number line

20C

Students relate a fraction to given endpoints in this open-ended task.

20

Fractions greater than 1 on a number line

20D

Students find missing endpoints by working with mixed numbers on a number line.

21

Decomposing fractions

21A

Students decompose fractions in different ways using representations.

21

Decomposing fractions

21B

Students compose a fraction and then decompose it in two different ways.

21

Decomposing fractions

21C

Students consider different decompositions and compare them to a given fraction.

21

Decomposing fractions

21D

Students decompose a fraction in two different ways.

22

Equivalent fractions on a number line

22A

Students use a number line to justify the equivalency of two fractions.

22

Equivalent fractions on a number line

22B

Students create two equivalent fractions and then make use of different representations to justify their equivalency.

22

Equivalent fractions on a number line

22C

Students justify equivalency by using a number line.

22

Equivalent fractions on a number line

22D

Students consider if fractions with the same numerator are equivalent and justify their reasoning.

23

Comparing fractions

23A

Students compare fractions in context.

1

Big Ideas & Tasks at a Glance

xvii

Big Idea No.

xviii

Big Idea

Task No.

Description

23

Comparing fractions

23B

Students use number lines to compare fractions.

23

Comparing fractions

23C

Students compare fractions and then explain their approach.

23

Comparing fractions

23D

Students use representations to compare fractions to one half and a whole.

24

Reasoning about fractions

24A

Students compare fractions without using common denominators.

24

Reasoning about fractions

24B

Students order fractions and explain their approach to doing so.

24

Reasoning about fractions

24C

Students create fractions relative to

24

Reasoning about fractions

24D

25

More reasoning about fractions

25A

Students work with fractions as the size of the whole is changed.

25

More reasoning about fractions

25B

Students reason about the size of pieces and their relationship to the whole.

25

More reasoning about fractions

25C

Students use pictures, numbers, or words to tell if each fraction is closer to 0, 1 , or 1.

25

More reasoning about fractions

25D

Students consider fraction relationships and their proximity to benchmarks.

26

Representing decimals

26A

Students connect representations of eight tenths.

26

Representing decimals

26B

Students examine representations of the same decimal.

26

Representing decimals

26C

Students work with representations of equivalent decimals.

26

Representing decimals

26D

Students compare how tenths and hundredths are related.

27

Representing decimals as numbers

27A

Students find missing values on number lines featuring decimal endpoints.

27

Representing decimals as numbers

27B

Students identify the location of a decimal on a number line with only one endpoint identified.

27

Representing decimals as numbers

27C

Students identify locations of decimals on number lines with modified intervals.

27

Representing decimals as numbers

27D

Students work with decimals on a modified hundred chart.

28

Estimating and rounding decimals

28A

Students relate a decimal to a benchmark of 0, 1 , or 1 and justify why their estimation makes 2 sense.

Mine the Gap for Mathematical Understanding

1 . 2 1

Students create two fractions greater than 4 and then tell how they picked and compared their fractions.

2

Big Idea No.

Big Idea

Task No.

Description

28

Estimating and rounding decimals

28B

Students extend understanding of familiar benchmarks of 0, 0.5, and 1 to 31, 31.5, and 32.

28

Estimating and rounding decimals

28C

Students place decimals on empty or open number lines.

28

Estimating and rounding decimals

28D

Students generate numbers that will round to given numbers.

29

Decomposing decimals

29A

Students find a missing part of a decimal that has been broken into two numbers.

29

Decomposing decimals

29B

Students decompose a decimal into two different numbers.

29

Decomposing decimals

29C

Students break apart a decimal into three component numbers.

29

Decomposing decimals

29D

Students consider decomposition of decimals and their relationship to fractions.

30

Comparing decimals

30A

Students determine equivalency of two decimals through representations.

30

Comparing decimals

30B

Students consider common misconceptions when comparing decimals in context.

30

Comparing decimals

30C

Students place decimals, including hundredths, on a number line partitioned by tenths.

30

Comparing decimals

30D

Students consider how the decimal point affects the value of a number as well as common misconceptions.

31

Addition with decimals

31A

Students add decimals on an empty or open number line.

31

Addition with decimals

31B

Students use decimal grids to represent addition of decimals.

31

Addition with decimals

31C

Students create different addition expressions to find the same decimal sum.

31

Addition with decimals

31D

Students react to a procedural misconception about adding decimals.

32

Subtracting with decimals

32A

Students find decimals that have a given difference.

32

Subtracting with decimals

32B

Students subtract decimals on an open number line.

32

Subtracting with decimals

32C

Students create subtraction equations that have a determined difference.

32

Subtracting with decimals

32D

Students place missing decimal points in a subtraction equation by reasoning about the minuend, subtrahend, and difference.

33

Problem solving with decimals

33A

Students solve problems with decimals presented in a data table.

Big Ideas & Tasks at a Glance

xix

Big Idea No.

xx

Big Idea

Task No.

Description

33

Problem solving with decimals

33B

Students add and subtract to solve problems with decimals presented in a data table.

33

Problem solving with decimals

33C

Students solve problems of different problem types and justify their thinking.

33

Problem solving with decimals

33D

Students find sums greater than 2.0 when given an addend and represent their thinking on number lines.

34

Addition with fractions on number lines

34A

Students add fractions on an empty or open number line.

34

Addition with fractions on number lines

34B

Students add decimals using a created number line.

34

Addition with fractions on number lines

34C

Students complete a pattern that adds fractions.

34

Addition with fractions on number lines

34D

Students create addition expressions that have a given sum using number lines to represent their thinking.

35

Reasoning about addition with fractions

35A

Students reason about fractions to find pairs that have sums greater than 1.

35

Reasoning about addition with fractions

35B

Students compare sums of fractions by reasoning about the addends and justify their solutions.

35

Reasoning about addition with fractions

35C

Students compare sums of fractions by reasoning about the addends.

35

Reasoning about addition with fractions

35D

Students reason about the sum of mixed numbers and compare their solutions to a specific outcome.

36

Subtraction with fractions

36A

Students solve word problems with mixed numbers and represent their solutions with number lines.

36

Subtraction with fractions

36B

Students subtract fractions using empty or open number lines.

36

Subtraction with fractions

36C

Students find pairs of fractions that have a targeted difference.

36

Subtraction with fractions

36D

Students react to a common misconception of subtraction with fractions.

37

Reasoning about subtraction with fractions

37A

Students reason about differences and compare

37

Reasoning about subtraction with fractions

37B

Students create fractions that match a provided difference.

37

Reasoning about subtraction with fractions

37C

Students react to reasoning about subtraction with mixed numbers.

37

Reasoning about subtraction with fractions

37D

Students create fractions that have a difference

Mine the Gap for Mathematical Understanding

them to 1 . 2

of

1 4

.

Big Idea No.

Big Idea

Task No.

Description

38

Problem solving with addition and subtraction of fractions

38A

Students use addition or subtraction of fractions to solve less unconventional problems.

38

Problem solving with addition and subtraction of fractions

38B

Students use addition or subtraction to solve multi-step problems and justify their solutions with representations.

38

Problem solving with addition and subtraction of fractions

38C

Students solve problems with mixed numbers featured in varied problem-solving structures.

38

Problem solving with addition and subtraction of fractions

38D

Students solve a problem and then apply their solution to an extension of the problem.

39

Addition and subtraction of mixed numbers

39A

Students add mixed numbers on an empty or open number line.

39

Addition and subtraction of mixed numbers

39B

Students subtract mixed numbers on an empty or open number line.

39

Addition and subtraction of mixed numbers

39C

Students solve problems with mixed numbers presented in different problem-solving structures.

39

Addition and subtraction of mixed numbers

39D

Students consider the results of their solution to a word problem in two different contexts.

40

Multiplication with fractions

40A

Students determine the meaning of multiplication with unit fractions.

40

Multiplication with fractions

40B

Students react to misconceptions about multiplication with fractions.

40

Multiplication with fractions

40C

Students solve problems involving multiplication with fractions.

40

Multiplication with fractions

40D

Students solve problems with multiplication of fractions and justify their solutions.

41

More with multiplication of fractions

41A

Students consider if the distributive property can be applied to multiplication with mixed numbers.

41

More with multiplication of fractions

41B

Students multiply mixed numbers and explain their thinking.

41

More with multiplication of fractions

41C

Students represent multiplication of mixed numbers on an empty number line.

41

More with multiplication of fractions

41D

Students solve a multi-step problem that includes addition and multiplication of fractions.

42

Division with fractions

42A

Students reason about quotients of a number divided by different fractions.

42

Division with fractions

42B

Students react to a common misconception about division with fractions.

Big Ideas & Tasks at a Glance

xxi

Big Idea No.

xxii

Big Idea

Task No.

Description

42

Division with fractions

42C

Students reason about quotients of whole numbers and fractions.

42

Division with fractions

42D

Students solve problems with division of fractions.

43

Problem solving with multiplication and division of fractions

43A

Students use multiplication and division of fractions to solve problems with mixed numbers and data tables.

43

Problem solving with multiplication and division of fractions

43B

Students solve problems with division of fractions.

43

Problem solving with multiplication and division of fractions

43C

Students solve problems with division or multiplication of fractions and justify their solutions.

43

Problem solving with multiplication and division of fractions

43D

Students use models to represent division of whole numbers by fractions.

44

Multiplication with decimals

44A

Students compare estimated products of decimal factors with another value.

44

Multiplication with decimals

44B

Students apply multiplication of decimals to a problem in a real-world context.

44

Multiplication with decimals

44C

Students connect multiplication of decimals to an area model used for multi-digit wholenumber factors.

44

Multiplication with decimals

44D

Students use different data points to solve a multi-step problem.

45

Division with decimals

45A

Students reason about quotients when dividing by decimals.

45

Division with decimals

45B

Students work with a common misconception that division yields a quotient smaller than the dividend.

45

Division with decimals

45C

Students consider patterns in quotients when dividing with decimals.

45

Division with decimals

45D

Students work with division of decimals in context.

Mine the Gap for Mathematical Understanding

ACKNOWLEDGMENTS

Mine the Gap for Mathematical Understanding has been a collaborative effort. I am grateful to Corwin for making this project a reality. I am thankful that they recognize the importance of quality mathematics tasks, anticipating student thinking, and considering what we do next. I appreciate that they too know that student answers aren’t random, that flawed answers provide great insight, and that seemingly correct answers don’t always tell the whole story. I would like to especially thank the following teachers and coaches who embody the best of mathematics education. They were more than happy to try new tasks and gather examples of student thinking. Those colleagues are Jodi Aikens, Cheryl Akers, Randi Blue, Denise Bogart, Beth Cayer, Claudia Eckstrom, Kim Filler, Jason Fischer, Treva Hilliard, Susan Jensen, Asha Johnson, Kendra Johnson, Kelly Krownapple, Kristen Mangus, Maria Merrill, Colleen Pollitt, Danielle Provance, Kim Quintyne, and Joan Tellish. Thanks to my many other “math friends” who make me better. This includes the colleagues that I work with closely every day, including Connie Conroy, Heather Dyer, and Caroline Walker. It also includes friends and mentors, including Kay Sammons and Skip Fennell. Thanks to the staff at Corwin for transforming a featureless document into such an appealing, practical tool for teaching and learning. Special thanks to Erin Null for her enthusiasm, partnership, thoughtful questions, and insight. Lastly and most important, many thanks to my family who saw me less than they might like and tolerated me when I grumbled. I cannot thank them enough, but I can return to them our desk, our kitchen table, and a much, much less cluttered dining room.

xxiii

PUBLISHER’S ACKNOWLEDGMENTS

Corwin gratefully acknowledges the contributions of the following reviewers: Paulette Moses Fifth Grade Educator, NBCT Ballentine Elementary Irmo, SC Natalie Crist Elementary Mathematics Specialist Baltimore County Public Schools Baltimore, MD Tamara Daugherty Elementary School Teacher Lakeville Elementary, Orange County Public Schools Orlando, FL Jennifer Sue Flannagan Elementary Science Coordinator Virginia Beach City Public Schools Virginia Beach, VA

xxiv

ABOUT THE AUTHOR John SanGiovanni is a mathematics supervisor in Howard County, Maryland. There he leads mathematics curriculum development, digital learning, assessment, and professional development for 41 elementary schools and more than 1,500 teachers. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leader graduate program at McDaniel College. He is an author and national mathematics curriculum and professional learning consultant. John is a frequent speaker at national conferences and institutes. He is active in state and national professional organizations and currently serves on the Board of Directors for the National Council of Teachers of Mathematics.

   xxv

INTRODUCTION

MINING THE GAP The pursuit of natural resources and precious gems is complex and hazardous. Yet countless men and women pursue the work because the reward is so great. These individuals rely on investigation and research to determine when and where to strike. They make use of practical tools and innovation. They anticipate what will happen and plan accordingly. They are careful of unstable footing and faulty supports or bracing. In some ways, we might think of teaching and learning mathematics similarly. Our students’ success is precious to us. Achieving it is complex, and some days it even feels hazardous. We too make use of investigation and research to identify when and what to teach. Gaps in student understanding can create considerable consequences. Mathematics tasks are the tools that enable us to uncover those gaps. And high-quality mathematics tasks provide even greater insight into the quality, depth, and complexion of our students’ understanding. Misconceptions and seemingly correct answers can undermine our stable footing and consistent progress. Essentially, we must mine the gaps in our students’ understanding so that we can achieve the goals we set for them. We must select and implement high-quality mathematics tasks to uncover student reasoning. We must anticipate what might happen, why it might happen, and what we will do about it when it happens.

ABOUT THIS BOOK Our mathematics instruction must be vibrant and engaging. We must go beyond direct instruction of rules and procedures. We must make use of much more than abundant practice and tedious worksheets. We can realize this vision with high-quality tasks that promote reasoning and representation. Our students have their own ideas about mathematics. Like their fingerprints, their ideas are unique to them. Sometimes their reasoning is sophisticated. Other times, their reasoning is faulty. Their mistakes are not random. Instead, their mistakes are grounded in incomplete understanding, skewed observations, or flawed logic. We must uncover their confusion so that we can make informed decisions about our instructional next steps. High-quality tasks enable us to do so.

xxvi

This book is a collection of high-quality tasks aligned to the big ideas of elementary mathematics. It shares perspective about what we might anticipate before our students work with specific tasks. It uncovers misconceptions, incomplete understanding, and unique student perspectives through multiple student work samples for each task. It offers what we might do next for the student samples. Three additional tasks are provided for each topic. The result is more than 160 examples of student work and more than 180 quality tasks for classroom use. All of the tasks are provided electronically for use in the classroom. Options for modifying the tasks are provided so that any task can be used in any intermediate classroom.

WHO IS THIS BOOK FOR? This book is designed for any stakeholder. Classroom teachers can use the tasks in this book for instruction or assessment. Student work samples can be reviewed to better understand what might happen in the classroom as well as what might be done next. Mathematics coaches and curriculum specialists can use this book to support the design of instruction or assessment resources. It can serve as a core resource to develop professional learning around characteristics of high-quality tasks, anticipating student thinking, identifying student misconceptions, and planning instructional next steps. This book can be a centerpiece for a school-wide professional learning community, an after-school workshop series, or collaborative planning meetings. Principals and school administrators may also use the book to guide collaborative reviews of student work by grade-level teams. Administrators may rely on it to guide the development of common assessments for benchmarking student growth. Teacher preparation programs might use it for elementary mathematics methods or diagnosis and intervention courses.

WHAT IS THE PURPOSE? WHAT IS THE PROBLEM? Teaching elementary mathematics is a complicated endeavor. As teachers, we must understand the concepts, procedures, and application of seemingly “basic” mathematics. We must understand and apply research-informed pedagogy. We must also understand mathematical misconceptions and apparently correct answers as well as the “logic” behind these answers. Misconceptions and incomplete mathematical thinking may go unnoticed because we are trained to think about mathematics in black-and-white or correct and incorrect. As teachers, we find incorrect answers and often react with steps to reteach the correct processes. However, student misconceptions are not random. Often, we may overlook why or how the answer was generated and the underlying problem persists. To complicate matters, even our students’ correct answers do not always indicate accurate reasoning. For example, a student might correctly compare 3 and 1 . Yet, 2 4 that student might select 3 because both digits are larger rather than reasoning 4 about the meaning of fractions and comparison.

Introduction   xxvii

As teachers, our lack of training for diagnosis is only part of the challenge. Teacher access to high-quality tasks that provide rich insight into student thinking can be limited. It can be a challenge to identify these tasks and even more challenging to create them. Yet such tasks are necessary because of another complication with diagnosing student thinking. Unlike the perception of black-and-white answers in mathematics, student thinking and reasoning is highly variant. Simply, mistakes and misunderstandings do not always occur in the same ways or for the same reasons. They can be simplistic or complex. They can be independent of or connected to other mathematical skills and concepts. There are two long-term ramifications of limited insight into student understanding. First and foremost, unrecognized student misconceptions become unchanged misconceptions. These misconceptions can become permanent ways of thinking. Each new layer of mathematics knowledge is then built on flawed foundations. These students are then likely to develop other misconceptions and forced to rely on rules and procedures that are lost over time. The other ramification is that the pattern continues in this teacher’s classroom year after year, affecting large numbers of students.

HOW DOES THIS BOOK SOLVE A PROBLEM? As teachers, we need easy access to practical, quality tasks that uncover student thinking. We need multiple examples of tasks to support instruction throughout the year. We also need help thinking about how tasks are selected, what students might do, and what to do with student responses. Low-level tasks featuring simple recall, procedures, or algorithms are often found in textbooks. Often, these tasks yield little more than correct or incorrect answers. They provide limited insight into our students’ thinking. These shortcomings can challenge us to make informed instructional decisions. Moreover, inaccurate perceptions of our students also lead to instructional missteps. Student progress and long-term retention of skills and concepts are affected. Conversely, the rich tasks featured in this book can be used for instruction or formative assessment. They will provide opportunities for teachers to go deeper with student performance by zz

considering what students do and don’t know about the content,

zz

describing misconceptions and limited understanding through incorrect and correct responses,

zz

anticipating what might happen with a specific task, and

zz

identifying possible instructional next steps.

As a result of using this book, teachers will be able to

xxviii   

zz

identify and select rich tasks for instruction or assessment,

zz

consider what counts as mathematical understanding,

zz

anticipate and plan for student misconceptions,

zz

make instructional decisions based on specific misconceptions or incomplete understandings, and

zz

access a substantial collection of rich tasks for classroom use.

Mine the Gap for Mathematical Understanding

ORGANIZATION OF THE BOOK This book is organized around the big ideas of mathematics in Grades 3 through 5, including zz

addition and subtraction of multi-digit numbers,

zz

multiplication and division of single and multi-digit numbers,

zz

foundational fraction concepts,

zz

foundational decimal concepts, and

zz

computation with fractions and decimals.

Each chapter provides zz

a collection of tasks aligned to the subtopics of the big ideas,

zz

a brief description of each task and its importance,

zz

ideas about what we might anticipate our students will do with the task,

zz

samples of student work with descriptions of what they did with the task,

zz

considerations for next steps with the highlighted student work,

zz

three additional tasks aligned to the mathematics topic, and

zz

ideas about what students might do with these additional tasks.

THE APPROACH TO STUDENT WORK This book is a guide to rich tasks, student understanding, and misconceptions. It is shaped by authentic student work and reasoning. The student work samples are from real students in real mathematics classrooms. The tasks were collected from random classrooms. This was done so that we could see what students do when working with these tasks. Tasks were not continuously distributed until just the right samples could be found. The tasks were provided after concepts had been taught, although in many cases, it had been weeks since the concept had appeared in the classroom. None of the student samples were collected within the same period as the concept was taught. Specific student samples were selected because they represent what our students frequently do or think about the mathematics. COMPANION WEBSITE All of the tasks are provided electronically on the book’s companion website at resources.corwin.com/minethegap/3-5. Note that the display of some tasks have been modified to fit the book’s layout, but the full and complete version can be found online. They can be reproduced for instruction, assessment, independent practice, or possibly for homework. Suggestions for modifications are made throughout the book. THREE REASONS YOU NEED THIS BOOK 1. This book provides a wealth of high-quality mathematics tasks. Identifying high-quality mathematics tasks can be quite difficult. We can search online for hours trying to find just the right combination of rigor, relevance, and interest. Even then, we may not find what is best for our students. Introduction   xxix

Writing or creating these tasks is even more difficult. This book provides more than 180 tasks electronically that can be modified, enhanced, or replicated for countless possibilities. 2. This book provides insight into common approaches and misconceptions students have. Anticipating and identifying what students might do and the misconceptions they have can be acquired with years of experience. Yet, experience alone may not provide enough insight into what might happen with the students we’re teaching this year. This book provides diverse student samples for more than 40 tasks related to the big ideas of intermediate mathematics (Grades 3–5). 3. This book offers ideas about what we might do next. We know that “louder and slower” is not the solution to incorrect student thinking. Instead, we have to consider what students know and how well they understand it. We need to pinpoint where the mathematics falls apart for our students and determine what to do next. This book highlights where things go right and where they go wrong for students. It also gives ideas about next steps for reteaching, enriching, or extending our students’ thinking.

xxx   

Mine the Gap for Mathematical Understanding

CHAPTER

1

RICH MATHEMATICS TASKS, STUDENT MISCONCEPTIONS, USING TASKS PROMPTS WITH PURPOSE: USING HIGH-QUALITY TASKS The quality of our mathematics tasks directly affects the learning of our students (Stein & Lane, 1996). For years, there has been a belief that the quantity of mathematics trumps the quality of mathematics. And so, it was likely believed that a practice sheet with 20 or more problems was a more effective way to learn mathematics than one or two high-quality prompts or tasks. This notion of quantity creates a myriad of problems. For one, a student who practices a skill or concept incorrectly over and over again ingrains a misconception that can be extremely difficult for us to correct. Copious amounts of low-level practice become mundane and can cause students to fall out of love with mathematics. Often, these low-level tasks don’t further one’s learning. They don’t always, if ever, promote reasoning. The procedural focus on isolated concepts may limit our students’ ability to transfer mathematical ideas to new situations. Most important, low-level tasks don’t provide opportunities to engage in mathematics in the same ways we encounter it in day-to-day life. In other words, mathematics in the real world isn’t scripted. We don’t continue to do or use the same skill over and over in a short amount of time. In the real world, mathematics isn’t isolated and certainly isn’t contrived. Selection of high-quality mathematics tasks is a foundational part of exemplary mathematics instruction. After selecting tasks, we plan for our students to work with partners or small groups. Then, we must anticipate what will happen. We must consider the questions we will ask. We must think about how we will facilitate meaningful discourse and close the lesson. All of this builds from and with conceptual understanding and mathematics vocabulary. Simply, quality mathematics tasks alone won’t produce proficient students. However, proficiency can’t be developed without quality tasks.

Tools of the Trade: Qualities of High-Quality Tasks We might think of mathematics tasks as tools of our trade. Like tools designed for other jobs, we want the highest quality. We look for precision, craftsmanship, effectiveness, and practicality. The idea of high-quality mathematics tasks means different things to different people. There are all sorts

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   1

of tools and rubrics for identifying a quality mathematics task. These tools usually identify that high-quality mathematics tasks zz

align to mathematics content standards and/or significant mathematical ideas.

zz

make use of representations.

zz

provide students with opportunities for communicating their reasoning.

zz

can be modified for multiple entry points.

zz

create opportunities for different strategies for finding solutions.

zz

allow students to make connections between concepts.

zz

require cognitive effort.

zz

are problem based, authentic, or interesting.

Selecting High-Quality Tasks We can find mathematics tasks in textbooks, supplemental resources, and of course online. But how do we know if they are high quality? A rating or review tool can help us develop our “high-quality filter” for selecting good tasks. We can apply it to those tasks we find in print resources and online. It is important to keep in mind that there is no perfect task. Every task can be improved. This tool aligns to the characteristics of high-quality tasks described here.

Identifying High-Quality Tasks The purpose of the task is to teach or assess:

 Conceptual

 Procedural skill and

understanding

 Application

fluency

Rating Scale: Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for downloadable version of this chart.

2 - Fully Meets the Characteristic 1 - Partially Meets the Characteristic 0 - Does Not Meet the Characteristic The mathematics task Aligns to mathematics content standards I am teaching. Encourages my students to use representations. Provides my students with an opportunity for communicating their reasoning. Has multiple entry points. Allows for different strategies for finding solutions. Makes connections between mathematical concepts, between concepts and procedures, or between concepts, procedures, and application. Prompts cognitive effort. Is problem-based, authentic, or interesting.

2   

Mine the Gap for Mathematical Understanding

Rating

The Purpose of the Task Mathematical rigor promotes instructional balance between concepts, procedures, and applications of mathematics. So our initial consideration for selecting a task is to determine the purpose of the task. How does it connect with one of these components of rigor? Does the task engage students in concepts? Does it build procedural fluency? Does it apply concepts and procedures to problems and real-world contexts? There are other important considerations to determine the quality of the task.

The task aligns to the mathematics standards I am teaching. Tasks must be worthwhile and aligned to the skills and concepts in our curriculum. Tasks that fully meet this characteristic align directly to standards in my curriculum.

Tasks that partially meet this characteristic align to a standard in an adjacent grade level but are important and necessary.

Tasks that do not meet this characteristic do not connect with standards in my curriculum.

The task encourages my students to use representations. Representations help students make sense of and communicate mathematical ideas. Tasks that fully meet this characteristic explicitly direct students to use representations.

Tasks that partially meet this characteristic imply or provide space for representations.

Tasks that do not meet this characteristic are clearly procedural with no reference or space for representations.

The task provides my students with an opportunity for communicating their reasoning. Students can communicate their reasoning with models or pictures, numbers, and words. Tasks that fully meet this characteristic explicitly direct students to communicate their reasoning.

Tasks that partially meet this characteristic imply that students should communicate their reasoning.

Tasks that do not meet this characteristic do not require students to explain or justify their thinking.

The task has multiple entry points. Students can approach a problem from various perspectives, using diverse strategies and/or representations. Tasks that fully meet this characteristic are open to many possible solution paths and representations.

Tasks that partially meet this characteristic can be approached in different ways but may provide an example or prompt to direct students to an approach.

Tasks that do not meet this characteristic have a specific solution path intended or directed.

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   3

The task allows for different strategies for finding solutions. Students can solve a problem in various ways. Tasks that fully meet this characteristic are open to any strategy regardless of the efficiency of the strategy.

Tasks that partially meet this characteristic can be approached in different ways but imply a specific strategy for students to use.

Tasks that do not meet this characteristic direct students to a specific solution path or calculation.

The task makes connections between mathematical concepts. Mathematics ideas are related. We can also connect them to representations, procedures, and applications. Tasks that fully meet this characteristic connect mathematical ideas or connect concepts/ procedures/applications within a topic.

Tasks that partially meet this characteristic allow for connections but do not call for them directly.

Tasks that do not meet this characteristic make no connections. They focus on a single procedure or recall.

Task prompts cognitive effort. High-quality tasks should generate some amount of struggle. Students should have to make sense of the prompt, the problem, or the representation. Tasks that fully meet this characteristic offer no obvious solution path. Or they require concepts and procedures to be applied to new situations or contexts.

Tasks that partially meet this characteristic are problem based but indicate how they can be solved.

Tasks that do not meet this characteristic provide no cognitive resistance. Students are directed to do something exact or recall a skill or concept.

Tasks are problem based, authentic, or interesting. High-quality tasks are problem based. They can reflect real-world, authentic applications of mathematics. They should have interesting or novel prompts that grab students’ attention. Tasks that fully meet this characteristic are problem based and authentic or interesting.

Tasks that partially meet this characteristic are problem based.

Tasks that do not meet this characteristic are not problem based.

Hazards of Low-Level Tasks Low-level tasks are typically grounded in recall. They might only require procedure for completion. They lack the characteristics that engage students in reasoning about mathematical ideas. They don’t provide opportunity for discussion. These tasks present mathematics concepts in isolated fashions. 4   

Mine the Gap for Mathematical Understanding

Low-level tasks create other challenges that are difficult for us to overcome. Most important, these tasks provide a limited, if any, window into student understanding. This happens because they generally require a single, correct answer. They require a specific strategy or process for successful completion. Because of this, we aren’t always able to see our students’ strategies, partial understanding, or misconceptions. Without this understanding, we are left to make guesses about what to do next. We are unable to directly address specific mathematics needs. We are left to reteach or provide interventions that don’t necessarily address the problem. Low-level tasks may also yield correct answers with flawed or incomplete understanding of the mathematics. A student may find the product of 5 × 6 by skip-counting by 5. He may apply this strategy to every multiplication expression he encounters. It may be his only strategy. If we consistently use low-level, basic multiplication prompts, we may only recognize his correct answers without learning of his only strategy. In time, his limited strategy will create considerable challenges because of the difficulty of efficiently and accurately skip-counting with larger, multi-digit numbers. Low-level tasks zz

make use of simple recall or procedure,

zz

require one answer that is found with a specific strategy or pathway,

zz

do not feature opportunities for representing mathematics,

zz

do not prompt for reasoning and justification, and

zz

lack connections within and between mathematics concepts.

Is It REALLY a Good Task? Some tasks are quite easy to identify as low level. Typically, these tasks are pure computation. But some tasks can be quite misleading. We must keep in mind that a task that makes use of representations does not automatically qualify as a quality task. Consider prompts that ask students to identify a fractional piece of a rectangle or circle. Sure, this presents the concept with a representation. But it is simply recall of what a fraction is. In other situations, we may see a representation that is different from those we encountered as students. The following task is an example. Many of us did not shade grids to represent decimals. This “new” representation may mislead us to believe that the task is of high quality. Shade the grid to show 0.4.

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   5

Another faulty indicator of quality is relying on tasks that require more than one right answer. Consider the following prompt. Identify all of the numbers below that are greater than 48. A. 19 B. 37 C. 49 D. 60 E. 25 F. 100

The prompt above does ask for more than one right answer. However, the cognitive demand required to complete the task is quite low. We should look for tasks that have more than one strategy or solution path rather than more than one correct answer. Context can also mislead us about the quality of a task. The purpose of learning mathematics is to apply it to the problems we face in the real world. So it makes sense that high-quality tasks have a real-world connection. But when considering a task, we need to keep in mind if the real-world application makes sense. Is it contrived? Is it possible? Is the problem worth solving? Consider this problem:

32 ice skaters broke their arms at a figure skating competition. A hospital uses 1 5 feet of fabric to make a cast. The hospital has 300 yards of fabric available. 2 Does the hospital have enough fabric for all of the skaters?

The task certainly provokes questions. Our students might ask how many people were in the competition? Did their parents have to sign a waiver? Why does every cast get the same amount of fabric when each person’s arm is different? Who measures out the fabric exactly like that? Why did they all go to the same hospital? Yet, none of these questions are about the mathematics. Why should they be? The problem is a silly fabrication so that fractions, measurement, and multi-step problem solving can be “applied” to an authentic situation. It reminds us of the classic “two trains leaving Chicago” problem. Some believe that manipulatives and tools are a “must-have” to determine the quality of a task. Others believe that the use of manipulatives and other tools, including calculators, cheapens or lessens the quality of a mathematics task. Neither is true. The quality of a task is determined by what you do with tools or representations rather than if you have access to them. It is important for us to remember that meaning is in the mathematics, not the manipulative. USING QUALITY TASKS We can use the quality tasks we select in different ways. We might choose to use them as our instructional centerpiece during a lesson. When doing this, we must

6   

Mine the Gap for Mathematical Understanding

consider if students will engage in the task independently or cooperatively. We must think about what our students might do and why they might do it. We also have to consider how we will debrief the task and facilitate discussion about the task, the solution, and strategies for solutions. We can also use these tasks for formative or summative assessment. When assessing with the tasks, we should reflect on what will satisfy the prompt. We should think about what ideas, representations, or strategies will count as evidence of understanding. We also want to begin to think about how we will use the information for our instructional next steps.

Using Tasks Instructionally Use of quality tasks for instruction, from selection to implementation, must be intentional. An effective way to implement these tasks is through three stages.

Launch

Engage

Debrief

During the first stage, we set the context for the problem, revisit skills or concepts that students might use with the task, and convey our expectations for quality work and collaboration. In the second stage, students engage collaboratively with partners or small groups to exchange ideas, apply strategies, and adjust thinking. During the second stage, we circulate to monitor student thinking. We ask questions to focus student thinking and make note of work that we want to highlight during the debrief. At this time, we might begin to think about the sequence of student sharing. We can give students numbers written on sticky notes to help organize our sequence. As we sequence their work, we want to consider how their relationships, representations, strategies, misconceptions, and errors are connected. We want to sequence so that each new sharing builds from or contrasts with the previous idea. During the third stage, we facilitate a discussion or gallery walk so that groups can share their solutions and strategies. In this stage, students construct their meaning of the mathematics. They share their thinking and push back on the thinking of others. At this time, we facilitate a discussion, being careful to avoid influencing or even contaminating student thoughts by offering or dismissing specific strategies that we would use or prefer. In some situations, we might modify the approach so that students engage with tasks independently before sharing ideas with partners and eventually the whole group. In this sequence, students work with the task independently, they share their ideas with a partner, and then the whole class comes together to share strategies and insights.

Independent Student Work

Partners Discuss

Group Shares and Teacher Charts

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   7

Anticipating Student Responses We must anticipate what our students’ responses may be so that our tasks, questions, and discussion will be most effective. It is impossible to anticipate every strategy or misconception that our students will have. We can’t expect to know every mistake they will make. Even so, anticipating possibilities prepares us for the next instructional steps we might take during the discussion, later in the lesson, or the next day. We can develop our ability to anticipate student responses by working with other teachers to select and plan tasks. We can work to complete the problem in as many different ways as possible. We can even attempt to apply the misconceptions we think our students might have. Anticipating student thinking is highlighted throughout this book.

Misconceptions Anticipating student work enables us to imagine the misconceptions. Misconceptions are not random. They happen when students apply faulty logic. Often, misconceptions can be explained. They occur when students make improper connections between skills or concepts. Sometimes this happens as students look to find patterns and connect them to seemingly similar procedures. For example, when adding fractions, we add numerators while the denominators remain unchanged. Our students might then do something similar when multiplying fractions by multiplying the numerators and keeping the denominators unchanged. So what is a misconception? A misconception is any idea that is grounded in some degree of understanding but is mathematically flawed. Misconceptions can also be rules or strategies that work in certain situations. For example, we might compare 2

55 and and , 86

5

noting that is greater because we can think about the number of pieces 6 missing from the whole. Yet that strategy doesn’t work when two fractions are miss8

ing the same number of pieces

3   e.g., and 4 

 . 10  9

Misconceptions are developed through passive, independent observation and incomplete understanding. They can be taught by sharing chants and tricks or even simple reminders. For example, third-grade teachers might “help” their students remember how to record a fraction by saying that the larger number is always on the bottom. This “tip” then causes considerable challenges as students begin to work with fractions greater than 1. If misconceptions go unnoticed, we are essentially reinforcing them. As teachers, we have to constantly be on the lookout for misconceptions. We have to probe thinking to be sure it is legitimate and complete. We have to be cautious of overreliance on correct answers. As we know, students can arrive at correct answers for the wrong reasons. Yet correct answers can also be found with degrees of correct thinking.

MINING HAZARD Look for this icon throughout the book. It highlights where students—and sometimes teachers— go awry.

8   

Incomplete thinking can be just as hazardous as a misconception. We might think of incomplete thinking as a mining hazard. We assume that everything is fine because student answers and representations seem to indicate understanding. But in fact, our students have found a correct answer for the wrong reasons. Consider a student who always works with region models of fractions. He consistently finds the shaded part and counts the total pieces. He may begin to associate the idea of a fraction as shaded over total. But in fact, the unshaded portion is also a fraction. He will give correct answers to shaded fraction problems. But he may not have a deeper understanding of fractions. Because of this, he may struggle ­tremendously

Mine the Gap for Mathematical Understanding

with other models, including number lines, computation with fractions, problem solving with fractions, and eventually ideas about ratio and proportion. In other situations, a student might rely on repeated addition to find products of multiplication expressions. He seems to understand the meaning of multiplication. Yet when he begins to multiply multi-digit numbers, he is unable to separate from repeated addition in order to apply more efficient methods such as the area model, partial products, or even an algorithm. Also consider working with fractions on number lines. Students who consistently work with endpoints of 0 and 1 may develop incomplete understanding of fractions and have considerable difficulty as fractions greater than 1 are introduced.

Facilitating Discourse Discourse about our students’ ideas and strategies is essential for maximizing the instructional potential of these tasks. These tasks can be applied seamlessly to the five practices Smith & Stein (2011) describe for orchestrating productive discussion: 1. Anticipating student responses 2. Monitoring student work, engagement, and reasoning 3. Selecting student work for discussion 4. Sequencing student responses during discussion 5. Connecting responses and mathematical ideas

Using High-Quality Tasks for Assessment Quality assessments yield quality information about our students’ understanding. These high-quality instructional tasks can easily be used for assessment purposes. We have to keep the purpose of assessment in mind as we use the task. Essentially, we have to know if we will use the task formatively or summatively. In other words, will we use it for instruction (formatively) or for evaluation instruction (summatively)?

What Counts as Evidence? Regardless of the assessment purpose, we must determine what will count as evidence of understanding before our students work with the task. We might ask if our students will have to compute. Will they be able to justify with a model or drawing? What might their drawings look like? What strategies might they use? Will they have to write sentences to convey understanding? All of these questions represent the thoughts we must consider when determining what counts as evidence of understanding. This process is similar to anticipating what students will do with a task during instruction. Essentially, we want to determine what will constitute evidence of student understanding. We should make note of specific answers as well as the various strategies our students might use. It is important that we do not confine student responses to what we determine to be evidence.

Understanding Student Thinking and Inference We can be tempted to infer what students mean when we review student performance on assessments or assessment tasks. This can be hazardous. High-quality tasks don’t provide opportunities for random answers to be correct. But, as noted,

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   9

correct answers are not always the result of correct or complete mathematics ­understanding. A good rule of thumb is to consider the question, “What would I ask [the student] if she were here right now?” In other words, if we need to ask the student something about her work, her response, or her calculation, it is probably not complete. This doesn’t necessarily have to affect a student’s score or grade. Instead, it can be an indication of teaching or reinforcing what we need to do with the student.

Determining Student Performance We might think of our students’ performance results in two distinct ways. In one instance, our students demonstrate understanding of the skills and concepts. These students are ready for more opportunities to reinforce their understanding or to advance it to more complex situations. Extending and enriching are other ways to think about advancing. In the other instance, our students demonstrate the need for reteaching.

Reinforce and/or Advance Student demonstrates full understanding of the concept. The solution is correct. Reasoning is provided through pictures, words, or numbers/equations. Justification is complete. Minor errors may be present but do not affect the response.

MINING TIP We can snap photos with our phones or tablets as students work on tasks to accompany our observation data. This

Student demonstrates understanding of the concept. The solution may be incorrect but this can be attributed to a computational error rather than flawed logic. Reasoning is provided but may not be complete.

Reteach Student demonstrates flawed logic or misconception. The solution may be correct, but it is coincidental.

Student demonstrates no understanding of the concept. The solution is incorrect. There is no justification or reasoning. Numbers or terms are disconnected from the prompt or a restatement of the prompt.

A General Rubric We know that assessment is much more than counting the number of correct answers on a page. We know that there are layers to every answer. Rubrics are useful because they delineate the layers of understanding and performance. We can design specific rubrics for every task we use. Doing so can be quite daunting. Instead, we may choose to make use of a general rubric that can be applied to most if not all tasks. We can also connect these rubrics with class observation sheets that capture evidence of student understanding during performance-based or hands-on tasks.

can be helpful for communicating with parents.

Using More Than One Task As we know, it is important to triangulate data points to get a clear picture of where a student is mathematically. This is also true when using high-quality mathematics tasks. These tasks will give better insight into what students know, their partial understanding, and the misconceptions they have. Even so, we should be careful to avoid relying on one task as evidence of student understanding.

10   

Mine the Gap for Mathematical Understanding

Student Performance Recording Sheet Use this with formative assessment tasks, classroom activities, and observations. Date

Date

Date

Date

Date

Student demonstrates understanding of the concept. The solution may be incorrect but this can be attributed to a computational error rather than flawed logic. Reasoning is provided but may not be complete.

Student demonstrates no understanding of the concept. The solution is incorrect. There is no justification or reasoning. Numbers or terms are disconnected from the prompt or a restatement of the prompt.

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for downloadable versions of this chart.

Reteach

Student demonstrates flawed logic or misconception. The solution may be correct but it is coincidental.

Reinforce and Advance

Student Names

Student demonstrates full understanding of the concept. The solution is correct. Reasoning is provided through pictures, words, or numbers/ equations. Justification is complete. Minor errors may be present but do not impact the response.

Other Assessment Tips: Erasing Erasing is a common practice. We make a mistake and we erase. It may be unavoidable. Yet, we may want to rethink why we erase and if it is always a good idea. When our students erase, they remove a strategy, diagram, or calculation that can provide good insight into their thinking. Knowing what students erase can be just as important as what they didn’t erase. Granted, limited space and other confines may affect when our students have to erase. That being said, we may reconsider encouraging students to erase. Instead, we can include their errors as part of the written record.

Other Assessment Tips: Hands-On Tasks This book provides a collection of paper-and-pencil tasks for instruction or assessment. This is a limitation of the format and medium rather than a message about what we should value in a classroom or what makes a high-quality mathematics task. It is critical that our students have hands-on, concrete experiences with tools, manipulatives, and other resources to learn mathematics. A hands-on task might be an opportunity when students are asked to compare two different fractions using two different tools such as fraction tiles and fraction circles. It’s also important to keep in mind that the quality of hands-on tasks can vary wildly as well. The high-quality task identification tool presented earlier in this chapter can be used with hands-on experiences as well.

MINING TIPS Other tips about the mathematics content, misconceptions, or implementing the tasks are sprinkled throughout the book. This icon signals those ideas.

Chapter 1: Rich Mathematics Tasks, Student Misconceptions, Using Tasks   11

Hands-on tasks can also be applied to assessment situations. We can use observation rubrics, take notes of student statements and performance, and take pictures of what they do during the activity to document student understanding.

MODIFYING THE TASK Look for this icon throughout the book. It calls attention to

Modifying Tasks Miners modify or customize their tools as the mining conditions change. Each of the provided tasks can be modified. We might do this so that they better align with our grade-level standards or for a different perspective on student thinking. Each task is provided electronically, and ideas for modification are sprinkled throughout.

ideas for modifying the associated task.

REFLECTING ON CHAPTER 1

12   

zz

How do I typically find and select the mathematics tasks I use in my classroom?

zz

What are the characteristics of quality mathematics tasks?

zz

How do I think about the misconceptions my students have? Do I think about them before, during, and after a lesson?

zz

What misconceptions do I frequently encounter?

zz

How do I structure my mathematics tasks? Are they teacher centered or student centered? Are there opportunities for collaboration and discussion on a daily basis?

zz

What are my mining hazards? What do the skills, concepts, or topics that I teach that students seem to understand but do not fully understand?

zz

What experiences have I had modifying tasks when the topic is related but not completely aligned to what I am teaching?

Mine the Gap for Mathematical Understanding

notes

CHAPTER

2

ADDITION AND SUBTRACTION WITHIN 1,000 THIS CHAPTER HIGHLIGHTS HIGH-QUALITY TASKS FOR THE FOLLOWING: zz

Big Idea 1: Adding Within 1,000 Multi-digit addition can be represented with different models, including place value models and number lines. Work with these models builds understanding and lays the foundation for flexible strategies for addition.

zz

Big Idea 2: Reasoning About Addition Within 1,000 There is a relationship between addends and sums. Sums change as addends are changed. We can manipulate addends to make addition more friendly. Although addition strategies always work, the efficiency of the strategy relates to the numbers in the situation and the individual’s own number sense.

zz

Big Idea 3: Subtraction Within 1,000 Multi-digit subtraction can also be represented with place value models and number lines. Subtraction can be thought of as taking away, breaking apart, or comparing two values. We can count back (subtract) or count up (add) to find differences.

zz

Big Idea 4: Reasoning About Subtraction Within 1,000 Reasoning about subtraction situations can help us determine accurate differences. To do this, we need to understand the relationship between the minuends, subtrahends, and differences. As with addition, we can manipulate numbers to subtract more efficiently.

zz

Big Idea 5: Problem Solving With Addition and Subtraction Problems can be thought of as any situation in which the solution path isn’t apparent. We use addition and subtraction to solve problems. Story problems are one small subset of the types of problems we encounter in mathematics or everyday life. Problem solving requires making sense of the problem, knowledge of strategies, reasoning, and justification.

14   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 1 Adding Within 1,000

11

TASK 1A Use a number line to show how you add 358 + 453 and 371 + 361.

MINING TIP It is important to

About the Task We can represent addition with larger numbers on number lines. However, the size of these numbers limits the model to an open or empty number line. These number lines do not have tick marks for each number. In some cases, these number lines do not have defined endpoints either. In this task, students add different three-digit numbers on an open number line. The openness allows them to apply flexible strategies to the computation.

connect ticked number lines with open number lines to support our students’ transition to open number lines. We can adjust the intervals of tick marks to support the transition. For example, we can

Anticipating Student Responses Students are likely to decompose one or both numbers by their place value. For 358 + 453, these students may begin with 358 and make a jump of 400 (to 758), a jump of 50 (to 808), and then of 3 (to 811). Some students may decompose an addend and then make repeated jumps of the place value. In other words, a jump of 400 would be represented by four jumps of 100. Other students may jump by place value but begin with the ones place. They would first make a jump of 3, then 50, and then 400. Some students may make endpoints of 0 and 1,000 on their number line. Other students may assign one of the addends to the left endpoint and then jump/count on from there. Some of our students may find the sum of the numbers with an algorithm or similar procedure and then represent the addends and sum on the number line.

change the intervals from 1, 2, 3 to 10, 20, 30 or 100, 200, 300.

MINING HAZARD Students who make a jump of 400 as four jumps of 100 are mathematically accurate. However, this strategy is less efficient than making one jump of a larger amount.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 15

MINING TIP We sometimes marvel at students who offer creative, even complicated, jumps on the number line. It’s

WHAT THEY DID

Student 1 Student 1 shows that he doesn’t understand the meaning of the equations. He adds up from one addend to the other. This would be a viable strategy for finding the difference between two numbers on the number line. We can be encouraged that he makes use of friendly numbers in the first prompt.

important to remember that we want our students to work toward efficient and accurate strategies.

Student 2 Student 2 decomposes the addends into unique chunks. For the first prompt, he jumps by 50, then 40, and then two jumps of 5. The sequence is equivalent to 100 but slightly more complicated. We can also note that he mixes in two jumps of 5 before then adding a large jump of 300. In the second prompt, he jumps by 200 before breaking apart the remaining 100 to smaller jumps. His mathematics is accurate but inefficient.

USING EVIDENCE

What would we want to ask these students? What might we do next?

MINING TIP We may need to work with physical models and an open number line with two-digit addends before moving to three-digit numbers.

MINING HAZARD We want our students to be efficient mathematicians. The strategies that we develop in them should support their efficiency. We have to be mindful of students who apply strategies both

Student 1 Our first action with Student 1 is to ask him to describe the meaning of the expressions. It is possible that he misread the problem, thinking subtraction instead of addition. Assuming that he did read it correctly, we know that we have work to do to develop understanding of the expressions and the operation. It would be wise to put the expression into context and work with models of the quantities with base ten blocks or similar models. We can work to count up by place values, making use of expanded form. We can compare the sum of the base ten blocks with the representation and location on the number line.

Student 2 It is likely that Student 2 is quite comfortable manipulating numbers. His complicated jumping may be a “look what I can do” statement. It’s also possible that he has a notion of benchmarks and is trying to navigate them through the computation. For example, on the first number line, he jumps to 408, which is close to the 400 benchmark. His next jump of 40 lands him at 448, which is also close to the benchmark of 450. Student 2 serves as a reminder that our students can exaggerate ideas of our mathematics instruction. We should discuss with him how to add parts of the addend with fewer jumps. We may also want to shift focus to decomposition of numbers to friendly chunks. For example, 453 might be thought of as 400 + 50 + 3 or 450 + 3.

inappropriately and unnecessarily. Student 2 is a good example of the latter.

16   

Mine the Gap for Mathematical Understanding

TASK 1A: Use a number line to show how you add 358 + 453 and 371 + 361.

Student Work 1

Student Work 2

Chapter 2: Addition and Subtraction Within 1,000   17

WHAT THEY DID

Student 3

MINING TIP Student jumps on number lines, especially when using open number lines may not be proportionate. This is understandable because there are no guiding tick marks. It does not necessarily indicate

Student 3 is accurate in his jumps. Unlike Student 2, he decomposes the addends into reasonable chunks. His first prompt shows reliance on skip-counting by hundreds before adding a multiple of 10. His work with the second prompt shows a jump of a multiple of 100 (300) and then a multiple of 10 (60). The jump of 60 results in a miscalculation.

Student 4 Student 4 decomposes by place value. He adds the partials accurately. The first prompt is likely less complicated because 50 is added to 758. However, he is equally successful when he wraps around the century by adding 60 to 371. We can see that his jumps are not proportionate, especially with the jumps of 3 and 1 in the first and second prompts, respectively.

misunderstanding.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3

MINING TIP Student miscalculations are not necessarily a result of misunderstanding. Student 3 is a wonderful example of a student who understands the mathematics.

18   

Student 3’s work in the first prompt shows comfort, possibly preference, for counting on by hundreds. His next jump of 50 is likely connected to his recognition of 758 and understanding of adding fifties (758 and 50). This is evidence of comfort with making hundreds. We should connect this idea with his response in the second prompt, where he tries to “wrap around” the century (671 + 60). Instead, we might help him think about how many are needed to get to the next hundred (30) and then add the rest of the tens from that point.

Student 4 Student 4 shows proficiency with adding on a number line. He also shows proficiency with breaking apart addends to add more efficiently. We can begin to challenge him with more advanced ways of decomposing numbers as he shows readiness. For example, he may be ready to think of 361 + 371 as 350 + 350 + 11 + 21, which becomes 700 + 32 by redistributing the amount within each addend. In the first prompt, he might think of 358 + 458 as 350 + 350 + 100 + 16 or 350 + 450 + 16.

Mine the Gap for Mathematical Understanding

TASK 1A: Use a number line to show how you add 358 + 453 and 371 + 361.

Student Work 3

Student Work 4

Chapter 2: Addition and Subtraction Within 1,000   19

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 1B: Dennis got out some base ten blocks (235). Jackie got out some base ten blocks (137). How many blocks did Dennis and Jackie get out all together? Use pictures, numbers, or words to explain your answer.

MODIFYING THE TASK We can extend the task by asking students to create an addition problem and then represent it with base ten blocks. Students can draw/represent base ten blocks with dots (ones), lines (tens), and squares (hundreds).

Addition within 1,000 should build from conceptual understanding of numbers and the operation as it does with smaller addends. We can use similar tools to develop this understanding. This problem prompts students to add two three-digit numbers represented by base ten blocks. The blocks are not arranged in order of place value. Students will need to show that they have made sense of each number. They will also need to show how they found their sum. Some students may simply count all of the blocks for each place value to find the sum. Doing so shows that they understand the meaning of addition, but it also shows that they rely on a lower-level strategy (counting on) and physical models or drawings to combine larger numbers. It will be important to explicitly connect the representations with computation on number lines and equations.

TASK 1C: Kelly added 348 + 256 by breaking both numbers apart. She created 300 + 40 + 8 and 200 + 50 + 6. She said she could then just add the hundreds, tens, and ones to get the sum. Do you agree? Will this always work? Create a new equation to show if it will or won’t work. Flexibly decomposing numbers enhances our ability to compute efficiently and mentally. In this task, students are asked to consider if we can decompose two addends by their place values and then add by their place values. Student understanding of decomposition, in this case expanded form, may be their greatest challenge. Students who understand this will note that you are still adding the same numbers, so it does and always will work. Others will state that there is a difference between the number (348) and the expanded form of it (300 + 40 + 8). The task notes that the numbers are then added by hundreds, then tens, and lastly ones. Students may believe that you can only add by starting with the ones place. Yet, this is only necessary when applying a traditional algorithm. For these students, we can add using the expanded form of two addends starting with ones and then add again starting with hundreds, noting that the sum remains the same regardless of which place value we begin with.

TASK 1D: Use a hundred chart (701–800) to add 732 + 59. Use a hundred chart (501–600) to add 514 + 77. Hundred charts are quite useful for adding within 100. We can modify them to model addition within different centuries. In this task, students use a 700 chart and 500 chart to add numbers within the respective century. Some students may count on from one addend by ones. Many students are likely to add by tens and then ones or by ones and then tens. We should look for students who make a jump of a multiple of ten. Doing so shows a more refined approach to computation. We can connect the computation represented on hundred charts with it represented on number lines. This helps students transfer their understanding to new models. We can also connect it to equations to lay the groundwork for developing understanding of symbolic representations and eventually an algorithm. 20   

Mine the Gap for Mathematical Understanding

BIG IDEA 2 Reasoning About Addition Within 1,000

BIG BIGIDEA IDEA

21

TASK 2A Amy is stuck with 234 + 599. Lisa says she can just think of it as 233 + 600. Do you agree with Lisa? Tell why you agree or disagree. Does Lisa’s idea always work? Give another example to prove why it does or doesn’t work.

About the Task Thinking flexibly about addends improves our efficiency and accuracy. Some addends make computation more challenging. However, we can manipulate those numbers to make them more friendly. Compensation is a strategy for manipulating addends. With this strategy, we can adjust one addend so long as we compensate for the change by changing the other addend by the same amount. In this task, students are presented with 234 + 599 and asked if they can use a compensation strategy to make the expression 233 + 600. The task also asks students if they can generalize this strategy to other expressions.

MODIFYING THE TASK We can modify the numbers in the task to work with other compensation situations. Different expressions might include 375 + 433, where we would give

Anticipating Student Responses

25 from 433 to 375

Some of our students will be challenged by the computation. Some students will disagree with the prompt because the compensated addends (233 + 600) are different from the original addends (234 + 599). We will have students who justify their agreement by finding the sum of both expressions and then noting that these two sums are equal. Other students will note that they are equal because an amount (1) was taken from one addend (234) and given to the other (599). Students’ extended responses will show if they have generalized the concept of compensation and understand it. Some students may provide extended examples that simply change the value in the hundreds place of each addend (i.e., change 234 + 599 to 134 + 499). This does not necessarily show understanding of compensation because it is a simple modification.

Another expression

to make 408 + 400. might be 515 + 388, where we would give 12 to 388 to make 503 + 400.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 21

WHAT THEY DID

Student 1 Student 1 agrees that 234 + 599 = 233 + 600. She explains that one has been taken from an addend and given to another addend. Student 1 explains the situation but doesn’t understand that it always works. Her understanding seems to be limited to three-digit numbers that contain 99.

MINING HAZARD Students who agree with prompts or

Student 2 Student 2 answers the prompt correctly. She also uses computations to justify her solution. She conveys that this compensation strategy always works but doesn’t offer a new situation to support her justification.

situations without explanations or justifications may not fully understand the generalization.

USING EVIDENCE

What would we want to ask these students? What might we do next?

It is important that we don’t infer their understanding.

MINING TIP Subtracting across zeroes can be challenging when using an algorithm. This compensation strategy is sometimes used to ease that challenge.

Student 1 Student 1’s misconception that compensation can only be applied to certain situations might be connected to the limited examples used during instruction. In previous lessons, she may have learned that we can compensate when subtracting across zeroes. For example, we can change 400 – 238 into 399 – 237 to make the subtraction easier. Or her compensation strategies with addition have been situations that easily convert to the next hundred, such as the example in this prompt. To further refine her thinking, we can begin to compensate with other values. For example, we might consider how 234 + 590 is similar to 224 + 600. We can then move to other compensation sizes. In these examples, we have compensated to add to a multiple of 100. In time, we might want to work with benchmarks of multiples of ten, fifty, or other friendly numbers.

Student 2

MINING HAZARD As students develop strategies, such as compensation, they may make inaccurate generalizations about the strategy.

Like Student 1, Student 2 shows how the compensation of one works in the prompt. Unlike Student 1, she explains that this always works regardless of the numbers involved. We want to begin to extend her compensation strategies to greater intervals. We may have to provide explicit examples before she can begin to generalize the idea to any and all situations. We should also develop understanding that compensation strategies can be applied to either addend. For example, we might ask if 347 + 382 is the same as 329 + 400 or if it is the same as 350 + 379. As we know, both are true. However, the first situation (329 + 400) may be more efficient.

For example, they may think that only one addend can be adjusted.

22   

Mine the Gap for Mathematical Understanding

TASK 2A: Amy is stuck with 234 + 599. Lisa says she can just think of it as 233 + 600. Do you agree with Lisa? Tell why you agree or disagree. Does Lisa’s idea always work? Give another example to prove why it does or doesn’t work.

Student Work 1

Student Work 2

Chapter 2: Addition and Subtraction Within 1,000   23

WHAT THEY DID

Student 3 Student 3 also agrees with the compensation in the prompt. She notes that one addend (599) “is closer” to a friendly number or benchmark (600). She notes that helps determine which addend should be adjusted. She doesn’t share if the approach always works. However, she does offer a new situation 224 + 699 that is equal to 223 + 700.

Student 4 Like the other students, Student 4 agrees with the prompt. She notes that the n ­ umbers (sums) are the same, although the addends are different. She then extends her argument with a new situation. Interestingly, she compensates by a three (283 → 280 and 397 → 400). Yet, she aligns the computations in an algorithmic way similar to Student 3’s approach.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 Like the others, Student 3 needs opportunities to further develop her understanding with other situations. This may be most evident through her extended response. In it, she adjusts the hundreds but essentially provides the same expression. We might also notice in the lower left-hand corner that she uses a traditional algorithm to confirm that her created expressions are equivalent. These two points signal understanding that may not be as solid as we first assumed.

Student 4 Student 4 does rely on exact computations to justify her thinking. We can’t be sure if these computations are to prove the equivalence to herself or to her reader. It would have been interesting to see her complete the task. We could have looked to see if she used the algorithm to regroup and find the sum before simply reproducing the expression and equation because she understood why the two equations were equal. It is noteworthy that she extended the compensation to a much different expression (283 + 397) rather than just adjusting the hundreds place in the provided prompt. She too needs more opportunities to develop fluency with the concept of compensation. We can incorporate this into our daily number routines. We can also highlight possibilities as they come up naturally in the mathematics problems we work with daily.

24   

Mine the Gap for Mathematical Understanding

TASK 2A: Amy is stuck with 234 + 599. Lisa says she can just think of it as 233 + 600. Do you agree with Lisa? Tell why you agree or disagree. Does Lisa’s idea always work? Give another example to prove why it does or doesn’t work.

Student Work 3

Student Work 4

Chapter 2: Addition and Subtraction Within 1,000   25

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 2B: Look at each. Circle the expression that has the GREATEST sum. • 530 + 200 or 628 + 30 • 16 + 790 or 310 + 350 • 720 + 100 or 400 + 25 • 25 + 600 or 200 + 275 Choose one of your comparisons. Tell how you found your answer. Use pictures, numbers, or words to explain. Sometimes exact calculations are needed. Other times, we can estimate our sums and differences. In this task, students are asked to compare sums of different expressions. Some students will find exact sums for each expression. Students with advanced reasoning strategies will consider the addends. They might reason about the first prompt by thinking that 530 + 200 is greater than 700, whereas 628 + 30 is less than 700. In the second row, 310 + 350 is less than one of the addends (790) in the other expression. A similar approach can be applied to the third and fourth rows. We can help our students think critically about their calculations and results by estimating sums before adding. Doing this routinely develops our students’ reasoning about addends, sums, and comparisons.

MODIFYING THE TASK This task can be modified to feature various three-digit addends or addends of other sizes including two-digit addends or larger situations like four-digit addends.

TASK 2C: Write two different addition equations that have the same sum as 340 + 565. In Task 2C, students have to find a sum (905) and then create two new expressions that have the same sum. This can be particularly challenging for students who see computation in a procedural way. These students will work to create various expressions before finding a pattern or creating quite a simplistic equation that adds 0 or 1 to 905 or 904, respectively. Other students will demonstrate ideas about compensation by adding 1, 10, or 100 to one addend while taking the same amount away from the other. Their work and strategies can serve as good discussion starters in whole-group settings.

TASK 2D: Billy added two three-digit numbers. The sum was 615. What might have been Billy’s numbers? Use pictures, numbers, or words to explain your thinking. This task asks students to reason about a sum differently. We may see students who incorrectly find their sum of 615 by writing an equation of 6 + 1 + 5. We may see students who show it as the sum of decomposed place values (600 + 10 + 5), which yields three addends rather than two. In other cases, students may call on their understanding of decomposition to simply break apart 615 into two friendly numbers such as 600 and 15 or 610 and 5. In similar situations, students will break 615 into 300 + 315 or 310 + 305. Like other tasks, this task is a good opportunity for students to exchange ideas about their strategies and solutions. In doing so, our students are able to acquire a wide perspective about sums and possible addends as well as thoughts about breaking apart numbers.

26   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 3 Subtraction Within 1,000

31

TASK 3A Represent 755 – 435. Show it on a number line. Write an equation to show it. Show it with base ten blocks. Show it on a part-part-whole chart.

About the Task Our students’ mathematical understanding is strengthened as they make connections between representations. This task is an example of a Frayer Model graphic organizer. Students are provided with a subtraction expression (755 – 435) and have to represent it in four different ways. These types of tasks are useful because they help students reinforce understanding by connecting models of concepts. We can use these tasks for small-group instruction where two or three students work together to complete the organizer on a recording sheet or poster paper. Groups can then merge ideas to create a class anchor chart. These tasks can also help us identify aspects of a concept that students don’t fully understand.

Anticipating Student Responses Students are likely to use one of the sections to find the difference. After finding it, students are likely to simply transfer the numbers to the other models or representations. It is important to look for the section that our students begin with. This can tell us which model they prefer when subtracting. Some students may write the traditional algorithm in the equation section. This may be the initial strategy that they use to create the other models. Students may show counting back or counting up with the number line. The part-part-whole model may represent the problem incorrectly because they will attempt to replicate the equation from left to right in the sections rather than reason about the relationships between the numbers.

MODIFYING THE TASK We can modify the sections to examine other ideas about subtraction. For example, we might change a section to prompt for a related addition equation or for a word problem that could be solved with the identified expression.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 27

WHAT THEY DID

Student 1

MINING HAZARD Each student appears successful with this task. Yet each gives us perspective about ideas that need to be further developed. These students are reminders that right answers do not always indicate full and complete understanding.

All four students in this task are able to find the difference of 755 and 435. Yet each shows different levels of understanding. Student 1 shows that he can use the number line to decompose the subtrahend and count back by the parts. He doesn’t offer the standard algorithm in the equation section, so it is likely that he relied on base ten blocks or the number line to find his difference. His part-part-whole model is incorrect as it reads in a similar order as his equation.

Student 2 Student 2’s work offers interesting insights. Like Student 1, he decomposes 435 and takes away chunks from 755. His number line decreases from left to right, which we know to be incorrect. However, it may show how he thinks about counting down from a number as he starts with a number, and decreases as he moves “forward” on the number line. His part-part-whole model is labeled, which helps us see that he has confused the idea with decomposing the subtrahend. His idea is that each decomposition is a “part” that is taken away, which leaves a “whole.”

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 Student 1 shows an effective strategy for subtracting by using partial differences on the number line. We can use it to develop other strategies, mental mathematics skills, and understanding of the traditional algorithm in time. We may be able to leverage his understanding of base ten blocks to build proficiency with part-part-whole models. We might do this by using physical models on a part-part-whole workmat before moving to drawings of the blocks on a recording sheet. When doing both, we can also record the numbers on the model and discuss the connections between the parts and the whole as well as the connection with them and the equation.

Student 2 Student 2 also shows a sound strategy for approaching subtraction. His ideas about partial differences will be useful as he develops mental mathematics ideas. As with Student 1, we need to reteach the meaning of part-part-whole relationships. This understanding will help each of these students with problem solving and algebraic thinking connected to part-part-whole relationships, bar diagrams, and equations. We first need to address his number line representation. We might ask him to compare his number line with others, to describe how they are different. We might ask him why his numbers decrease as the line moves to the right. We can make use of any number line activity to reinforce this fundamental concept.

28   

Mine the Gap for Mathematical Understanding

TASK 3A: Represent 755 – 435. Show it on a number line. Write an equation to show it. Show it with base ten blocks. Show it on a part-part-whole chart.

Student Work 1

Student Work 2

Chapter 2: Addition and Subtraction Within 1,000   29

WHAT THEY DID

Student 3 Student 3’s work is unique because he offers a word problem instead of an equation. His word problem is a take-away situation. It’s possible that he doesn’t know what an equation is, although that seems unlikely as the other models are accurate, indicating that he’s likely had experience with equations. Instead, it’s possible that he often works with Frayer models or similar organizers that include a word problem section. It’s also possible that he wanted to show what else he knows about the subtraction expression.

Student 4 Student 4’s work on the number line is an indication of how he thinks about computation. It shows that he decomposes the subtrahend into comfortable numbers. He counts back by individual hundreds before counting back by groups of tens and ones. His part-part-whole provides an interesting clue that we might easily overlook. In the upper left section, he adds a question mark. It notes the unknown in the equation.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 Obviously, Student 3 doesn’t include an equation. We should be able to confirm quickly that he understands equations and how they connect to his models. Student 3 also serves as a reminder that our students have all sorts of ideas about mathematics. Their ideas sometimes go beyond what we might be looking for. A narrow view of his work tells us he doesn’t read or follow directions. But with a different perspective, we may see the signal that he needs extensions, enrichments, or other challenges.

Student 4 Student 4 provides accurate models. His work on the number line shows comfort with decomposition and preference for multiples of tens rather than multiples of hundreds. Although his strategy is accurate, we want to help him advance to more efficient strategies. One way to do this is to connect counting back by single hundreds with counting back by groups of hundreds. We might show him this by pairing his subtraction on a number line with another example that takes away larger amounts. We can also record the results of subtracting hundreds and subtracting multiple hundreds using a calculator. For example, we could compare the difference of 735 – 100 – 100 – 100 – 100 and the difference of 735 – 400. Fortunately, he shows some comfort with this idea as he counts back by 30 rather than three tens. We may also be able to build from or connect to his ideas about groups of ones and groups of tens to groups of hundreds.

30   

Mine the Gap for Mathematical Understanding

TASK 3A: Represent 755 – 435. Show it on a number line. Write an equation to show it. Show it with base ten blocks. Show it on a part-part-whole chart.

Student Work 3

Student Work 4

Chapter 2: Addition and Subtraction Within 1,000   31

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 3B: Use a number line to subtract 600 – 260. Then, use a different number line to subtract 728 – 188.

This task asks students to represent subtraction on an open number line. Students are likely to approach the computation by counting back. They may decompose their jumps by place value or friendly numbers. For example, they may count back from 600 by jumping back by 200 (landing at 400) and then make another jump of 60, showing a result of 340. Some students may count up or jump forward. In doing so, students will begin with the subtrahend and jump up to the minuend. The distance of the jump represents the difference of the expression. Like those who jump back, these students may add on by single numbers, place values, or friendly numbers. It’s also possible that these students will show the minuend and subtrahend as the endpoints on the number line.

TASK 3C: Break apart one or both numbers to make the subtraction friendlier. 815 – 719 = ________

650 – 475 = ________

Breaking apart or decomposing numbers makes computation friendlier. There are all sorts of ways that these numbers may be decomposed. For 815 – 719, students may decompose by place value similar to expanded form. But with 815 – 719, this decomposition does not necessarily make computation easier because they will have to make sense of 5 – 9. The most efficient decomposition may be to break 719 into 715 and 4. We can then subtract 815 – 715 (100) and then subtract 4 more (96). The numbers do not have to be decomposed into two parts. 719 could be broken into 500, 200, 10, 5, and 4, for example. Breaking 719 into more parts can make subtracting easier for students. In this case, they can think about 815 – 500 (315) – 200 (115) – 10 (105) – 5 (100) – 4, which is 96.

TASK 3D: Use a hundred chart (701–800) to subtract 786 – 62. Use another hundred chart (501–600) to subtract 551 – 39.

701

702

703

704

705

711

712

713

714

715

716

721

722

723

724

725

726

731

732

733

734

735

736

741

741

743

744

745

746

751

752

753

754

755

756

761

762

763

764

765

766

771

772

773

774

775

776

781

782

783

784

785

786

791

792

793

794

795

796

32   

706

As with addition within 1,000, we can represent subtraction within 1,000 on a modified hundred chart. In Task 3D, students are asked to subtract 786 – 62 and 551 – 39 on a 700 chart and a 500 chart, respectively. We can expect stu707 708 709 710 dents to count or jump back by tens and ones or groups of tens 717 718 719 720 727 728 729 730 and ones. Both strategies are reasonable, although subtracting by 737 738 739 740 groups of tens or ones is more efficient. This model doesn’t transfer 747 748 749 750 well to adding up because the subtrahend (62) is not represented 757 758 759 760 on the chart. We may also find students who subtract procedurally 767 768 769 770 and then show the minuend and difference on the chart. In all cases, 777 778 779 780 787 788 789 790 it is important for students to examine the count-back strategies 797 798 799 800 that others used to begin to develop their own efficiencies.

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 4 Reasoning About Subtraction Within 1,000

41

TASK 4A Compare the subtraction expressions in each row. Circle the expression that has the GREATEST difference. • • • •

900 – 480 or 500 – 380 600 – 199 or 900 – 799 850 – 750 or 400 – 200 753 – 300 or 598 – 501

Choose one row. Tell how you found your answer. Use pictures, numbers, or words to explain.

About the Task We reason about computation frequently in our daily lives. We may quickly compare prices in a store and the amount of money in our wallet. In many situations, exact answers are not needed. We reason or estimate the quantity, sum, or difference. In this task, students compare differences of two expressions. They then explain how they reasoned about their comparison. The task can be easily modified and can be used as a daily number routine. When doing this, we can provide one or two comparisons for students to complete mentally and independently. After, we can bring the class together to discuss.

Anticipating Student Responses Some students will find exact differences and use them to justify each of their comparisons. Students may use mental strategies for computations. They too will then compare the differences. Other students will estimate and reason about the expression. For example, when comparing 900 – 480 with 500 – 380, they may reason that the first is about 400 and the second is much less, about 100. These students might say the difference of 600 – 199 is greater than 900 – 799 because the latter is about 100 and the former is clearly much more than 100. Flawed reasoning is also likely. For example, 850 – 750 may be thought of as greater than 400 – 200 because both the first minuend (850) and subtrahend (750) are greater than those in the second expression.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 33

WHAT THEY DID

Student 1 Student 1 relies on computation to compare the differences in the prompt. Her computation raises concern. She seems to apply ideas about partial differences to her computation. She takes away hundreds and then tens in two rows (900 – 480 and 900 – 799). Each time she subtracts hundreds and adds the rest to that difference. Oddly, she doesn’t repeat this error with 500 – 380. It is difficult to decipher how she found the difference 600 – 199 = 106.

Student 2 Student 2 explains that she used the algorithm to find differences and then compared them. She does not provide any examples of the algorithms. Yet, all of her differences are accurate. This seems to indicate that she did all of the computations in her journal or on scratch paper.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1

MINING HAZARD Reliance on procedure is not a terrible thing. However,

Student 1’s work reminds us that our students must have understanding of the concepts to reason about situations and comparisons. We want accurate understanding of subtraction and alternate strategies for subtraction. We may first need to reteach three-digit situations before moving to reasoning activities with three-digit subtraction. We can adjust these tasks to make use of two-digit situations to work with reasoning about subtraction situations.

written procedures can be inefficient. Procedural approaches can undermine the development of flexibility. Consider computing 950 – 849. It will take much longer doing it with pencil and paper than just recognizing that it is 1 more than 100.

34   

Student 2 We should first confirm with Student 2 that she used an algorithm for each comparison. Using the algorithm for each one tells us something about her approach to subtraction or mathematics in general. By using it, she may signal a preference for procedure when calculating and even solving problems. She successfully completes the task. But her reasoning about the relationship between subtrahends and minuends and between expressions can be developed. We want to help her think critically about the problems she encounters. We want to encourage her to develop ideas about mental mathematics and reason as she works with it. This takes time, exposure, and discussion.

Mine the Gap for Mathematical Understanding

TASK 4A: Compare the subtraction expressions in each row. Circle the expression with the GREATEST difference. Tell how you found your answer.

Student Work 1

Student Work 2

Chapter 2: Addition and Subtraction Within 1,000   35

WHAT THEY DID

Student 3 Student 3 presents a simple justification. Her computation satisfies the prompt as she justifies her differences. She doesn’t fully explain that 200 is greater than 100, but we can confidently assume she understands. All of her other comparisons are correct. We are left to wonder if she found each difference mentally or with paper and pencil.

Student 4 Student 4’s justification is very telling. She compares 753 – 300 with 598 – 501. She describes the “size” of the difference when subtracting within a century (598 – 501) with subtracting across hundreds (753 – 300). Her wording, “you will [only] have [your] tens and ones,” insinuates this comparison. It is also noted that 900 – 480 and 500 – 380 are computed with a traditional algorithm on the back of her paper. The other expressions are not computed on the back.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3

MINING HAZARD Student 4 could overgeneralize the idea. We may ask her to compare subtraction within a century (472 – 401) and across centuries (610 – 599) to confirm that she isn’t developing a misconception about “taking away

Our first action with Student 3 is to determine how she compared the other rows. Let’s assume that we know her well and know that she calculated each with paper and pencil. Our next step will be to provide opportunities to develop mental mathematics skills and critical thinking skills similar to Student 2. On the other hand, let’s assume she did each mentally. We want to acknowledge and encourage her mental mathematics development. We want to give her opportunities to explain how she found sums or differences. We also want to be sure that we don’t constantly ask how she found her solutions as this becomes a tedious defense and may discourage her from using mental mathematics strategies.

Student 4 Student 4 shows us that she can reason about subtraction in these situations. She shows that the difference within a century is less than the difference across centuries. Her computation on the back is also telling. It shows that she can compute some numbers mentally and others still require pencil and paper. This shouldn’t discourage us. Instead, it provides a clear path for moving forward with Student 4. We want to encourage mental mathematics and share ideas about how written computations may be thought of with more efficient methods. Although we can share these ideas directly with her, it is likely more beneficial for her to develop them through discourse with classmates during number discussions.

hundreds.”

36   

Mine the Gap for Mathematical Understanding

TASK 4A: Compare the subtraction expressions in each row. Circle the expression with the GREATEST difference. Tell how you found your answer.

Student Work 3

Student Work 4

Chapter 2: Addition and Subtraction Within 1,000   37

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 4B: Kim is trying to subtract 600 – 346. Connie says she can change it to 599 – 345. Do you agree with Connie? Does Connie’s idea always work? Give another example to prove why it does or doesn’t work. The concept of compensation tells us that we can adjust the minuend or subtrahend by any amount so long as we do the same to the other number. Subtracting across zeroes can be quite challenging for students when attempting to apply the traditional algorithm. If they understand compensation, they can take one from the minuend to eliminate subtraction across zeroes. In this task, students may disagree because they make a computational error. This is a telling mistake because it tells us that they focus on the result more so than the relationship between the numbers. Other students may disagree with the prompt because the minuends and subtrahends are different. In either case, it is important to model both expressions with a number line, base ten blocks, or similar tool. When we do this, we want to show both representations side by side and talk about the similarities and differences between them. It will be critical to highlight the same difference as well as the same adjustment to minuends and subtrahends. From there, we can investigate other expressions to see if the concept of compensation always works.

TASK 4C: Subtract. 662 – 400 = 662 – 350 = 662 – 300 = 662 – 250 = 662 – 200 = What are two things you notice about the equations? Task 4C asks students to focus on the patterns of relationships within computations. Each new subtraction problem takes away 50 more. Our students may approach each new problem as an isolated calculation. Others will see the pattern in the subtrahends or the relationship between the minuends and subtrahends. We want them to recognize more than the fact that each new problem takes away 50 more. It is also important for them to note that the minuend is the same in each problem. With the same minuend (662), the difference increases by the same amount (50) as the subtrahend decreases. In other words, as we take less away from a number, the difference then increases.

38   

Mine the Gap for Mathematical Understanding

TASK 4D: Mila knows that 558 – 338 = 220. What is the difference of 559 – 339

MINING HAZARD We want our students

568 – 348

to identify patterns

548 – 338

in mathematics. But

658 – 438

we also want them to understand the

Task 4D examines the idea of compensation more generally. Students are provided with an equation and asked to find differences of similar expressions. The equation is provided so that reasoning can be used rather than calculation. The compensated expressions add 1, add 10, add 100, and subtract 10 from both the minuends and the subtrahends. Students may justify their solution because they find an exact answer with an algorithm, number line, or picture. These students will pay little if any attention to the related equation. Others may say that the equation has a difference of 220 because they all have a difference of 220.

patterns. Saying that they all have a difference of 220 doesn’t establish WHY they all have a difference of 220.

notes

Chapter 2: Addition and Subtraction Within 1,000   39

BIG IDEA

5

MINING TIP We can collect data from cereal boxes, student magazines, or newspapers. We can ask students to gather examples from home. Students may be explicitly taught about certain types of graphs, but they can also have experience

BIG IDEA 5 Problem Solving With Addition and Subtraction TASK 5A The table shows the weight of animals at the zoo. Animal Tiger Gorilla Polar bear Zebra

Weight (in pounds) 667 408 437 568

How much more does the heaviest animal weigh than the gorilla? The zoo is getting a new hippo that weighs 240 pounds more than the zebra. How much does the new hippo weigh?

with less traditional, print representations of data.

MODIFYING THE TASK We can ask different questions about the data in this task. We can ask students

About the Task As we know, problem solving is much more than solving a story problem. In the real world, we solve problems when gathering data from investigations or graphs and tables. These situations often lack the familiar structure or phrasing of a word problem. They may offer much more than two data points. We can make use of varied data structures to engage our students in less traditional problem solving. In this task, students do just that. They answer questions about data presented in a table.

to create their own questions about the data. We may bring new information to the data similar to the second prompt in the task.

Anticipating Student Responses The first problem may challenge our students because they have to determine that the greatest value represents the heaviest animal. Some students may read the table incorrectly, and there is always the possibility of a calculation

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks.

40

error. Some students may simply choose an operation and repeat that choice for both prompts. Accurate students may subtract or add up from the gorilla to the tiger to find the difference. Like the first prompt, the second prompt may be tricky for students who rely on two words and an action or keyword to solve a problem. Students may misunderstand the relationship stated in the prompt and subtract 240 rather than add 240. We should look for students who justify their solutions with pictures and models alone, calculations and models together, or calculations alone. Each of these three possibilities gives insight into how the student makes sense of and solves problems.

MODIFYING THE TASK The solution to the first prompt may be correct because the data points are already aligned vertically. We may choose to move the gorilla to another location in the table.

notes

Chapter 2: Addition and Subtraction Within 1,000   41

WHAT THEY DID

Student 1 Student 1 successfully identifies the value of the heaviest animal and the gorilla. He subtracts with an equation. His equation indicates some understanding of regrouping, but the procedure is obviously flawed. In the second prompt, he successfully combines the weight of the animals.

Student 2 Student 2’s work shows strategies for subtracting. We can see that he uses base ten blocks. In the first prompt, he hadn’t accounted for a regrouping situation but compensated by taking one away from one hundred. He didn’t have a regrouping situation in the second prompt. His calculations are correct, but his solutions to the problems are not. In the first prompt, he selects the two largest numbers, showing some regard for the term heaviest, but overlooks the gorilla’s weight. In the second prompt, he uses the correct data point but replicates the operation.

USING EVIDENCE

What would we want to ask these students? What might we do next?

MINING HAZARD Students who only offer calculations do not necessarily misunderstand how to solve a problem. We may be tempted to require them to draw

Student 1 Student 1’s equations don’t necessarily indicate understanding of the problem. In this example, both operations are the correct choices, but we can’t be sure that he intentionally selected those operations. We should speak with him to confirm his understanding. We should also consider providing Student 1 and others like him with tools to reinforce his accuracy. We might ask him to first solve the problem with paper and pencil. After doing so, he can use a calculator or other tool to confirm his accuracy. This helps us determine if errors are due to miscalculations or undeveloped problem-solving skills.

a picture or diagram of their thinking. This is a good instructional strategy. However, we should also recognize that simply calculating with the correct operation for a situation can also indicate understanding.

42   

Student 2 Student 2 shows that he can use these models to find his differences. He shows that he is clever by taking 1 away from 100 rather than decomposing the tens and hundreds. We have two areas to develop with Student 2. First, we want to help him develop other strategies for computing. We can connect his models to number lines, partial sums and differences, and other models or strategies to do this. We also need to help develop his ability to make sense of problems. We can encourage him to use bar diagrams or pictures. As he works with these models, we want to reinforce the relationships between “parts” and the “whole.” We also want to work with situations that compare values. Student 2 will benefit from partner work and discussions with others about how problems are solved and why certain operations may be selected.

Mine the Gap for Mathematical Understanding

TASK 5A: The table shows the weight of animals at the zoo. How much more does the heaviest animal weigh than the gorilla? The zoo is getting a new hippo that weighs 240 pounds more than the zebra. How much does the new hippo weigh?

Student Work 1

Student Work 2

Chapter 2: Addition and Subtraction Within 1,000   43

WHAT THEY DID

Student 3 Student 3 successfully uses a number line to find the difference between values. He also indicates decomposition strategies that support his calculations. He correctly solves the first problem. In the second prompt, he either doesn’t understand the problem or simply recycles the operation from the first problem without giving the second problem any thought.

Student 4 At first glance, we may think that Student 4 doesn’t understand the problem in the first prompt because he adds. However, we can see that he adds up from the weight of the gorilla (408) to the heaviest animal (667). He also circles this difference to imply that this amount is the difference. It is likely that he wrote his equation after completing his work on the number line because of the unfriendly difference caused by a regrouping situation. In the second prompt, he correctly finds the solution. Although the “answer” is not circled in this prompt, its location in the equation indicates that he added the numbers and the sum is the combined amount.

USING EVIDENCE

MINING TIP It is important for students to have problem-solving strategies and models to rely on, even when they are computationally fluent. These strategies will support them as they encounter unfamiliar contexts, unique problems, and other situations that are less straightforward.

44   

What would we want to ask these students? What might we do next?

Student 3 Student 3 shows a more refined approach to computation than Student 2. However, like Student 2, he shows difficulty in making sense of the problems. We should recognize his computation prowess. Then, we want to ask him about the numbers he selected and why he subtracted. We should do this for both problems as we may be able to help him make meaning by comparing and contrasting the situations. Like Student 2, Student 3 also needs more opportunities to represent problems with models until he shows greater understanding more consistently. Then, he can be encouraged to move to equations and computations if preferred.

Student 4 Student 4 serves as a reminder of caution. We may inadvertently dismiss his solution because we expect to see subtraction rather than addition. Student 4 would also be a good example of student work to share with the group after others have shared because he presents a rather unique approach. We can confirm his understanding by asking, “Why did you add?” “How did you know to add?” or “How did you select your numbers?” His responses can help other students make sense of problems and consider solution pathways and computation strategies.

Mine the Gap for Mathematical Understanding

TASK 5A: The table shows the weight of animals at the zoo. How much more does the heaviest animal weigh than the gorilla? The zoo is getting a new hippo that weighs 240 pounds more than the zebra. How much does the new hippo weigh?

Student Work 3

Student Work 4

Chapter 2: Addition and Subtraction Within 1,000   45

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

MODIFYING THE TASK We can adjust the task to make use of varied addends. We can widen the range of the sums. We might even ask students to create similar prompts to exchange with classmates for solving.

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 5B: Robert added a number to 368. His sum was greater than 700 but less than 750. What might his number have been? Use pictures, numbers, or words to explain your thinking. Task 5B provides students with an addend and range for the sum. Our students may reason that the missing addend has to be less than 400 because the sum of the new number and 368 is less than 750. These students may even notice that 68 is 18 greater than 50 and adjust their addend by a similar amount. Others might consider that adding 300 to 368 will fall short of 700 and compensate by adding 40 or 50 more. Some students may use a trial-and-error or a guess-and-check approach to solving the problem. These students may try several possibilities before finding their solution.

TASK 5C: Derrick had 1,000 blocks. He gave some to Beth. The blocks he has left are shown below.

How many blocks did Derrick give to Beth? 46   

Mine the Gap for Mathematical Understanding

This task features a change unknown problem situation. The picture represents the amount of blocks remaining after some are given away. The blocks are not grouped by place value, and there are 11 tens, causing a necessary place value regrouping. Students may be challenged by the configuration of the blocks because they are not in grouped order. For these students, we might revisit number and place value concepts. Some students will count on or count up from the value (318), drawing representations of the added blocks, while others will add up by using a number line or with numeric representations. Subtracting the amount represented from 1,000 is also a viable strategy.

TASK 5D: Each problem adds two three-digit numbers. Find the missing digits.

MODIFYING THE TASK We can change the

75_ + 2_0 = 967

number of digits in

590 + _85 = 975

the addends or sums.

815 = 30_ + _07

We can also change the operation to

500 = 2_0 + 260

subtraction.

Finding missing digits in addends or sums is also a good example of a non-storytype problem. The sums in the prompt provide clues about the missing digits. In the first and third prompts, students can use the tens place from the first addend and sum to find the missing 2 in the second addend. They can repeat this strategy for the first addend’s ones place. The second prompt may be more challenging, although just one digit is missing. Students may quickly write “4” because 500 + 400 equals 900. But the problem requires regrouping of tens, so there are actually only three hundreds in the second addend. Some students may use guess-and-check strategies to find their solutions. Others may rely on their number sense and reasoning to find the missing numbers. This task might be a good opportunity for a four corners rotation activity where groups of students visit equations and find solutions. After visiting all four equations, the class would come together to share their solutions and strategies.

MODIFYING THE TASK This task doesn’t prompt for students to share how they found their solutions. We could add that condition to the prompt.

notes

Chapter 2: Addition and Subtraction Within 1,000   47

CHAPTER

3

MULTIPLICATION AND DIVISION THIS CHAPTER HIGHLIGHTS HIGH-QUALITY TASKS FOR THE FOLLOWING: zz

Big Ideas 6 and 13: Representing Multiplication and Multi-Digit Multiplication Multiplication and division can be represented with arrays, area models, repeated addition or subtraction, and on number lines. Representing these concepts is foundational understanding. These models and representations also hold true for multi-digit factors.

zz

Big Ideas 7 and 14: Reasoning About Multiplication and Multi-Digit Multiplication There is a relationship between factors and products. The product changes as one or both factors change. We can use this relationship to leverage known products for unknown products. There are similar relationships in division that help us reason about our quotients.

zz

Big Idea 8: Properties of Multiplication The properties of multiplication help us develop fact fluency and computational fluency in general. They are algebraic in nature. Understanding a property is much more than identifying it (e.g., a × b = b × a).

zz

Big Ideas 9 and 15: Representing Division and Multi-Digit Division We multiply and divide to solve problems. Multiplication and division can be represented differently depending on the problem structure or situation. Representations for division work regardless of the size of the dividends, divisors, and quotients.

zz

Big Ideas 10 and 16: Reasoning About Division and Multi-Digit Division Reasoning about the relationship between dividends, divisors, and quotients helps us determine the reasonableness of our answers. We can reason about multi-digit division in the same ways we reason about single-digit multiplication and division.

zz

Big Idea 11: Problem Solving With Multiplication and Division Problem solving with multi-digit multiplication and division is similar to that with single-digit multiplication and division.

zz

Big Idea 12: Connecting Multiplication and Division Multiplication and division are connected; we can use one to find solutions for the other.

48   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 6 Representing Multiplication

61

TASK 6A Show 7 × 3 in two different ways. You can use arrays, jumps on a number line, equal groups, repeated addition, or the area of a rectangle.

MINING HAZARD We feature models or representations that we prefer or those that our

About the Task Students are often asked to identify representations of multiplication. This prompt reverses the situation. In this task, students are asked to create two different representations for a multiplication fact (7 × 3). The models our students select likely indicate those that they are most comfortable with. They may also signify the models we most regularly feature during our instruction. A comparison model is not included in the prompt. It could be added to modify the prompt in fourth or fifth grade.

Anticipating Student Responses Some students will represent 7 × 3 correctly but show an incorrect product due to a computational error. Their models will include equal groups, equal jumps, area models, and/or repeated addition. Some students will show the commutative property of 7 × 3 with the same model. For example, these students might show seven groups of 3 and three groups of 7 as representations for 7 × 3. These are two examples of equal groups, but they are not different representations. This shows a limited understanding of the interpretations and representations of multiplication. Equal groups and repeated addition are likely to be the most frequently used representations. Use of jumps on a number line and area models are less likely to appear. Note that these anticipated models do not follow the sequence of examples in the prompt, eliminating the notion that students simply select the first two examples.

students are successful using. But our students need work with various models to prepare for different situations and contexts. Students should be proficient with all models of multiplication.

MODIFYING THE TASK Consider taking away the representation examples in the prompt to get a better sense of student thinking about multiplication. This new prompt might read “Show 7 × 3 two different ways.”

MINING HAZARD We are sometimes led to believe that

PAUSE AND REFLECT

miscalculations signal

zzHow

concepts.

does this task compare to tasks I’ve used?

zzWhat

misunderstanding of

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 49

WHAT THEY DID

MINING HAZARD Students who recall basic facts may not necessarily understand the meaning of the operation or the equation. For example, recalling 9 ÷ 3 = 3 does not necessarily

Student 1 This student’s first representation is an equation with the correct product. Equations are also mathematical representations. By itself, we might think he has some understanding of multiplication. His second model is not a representation of multiplication. It seems to be an example of a part-part-whole model. Combining these two pieces of evidence tells us that the student may not understand multiplication as well as we would like. Eventually, we want our students to work with symbolic representations, including expressions and equations. Yet without the conceptual understanding of multiplication, this student may struggle with multi-digit multiplication and other computations in the future.

mean a student can put 9 beads into 3 groups of 3.

Student 2 In both examples, this student shows a product of 31. At first glance, we may be quick to dismiss this student’s understanding of multiplication because of the error. But, Student 2 does well to represent multiplication as a repeated addition sentence. The repeated addition sentence correctly shows seven addends of 3. There is a calculation error. The second representation is an attempt at representing multiplication on a number line. He does show jumps of 7. Yet, there are only two jumps. This is because he begins with 7 instead of 0. He shows some understanding of multiplication with a number line that we can build on.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1

MINING TIP Calculation tools, like calculators, help promote accuracy. They can help students understand errors when used to check computations first completed with paper/ pencil or in a student’s head.

50   

Student 1’s work signifies that he knows 7 × 3 = 21. It is likely a basic fact that he can recall. His other representation is incorrect. We should consider asking him how he knows that 7 × 3 = 21. Assuming he hasn’t mastered the concept of multiplication, he would benefit from reteaching of representations and situations for multiplication. Knowing his facts, at least some, can serve as an anchor for making sense of the representations. We may begin with those facts for initial reteaching before moving to unknown facts to check for understanding of the concept. We should also keep in mind that mixing representations with this student shouldn’t happen until solid understanding of a specific representation is demonstrated.

Student 2 Clearly, this student would benefit from a tool to support accuracy. This could be an addition chart, a multiplication chart, a calculator, or something similar. Using one of these tools to check his accuracy, after completing a task, may make more sense. He shows that he can use repeated addition. We can connect this understanding to the number line representation to expand understanding. To do this, he can represent different basic facts with repeated addition and then represent them on a number line. We can ask him questions about what is happening with the representation and what that might look like on a number line. In time, we’ll connect familiar representations with new examples.

Mine the Gap for Mathematical Understanding

TASK 6A: Show 7 x 3 in two different ways. You can use arrays, jumps on a number line, equal groups, repeated addition, or the area of a rectangle.

Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   51

WHAT THEY DID

MINING HAZARD The order of the factors does not dictate what the model should look like unless clearly indicated. 7 × 3 can be

Student 3 This student shows understanding of the concept of 7 × 3. His first model is an example of repeated addition, although it is not represented in a traditional way. He establishes the sum of two 7s before adding a third. His second model is an array with seven columns of 3. Typically, we are likely think of 7 × 3 as “seven rows of 3” instead of “seven columns of 3.” However, the prompt does not label either value and so the student’s representation is fine.

represented with seven rows of 3 or seven columns of 3. However, 7 bags of 3 apples should clearly show seven groups of 3.

Student 4 Student 4’s work shows 7 × 3 with an array and repeated addition. He also shows 3 × 7 with an array and repeated addition. It is pretty clear that he understands these models. As previously noted, the order of the facts and their relationship to the rows and columns of the array are not problematic unless it is stated in the prompt. We notice that this student’s arrays are modified as the fact is reversed. This is also true with his repeated addition models. This suggests that he consistently processes multiplication expressions in a specific order of “blank rows (7) of blank columns (3)” or “something (7) added blank (3) amount of times.”

USING EVIDENCE

What might we ask these students? What might we do next?

Student 3 This student’s understanding of multiplication is likely in a good place. We’ll want to make sure that he can explain his repeated addition equation. It would be wise to have Student 3 compare the standard representation of repeated addition with his model. This will offer insight into his understanding. We also want to be sure that he can connect these representations with other models. In time, we’ll move him from representations to equations to improve efficiency.

Student 4 Student 4 clearly demonstrates understanding of these two models for multiplication. He also adds an equation to the diagrams. We want to be sure that students such as Student 4 also understand jumps on a number line, area models, and eventually comparison situations as well. Understanding different models helps make sense of the multiplication as contexts change. It’s also important that we don’t dwell with the representations of multiplication with students like him. He shows understanding. We will want to connect to these representations as numbers become more complex. However, this student should not be required to consistently “prove” or show his work with single-digit multiplication situations.

52   

Mine the Gap for Mathematical Understanding

TASK 6A: Show 7 x 3 in two different ways. You can use arrays, jumps on a number line, equal groups, repeated addition, or the area of a rectangle.

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   53

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

MODIFYING THE TASK The multiplication expressions can be adjusted to explore other situations in which a number is the product of two different expressions.

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 6B: Jenny has 3 books on 6 shelves. Holly says she has the same number of books but Holly has 9 books on 2 shelves. Use pictures, numbers, or words to explain if you agree or disagree with Holly. This task establishes that different factors can produce the same product. Successful students will represent 9 × 2 and 6 × 3 in different ways but should justify that the two situations are equal. These students would benefit from open-ended prompts that explore different ways to make products such as 12 or 24. They may benefit from investigating variations for making even-number products versus variations for making odd-number products. Some students may disagree that the two expressions are equal because they use a different set of numbers. These students need additional work with multiplication focused on different factors yielding the same products as well as representations and contexts for multiplication.

TASK 6C: Annie made this array with counters.

Write a multiplication equation that describes Annie’s array. Ben made a different array that shows the same number. Draw a picture of an array that Ben might have made. Tell how two different arrays could show the same number. This task is a variation of Task 6B with a provided array. It is intended to identify students who recognize that some products can be made with different factors. Students who understand this concept will note the size of the rows and columns to verify the same product. Look for students who only count to find the total number of circles in the provided array and the array they create. Also look for students who create a new array that is incorrectly formed (e.g., uneven number of circles in a row). In both cases, these students indicate incomplete understanding of multiplication.

54   

Mine the Gap for Mathematical Understanding

TASK 6D: Write an equation if the words describe a multiplication situation. Write “no” if they do not describe multiplication. zz

6 boxes of toys with 3 toys in each box

zz

3 pieces of candy eaten from a bag of 4 pieces

zz

3 dogs have 3 bones

zz

3 rows of 6 flowers

zz

2 songs played then 4 songs played

MODIFYING THE TASK The factors can be

Choose one of the examples above. Use pictures, numbers, or words to tell how you know it is multiplication.

modified to make use

This task is designed to uncover student thinking about multiplication situations. Proficient students will note that the second and last situations are not multiplication contexts. However, some students who understand the contexts may not represent the situation with a corresponding model. For example, the 3 rows of 6 flowers situation lends itself to an array model more so than jumps on a number line, if it does at all.

We may consider

of multi-digit numbers for later grades. limiting the number of situations to consider or provide them one at a time for some students.

Notes

Chapter 3: Multiplication and Division   55

56   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 7 Reasoning About Multiplication

71

TASK 7A Danny knows 5 × 7 = 35. How can he use that to find 7 × 4? Use pictures, numbers, or words to explain your thinking.

About the Task Computational fluency is grounded in reasoning about numbers, operations, and properties of operations. Our students apply this reasoning as they develop fact fluency and strategies for mental math with larger numbers. This task examines our students’ notions about the relationship between factors and products. It investigates if students recognize that known products or equations can be used to find unknown products. In the past, work with basic facts hasn’t always examined the relationship between facts or patterns within fact sets.

Anticipating Student Responses

MINING TIP Helping students recognize patterns and connections is useful for developing computational fluency. Arming students with these strategies supports long-term retention.

This task requires a different understanding of the relationship between factors and products. It relies on the meaning of multiplication. Knowing 7 × 5 helps with 4 × 7 because it is one less group of seven. Simple recall of 7 × 4 does not justify the relationship, although some students may attempt to use this as their justification. Students may note that the significant difference between the expressions is that the 7 has moved locations in the equation. This may happen because students see the relationship in the factors but haven’t connected that relationship to the product. This too does not satisfy the prompt. Some students may create a representation to prove 4 × 7 without making the connection to 7 × 5. Full understanding will note one less group of seven. Some students may extend the task to show how the pattern can be used to find other facts.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 57

WHAT THEY DID

Student 1

Student 3

Student 1 has a minimal, incorrect response. Yet, his response is telling. He likely subtracts 1 from 35 because 4 is one less than 5. This means that he notices that both expressions have 7 as a common factor. He applies the difference of the factors (5 – 4) to the difference of the products. He provides no other work to support his reasoning.

Student 3 doesn’t clearly communicate his understanding with words (subtract 35 from 7). However, his equations alone do well to communicate his thinking. In addition, he doesn’t clearly connect why 7 can be subtracted from 35. We can surmise that he understands the relationship, although his communication is flawed and not complete.

Student 2 Student 2’s work is encouraging because he can model the expression 7 × 4. He also demonstrates a doubling and double-double strategy for finding the product of 28. He doesn’t connect the two expressions or explain how the common factor can be used to find an unknown product. It is likely that he hasn’t developed a sense of relationships between expressions with similar factors.

Student 4 Student 4 understands the impact of changing a factor. He proves that 7 × 4 is 28 with his representation. He notes that you can use a representation but don’t need to because 5 × 7 is already known (“7 × 5 and that’s the same as 7 × 4”). He also notes that changing the order of the expressions doesn’t affect this relationship. Student 4 also adds two different representations to justify 7 × 4. But does he recognize that 7 × 4 is one group of seven less than 5 × 7?

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 We need to determine if Student 1 understands the meaning of 7 × 5 = 35. In other words, does he understand that there are seven groups of 5? If he doesn’t, we must reteach the concept of multiplication before comparing factors in expressions. We have different options if he does understand multiplication. We could have him find the product of 7 × 4 and compare that to both 35 and his response of 34 (35 – 1). We can also create a string of expressions with one common factor. Then, we can have him describe the patterns that he notices as well as the relationship to the products. For example, we can provide the string 3 × 1, 3 × 2, 3 × 3, and so on. Ultimately, we will develop the notion that one expression can be used to find the product of another by adding or subtracting a number of groups.

Student 2 Student 2 demonstrates that he uses patterns, such as doubling, to find products. He also uses an equal group representation to find the product. We can use these understandings to help him make connections between

58   

Mine the Gap for Mathematical Understanding

the two expressions (7 × 5 and 7 × 4). Like with Student 1, we can use a string of equations with a common factor to discuss how changing a factor affects the product. In this case, we can also use the equal group representation to compare the number of groups and the product.

Students 3 and 4 Both students insinuate that they understand the impact of changing a factor in this situation. But it isn’t necessarily clear that there is one less group of seven, so we can subtract 7 from 35. We may ask them to explain what the factors mean, eliciting five groups of 7 or five jumps of 7 and so on. We want to reinforce this concept with new factors, products, and contexts. We can make note when they appear naturally in problems or intentionally during a warm-up, routine, or independent work. Both students can benefit from comparing each other’s writing. In time, we’ll extend this work to multi-digit numbers and fractions as factors. In those cases, it will be important to clearly connect the two expressions and the impact of changing a factor. Students such as these will need to work to explicitly communicate this connection when asked to do so.

TASK 7A: Danny knows 5 x 7= 35. How can she use that to find 7 x 4? Use pictures, numbers, or words to explain your thinking.

Student Work 1

Student Work 3

Student Work 2

Student Work 4

Chapter 3: Multiplication and Division   59

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

MODIFYING THE TASK Tasks built on the relationship between factors (doubling a factor, a factor that is two more than another, etc.) can be easily modified to feature multi-digit numbers for use in upper grades.

MINING HAZARD

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 7B: Oscar knows 3 × 4 = 12. How can he use that to help him solve 6 × 4 and 6 × 8? Use pictures, numbers, or words to explain your thinking. Students who see the relationship between 3 × 4 and 6 × 4 will note that the product doubles because 6 is double 3. They will extend this generalization to find 6 × 8 by doubling 6 × 4. They may quadruple 3 × 4 because each factor is half of the corresponding factor in 6 × 8. They may note that they know their basic facts as proof of understanding. Solely noting basic fact recall does not meet the intent of the task. Other students may note the relationship between factors but provide an incorrect product. This may happen because of doubling error. It may also happen because students see the relationship in the factors but haven’t connected that relationship to the product. It is critical that we determine which reason is responsible for the error.

Students who can

TASK 7C: Compare the two multiplication facts on each line. Circle the

recall facts will show

fact that has the greater product.

correct products in reasoning activities but may not understand the relationships and connections within expressions.

6 × 6

6×4

9 × 1

5×8

10 × 2

6×9

8 × 9

5×7

Choose one of the lines you compared. Use models, numbers, or words to tell how you compared the products. Considering the size of factors in different problems helps us reason about our products. 6 × 6 is greater than 6 × 4 because we have two more groups of 6 in 6 × 6. 9 × 1 and 5 × 8 have completely different factors. However, we instantly know that 9 × 1 = 9, so the other should be larger even if we don’t know that it is  40. A similar rationale can be applied to the third line. In the last example, 8 × 9 and 5 × 7, we know the first product will be larger because both factors are greater. Representations as justifications do not necessarily prove reasoning in this task. Students who reason should speak to the size, similarities, or differences of the factors.

60   

Mine the Gap for Mathematical Understanding

TASK 7D: Look at each row. Which product is greater? How can you tell without actually multiplying? 3 × 9

9×6

8 × 8

7×4

Use pictures, numbers, or words to explain your thinking. This task also examines how changing factors affects products. Students should reason about how one factor (top situation) or both factors (bottom situation) change. Students who rely on representations or quick recall of basic facts do not necessarily demonstrate understanding in this or other tasks. All students can benefit from number computation and reasoning conversations. These can be classroom routines or journal writing opportunities in the classroom.

Notes

Chapter 3: Multiplication and Division   61

62   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 8 Properties of Multiplication TASK 8A

MODIFYING THE TASK

Look at each row. Circle agree or disagree. 6 × 4 = 4 × 6 6 × 3 = (6 × 2) + (6 × 1) 8 × 7 = (8 × 5) + (8 × 2) 7 × 2 × 2 = 7 × 4

AGREE AGREE AGREE AGREE

81

The grouping symbols

DISAGREE DISAGREE DISAGREE DISAGREE

Choose one of the statements above. Use models, numbers, or words to explain why you agree or disagree.

About the Task Deep understanding of properties of multiplication is essential for flexibility and efficiency when learning basic facts or multiplying multi-digit numbers. This task asks students to consider equivalence of expressions. They represent the commutative, distributive, and associative properties of multiplication. Students are also asked to justify why two of the expressions are equal. It is intentional that all are equal. This task is different because students are asked to apply their understanding of properties rather than simply identify an abstract generalization of a property (a × b = b × a).

Anticipating Student Responses Students are likely to recognize the commutative property in the first prompt. However, the others may not be as recognizable. Look for a preponderance of justifications for the commutative property even with students who agree with other situations. Justification of the commutative property may be more than a statement that both expressions equal 24. Students could use a representation of multiplication, including arrays, number lines, and repeated addition to justify equivalence. The other properties naturally lend themselves to a more developed justification.

(parentheses) may confuse students who have not been exposed to these concepts. They can be removed without compromising the mathematics.

MODIFYING THE TASK We can pose fewer statements for students to consider but require that they justify their thinking for each.

MODIFYING THE TASK We can use multi-digit numbers to align with concepts taught in fourth and fifth grades.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 63

WHAT THEY DID

Student 1

Student 3

Student 1 correctly agrees with each equation. She attempts to justify the first example of the distributive property. We may be led to think she fully understands. But, we should note that her explanation is not fully complete. It does add new information (6 × 2 = 12 and 6 × 1 = 6) to make it relevant. She needs to connect those products to the product of 6 × 3.

Student 3 explains her rationale for agreeing that 6 × 3 = (6 × 2) + (6 × 1). She brings new information to the response. She states that 6 × 3 = 18, 6 × 2 = 12, and 6 × 1 = 6. She goes on to say that 6 + 12 = 18 and that 18 = 18. It is also worth noting that her computations between the second and third prompts, although not clearly connected, verify the agreement of the third prompt.

Student 2 Student 2 uses a representation to justify her agreement with 6 × 4 = 4 × 6. Her model proves that six groups of 4 is equal to four groups of 6. She maintains consistency in her models, meaning that the first factor in each identifies the number of groups while the second factor identifies the number in each group. She also adds that the products of both are 24 and her arrows show that 24 is 24.

Student 4 Student 4 defends her reasoning for the fourth prompt, which is an example of the associative property. She explains the equivalence of the factors in each expression. She shares that 2 × 2 is the same as 4 and that 7 equals 7.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1

Student 3

We can be confident that Student 1 understands the meaning of these properties. We need to work with her to develop clear, complete justifications. This is not to say that she has to write a paragraph. In fact, this type of prompt can easily be justified with equations, computations, or representations. She might write the products above each component of the equation to share her reasoning. For example, she could write 18 above 6 × 3, 12 above 6 × 2, and 6 above 6 × 1. This then shows that 18 = 12 + 6. We might not expect a student to write 18 = 18 above that equation.

Student 3 uses basic equations to justify her response, similar to what was discussed with Student 1’s work. We want to continue to provide opportunities for Student 3 to reinforce her ability to apply understanding of properties. We can encourage her to share her thinking in simple, elegant ways as she did in this prompt. In time, we will apply this understanding to larger, multi-digit numbers.

Student 2 It can be especially tricky to determine if students understand the commutative property when basic facts are used. For them, they may simply write 24 above each expression. This is a mathematically accurate and reasonable proof. When students do this, we should connect their responses with work similar to Student 2 as it justifies that both are expressions equal 24.

64   

Mine the Gap for Mathematical Understanding

Student 4 Like the other students, Student 4 shows understanding of the properties. She brings new, relevant information to the explanation. She would benefit from working with Student 3 so that she can begin to think about other ways to justify her thinking. She too needs opportunities to apply her understanding and will apply this to larger numbers in the future.

TASK 8A: Tell if you agree or disagree with each statement. Use models, numbers, or words to explain.

Student Work 1

Student Work 3

Student Work 2

Student Work 4

Chapter 3: Multiplication and Division   65

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 8B: Kim showed the commutative property on a number line. Which multiplication facts does she show?

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

Create two other examples of the commutative property with different multiplication facts. Show them on a number line. This task makes use of both sides of the same number line to show the commutative property. Students may show understanding of equal jumps on a number line. Some students may not think this is the commutative property because the jumps are on both sides of the number line rather than one (top) side. This may be an overly literal interpretation of this representation of multiplication. Some students may show a new example of the commutative property by separating the expressions, placing each expression on a separate number line. The intent of the task is to use a single number line to justify why the commutative property works. Students who make use of separate number lines likely demonstrate understanding as well. Students who are challenged to complete the task need mini-lessons designed to develop understanding of multiplication as represented on a number line. After reinforcing this understanding, we can move them to compare expressions that are examples of the commutative property.

TASK 8C: Jake said it’s easier for him to think about 5 × 7 as 5 × 5 + 5 × 2. Do you agree that 5 × 7 is the same as 5 × 5 + 5 × 2? Use models, numbers, or words to explain your thinking. This task should be given after students have had extensive work with breaking apart factors through the distributive property. Students who disagree clearly show a need for more work with decomposing factors. Some students may justify that it works using arrays or jumps on a number line. This is satisfactory, although not as efficient as students who simply state the product of 5 × 7 (35) and the partial product of 5 × 5 (25) added to the product of 5 × 2 (10) are equal. Students who rely on diagrams or drawings need more opportunities to connect diagrams with symbolic representations.

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TASK 8D: Deron and Scott are learning about the associative property. They disagree when they work with 8 × 6. zz

Deron says 8 × 6 = (2 × 2 × 2) × 6 because you have to break apart the first factor.

zz

Scott says 8 × 6 = 8 × (2 × 3) because you have to break apart the second factor.

Who do you agree with? Use pictures, numbers, or words to explain your thinking. This task is likely to show different degrees of understanding about the associative property. Students may agree with Deron or Scott from the prompt because they have a misconception that only the first or second factor can be decomposed. Other students may agree with Scott because they think a factor can only be decomposed into two smaller factors. Obviously, some students may disagree with one of the examples due to a computational error. Any of these situations are cause for additional work with the associative property, especially multiplication expressions that can be decomposed in different ways. Students who justify with representations are not incorrect but should work with connecting pictures and symbols so that they can transition to more efficient strategies.

Notes

Chapter 3: Multiplication and Division   67

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Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 9 Representing Division

91

TASK 9A How many groups of 4 are in 16 star stickers? Write a division equation that goes with the picture.

Meg has 20 star stickers to put in groups. How many groups could she make? How many star stickers would go in each group? Draw a picture and write an equation to show what Meg could do.

About the Task Our students’ understanding of division is developed through the ability to represent division. Students who recall that 16 ÷ 4 = 4 but are unable to show it with models or pictures may have considerable difficulty with multi-digit division. In this task, students identify groups and connect them with an equation before working with an open-ended prompt. The second part of the task intentionally avoids prompting for division. This is done to see if the students make use of the prompt above or if they prefer a multiplication situation.

Anticipating Student Responses The first prompt enables us to see if our students can connect representations and equations for division. Do our students make equal groups? Do our students make groups of 4? Do they find four groups? Does their equation answer the question about how many groups of 4 are in 16 (16 ÷ 4)? We may have students who know that 16 ÷ 4 is 4 and so they will circle only four stars. This is because they know the statement but not the concept. The second prompt will have different solutions. Some students will create models that don’t represent division. Others will create an accurate representation but have a disconnected equation. Some students may recognize more than one possible solution with multiple examples and equations. In the second prompt, students may connect with multiplication to find a solution for the number of stickers in a group.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 69

WHAT THEY DID

Student 1

MINING HAZARD Our understanding of mathematics may lead us to make connections that students haven’t explicitly communicated.

This student shows significant misunderstanding about division. Clearly, her quotient is incorrect. In other tasks, we may have considered that to be a computational error. Yet, in this prompt, the size of the group (four stars) is provided, and there is a model ready for grouping. However, she doesn’t group them. Instead, she crosses out all 16. We can see two groups of 10 in her subtraction equation below. But that is because as adults, we understand the division situation. The student’s work only shows an understanding of subtraction as there is no explicit mention or model for division.

Student 2 Student 2 writes an incorrect division equation. In fact, it appears that she is writing an addition equation. Both eights seem irrelevant at first. However, we can see where these ideas come from below. The student establishes that there are two groups of 8 in 16. Her model reinforces her statement, yet she does not make use of the 16 stars provided in the prompt. It is possible that she misread the prompt. It is also possible that she is unsure how to show 16 ÷ 4 because she doesn’t know the quotient, but she does know 16 ÷ 2.

USING EVIDENCE

What might we ask these students? What might we do next?

Students 1 and 2 These students’ work with the task shows that they don’t fully understand division yet. However, there are ideas that we can build on. Student 1 shows that she knows what a division equation looks like in the first part of the prompt. In the second part, she shows that there are two tens in 20, although she indicates this indirectly through subtraction. Student 2 shows that she can break 16 into two groups in the second part of the task. Her equation appears to be an attempt to communicate this as well. We want to revisit the meaning of division with both students. We may first want to confirm that they understand equal groups and multiplication. We can begin with physical models before moving to pictures and diagrams. We must be sure to write equations to accompany the models that we use and connect the numbers in those equations with models or pictures we use.

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TASK 9A: How many groups of 4 are in 16 star stickers? Write a division equation that goes with the picture provided. Meg has 20 star stickers to put in groups. How many groups could she make?

Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   71

WHAT THEY DID

MINING HAZARD We may overlook students who offer multiplication in division prompts. In fact, we may even dismiss their thinking or mark it incorrect.

Student 3 This student does well to identify that there are four groups and that there are four stars in each group. She writes an equation that aligns with her work. Her answer in the second part of the prompt is accurate. She shows four groups of 5. Her multiplication equation reverses the natural interpretation of four groups of 5. In addition, she doesn’t note how many stars are in each group, although her model or circling of stars clearly indicates four in a group. Most important, her extension indicates that she appears to prefer to use or think about multiplication when putting stars into groups.

Student 4

MINING HAZARD Sometimes we see an incorrect equation or model and assume the student doesn’t understand the skills or concepts.

Student 4 is a wonderful example of what might happen if we use closed questions and prompts to determine our students’ understanding. Student 4 writes an incorrect equation for the division situation. It is likely that she simply used the order of the numbers in the prompt to write her equation. Her model doesn’t connect with her equation. Her work in the extension is promising. She shows that there is more than one possible size of the group and all are accurate. She doesn’t provide an equation, so we need to ask her if she used multiplication or division to find her solutions.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 3 Student 3 demonstrates understanding of grouping in both representations. She can write a division equation but appears to prefer multiplication, as indicated in the work at the bottom of the task. We should be sure that she clearly understands the connection between multiplication and division. We shouldn’t compel her to use division. Instead, we should consistently connect the two operations when working with other tasks.

Student 4 We can determine next steps with this student after determining her strategy for finding her solutions in the second prompt of the task. We should redirect the conversation to compare her strategies in the second prompt with her approach in the first prompt. It may be that she needs practice writing and matching equations to converse operations as well as representations of contextual problems. It’s possible that her understanding of grouping and making groups is much more limited or disconnected from full understanding of multiplication.

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TASK 9A: How many groups of 4 are in 16 star stickers? Write a division equation that goes with the picture provided. Meg has 20 star stickers to put in groups. How many groups could she make?

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   73

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 9B: Show 28 ÷ 4 in two different ways. You can show it with an array, equal groups, repeated subtraction, jumps on a number line, or with the area of a rectangle. Like multiplication, there are different representations for division. Some students may show two partitions of equal groups or arrays (28 with four groups and 28 with seven groups). Other students may show two different repeated subtraction examples (subtracting 4 and subtracting 7). In both cases, students are relying on a specific model. We should also look for diversity of models in a class sample. In the class sample, all of the models should be represented, even though some models may be represented more frequently. If certain models are not evident in student work, we should make an effort to incorporate those missing models in upcoming lessons.

MODIFYING THE TASK 24 is a convenient number to use in mathematics problems. It has many factors. It is rather small and easy to represent. If not careful, we can rely overly on using 24 in multiplication and division situations.

TASK 9C: Smitty’s Bakery has 24 cookies to put in boxes. Each box will have the same number of cookies. How many cookies could they put in each box? Use pictures, numbers, or words to explain the amount. What is another amount of cookies they could put in each box? Use pictures, numbers, or words to explain the different amount. This task reinforces the notion that an amount can be broken into different groups of different sizes. It is open-ended in both prompts to avoid cueing student responses. Students who are unable to show one grouping option need reteaching of the concept of division. Some students may show the related grouping (24 ÷ 4 and 24 ÷ 6). Although accurate, we should investigate further to confirm that students understand that quantities can be grouped in varying ways (24 ÷ 3).

TASK 9D: Annie solved these problems: 36 ÷ 6 = 6    42 ÷ 7 = 6    60 ÷ 10 = 6 She isn’t sure if different division facts can have the same quotient. How could she prove that different division facts can have the same quotient? Some students have the misconception that two different division facts cannot have the same quotient. They know that 16 ÷ 4 = 4, so they believe that 24 ÷ 6 cannot equal 4. This may be evident of or connected to an incomplete understanding of multiplication. Clearly, these students need additional division instruction. Proficient students may justify the same quotient in different ways. Some may use a picture of each expression while others may look to connect the division fact to a related multiplication fact.

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BIG BIGIDEA IDEA BIG IDEA 10 Reasoning About Division

10 1

TASK 10A Luke believes that when you divide, you always get a quotient with a smaller number. He uses the equations below to prove his thinking. 20 ÷ 10 = 2    8 ÷ 4 = 2    12 ÷ 4 = 3    30 ÷ 6 = 5 Do you agree with Luke? Use pictures, numbers, or words to prove your thinking.

About the Task Students’ misconceptions about mathematics present many challenges. Their misconceptions are not random. Instead, they are the result of a limited understanding of concepts and flawed connections between concepts. This task highlights a misconception that dividing always yields a smaller number. This misconception can also be evident when subtracting. In these situations, students think of both operations as getting smaller as opposed to thinking about the relationship between the size of the group and the number of groups. In the past, students may have been simply asked to find a quotient rather than consider how it relates to the divisor and dividend within the equation and between other equations.

Anticipating Student Responses Some students will agree with Luke in the problem because they too hold the same misconception. They may be making assumptions based on patterns they have observed or beliefs that they have about other operations such as subtraction. Others will disagree and use fact recall with equations to disprove the situation. Some may create a representation to justify their thinking. Students with full understanding will note how the size of groups changes as the number of groups changes.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 75

WHAT THEY DID

Student 1

Student 3

Student 1 conveys his misconception that both subtraction and division “go down.” He provides an equation that can be interpreted as a contradiction or confirmation of his thinking. His example contradicts the relationship between divisor and quotient (because they are both 10). His argument may be that division goes down if we consider the relationship between the dividend and the quotient. This too is a misconception held with division and subtraction.

Student 3 correctly disagrees with the prompt. He stresses that you don’t always find quotients that are less than the divisor. He notes that you can also divide “the other way.” This insinuates the connection between divisor and dividend similar to the connection in the commutative property of multiplication. He provides an accurate example (20 ÷ 2 = 10) to support his argument. He also shares that division doesn’t always occur with one-digit, “small” numbers.

Student 2 Student 2 also agrees with the misconception. He uses representations to justify each of the equations in the example. He notes that it (the quotient) gets a smaller number every time he divides. Each of his examples justifies his conclusion because he divides with the larger number in the fact family (i.e., 32 ÷ 8 rather than 32 ÷ 4).

Student 4 Student 4 doesn’t provide a litany of evidence for his argument. Instead, he provides one counterexample. He states that 100 ÷ 1 = 100. He explains that the dividend is equal to the quotient.

USING EVIDENCE

What might we ask these students? What might we do next?

Students 1 and 2 Students 1 and 2 share the common misconception about division yielding smaller numbers. We should praise them for using representations and equations to support their arguments. It may be that they misinterpreted the prompt and related the argument to the four equations presented. This is more likely with Student 2 as he represents these equations specifically. We can work to undo these misconceptions in a few ways. First, we can provide counterexamples and ask them to use equations and representations to justify the division before comparing the divisor and quotient. We can pair these students with others who don’t hold the misconception for upcoming investigations about the idea. We should also be sure to take note of computations that appear naturally during instruction. We

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can make a subtle note of our observations about the relationship when debriefing naturally occurring problems and/or computations.

Students 3 and 4 Students 3 and 4 do not acknowledge the misconception. They provide evidence to disprove it. Student 3 cites the relationship between divisor and quotient, reversing them as evidence. Student 4 provides a counterargument with 100. Like Student 4, Student 3 goes on to say that this doesn’t apply to larger numbers either. Both are able to generalize their arguments. These students would be good partners for Students 1 and 2 during upcoming investigations related to the concept. It is also noteworthy that Student 4 does well to use division vocabulary.

TASK 10A: Luke believes that when you divide you always get a smaller number. Tell if you agree or disagree with Luke.

Student Work 1

Student Work 3

Student Work 2

Student Work 4

Chapter 3: Multiplication and Division   77

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 10B: Emma knows 60 ÷ 6 = 10. How can she use it to find 54 ÷ 6? How can she use it to find 48 ÷ 6? How can she use it to find 72 ÷ 6? We can use known multiplication facts or equations to find unknown equations. We can reason about division equations in a similar fashion. In this task, knowing 60 ÷ 6 = 10 helps us with 54 ÷ 6 because it is one group of 6 less than 60. This reasoning can be applied to 48 ÷ 6 (12 or two groups less) and even 72 ÷ 6 (12 or two groups more). Students who can’t fully describe these relationships need more work with the concepts. Additional work should occur with brief activities over a few weeks or months that focus on discussion and modeling.

TASK 10C: Oscar wrote these division facts. 50 ÷ 10 = 5 40 ÷ 10 = 4 30 ÷ 10 = 3 20 ÷ 10 = 2

50 ÷ 5 = 10 40 ÷ 5 = 8 30 ÷ 5 = 6 20 ÷ 5 = 4

What patterns do you notice in the division facts? There are different patterns in these problems. Specify that the statements are about the first two rows. Students might look for patterns horizontally. These students might note that the equations in the first row are part of a fact family. Students might look horizontally and see that the quotient doubles as the dividend is halved. Other students may look at the equations vertically and see similar relationships. Although recognizing fact families is helpful, understanding the relationship of divisors and quotients in this way of doubling and halving can be more useful, especially when working with division problems beyond basic facts. For example, we could use 420 ÷ 10 = 42 to find that 420 ÷ 5 = 84 because 5 is half of 10 and 84 is double 42.

TASK 10D: Circle the greater quotient on each line.

MINING HAZARD Students may accurately compare the expressions in rows A, C, and D for the wrong reasons. In them, they may select the expression with the greater quotient because each also has the greatest number (dividend) in the row.

78   

A.  B.  C.  D. 

72 ÷ 8 48 ÷ 8 16 ÷ 2  16 ÷ 8 40 ÷ 4 42 ÷ 7   5 ÷ 1 63 ÷ 7

Tell how you knew which quotient was greater in Problem A. Choose another problem to explain how you knew which quotient was greater. In this task, students should reason about the greater quotient rather than find the exact quotient. Justifications will vary. Justification for the first comparison may be that the size of the group is unchanged and 72 is greater than 48, so there will be more groups of 8 in 72. The second prompt lends itself to considering the relationship between the dividend and the divisor. In the third prompt, our students might know that 40 ÷ 4 is 10 and 42 ÷ 7 has to be less because 70 ÷ 7 is 10. Similarly, the last prompt yields quick recall of one quotient while the other is clearly larger.

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 11 Problem Solving With Multiplication and Division

11 1

TASK 11A Solve the problems. Use pictures, numbers, or words to justify your thinking. A restaurant needs 8 minutes to bake a pizza. How many pizzas can they bake in 32 minutes? 40 books are on 5 shelves, with each shelf holding the same number of books. How many books are on each shelf?

About the Task Story problems are one example of problem solving. They are a predominant feature in mathematics classes. Many of us may fall into the habit of featuring equal group, result unknown problem types or situations. In addition, we may not lift up both types of division (group size and number of groups). We can complicate matters even more when we guide students to look for numbers and keywords instead of making sense of the problem. In the first problem of this task, we know the size of the group (8 minutes) and need to find how many groups. In the second problem, we know how many groups (5 shelves) and need to find the size of the groups. Story problems can also be an opportunity to connect multiplication and division through context.

Anticipating Student Responses Both division and multiplication are viable for solving these problems. Students will represent these problems in different ways. Some students will include an equation with their models. Some will only use an equation. Both problems will likely be represented with equal group models. However, a linear model (groups on a number line) could be quite useful for the first problem as we can represent elapsed time on a number line. An array or area model is a natural representation for a bookshelf in the second problem. Students with incorrect answers but complete strategies show a need for computation practice. Conversely, some students may quickly write an equation because they connect the numbers in the problem with the operations but do not understand why these operations make sense.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 79

WHAT THEY DID

Student 1

MINING HAZARD Student 1 is a classic example of a student who offers correct answers but doesn’t understand the problem. Having students justify their thinking helps us determine their solution paths. We may think that the correct answer is all that matters. However, reasoning is essential

Student 1 creates equal group representations for both problems in the task. Her models evolve from representing every pizza in the first prompt to simply writing a number (5) in the second prompt. She has the correct equation in the first problem, which may lead us to think she understands the problem. However, her representation doesn’t match the equation or the problem. In the second problem, she has the correct answer with a supporting representation. Again, her representation doesn’t correctly connect with the problem. We notice she has put five books on a shelf, creating eight shelves.

Student 2 Student 2 uses think-multiplication to solve the first problem. She skip-counts by 8 and counts the number of skip-counts or groups of 8. With an isolated story problem, we may be misled to think she fully understands and makes use of multiplication. Yet, the second problem contradicts this notion. Again, she multiplies. But in this problem, she just multiplies the numbers without regard for how they are connected in the problem.

as problems become more complicated.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 Student 1 makes use of equal groups, although they are not accurate in relation to the problems. We must determine if the disconnect is from her problem-solving strategies or her understanding of multiplication and division. Our first step is to talk with her about her representations. With the first problem, we might ask, “What does your equation say?” or “How does your equation connect with the problem?” We can ask similar questions about the model she drew. We might ask, “Where are the pizzas in your drawing?” or “How did you show the minutes in your problem?” We can ask similar questions for the second prompt. Encouraging Student 1 to label her diagrams can also help her accurately represent the problem.

Students 2 and 3 Students 2 and 3 show some understanding for solving problems. They need more work with different problem types. When working with these different types, we want to reinforce modeling the problems to help them make sense of situations. Labeling the pictures, skip-counting, or making use of other models will help us see their thinking. We can then connect the representations to multiplication or division equations. We will also want them to develop metacognitive thinking while attacking problems. To do this, we can frame questions such as, “How many are there in all?” “How are they arranged or grouped?” or “What does this problem look like?” These and similar questions may make useful classroom anchor charts. After completing problems, we also want to help these students make sense of their answers.

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TASK 11A: Solve the problems. Use pictures, numbers, or words to justify your thinking. A restaurant needs 8 minutes to bake a pizza. How many pizzas can they bake in 32 minutes? 40 books are on 5 shelves with each shelf holding the same number of books. How many books are on each shelf? Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   81

WHAT THEY DID

Student 3 Student 3 counts the number of 8-minute groups. Her strategy is logical, but she doesn’t count the first group of eight. This yields an incorrect answer. She shows that she can build up toward a total when the number of groups is unknown. She attempts to multiply the numbers but notes that she doesn’t understand the problem. This may be because she hasn’t multiplied large numbers before or because she doesn’t understand how to represent the problem.

Student 4

MINING HAZARD In an effort to understand student thinking, we may discount work

This student offers ideas that we may not have expected. She shows that she makes use of the relationship between multiplication and division for solving problems. She thought of the first problem as an unknown factor problem. She counts up in groups of eight toward the 32-minute total. She then counts the groups. In her second problem, she simply divides. She does not support her second solution with a problem, although her equation is an accurate representation.

that doesn’t have pictures or words for justification. Equations, especially accurate equations, can justify thinking.

USING EVIDENCE

What might we ask these students? What might we do next?

Students 2 and 3 Students 2 and 3 show some understanding for solving problems. They need more work with different problem types. When working with these different types, we want to reinforce modeling the problems to help them make sense of situations. Labeling the pictures, skip-counting, or making use of other models will help us see their thinking. We can then connect the representations to multiplication or division equations. We will also want them to develop metacognitive thinking while attacking problems. To do this, we can frame questions such as, “How many are there in all?” “How are they arranged or grouped?” or “What does this problem look like?” These and similar questions may make useful classroom anchor charts. After completing problems, we also want to help these students make sense of their answers.

Student 4 These two problems indicate understanding of one-step, single-digit multiplication or division problem solving. It is interesting to note she uses a different strategy for two different division contexts or problem types. She uses think-multiplication when she knows the size of a group (8 minutes) and division when the number of groups is known. It may be that she simply thinks about “building up” or “counting groups” in some situations and “distributing” or “passing out” in others. There is no flaw in her approaches, nor is there cause for concern. Additional work with diverse problem types will provide better insight about her approaches.

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TASK 11A: Solve the problems. Use pictures, numbers, or words to justify your thinking. A restaurant needs 8 minutes to bake a pizza. How many pizzas can they bake in 32 minutes? 40 books are on 5 shelves with each shelf holding the same number of books. How many books are on each shelf?

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   83

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 11B: Solve the problems. Use pictures, numbers, or words to justify your thinking. 9 inches of ribbon is needed for a bow. A worker has 27 inches. How many bows can he make?

MODIFYING THE TASK Modify the factors to relate to the factor size that students are comfortable using.

Nina can practice a song 6 times in an hour. If she wants to practice the song 30 times before the recital, how many hours does she need to practice? This task is another example of both types of division situations. The problems can be solved with multiplication or division. Students who use multiplication to solve the problem should be able to explain how their approach connects to division but shouldn’t be redirected to use division. Students who struggle with computational accuracy may be supported with multiplication charts or calculators. We must be sure that representations closely match the problem and the equation. We must also be sure that there is reasoning behind the equations our students write.

TASK 11C: 6 inches of yarn are needed to make one bracelet. Lin has 27 inches of yarn. How many bracelets can Lin make? Before she begins, Lin measures again and finds she only has 25 inches of yarn. She knows that this won’t change the number of bracelets she can make. Tell why you agree or disagree with Lin. Simple one-step story problems can yield false positives about student reasoning and understanding. This happens when students simply connect numbers with the operations that they have been working with recently. In this task, students have to reason about the relationship between the total and the size of the group when the total changes. Students with more developed thinking will also note the size or amount of the remainder. Noting the remainder shouldn’t be a requirement for complete success. Some students will be convinced that the number of bracelets has to change because the total amount of yarn has changed. These students benefit greatly from group discussion, modeling of the problem, and additional work with similar problems.

TASK 11D: Write a word problem for each equation. 6 × 5 = 30 18 ÷ 6 = 3 Creating context for abstract symbols develops mathematical proficiency. In addition, our students become better problem solvers when tasked with creating problems for expressions or equations because they are forced to consider number relationships, operations, and question construction. In this task, students may create multiplication situations for both problems. Be sure that these students can explain how their problem could connect to division. We must also consider if the created division problem matches the equation. After all, there is a difference between 18 stamps in three boxes and 18 stamps in six boxes, although both equations have the included numbers. 84   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 12 Connecting Multiplication and Division

12 1

TASK 12A Write the related multiplication or division fact for each expression below.  7

× 3

 40

÷5

 2

× 9

 27

÷3

Choose one expression above. Explain how you found the related fact. Use, pictures, numbers, or words to explain your thinking.

About the Task Many of us see division as an unknown factor situation. Simply, we think multiplication when we see division. Yet, this can only happen when we understand the relationship and work with it in meaningful ways beyond memorizing fact families. In this task, students are prompted to create connected examples of multiplication and division expressions. They are then asked to justify one of the relationships. The products and quotients are intentionally omitted so that students cannot simply rearrange the numbers to find the inverse operation.

Anticipating Student Responses Students may demonstrate understanding in varying ways and complexities. Some students will represent both situations with a picture or diagram establishing equal groups and groupings. Other students will show how factors and products connect with divisors and quotients. They may provide a complete fact family for the example cited. Others may extend their argument to other examples of fact families. They may write things like “what times × equals y.” Be sure that students who rely on fact families or rearranging numbers truly understand how the two operations connect.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 85

WHAT THEY DID

Student 1

Student 3

This student looks for a pattern within the columns. Her numbers are not written at random. For the first unknown, she explains that she found the product of 7 × 3 and doubled it so she could divide by 2. Her other expressions are related to either the quotient or the product in the adjacent cell. She shows some proficiency with multiplication and division but doesn’t establish how the two are explicitly connected.

All of Student 3’s calculations are accurate. She proves that 8 × 5 = 40 by giving three different representations of 8 × 5 to prove her factoring. These representations are striking and may quickly lead us to believe she understands the connection between the operations. Similar to Student 2, she likely understands the relationships at least between basic facts.

Student 2

Student 4

This student appears to successfully connect multiplication and division expressions. She brings new, accurate information to each row. This leads us to believe she can connect multiplication and division. But does she understand why? We might ask her where or how she found the missing expressions. The prompt asks her to do that, but she communicates justification for the product of 2 × 9. She doesn’t explain how one expression is connected to the other.

This student shows clear understanding of the connection between multiplication and division. She fully explains that connection. She shares her thinking for rows 1 and 2. She uses language such as “what times five equals forty” to communicate how she found 8. She also shares how the new numbers are associated with the new expression.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 This student’s computation seems somewhat sound. The given expressions are evaluated accurately. She doubles 21 (in the first row) and 18 (third row). She has a clear understanding of multiplication and division but is unable to connect these two operations in this task. She will benefit from instructional mini-sessions that reinforce the connection. She may also benefit from matching activities (matching multiplication and division facts). After making pairs, she should be asked to explain how the two equations are connected.

Students 2 and 3 Students 2 and 3 seem to show an understanding of the relationship between multiplication and division. Their accurate calculations may lead us to think they have a better understanding of the relationship than they really do. We can simply ask how multiplication and division are related, such as the prompt in Task 12D. We may give them exact expressions and ask how the two are related. Another option is to work with

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different products or factors. For example, we might ask them what multiplication expression is related to 84 ÷ 4 = 21. We might ask them what division expression is related to 4 × 16 = 64. As students explain, we listen to identify if they just move the numbers around or if they talk about number of groups, size of groups, or how many in all. We may even need to overtly ask about those features in each equation. We might ask, “What number tells how many in all?” or “What number tells us how many are in a group?”

Student 4 Student 4 shows a developed understanding of the relationship between multiplication and division. She should be able to apply this understanding to new situations that multiply and divide with multidigit numbers. Occasional practice with this concept will prepare her to apply the idea to expressions with one or two multi-digit factors. This will position her to be successful with partial products, partial quotients, and standard algorithms. For example, she would benefit from considering how 60 × 3 is related to 180 ÷ 3.

TASK 12A: Write the related multiplication or division fact. Explain how you found the related fact.

Student Work 1

Student Work 3

Student Work 2

Student Work 4

Chapter 3: Multiplication and Division   87

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 12B: Write the missing numbers to make the statement true. 63 ÷ 7 = __________ is the same as 9 × __________ = 63. 32 ÷ 4 = __________ is the same as 8 × __________ = 32. Explain how you found the missing numbers. 42 ÷ __________ = 7 is the same as 6 × __________ = 42. 15 ÷ __________ = 3 is the same as 3 × 5 = __________. Explain how you found the missing numbers. This task is intentionally designed with complementing missing numbers (e.g., 7 is provided in the first part of the prompt and missing in the second prompt). It is designed in this way to see if students quickly see the connection and then communicate it. Do students model and diagram the situation to establish the relationship? Do other students need to rely on or even write out fact families to see the relationship between factors and divisors? Do students simply know the relationship between multiplication and division? In each case, students demonstrate understanding. However, there is a progression of sophistication in strategy and reasoning about these relationships.

TASK 12C: Kim knows 6 × 5 = 30. How can that help with 30 ÷ 5? Use pictures, numbers, or words to explain your thinking. This task provides insight into how students process the relationship between multiplication and division. As noted, some students recite fact families as the relationship between multiplication and division but are unable to extend this thinking to multi-digit number computation. Insight into their thinking, through tasks like this, helps us determine the evolution of their thinking. Consider modifying this task to make use of larger numbers. It could be rewritten as “Kim knows 6 × 50 = 300. How can that help her with 300 ÷ 50?”

TASK 12D: How could someone prove that multiplication and division are related? Use models, numbers, or words to explain your thinking. How could it be helpful to know that multiplication and division are related? This task is the most open of the four tasks. In some ways, it may be the best place to begin because it may provoke all sorts of reasoning. Models and equations are likely justifications. Take note of the second prompt in the task. It is open-ended with no “correct” answer. Yet, it serves as a window into how students think about and may leverage the relationship between multiplication and division. Be sure to look for students who relate that each operation is useful for working with the other. 88   

Mine the Gap for Mathematical Understanding

BIG IDEA 13 Representing Multi-Digit Multiplication

BIG BIGIDEA IDEA

13 1

TASK 13A Kelly used an area model to multiply 22 × 39. Draw lines on the rectangle and write numbers to show how she might have multiplied these numbers.

Tell how your model shows the product of 22 × 39.

About the Task An algorithm is not always the most efficient strategy for multiplying multi-digit numbers. In fact, it is critically important that our students understand what is happening with the mathematics before working with an algorithm. Many of us mentally decompose and recompose factors by using the distributive property. These ideas are grounded in understanding multiplication and partial products. This task represents partial products with an area model. In it, students decompose 22 × 39 in varying ways to represent and eventually enhance the efficiency of multi-digit multiplication.

Anticipating Student Responses Students may not successfully complete the task because they don’t understand multiplication. Some will use the traditional algorithm to multiply the numbers. Students who decompose will do so in diverse ways. Some may decompose by place value (22 × 39 = 20 × 30 + 2 × 30 + 30 × 20 + 9 × 30). Others may decompose by friendly numbers or benchmarks (22 × 39 = 10 × 39 + 10 × 39 + 2 × 39). There may even be some students who extend the model to create 22 × 40 before removing a group of 22. Students are not likely to proportionally decompose the area of the rectangle.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 89

WHAT THEY DID

Student 1 Student 1 breaks apart both factors. She partitions the rectangle into a familiar four-panel representation. She records one factor (39) on the outside of the area model and the other factor (22) on the inside. She multiplies the factors by place value (tens × tens and ones × ones). She then adds those two partials together. Her flawed representation leads to her error. She doesn’t multiply both place values of each factor. Instead, she multiplies tens and tens and ones and ones. This may be a simple representational error. It’s also possible that she is transferring an idea about addition (adding ones and ones and tens and tens) to multiplication.

MINING HAZARD Well-written responses may not provide the evidence that we need. Students who describe “what” they did rather than “why” they did may not have

Student 2 Student 2 is another example of a student who appears to understand concepts but calculates incorrectly. We notice that her writing conveys the procedure that she used to complete the task. She notes drawing the rectangle, “splitting” the factors, multiplying, and then adding. Her writing may be more telling than we think at first. An estimation of 22 × 39 as 20 × 40 tells us that our product will be in the neighborhood of 800. Student 2’s product is considerably off. This may be due to her processing of the problem. Her writing seems to indicate a procedural approach to computation. Her focus on procedure may inhibit her understanding of the concept.

full understanding of a concept.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 Student 1 shows some understanding of partial products and area models. We might ask her why she placed 39 outside the model and 22 inside the model. This may trigger her to recognize the error. We may also want to ask her why she multiplied 20 × 30 but did not multiply 20 × 9. This too may help her identify her flaw. It’s also possible that she won’t recognize the problem because she believes that we only multiply common place values. If so, we will need to return to multi-digit multiplication starting with two-digit by one-digit multiplication. We will need to begin with models and move to other representations, including area models and partial products equations. It’s also worth noting that Student 1 is a good example of why we want to work with estimating products, and answers in general, before computing. In this case, 22 × 39 is about 20 × 40. The product of 20 × 40 is 800, but Student 1’s product is 618. Estimation can serve as a signal that something was overlooked or done inaccurately.

Student 2 Student 2 seems to have procedural understanding for multiplying two multi-digit numbers. We shouldn’t overemphasize the miscalculation. We can provide other tasks and opportunities to be sure the error is an anomaly. We do want to talk with her about why we can break apart the numbers and put them back together. She should describe the distributive property, although she may not use the exact term. We also want to confirm that she recognizes that it (the distributive property) works regardless of the number of digits in the factors.

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TASK 13A: Partition the rectangle to model 22 x 39. Tell how your model shows the product of 22 x 39.

Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   91

WHAT THEY DID

MINING HAZARD We can be distracted by students who don’t follow directions

Student 3 Student 3 successfully breaks apart the factors by place value. She multiplies and adds the partial products. She did not partition the large rectangle but she did create her own area model. This is fine. We should note that she doesn’t explain why she added the partials.

exactly. It’s more important to look for mathematical understanding.

Student 4 Like Student 3, Student 4 creates her own area model. She finds the partial products and adds them back together. She explains that the factors are “still the same numbers” when we break them down. She notes that she can add the products back together. She too shows full understanding.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 3 As with Student 2, we should talk with Student 3 about why she added the partial products together. She should be able to note that they are added together because we broke apart the factors and now we’re putting the factors back together. Her work indicates that she understands what to do and why to do it.

Student 4 Like Student 3, Student 4 also shows complete understanding. Using area models to multiply multi-digit factors works to develop the meaning of what is happening. We will want to move toward partial product algorithms by connecting them to this model next. Using the partial product algorithm sets the stage for teaching the standard algorithm for multi-digit multiplication. That algorithm becomes more efficient as the number of digits in a factor increases beyond two or three.

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TASK 13A: Partition the rectangle to model 22 x 39. Tell how your model shows the product of 22 x 39.

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   93

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 13B: Show two different ways to multiply 54 and 28. This task is likely to yield two groups of representations. There is likely to be a group of students who multiply 54 and 28 with an algorithm and a partial products model. There may also be another group of students who multiply 54 and 28 with an area model and a partial products model. This grouping depends on exposure and understanding of multiplication and procedure. Next steps for instruction are dependent on the scope and sequence of our curriculum. Partial products representations are likely to yield more diversity in responses. In some cases, both factors will be decomposed into place value. In other cases, factors may be decomposed into friendly numbers (e.g., 54 is decomposed into 50 and 4). Students who use partial products and an area model shouldn’t be required to decompose the factors in two different ways. We should also look for students who attempt to use repeated addition or drawing equal groups. Although these strategies may be mathematically accurate, they are terribly inefficient with large factors.

TASK 13C: This is an area model of a multiplication problem. Some of the numbers are missing. X ??

30 +

?

?

?

210

?

+ 7

What could be the factors in the problem? What could be the product? How did you find the missing numbers? This task is an open-ended task that requires students to make sense of the relationship between the partial factors and partial products. Students can select any value for the missing factors. Yet, the factors they select affect the partial and final product. Some students may use a single digit in both missing factors. This doesn’t necessarily indicate misunderstanding. However, we would want to talk with them about why the original factor would be decomposed in such a way. For example, placing an 8 in the left factor creates a composite factor of 15. It would be more useful to decompose 15 into 10 and 5.

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TASK 13D: Dennis and Sorsha multiplied 28 × 17. Dennis multiplied

Sorsha multiplied

20 × 10 = 200

28 × 10 = 280

8 × 10 = 80

28 × 5 = 140

20 × 7 = 140

28 × 1 = 28

8 × 7 = 56

28 × 1 = 28

then he added

then she added

200 + 80 + 140 + 56 = 476

280 + 140 + 28 + 28 = 476

Who do you think multiplied correctly? How do you know? Both examples in this task make use of partial products accurately. Students who agree with one example, even with an accurate explanation, do not show full understanding of partial products. These students demonstrate a need for more work and discussion around decomposition and partial products. Students may agree with both examples for flawed or incomplete reasons. For example, they may agree with both examples because both equal 476. Other students may disagree with both examples because they don’t resemble recognizable algorithms. Or students who rely on an algorithm may find the product for 28 and 17 and note that both examples have the same product. A better justification also connects the partial products to the numbers embedded in the algorithm.

Notes

Chapter 3: Multiplication and Division   95

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Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 14 Reasoning About Multi-Digit Multiplication

141

TASK 14A Tell if the product of each will be greater or less than 2,000 without multiplying. 39 × 48

is greater than 2,000

is less than 2,000

12 × 20

is greater than 2,000

is less than 2,000

410 × 8

is greater than 2,000

is less than 2,000

99 × 78

is greater than 2,000

is less than 2,000

Choose one above. Tell how you knew it was greater or less than 2,000.

About the Task Considering the size of the product helps us reason about the results of multiplication. We can relate one or both factors to a friendly number or benchmark. Rounding is another strategy although procedural and not always as accurate. In this task, students estimate a product and relate the estimate to a known quantity, which in this case is 2,000.

Anticipating Student Responses Students will find their solutions in different ways. Some may find the exact product and then relate that to 2,000, although they are prompted not to multiply. This strategy falls short of the reasoning expectation intended in the task. Some students may round one or both factors and then multiply. This strategy is acceptable, although rounding may occasionally yield incorrect results. Other students will find friendly numbers or benchmarks. Even after rounding or estimating, students will need to recognize how that affects the exact product. For example, the first prompt can be estimated to exactly 2,000, but the actual product will be less because both factors are less than the rounded or estimated factors.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 97

WHAT THEY DID

Student 1 This student has found the correct solution for each prompt in the task. Her reasoning with repeated addition is accurate but incomplete. She makes the claim that 410 added eight times is more than 2,000 but doesn’t establish how she knows that the sum is greater than 2,000. More important, her reasoning is considerably inefficient. In fact, repeated addition would be quite problematic if each of the factors in the task were multi-digit.

MINING HAZARD As teachers, we read student ideas and make mathematical connections that they do not recognize themselves or

Student 2 Student 2’s writing appears accurate if we make inferences. But we can note that Student 2 is starting down the right path for justifying her reasoning about the product. She may have complete reasoning but she doesn’t fully communicate it. She talks about 39 × 48 rounding to 40 × 50. She correctly notes that this product is the benchmark for comparison. However, she doesn’t note that increasing both factors increases the product, and therefore the actual product is less than the rounded product.

connections that they haven’t fully developed.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 It may be that this student is simply more comfortable with the repeated addition representation of multiplication. It is also possible that it is her only strategy for multiplication. We must confirm that she has other strategies. We would develop those strategies if they are not understood. We also want to be sure she can multiply with multiples of 10 and 100. Reinforcing these ideas will enable her to participate in discussions about reasoning. We should also consider working with reasoning tasks that make use of smaller factors or expressions with a two-digit and a one-digit factor.

Student 2 It may be that Student 2 does fully understand the impact on products when we round both factors up. We can determine this by asking her follow-up questions such as “How is 40 × 50 related to 39 × 48?” or “How does the product change when you round both factors up?” We may even have to directly ask, “But how did you know 39 × 48 is less?” We can prompt her in similar ways for other situations to reinforce the connection assuming she does have this reasoning in place.

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TASK 14A: Tell if the product in each row is greater or less than 2,000 without multiplying.

Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   99

WHAT THEY DID

Student 3 Student 3 defends her reasoning about 12 × 20. This expression is intentionally included because it is the one that students are likely able to find the exact product. Her computation accuracy indicates that she knows how 240 is less 2,000, although she doesn’t explicitly state the comparison. We can’t be sure that she can reason about factors by rounding them or estimating them to reasonable, friendly numbers. We also notice this in her first response comparing 39 × 48 to 2,000 incorrectly.

Student 4 Student 4’s reasoning is sound. She rounds both factors. She compares the result to the benchmark of 2,000 as noted in the prompt. The strategy works well when the factors yield products clearly greater or less than a benchmark (8,000 is clearly greater than 2,000). But what might happen if she applied her strategy to the first expression, 39 × 48?

USING EVIDENCE

What might we ask these students? What might we do next?

Student 3 Student 3’s work is interesting because it shows that she knows that factors can be decomposed. This foundational understanding will be important as she works with various factors. A natural first step with her, as well as other students who prove 12 × 20, is to have her explain another expression in the task. We’ll look to see if she makes sense by rounding or estimating the factors or if she finds exact products. If the latter happens, we will want to connect the expressions to related expressions with close friendly numbers. A proactive measure for developing reasoning when multiplying is to have students estimate products before computing in situations that ask for precise answers. After doing so, we can have conversations that compare their estimates and exact answers. Multiple exposures and experiences as well as conversation will develop their ability to identify reasonable answers.

Student 4 Again, Student 4’s strategy and reasoning are sound. We might ask her if both factors in 99 × 78 needed rounding once she established that 99 rounded to 100. She, like Student 2, wil benefit from additional practice opportunities. It is important for her to explore diverse situations so that she can refine her approaches and recognize what situations require which strategies. She, like all students, will develop her proficiency by exchanging ideas and strategies through discussion with the class. We may extend her situations to include three- or four-digit factors or factors that yield products much closer to the benchmark comparison.

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Mine the Gap for Mathematical Understanding

TASK 14A: Tell if the product in each row is greater or less than 2,000 without multiplying.

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   101

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 14B: How does 30 × 30 = 900 help you solve 29 × 30? How does 614 × 10 = 6,140 help you solve 614 × 5? These two prompts ask students to rely on their foundational understanding of multiplication. Some students will find their products with an algorithm, partial products, or area model. These strategies don’t illuminate student understanding about the size or number of groups that the first prompt promotes. In the second prompt, we can see that one factor (5) is exactly half of the other (10). This means that the size of the product will be exactly half as well. Students who demonstrate understanding of these multiplication concepts with smaller factors should continue to work with larger factors that explicitly model similar conditions. Other students who rely on computation need additional work discussing and examining reasoning strategies.

MODIFYING THE TASK

TASK 14C: Circle yes if the product is between 4,000 and 6,000. Circle no if it is not. 8 × 703

yes

no

of how tasks can be

510 × 9

yes

no

modified according

81 × 73

yes

no

students. It includes

62 × 60

yes

no

three-digit factors

48 × 74

yes

no

350 × 49 yes

no

Task 14C is an example

to the needs of our

which can be changed as needed.

Choose TWO from above. Explain how you know that the product is or is not between 4,000 and 6,000. TASK 14D: Which of these have a product that is about 240? 38 × 6   is about 240   is not about 240 50 × 9   is about 240   is not about 240 3 × 78   is about 240   is not about 240 30 × 81   is about 240   is not about 240 22 × 11   is about 240   is not about 240 43 × 5   is about 240   is not about 240 Choose TWO from above. Explain how you know that the product is about or not about 240. 102   

Mine the Gap for Mathematical Understanding

The prompts in Tasks 14C and 14D allude to estimating or rounding of factors. Again, some students will complete the multiplication to consider the size of the product. In all situations, friendly numbers can be used for an accurate answer. For example, the first prompt in Task 14C features 8 × 703, which may be thought of as 8 × 700. That product is between 4,000 and 6,000. But if rounded, the expression becomes 10 × 700, which is outside of the range. In another line, 62 × 60 creates a product less than the 4,000 to 6,000 range. The expression may be rounded to the nearest hundred. In doing so, the result (100 × 100) is greater than the range.

Notes

Chapter 3: Multiplication and Division   103

BIG IDEA

15

BIG IDEA 15 Representing Multi-Digit Division TASK 15A Alexis is planning a bike tour. She is planning a 357-mile trip in 7 days. She is planning to bike the same number of miles each day. She wants to know how many miles she will bike each day. She sketched the model below. 357 miles

Tell why you think her model makes sense. How many miles will she bike each day? ___________

MODIFYING THE TASK The task intentionally makes use of a three-digit dividend and friendly divisor. The numbers in the problem can be altered to align with any classroom. The part-to-whole or bar diagram will need to be removed for large divisors.

About the Task Our students need to understand multi-digit division before making use of the standard long division algorithm. This task and the related tasks make use of different representations of division situations. In this task, students are prompted to make sense of a part (7 days represented with boxes) to whole (357 miles) model. This model is similar to a bar diagram.

Anticipating Student Responses We can anticipate students who understand the problem and students who do not. The latter may attempt the problem by drawing groups or finding a similar division situation. They may have been successful with smaller numbers, but that may not have indicated full understanding of division. We should be sure to look for students who find the solution but cannot represent or explain how they found it. This is indicative of a procedural understanding that is useful but

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 104

may lead to challenges with problem solving or application situations. Successful students may build groups by divvying 10 miles at a time followed by a group of 1. Others may give out larger amounts at a time, such as 50. Students with advanced number sense may recognize 35 tens in 357 and quickly recognize the quotient.

Notes

Chapter 3: Multiplication and Division   105

WHAT THEY DID

Student 1

MINING HAZARD We must review apparent calculation errors carefully. Sometimes they are just that. In other cases, they may be a misunderstanding of the procedure being applied, as is the case with Student 1.

MINING TIP We must be careful to avoid discounting student responses

Often, we recognize incorrect answers as simple miscalculations. Student 1 is a good example of this. He acknowledges equal groups. He identifies 61 miles for each day. It seems likely that he miscalculated because his result of 61 miles per day is similar to the actual answer of 51 miles per day. Below, we see his partial quotients algorithm where he repeatedly subtracts 20 groups of 7. His miscalculation is that he finds 20 groups of 7 to equal 114 instead of 140. His subtraction leads him to 15. There he finds that there is one group of 7 in 15. There is no subtraction to show that there are 18 left over. Are we sure that his 61 is the sum of partials along the right side of his algorithm? Is his 61 the sum of the 20s and the 1 at the bottom of the algorithm? If not, how does he account for that extra 1?

Student 2 Student 2 communicates understanding that the rectangles in the bar diagram represent the days biking. Placing 51 in each box is mathematically correct. However, we can see that he adds 51 each time and records the “running total” in the rectangles. He explains that each rectangle represents one day. Interestingly, he says that it (the amount) doubles each time, which is incorrect. Student 2 states that Alexis bikes 51 miles a day but doesn’t fully justify how he found the amount.

when flawed or incorrect statements are included with mathematically correct ideas. Student 2 uses the “double” incorrectly but does understand what the problem is asking and how the representation connects to it.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 1 Student 1 seems to show understanding of the partial quotients division algorithm. He connects it with the bar diagram included in the prompt. We need to ask him about the 1 in the lower part of the algorithm. This will help us determine his proficiency with the algorithm as well as his ability to recognize remainders. It’s possible that his incorrect product (114) for 20 × 7 is a simple miscalculation. He will benefit from using tools to confirm his accuracy. He could use a multiplication chart to support accuracy during the problem or he could use a calculator after he completes the problem.

Student 2 We should talk with Student 2 to confirm that he understands the rectangle represents 51 and that he is simply adding 51 to the total. We need to ask because we can’t be sure how he found the solution of 51. There is no computational record. It’s possible that he was able to determine 357 ÷ 7 = 51 mentally. Knowing our students helps in these situations. We have to be careful to avoid unnecessarily requiring students with advanced mental mathematics skills to consistently prove their calculations.

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Mine the Gap for Mathematical Understanding

TASK 15A: Alexis is planning a bike tour. She is planning a 357-mile trip in 7 days. She is planning to bike the same number of miles each day. She wants to know how many miles she will bike each day. She sketched the model provided. Tell why you think her model makes sense. How many miles will she bike each day? Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   107

WHAT THEY DID

Student 3 Student 3 explains that multiplication via repeated addition is a strategy for finding the number of miles biked each day. It’s possible that, like Student 2, he used his mental mathematics skills in this think-multiplication situation. We should also note that there is a partial quotients algorithm in the upper right-hand corner that he has attempted to erase.

Student 4 Student 4 finds the correct solution by making use of the bar diagram. He notes that the diagram allows you to try different equations. He says that he started by giving 50 to each rectangle, arriving at a total of 350. He then added 1 to each rectangle. Below, he has a partial quotients algorithm and area model to justify his calculations.

USING EVIDENCE

What might we ask these students? What might we do next?

Students 3 and 4 Both Student 3 and Student 4 show understanding of the model provided in the prompt. They also demonstrate skill with the partial quotients algorithm and possibly capability with mental mathematics. Student 3’s area model conveys a connection with multiplication as Student 4’s writing does. Both students will benefit from more opportunities with division problems. We can begin to remove specific representational prompts. Yet, when debriefing those tasks, we want to be sure to connect different models with the problem as well as division and multiplication equations. It will also be interesting to see how these students perform with computational situations that may be less obvious than 357 ÷ 7.

108   

Mine the Gap for Mathematical Understanding

TASK 15A: Alexis is planning a bike tour. She is planning a 357-mile trip in 7 days. She is planning to bike the same number of miles each day. She wants to know how many miles she will bike each day. She sketched the model provided. Tell why you think her model makes sense. How many miles will she bike each day? Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   109

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 15B: The school cafeteria needs to be set up for a concert. There are 450 chairs to place in 6 long rows. How many chairs will be placed in each row? ? 450 chairs

6 rows

Use the map above to find the number of chairs in each row. This task makes use of an array/area model with a missing dimension. Some students will represent every chair in the array. Other students will subdivide the rectangle, noting the number of chairs in each subdivision. Look for students who complete the division problem procedurally or make use of a different representation such as repeated subtraction. These students will benefit from additional work with similar problems and the opportunity to connect representations. Some students will make use of the relationship between multiplication and division. They are likely to think multiplication and build up to the dividend.

TASK 15C: Use an array, area model, or partial quotients to solve 546 ÷ 6 and 790 ÷ 80. Task 15C is a straightforward prompt to discover which model students prefer for dividing with multi-digit numbers. Understanding the models they understand well provides us with useful information. First, it may help us understand models or strategies that need further refinement. It also provides us with an anchor for developing new representations or strategies as well as procedures and algorithms. We can look for common representations in a student’s work or the class as a whole.

110   

Mine the Gap for Mathematical Understanding

TASK 15D: A worker is checking to see how many boxes are needed. She knows a machine packs 420 candy bars. It puts 6 candy bars in a box. She thinks 70 boxes are needed. The picture shows how she found her answer.

)

6 420 - 300 120 - 120 0

6 × 50 6 × 20 70

Explain what she did. Another machine packs 420 granola bars. It packs 5 granola bars in a box. How many boxes will it need? Use the worker’s strategy to find how many boxes are needed for granola bars. This task models a partial quotient strategy. This strategy is a bridge to working with the long division algorithm. It also supports development of mental division. The second portion prompts students to make use of the model in a new situation. Successful students may take away larger groups similar to the example. Other students may take smaller groups away. In this task, we’re looking for students who understand what is happening even though they may make minor computational errors.

Notes

Chapter 3: Multiplication and Division   111

112   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 16 Reasoning About Multi-Digit Division

16 1

TASK 16A Estimate each quotient. Tell how you found your estimate. 300 ÷ 49 is about ___________ 297 ÷ 15 is about ___________ 681 ÷ 94 is about ___________

About the Task Similar to multi-digit multiplication, reasoning about multi-digit division supports the development of algorithms as well as the reasonableness of our solutions. In this task, students can use benchmarks to make the division “more friendly.” For example, we might think of 300 ÷ 49 as 300 ÷ 50. This tells us the actual quotient is about 6. We can think of the quotient of 297 ÷ 15 as something close to 300 ÷ 15, or 20.

MINING HAZARD The intent of these reasoning types of tasks is to examine

Anticipating Student Responses

student reasoning.

As always, some students will rely on exact computations. They will find a quotient and ignore the remainder to find their “about” situation. This may be most evident in the problem 297 ÷ 15. These students may find the quotient 19 with a remainder of 12. Their answer for the task will be “about 19.” Even students who consider sensible friendly numbers may divide in different ways. Again, consider 297 ÷ 15 as 300 ÷ 15. Some students may quickly recognize that there are two 15s in 30 so the solution is about 20 tens or 200. Other students may think about double 15 to consider a related problem 300 ÷ 30. This quotient is 10. These students then double the quotient because they doubled the original divisor (15).

manipulating exact

Finding and answers does not meet the intention. It provides insight into their procedural fluency but doesn’t help us determine their number sense or reasoning about the division situation.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 113

WHAT THEY DID

Student 1 Student 1’s work is quite interesting. Her estimated quotients are reasonable. It appears that she relies on rounding instead of estimating or making friendly numbers. Evidence of this is in the second prompt where she rounded 15 to the nearest 10 when 300 ÷ 15 could have been a more friendly computation. We also see that she used paper/pencil computation to find her quotients of those friendly numbers, possibly indicating that she is hesitant to rely on mental computation.

Student 2

MINING HAZARD Friendly numbers are not always benchmark numbers (multiples of 5, 10, or 100). In this example, 90 is a

This student’s work reveals some interesting perspectives, even contradictions. In the first prompt, she likely misread 300 as 200. She then divided 200 by 50. She noted that she had to add the zeroes, resulting in a quotient much larger than the original dividend. Her second response shows reasoning about friendly numbers or estimates. Her third prompt shows even more sophisticated thinking. Instead of relating the problem 681 ÷ 94 to 700 ÷ 100, she relates it to 720 ÷ 90 because she knows that 72 ÷ 9 = 8.

friendly number in the context of a dividend 681 that is close to a

USING EVIDENCE

multiple of 90 (720).

What might we ask these students? What might we do next?

Student 1

MINING HAZARD Students confuse rounding with estimation. These are two different ideas. For example, 25 rounds to 0 (nearest hundred) or 30 (nearest 10). But 25 itself is a useful benchmark number. Depending on the calculation (i.e., 138 ÷ 25), we may want to think of it as 20. In other calculations (i.e., 241 ÷ 25), we would want to think of

Student 1 will benefit from separating the procedure of rounding from the reasoning of estimation. Activities and discussion that highlight estimation strategies will reinforce this difference. She finds estimated quotients with computational procedures (long division) even when the computation is fairly basic (700 ÷ 100). We have work to do to develop mental computation strategies. We can do this with frequent mental mathematics routines that highlight friendly computations. It is also helpful to have students estimate answers before doing precise calculations and to make sense of the difference between estimated and actual results.

Student 2 It may be that Student 2’s first response is a simple oversight. It may be more. This is why knowing our students plays a critical role in making decisions about their written responses. Let’s assume she demonstrates inconsistency with multiplication and division of numbers that are multiples of 10 and 100. In this case, we would create equation strings with changing factors and discuss the pattern. For example, we might offer the string 13 × 2 = 26, 13 × 20 = 260, 13 × 200 = 2,600, 13 × 2 = 26, 130 × 2 = 260, 1,300 × 2 = 2,600, and so on. In this example, we are working to develop her understanding of the pattern and impact of increasing factors. We would move from multiplication to division before mixing the equations.

it as 24.

114   

Mine the Gap for Mathematical Understanding

TASK 16A: Estimate each quotient. Tell how you found your estimate. 300 ÷ 49 is about ___________ 297 ÷ 15 is about ___________ 681 ÷ 94 is about ___________

Student Work 1

Student Work 2

Chapter 3: Multiplication and Division   115

WHAT THEY DID

Student 3 Student 3 rounds the dividend and divisor in all three prompts. In the second prompt, she gives an estimated product of 20. It is correct but the reasoning is wrong. In that prompt, she rounds 297 to 300 and 15 to 20, yielding 300 ÷ 15. She then uses 20 × 20, likely because it is a product she knows, and she explains that is close to the dividend of 300. The thinking is similar, although less precise, in the third prompt. In that prompt, she describes how she estimates the product of 94 and different numbers (10, 9, 8, and 7) and compares them to the dividend.

Student 4 Student 4 thinks about mental computation in a different fashion. She reasons about the quotient by multiplying by the divisor instead of estimating the dividend and/ or divisor to create a friendly division expression. Her estimation in the first prompt is reasonable, although she communicates a strategy that is flawed or incomplete. However, if we look more closely at the second and third prompts, we discover this reasoning about the quotient. In the second prompt, she states that she knows 15 × 10 is only 150. She doubles it to make 15 × 20 and notes that product of 300 is only 3 from the dividend of 297. In the third prompt, she rounds 94 to 90. She uses multiplication to get close to the dividend. She then states that 9 × 70 is 630. Here she clearly loses track of the place values being multiplied.

USING EVIDENCE

What might we ask these students? What might we do next?

Student 3 Student 3 is a perfect example of reasons why students need opportunities to estimate expressions and share their thinking with others. In the third prompt, she does this again rounding 94 to 90 when it may be more effective rounding it to 100. Her strategy in the third prompt, rounding the divisor and multiplying to get close to the dividend, is a sound approach for mental computation. Moreover, multiple exposures and discussion help students develop diverse strategies as well as the different situations in which they are useful. Her work in the first prompt reminds us that we also want to be sure that Student 3 works to develop and communicate complete reasoning. This too can be improved through discussion and our questioning.

Student 4 Student 4 uses multiplication to solve division problems in all three prompts. She also seems to round numbers relative to the greatest place value they have (49 to the nearest 10, 297 to the nearest 100). Her strategy for estimating products and comparing them to dividends is unique compared to the work of others. It is a viable strategy that others may acquire and apply if they are exposed to it. Her work shows that she has some good ideas for reasoning about division. Like other students, she needs opportunities to compute mentally, record her thinking, and discuss it with others. During discussion, we should be sure to lift up the ideas of estimating numbers to close benchmarks versus rounding to a specific place value. We should also lift up unique ideas like estimating or rounding the divisor even if the dividend is unfriendly. 116   

Mine the Gap for Mathematical Understanding

TASK 16A: Estimate each quotient. Tell how you found your estimate. 300 ÷ 49 is about ___________ 297 ÷ 15 is about ___________ 681 ÷ 94 is about ___________

Student Work 3

Student Work 4

Chapter 3: Multiplication and Division   117

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 16B: Tell how you know the quotient of each is greater or less than 50. 96 ÷ 2   452 ÷ 8   563 ÷ 11   810 ÷ 15 This task draws our students’ attention to a specific benchmark, 50. Their reasoning for each situation will differ, although there are some likely approaches. For example, 96 ÷ 2 must be less than 50 because 100 ÷ 50 is 2, so dividing less by 2 will result in something less than 50. Others may reason that there are two groups of 50 in a hundred to give them a sense of how many fifties might be in the dividend. Our strongest “number crunchers” may struggle with these sorts of tasks. It is important to recognize their procedural fluency but develop their reasoning. Intentional tasks with group discussion around these and similar problems will lead to refined reasoning. We do need to keep in mind that this takes time, and one or two problems on a consistent basis outweigh many problems in a day or two of specific instruction.

TASK 16C: Emily knows that 56 ÷ 8 = 7. Tell how she can use it to solve these problems. 560 ÷ 8 560 ÷ 80 568 ÷ 8 Jake knows that 7 × 9 = 63. Tell how he can use it to solve these problems. 630 ÷ 70 630 ÷ 7 623 ÷ 7 These prompts make use of multiplication and division with multiples of 10. Look for students who simply “add zeros.” These students may not process that 560 ÷ 8 results in 7 tens because they see the 560 ÷ 8 as 7 and “add a zero.” Students who only “add” or “subtract zeros” are likely to have difficulty with related problems that feature one less or one more group of 8 (568 ÷ 8). The second prompt provides added rigor as students need to identify the relationship between multiplication and division. In both cases, students who just manipulate zeros should revisit work with multiplying and dividing these large numbers using base ten blocks and other representations.

118   

Mine the Gap for Mathematical Understanding

TASK 16D: Tell if the quotient in each row is greater than 25. 256 ÷ 25

quotient is greater than 25

quotient is less than 25

1,320 ÷ 130

quotient is greater than 25

quotient is less than 25

899 ÷ 91

quotient is greater than 25

quotient is less than 25

1,508 ÷ 29

quotient is greater than 25

quotient is less than 25

1,627 ÷ 410

quotient is greater than 25

quotient is less than 25

Choose one of the problems. Use models, numbers, or words to explain how you knew the quotient was greater or less than 25. Similar to Task 16B, this task makes use of an important benchmark, 25. In the first three situations, the quotient is about 10 because we can find a friendly dividend or divisor. The last two situations can also make use of friendlier numbers. 1,508 ÷ 29 might be thought of as 1,500 ÷ 30, while 1,627 ÷ 410 might be thought of as 1,600 ÷ 400. In this and other reasoning tasks, it is important that we remember there are diverse strategies that our students might use. We should honor and accept all while encouraging less efficient strategies to evolve.

Notes

Chapter 3: Multiplication and Division   119

CHAPTER 

4

FOUNDATIONAL FRACTION CONCEPTS THIS CHAPTER HIGHLIGHTS HIGH-QUALITY TASKS FOR THE FOLLOWING: zz

Big Idea 17: Representing Fractions Fractions are much more than brownies, cakes, and pizzas. Fractions can be represented in many ways. Fractions are not dictated by the number of shaded pieces. Fractions are the relationship between a specified amount and the number of same-sized pieces in the whole.

zz

Big Idea 18: Connecting Representations of Fractions Fractions can be represented in many ways. Deep understanding of fractions helps us make connections between these representations.

zz

Big Idea 19: Fractions on a Number Line Fractions are numbers. They can be represented on number lines. The number line shows fractional intervals. Placement on the number line is relative to the endpoints on the number line.

zz

Big Idea 20: Fractions Greater Than 1 on a Number Line Fractions greater than 1 are similar to fractions less than 1. We can represent them in the same ways, including number lines. We can reason about them and decompose them into their whole-number and fraction components.

zz

Big Idea 21: Decomposing Fractions Fractions can be decomposed much like whole numbers. We can decompose a fraction into a sum of unit fractions. But, we can decompose fractions in other ways as well. Decomposing fractions is useful for comparing and computing with fractions.

zz

Big Idea 22: Equivalent Fractions on a Number Line There are many approaches to justifying that fractions are equivalent. Two fractions are equivalent when they occupy the same location on a number line.

zz

Big Idea 23: Comparing Fractions There are many strategies for comparing fractions. Procedural approaches, including common denominators and cross-multiplication, may be performed without understanding. These strategies may also be inefficient. Other strategies for comparing fractions include the use of benchmarks, reasoning about the size of the fractional pieces, distance from 1, common numerators, and more.

zz

Big Idea 24: Reasoning About Fractions When we deeply understand fractions, we can reason about them to make comparison, problem solving, or computation more efficient. Often, relating fractions to benchmarks such as zero, one-half, or one whole is a useful way to reason about them.

zz

Big Idea 25: More Reasoning About Fractions A fraction relates to a specific whole. The fractional value of a quantity changes as the size of the whole changes. For example, a can of soda is 1 of a six-pack, 1 of a twelve-pack, and 1 24

120   

6

12

of a case. The soda can has not changed, but the size of the whole has.

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA

171

BIG IDEA 17 Representing Fractions TASK 17A

Circle the figures that show

1 . 4

Tell how you know if each of the figures below show

1 . 4

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 121

MINING HAZARD Overuse of a model may lead us to think students understand a concept. Yet, they may only be familiar with the representation or model and cannot transfer their understanding to new

About the Task Representing fractions with models lays the foundation for understanding fractions as numbers and quantities. Traditionally, we have shaded a portion of a figure to identify the fraction. Overuse of this approach may lead to false indication of student understanding. Full understanding of fraction concepts includes knowledge that 1 4

can describe the shaded part of a square but

3 4

can describe the unshaded part.

This task makes use of shaded and unshaded representations of fractions as well as diverse figures and partitions. The extension of the task provides insight into our students’ ability to transfer their understanding of fractions to different situations and unconventional models.

situations, models, or contexts.

Anticipating Student Responses Students are likely to identify the far-right square in the first row because it is a traditional representation. Some students may also identify the middle square of the second row because of the one shaded part, although it is not partitioned equally. The figures on either end of the bottom row will challenge students’ reasoning. Many students just beginning their work with fractions will not identify either representation of the prompt’s extension. Students with more refined understanding of fractions will identify that both are examples of 1 . They may reason that the first figure 4 can be justified by extending the left diagonal. The second figure can be justified by noting two rows of 1 . 4

Notes

122   

Mine the Gap for Mathematical Understanding

Chapter  4: Foundational Fraction Concepts   123

WHAT THEY DID

Student 1 This student correctly identifies one model. We can see that he considers the number of shaded parts compared to the number of parts and he is only looking for one of four parts shaded. He is simply counting the parts rather than thinking about their relationship to the whole. He shows this by selecting the middle square of the bottom row, which is not partitioned into equal parts. His ideas at the bottom reinforce our suspicions. In both models, he notes that he is counting the number of pieces in the whole or the number of pieces shaded. We should ask him about identification of the partial circle. It may be that he notes 41 is missing from the model.

Student 2 The models that Student 2 identifies are correct. Interestingly, he identifies a 1 4 unshaded example and overlooks a similar model directly to the left of it. Like many students, he also discounts the lower left-hand square. He likely sees it as 4 rather 16 than 41 because he hasn’t yet worked with equivalent fractions. His reasoning in the extension clearly notes that he is counting pieces without considering their relationship to the whole.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1

MINING TIP Providing advanced representations provides insight into students’ thinking. This task can be given to students who haven’t worked with equivalent fractions yet. Those who successfully identify the

  4 16

model

indicate deeper understanding of the concept, which may predict their success with equivalency.

124   

We have work to do with Student 1. He shows that he is counting pieces and only considers the shaded portion. It would be interesting to extend the diagonal in the first square of the extension. Would he then see 1 ? It would also be interesting to 4 compare the top-right square he correctly identified with the bottom square of the extension after rotating it 90 degrees. Before working with complicated models, we should focus on equal partitions of diverse shapes and discuss the relationship of the pieces to the whole, both shaded and unshaded.

Student 2 Student 2 considers the size of the pieces when identifying fractions. He also seems to recognize that 41 can identify the shaded or unshaded part of a model. It is worth noting that he identifies the triangle as 41 unshaded but not the adjacent square that is 41 unshaded. It may be an oversight or that he doesn’t have full understanding. We can ask Student 2 to compare the first and second models of the first row. We might ask, “How are these different?” or “How are they the same?” We might also 1 ask, “How do you know that this (triangle) shows 4 but the square doesn’t?” We can assume he hasn’t worked with equivalent fractions yet so he doesn’t identify the other models.

Mine the Gap for Mathematical Understanding

TASK 17A: Circle the figures that show 14 . Tell how you know if each of the 1 4

figures show .

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   125

WHAT THEY DID

Student 3 Student 3 identifies each model in the top row correctly. He may dismiss the partial circle for a number of reasons, the most likely being that there is no part shaded. 4 The 16 square is a reasonable oversight. His reasoning in the extension prompts shows evidence of counting parts.

Student 4

MINING HAZARD As teachers, we have to be careful to consider the entire body of work to confirm understanding. This student’s reasoning seems to show a more complex strategy than

Student 4 identifies four of the representations for 41 . He considers more than the amount shaded to identify a fraction. His reasoning about the first extension representation seems to indicate that he considers more than just the number of pieces as he notes that one part “is shaded and the others are not.” Moreover, he doesn’t mention that there are “3” unshaded parts. It is more likely that this is a coincidence when we consider that he didn’t identify the 4 model from the first prompt 16 and he mentions that the 2 model doesn’t show 1 “because there are eight fractions 8 4 (pieces).”

counting. But other evidence (the

  4 16

and

2 8

models) indicate that he does simply count pieces.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 We first want to understand why Student 3 disregarded the partial circle. We want to be sure that the lack of shading didn’t influence his thinking. It can be argued that this model doesn’t show 41 . He has a foundation for fractions. Like Student 1, we may ask him to reconsider extending the diagonal of the square in the extension. After doing so, we can discuss the relationship between the size of the part to the number of parts in the whole. His foundational understanding can also be applied to the second square in the extension.

Student 4 Like Students 2 and 3, Student 4 has foundational understanding that we can build on. Like the others, we want to reinforce the concept that the fractional pieces are the same size. He doesn’t necessarily count shaded and unshaded pieces but may be counting pieces in all. It makes sense that Student 4, like the others, doesn’t consider the 4 and 2 models because their ideas about fractions are still develop16

8

ing. We can shift to these models and similar representations after we are sure that students understand the meaning of fractions. It may be quite helpful to students if we enlarge these models and encourage students to cut them apart to compare the shaded and unshaded portions.

126   

Mine the Gap for Mathematical Understanding

TASK 17A: Circle the figures that show 14 . Tell how you know if each of the 1 4

figures show .

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   127

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

TASK 17B: Show

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

3 4

in three different ways. Tell how each model could look different but still show 34 .

This open-ended task allows students to create different representations of 43 . Students are likely to create area or regional models. Look for students who create the same model and simply shade different parts in each representation. Encourage those students to use different shapes or figures for each model. Other students may show 41 with parts of a set, on a number line, or through authentic situations such

as one of four pets is a dog. Their justification should explain the meaning of 43 and that it can be represented in many different ways so long as three of four equal parts are identified. It is wise to have a group discussion about the varied representations and how they are all similar and different. This will help students make meaning of the concept rather than associate it with a picture or model.

TASK 17C: Carlos made a design with triangles and shaded three of the triangles.

Carlos says his design shows

3 8

. Amy says the design shows

5 8

.

Who do you agree with? Explain your thinking. This task entertains the notion of recognizing shaded parts as the only representation of a fraction. In this task, both Carlos and Amy are correct. Carlos is speaking to the shaded portion of the model while Amy is speaking to the unshaded portion. Student writing should specifically note eight equal parts and the number of parts identified as shaded or unshaded.

128   

Mine the Gap for Mathematical Understanding

TASK 17D: Chonda shaded 4 different pieces of paper. This is what they looked like. A

B

Which of her papers show

C

1 4

D

?

Explain your thinking. Draw two different ways she could have shaded her paper to show

1 4

.

This task addresses the notion of shaded and unshaded parts as well as equal partitions. Some students will only identify Paper A because it shows one fourth shaded. Some will note Papers A and B, although Paper B does not have equal parts. Some will note all four papers, recognizing that all have four parts with one unique part. In the extension, look for new student models to look different than the models in the prompt. Students might use circles or triangles. Others might subdivide the rectangles above to show more parts or divide them horizontally rather than vertically.

Notes

Chapter  4: Foundational Fraction Concepts   129

BIG IDEA

18

BIG IDEA 18 Connecting Representations of Fractions TASK 18A Shade the rectangles to show the fraction in the box. Then, put the fraction on the number line.

2 4

1 4

3 4

3 8

1 3

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 130

About the Task For many of us, pizzas, pies, brownies, and cakes were the premiere models for fraction concepts. Today, we know that our students must understand these models but also understand that fractions are numbers that can be represented on number lines. Representing fractions on number lines establishes this foundational understanding that can be leveraged for comparisons and computations. In this task, students connect area of region representations of fractions with representations of fractions on the number line.

MODIFYING THE TASK We can change the fractions in this task, including use of fractions greater than 1. We can modify the regions that we ask students to partition or

Anticipating Student Responses

leave the task open so

Experience tells us that students are likely to show equal parts of a rectangle more easily than that same fraction on a number line. Some students may partition the fractions with fourths differently because one is 41 and the other is 42 . Some students may understand the relationship between fourths and eighths so they make use of fourths partitioning above, essentially splitting the partitions in half. Partitions of models, especially number lines, should be close but may not be exact. We should accept reasonable partitions. However, placement should be justified by establishing other partitions.

the region model of

that they can choose their choice.

Notes

Chapter  4: Foundational Fraction Concepts   131

WHAT THEY DID

Student 1 Interestingly, Student 1 partitions the area model well when working with fourths or halves. His partition of 31 is inaccurate although similar to how we partition circles into thirds. It may be that he has only worked with thirds represented by circles. He partitions the number line accurately in all five models. He shades the interval to show the amount. He doesn’t label the tick marks to indicate the interval.

MINING HAZARD We may tend to think that partitioning on a number line must occur with endpoints of 0 and 1. This isn’t so. Instead, we can place

Student 2 Student 2 shows an interesting contradiction. He partitions the area models with equivalent partitions. His partitioning of the number line is not equally partitioned. Instead, he appears to simply count tick marks along the number line. His placement of 31 could be considered accurate as the space between 0 and 33 is equally partitioned. However, he doesn’t establish where 0 is on the number line.

0 and 1 anywhere on the number line. The partitions must remain equal, although their placement shifts to the

USING EVIDENCE

What would we want to ask these students? What might we do next?

left as the endpoint does too.

Student 1 Student 1 demonstrates good understanding of partitioning number lines. His partitioning of area or region models is accurate with fourths. We should have him 1 explain his model for 3 . We can ask him to prove how he knows it shows 31 . It’s likely a misconception about models for 31 . It could be that he hasn’t fully developed the concept that the fractional parts must be equal. We could have him cut out his model and the pieces to compare their sizes if he doesn’t see that they are unequal.

Student 2 Student 2 has some basic understanding of partitioning. He clearly has difficulty establishing these fractions on the number line. This may be even more challenging because he doesn’t show where 0 and 1 are on the number line so it is hard for him to think about the partitions. We can use number lines with endpoints separated by the distance of one whole fraction tile. We can then use fraction tiles to establish partitions as well as the tick marks that signify those distances. This will also help him connect representations.

132   

Mine the Gap for Mathematical Understanding

TASK 18A: Shade the rectangles to show the fraction in the box. Then, put the fraction on the number line.

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   133

WHAT THEY DID

Student 3

MINING HAZARD

While Student 3’s area models are accurate, his placement of fractions on the number line is problematic. He doesn’t establish where 0 and 1 are on the number line. He places the fractions but does not relate their placement to partitioned intervals within the space between 0 and 1. It may be noteworthy that his placement of 41 is 2 not half of 4 . This disconnect may signify that he does not relate the two fractions or understand the relationship of fractions with like denominators.

Students placing

Student 4

fractions reasonably

Student 4’s shading of the area models is interesting because he doesn’t shade consecutive pieces in both 42 and 83 . His placement of the fractions on the number line is reasonable. He doesn’t provide justification of their placement, although the placements shouldn’t be considered random. He doesn’t show endpoints or partitions.

accurate on a number line does not necessarily show full understanding without endpoints and partitions to show placement.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Students 3 and 4 Student 3 and 4 place the fractions on the number line differently. Yet, neither student provides endpoints or partitions to justify the placement. Like Student 2, these students need work developing understanding of fractions as numbers on a number line. They too can use fraction tiles. We can use sentence strips as another model for fractions on number lines. To do this, they can fold sentence strips using the folds to show partitions. One side of the sentence strip can model a number line while the other side represents a partitioned rectangle to connect the representations.

134   

Mine the Gap for Mathematical Understanding

TASK 18A: Shade the rectangles to show the fraction in the box. Then, put the fraction on the number line.

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   135

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

TASK 18B: Write the fraction that each

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

•

shows on the number line.

0

1

0

1

0

1

0

1

This task features locations on number lines that aren’t identified with exact partitions or tick marks. In the first two and last representations, the points are exactly halfway between tick marks that represent unit fractions. Students might notice this improving their accuracy for finding the fractional parts. Some students may simply add a tick mark at the same location as the point without considering that the partitions must be equivalent. For example, in the first representation, these students will add a third tick mark and note the fraction as 31 . Instead, there should also be a tick mark placed halfway between “half” and “one,” creating fourths. The third number line features a model with a point at the same location of a tick mark in the model above. Students 2 should be able to quickly note that it is 3 . 136   

Mine the Gap for Mathematical Understanding

TASK 18C: Place each fraction on the number line 3 4 0

1

2 3 0

1

8 8 0

1

1 5 0

1

2 6 0

1

The tick marks that represent 1 in this task are intentionally scattered on each number line. Often, students work with consistent models with static endpoints. This may establish a false sense of understanding. This task is intended to provoke student understanding of the relationship between the fractions, 0, and 1 on the number line regardless of where 1 is located on the number line. We might think of it as a number line version of “changing the size of the whole.”

Chapter  4: Foundational Fraction Concepts   137

TASK 18D: Figure out what the shaded part of each fraction is. Then show the fraction on a number line.

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

This task asks students to interpret representations of fractions and transfer that knowledge to a different model. The first two models are intentionally set as fourths. Look for students who notice this and quickly replicate the number line. The third model shows eighths. The models of one fourth above should help accuracy for creating eighths. The last model uses sixteenths and can be built from fourths or eighths models. Consider the strengths and needs of your students. We may consider changing the last model to a different number of parts.

Notes

138   

Mine the Gap for Mathematical Understanding

Notes

Chapter  4: Foundational Fraction Concepts   139

BIG IDEA

19

BIG IDEA 19 Fractions on a Number Line TASK 19A The point shows 43 on each number line. Write the missing endpoint for each number line in the box. Explain how you found your endpoints.

0

0

0

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 140

About the Task Placing fractions on a number line is one way to demonstrate understanding of fractions as numbers. Number lines also reinforce relationships between fractions and whole numbers. These representations are useful for comparisons, finding equiva3 lency, and computation. In this task, students are asked to identify how is rep4 resented on a number line when the endpoints change. Often, tasks have students place fractions on number lines with endpoints of 0 and 1 or 0 and 2. This task asks 3 students to find different endpoints as the location of changes. 4

MINING TIP Students may show fourths on a number line by providing four

Anticipating Student Responses

tick marks. As we

In this task, students may show the misconception that number lines always have endpoints of 0 and 1. Students may misunderstand how the tick marks establish fractional segments. In other words, they may count tick marks rather than intervals. Others may show incomplete understanding of fractions as parts of a whole and their relationship to a whole. To complete the task, students are likely to establish the location of one whole and use that as a reference for finding the endpoint.

intervals or regions.

know, this shows five Essentially, they are incorrectly transferring ideas about wholenumber tick marks to this fraction model.

Notes

Chapter  4: Foundational Fraction Concepts   141

WHAT THEY DID

Student 1 Student 1 shows that he doesn’t understand the meaning of fractions and cannot represent them on the number line. It is encouraging that he writes the first endpoint as 44 . This is likely because he has worked with fractions on number lines with endpoints of 0 and 1. In those cases, we may have interpreted his work as understanding. As he works with different endpoints, we find that he doesn’t fully understand. His second endpoint seems to indicate that he is counting the tick marks, including 0. However, the last number line is a similar representation, and he counts a different amount.

Student 2 Like Student 1, Student 2 has a correct endpoint for the first number line. He seems to have a better understanding of the meaning of the tick marks. He notes that he counted the lines (line segments), and there were eight. He then counted how many line segments were in the “whole” because he interpreted the last endpoint to be equivalent to 1. It is good that the last two number lines were consistent. Unfortunately, he doesn’t recognize that the dot represents 43 or that the endpoints can be something other than 0 and 1.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 We might ask Student 1 what each line segment represents. Does he know that each is part of a whole? Does he see that there are four of them in the first number line? He may be struggling with the meaning of fractions. He would likely benefit from working with representations of fractions to reestablish the meaning of fractions and their relationship to the whole. More work is also needed with number lines. In his case, work with common endpoints of 0 and 1 would be good before changing endpoints.

Student 2 Student 2 shows understanding of fractions equivalent to one whole. He can find the number of parts in one whole. It may be that he misunderstood the prompt. A follow-up question may redirect him to the significance of the point on the number line representing 43 . Or, he may need work with fractions on diverse number lines, including representations of fractions greater than 1 on a number line.

142   

Mine the Gap for Mathematical Understanding

TASK 19A: The point shows 34 on each number line. Write the missing

endpoint for each number line in the box. Explain how you found your endpoints.

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   143

WHAT THEY DID

Student 3 3

Student 3 shows a good understanding of the relationship between and 1. He 4 too has likely worked only with number lines with endpoints of 0 and 1. Or, he has a flawed understanding of number lines in general. He doesn’t recognize or ignores 3 that the dot represents on each line in a different place. 4

Student 4 Student 4 shows full understanding of the task. He clearly understands that the point 3 represents . He also shows that the endpoints in the second and third number 4 3 lines change because the location of has changed. 4

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 3

Student 3 shows a different understanding of fractions. He places on the number 4 line each time in relationship to an endpoint of 1. He too may show understanding with a redirect to the prompt. Like Student 2, Student 3 seems to indicate limited experience with fractions on diverse number lines and fractions greater than 1 on a number line.

Student 4 Student 4 shows complete understanding of the task. He makes sense of the location and relates it to one whole. Student 4 can benefit from similar tasks occasionally to maintain his understanding. In time, we will introduce fractions greater than 1 to see if he can apply the same logic about fractions on number lines.

144   

Mine the Gap for Mathematical Understanding

TASK 19A: The point shows 34 on each number line. Write the missing

endpoint for each number line in the box. Explain how you found your endpoints.

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   145

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

•

TASK 19B: The shows where the fraction belongs on the number line. Show where 1 belongs on the number line.

•

1 2 0

•

1 3 0

•

1 4 0

2 4

• 0

This task is useful for reinforcing the idea that placement changes as endpoints change. It also reinforces that points may appear to have the same location on a number line, although they have a different value. The first three number lines are unit fractions. Students are likely to see that the distance between 0 and the point is one of the identified parts. Therefore, they can reiterate that distance in number of times relative to the value of the denominator. The last number line can use the number line directly above as a benchmark for fourths. Some students may know that 42 and 21 are equivalent so they can replicate the tick marks from the top number line. 146   

Mine the Gap for Mathematical Understanding

TASK 19C: Look carefully at the endpoints. Place each fraction on the number line.

3 4

1 8

2 3

4 4

0

2

0

1

0

2

0

4

This task is a variation of the changing endpoints task. It approaches the notion that a fractional part looks different or is placed on a number line differently based on the size of the whole. Students are likely to establish the location of one as the midpoint for the first and third number lines. The second number line is a traditional representation. The last fraction requires students to process the meaning of 44 as one whole and relate that to four wholes. Plotting 1 on a number line relative to other whole numbers is a skill students have worked with since kindergarten or first grade.

on a number line. Show 31 on a new number line with different endpoints. Tell how you knew where to place 31 on the second number line.

TASK 19D: Show

1 3

1

In this task, students are asked to place 3 on a number line. Then, students are 1 asked to place 3 on another number line with different endpoints. Students may misunderstand that the placement of 1 is relative to the placement of 1 or other 3 endpoints on the number line. Some may change the value of the endpoints but keep the location of the fraction. Chapter  4: Foundational Fraction Concepts   147

BIG IDEA

20

BIG IDEA 20 Fractions Greater Than 1 on a Number Line TASK 20A Place the fraction on the number line.

5

5

5

5

3 4 5

6

5

7

5

8

3 4

3 4

3 4 4

3 4

6

3 4

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 148

About the Task Students should work with fractions greater than 1 in similar ways to fractions less than 1. They also need to develop understanding of the placement of these numbers on a number line. Work with fractions greater than 1 should be concurrent with or shortly after basic understanding of fractions is established. This task requires students to consider the relationship of a fraction with other whole numbers.

Anticipating Student Responses The first number line is a rather typical number line representation applied to mixed numbers. Students who recognize that 5 3 is between 5 and 6 in a similar way that 4 3 4 is between 0 and 1 are likely to transfer their knowledge rather easily. The second and third number lines are more challenging as the endpoints have shifted. It is helpful to establish where 6 is on these number lines before finding the fourths between 5 and 6. Some students may show fourths throughout the number line. Others will only show fourths between 5 and 6. The last number line may be more challenging to partition. However, the problem becomes much easier if one reasons about the relationship between the fractions and the endpoint.

notes

Chapter  4: Foundational Fraction Concepts   149

MINING HAZARD We may look at students’ work and make an assumption about what they know as well as their errors. Yet, in some cases (as with Student 1), there are even more errors or misunderstandings than we first notice.

WHAT THEY DID

Student 1 Student 1 shows that she knows 43 of an interval. Her first three responses are approximately three fourths of the way between endpoints. As we know, this works well for the first number line. However, for the next two number lines, it is problematic. Clearly, she doesn’t recognize the endpoints. It is also possible that she doesn’t understand how the changing endpoints affect the location of 5 3 . Yet, if we look 4 more closely, we might note that her placement of 43 is relatively accurate to the actual location of 43 . But we will also notice that she creates four tick marks or five intervals, which inaccurately establishes 43 .

Student 2 Student 2 creates equal partitions between endpoints. She counts by fourths. Her first placement is accurate. Interestingly, using the idea of counting on by fourths   from 5, she finds 5 3  3  if we think of 6 as 5 4 . It is unlikely that she was applying 4 4 4   this logic. The last two number lines speak to her challenges. In the third number line, she counts by fourths. In the last number line, she counts by whole numbers and fourths.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 3

We first want to revisit Student 1’s understanding of 4 and its location on the number line. We need to make sense of her four tick marks and five intervals. After reestablishing her understanding of 43 and other fractions, we can move to fractions greater than 1. First, we want to work with finding missing whole numbers between various endpoints. For example, what whole numbers are between 5 and 7? Where are they located on the number line? What numbers are between 5 and 8 (third number line), and where are they located on the number line? Students like Student 1 can then begin to work with establishing points between whole numbers (mixed numbers).

Student 2 Student 2 shows that she can continue to count by a specific unit fraction. We can use this to develop her understanding of mixed number locations on number lines. We may choose to extend number lines as needed instead of placing mixed numbers on number lines with fixed endpoints. In other words, we can work with number lines that have endpoints of 0 and 1, and as the counting goes beyond 1, we can extend the number line to the next whole-number endpoint. We also want to be sure that Student 2 understands when fractions are equivalent to whole numbers. An example is that 84 is equivalent to 2 and 12 is equivalent to 3. 4

150   

Mine the Gap for Mathematical Understanding

TASK 20A: Place the fraction on the number line.

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   151

WHAT THEY DID

Student 3 Student 2’s location of 5 3 moves on each number line as we might expect. She 4 establishes 5 3 on the first number line. We can assume she understands the chang4 ing endpoints as the location of 5 3 shifts to the left on each new number line. Did 4 she estimate where 5 3 might be? Are her placements precise? She provides tick 4 marks, but they are not labeled, so we are unable to determine how they supported her placement. We also should note that the tick marks, even with inference, are imprecise. This is most clear with the last number line.

Student 4 Student 4 shows full understanding. She establishes 5 3 on each number line. She 4 creates benchmarks for endpoints that are not consecutive whole numbers. In other words, she finds 6 as the midpoint between 5 and 7 and locates both 6 and 7 between the endpoints on the third number line. Like Student 3, she doesn’t label her tick marks in the third number line. However, she does establish this idea in the first two number lines. We can also note that she labels the major (whole-number) tick marks between endpoints on this number line. Student 4 also shows advanced thinking in the last number line as she clearly communicates that 5 3 is halfway 4 3 between 4 43 and 6 4 .

USING EVIDENCE

MINING TIP It is important to include students with advanced understanding in our conversations. However, as we sequence student work during the discussion, we must be careful when we include these students and their work. Doing so prematurely can limit the sharing and discovery of other students.

152   

What would we want to ask these students? What might we do next?

Student 3 Student 3 clearly has some ideas about partitioning spaces between whole numbers and endpoints. Her first two placements are fairly accurate. The others are not. She may need to develop strategies for reasoning about and finding midpoints as well as other benchmarks. She can extend this strategy to partition between midpoints or benchmarks. We also want to reinforce labeling tick marks to better communicate her thinking. Doing so may also help her refine her accuracy and precision. Like Student 2, she may benefit from extending number lines.

Student 4 As noted, Student 4 shows good understanding of the relationship of points on a number line. We should continue to revisit this idea with her as a center, warm-up, or occasional homework activity. We can begin to modify the task to create endpoints that are farther apart. We want to be sure that we include her thinking during group discussion of this and similar tasks. Doing so can help other students develop reasoning.

Mine the Gap for Mathematical Understanding

TASK 20A: Place the fraction on the number line.

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   153

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 20B: Create two numbers or fractions that can be placed on this number line.

2

1

4

2

1 2

This open-ended task allows students to place any two fractions on the number line. Counting by halves is a useful strategy for establishing tick marks on the number 1 1 line. Some students may also recognize that 3 2 is halfway between 2 2 and 4 21 . This benchmark can be used to locate other points such as 3 because it is halfway between 2 21 and 3 1 . Or 3 21 can be used to locate 4 because it is halfway between 2

3 21

and

4 21 .

TASK 20C: What fraction might the point represent? Explain your thinking.

5

•

In this task, there aren’t two endpoints to use as references for the fraction. Some students may create an endpoint of 6 with the point representing 5 1 . However, any 4 value greater than 5 can be used. It would be wise to show work of students with diverse answers to help build understanding of how and why the value can change.

154   

Mine the Gap for Mathematical Understanding

TASK 20D The point shows 7 41 on each number line. Write the missing endpoint for each number line in the box. Explain how you found your endpoints.

7

7

7

This task is modified from the “Fractions on a Number Line” task (19A) earlier in the chapter. In this task, students work with the changing placement of a mixed number. Students have to use the tick marks to make sense of the relationship between the mixed number 7 41 and the first endpoint. The first number line is rather straightforward. However, the second and third require deeper thinking about the tick marks and the meaning of 41 more than 7 wholes.

Notes

Chapter  4: Foundational Fraction Concepts   155

BIG IDEA

21

BIG IDEA 21 Decomposing Fractions TASK 21A Emmett made a rod with cubes.

He had 12 cubes in the rod. Each cube was

1 12

of the rod.

He broke it apart into smaller pieces. He wrote the equation 3 + 1 +4 + 2 + 2 12

12

12

12

12

What are TWO other ways he could break apart his rod? Draw a picture and write an equation for each.

MODIFYING THE TASK We can provide students with concrete linker cubes to support their thinking as they work with this task.

About the Task Decomposing fractions is useful for developing reasoning about comparison and computation with fractions. We can decompose fractions similar to the ways we decompose whole numbers. We might decompose fractions into unit fractions or some other combination. In this task, students are prompted to decompose a fraction in two different ways. Students are provided with a model to support their work. This tasks reinforces the notion of flexible thinking about decomposition by asking students to decompose the same amount in different ways.

Anticipating Student Responses The model shows one whole composed of 12 twelfths broken into five parts or addends. Students are prompted to break it apart in two different ways. Students must first recognize that there are 12 parts in the whole. This information is provided in the prompt, the picture, and the expression. This may

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 156

be problematic for some students. Some students will treat each piece as a whole, thus adding whole numbers for a whole-number sum (12). Although addition with fractions is provided, some students may show the classic misconception of adding numerators and denominators. Others with better understanding may break it solely into two parts or addends. They may use the commutative property to show the second possibility or may provide another example of two addends. More advanced students will break it into any number of parts.

Notes

Chapter  4: Foundational Fraction Concepts   157

WHAT THEY DID

Student 1 This student’s work is problematic for different reasons. She seems to understand that the task is asking her to break apart the rod in different ways. She treats each block as an individual whole. Her first example has a different number of pieces than the prompt. Her equation doesn’t match her picture. Her second picture and equation do not match each other or the amount in the first decomposition.

Student 2 This task models an expected approach for decomposing the rod into fraction addends. Without this model in place, we may think that Student 2 understands the mathematics. She decomposes the rod while sustaining the sum of 12. Like Student 1, she considers each block as a whole number rather than 1 of the rod. 12 She also breaks the rod into equal-size pieces (6 + 6 and 4 + 4 + 4). The latter is not necessarily problematic, although it may speak to limited understanding about decomposing numbers.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 This student has clear misunderstanding of number concepts as well as fractions. It may be possible that she didn’t understand the prompt or context. A first step would be to confirm that she understands how a whole number can be decomposed into varied parts. It would be best to do this with a physical model and record the resulting equation. After doing so, she should be prompted to decompose the model in different ways before recording the equations. We can then shift our work to fractional parts after we are sure that she understands decomposition of whole numbers.

Student 2 We need to help Student 2 recognize how many cubes are in the whole rod. Student 2 should work with rods of varied lengths. At first, she can break them into two pieces and record the fractional size of each piece and the addition equation. She can then work to break them into three or more pieces, also recording the size of the pieces and the equation. She can move to open decomposition after demonstrating proficiency with these.

158   

Mine the Gap for Mathematical Understanding

TASK 21A: Emmett made a rod with cubes. He had 12 cubes in the rod.    1 of the rod. He broke it apart into smaller 12 3 1 4 2 2 + + + + . What are TWO other ways 12 12 12 12 12

Each cube was

pieces. He wrote

the equation

he could break

apart his rod? Draw a picture and write an equation for each.

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   159

MINING HAZARD Fractions can be decomposed in many ways. Students who consistently decompose fractions into two parts may signal limited

WHAT THEY DID

Students 3 and 4 These students decompose the rod into fractional pieces or addends. Their decomposition is accurate but may lead us to think they fully understand how a fraction can be decomposed. In both cases, Student 3 decomposes the rod into two addends. 6 2  Student 4 decomposes the rod into equal-size pieces  12 and 12  . Both students   show some understanding of decomposition. Is their understanding complete?

understanding of decomposition. In this case, both students are correct, but their understanding may not

USING EVIDENCE

What would we want to ask these students? What might we do next?

be full and complete.

Students 3 and 4 Both students show that they have some understanding of decomposition of fractions. It may be that simple prompting or redirecting to the model in the task will provoke more complex decompositions. It’s also possible that the students have misconceptions about the number of parts a fraction can be decomposed into or that the size of the decompositions must be equivalent

6 6  + .  12 12 

In either case, we

should honor their decompositions and ask them to consider less “regular” ways similar to the example. We can provide specific compositions and ask them to find the sum or provide the sum and ask them to break it into a specific number of decompositions. As students show proficiency with the concept, we want to move them away from the concrete model or drawing to purely symbolic representations of decomposition.

160   

Mine the Gap for Mathematical Understanding

TASK 21A: Emmett made a rod with cubes. He had 12 cubes in the rod.    1 of the rod. He broke it apart into smaller 12 3 1 4 2 2 + + + + . What are TWO other ways 12 12 12 12 12

Each cube was

pieces. He wrote

the equation

he could break

apart his rod? Draw a picture and write an equation for each.

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   161

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 21B: Kristen decomposed a fraction into the fractions below. What is the fraction? 3 8

+

1 8

+

2 8

+

1 8

What are two different ways that Kristen could decompose this fraction? Use pictures, numbers, or words to explain your thinking. In this task, students are asked to recompose a fraction before breaking it apart into different pieces. They are asked to decompose it in two different ways and justify   their thinking. Look for students who find the fraction  87  and decompose it into   the same number of addends as the prompt. These students may simply reverse or reorder the addends in the prompt. It may signal limited understanding, although it is mathematically correct. The writing doesn’t need to be extensive to explain their approach to the prompt. Students may also use a model instead of words to justify their solutions. 9 TASK 21C: Circle the expressions that show ways to break apart 12 . 1 12 3 6

+

+

3 12

1 12

4 2

+

+ 4

12

+

6

1

12

5

+

2

2

4

12

+

2

4

12

12

1 5

+

+

+ 2

12 4 12

7

1

2

12

+

+

5

2

12

12

1

2

12

3

+

+

+

2 3

4 12 2 12 +

+

+ 2 3

3 12 2 12 +

+

+

1 12 2 12

2 3

Create two different expressions that show ways to break apart 65 . 9

In this task, students are asked to recognize different ways to decompose . The 12 expressions pose various ideas to consider. Some have twelfths that, when added, 9 don’t equal . Others have denominators that, when added, equal twelfths. Look for 12 students who add both numerators and denominators. Look for students who are satisfied with any expression that has twelfths. Look for others who find only one 9 decomposition of . The second prompt in the task asks students to then decom12   pose a different fraction  5  . This open-ended portion of the task will reinforce the 6 misconception or incomplete understanding uncovered above. 162   

Mine the Gap for Mathematical Understanding

TASK 21D: Decompose the fraction 7 the fraction 8 two different ways.

10 16

two different ways. Decompose

This task is completely open-ended. It prompts students to decompose a fraction in two different ways. It could be modified to prompt for more possibilities. Regardless of the fractions used in the task, look for students who understand the size of the pieces (denominator) remains constant in the expression. Look for students who decompose it into two parts versus those who decompose it into many parts. Also be on the lookout for any students who might decompose it into pieces with equivalent fractions. For example,

10 16

could be decomposed into

1 2

+

1 + 1 16 16

or 1 + 4

3 16

+

3 16

. This

is mathematically accurate and provides insight into more complex thinking about fractions.

Notes

Chapter  4: Foundational Fraction Concepts   163

164   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 22 Equivalent Fractions on a Number Line

221

TASK 22A Create two equivalent fractions. Use number lines to show that they are equivalent.

About the Task Equivalent fractions are often developed with region models using pattern blocks, circles, squares, and rectangles. Cuisenaire rods and fraction tiles are also useful. These models lay the groundwork for understanding equivalency. This understanding of equivalency should also be connected to number lines. This task makes use of number lines to verify equivalency. We may be accustomed to number line tasks that are partitioned and prompt students to identify if points are equivalent. This task asks students to establish where they think equivalent fractions are represented. The number lines do not have specific endpoints, so reasoning about the fraction and its relationship to one whole or other fractions is necessary for success.

Anticipating Student Responses The task is open so that students can generate their own equivalent fractions. Students are likely to work with halves, fourths, and eighths. Some students may show misunderstanding about the numerator. They will share that 1 2

and 1 or 1 and 1 are equivalent because the numerators are equal. Other 2

4

3

students may identify equivalents accurately but fail to partition the number lines accurately. This provides some insight into understanding equivalency but also shows flaws with thoughts about number lines. For example, a student 2 may show 1 on the top number line and 4 on the bottom number line at a 2 similar location yet without justification of fourths on the second number line.

MINING HAZARD Partitioned number lines may mislead us to think that students understand fractions on number lines better than they actually do. Using open number lines for them to partition provides better insight into their understanding.

MINING HAZARD The essential understanding of equivalency on a number line is that two fractions are equivalent

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

if their positions are the same. We have to be cautious of students who show

1 2

and

2 4

at

the same, incorrect location.

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 165

WHAT THEY DID

Student 1 At first glance, Student 1 appears to have good understanding of equivalence on 63 6 number lines. Although slightly misaligned, his locations for 3 and are close. After 4 8 4 8 a closer look, we notice a significant problem. The endpoints for the number lines are not the same. In the first number line, the endpoints are 4 and 5, so the identified 3

point is 4 4 . The bottom number line has the endpoints of 8 and 9 with an identified point of 8 6 . We know that these two values are not equivalent. It seems as though 8

he uses the denominator to determine the first endpoint on the number line. He may be doing this because all of his number line work has featured the same endpoints and so he has never given them much thought.

Student 2 Student 2 has tick marks that are equally partitioned. He has the same number of tick marks on both number lines. The first five or six tick marks align well. He identifies that

5 5

and

5 10

have the same location, which they do. This is because he

isn’t using the same size whole to compare the fractions on the number line. This is evident because

5 10

and

10 10

are not in the same location. It is clear that he is using

the numerator to establish equivalency of fractions on number lines. USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 We want to ask Student 1 how he knows the two fractions are equivalent. It is likely that he will note that 3 and 68 are equivalent because they have the same spot on 4 the number line. If so, we want to ask him about the endpoints. We also want to determine his understanding about equivalency due to the location of points on the number line. If he has that understanding, we’ll then need to work to help him understand that these two points aren’t in the same location. To do this, we would be wise to extend both number lines and then align them with common endpoints. For the top number line, we would extend it to the right until we have an endpoint of 9. The bottom number line would be extended to the left until we have an endpoint of 4. After doing so, we can see that these two fractions are not the same or that these two similar locations do not identify the same fraction.

Student 2 It may be best to divorce these fractions from a number line for a short time. Instead, 5 we might have him build a model or draw a picture of both 55 and 10 . After doing so,

we can ask Student 2 if the two fractions are the same. It’s possible that Student 2 considers the fractions symbolically. We can ask, “Is 5 greater, less, or equal to 5 one whole?” We can also ask, “Is 5 greater, less, or equal to one whole?” Hopefully, 10 these questions elicit answers that note 55 is equal to one whole and that 5 is equiv10 alent to 21 . With this information, we can discuss if 21 is equal to 1. Moving forward, we’ll want to reconnect representations with points on a number line as well as revisit partitioning and placement of fractions on a number line. 166   

Mine the Gap for Mathematical Understanding

TASK 22A: Create two equivalent fractions. Use number lines to show that they are equivalent.

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   167

MINING HAZARD

WHAT THEY DID

Our students

Student 3

need procedural

It’s interesting to note that Student 3 relates

understanding as well as conceptual understanding. Students who show procedural understanding may not have understanding of the concepts. This can be problematic because over time, they forget

1 2

tions to

proof? His 4 8

÷

4 4

=

1 2

. It is clear that he knows that 4 8

5 are not 10 5 ÷5 = 1 . 10 5 2

to

and

4 8

4 8

to

5 10

rather than one of these frac-

is equivalent to

5 . 10

But what is his

aligned terribly well on the number line. He does note that His procedural understanding is clear. Is his conceptual

understanding of equivalency clear?

Student 4 Student 4 also works with 5 . He shows that it is equivalent to 10 not exactly aligned but close enough to show understanding.

1 2

. His locations are

the steps or rules for a procedure and then have no conceptual understanding to rely on.

MINING HAZARD Procedural proof is important. We have to be careful to avoid using this proof as evidence of conceptual understanding as well.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 Student 3 is in a decent place because he shows an ability to find equivalent fractions procedurally. This works well to prove equivalency. We want to ask him about his number line representation. It may be a simple oversight or lack of precision. But we must be careful not to overly discount the mistake. His calculations show that both fractions are equal to 1 . A half or midpoint is fairly easy to find on a number 2 line. With this in mind, it would make more sense that both fractions have the same place on the number line and the partitions between these locations might be off. We want to work to develop or reinforce his understanding of fractions on number lines. We also want to be sure that we avoid any procedural approaches to finding those locations.

Student 4 Student 4 is a good example of how a task may be considered differently relative to its intended use. If this task is used as an assessment, we would give Student 4 full or close to full (because of slight misalignment) credit. However, if we are using the task for instruction, we would want to discuss how he knows that 5 is one half 10 of 10 10 . This discussion would help us better understand his thinking about fractions, number lines, and equivalence.

168   

Mine the Gap for Mathematical Understanding

TASK 22A: Create two equivalent fractions. Use number lines to show that they are equivalent.

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   169

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

TASK 22B: Write two equivalent fractions. __________   ____________ Use the rectangles to prove that they are equivalent. Use the number line(s) to prove that they are equivalent. This task provides an opportunity to prove equivalency through different representations. It allows students to connect symbolic representations with region models and number line models. The task is open-ended. It requires students to partition regions and number lines accurately. We may see students who are able to partition the regions accurately but struggle to do so with the number line. In these cases, more work with connecting concept and different representations is needed.

TASK 22C: The point shows

2 4

and

4 8

.

• Tell how this is possible. This task is useful for establishing fractions and equivalent fractions on a number line. Strategies for solving the problem will likely include finding where to place 0 and 1. Students may use 2 to find these endpoints and before further partitioning 4 shows eighths. Look for students who do not make use of tick marks to establish equivalency. Instead, they use computation and procedure to prove equivalence.

TASK 22D: Do you agree or disagree that

3 4

3 8

are equivalent? Use models, numbers, or words to explain your thinking. and

Task 22D asks students to agree or disagree about the equivalency of

3 4

and

3 8

. This task doesn’t require a number line for reasoning. However, students are asked to justify why they think the fractions are or are not equivalent. Some students will say that they are because they misunderstand the meaning of the numerator. Others will disagree and accurately describe why. The task is included here because it offers insight into the models students prefer or access most readily for thinking about fractions. It informs us about work with number line representations if region models are the predominant or sole representations for justification.

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Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 23 Comparing Fractions

231

TASK 23A Erin ran

4 10

of a mile. Kristen ran

5 8

of a mile. Who ran farther?

Explain how you know who ran farther. Jon ran

3 4

of a mile. How does he compare to Kristen and Erin?

About the Task We have often prompted students to compare fractions. From time to time, we have asked students to represent and justify their comparison. This task asks students to compare and justify as well. The context of the problem aligns with a linear (number line) model rather than a regional model. After justifying the comparison of two fractions, students are asked to consider how a third fraction compares. This natural extension provides insight into how students compare more than one fraction.

Anticipating Student Responses Students will compare and justify the fractions in different ways. Some students will use parts of a whole to show the relationship. We might ask these students how the circles or rectangles compare to a mile. Some students may use a number line. Some students may procedurally compare the fractions by finding common denominators while others will reason that Erin ran less than 1 of a mile and Kristen ran more than 21 of a mile. When comparing the third 2 runner, students are likely to incorporate the model that was used for the first prompt. Look for students who compare the third runner to both runners and students who compare it to only one runner because they have already established greater and less than above.

PAUSE AND REFLECT zzHow

does this task compare to tasks I’ve used?

zzWhat

might my students do in this task?

Visit this book’s companion website at resources.corwin.com/minethegap/3-5 for complete, downloadable versions of all tasks. 171

MINING HAZARD Rationalizing about the size of pieces and the number of missing pieces is a strategy for comparison. Students, like Student 1, can misinterpret the strategy or overly generalize it.

MINING HAZARD Student work with lots of writing and diagrams can be overlooked. We have to be careful that their ideas are clear and accurate.

WHAT THEY DID

Student 1 Making use of a benchmark like 1 is an efficient strategy for comparing fractions. 2 Student 1 does this for the first prompt in this task. She states that 4 is less than 10 1 and that 85 is greater than 1 . She doesn’t reference the one-half equivalents for 2 2 those denominators. Her comparison of the extension shows a misconception about comparing fractions. She notes that 3 is 1 away from a whole, whereas 4 and 5 4

are 6 and 3 away from the whole, respectively. She states that because of this.

3 4

10

8

is then greater

Student 2 Student 2 writes a considerable amount and includes pictures. If we are not careful, we may assume that the student understands the task, especially when the first prompt is correct. Like Student 1, she compares the first set of fractions using benchmarks. She also justifies this comparison. Her models are accurate in the second prompt. She compares the third fraction correctly to the previous two. Yet, as we continue to read, we notice that her additional thinking represents the same misconception about the number of missing pieces that Student 1 holds.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 1 We would want to ask Student 1 why she compared the fractions in the first prompt using a benchmark and used a different strategy for the second prompt. We would also want her to compare 9 and 3 . She might say the two are equivalent because 10 4 they are both one piece from the whole. With this in mind, it will be important to provide opportunities to compare fractions that have a similar number of pieces so that she develops the understanding that this isn’t a viable strategy for comparing fractions.

Student 2 We want to discuss Student 2’s pictures. We want her to tell us about the size of the pieces. We want to ask her if the size of the piece makes a difference when we compare and how she knows. Like Student 1, it will be important for her to have opportunities to compare fractions that are missing the same number of pieces. Another option is to compare fractions with the same numerator. In both instances, we will need to encourage her to make models of the fractions while developing her reasoning about the size of the pieces.

172   

Mine the Gap for Mathematical Understanding

   4 5 TASK 23A: Erin ran 10 of a mile. Kristen ran of a mile. Who ran farther? 8

Explain how you know who ran farther. Jon ran

compare to Kristen and Erin?

3 4

of a mile. How does he

Student Work 1

Student Work 2

Chapter  4: Foundational Fraction Concepts   173

WHAT THEY DID

MINING HAZARD Pictures and diagrams that accurately support mathematical ideas are valuable. They should be used to make sense and justify ideas. However, drawing pictures can be inefficient. We want to advance student strategies to improve efficiency.

MODIFYING THE TASK Students may be inclined to use benchmark comparisons because

Student 3 Student 3 makes use of a representation to compare the two fractions. She uses the same-size whole for both fractions and partitions them fairly accurately. This strategy for comparison isn’t as efficient as reasoning about benchmarks or the size of pieces, but it is satisfactory. Her model in the second prompt helps us see her thinking. Her justification isn’t clear, even though she notes the correct answer. She does talk about the size of pieces but doesn’t connect how that affects the comparison between the three fractions.

Student 4 Like other students, Student 4 justifies her comparison using the benchmark of 1 . 2 Student 4 adds a different twist to comparing the fractions. She alludes to the idea that she only has to compare to Kristen because she compared her to Erin in the first prompt. She makes use of a common denominator most likely because she easily   recalls that 4 is half of 8. She notes that you can multiply Jon’s amount  43  by 22     but doesn’t explain how that will compare to Kristen’s distance  85  . It is also possi  ble that she only compared to Kristen because it was the only common denominator she could find easily.

both numerators are related to a clear half of the denominator. We can change the fractions to elicit other comparison strategies. For example, we might adjust Erin to 4 5

and Kristen to

7 8

to

provoke thoughts about distance from the whole.

USING EVIDENCE

What would we want to ask these students? What might we do next?

Student 3 In time, we’ll want to move Student 3 from drawing pictures or diagrams to finding her solutions by reasoning about the fractions. She shows some ideas about the size of pieces in the fractions but seems to rely on her pictures. With that in mind, we can leverage her pictures and other models to develop those strategies. We can add 1 as 2 a third fraction to compare the first two distances in the first prompt. Doing so will help establish the idea of using benchmarks for comparison. In the second prompt, we want to have conversations about the size of the pieces, the distance between the fraction and the whole. She lines up her area models, so making use of number lines may also be advantageous. Essentially, Student 3 needs opportunities to work with fractions to develop reasoning strategies grounded in models and pictures.

Student 4 We need to understand why Student 4 only compared Jon’s distance to Kristen’s distance. To do this, we might ask, “How does Erin compare to Jon?” or “Why didn’t you include Erin’s distance in the comparison?” We also want to understand how   having the same denominator is helpful and what happens to Jon’s distance  43    when we find the equivalent in eighths. We want to work with Student 4 to determine when certain strategies, including finding common denominators, are more useful. We also want to work with her to develop clear, complete justifications. To do this, we can provide opportunities to exchange ideas during discussion and make use of teacher questions that focus student thinking. 174   

Mine the Gap for Mathematical Understanding

   4 5 TASK 23A: Erin ran 10 of a mile. Kristen ran of a mile. Who ran farther? 8

Explain how you know who ran farther. Jon ran

compare to Kristen and Erin?

3 4

of a mile. How does he

Student Work 3

Student Work 4

Chapter  4: Foundational Fraction Concepts   175

OTHER TASKS zzWhat

will count as evidence of understanding?

zzWhat

misconceptions might you find?

zzWhat

will you do or how will you respond?

TASK 23B: A fraction is less than

Visit this book’s companion website at resources.corwin.com/ minethegap/3-5 for complete, downloadable versions of all tasks.

5 8

. What might the fraction be? Use a number line to prove that one fraction is greater than the other fraction. Tell how a number line can help you compare fractions.

MINING HAZARD We may incorrectly assume that our students will take the least difficult path when presented with an open-ended task for all sorts of reasons. It’s important to keep in mind that students may choose paths we perceive to be less challenging because they do have limited understanding, lack confidence in their understanding, or are simply unaware of more complex possibilities.

This task provokes many possibilities. Students can write a variety of fractions that are less than 5 . Some students will write fractions with different denominators such as 8 1 1 or . Keep in mind that 08 or 0 parts of any other fraction also qualifies. Look for 2 4 students who rely on the same denominator versus those who do not. Using the same denominator, although accurate, might indicate limited confidence with or understanding of fraction comparison. Some students may recognize that 85 is a bit more than 1 and quickly create fractions less than half. The power of this task is in the second 2 prompt. Look for students to accurately describe that the value of a fraction increases as we move to the right on a number line and decreases as we move to the left.

TASK 23C: Circle all the inequalities that are true. 1 < 2 4 3

1 < 1 5 6

4 > 1 3 5

2 > 3 5 4

3 7 < 4 8 6 3 > 8 8

Choose one inequality to tell how you compared the fractions. This task offers different comparisons to provide insight into student strategies for comparison. Students who successfully compare with common denominators throughout the task show readiness to develop reasoning strategies. These strategies  1  4 include using benchmarks  0 , , 1  for comparing fractions like and 1 or 2 and 3 . 5 5 3 4  2  Another strategy is to reason about the size of the pieces with like numerators for 1 1 fractions such as 5 and 6 . Reasoning about the size of pieces is also useful when the 7 3 same number of pieces is missing for comparing fractions such as and . 8

TASK 23D: Two fractions are greater than

4

1 2

and less than 1. They are not equivalent. What could the fractions be? Represent the fractions on a number line and with area models to prove that they are greater than 21 but less than 1. This problem-based task has many possibilities. Students are asked to represent these fractions in two different ways. It is useful for developing reasoning strategies by visual1 izing the fractions. It also develops relationships to benchmarks of 2 and 1. Successful students demonstrate understanding needed for developing these strategies. 176   

Mine the Gap for Mathematical Understanding

BIG BIGIDEA IDEA BIG IDEA 24 Reasoning About Fractions

241

TASK 24A Write >, , ,