Elementary and Middle School Mathematics: Teaching Developmentally (10th Edition) [10 ed.] 013480208X, 9780134802084

Table of contents :
Elementary and Middle School Mathematics, 10th Edition [John A. Van de Walle]
Title Page
Copyright
About the Authors
About the Contributors
Brief Contents
Contents
Preface
Chapter 1 Teaching Mathematics in the 21st Century
Chapter 2 Exploring What It Means to Know and Do Mathematics
Chapter 3 Teaching through Problem Solving
Chapter 4 Planning in the ProblemBased Classroom
Chapter 5 Creating Assessments for Learning
Chapter 6 Teaching Mathematics Equitably to All Students
Chapter 7 Developing Early Number Concepts and Number Sense
Chapter 8 Developing Meanings for the Operations
Chapter 9 Developing Basic Fact Fluency
Chapter 10 Developing Whole-Number Place-Value Concepts
Chapter 11 Developing Strategies for Addition and Subtraction Computation
Chapter 12 Developing Strategies for Multiplication and Division Computation
Chapter 13 Algebraic Thinking, Equations, and Functions
Chapter 14 Developing Fraction Concepts
Chapter 15 Developing Fraction Operations
Chapter 16 Developing Decimal and Percent Concepts and Decimal Computation
Chapter 17 Ratios, Proportions, and Proportional Reasoning
Chapter 18 Developing Measurement Concepts
Chapter 19 Developing Geometric Thinking and Geometric Concepts
Chapter 20 Developing Concepts of Data and Statistics
Chapter 21 Exploring Concepts of Probability
Chapter 22 Developing Concepts of Exponents, Integers, and Real Numbers
Appendixes
References
Index
Credits

Citation preview

TENTH

EDITION

Elementary and Middle School Mathematics Teaching Developmentally

John A. Van de Walle Late of Virginia Commonwealth University

Karen S. Karp Johns Hopkins University

Jennifer M. Bay-Williams University of Louisville

With contributions by

Jonathan Wray Howard County Public Schools

Elizabeth Todd Brown University of Louisville (Emeritus)

@ Pearson

330 Hudson Street, !XV NY 10013

Vice P resident and Editor in Chief: Kevin M. Davis Portfolio Manager: Drew Bennett Managing Content Producer: Megan Moffo Content Producer: Yagnesh J ani Portfolio Management Assistant: Maria Feliberty Executive Product Marketing Manager: Christopher Barry Executive Field Marketing Manager: Krista Clark D evelopment Editor: Kim Norbura D igital Studio Producer: Lauren Carlson Senior Digital Producer: Allison Longley Procurement Specialist: Deidra Sm ith Cover Designer: Studio Montage Cover Art: shutterstock.com/Petr Vaclavek Editorial Production and Composition Services: SPi-Global Full-Service and Editorial Project Manager: Jason Hammond/Kelly Murphy, SPi-Global Text Font: Janson Text LT Pro 9.75/12 Credits and acknowledgments for materials borrowed from other sources and reproduced, with permission, in this textbook appear on the appropriate page within the text or on page 697. Every effort has been made to provide accurate and current Internet information in this book. However, the Internet and information posted on it are constantly changing, so it is inevitable that some of the Internet addresses listed in this textbook 'viii change. Copyright© 2019,2016,2013,2010,2007,2004, by Pearson Education, Inc. All rights reserved. Printed in the U nited States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or like,vise. To obtai n permission(s) to use material from this work, please visit http://www.pearsoned.com/permissions/ Library of Congress Cataloging-in-Publication Data Names: Van de Walle, John A. I Karp, Karen, 1951 - author. I Bay-W.lliams, Jennifer M., autlwr. I Wray, Jonatl1an A., author. T itle: Elementary and middle school mathematics : teachi ng developmentally. Description: Tenth edition I John A. Van de vValle, late of Virginia Commonwealth University, Karen S. Karp, University of Louisville, Jennifer M. Bay-vVilliams, University of Louisville; with contributions by Jonathan Wray, Howard County Public Schools. I Boston: Pearson, [2019) I Includes bibl iographical references and index. Identifiers: LCCN 2017037185 I ISBN 9780134802084 Subjects: LCSH: Mathematics--Study and teaching (Elementary) I Mathematics--Study and teaching (Middle school) Classification: LCC QA135.6 .V36 2019 I DOC 510.7112--dc2 3 LC record available at https://lccn.loc.gov/201703 7185

10 9 8

7 6 5 4 3 2

@Pearson

ISBN 10: 0-13-480208-X ISBN 13: 978-0-13-480208-4

About the Authors John A. Van de Walle The late John A. Van de Walle was a professor emeritus at Vi rginia Commonwealth University. He was a leader in mathematics education who regularly gave professional development workshops for K-8 teachers in the United States and Canada focused on mathematics instruction that engaged students in mathematical reasoning and problem soh~ng. He visited and taught in many classrooms and worked with teachers to implement student-centered mathematics

lessons. He co-authored the Scott F01·es111an-Addison Wesley Mathematics K-6 series and con tributed to the original Pearson School mathematics program enVisionMATII. Additionally, John was very active in the National Council of Teachers of Mathematics (NCTM), writing book chapters and journal articles, serving on the board of directors, chairing the educationa l materials committee, and spea king at national and regional meetings.

Karen S. Karp Karen S. Karp is a visiting profe.%or at Johns Hopkins University (Maryland). P reviously, she was a professor of mathematics education at the University of Louis,~lle for more than twenty years. Prior to entering the field of teacher education, she was an elementary school teacher in New York. She is the coauthor of Developing Essential Undentlmding ofAddition and Subtmction jo1· Teacbing M11the1!1atics in PreK-Grade 2, Discove1·ing Lessons fm· tbe Common Co1·e St11te Strmdtwds in Gmdes K-5 and Putting Essential Understmrding of Addition and Subtmction into Pmctice PreK-Gr11de 2. She is a former member of the boa rd of directors for the National Council of Teachers of Mathematics (NCTi'vl) and a forn1er president of the Association of Mathematics Teacher Educators. She continue.s to work in classrooms to support teachers in ways to instruct students with disabilities.

Jennifer M. Bay-Williams Dr. Jennifer Bay-Williams is a professor at the University of Louisville. She has written many article.s and books around K-12 mathematics education, including the three book series related to this book- Te~tcbing Student Cente1·ed Mathmwtics-and various other books, including Eve1ything Yon Need jo1· Mllthematics Coacbing, Developing Essential Understanding ofAdditionmrd Subtraction in P1·ekinde1-gmten-Gmde 2, On the Money (financial literacy book series), and iVf11th and Litemture: Gmdes 6-8. Jennifer taught elementary, middle, and high school in Missouri and in Peru, and continue.s to learn and work in K-8 classrooms in rural and urban settings. Jennifer has been a member of the NCTM board of directors, AMTE secretary, president, and lead writer for Stmrdan/s for tbe Prep11mtion ofTeachers ofMathematics (AiV1TE, 20 17), and is cu rrently on the TO DOS: Mathematics for All Board of Directors.

Ill

About the Contributors Jonathan Wray is the technology contributor to Elementmy and Middle Scbool Matbematics, Teacbing Developmentally (sixth-tenth editions). He is the acting coordinator of secondary mathematics curricula r programs in the Howa rd County public school system. He has served as the president of the Association of Maryland Mathematics Teacher Educators (AiV1MTE) and the Maryland Council of Teachers of Mathematics (MCTM) and currently is manager of the E lementary Mathematics Specialisrs and Teacher Leaders (ems&tl) Project. Jon also served on the NCTM Boa rd of D irectors (2012-2015). He has been recognized for his expertise in infusing technology in mathematics teaching and was named an outStanding technology leader in education by the Maryland Society for Educational Technology (MSET). He was a primary and intermediate grades classroom teacher, gifted and talented resource teacher, elementary mathematics specialist, curriculum and assessment developer, grant project manager, and educational consultant.

Elizabeth Todd Brown is the assessment and media contributor to Eleme-ntmy and Middle Scbool Matbe111atics, Teacbing Developmentally (tenth edition). She is a professor emeritus from the University of Louisville. She was also an elementary and middle school teacher in Iowa, Missouri, and Kentucky and continues to volunteer weekly at a loca l elementary school in Louisville. Todd coauthored the book, Feisty Females, and has written articles on the importance of early mathematics learning. She served as president of the local NCTM affiliate Greater Louisville Council of the Teachers of Mathematics and was the cha ir of the volunteers for the NCTM Regional in Louisville in 1998 and 2013. Todd was the 1998 Kentud:y Presidential Awardee for Excellence in Teaching Mathematics and Science.

iv

Brief Contents PART I

Teaching Mathematics: Foundations and Perspectives

CHAPTER 1

Teaching Mathematics In the 21st Century

CHAPTER 2

Exploring What It Means to Know and Do Mathematics

CHAPTER 3

Teaching through Problem Solving

CHAPTER 4

Planning In the Problem-Based Classroom

CHAPTER 5

Creating Assessments for Learning

CHAPTER 6

Teaching Mathemat ics Equitably to All Students

PART II

1 13

30 55

83 103

Development of Mathematical Concepts and Procedures

CHAPTER 7

Developing Early Number Concepts and Number Sense

125

CHAPTER 8

Developing Meanings for the Operat ions

CHAPTER 9

Developing Basic Fact Fluency

CHAPTER 10

Developing Whole-Number Place-Value Concepts

CHAPTER 11

Developing Strategies for Addition and Subtract ion Computation

CHAPTER 12

Developing Strategies for Multiplicat ion and Division Computation

CHAPTER 13

Algebraic Thinking, Equations, and Functions

CHAPTER 14

Developing Fraction Concept s

CHAPTER 15

Developing Fraction Operations

CHAPTER 16

Developing Decimal and Percent Concepts and Decimal Computation

CHAPTER 17

Rat ios, Proportions, and Proport ional Reasoning 435

CHAPTER 18

Developing Measurement Concepts

CHAPTER 19

Developing Geometric Thinking and Geometric Concepts

CHAPTER 20

Developing Concepts of Data and Statistics

CHAPTER 21

Exploring Concept s of Probability

CHAPTER 22

Developing Concepts of Exponents, Integers, and Real Numbers

APPENDIX A

St andards for Mathematical Practice

APPENDIX B

NCTM Mathematics Teaching Practices from Principles to Actions

APPENDIX C

Guide to Blackline Masters

APPENDIX D

Activities at a Glance

153

183 211 238 273

299

337 373 405

460 500

543

580 604

632 635

637

643

v

Contents Preface

xiii

PART I

Teaching Mathematics: Foundations and Perspectives

The fundamental core of effective teaching of mathematics combines an understanding of how students lea rn, how to promote that learning by teaching through problem solving, and how to plan for and assess that learning da ily. That is the focus of these first six chapters, providing discussion, examples, and activities that develop the core ideas of learning, teaching, planning, and assessment for each and every student. CHAPTER 1 Teaching Mathematics In the 21st Century 1

CHAPTER 3 Teaching through Problem Solving

Becoming an Effective Teacher of Mathematics A Changing World 2 Factors to Consider 3 The Movement t oward Shared Standards Mathematics Content Standards 5

4

Resources for Chapter 1

Tasks That Promote Problem Solving High·Level Cognitive Demand 37 Multiple Entry and Exit Points 38 Relevant Contexts 40 Evaluating and Adapting Tasks 42

10

12

Exploring What It Means t o Know and Do Mathemat i cs 13

How Do Students Learn Mathematics? 25 Constructivism 25 Sociocultural Theory 25 Implications for Teaching Mathematics 26 Connecting the Dots

28

Resources for Chapter 2

vi

29

Orchestrating Classroom Discourse Classroom Discussions 48

13

What Does It Mean to Know Mathematics? Relational Understanding 20 Mathematical Proficiency 22

37

Developing Procedural Fluency 45 Example Tasks 45 What about Drill and Practioe? 47

CHAPTER 2

What Does It Mean to Do Mathematics? Goals for Students 14 An Invitation to Do Mathematics 14 Where Are the Answers? 19

Problem Solving 30 Teaching for Problem Solving 31 Teaching about Problem Solving 31 Teaching through Problem Solving 34 Teaching Practices for Teaching through Problem Solving 35 Ensuring Success for Every Student 35

The Process Standards and Standards for Mathematical Practice 6 How to EffectivelyTeach the Standards 8 An Invitation t o Learn and Grow 9 Becoming a Teacher of Mathematics

1

30

48

Questioning Considerations 51 How Much to Tell and Not to Tell 52 Writing 52

19

Resources for Chapter 3

54

CHAPTER 4 Planning In t he Problem-Based Classroom A Three·Phase Lesson Format 55 The Before Lesson Phase 55 The During Lesson Phase 58 The After Lesson Phase 60

55

Contents Process for Preparing a l esson 62 Step 1: Determine the learning Goals 62 Step 2: Consider Your Students' Needs 63 Step 3: Select, Design, or Adapt a Worthwhile Task Step 4: Design l esson Assessments 64 Step 5: Plan the Before Phase 64 Step 6: Plan the During Phase 65 Step 7: Plan the After Phase 65 Step 8: Reflect and Refine 66 High-leverage Routines 66 3-Act Math Tasks 67 Number Talks 67 Worked Examples 67 Warm-ups and Short Tasks 68 learning Centers 69 Differentiating Instruction Open Questions 70 Tiered l essons 70 Parallel Tasks 73 Aexible Grouping 73

69

Planning for Family Engagement 74 Communicating Mathematics Goals 74 Family Math Nights 75 Homework Practices 78 Resources for Families 79 Involving A// Families 80 Resources for Chapter 4

81

CHAPTER 5 Creating Assessments for Learn ing 83 Integrating Assessment into Instruction 83 What Are the Main Assessment Types? 84 What Should Be Assessed? 85 Assessment Methods Observations 86

86

Questions 88 Interviews 88 Tasks 91 Rubrics and Their Uses 95 Generic Rubrics 95 Task-Specific Rubrics 96

PART II

Student Self-Assessment

63

98

Tests 99 Expanding the Usefulness atTests 99 Improving Performance on High-Stakes Tests

100

Communicating Grades and Shaping Instruction Grading 100 Shaping Instruction

vii

100

101

Resources for Chapter 5

102

CHAPTERS Teaching Mathematics Equitably t o AU Students 103 Mathemat ics for Each and Every Student

104

Providing for Students Who Struggle and Those with Special Needs 105 Multitiered System of Support: Response to Intervention 105 Implementing Interventions 106 Teachi ng and Assessi ng Students with learning Disabilities 110 Adapting for Students with Moderate/ Severe Disabilities 112 Cult urally and Unguistically Diverse Students 112 Funds of Knowledge 113 Mathematics as a l anguage 113 Culturally Responsive Mathematics Instruction 114 Teaching Strategies That Support Culturally and linguistically Diverse Students 116 Focus on Academic Vocabulary 116 Foster Student Participation during Instruction 118 Implementing Strategies for English learners 120 Providing for Students Who Are Mathematically Gifted Acceleration and Pacing 122 Depth 122 Complexity 122 Creativity 122 Strategies to Avoid 123 Reducing Resistance and Building Resilience Give Students Choices That Capitalize on Their Unique Strengths 123 Nurture Traits of Resi lience 123 Make Mathematics Irresistible 124 Give Students l eadership in Their Own l earning Resources for Chapter 6

121

123

124

1.2 4

Teaching Student-Centered Mathematics

Each of these chapters applies the core ideas of Part I to the content taught in K-8 mathematics. Clear discussions are provided for how to teach the topic, what a learning progression for that topic might be, and what worthwhile tasks look like. Hundreds of problem-based, engaging tasks and activities are provided to show how the conceptS can be developed with students. T hese chapters are designed to help you develop pedagogical strategies now, and serve as a resource and reference for your teaching now and in the future.

viii

Contents

CH PTER 7 Deve.l oplng Early Number Concepts and Number Sense 125 Promoting Good Beginnings 126 The Number Core: Quantity, Counting, and Cardinality Quantity and the Ability to Subitize 127 Counting 128 Cardinality 130 Thinking about Zero 131 Numeral Writing and Recognition 131 Counting On and Counting Back 133 The Relations Core: More Than, Less Than, and Equal To 134 Developing Number Sense by Building Number Relationships 136 Relationships between Numbers 1 through 10 136 Relationships for Numbers 10 through 20 and Beyond Number Sense In Their World 147 Calendar Activities 14 7 Estimation and Measurement 148 Represent and Interpret Data 149 Resourtes for Chapter 7

127

145

One More Than and Two More Than (Count On) 190 Adding Zero 191 Doubles 192 Combinations of 10 193 193 10 + Making 10 193 Use 10 195 Using 5 as an Anchor 195 Near-Doubles 195 Reasoning Strategies for Subtraction Facts 196 Think-Addition 196 Down under 10 198 Take from 10 198 Reasoning Strategies for Multiplication and Division Facts 199 Foundational Facts: 2, 5, 10, 0, and 1 199 Nines 201 Derived Multiplication Fact Strategies 202 Division Facts 203 Reinforting Basic Fact Mastery 205 Games to Support Basic Fact Auency 205 About Drill 207 Fact Remediation 208

151

Resourtes for Chapter 9

210

CHAPTER 8 CHAPTER 10

Developing Meanings for the Operatlon.s 153 Developing Addition and Subtraction Operation Sense Addition and Subtraction Problem Structures 155 Teaching Addition and Subtraction 158 Properties of Addition and Subtraction 164 Developing Multiplication and Division Operation Sense Multiplication and Division Problem Structures 166 Teaching Multiplication and Division 169 Properties of Multiplication and Division 174 Strategies for Teaching Operations through Contextual Problems 175 Resourtes for Chapter 8

182

Pre-Place-Value Understandings 166

212

Developing Whole-Number Place-Value Concepts 213 Integrating Base-Ten Groupings with Counting by Ones 213 Integrating Base-Ten Groupings with Words 214 Integrating Base-Ten Groupings with Place-Value Notation 215 Base-Ten Models for Place Value Groupable Models 216 Pre~uped Models 216 Nonproportional Models 217

215

Activities to Develop Base-Ten Concepts Grouping Activities 218 Grouping Tens to Make 100 220 Equivalent Representations 22 1

CHAPTER 9 Developi ng Basic Fact Fluency 183 Teaching and Assessing the Basic Facts 184 Developmental Phases for learning Basic Facts 184 Approaches to Teaching Basic Facts 184 Teaching Basic Facts Effectively 186 Assessing Basic Facts Effectively 188 Reasoning Strategies for Addition Facts

154

Developing Whole-Number Place-Value Concepts 2U

189

211

Reading and Writing Numbers 223 Two-Digit Number Names 223 Three-Digit Number Names 225 Written Symbols 225 Place Value Patterns and Relationships-A Foundation for Computation 227 The Hundreds Chart 227 Relative Magnitude Using Benchmark Numbers 230

Contents Approximate Numbers and Rounding 232 Connections to Real-World Ideas 232

Invented Strategies for Division

Numbers Beyond 1000 232 Extending the Place-Value System 232 Conceptualizing Large Numbers 234 Resources for Chapter 10 237

Computational Estimation 292 Teaching Computational Estimation 292 Computational Estimation Strategies 293 Resources for Chapter 12 298

CHAPn:R 11 Developing Strategies for Addition and Subtraction Computation 238 Toward Computational Fluency 239 Connecting Addition and Subtraction to Place Value

240

Three Types of Computational Strategies 246 Direct Modeling 246 Invented Strategies 24 7 Standard Algorithms 249 Development of Invent ed Strategies In Addition and Subtraction 251 Creating a Supportive Environment for Invented Strategies 251 Models to Support Invented Strategies 252 Adding and Subtracting Single-Digit Numbers 254 Adding Multidigit Numbers 256 Subtraction as "Think-Addition• 258 Take-Away Subtraction 259 Extensions and Challenges 260 Standard Algorithms for Addition and Subtraction 261 Standard Algorithm for Addition 261 Standard Algorithm for Subtraction 263 Introducing Computational Estimation 265 Understanding Computational Estimation 265 Suggestions for Teaching Computational Estimation 265 Computational Estimation Strategies 267 Front-End Methods 267 Rounding Methods 267 Compatible Numbers 268 Resources for Chapter 11 272

CHAPTER 12 Developi ng Strategies for Mult iplication and Division Computation 273 Invented Strategies for Multiplication 274 Useful Representations 274 Multiplication by a Single-Digit Multiplier 275 Multiplication of Multidigit Numbers 276 Standard Algorithms for Multiplication Begin with Models 2 79 Develop the Written Record 282

279

283

Standard Algorith m for Division 286 Begin with Models 287 Develop the Written Record 288 Two-Digit Divisors 290 A Low-Stress Approach 291

CH p· ER13 Algebraic Thinking, Equations, and Functions 299 Strands of Algebraic Thinking

300

Connecting Number and Algebra 300 Number Combinations 300 Place-Value Relationships 301 Algorithms 303 Properties of the Operations 303 Making Sense of Properties 303 Applying the Properties of Addition and Multiplication Study of Patterns and Functions 307 Repeating Patterns 308 Growi11g Patterns 309 Relationships in Functions 310 Graphs of Functions 312 Unear Functions 314 Meani ngful Use of Symbols 317 Equal and Inequality Signs 318 The Meaning of Variables 325 Mathematical Modeling 330 Algebraic Thinking across the Curriculum 332 Geometry, Measurement and Algebra 332 Data and Algebra 333 Algebraic Thinking 334 Resources for Chapter 13 335

CHAPTER 1.4 Developing Fraction Concepts 337 Meanings of Fractions 338 Fraction Constructs 338 Fraction Language and Notation Fraction Size Is Relative 340 Models for Fractions 341 Area Models 341

339

306

ix

X

Contents The Role of the Decimal Point 407 Measurement and Monetary Units 408 Precision and Equivalence 409

Length Models 343 Set Models 344 Fractions as Numbers 346 Partitioning 346 Iterating 354 Magnitude of Fractions 35 7 Equivalent Fractions 359 Conceptual Focus on Equivalence 359 Equivalent Fraction Models 360 Fractions Greater than 1 363 Developing an Equivalent-Fraction Algorithm

Connecting Fractions and Decimals 409 Say Decimal Fractions Correctly 409 Use Visual Models for Decimal Fractions 410 Multiple Names and Formats 412 Developing Decimal Number Sense 414 Familiar Fractions Connected to Decimals 414 Comparing and Ordering Decimal Fractions 418 365

Computation with Decimals 421 Add ition and Subtraction 421 Multiplication 423 Division 42 6

Comparing Fractions 367 Comparing Fractions Using Number Sense 367 Using Equivalent Fractions to Compare 370 Teaching Considerations for Fraction Concepts Fraction Challenges and Misconceptions 3 70

370

Resources for Chapter 14 372

Introducing Percents 428 Physical Models and Terminology 428 Percent Problems in Context 430 Estimation 432 Resources for Chapter 16

434

CHAPTER 15 Developing Fraction Operations 373 Understanding Fraction Operations Effective Teaching Process 3 74

CHAPTER 17 Ratios, Proportions, and Proportional Reasoning 435

374

Addition and Subtraction 376 Contextual Examples 376 Models 377 Estimation 380 Developing the Algorithms 381 Fractions Greater Than One 384 Challenges and Misconceptions 384

Ratios 436 Types of Ratios 436 Ratios Compared to Fractions 437 Two Ways to Think about Ratio 437 Proportional Reasoning 438 Types of Comparing Situations Covariation 443

Multiplication 386 Contextual Examples and Models 386 Estimation 392 Developing the Algorithms 393 Factors Greater Than One 393 Challenges and Misconceptions 394

Strategies for Solving Proportional Situations Rates and Scaling Strategies 450 Ratio Tables 453 Tape or Strip Diagram 454 Double Number Une Diagrams 455 Equations (Cross Products) 456 Percent Problems 457

Division 394 Contextual Examples and Models 395 Answers That Are Not Whole Numbers 400 Estimation 401 Developing the Algorithms 401 Challenges and Misconceptions 403 Resources for Chapter 15

439

Teaching Proportional Reasoning Resources for Chapter 17

449

458

459

404 CHAPTER 18 Developing Measurement Concepts 460

CHAPTER 16 Developing Deci mal and Percent Concepts and Decimal Computation 405 Extending the Place-Value System 406 The 10-to-1 Relationship-Now in Two Directions!

406

The Meaning and Process of Measuring Concepts and Skills 462 Introducing Nonstandard Units 463 Introducing Standard Units 464 Developing Unit Familiarity 464

461

xi

Contents Elqlloring Properties of Quadrilaterals 519 Exploring Polygons 521 Circles 521 lnvestlg;Jtions, Conjectures, and the Development of Proof 522

466

Measurement Systems and Units

The Role of Estimation and Approxlmation 466 Strategies for Estimating Measurements 467 Measurement Estimation Activities 468 Length 469 Comparison Activities

Area 474 Comparison Activities

Transformations 523 Symmetries 525 Composition ofTransformations 526 Congruence 528 Similarity 529 Dilation 529

469

Using Physical Models of Length Units Making and Using Rulers 4 71 Conversion 4 73

4 70

474

Location 530 Coon11nate Plane 531 Measuring Distance on the Coordinate Plane

Using Physical Models of Area Units 4 75 The Relationship between Area and Perimeter 4 77 Developing Formulas for Perimeter and Area 4 79 Volume and Capacity 484 Comparison Activities 485 486

Using Physical Models of Volume and Capacity Units Developing Formulas for Volumes of Common Solid Shapes 487

534

Visualization 534 l'wo-Dimensionallmagery 535 Three-Dimensional lmagery 536 Resources for Chapter 19

542

Weight and Mass 489 Comparison Activities 489 Using Physical Models of Weight or Mass Units 489 Angles 490 Comparison Activities 490 Using Physical Models of Angular Measure Units 490 Using Protractors 491 Time 492 Comparison Activities 492 Reading Clocks 493 Elapsed Time 494 Money 495 Recognizing Coins and Identifying Their Values Resources for Chapter 18

CHAPTER 20 Developing Concepts of Data and Statist ics What Does It Mean to Do Statistics? 544 Is It StatiStics or Is It Mathematics? 545 The Shape of Data 545 The Process of Doing Statistics 546 Formulating Questions 549 Classroom Questions 549 Questions beyond Self and Classmates

495

499

Data Collection 552 Sampling 552 Using Existing Data Sources Data Analysl.s: Classification Attribute Materials 554

Geometry Goals for Students

501 501

The van Hiele Levels of Geometric Thought Implications for Instruction 506

553 554

Data Analysis: Graphical Representations Creating Graphs 55 7 Bar Graphs 558 Pie Charts and Circle Graphs 560 Continuous Data Graphs 561 Bivariate Data 564

CHAPTER 19 Developing Geometric Thinking and Geometric Concepts 500

Developing Geometric Thinking

549

502

Shapes and Properties 507 Sorting and Classifying 508 Composing and Decomposing Shapes 510 categories ofTwo- and Three-Dimensional Shapes Construction Activities 515 Applying Definitions and categories 516 Elqlloring Properties ofTriangles 516 Midsegments of a Triangle 518

512

557

Data Analysis: Measures of Center and Variability Measures of Genter 568 Understanding the Mean 569 Choosing a Measure of Center 572 variability 514 AnalyzJng Data 576 Interpreting Results

577

Resources for Chapter 20

578

568

543

xii

Contents

CHAPTER 22 Developing Concepts of Exponents, Integers, and Real Numbers 604

CHAPTER 21 Exploring Concepts of Probability 580 Introducing Probability 581 Li kely or Not Likely 581 The Probability Continuum 584 Theoretical Probability and Experiments Process for Teaching Probability 586 Theoretical Probability 587 Experiments 589 Why Use Experiments? 592 Use ofTechnology in Experiments 592

Exponents 605 Exponents in Expressions and Equations 605 Order of Operations 606 Exploring Exponents on the Calculator 609 Integer Exponents 610 Scientific Notation 611

585

Sample Spaces and the Probability of Compound Events Independent Events 593 Area Representation 596 Dependent Events 597 Simulations 598 Student Assumptions Related to Probability Resources for Chapter 21

593

Positive and Negative Numbers 614 Contexts for Exploring Positive and Negative Numbers 614 Meaning of Negative Numbers 616 Tools for Illustrating Positive and Negative Numbers 618 Operations with Positive and Negative Numbers Add ition and Subtraction 619 Multiplication 623 Division 624 Real Numbers 626 Rational Numbers 626 Irrational Numbers 627

601

602

Supporting Student Reasoning about Number Resources for Chapter 22

APPENDIX A Standards for Mathematical Practice

631

632

APPENDIX B NCTM Mathematics Teaching Practices from Principles to Actions APPENDIX C Guide to Blackline Masters APPENDIX D Activities at a Glance

References 662 Index 687 Credits 697

643

619

637

635

629

Preface All students can learn mathematics with understanding. It is through the teacher's actions that every student can have this experience. vVe believe that teachers must create a classroom environment in which students are given opportunities to solve problems and work together, using their ideas and strategies, to solve them. Effective mathematics instruction involves posing tasks that engage students in the mathematics they are expected to learn. Then, by allowing students to interact with and productively struggle with tbei1· O'IVIl ?JifJtbematical ide11s and tbei1· fYIVn str11tegies, they will learn to see the connet.-rions among mathematical topics and the real world. Students va lue mathematics and feel empowered to use it. Creating a classroom in which students design solution pathways, engage in productive struggle, and conne tlonal~bieS..

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To help readers know what they should expect to learn , each chap. ter begins with learning outcomes. Self-checks are numbered to cover and thus align with each learning outcome.

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Much of the research and literature espousing a student-centered approach suggests that teachers plan their instruction around big ideas rather than isolated skills or concepts. At the beginning of each chap. ter in Part II, you will find a list of the big mathematical ideas associated with the chapter. Teachers find these lists helpful to quickly envision the mathematics they are to teach.

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Books Carpenter, T P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. (2014). Childnmbnnrhm111tirs: Cognitively guided instruction. (2nd Edition) Portsmouth, N H: Heinemann.

This is the clns:rir book for tmdtrftnruling wo•·d-pr-oblem structures fo•· all oprratior1s. The srructm-rs nr·e explained in detail along with 11mhods for t1Si11g thm problems with mtdmts. Vidtos ofclas:rrtX1111S and rhildrtn mod~li11g strategies arr includtd. Caldwell, J.. Koben, B., & Karp, K. (20 14). Puning essmtial undtrrtanding ofaddition and sltbtraction mto /Traerire i11 grad~s f1rrkindrrganm-2. Reston, VA: CTM. Lannin, J.. Ch,'lll, K., & Jones, D. (2013). Putti11g ~ssmtial undtrrtanding of multipliratiofl and divisio11 mto practice in grades J-5. Reston, VA: NCrM. Thm ru:o books offn· th~ big idtas abollt hrr

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FIGURE 10.15

The triples system for nam ing large numbers.

to name the unit for that triple (see Figure I0. 15). Leading zeros in each triple are ignored. If students can learn to read numbers like 059 (fifty-nine) or 009 (nine), they should be able to read any number. To write a number, use the same scheme. If first mastered orally, the system is quite easy. Remind studentS not to use the word "and" when reading a whole number. For example, 106 should be read as "one hundred six," not "one hundred and six." The word "and" will be needed to signify a decima l point. Please make sure you read numbers accurately. It is important for studentS to realize that the system does have a logical structure, is not totally arbitrary, and can be understood.

Conceptualizing Large Numbers The ideas just discussed are only partially helpful in thinking about the actual quantities involved in very large numbers. For example, in extending the paper square-strip-squarestrip sequence, some appreciation for the quantities of 1000 or of I00,000 is acquired. But it is hard for anyone to translate quantities of small squares into quantities of other items, distances, o r time.

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Mylab Education Video Example 10.4 Explore this video of Donny reasoning about 192,000 as 1 9,200 tens or 1 920 hundreds.

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....

VIDEO EXAMP\.E

These ideas are important to discuss as a class as students need to explore these relationships.

WPause & Reflect How do you think about 1000 or 100,000? Do you have any real concept of a million? •

In the following activities, numbers like 1000, I 0,000 (see I0,000 Grid Paper), or even I,000,000 are translated literally or imaginatively into something that is more meaningful or fun to think about. Interesting quantities become lasting reference points or benchmarks for large numbers and thereby add meaning to numbers encountered in real life.

Mylab Education

Blackline Master: 10,000 Grid Paper

Numbers Beyond 1000

Activity 10.21

235

CCSS-M: 2 .NBT.A .2 ; 2.NBT.A.3

Collecting 10,000 As a class or grade-level project, collect some type of object with the objective of reach ing some specific quantity-for example, 1,000 or 10,000 bread tabs or soda can pop tops. If you begin aiming for 100,000 or 1,000,000, be sure to think it through. One teacher spent nearly 10 years with her classes before amassing a million bottle caps. It takes a small dump truck to hold that many!

Activity 10.22

CCSS-M: 2 .NBT. A.2 ; 2.NBT.A.3

Showing 10,000 Sometimes it is easier to create large amounts than to collect them. For example, start a project in which students draw 100 or 200 or even 500 dots on a sheet of paper. Each week, different students contribute a specif ied num ber. Another idea is to cut up newspapers into pieces the same size as dollar bills to see what a large quantity would look like. Paper chain links can be constructed over time and hung down the hallways with special num bers marked. Let the school be aware of the ultimate goal.

Activity 10.23

CCSS-M: 2 .NBT. A.1a; 2 .NBT.A .3

How Long?/How Far? In this activity, talk about real or imagined distances with students by posing investigations for them to consider such as, How long is a million baby steps? Other ideas that address length include a line of toothpicks, dollar bills, or energy bars end to end; students holding hands in a line; blocks or bricks stacked up; or students lying down head to toe. Standard measures-feet, centimeters, meters-can also be used with students noting that larger numbers emerge when the smallest units are used.

Activity 10.24

CCSS-M: 3. MD.A.1

~

lili!J

A Long Time

How long is 1000 seconds? How long is a m illion seconds? A billion? How long would it take to count to 10,000 or 1 ,000,000? (To make the counts all the same, use your calculator to do the counting. Just press the @l How long would it take to do some task like buttoning a button 1000 times?

Activity 10.25

CCSS-M: 2. MD.A.3; 3.MD.A.1; 3 .MD.C.5

Really Large Quantities Ask how many: •

Energy bars would cover the floor of your classroom

•

Steps an ant would take to walk around the school building

•

Grains of rice would fill a cup or a gallon jug

•

Quarters could be stacked in one stack from floor to ceiling

•

Pennies can be laid side by side down the entire hallway

•

Pieces of notebook paper would cover the gym floor

•

Seconds you have lived

Big-number projects need not take up large amounts of class time. T hey can be explored over several weeks as take-home projects, done as group projects, or, perhaps best of all, translated into great schoolwide estimation events.

236

C h apt er 10 Developing W h ole-Nu mber Place-Valu e Concept s

Mylab Education

Self-Check 10.7

Mylab Education

Blackline Master: Base-Ten Materials

Now that you've explored many of the main ideas in this chapter, look at Table I0. 1 for some of the possible common challenges and misconceptions your students may face. Suggestions on what you might notice and how to help are included.

TABLE 10.1

COMMON CHALLENGES AND MISCONCEPTIONS IN PLACE VALUE AND HOW TO HELP

Common Challenge or Misconcepti on

1. Students lose track of the fact that each digit In a mult l dlglt numeral carries a val ue dependent on Its position In t he number.

What It Looks Uke

How to Help

When students are asked to compare the numbers (2) bolded in t he two amounts that follow t hey will say they are the same. 2357 and 49,992 .

• • • • •

2. Student reverse the d lglts when writing twCHIIglt numbers.

Writ es "53" when should write "35."

3. Student represents

When asked to represent 13 with base-ten mat erials, the student uses one piece for the "1" and three pieces for the "3" as shown

a number with base-ten materials using t he face value of the d lglts.

1

3

D

DD D

• • • •

Use Base-Ten Materials and have student show with materials the value of these two numbers. The reading of numbers in addit ion or subtraction problems as digits (saying 5 inst ead of 5 tens or fifty) conf uses students. Use the place value cards discussed previously to reinforce how numbers a re built . Student s hear numbers like 2357 read as two, three, five, seven- when they should always be read two thousand, three hundred, fifty-seven. Use the d igit correspondence task described in this chapter to identify which of the five levels o f understanding matches your student's performance. Reinforce that the value of an individual d igit in a multidig it number is the product of that digit mult iplied by the value assigned to it s posit ion in the number Have students use virtual base-ten mat erials that d isplay the corresponding number to check their answer. Have the child model both 53 and 35 with base-ten mat erials and describe how the numbers are similar and different. Have the student use base-ten materials t o count out

13 single units. Then ask them to compare that amount to what they previously showed. Have the st udent build the number wit h the place value cards and then use base-ten materia ls t o represent the corresponding amounts. Again, compare t o the amount originally shown.

4. Students put the word "and" In a number when t hey read It aloud.

When reading 1016 students will say "one thousand and sixteen."

•

St udent s must practice reading numbers without using the word •and." The only t ime the word ·and" is used is to represent a decimal point.

S. Students use a form of "expanded number writing" (Byrge, Smith, & M ix, 2013).

Students write "three hundred eightyfive" as something like 300805, 310085 or 3085.

•

Provide examples of the act ual mat erials on a place value mat and use t he p lace value cards to show how the mat ching number is built.

6. When shown a collection of base-ten materials where there Is an Internal zero the students Ignore the zero or mlsunderstand the zero.

Given 5 hundreds and 8 units in baseten materials, st udent s will write that number as 85. The student believes that 802 and 8002 represent the same amount .

•

Focus on the meaning of a zero in any number by starting with a number like 408 and asking how that would be shown with materials. Explicitly discuss the role of an internal 0 in the number. Never refer to 0 as a "placeholder.• This terminology gives the impression t hat it is not a numerical value and it is there just as a way t o fill a space. Never read or refer to 0 as oh or zip. Say •zero• as it is a number.

7. If students are g iven the p lace values of numbers out of order they write the number as given left to right regardless of the pl ace value.

When st udents are asked to write the number that represent s: 7 ones, 4 tens, 1 thousand, and 3 hundreds. They write 7413.

8. Students misinterpret the val ue of the base-ten materials.

Students think the value of t he large 1000 place value block is actually 600 by just calculating the number of squares on each face of t he cube.

• • • •

Go back to the base-ten materials and use the place value mat to take out the same amount of base-ten mat erials as in the number. Then have the student writ e the number of base-ten materials. The st udent should then compare the two answers to consider which one is accurate Particularly with the 1000 place value b lock, if students don't see the building of t he block (grouping into a unit), they may confuse the value. So, explicitly show the bui lding of the cube by taking ten hundreds blocks and forming a cube with them (holding them together with elastic bands.)

Numbers Beyond 1000

237

RESOURCES FOR CHAPTER 10 LITERATURE CONNECTIONS

If I Had a Million Bucks

Books that emphasize groups of things, even simple counting books, are a good beginning to the notion of ten things in a single group. .\lany books have wonderful explorations of large quantities and how the numbers can be composed and decomposed.

Johnson (2012) Tills story is about Ada, a girl who likes to plan. Ada is thinking about what she could do if she had one million dollars. Tills planning is a fun way to have students think about what could be purchased "~th large amounts of money.

1.00th Day Worries

RECOMMENDED READINGS

Cuyler (2005)

The 1.00th Day of School from the Black Lagoon Thaler, 2014 Both of these books focus on the IOOth day of school, wruch is one way to recognize the benchmark number of I 00. Through a variety of ways to think about I00 (such as collections of I 00 items), students will be able to use these stories to think about the relative size of 100 or ways to make 100 using a variety of combinations.

How Much Is a Million? Schwartz (2004)

If You Made a Million Schwartz ( 1994)

On Beyond a Million: An Amazing Math Journey

Articles Burris, J. T. (2013). Virtual place \>alue. Ttachi11g Childnm Mathematics, 20(4), 228-23 6. Tbis interesting study explo1·es tencbing plate value using several activities witb techuology. Studmts wo,·ked wirb botb conn-ere bose-ten materials 1111d virrual vmions to belp reinfone coucepntal srmctm·es, particulltdy fo,· sbowing equivalem ''·ep,·esentntions. Kari, A. R. , & Anderson, C. B. (2003). Opportunities to develop place value through student dialogue. uaching Childtro Mathematics, 10(2), 78-82. r or < ), using the symbols and saying verbally what they mean. Include examples for which students can make the determination by analyzing the relationships on the two sides rather than by doing the computation. Visuals and manipulatives can be a support for students with disabilities. Also, use m issing-value expressions (see Figure 13.14[b)). Ask students to find a number that will result in one side tilting downward, a number that will result in the other side tilting downward, and one that will result in the two sides being balanced. (See Expanded Lesson: Tilt or Balance for more a full lesson.)

Mylab Mylab Mylab Mylab

Education Education Education Education

D

Standards for Mathematical Practice MPS. Use appropriate tool s strategically.

Activity Page: Tilt or Balance Equation Cards Set A Activity Page: Tilt or Balance Equation Cards Set B Activity Page: Balance Scales Placemat Expanded Lesson: Tilt or Balance

TECHNOLOGY Note. T here are severa l excellent virtua l balance activities:

• PBS Cyberchase Poddle Weigh-in: Shapes are balanced with numbers between 1 and 4. • Agame Monkey Math Balance: Students select numbers for each side of a balance to make the two sides balance (level of difficulty can be adapted). • NCTM Illuminations Pan Balance-Numbers or Pan Balance-Expressions provide a virtual balance where students can enter what they bel ieve to be equivalent expressions (with numbers or symbols). •

True/ False Sentences. True or false statements can help bui ld relational understand ing and address student misconceptions (Carpenter et al., 2003). Any of these could be the focus of a N umber Talk. And, the concrete idea of a ba lance (or seesaw) can support student reasoning. Students will gen erally agree on equations when there is an expression on one side and a single number on the other, although initially the less familia r form of7 = 5 + 2 may generate discussion. For an equation with no operation (8 = 8), the discussion may be lively. Reinforce

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Activity 13.14

323

~ ~

CCSS-M : 1.0A.B.3 ; 1 .0A.D.7 ; 1 .NBT.B. 4 ; 2 .NBT.B.5; 3.0A.B .5 ; 4 .NBT.B.5 ; 5.NF.A.1

True or False?

ENGLISH LEARNERS

Introduce true/ false sentences or equations with simple examples to explain what is meant by a true equation and a false equation. Then put several sim ple equations on the board, some true and some false. The following are appropriate for primary grades:

7 = 5 + 2

4 + 1 = 6

4 + 5 = 8 + 1

8 = 10 - 1

Your collection might include other operations, but keep the computations simple. Ask students to talk to their partners and decide which of the equations are true equations (and why) and which are not (and why not). For older students, use fractions, decimals, and larger numbers.

120 = 60 X 2

STUDENTS with SPECIAL NEEDS

31 8 = 318

3 2 1 =-+4 1

1 1 1 -=- + 2 4 4

345 + 71 = 70 + 344

1210 - 35 = 1310 - 45

0.4 X 15 = 0.2 X 30

listen to the types of reasons that students use to j ustify their answers, and plan additional equations accordingly. Els and students with disabilities will benefit from first explaining (or showing) their thinking to a partner (a low-risk practice) and then sharing with the whole group.

that the equal sign means "is the same as" by using that language when you read the symbol. Inequalities should be explored in a similar manner.

Solving equations. T he ba lance is a concrete tool that can help students understand that if you add or subtract a value from one side, you must add or subtract a like value from the other side to keep the equation balanced. Figure 13.1 5 shows solutions for two equations, one in a balance and the other without. The notion of preserving balan ce also applies to inequalities-but what is preserved is imbalance. In other words, if one side is more than the other side and you subtract 5 from both, the one side is still more. Even after you have stopped using the balance, the mental

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1....

Standards for

Mat hemat ical Practice MP3. Construct viable arguments and critique the reasoning of others.

Check:

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1.... 4-6x J

II

Multtply both sides by 2. )

8.4N = N - 126

Subtract N from both sides. 7.4N = - 126 Divide both sides by 7 .4. (Use a calculator!) N = - 17.03 (about)

FIGURE 13.15

Using a balance scale to build understand ing of solving equations.

324

Chapt er 13 Algebraic Thinking, Equations, and Functions

idea of balance can be used to reinforce the concept of equivalent expressions. For example, open sentences can be explored using reasoning Strategies. A~ early as first grade studentS ~-an understand and benefit from using variables (Blanton et al., 2011). Encourage studentS to look at the number sentence holistically and discuss in words what is missing (and how they reasoned to figure it out).

Activity 13.15

CCSS-M : 1..0A.D.7 ; 1..0A.D.8; 2 .0A.A .1.; 3.0A.A. 4 ; 5 .0A.A.2; 6 .EE.A.3; 6 .EE.B.5

What's Missing? Prepare a set of Missing-Value Equations. Ask students to figure out what is missing and how they know. Notice that the equations are set up so that students do not always have perform t he operations to figure out what is missing. Encourage them to look at the equation and see if they can figure out what is missing without solving it. Probe to see if there is more than one way to find what is m issing. Here is a sampling of ideas across the grades:

4 +0 = 6

4 + 5 =0- 1

15 + 27 = n + 28

12

0.5+a= 5

4.5 + 5.5 = a + 1

X

n

=

24

X

5

0+ 5 = 5 + 8

3 X 7 = 7 X0

6 Xn = 3 X8

15

a x 4 = 4.8

2.4 .,. a = 4.8 .,. 6

X

27 = n

X

27 + 5

X

27

Using "messy" numbers further encourages applying relational understanding to solve:

x 18 .,. 37 = n

126 - 37 = n - 40

37

68 + 58 = 57 + 69 + n

7.03 + 0.056 = 7.01 + n

MyLab Education

20 X 48 = n X 24

Activity Page: Missing-Value Equations

Molina and Ambrose (2006), used true/false and open-ended promptS with third graders who did not have a relational understanding of the equal sign. For example, all 13 students answered 8 + 4 = _ + 5with 12.They foundthataskingstudentsto wt·itetbeir ownopensentenceswas particularly effective in helping students solidify their understanding of the equa l sign.

Activity 13.16

CCSS- M: 1..0A.D.7 ; 1..0A.D.8; 2 .0A.A .1.; 3.0A.A. 4 ; 4 .NF.8 .3a ; 5 .0A.A.2 ; 6 .EE .8 .8

Make a Statement! Ask students to write their own true/ false and open sentences that they can use to challenge their classmates. This works for both equations and inequalities! To support student t hinking, provide dice with numerals on them.

STUDENTS with SPECIAL NEEDS

They can turn the dice to different faces to try different possibilities.

Ask students to write three equations (or inequalities) with at least one true and at least one false sentence. For students who need additional structure, in particular students with disabilities, provide the Make a Statement Recording Page, which provides options for writing equations and inequalities. St udents can t rade their set of statements with other students to find the False Statement. Interesting equations/ inequalities can be the focus of a follow-up whole-class discussion.

Mylab Education

Activity Page: Make a Statement Recording Page

Meaningful Use of Symbols

325

When students write their own true/false sentences, they often are intrigued with the idea of using large numbers and lots of numbers in their sentences. Trus encourages them to create sentences involving relational-structural thinking. In middle school, students begin to manipulate equations so that they are easier to graph and/or to compare to other equations. Students need many and ongoing opportunities to explore problems that encourage relational trunking (Stephens et al., 2013). Balance activities, true/false, and open sentences help ground a student's understanding of how to preserve equivalence when "mO\~ng" numbers or variables across the equa l sign (or inequality symbols).

n

FORMATIVE ASSESSMENT Notes. As students work on these types of tasks, you can

LJ interview them one on one (though you may not get to everyone). Listen for whether they are using relational-structural thinking. If they are not, ask, "Can you find the answer without actually doing any computation?" This questioning helps nudge students toward relational thinking and helps you decide what instructional steps are next. •

The Meaning of Variables Variables can be interpreted in many ways. 'W hile the use of variables in elementary grades has been limited, researchers suggest elementary students can understand variables and such experiences are important preparation for the more complex mathematical situations they will encounter in middle school (Blanton et al., 201 1). Variables can be used to represent a unique but tmlmiYWn qtumtity (e.g., missing number) or 1·ep1·esent tl quantity tbat varies (e.g., output of a function). Unfortunately, students often think of the former and not the latter. Experiences in elementary and middle school must focus on both, the focus of the next two sections.

Variables Used as Unknown Values. The missing-value explorations (Acti,~ty 13.15) are examples of variables as unknowns in equations. Many story problems involve a situation in which the variable is a specific unknown, as in the following basic example: Gary ate 9 strawberries and Jeremy ate some, too. The container of 25 was gone! How many did Jeremy eat?

Mylab Education Application Exercise 1 3 .2: Obse rving and Responding to Student Thinking Click the link to access this exercise, then watch the video and answer the accompanying questions. Although students can solve this problem without using algebra, they can begin to learn about variables by expressing it in symbols: 9 + s = 25, 25 - 9 = s or 25 - s = 9. These problems can grow in difficulty over time. Illustrations can help students reason about the meaning of the story and serve as a way to help write the related equation (Ga,~n & Sheffield, 2015). For this task, these two illustrations might be created by students:

Number Line

Bar Diagram

25 9

9

s

s

Translating stories to equations invoh~ng variables are particularly challenging for students, in particular when the word order does not match the symbol order. For example, students

m

Standards for

Mathematical Practice

MP1. Make sense of

problems and persevere in solving them.

326

Chapt er 1 3 Algebraic Thinking, Equations, and Functions

incorrectly write equations 6S = T for a problem that says, The number of students (S) was six times the number of teachers (T), (Clement, 1982). A bar diagram for this situation looks like this: 6 .----

5+--4-l-- -

3+---2-l---

• The number of students (S) • The number of teachers (T)

Another strategy that is helpful is to have student~ compare the variables (without the numbers) before the solve a problem. In this case, to reason whether there are more students or more teachers. In Figure 13.16, a series of examples shows problems in which each different shape on the scales represents a different value. "W hen no numbers are involved, as in the top t:wo examples of Figure 13 .16, students can find combinations of numbers for the shapes that make the balances balance. If an arbitrary value is given to one of the shapes, then values for the other shapes ~-an be found accordingly. Problems of this type can be adjusted in difficulty for students across the grades. The last three problems in F igure 13. 16 are on scales, and therefore the shapes have an unknown va lue that ~-an be determined through reasoning. As noted above, severa l apps are available that can support solving for va riables. These include NCTM's Illumination scales and NLVM Algebra Balance Scales and Algebra Balance Scales-Negative.

Activity 13.17

CCSS-M : 6 .EE.A.4 ; 7.EE.A.2; 8 .EE.C.8b

Which shape weighs the most? Explain. Which shape weighs the least? Explain .

Ball Weights Students will figure out the weight of three balls, given t he following three facts:

1.

Q..C

+

2.

ft

+

3.

~

+

-

~

1.25 pounds 1.35 pounds = 1.9 pounds

Ask students to look at each equation and make observations that help them generate relationships. For example, they might notice that the soccer ball weighs 0.1 pound more than the football. Write this in the same fashion as the other statements. Continue until these discoveries lead to finding the weight of each ball. One possible approach : Add equations 1 and 2:

{C

+

C'

+

-

+

How much does each shape weigh? Explain.

How much does each shape weigh? Explain.

~ = 2.6 pounds

Then take away the football and soccer ball, reducing the weight by 1.9 pounds (based on the information in equation 3), and you have two baseballs that weigh 0. 7 pound. Divide by 2, so one baseball is 0.35 pounds.

What will balance 2 spheres? Explain.

How much does each shape weigh? Explain.

FIGURE 13.16 Exam ples of problems with multiple variables and multiple scales.

Meaningful Use of Symbols

327

Mylab Education Application Exercise 13.3: Observing and Responding to St udent Thinking Click the link to access this exercise, then watch the video and answer the accompanying question s.

As you can see in Activity 13.17, students can explore equations with three variables. T he variables, or missing values, can be determined through reasoning (rather than formal strategies for solving systems of equations). Solving systems of equations has traditiona lly been presented as series of procedures with little attention to meaning (e.g., solve by graphing, by substitution, and simultaneously). Mathematically proficient studen ts should have acce.ss to multiple approaches, including these three but also to reasoning strategies. Rather than learn one way by rote each day o r be tested on whether they can use each approach, students s hould cboose a method that fits the situation, using appropriate tools. And, they need to understand what that point of intersection means in the context of the problem given. Among the strategies that s tudents must try is obsen111tion. Too often, students leap into solving a system algebraically without stopping to observe the va lues in the two equations. Look at the systems of equations here, and see which o ne.s might be solved for x or y by observation or mentally (without employing one of the three strategie.s listed earlier). x+ y Sx

+

= 25 and x + 2y = 25 = 82 and 4x + 3y = 41

m

Stand ards for

Mathematical Practice MP1. Make sense of problems and persevere in solving them.

= 20 and x + 2y = 10 !. = 5 and y + 5x = 60 3

3x

6y

+y

Simplifying Expressions. Simplifying expressions can be c hallenging for students because they are accustomed to solving equations and finding an answer, typically a number. Knowing how to simplify and recognizing equiva lent expressions are essential skills for working algebraically, and are based on the properties of the operations. Students are often confused about what the instruction "simplify" means. The Activity 13 .I 0 (Border T iles Expressions) can be extended to variables, as described here.

Activity 13.18

CCSS-M : 6 .EE .A.1 ; 6.EE.A.2a ; b , c, 6 .EE .A .3 ; 6.EE.A.4

Border Tiles Expressions-Part 2 Distribute t he Border Tiles Expressions- Part 2 Activity Page and color tiles (optional, but recommended). Explain that Marianna's pool patio work has expanded-she is building square pools of different side-lengths. Ask students to prepare variable expressions describing efficient ways for her to determ ine how many patio tiles are needed for any square-shaped pool. If the square had a side of length p , the total number of tiles could be found in similar ways: (2 X 10)

10 + 10 + 8 + 8 (p + 2)

+ (p + 2) + p + p

2 X (p

+

+ (2 X 8) 2)

+

(2 X p)

4 X9

100 - 64

4 X (p + 1)

(p + 2? - l"

Finally, challenge students to prepare a variable expression for any rectangular-shaped pool. Compare each expression and have students j ustify why any two are equivalent to each other.

Mylab Education

Activity Page: Border Tiles Expressions-Part 2

Invite students to enter the.se expressions into the Table function on their graphing calculator and graph them to see whether they are equivalent (Brown & Mehi los, 20 I 0). Looking at these options, the connection can be made for which one is stated the most simply. O ne effective way to help students understand the importance of preserving equivalence is to look at wod~ed examples and determine if they are correct or not, and if there are errors, explain how to fix the errors (Hawes, 2007; Renkl, 2014; Star & Verschaffel, 2016). Figure 13.17

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iii

Stand ards for

Mathematical Practice MP7. Look for and make use of structure.

328

Chapt er 13 Algebraic Thinking, Equations, and Functions

Explain how to fix this simplification. Give reasons. (2x + 1) - (x + 6) = 2x + 1 - x + 6 Gabrielle's solu1ion

~ y. ::::-3 ~1--e-n ~e.. o..-~.- o~ o~«*··~ ~\c 'M>-¥-,e ~\aGe so \\-.to- "?(oblem 1.0CV\o \cx:lz \,\ not {,.

shows how three students have corrected the simplification of (2x + I) - (x + 6). You can create your own worked examples of simplified expressions, or post correct and incorrect examples from your students. Activity 13. 19 provides an engaging way for students to explore properties and equivalent expressions.

nC>t \-~ . i.:l,(n ~OU 1im

.-.....;;... ....

V1DEO EXAMPLE

Counters

FIGURE 14.8 Which of these shapes are partitioned into sixths? Explain why or why not for each.

Some manipulatives, like fraction bars or fraction circles, can mislead students to believe that fractional parts

Fract ions as Numbers

must be the same sbape as well as the same size. Color tiles can be used to create rectang les that address this misconception, such as the o ne illustrated here:

•••••• ·· -

Students who recogn ize that each color represents thirds understand that fractional parts must be the same size, and that the shape may be different. Activity 14. 1 (P layground Fractions) offers an example of using pattern blocks to focus on equal-sized parts, which can be repeated by building other shapes that use different pattern block pieces. Activity 14.5 offers a similar experience with partitioning va rious shapes.

Activity 14.5

CCSS- M: 1 .G.A .3 ; 2.G .A.3 ; 3 .NF.A.1

Partitioning: Fourths or Not Fourths? Use Fourths o r Not Fourths Activity Page showing examples and nonexamples (which are very important to use with students with disabilities) of fourths (see Figure 14.9). Ask students to identify t he wholes that are correctly divided into fourths (equal shares) and those that are not. For each response, have students explain their reasoning. Repeat with other fractional parts, such as thirds or eighths. To challenge students, ask them to draw shapes that fit each of the four categories listed on the next page for other fractional parts, such as sixths. (See Sixths or Not Sixths Activity Page.)

MyLab Education Activi ty Page: Fourths or Not Fourths MyLab Education Activi ty Page: Sixths or Not Sixths

In the preceding activity, the shapes fall in each of the following categories: 1. 2. 3. 4.

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Same shape, same size (equivalent]: (a) and (f) D ifferen t shape, same size [equiva lent]: (e) and (g) D ifferent shape, different size [not equivalent): (b) and (c) Same shape, different size (not equivalent]: (d) FORMAnVE ASSESSMENT Notes. Activity 14.5 is a good diagnostic inten>iew to asse.ss

LJ whether students understand that it is the size that matters, not the shape. Ifstudents get (a)

~ (e)

(b)

(c)

(d)

rr:g L111 DE (f)

(g)

BJ] ® FIGURE 14.9 show fourths?

Wh ich of these part itioned regions

STUDENTS with SPECIAL NEEDS

351

352

Chapt er 14 Developing Fraction Concepts

all correct except (e) and (g), they hold the misconception that parts should be the same shape. Future task~ are needed that focus on equivalence. For example, you can ask studentS to ta ke a square and partition it themselves in as many ways as they can. • Pa rtitioning with length models is critical to understand ing fractions as numbers. For example, students need to be able to partition a number line into fourths and realize that each section is one-fou rth:

0

4

•I

4 1

4 I

4

3 4

I

I

2

0

~

4

5

4 I

6 4 I

1

7

8 4

4 I

9

10

4

4

I

I

11

4

I )I

2

s·1ze o1·1nterval: 43

'With number lines, students may ignore the size of the interva l (McNamara & Shaughnessy, 20 10; Petit et al., 20 10). A good way to help studentS understand the number line is to create and use paper strips. P rovide examples where the shaded sections are in different positions and where partitioning isn't already shown to strengthen students' understanding of equal pans. Activity 14.6 and Activity 14.7 provide such opportunities, one with paper strips and one with nun1ber lines.

Activity 14.6

CCSS-M: 3 .NF.A.1; 3.NF.A.2a , b

What Fraction Is Colored? Prepare a set of paper strips prior to doing this activity (you can cut 1-inch wide pieces of 8.5" by 11.• paper and shade, or cut pieces of adding machine tape). Color the strips so that they have a fractional amount shaded in various positions ( not just left j ustified !) (Sarazen, 2012). Here are a few examples:

L_____J. ______, Explain that the strip represents one whole. Give each student a paper strip. Ask students to explain what fraction is colored and explain how they know. A common misconception is for students to count parts and call each of these o ne-third. If a student makes this error, ask if the parts are the same-sized and if not to partition to make same-sized parts. Use toothpicks or uncooked spaghetti to illustrate the partitions:

Students can also justify their reasoning by measuring the length of each partition.

Using paper strips can help studentS better understand the number line, the focus of the next activity.

Fractions as Numbers

Activity 14.7

CCSS- M: 3 .NF.A.1; 3.NF.A .2a, b

How Far Did Nicole Go? Give students number lines partitioned such that only some of the partitions are showing. Use a context such as walking to school. For each number line, ask, •How far has Nicole gone? How do you know?"

t

0

Nicole's distance I

I 0

t

Nicole's distance I

I 0

t

Nicole's distance Students can justify their reasoning by measuring the length of each partition.

Locating a fractional value on a number line is particularly challenging but very important. Shaughnessy (20 I I) found four common errors students make in working with the number line: They (I) use incorrect notation, (2) change the unit (whole), (3) count the tick marks rather than the space between the marks, and (4) count the ticks marks that appear without noticing any missing ones. This is e'~dence that we must use number lines more extensively in exploring fractions, along with formative assessment tools to monitor students developing conceptions. Partitioning using a length diagram is a strategy commonly used in Singapore (a highperfomling country on international mathematics assessments) as a way to solve story problems. Consider the following story problem (Englard, 2010): A nurse has 54 bandages. Of those,~ are green and the rest are blue. How many of them are blue?

To solve, a student first partitions a strip into nine parts and then figures out the equal shares of bandages for each partition: Standards for Mathematical Practice MP4. Model with Mathematics.

II D id you notice that this is an example of fraction as operator? These types of partitioning tasks are good bu ilding blocks for mu ltiplying with fractions. Students can partition with set models such as coins, counters, or baseball cards. In the example in Figure 14.8, the 12 counters are partitioned into 6 sets-sixths. Each share or part has two counters, but it is the number of sha res that makes the partition show sixths. As with the other models, when the equal parts are not already figured out, then students mal not see how to partition. Students seeing a picture of two cats and four dogs might think 4 are cats (Bamberger, Oberdorf, & Schultz-Ferrell, 2010). Consider the following problem: Eloise has 6 trading cards, Andre has 4 trading cards, and lu has 2 trading cards. What fraction of the trading cards does lu have?

353

354

Chapt er 14 Developing Fraction Concepts

m

Standards for Mathematical Practice MP7. Look for and make use of structure.

A student who answers "one-third" is not thinking about equal sha res but about the number of people with trading cards. Understand ing that pa rtS of a whole must be partitioned into equal-sized parts across different models is an important step in conceptualizing fractions and provides a foundation for exploring sharing and equivalence tasks, all of which are prerequisites to performing fraction operations (Cramer & 'Whitney, 2010).

Iterating In whole-number learning, counting precedes and helps students to add and later subtract. T his is also true with fractions. Counting fractiona l parts, or itemting, helps students understand the relationship between the parts (the numerator) and the whole (the denominator) and to understand a fraction as a number. After experiences iterating, students should be able to answer the question, "How many fifths are in one whole?" just as they know how many ones are in ten. However, the 2008 NAEP results indicated that only 44 percent of students answered this question correctly (Rampey, D ion, & D onahue, 2009). The purpose of iterating activities is for students to develop the measurement interpretation of fractions, specifically that i, for example, can be thought of as a count of three partS called fourtbs (Post, Wachsmuth, Lesh, & Behr, 1985; Siebert & Gaskin, 2006; Tzur, 1999). The iterative concept builds meaning of fraction symbols: • T he numerator counts. • T he denominator tells wbat is being counted. Iterating can be done concretely by using a single piece, representing one unit or one share, and repeating it a number of times to "measure" how many are in the whole. 'W hile equal shares can be found by partitioning, estimating one share and seeing if it is accurate builds understanding of the relative size of unit fractions (Tzur & Hunt, 20 15).

Activity 14.8

CCSS- M: 3 .NF.A.1; 3 .NF.A .2a , b

e•

Estimating and Counting Fair Shares Select a length context such as pretzel sticks, licorice, o r ribbon. Distribute several paper strips to each pair of students, one of which in a d ifferent color to represent one whole (pretzel). Ask students to estimate how long they think a fair share is for each of the following situations, test their estimates, and try again. Special

ENGLISH LEARNERS

r ule: no paper foldin~ Shares for 2 friends

Shares for 3 friends

Shares for 4 friends

Shares for 6 friends

Shares for 8 friends

Shares for 12 friends

For example, for shares with 3 friends, students will cut a piece from one of their additional strips that they think represents one-third and iterate it (lay it down three times or cut it out three times and lay each piece next to each other) to see if it reaches one whole. If it is too short or too long, they try again until they find a piece that is approximately accurate. Label it as "one-third" and/or ~· As you observe, ask students to count the shares to show you the fair shares (one-third, tw~thi rds, three-thirds). Ask questions such as, •w hat strategies are you using to see if your piece is a fair share?" •will your next try to a longer piece or a shorter piece? (Why?) •w hat strategies are you using to estimate the fair share/ unit length?" As you and the students describe the shares as fractional parts (e.g., fourths) emphasize the 'th' sound to distinguish from counting numbers (e.g., fours). This is particularly important for Els.

Iterating makes sense with length models because iterating is like measuring. And, the number line is a good model for iterating with mixed number.

Mylab Education

Video Example 14.7

Watch this classroom video of a teacher building on students' understanding of fractions to iterate with mixed numbers.

355

Fractions as Numbers

Emily has 2 ~ yards of ribbon and needs pieces that are one-fourth yards for making bows. How many pieces can she cut?

To begin, students draw a strip (or line) representing 2! yards:

1 yard

1 yard

Next, the strip is partitioned into fourths (the unit) and counted-10 fourths:

Activity 14.9

CCSS-M: 3 .NF.A.1; 3.NF.A .2a , b

More, Less, or Equal to One Whole Give students A Collection of Fractional Parts Activity Page and Cuisenaire rods. Indicate the kind of fractional part they have.

l:

For example, a collection m ight have seven light green rods/ strips with a caption or note indicating •Each light green rod is The task is to decide if the collection is less than one whole, equal to one whole, or more than one whole. Ask students to draw pictures or use symbols to exp lain their answer. Adding a context. such as people sharing candy bars, pizzas, or sticks of clay, can help children understa nd and reason t hrough t he problem. Ask, •why does it take more fourths to make a whole than thirds?" •why did we get two wholes with four halves but just a little more than one whole with four t hirds?" Take this opportunity to con· nect this language to mixed numbers. •what is another way we could say ten-fourths?" (Possible responses are two wholes and two-fourths, or two wholes and one-half, or one whole a nd six-fourths.)

Mylab Education

Activity Page: A Collection of Fractional Parts

Students can participate in many tasks that involve iterating lengths, including ones where they are asked to find what the whole or unit is.

Activity 14.10

CCSS-M: 3 .NF.A.1; 3.NF.A.2a , b

A Whole Lot of Fun Use A Whole lot of Fun Activity Page and t he Cutouts for Fraction Strips Activity Page like the one here:

Tell students that t his strip is three-fourths of one whole (unit). Ask students to sketch strips of the other lengths on their paper

(e.g.,~). You can repeat this activity by selecting other values for the starting amount and selecting d ifferent fractional values to sketch. A context, such as walking, is effective in helping students make sense of the situation. Be sure to use fractions less than and greater than 1 and mixed numbers.

Mylab Education Mylab Education

Activity Page: A Whole Lot of Fun Activity Page: Cutouts for Fraction St rips

356

Chapt er 14 Developing Fraction Concepts

Notice that to solve the task in Activity 14.10, students first partition the piece into three sections to find~ and then iterate the to find the other lengths. Iterating can also be done with area models, for example the Circular Fractional Pieces in Figure 14.10 using the context of quesad illas. Or, engage studentS in creating their own fractional pieces using paper plates of various colors and use those plate pieces to bui ld various fractional amountS (McCoy, Barnett, & Stine, 2016)! For each collection, tell studentS what type of piece (portion) is being shown and simply count them together: "One-fourth, two-fourths, three-fourths, four-fourths, five-fourths." Ask, "If we have five-fourths, is that more than one whole, less than one whole, or the same as one whole?" To reinforce the piece size even more, you can slightly alter your language to say, "One one-fourth, two one-fourths, three one-fourths," and so on. See Figure 14.10 for some great questions to ask as your students are counting each collection of parts. Make comparisons between different collections: "Why did we get more than two wholes with ten-fourths, and yet we don't even have one whole with ten-twelfths?"

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t

5 fourths

10 fourths

Myl ab Education

10 twelfths

Activity Page: Circular Fracti onal Pieces

FIGURE 14.10 Iterating fractional parts in an area model. Activity 14. 11 returns to using pattern blocks (See Activity 14.1) to help students focus on the size of the partS, not the number of pieces or partitions (Champion & 'Wheeler, 2014; Ellington & 'Whitenack, 2010). In bui lding creatures, students are partitioning and iterating.

Activity 14.11

CCSS-M: 3.NF.A.1 ; 3.NF.A.2a, b

Pattern Block Creatures Ask students to build a Pattern Block Creature that fits with a set of rules (a creature represents one-whole). These rules can begin with just stating a fractional quantity for a color, such as •The red trapezoid is o ne-fourth of the creature." But, more con· straints can be added to the rules. For example: •

The blue parallelogram is one-sixth of the creature. Use at least two colors to build your creature.

•

The yellow hexagon is one-half of the creature. Use three colors to build your creatu re.

•

Green triangles are o ne-third of your creature. Use four different colors to build your creature.

After a student creates t heir creature based on these r ules, have them count the fractional parts to make sure t hey have a whole creature. Then have students write their own rules for their own pattern block creature. Instruct t hem to trace the outline of t heir creature on a piece of paper and write a rule, such as •The red trapezoid is __ of my creature." Students can trade with other students to see if they can figure out the fractional amounts for each other's creatures.

Iteration can also be done with set models, but it is more challenging. For example, show a collet:tion of two-color counters and ask questions such as, "If 5 counters is one-fourth of the whole, how much of the whole is 15 counters?" These problems can be framed as engaging puzzles for studentS. For example: "Three counters represent kof my set; how big is my set?" If the fraction is not a unit fraction, then students first partition and then iterate. For example, "'T\venty counters representS of my set; how big is my set?" first requires finding find

f

t

Fract ions as Numbers

357

(10 counters), then iterating that three rimes to get 30 counters in three-thirds (one whole).

Counting Counters: Find the Part Activity Page and Counting Counters: Find the Whole Activity Page provides such problems for students to solve.

Mylab Education

Activity Page: Counting Counters: Find the Part

Mylab Education

Activity Page: Counting Counters: Find the Whole

Finally, iterating can be done on a calculator, transirioning students to mentally visualizing fractional partS.

Activity 14.12

CCSS-M : 3.NF.A.1 ; 3 .NF.A.3a, c

Calculator Fraction Counting Many calculators display fractions in correct fraction format and offer a choice of showing results as mixed

il

STUDENTS

with numbers or simple fractions. Ask students to type in a fraction (e.g., and then + and the fraction again. To SPECIAL count, press 0 ~. ~ , ~. repeating to get the number of fourths wanted. The display will show the counts NEEDS key has been pressed. Ask students questions such as by fourths and also the number of times that the the following; •How many fourths to get to 3?" "How many fifths to get to 2?" These can get increasingly more challenging;

ee:!l

"How many fourths to get to 4t?" "How many two-thirds to get to 6? Estimate and then count by two-thirds on the calculator." Students, particularly students with disabilities, can coordinate their counts with a concrete fraction model (e.g., Cuisenaire Rods).

[J Figures

FORMATIVE ASSESSMENT Notes. T he tasks in 14. 11 and 14. 12 can be used as perfom1ance

If this rectangle is one whole, - find one-fourth. - find two-thirds. - find five-thirds.

assessments. If students are able to solve these types of tasks, they can partition and iterate. T hat means they are ready to do equivalence and comparison tasks. If they are not able to solve problems such as these, provide a range of similar tasks, usi ng real- life contexts and involving area, length, and set models. •

If brown is the whole, find one-fourth. If dar!< green is one whole, what rod is two-thirds? If dar!< green is one whole, what rod is three-halves?

Magnitude of Fractions N umber sense with fractions requires that students have some intuitive feel of the size (or magnitude) of fractions. T his requires that students see a fraction as a number, which helps them develop "fraction sense" (Fennell, Kobett, & W ray, 201 4). T he use of contextS, models, and mental imagery can help students build a strong understanding of the relative size of fractions (Bray & Abreu-Sanchez, 2010; Petit et al., 2010; Tobias, 20 14). As with who le numbers, s tu dents are less confident and less capable of esti mating tha n they are at computing exact answers, yet a focus on estimation can strengthen their understanding of fractions (Clarke & Roche, 2009). Therefore, you need to provide many opportunities for students

·.

(·:. ·. ) ••••• ••••• ••••• •••• •• • • • FIGURE 14.11 part.

If 8 counters are a whole set, how many are in one-fourth of a set? If 15 counters are a whole, how many counters make three-fifths? If 9 counters are a whole, how many are in five-thirds of a set?

Given the whole and the fraction, find the

358

Chapt er 14

-

Developing Fraction Concepts

If this rectangle is one-third, what could the whole look like? If this rectangle is three·fourths, draw a shape that could be the

whole. If this rectangle is four·thirds, what rectangle could be the whole?

to reason about the relative size of a fraction. Ask questions like "About what fraction of your classmates are wearing sweaters?" Or after tallying survey data about a topic like favorite dinner, ask, "About what fraction of our class picked spaghetti?" Additionally, have students approximate fractions for shaded regions and points on the number line (See Figure 14.13).

If purple is one·third, what rods are the whole? If dar!< green is two·thirds, what rod is the whole? If yellow is five-fourths, what rod is one whole?

J

If 4 counters are one-half of a set, how big is the set? If 12 counters are three·fourths of a set, how many counters are in the full set?

0

If 10 counters are five·halves of a set, how many counters are in one set?

FIGURE 14.12 Given the part and the fraction, find the whole.

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0

2

FIGURE 14.13 About how much? Name a fraction for each drawing and explain why you chose that fraction.

The number line is a good model for helping students develop a better understanding for the relative size of a fraction (Petit, La ird, & Marsden, 201 0) and is therefore effective for estimating about the relative size of fractions.

Activity 14.13

CCSS-M : 3.NF.A .2a ; 3 .NF.A.2b

Fractions on the Number Line Use a piece of rope as long as the room (this works well in the hall or gym, too!) Have t wo students stand and hold the end of the rope and also hold nu mber cards. For example, the student on the left end can hold 0 and the student on the right end ca n hold 1 (these end points ca n change to other values). Give out a variety of Fraction Cards (appropriate for your group of students) with clips or clothespins. Ask a volunteer to place their

STUDENTS with

SPECIAL NEEDS

fraction on the number line. Ask, •1s this fraction in the approximate correct location?" •How do you know?" Students can fold the rope to determine benchmark fractions such as}. ~. or~) and use other strategies to determ ine approximate locations on the number line. Continue to have students place their fractions on the num ber line. This activity can be played repeatedly wit h cards t hat include equivalent fractions (placed at t he exact same point), cards with fractions greater than one (use larger pieces of rope and more holding points), and eventually decimals and variables (see Bay, 2001).

Mylab Education

Activity Page: Fract ion Cards

After experiences with number lines and other models, students need opportunities to reason about the relative size of fractions using mental strategies considering what benchmark they are near. For example, if students think about where might be by partitioning a number

k

Equivalent Fractions

fo

35 9

line between 0 and I, they will see that is close to 0, w hereas ~ is quite close to I . And this applies to fractions greater than one-for example, is close to t he benchmark

Activity 14.14

3t

3t.

CCSS-M: 3.NF.A.3d; 4.NF.A.2

Zero, One-Half, or One Create sets of Fraction Cards for each small group of students. A few should be greater than 1~ or~. with the others ranging from 0 to 1. Let students sort the fractions into three groups: those close to 0 , close to~. and close to 1. For those close to~. have them decide whether the fraction is more or less than half. The difficulty of this task largely depends on t he fractions you select. The first time you try this, use fractions that are very close to the three benchmarks, such as if;, ,Sg0 , or f.>. On subsequent days, mostly use fractions with denominators less than 20. You might include a few fractions that are exactly in between the benchmarks, such as~ or i. Ask students to explain how they are using the numerator and denominator to decide. For Els, be sure the term benchmark is understood and encourage illustrations as well as explanations. As an extension or alternative to differentiate this activity, ask students to create their own fractions close to each benchmark fraction.

Mylab Education

Mylab Education

Activity Page: Fraction Cards

Self-Check 14.2

Equivalent Fractions As discussed in Chapter 13, equivalence is a critical but often poorly understood concept. T his is particularly true with fraction equiva lence. In the CCSS-M, fraction equiva lence and comparisons are emphasized in grade 3 and applied in grade 4 (and beyond) as students engage in computation with fractions. Students cannot be successful in fraction computation without a str ong understanding of fraction equivalence.

Mylab Education

Video Example 14.8

Watch this video wherein John Van de Walle describes important idea s, which a re a lso illustrated in a whole class les son.

Conceptual Focus on Equivalence

WPause & Reflect

How do you know that ~ = ~? Before reading furt her, think of at least two different explana· tions. •

H ere are some possible answers t o the preceding question:

t

1. T hey ar e the same because you can sim plify an d get~· 2. If you have a set of 6 items and you ta ke 4 of them, that would be~· But you can ma ke the 6 into 3 groups, and the 4 would be 2 groups o ut of the 3 groups. T hat means it's~·

360

Chapt er 14 Developing Fraction Concepts

J,

3. If you start with you can multiply the top and the bottom numbers by 2, and that will give you £,so they are equal. 4. If you had a square cut into 3 partS and you shaded 2, that would be~ shaded. If you cut all 3 of these parts in half, that would be 4 parts shaded and 6 partS in al l. That's £, and it would be the same amount.

m

Standards for Mathematical Practice MP2. Reason abstractly and quantitatively.

All of these answers are correct . But let's think about what they tell us. Responses 2 and 4 are conceptual, although not as efficient. The procedural responses, 1 and 3, are efficient but do not indicate conceptual understanding. Consider how di fferent the procedu re and the concept appear to be: C!mcept: 'T\vo fractions are equivalent if they are representations for the same amount or quantity-if they are the same number. Pmcedm·e: To get an equivalent fraction, multiply (or divide) the top and bottom numbers by the same nonzero number.

Rushing too quickly to the algorithm can impede students' conceptual understand ing of fractions and fraction equivalence. Students need to see the connections between the concept and the procedure, and this requires multiple experiences (and time). Be patient!

Equivalent Fraction Models To help studentS build a strong and flexi ble understanding of equiva lent fractions requires the use many contexts and models in which the goal is to rename a fractional amount. T his is the first time in students' experience that they are seeing that a fixed quantity can have more than one name {actually an infinite number of names). Area models are a good place to begin understanding equivalence (see Figure 14.14).

Activity 14.15

CCSS-M: 3 .NF.A.1.; 3.NF.3a , b, c

Making Stacks Select a manipulative that lends to exploring fractions (e.g., pattern blocks, tangram s, fraction strips, or fraction circles). Prepare

t

Fraction Cards with different fractional amounts, such as ~. ~. 1 , ~. ~. and 2. ( Note: you may want to begin with fractions less than 1, then move to fractions equal to and greater than 1 .) Select a context that fits the manipulative (e.g., pancakes, quesadillas, pizza, or cookies for a circle model). Students use their manipulative to (1) determine a whole, (2) build the fraction on their card with pieces, stacking it on top of the whole, and (3) find as many equivalent fractions as possible, stacking each new equiv· alency on top. For example, with fraction circles, and a card of ~. a student picks the whole circle as the whole, places 3 of the fourths pieces on top of the whole, and then looks for other fractional pieces (same size) to stack. In this case, 6 eighths can stack on top. Ask students to record all the possibilities they find, drawing a sketch and writing the equation. Ask for students to use the context to explain the equivalency. For example, if they ate 3-fourths of a pizza, it is the same as EH!ighths, j ust cut in different sized pieces. Repeat with different fraction cards, including fractions that are not in sim plest form.

Mylab Education

Activity Page: Fraction Cards

361

Equivalent Fractions

Filling in regions with fraction pieces

Paper folding

1

4 1

3

first fold

1

8

%second fold 4

12

6

tz

2

8

3

Grid paper

third fold

Rectangle only partially subdivided

Dot paper 1

9

1

6

~ : ~ Ll_ll__ l... =_k___J FIGURE 14.14

~ I D= ~

I

rz l ~orp: ~ 1

Area models for equivalent fractions.

In a follow-up classroom d iscussion, ask students to consider what other equivalencies are possible (and justify thei r thinking) beyond what they could build with their manipulative. For example, ask, "What equivalent fractions could you find for ~ if we had sixteenths in our fraction kit? If you could have a piece of any size at all, what other fraction names are possi ble?" or "If the pizza was cut into 16 equal-sized pieces, what would your equivalent portion be?" T he next activity moves from using manipulatives to sketches on paper, opening up more fraction equivalencies for each illustration.

Activity 14.16

CCSS-M : 3.NF.A.1 ; 3 .NF.3a , b, c

Dot Paper Equivalencies Use Fraction Names Activity Page, which includes three different grids with a fraction shaded (each enclosed area represents one whole). Ask students how many fraction names they think the first problem has. Then ask them to see how many they can find (working individually or in partners). Invite students to share and explain the fraction names they found for problem 1. Repeat for the next two problems. Alternatively, cut this page into three task cards, laminate the cards, and place each at a station along with a n overhead pen. Have students rotate in partners to a station and see how many fraction names they ca n find for that shape (using the pen as needed to show their ways). Rotate to the next station. See Expanded lesson: Dot Paper Equivalencies for details. To make additional pictures, create your own using your choice of Grid Paper or Dot Paper (see Blackline Masters 5 -11). ( Figure 14.14 includes an example drawn on an isometric grid).

MyLab MyLab MyLab MyLab

Education Education Education Education

Activity Page: Fraction Names Expanded Lesson: Dot Paper Equivalencies Blackline Master: 1-Centim eter Square/ Diagonal Grid Paper Blackline Master: 1-Centim eter Isometric Dot Paper

362

Chapt er

14 Developing Fraction Concepts

Cuisenaire rods One whole • • • • • • • Blue=

f2 = ~

The Dot Paper Equivalencies activity involves what Lamon (2012) calls unitizing-that is, given a quantity, finding different ways to chunk the quantity into parts in order to name it. She points out that trus is a key ability related not only to equivalent fractions but also to proportional reasoning. Length models must be used to develop understanding of equivalent fractions. Asking students to locate and i'o on a number line, for example, can help them see that the two fractions are equivalent but the line is partitioned differently (Siegler et al., 2010)(see At.'tivity 14.13 for a life-sized number line activity). Cu iseniare rods can be used to designate both a whole and a part, as illustrated in F igure 14. 15. Students use smaller rods to find fraction names for the given part. To have larger who les or values greater than one whole, use a train of two or three rods. Folding paper strips is another method of showing equivalent fractions. In the example shown in F igure 14.15, one-half is subdivided by successive folding in ha lf. Other folds would produce other equ ivalent fractions. Legos, a rughly motivating manipulative, can help studentS learn to write fraction equivalencies (for an excellent elaboration on this idea, see Gould, 2011). Lego brick~ can be viewed as an area (array) or as a set (studentS can count the studs).

f

Folding paper strips

D

First fold (A)

-

Second fold (B)

FIGURE 14.15

III II

c

B

A 1

2 2

-4

Third fold (C) -

~

Last fold (D) -+

f's

Length models for equivalent fractions.

Activity 14.17

CCSS-M: 2 .G .A.3; 3 .NF.A. 1 ; 3.N F.3a , b, c; 4.NF.B.3a, b

LEGQ®Land: Building Options Ha nd out o ne 2-by-6 LEGO to each student. Ask them to describe it (t here are 12 studs, two rows of 6). For seco nd grade or as a warm-up, ask students what same-colored pieces could cover their land (e.g., 6 of the 2-by-1 pieces). Ask students to imagine that 12 pieces of LEGO blocks represents one plot of land. It can be covered with various smaller pieces as shown here: Ask students to build the plot of land using different Lego pieces (1-by-2, 1-by-3, 2-by-2, 1-by-1, 2-by~. etc.). After they have completed their plot of land, ask students to tell the fraction

il-

of their land that is represented by a particular piece (e.g., the 2-by~ is f2 as well as~ and To focus on iteration and to build connections to addition (grade 4), students can write equations to describe their LEGO Land. In the one pictured here, that would be

~ + 2_ + ~ + _.!. 12

12 1 2 1 2

OR

_! + _! + _! + _.!._ 24612

Note t hat students have m isconceptions about how to name fractions parts, naming the blue part as} rather than

i because

they see three pieces (Wilkerson, Bryan, & Curry, 2012). This becomes a good topic for a classroom discussion: What is the fractional value of the blue pieces (simplified)?

Two color counters (set model) are an effective tool for fraction equiva lencies.

Activity 14.18

CCSS-M: 3 .NF.A.1; 3.NF.3a, b, c

Apples and Bananas Use Apples and Bananas Activity Page or j ust have students set out a specific number of counters in two colors-for example, 24 counters, with 16 of them red (apples) and 8 of them yellow (bananas). The 24 counters make up t he whole. Ask students to group the apples a nd bananas into equal·sized groups, such that a group has to have only apples or only bananas. Cha llenge them to find as many ways as they can. For each that they find, record the fraction names for apples and for bana nas. They can record each option

ENGUSH LEARNERS

Equivalent Fractions

36 3

they find in a table, which helps notice patterns. Ask questions such as, •1s it possible to have the fruit in groups of 4?" to encourage students to think of different ways to form equal-sized groups. Counters can be arranged in different groups or in arrays (see illustrations in Figures 14.16 and 14.17 on the next page). Els may not know what the term group means because when used in classrooms, the word usually refers to arranging students. Spend time before the activity modeling what it means to group objects.

Mylab Education

Activity Page: Apples and Bananas

In the activities so far, there has only been a runt of a rule for finding equivalent fractions. Activity 14.19 begins to build the connection between the concept and the procedure by using manipulatives.

Activity 14.19

CCSS-M : 3 .NF.3a , b, c; 4 .NF.A .1

Missing-Number Equivalencies Use Missing-Number Equivalencies Activity Page or give students an equation expressing an equivalence between two fractions, but with an unknown value. Ask students to use counters or rectangles to illustrate and find the equivalent fraction. Example equations:

5 3

2

6

3

6

8 12

3

9 12

3

The missing value can be in a numerator or a denominator; the missing number can be either larger or smaller than the corresponding part of the equivalent fraction. (All four possibilities are represented in the examples.) Figure 14.18 illustrates how Zachary represented the equivalencies with equations and partitioning rectangles. The examples shown above involve simple

f2 =

whole-number multiples between equivalent fractions. Next, consider pairs such as~ 12 or ~. In these equivalencies, one denominator or numerator is not a whole-number multiple of the other. Include equivalencies for whole numbers and frac-

=

lf =

tions greater than one: ~ = ~

Mylab Education

§

Activity Page: Missing-Number Equivalencies

When doing "Missing-Number Equivalencies" you may want to specify a particular model, such as grid paper (area), paper strips/number line Oength) or two-color counters (set). Students who have learning disabilities and other students who struggle with mathematics rna~ benefit from using clocks to do equivalence; for example, to find equivalent fractions for 4, ~. and so on (Chick, Tierney, & Storeygard, 2007). NCTM's Illuminations website offers an excellent set of three units called "Fun with Fractions." Each unit uses one of the model types (area, length, or set) and focuses on comparing and ordering fractions and equivalencies. The five to six le.~sons in each unit incorporate a range of manipulatives and engaging activities to support student learning.

f¥,

Fractions Greater than 1 Students mu~t be able to move flexibly between fraction~ greater than I (e.g.,~) and mixed numbers (e.g., 2 If students have counted fractional parts beyond one whole, they know how to write And, if you have encouraged them to al~o tell how many wholes they have counted, they will also know this is equivalent to 2 without having to apply an algorithm, but by knowing the meaning of the numerator and denominator. If they have not had these experience.~ with iterating, then you must provide such experiences. It is not appropriate to just tell students to "divide the numerator by the denominator and write the remainder as a fraction." As illustrated in Figure 14.19, connecting cubes are very effective in showing how fractions greater than one can be written as whole numbers or mixed numbers (Neumer, 2007). Students identify one cube as the unit fraction {!) for the problem ( lj-). They count out 12 fifths and build wholes. Conversely,

!f.

k).

k,

364

Chapt er 14 Developing Fraction Concepts

Apples and Bananas

~ - 3-

24 counters = 1 whole 16 • and8 •

• •

•

•

•

•

•

..!.§. apples

• • •• • • • • •

24

e e J!.. bananas '-----------------~·~ 24

I ..., .•

•

Make the 16 e into 4 groups of 4 ~ riel .

and 2 more sets of 4 makes 24.

[··.·1[··:1[·.-· 16 4

groups are apples

~ groups are bananas

FIGURE 14.18 A student illustrates equivalence fractions by partitioning rectangles.

16 is 2 groups of e.

(•.•:•: ][r·-.-.-:-•-:""""') ~groups (~• ~ ••• ) t

are apples

groups are bananas

8 groups of •

4 groups of

1••1 1••1 1••1 l••l 8 l••l l••l l••l l••l 12 groups are apples 1 •·1 ~ ~ .. 1

Apples are

.!§. = ~ = g .. J!.. 24

Bananas are

6

1~ groups are bananas

1

3

12

of the fruit

J!.. = g = l • ..!. 24 6 3 12

FIGURE 14.16

of the fruit

Set models for illustrating equivalent

fractions.

~:

•••••••• ••••••••

~ of the rows are apples

•• ••••• •••• •• •• •• ••

AGURE 14.17

are apples

16

i 16

~ or 2 wholes and ~

FIGURE 14.19

Connecting cubes are used to represent the equivalence of ~ and 2~-

they could stan with the mixed number, build it, and find om how many total cubes (or fifths) were used. Repeated experiences in building and soh-ing these tasks will help students to notice a pattern that acrually explains the algorithm for moving between mixed numbers and &actions greater than I.

Mylab Education Video Exam ple 14.9 Watch this video of a teacher moving from concrete experiences to mental images to help students see relationships between mixed numbers and fractions.

2

24. 3 ~~

Whole (5 cubes)

are apples of the columns are apples

4

24"i Arrays for illustrating equivalent fractions.

Help students move from physica l models to menta l images. Challenge students to figure out the two etjuivalent forms by just picturing the stacks in their heads. A good explanation for 3 ~might be that there are 4 fourths in one whole, so there are 8 fourths in two wholes and 12 fourths in three wholes. The extra fourth makes 13 fourths in all, or {Note the iteration concept playing a role.) Do not push the standard algorithm, as it can interfere "ith students making sense of the relationship between the two representations. Allow students to make this generalization through their e.~riences.

lf.

365

Equivalent Fractions

Developing an Equivalent-Fraction Algorithm

m

When studentS understand that fractions can have different (but equivalent) names, they are ready to develop a method for finding equivalent names for a particular value. An area model is a good visual for connecting the concept of equivalence to the standard algorithm for finding equivalent fractions (multiply both the top and bottom numbers by the same number to get an equivalent fraction). Ask students to look for a pattern in the way that the fractiona l parts in both the part and the whole are counted.

Activity 14.20

Standards for

Mathematical Practice MPS. Look for and express regularity in repeated reasoning.

CCSS-M : 3 .NF.3b; 4.NF.A.1

Garden Plots Have students draw a square "garden• on blank paper, or give each student a square of paper (like origami paper). Begin by explaining that the garden is divided into rows of various vegetables. In the first example, you m ight illustrate four rows (fourt hs) and designate

i as corn. Ask students to partition their square into four rows and shade three-fourths as in Figure 14.20.

Then explain that the garden is going to be shared with family and friends in a way that each person gets a harvest that is

i

corn. Show how the garden can be partitioned horizontally to represent t wo people sharing (eighths) . Ask what fraction of the newly d ivided garden is corn { ~ ) . Next, tell students to illustrate ways that more friends can share the garden (they can choose how many friends, o r you can). For each newly divided garden, ask students to record what fraction of the garden is corn, using a table, equations, or both: People sharing garden

1

2

Sections of corn

3

6

Sections of garden

4

8

After studentS have prepared their own examples, provide time for them to look at their fractions and gardens and notice patterns about the fractions and the diagrams. O nce they have time to do this individually, ask students to share. Figure 14.2 1 provides student explanations that illustrate the range of "noticing." As you can see, for some of these students more experiences are needed. You can also assist in helping students

Garden plots activity

'I ·I'

? L(

.

~) 't

t;

...

4

3

(a)

(b) ""l~o. ..,,.

"y;, 0

~ (c)

~0

w~. 4

}

:r

~ ~~

••

~'I"•

FIGURE 14.20 A third grader partition s a garden to model fraction eq uivalencies.

"•*·V +. f'

1\~-1. (.f .7

Y••

, ·"

.... ' "'~

•• '•lfl'"'bfl.

b

0

Yo "

...... ,...., r't

"'(fJ.o

fl\., P. I P'\1

~l.,..

l,y

..,.

FIGURE 14.2 1 Students explain what they notice about fraction equivalencies based on partitioning •gardens· in different ways.

366

Chapt er 14

Developing Fraction Concepts make the connection from the partitioned square to the procedure by displaying a square (for example, partitioned to show~) (see Figure 14.22). Then, partition the square vertically into six partS, covering most of the square as s hown in the figure. Ask, "vVhat is the new name for my~?" The reason for this exercise is that it helps students see the connection to multiplication. With the covered square, studentS can see that there are four columns and six rows to the shaded part, so there must be 4 X 6 parts shaded. Similarly, there must be 5 X 6 parts in the whole. Therefore, the new name for is 1~ or ~-

t

!,

Writing Fractions in Simplest Terms. The multiplication scheme for equivalent fractions produces fractions with larger denominators. But creating equivalent forms for ~ might involve multipl ication to get or di,~sion to get i. To write a fraction in simplest te17!/S means to write it so that numerator and denominator have no common whole-number factors. One meaningful approach to this task of finding simplest terms is to reverse the earlier proce.ss, as illustrated in Figure 14.23. The search for a common factor or a simplified fraction should be connected to grouping. Two additional things should be noted rega rding fraction simplification:

Ji

1. Notice that the phrase ,·educingfmctions was not used. Because this would imply that the fraction is being made smaller, this tem1inology should be avoided. Fractions are simplified, not reduced. 2. Do not tell students th at fraction answers are incorrect if not in simplest or lowest terms. This also works aga inst understanding equivalence: for k + botb and ~ are correct.

r

!,

[[0]- 0 4 5

4 5 =?

FIGURE 14.22 How can you count the fractional parts if you cannot see them all?

Simpler terms

,..----,

.a.= 23 :x 12 ;x

4: 4:

L- ---~

=g3

Higher terms

FIGURE 14.23 Using the equivalent-fraction algorithm to write fractions in simplest terms.

Multiplying by One. Mathematically, equivalence is based on the multiplicative identity (any number multiplied by 1 remains unchanged). Any fraction of the form ~ can be used as the identity element. Therefore,! = ~ X 1 = ! X } = ~- Furthermore, the numerator and X ( :~:) = denominator of the identity element can also be fractions. In this way, -& = Understanding this idea is an expectation in the CCSS-M in grade 4.

h

!.

r. TECHNOLOGY Note. The Equivalent Fractions tool from NCTM's Illuminations

W (http://illuminations.nctm.org/Activity.aspx?id=3 51 0) has studentS create equiva lent fractions by dividing and shading square or circular regions and then matching each fraction to its location on a number line. Students can use the computer-generated fraction or select their own. Once the rectangular or circular shape is divided, the student fills in the parts or fractiona l region and then builds two models equivalent to the original fraction. The three equivalent fractions are displayed in a table and on a number line. •

Comparing Fractions

Mylab Education

Self-Check 14.3

--

Mylab Education Application Exercise 14.3: Equivalent Fractions Click the link to access this exercise, then watch the video and answer the accompanying questions.

.,~~ ...

VIDEO EXAMPLE

Comparing Fractions When students are looking to see whether two or more fractions are equivalent, they are comparing them. If they are not equivalent, then students can determine which ones are smaller and which ones are larger. Smith (2002) suggests that the comparison question to ask is, "Which of the following two (or more) fractions is greater, or are they equal?" (p. 9). He points out that this question leaves open the possibility that two fractions that may look different can, in fact, be equal, as illustrated by Sean's thinking. Therefore, the activities for equivalence can (and should) be adapted and used for comparing fractions. For example, in Garden P lots, you might ask if these friends have the same share of the garden (if not, who has more?): Ghia has~ and N ita has~·

Mylab Education

Video Example 14.10

Watch this video of a student deciding whether two fractions are equivalent or not.

Comparing Fractions Using Number Sense As illustrated in Al ly's interview, limited understandings of fractions can make it difficult or impossible to compare the relative size of fractions. NAEP resu lts found that only 21 percent of fourth-grade students could explain why one unit fraction was larger or sma ller than another-for example, and~ (Kloosterman eta!., 2004). For eighth graders, only 4 1 percent were able to correctly put in order three fractions given in simplified form (Sowder, 'VVearne, Martin, & Strutchens, 2004). This bui lds a strong case for the need to teach students numbe1· seme strategies to compare fractions! That is the focus of the rest of this section.

k

Mylab Education

Video Example 14.11

Watch this video of a student who struggles with the meaning of fractions, which makes it difficult for her to compare fractions.

Using Contexts. As stated numerous times in this chapter, contexts and visuals are essential to building an understanding (Bray & Abreu-Sanchez, 2010; Petit et a!., 2010; Tobias, 20 14). Using elastic strips to measure length has been effective in helping students understand equiva lence of fractions and compare fractions (Harvey, 2012). Because circles are popu lar fraction tools, and because sharing brings to mind food, many contexts end up being about brownies, cookies and pizza. Using a more expansive collection of contexts helps students see fractions in their world. Additiona lly, because a number line is one-dimensional, it can make comparisons more visible. Therefore, more examples oflinear situations are needed (e.g., how far people have walked/run; length of hair/banner; height of plants growing).

367

368

Chapt er 14 Developing Fraction Concepts

Activity 14.21

CCSS-M : 3 .NF.A .3a, b, d; 4 .NF.A.2

Stretching Number Lines Cut strips of elastic (about 1 meter or yard in length). Hold the elastic taut and mark off ten partitions on each. Hand one out to each pair of students. Ask students to use their stretching number line to find the place on a table that represents the fraction of t he distance across the table. For each pair, ask: Ask "Which distance is greater, or are they equal?" la.

fa of the distance across.

lb. ~ of the distance across. 2a.

fli of the distance across.

2b. ~of the distance across. (Note: they have to rethink the whole as 8 sections.) 3a.

i of the distance across. (Note: they have to rethink the whole as 8 sections.)

3b. ~of the dis tance across. (Note: they have to rethink the whole as 8 sections.) For early finishers, invite them to find their own sets {that are and are not equivalent) using their elastic, then trade with someone to determine which fraction is greater (or are they eq uivalent).

Comparing Unit Fractions. As noted earl ier, whole-number knowledge can interfere with comparing fractions. Students th in k, "Seven is more than four, so sevenths should be bigger than fourths" (Mack, 1995). The inverse relationship bet ween number of parts and size of parts cannot be "told" but must be developed in each student through many experiences, including the ones described earl ier related t o partitioning, iterating, magnitude of fractions, and equivalence.

Activity 14.22

CCSS-M: 3 .NF.A.3d; 4 .NF.A.2

Ordering Unit Fractions l.

fo

list a set of unit fractions such as ~. ~. and (assume same size whole for each fraction). Ask students to use reasoning to put t he fractions in order from least to greatest. Challenge students to explain their reasoning with an area model (e.g., circles) and on a number line. Ask students to connect t he two representations. (•What do you notice about ~ of the circle and ~ on the number line?") Repeat with all numerators equal to some number other t han 1. Repeat with fractions t hat have different numerators and different denominators. You can vary how many fractions are being compared to differentiate the task. See also, Activity 14.2 (Who Is Winning?) a nd Activity 14.13 (Fractions on the Number line), both of which can include an area model as a strategy to locate t he fractions on a number line.

Students may notice that larger denominators mean smaller fractions (this is an important pattern to notice), but it only ho lds true when the numerators are the same and students can overgeneralize this idea. Therefore, it makes a good conjecture t o explor e as a class.

Comparing Any Fractions. You have probably lea rned specific algorithms for comparing two fractions, such as finding a common denominators or using cross-multiplication. Too often these procedures are done with no thought about the size of the fractions, and often they are not the most efficient strategy. If students are taught these rules before they have had the opportunity to th ink about the relative sizes of various fractions, they are less likely to develop number sense (conceptual knowledge) or fluenc:y (procedura l knowledge) to effectively compare fractions.

Comparing Fractions

369

WPause & Reflect Examine the pairs of fractions in Figure 14.24 and select the largest of each pair using a reasoning approach that a fourth grader might use. •

Activity 14.23

CCSS-M : 4.a.N F.2

Which Fraction Is Greater? Use the "Which Is Greater?" Activity Page (see Figure 14.24). Ask students to use a reasoning strategy to determine which fraction is greater. For selected problems, ask students to explain how they determined their answer. Discuss the different reasoning strategies, asking students "For what types of fractions does that strategy work?•

Here we share four different ways to compare. Students fluent in comparing fractions choose the best method, based on the fractional values being compared. l. Some-siz e dmominnturs. To compare ~ and ~' think about having 3 parts of something and

also 5 parts of the same thing. (This method can be used for problems Band G.) 2. Same numemturs. Consider the case of ~ and If a whole is partitioned into 7 parts, the parts will certainly be smaller than if partitioned into only 4 parts. (This strategy can be used with problems A, D, and H.) 3. Mo1·e thnn/less thrm a bmchmm·k. The fraction pairs versus ~ and versus do not lend themselves to either of the previous thought processes. In the first pair, is less than t and ~ is more than Therefore, ~ is the larger fraction. In the second pa ir, one fraction is greater than I and the other is le.ss than I . (This method could be used on problems A, D , F, G, and H .) 4. Closeness to tl bmclmuwk. 'W hy is greater than t? Each is one fractional part away from one whole. So, is only away from I, so it is closer to I than Similarly, notice that ~ is smaller than because it is only one-eighth more than a ha lf, whi le a sixth more than a ha lf. (This is a good strategy for problems C, E, I,J, K, and L.)

m

Standards for

M athematical Practice MP2. Reason abstractly and quantitatively.

t.

r

ri

t

t·

fo

Mylab Education

fo

t

to

t.

t is

Activity Page: Which Is Greater?

How did your strategies for comparing the fractions in Figure 14.24 reflect these strategies? Hopefully you employed all or most of these Strategies for at least one of the comparisons. Many comparisons, such as problems D and H, can be solved using several of the strategies listed, providing an excellent opportunity for students to share their thinking and to evaluate which reasoning strategie.~ work well for which types of fractions. Figure 14.25 illustrate.s two students' reasoning for A-F. These strategies should emerge from your students' reasoning; to teach "the four ways to compare fractions" defeats the purpose of learning to choose efficient Strategies. Instead, select pairs of fractions that will likely elicit desired comparison strategie.s. Use contexts such as racing distances and installing carpet (Freeman &Jorgensen, 2015). Use number lines and other visuals to help students reason mentally. For example, you might pose two pairs of fractions with the same numerators. Ask students to tell which is greater and why they think so. Ask them to convince their partner using a visual or explanation. Have some students share their justifications. Discuss wlmz this strategy works for comparing fractions.

Which Is Greater? Use an efficient strategy to compare. Tell how you know which fraction is greater. 4

4

4

5

A.

5 or 9

B.

7 or 7

c. 83

or

7

5

3

3

5

6

9

4

G.

i2 or i2

H.

5 or 7

iO

I.

8 or iO

4

D.

3 or 8

5

5

J.

8 or 3

E.

4 or iO

3

9

K.

6

F.

8 or 7

3

4

L.

9 or 8

4

8

or

7

i2 7

FIGURE 14.24 Comparing fractions using reasoning strategies .

370

(a)

Chapter 14 Developing Fraction Concepts

Using Equivalent Fractions to Compare Equivalent-fraction concepts can be used in making comparisons when the fractions don't lend to the strategies described above. This, roo, can be a reasoning acti,~ty by asking students how they might change one or both of the fractions so they can compare them. Consider how you might compare~ to~- Beyond changing both fractions to find a common denominator (and use strategy I above), they can change only the ~ to so that both fractions are two parts away from the whole {and use strategy 4 above), or change both fractions to a common numemto1· of 12 (and use strategy 2 above). Imagine the fraction number sense students will develop when this is how fraction comparisons are approached. As you will read in the next chapter this solid understanding and flexibi lity with fractions is essential to developing fluency with fraction operations.

fa

(b)

Teaching Considerations for Fraction Concepts Be~-ause the teaching of fractions is so important, and because fractions are often not well understood even by adults, a recap of the big ideas, as well as challenges with fractions, is needed. Hopefully you have recognized that one reason fractions are not well understood is that there is a lot to know about them, from part-whole relationships to di~sion constructs, and understanding includes representing across area, length, and set models and includes contextS that fit these models. Many of these strategies may not have been part of your own learning experience, but they must be part of your FIGURE 14.25 Two students explain how they compared teaching experience so that your students can fully underthe fractions in problems A through F from Figure 14.24. stand fractions and be successful in algebra and beyond. Iterating and partitioning must be a significant aspect of fraction instruction. Equivalence, including comparisons, is a central idea for which students must have sound understanding and skill. Connecting ~sua ls with the procedure and not rushing the algorithm roo soon are important aspectS of the process. Clarke and colleagues (2008) and Cramer and Whimey (20 10), resea rchers of fraction teaching and learning, offer resea rch-based recommendations that pro~de an effective summary of this chapter:

1. G ive a greater emphasis to number sense and the meaning of fractions, rather than rote procedures for manipulating them. 2. Pro~de a variety of models and contexts to represent fractions. 3. Emphasize that fractions are numbers, making extensive use of number lines in representing frat.'tions. 4. Spend whatever time is needed for students to understand equiva lencies (concretely and symbolically), including flexible naming of fractions. 5. Link fractions to key benchmarks and encourage estimation.

Fraction Challenges and Misconceptions StudentS bui ld on their prior knowledge. vVhen they encounter situations with fractions, they natura lly use what they know about whole numbers to solve the problems (Cramer & Whimey, 2010; Lewis & Perry, 2017; Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, & Wray, 20 10). The most effective way to help students reach higher levels of understanding is to

Teaching Considerations for Fract ion Concepts

371

use multiple representations, m ultiple approaches, and explanation and justification (Harvey, 2012; Pantziara & P hilippou, 2012). Common challenges and m isconceptions and how to help are presented in Table 14.3. Anticipatin g student challenges and misconceptions is a critical part of p lanning- it can greatly influence task selection and how the lesson is structured.

Mylab Education

Self-Check 14.4

Mylab Education Math Practice: Need t o pract ice or refresh your mat h content knowledge? Click to access practice exercises associated with t he content from this chapter.

TAB LE 14.3

COMMON CHALLENGES AND MISCONCEPTIONS RELATED TO FRACTION CONCEPTS

Common Challenge or Misconception

Wh t It L00 k L'1k a s e

H t H 1 ow o e P

1. Numerat or and denominator are separate numbers (not seeing a fraction as a number)

~ is seen as a 3 over a 4. Student cannot locate a fraction such as ~ on a number line because they don't think it is a number.

•

• •

i

2. Fractional parts do not need to be equal-sized

Student says (three-fourths) of the figure below is green.

•

3. Fractional parts must be same shape

Student says this square is not showing fourths:

•

•

•

4 . Fractions with Student thinks ~ is smaller than f,; because 5 larger denominators are is less than 10. bigger

5. Fractions with Student thinks ~ is more than larger denominators are fifths are bigger than t enths. smaller

ro because

6. Representing fractions greater than 1

Student cannot represent a fraction such as because it is more than one whole.

7. Determining a whole when glven a part

Student is given an illustration and told this represents ~ and they cannot build onto it to create one whole.

¥

Find fraction values on a number line (e.g., a warm·up activity each day where students place values on a classroom number line). Measure with inches to various levels of precision (e.g., to the nearest fourth, eighth, or sixteenth). Avoid the phrase "three out of four" (unless you are t alking about ratios or probability) or •three over four"; inst ead, say "three-fourths" (Siebert & Gaskin, 2006). Have students create their own representations of fractions across various types of models. Provide problems like the one illustrated here, in which all the partitions are not already drawn and have students draw or show the equal-sized parts.

Provide examples and non-examples with partitioned shapes (see Activity 14.5, for example). Ask students t o generat e as many ways as they can t o show fourths (or eighths).

For misconceptions 4 and 5: • Use contexts, For example, ask students whether they would rather go outside for ~ of an hour, of an hour, or f,; of an hour and to explain why. • Use visuals, such as paper strips or circles to visualize the approximat e size of each fraction. • Teach estimation and benchmark strategies for comparing fractions.

l

• • • • •

Infuse fractions of various magnitudes (less than, equal to, and great er than 1) from the beginning. Use iterating on a number line and count beyond 1 . Do more tasks involving this skill (see Figure 14.12 and Activity 14.11). Encourage students to draw a number line or use another tool. Focus on unit tractions (what does one-fourth looks like?), and then figure out the whole.

372

Chapter 14 Developing Fraction Concepts

RESOURCES FOR CHAPTER 14 LITERATURE CONNECTIONS The Doorbell Rang Hutchins (1986) Often used to investigate whole-number oper.ttions of multiplie~tion and division, this book is also an excellent early introduction to fractions. The story is a simple rale of two children preparing to share a plate of 12 cookies. J ust as they ha,·e figured out how to share the cookies, the doorbell rings and more children arri,·e. You can change the number of children to create a sharing situation that requires fr.lcrions (e.g., 8 children). The Man Who Counted: A Collection of Mathematical Adventures Tahan (1993) This book contai ns a story, "Beasts of Burden," about a wise mathematician, Beremiz, and the narr.ttor, who are traveling together on one camel. They are asked by three brothers to solve an argument: Tbtirfarbtr bas lift tbnn 15 camtls to dividt among tbnn: half to ont brotbtr, ont-tbird to anothtr, and ontnimh to tht third brotbtr. The story is an excellent context for fr.lcrional parts of sets (and adding fractions). Changing the number of camels to 36 or 34, does not solve the challenge bee~ use the sum oft, i, and ~ will never be one whole, no matter how many camels are involved. See Bresser (1995) for three days of activities with this book. Apple Fractions Pallotta (2002) This book offers interesting liters about apples while introducing fr.tcrions as fair shares (of apples, a healthier option than books that focus on chocolate and cookies!). Ln addition, the words for fr.lctions are used and connected to fr.lcrion symbols, making it a good connection for fractions in grades 1-3.

Ten excellt:nt rips fo•· teacbing fractions m·e discussed tmd favorite activities are .rhm·ed. An excel/em overvie-cJJ ofteaching fractio11S. Le,vis, R. M., Gibbons, L. K., Kazemi, E., & LindT. (20 15). Umnapping students ideas about fractions. Teaching Childrm Mathnnatia, 22(3), 158-168. This rxullmt nod pnwidn a h=-ro for hnplmtmting sharing tasks, induding srqumring of tasks, qutstitms to post, and fonnativts assrs-n11e11T roo/ to 711oniror studmt tmdtTStallding. Freeman, D. W., & Jorgensen, T. A. (1015). Moving beyond brownies and pizzas. Tracbing Cbild1·m Mathematics, 2/(7), 412-420. This m1icle describes mulenr tbinking as they compare fractions. In tbe m01'1!4U pages, they offe·r excellent sets of tasks with a range of romexts, each sn f1X'using on a diffennt •·easoning strategy for comparing fractions.

Books Lamon, S. (2012). Trachingfractiom and ratios for undu-standing: Essmtial rommt kntrU!lrdgt and i11Slruaional straugrn. Kew York, NY: Taylor & Francis Group. As the title imp/its, this hook bas a wealth of infomtation to help u:itb bmertllllltmandingfraaions and uachingfractions well. Many r·icb tasks and student work are p1·ovided tbrougbollt. McNamar.t, J., & Shaughnessy, M. M. (2010). Beyond pizuts and pies: I 0 essmtitrl strategies for Slipporting fraction smse (grades 3-5). Sausalito, CA: Math Solutions Publications. This book bas it trlf.-classrom11 vignmes, discussion of•·rsrarch on uaching fractions, and many activitits, induding srudmt work.

Websites RECOMMENDED READINGS Articles Clarke, D. M., Roche, A., & J\1itchell, A. (2008). ' len practical rips for making fractions come alive and make sense. Mathemmics Teaching in the Middle Scbool, 13(7), 373-380.

Rational umber Project (http://www.cehd.umn.edu/ci/ rationalnumberprojectlrnp l-09.httnl).

Tbis p1vject offers exullent lessons and teaching fraction concepts effmively.

otbtt~·

materials fo•·

CHAPTER

Developing Fraction Operations LEARNER OUTCOMES After reading this chapter and engaging in the embedded activities and reflections, you should be able to: 15.1 Describe a process for teaching fraction operations with understanding.

15.2 Illustrate and explain how to add and subtract fractions with different fraction models. 15.3 Connect whol~number multiplication to fraction multiplication , including connecting fraction multiplication to meaningful contexts. 15.4 Connect whol~number division to fraction division using both measurement and partitive real-life examples.

fifth-grade student asks, "vVhy is it when we times 29 times two-ninths that the answer goes down?" (Taber, 2002, p. 67). Although generalizations to fraction computation from whole-number computation can confuse students, they can also help students ma ke sense of operations involving fractions. We must build on students' prior understanding of the whole-number operations to give meaning to fraction computation. This, combined with a firm understanding of fractions (including relative size and equivalence), provides the foundation for understand ing fraction computation (Petit et al., 2010; Siegler et al., 2010). Fraction computation is too often taught without any meaning. Instead, fraction operations instruction must help students be able to answer questions like, "vVhen might we need to multiply by fractions?" and ''"Why do we invert and multiply when d ividing fractions?" Students will be able to answer these questions when these big ideas are the focus of teaching operations involving fractions.

A

BIG IDEAS •

The meanings of each operation with fractions are the same as the meanings for the operations with whole numbers. Operations with fractions should begin by applying these same meanings to fractional parts.

•

For addition and subtraction, the numerator tells the number of parts and the denominator the unit. The parts are added or subtracted.

•

Repeated addition and area models support development of concepts and algorithms for multiplicat ion of fractions.

373

374

Chapter 15 Developing Fraction Operations

•

Partition and measurement models lead to two different thought processes for diVision of fractions.

•

Estimation should be an integral part of computation development to keep students' attention on the meanings of the operations and the expected sizes of the results.

Understanding Fraction Operations Success with fractions, in particular computation, is closely related to success in Algebra I. If students enter forma l algebra with a weak understanding of fraction computation (in other words, they have only memorized the four procedures but do not understand them), they are at risk of struggling in algebra, which in turn can limit college and career opportunities. Build ing such understanding takes time! The Cmmon Core State Standards (NGA Center & CCSSO, 2010) recognize the importance and time commitment required to teach fraction operations well and suggest the following developmental process: Grade 4: Adding and subtracting of fractions with like denominators, and multiplication of fractions by whole numbers (p. 27). Grade 5: Developing fluenty with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions) (p. 33). Grade 6: Completing understanding of division of fractions (p. 39). Grade 7: Solve real-world and mathematical problems involving the four operations with rational numbers (p. 48). Teaching fraction operations with understanding may be a significant point of departure from what you may have experienced as a student-learning only one designated (standard) algorithm While teaching one algorithm for each operation seems quicker, it is not effective! F irst, none of the algorithms helps students think conceptually about the operations and what they mean. When students follow a procedure they do not understand, they have no means for knowing when to use it and no way of assessing whether their answers make sense. Second, mastery of poorly understood algorithms in the short term is quickly lost, particularly with students who struggle in mathematics. All too soon the different algorithms become a meaningless jumble. Students ask, "Do I need a common denominator, or do I just add or multiply the bottom numbers?" "vVhich one do you invert, the first or the second number?" Third, students can't adapt to slight changes in the fractions, such as decimals or mixed numbers. Commit to teaching fraction operations with understanding and avoid these ineffective practices.

Mylab Education

Video Example 15.1

Watch this video about teaching fraction operations with understanding.

Effective Teaching Process Students must understand and have access to a vrwiety of ways to solve fraction computation problems. In many cases, a mental or invented strategy can be appl ied, and a standard algorithm is not needed. Procedural fluency includes being flexible in how students approach fraction operations.

Understanding Fraction Operations

A report summarizing what works for teacrung fraction operations suggests teachers "help students understand why procedures for computations with fractions make sense" (Siegler et al., 2010). The report suggests four steps to effective fraction computation instrut.'tion. Each is briefly described here (and then used as headers within each section of this chapter): 1. Use context7wl tasks. This should seem like deja vu, as this recommendation appl ies to nearly every topic in this book. Contextual problems for fractions help students develop their own methods and build understanding (Cramer & Whitney, 2010; L ewis, Gibbons, Kazerni, & Lind, 2015). Problem contexts need not be elaborate, but need to be familiar to students and fit the type of fractions in the problem (e.g., using ha lves, fourths and eighths for measuring in inches or cups, not tiUrds or fifths). 2. Explm·e eacb opemtilllz witb a vm·iety of models. Area, length, and set models give different insights into fractions, and a stronger conceptual understanding (Zhang, Clements, & Ellerton, 2015). Importantly, the models must be connected to the context (e.g., use a length model with a story about walking). Additionally, the models must be connected to the symbolic operations. The visua ls will help students make sense of the symbols and related operations, but only when they are explicitly connected through repeated experiences. 3. Let estimation and invented metbods play a big 1·ole in tbe developmellt of stmtegies. "Should 2! X be more or less than I? More or Jess than 2 ?" Estimation keeps the focus on the meanings of the numbers and the operations, encourages reflective thinking, and helps build number sense with fractions. Can you reason to get an exact answer without using the standard algorithm? One way is to apply the distributive property, splitting the mixed

375

m

Standards for Mathematical Practice MP1. Make sense of problems and persevere in solving them.

!

number and multiplying both parts by !: (2 X

!} +

(~

X

!}.'T\vo h are i or ! and a

half of a fourth is k. So, add an eighth to a half and you have ~. 4. Address cballenges, Cll1Jtmon e1-rors, and misc1J11.ceptions regarding computational procedures. Students apply their prior knowledge-in this case whole-number computation-to new knowledge. Using whole-number knowledge can be a support to learning. For example, ask, "vVhat does 2 X 3 mean?" Follow this with "vVhat might 2 X 3! mean?" The concepts of each operation are the same, but the procedures are different. Trus means that whole-number knowledge also leads to errors (e.g., adding denominators when adding fractions). Teachers should present common misconceptions and discuss why some approaches lead to right answers and why others do not (Siegler et a!., 20 I 0). Ongoing connections must be made between the contexts, models, and processes. It takes multiple experiences and time for these relationships to be well understood. 'Wrule this chapter addresses operations in separate sections, students need opportunities to solve story problems to determine wbicb operation fits the story. Too often students are given subtraction story problems on a day they are learning how to subtract, wruch removes the central question of "Which operation is a match to this story situation?"

WPause & Reflect Which operation can be used to solve the story situations in Figure 15.1? •

Let's look at jeremy's situations in Figure 15. 1 (area situation). In (a) the answer is in the problem-Jeremy ate k of the cake. In the second case, you must figure out ~of!. which is of the cake. In the third case, it is not clear if the cakes are the same-sized whole, so there is no way to solve this problem. If it is adapted to clarify that the cakes are the same-size, then it could be solved using addition. In J e.ssica's first situation (linear), the d ifference is needed, comparing how far she has walked to how much she needs to walk. In the latter case, the question is "How many trips?" or "How many eighths are in three-fourths?" This can be solved by counting eighths or by division: + ~· A story problem like any one of these can be posted as a warm-up problem throughout year with the focus on what is happening in the story and then discussing which operation makes sense (and why).

rz

*

MyLab Education

Self-Check 15.1

-

Standards for M athematical Practice MP2. Reason abstractly and quantitatively.

376

Chapt er 15 Developing Fraction Operations

a. Jeremy's friends eat~ of his birthday cake. Jeremy eats kof the cake. How much did Jeremy eat? b. Jeremy's friends eat~ of his birthday cake. Jeremy ate k of the leftover cake. How much of the cake did Jeremy eat? c . Jeremy's friends ate~ of one birthday cake and kof another cake. How much cake was eaten? d. Jessica walked kof a mile. Her goal is~ of a mile. How much further to reach her goal? e. Jessica walks k of a mile to school. How many trips to and from school are needed for her to reach her goal of wal king~ of a mile? FIGURE 15.1

Mixed story problems involving fractions.

Addition and Subtraction Here we describe how the four steps described earlier are used to develop a strong understanding of addition and subtraction (usua lly both are taught together). Students need to find a variety ofways to solve problems with fractions, and their invented approaches will contribute to the development of standard algorithms (Clarke et al., 2008).

Contextual Examples There are many contexts that can be used for adding and subtracting fractions. Recall that the CCSS-M outline addition situations, wruch are referenced for whole-number operations, but that also need to be applied to fraction addition and subtraction. Those situations are join, separate, part-part-whole, and c::ompare (NGA Center & CCSSO, 2010; Chval, Lannin, &Jones, 2013). For fraction subtraction, t!Us means including take-away situations (separate), as well as difference situations (compare). See Chapter 8 for more on whole-number addition and subtraction situations. Consider the real-life context of measuring something in inches (sewing, t·utting molding for a doorway, hanging a picture, etc.). One inch is one unit or one whole. Measurements include halves, fourths, eighths, and/or sixteenths-fractions that can be added mentally by considering the relationship between the sizes of the parts (e.g., ~ = = ~ = ft). If you ~~n easily find the equivalent fraction (e.g., that one-fourth of an inch is also two-eighths), then you can add like-sized parts mentally. Contexts should vary and be interesting to students. Here are several good examples. As you read, think about how they differ from each other (beyond the story context):

i

Jacob ordered 3 pizzas. But before his guests arrived he got hungry and ate much was left for the party?

i of one pizza. How i

On Friday, Lydia ran 1~ miles, on Saturday she ran 2 ~miles and on Sunday she ran 2 miles. How many miles did she run over 3 days?

i

Sammy gathered pounds of walnuts and Chala gathered ~ pounds. Who gathered the most? How much more?

In measuring the wood needed for a picture frame, Elizabeth figured that she needed two pieces that were inches and two pieces that were 7 inches. What length of wood does she need to buy to build her picture frame?

Sl

i

Addition and Subtraction

Notice that these story problems (1) incorporate different addition siruations Goin, compare, etc.); (2) use a mix of area and linear contextS; (3) use a mix of whole numbers, mixed numbers, and fractions; (4) include both addition and subtraction siruations; and (5) sometimes involve more than two addends. With each story you use, it is very important to ask srudenrs to select a picture or tool to illustrate it and write the symbols that accurately model the siruation. Meaningful contextS are embedded throughout the next sections, connected to models and symbols.

[J

I

I

5

10

377

-+ - =

Estimate first by putting an X on the number line.

0

2

I 2

1-

2

FORMAnYE ASSESSMENT Notes. Any of the prob-

lems above can be used as a formative assessment with an observation checklist. On the checklist would be such concepts as: (1) can determine a reasonable estimate; (2) selects and accurately uses a manipulative or picrure; (3) recognizes equiva lences between fourths and eighths; and (4) can connect symbols and visuals. Sruden ts may be able to illustrate but then not find equivalences, or srudents may select a circle to illustrate and find it does not help them think about the siruation. The next steps in instruction cou ld include expl icit attention to connecting the picrure to the symbols, or to use other manipulatives or picrures to represent the siruation. •

Solve with Fraction Circles. D raw pictures of what you did with the circles below.

~ 'h•6l@ Record what you did with the circles with symbols.

Models Recall that there are area, length, and set models for illustrating fractions (see C hapter 14). Set models can be confusing in adding fractions, as they can reinforce the adding of the denominator. Therefore, instruction should initially focus on area and linear models.

FIGURE 15.2 A student estimates and then adds fractions using a fraction circle. Source: Cramer, K.. Wyberg, T., & Leavitt, S. (2008). "The Role of Representations in Fraction Addition and Subtractio n." Mathematics Teaching in the Middle School, 13(8), p. 495. Reprinted with permission. Copyright@ 2008 by the National Council of Teachers of Mathematics.

Area Models. Circles are an effective visual for adding and All rights reserved. subtracting fractions because they allow srudenrs to develop mental images of the sizes of different pieces (fractions) of the circle (Cramer, Wyberg, & Leavitt, 2008). F igure 15.2 shows how srudenrs estimate first (including marking a number line) and then explain how they added the fraction using fraction circles.

Mylab Education

Video Example 15.2

Watch this video as Felicia uses circles to add.

Mylab Education

Video Example 15.3

Watch this video as Felicia uses circles to subtract.

Consider this problem, in which the context is circular: Jack and Jill ordered two medium pizzas, one cheese and one pepperoni. Jack ate ~ of a pizza, and Jill ate ~ of a pizza. How much pizza did they eat together?

m

Standards for Mathematical Practice MPS. Use appropriate tools strategically.

378

Chapt er 1 5

Developing Fraction Operations

WPause & Reflect Think of two ways that students migllt solve this problem without using a common-denominator symbolic approach. •

If students draw circles as in the earlier example, some will try to fill in the ~ gap in the pizza. Then they will need to figure out how to get ~ from ~. If they can think of~ as ~. they can use one of the sixths to fill in the gap. Another app~oach, after drawing the two pizzas, is to notice that there is a half plus two more sixths in the~ pizza. Put the two halves together to equal one whole, and there are~ more- l ~. These are certainly good solutions that represent the type of reasoning you want to encourage. There are many other area models that can be used, such as rectangles and pattern blocks. Rectangles, like circles, can be partitioned in whatever way is useful for the problem being solved. Activity 15.1 uses a rectangular context and illustration.

Activity 15.1

CCSS- M: 4 .NF.B.3a, d ; 5 .NF.A.1 ; 5.NF.A.2

Gardening Together Distribute the Empty Garden Act ivity Page (or give each student a blank piece of paper to represent the rectangular garden). Exp lain the situation a nd ask each student to design the garden to illustrate these quantities: AI, Bill, Carrie, Danielle, Enrique, and Fabio are each given a portion of the school garden for spring planting. Here are the portions: AI =

~

Danielle =

f6

Bill =

~

Enriq u e = ~

Carrie =

fs

Fabio = ~

They decide to pair up to share the work. What fraction of the garden will each of t he following pairs or groups have if they combine their portions of the garden? Show your work. Bill and Danie lle

A I and Carrie

Fabio and Enrique

Carrie, Fabio, and A I

To challenge students, ask them to solve puzzle-type questions like: · w hich two people could work toget her and have the least amount of the garden? The most? Which combinations of friends could combine to work on one-half of the garden?"

Mylab Education

m

Standards for M athematical Practice MP4. Model with mathematics.

Activity Page: Empty Garden

Unear Models. Cuisenaire rods are linear models. Suppose that you had asked the students to solve the Jack and Jill pizza problem but changed the context to submarine sandwiches Oinear context), suggesting students use Cuisenaire rods or fraction strips to model the problem. The first decision that must be made is what strip to use as the whole. That decision is not required with a circular model, where the whole is already established as the circle. The whole must be the same for both fractions. In this case, the smallest rod that will work is the 6-rod or the dark green strip, because it can be partitioned into sixths (1 rod/ white) and into halves (3 rod/light green). Figure 15.3(a) illustrates a solution. What if you instead asked studentS to compare the quantity that Jack and Jill ate? Figure 15.3(b) illustrates lining up the "sandwiches" to compare their lengths. Recall that subtraction can be thought of as "separate" where the total is known and a part is removed, "comparison" as two amounts being compared to find the difference, and "how many more are needed" as starring with a smaller value and asking how much more to get to the higher value (think-addition). This sandwich example is a comparison situation-be sure to include more than "take away" examples in the stories and examples you create. An important model for adding or subtracting fractions is the number line (Siegler et al., 2010). One advantage of the number line is that it can be connected to the ruler, which is a familiar context and perhaps the most common real context for adding or subtracting fractions. The number line is also a more challenging model than an area model, because it requ ires that the student understand i as 3 parts of 4, and as a value between 0 and 1 (Izsak, Tillema, & Tunc-Pekkam, 2008). Using the number line in addition to area representations can strengthen student understanding (Clarke et al., 2008; Cramer et al., 2008; Petit et al., 2010).

Addition and Subtraction

(b)

5

1

6

2

Find a strip for a whole that allows both fractions to be modeled.

379

~Whole

5

6 -

t

Whole

t

i

ior~

The sum is 1 whole and a red rod more than a whole.

Compare the lengths of the two fractions.

Ared is5ofadarl< green. So ~+ ~ = 15.

~ is ~ longer than ~ . so the difference is ~ or

FIGURE 15.3

a.

Using Cuisenaire ro ds to add and subtract fractions.

Activity 15.2

CCSS- M: 4 .NF.B.3a , d ; 5.NF.A.1 ; 5.NF.A.2

Jumps on the Ruler Use Jumps on the Ruler Activity Page (or provide a ruler). Have students use jumps on the ruler to add or subtract (which does not require a common denominator, though students may naturally trade equivalent

ENGLISH LEARNERS

fractions to help them reason). Examples include:

4 ~ - 3~

~+ ~ I

I

I

I

I

I

0

I

I

I

I

I

I

I

I

I

I

I

I

2

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

4

3

I-.

5

Either add a linear context (length of grass in the yard, hair growing/getting cut), or ask students to create their own stories for each. These j umps encourage students to invent strategies without having to find a common denominator (Taber, 2009). When sharing strategies, listen for students iterating (counting) and using fraction equivalencies. In the first problem, students might use

i

i.

i

1 as a benchmark. They count up ~ to get to a whole, then have more to add on to 1 Or, they could use ~ from the to equal a whole with the!. then add on the }. Els may not be as familiar with inches, because most countries measure in metric. In this case, be sure to spend time prior to the activity discussing how the inch is partitioned and add labels for fourths as a reminder that the inch is different from metric system units.

Mylab Education

Activity Page: Jumps on the Ruler

Mylab Education Application Exercise 15.1: Observing and Responding to Student Thinking Click the link to access this exercis e, then watch the video a nd answer the accompanying questions.

T he selected context and quantities can be adapted to steer students towa rd specific strategies (difference or take-away). For example, consider two ways to pose the subtraction problem:

t

1. Desmond runs 2 miles a day. If he has just passed the I ~ mile marker, how far does he still need to go? 2. Desmond is at mile ma rker 2 and James is at mile marker I ! . How much farther has Desmond gone?

t•

380

Chapt er 1 5

Developing Fraction Operations

'W'ith many experiences with area and linear models, students are ready to connect those visuals to algorithms . . . TECHNOLOGY Note. Conceptua Math has free tools that help students explo re various fraction concepts using area, set, and length models (including the number line). These interactive tools allow students to explore all the operations with attention to the concepts and the commutative and distributive properties of add ition and multiplication. •

W

Estimation Estimation is one of the most effective ways to build understanding and procedura l fluency with fraction operations (Johanning, 201 1). A frequently quoted resu lt from the Second National Assessment (Post, 1981) concerns the following item: Estimate the answer toM

+ ~· You will not have time to solve the problem using paper and pencil.

Nearly two-thirds of middle school students could find the exact answer to this problem, but only one-fourth could correG"tiy estimate (Reys, 1998). Notice that computing this answer requires finding the common denominator of thirteenths and eighths, but to estimate requires no computation whatsoever-only a realization that each fraction is close to I, so the answer is dose to 2. T here are different ways to estimate fraction sums and differences (Siegler et al. , 2010):

J,

J,

1. Be-ncbmarks. Decide whether the fractions are closest to 0, or I (or to 3, 3 or 4-the closest whole numbers and the half in between them for mixed numbers). After making the determination for each fraction, mentally add or subtract . Example:

k + [a. Think, "k is close to 1, [a is close to 0, the sum is about 1

+

0 or close to 1."

2. Relative size ofunit ft-actions. D ecide how big the fraction is, based on its unit (denominator), and apply this information to the computation.

fa.

"k

1

Example: ~ + Think, is just away from a whole (one) and the sum will be close to, but less than, 1."

1 is close to (but bigger than) frs, so

T he following activity ~-an be done frequently as a routine, warm-up or be a focus activity for a full lesson.

Activity 15.3

CCSS-M: 4 .NF.B.3a; 5 .NF.A.2

Over or Under 1 Tell students that they are going to estimate sums or differences of fractions. They are to decide only whether the answer is more than 1 or less than 1. Using Estimating Addition and Subtraction of Fractions Problems Activity Page,

STUDENTS with

SPECIAL project a problem for no more than 10 seconds, then hide or remove it. Use Less Than One, More Than One cards, NEEDS m ini whiteboards to write "over" or "under; clickers, or thumbs up/ down to see each students estimate. Then discuss how students decided on their estimates (including connecting to mental models of number lines or other models to reason). Students with disabilities might use a number line to think about the relative size of the fractions. They may also need more than 10 seconds to think about the amounts. Repeat with other examples (Figure 15.4 offers several possible problems.)

Mylab Education Mylab Education

Activity Page: Estim ating Addition and Subt raction of Fractions Problems Activity Page: Less Than One, More Than One

Over time, Activity 15.3 can include tasks that are more challenging, or it can be differentiated with different groups of students working on di fferent over/under values. Con sider the following variations: • Use a target other than I. For example, estimate more or less than!. I~. 2, or 3.

Addition and Subtract ion • Choose fraction pairs in which the fractions are both less than I or both great er than I. Estimate sums o r differen ces t o the nearest ha lf. • Ask studentS to n-er1te equations that are slightly less than or sl ig htly more t han I (or other va lues). They can trade equations with other studentS, who in tum decide if the answer is over I or under I (or ot her values).

Mylab Education

Estimate

Activity Page: Estimate, Write, Explain

WPause & Reflect Test your own estimation skills with the sample problems in Figure 1 5.4 . Look at each computation for only about 10 seconds and write down an estimate. After writing down all six of your estimates, look at the problems and decide whether your estimate is higher or lower than the actual computation. •

1.

8 + 5

1

4

2.

!!.

+ 7

Number your papers 1 to 6. Write only answers.

8

10

1

3

3.

3 5 + 4 + 8

4.

3

1

5.

12 - 4

11

3

6.

9 ,! - ;o 2

Estimate! Use whole numbers and benchmark fractions.

4 - 3

FIGURE 15.4

Example of fraction es timation expressions .

A relevant and familiar context for fractions is cups, such as cups of water or milk (Fung and Latu lippe, 2010; Zhang, et al. , 2015). Activity 15.4 uses cups to engage students in estimating sums and differences.

Activity 15.4

381

CCSS- M: 4 .NF.B.3a; 5 .NF.A.2

Cups of Milk Ask students to fill in the missing numerator and denominator with a whole number that makes the equation approximately true. They cannot use any of the numbers already in the problem. The context of milk {or water) can help students reason about the qua ntities. Four examples are provided here:

LjF+lj}s[P Lj} - Lj} ~ @ Keep in mind that these are estimates, so many values are possible. Except in some cases, where there a re no solutions (can you spot the impossible one in the list here?).

Developing the Algorithms T he algor ithms develop side by side with the visuals an d situations. T he way fractions ar e notated can lead to errors when studentS compute fractions using symbols, even if they are able to add using models. T he way to prevent this is to ensure students are connecting fraction symbols with contextS and visuals.

Mylab Education Application Exercise 15.2: Developing the Algorith m Click the lin k to access this exercise , then watch the video and answer the accompanying questions.

-~ ~ ~-t-.! ....

VID£0 EXAMP\.£

Like Denominators. Fraction addition and subtraction begin with s ituations using li ke denominators. In the Common Core State Standards (NGA Center & CCSSO, 2010), this is

382

Chapter 1 5

Developing Fraction Operations

suggested for grade 4. "W hen working on adding with like denominators, however, it is important to be sure that studentS are focusing on the key idea-the unitS are the same, so they ~-an be combined (Mack, 2004). This is the iteration idea discussed in Chapter 14. In other words, the problem ~ + ~ is asking, "How many fourths altogether?" The equation 3 ~ - 1~ is counting back (taking away) eighths or finding the number of eighths between the two quantities (comparing). Iteration connects fraction operations to whole-number operations and explains why the denominator stays the same.

-

Standards for iiji Mathematical Practice MP7. Look for and make use of structure.

n

FORMATIVE ASSESSMENT Notes. The ease with which studentS can or cannot add

LJ like-denominator fractions should be viewed as an important concept assessment before moving students to an algorithm. A diagnostic interview can ask students to (1) explain the meaning of the numerators and denominators, (2) connect the addition problems to a situation, and (3) illustrate with a model, which can help you determine whether they have a deep understanding of the numerator and denominator. If student responses are rule oriented and not grounded in understanding parts and wholes, then encourage studentS to focus on the meaning of the fractions by emphasizing the unit: "Three fiftbs plus one fiftb is how many fiftbs?" This must be well understood before moving to adding with un like denominators. Otherwise, any further symbolic development will almost certainly be without understanding. •

Unlike Denominators. As d iscussed in Chapter 14, having a strong conceptual foundation of equivalence is critica l to operations of fractions. StudentS who have a level of fluency in moving between ~. §, and f7, or ~. and can adjust the fractions as needed to add or subtract fractions. For example, ask students to think about how far a person has walked, if the following equation represents their walking: + Given sufficient concrete experiences, a student should be able to readily trade out one-half for four-eighths to solve:

t.

These are the same.

So rewrite them.

~

+

~

is the

same as

5

fa + fa

1

6-2 Use the dark green as a whole. It is 6 whites long. In terms of the white strips, the problem

~

-t

is the same

as~-~

i

i,

Ji

t.

Begin add ing and subtracting fractions with unlike denominators with tasks where only one fraction needs to be changed, such as ~ + ~· A situation might be amount of pizza eaten. Ask studentS to estimate whether they think one or more pizzas were eaten altogether. Then, invite students to draw or select a tool to illustrate a solution, recording fractions to go with their drawings or manipulatives. As students explain how they solved it, write equivalent expressions on the board and ask if they are equal:

~+_g

4

3

Use sets of 12.

••• • ••• •• •••

•••• •••• ••••

3

2

4 FIGURE 15.5 ferent tools.

This is just t he same

9

8

as12+12 . Rewrite the problem.

3

Illustrating common denominators with dif·

Continue with examples in which both fractions need to be changed-for example, + (or !}, still using contextS, visua ls, and student explanations. In the discussion of student solutions, focus attention on the idea of ,.ew,·iting on equiv11/e-nt problem to make it possible to add or subtract equalsized ports. If students express doubt about the equivalence of the two problems ("Is -/1 + really the answer to + ~?"), that should be a cue that the concept of equiva lent fractions is not well understood, and more experience using examples, visuals, or concrete tools is needed (see Figure 15 .5).

r

f

fi

f-

f

Addition and Subtraction

383

T he three examples in F igu re 15.5 show how tools mig ht be used. Note that each tool requires students to think about the size of a whole that can be partitioned into the units of both fractions (e.g., fifths and halves requ ire tenths). Over time, the equations may be given without situations or visuals, but students should be able to create stories or visuals for any pro blem. As you may recall in the discussion of whole numbers and in algebra, it is important to not always have "result unknown." The parts can also be unknown. This helps students relate addition to subtraction. Any of the stories provided earl ier can be adapted to have a part unknown. For example: Sammy gathered some walnuts and Chala gathered~ pounds. Together they gathered 1~ pounds. How m uch did Sammy gather'?

Activity 15.5 provides an engaging opportunity to reason about fractions when the result is known.

Activity 15.5

CCSS-M: 4.NF.B.3a; 5.NF.A.2

Can You Make It True? Use the Can You Make It True? Activity Page, w hich includes list of equations with m issing values, such as t he ones here:

~+ ~

= 1 and

a- ~ = ~

To begln, post one example and invite students to share possible whole number solutions. If they have found one way, ask "Is there more than one way to m ake that equation true?" Include examples that are impossible for whole number m issing values, such as:

&-

~

=1

As student s work, ask them to explain the reasoning they are using. Encourage student s to use visuals {e.g., number line or fraction pieces) to support their think ing.

Mylab Education

Activity Page: Can You Make It True?

Equations such as the ones in Activity 15.5 require reasoning and thinking about the role of the numerator and denominator, as well as reinforcement of why the numerators are added and the denominators are not.

Are Common Denominators "Required"? No. Teachers and websites often explain, "To add or subtract fractions, find common denominators." T his is one strategy, but not the only strategy.

Mylab Education

Video Example 15.4

Watch Javier illustrate his more straightforward mental strategy.

•

.

Using reasoning strategies, students will see that many correct solutions are found without ever finding a common denominator (Taber, 2009). As described earl ier, number lines can be used to solve addition and subtraction situations without finding a common denominators (see Acti,~ty 15.2 Jumps on the Ruler). Importantly, fluency requires choosing a strategy, so encourage students to decide on a strategy and contin ue to encourage reasoning strategies even after students learn to find common denominators to add/subtract.

m

Standards for Mathematical Practice MP1. Make sense of problems and persevere in solving them.

384

Chapt er 1 5

Developing Fraction Operations

Fractions Greater Than One A separate algorithm for mixed numbers in addition and subtraction is not necessary even though mixed numbers are often treated as separate topics in traditional textbooks and in some lists of objectives (note that mixed numbers have been integrated throug hout the previous discussion). Include mixed numbers in all of your stories and examples and encourage students to solve them in ways that make sense to them. When adding mixed numbers, it makes sense to focus on whole numbers first. Consider this problem: You had 5 kya rds of fabric and used 3 ~ya rds. How much do you have left? What equation descri bes this situation? 5 k - 3 ~· It is a linear situation, so a number line is a good way to think about this problem. Given the context students are likely to subtract 3, leaving 2 and then need to subtract Students may count down (iterate), stopping at I ~· Another approach is to take from the whole part, 2, leaving I t, and then add the kback on to get I ~ or The standard algorithm, which is not as intuitive, is to trade one of the wholes for add it to the kto get I and then take away~· Students tend to make many errors when regrouping mixed nun1bers. An excellent alternative is to change the mixed numbers to fractions greater than I. You may connect this strategy with multiplying fractions, but it works for addition and subtraction too. Let's revisit 5 k - 3 ~· T his can be rewritten as~ (See C hapter 14 for conceptual ways for helping students do this.) Because 41 -29 is 12, the solution is 1f or I ~· T his is certainly efficient and will always work. Provide both options to studen ts and encourage them to decide which Strategy works the best in different situations.

k,

i

11.

i·

l

i

?t.

&-. TECHNOLOGY Note. The National Library of Virtual Manipulatives (NLVM) has three W different activities that can help reinfo rce fraction addition and subtraction across different models: Fractions-Adding: 1\\'0 fractions and an area model for each are given. T he user must find a common denominator to rename and add the fractions. Fraction Bars: T his applet places bars over a number line on which the step size ~-an be adjusted, providing a flexible model that can be used to illustrate addition and subtrat.'tion. Diffy: This puzzle asks you to find the differences between the numbers on the com ers of a square, working to a desired difference in the center. This a~'tivity encourages students to consider equivalent forms of fractions to solve the puzzle. •

Challenges and Misconceptions It is important to explicitly ta lk about common misconceptions with fraction operations because students naturally overgeneralize what they know about whole-number operations. Table 15 . I offers some common challenges and misconceptions, with suggestions for helping students overcome them.

TABLE 15.1

COMMON CHALLENGES AND MISCONCEPTIONS FOR ADDING AND SUBTRACTING FRACTIONS

Common Challenge or Misconception

Wh t It L00 k L'1k a s e

1. Apply whole number operation rules.

~

+

~ = ~

H t H 1 ow o e P • • • • •

Ask students to tell what the numerator and denominator mean. Tell stories with familiar contexts. Ask students to create a story using a familiar context to make sense of the operation. Use area and linear manipulatives and be sure students connect the visual to the steps in the algorithms. See Video Example 15 .5 for more ideas.

Addition and Subtract ion

Common Challenge or Misconception

Wh t It L00 k 5 L'1k a e

H

2. Get different answer with models than with symbols (and aren't bothered that the two answers are different (Bamberger et at., 2010).

Rlr a task such as:

•

Ms. Rodriguez baked a pan of brownies

•

for the bake sale and cut the brownies into 8 equal-sized parts. In the morning, three of the brow nies were sold; in the

• •

afternoon, two more were sold. What

385

t H 1 ow 0 e P Ask students to decide whether both answers could be right. Ask students to defend which is right and why the other answer is not right. Ask students to use the same model to show the solution. Encourage students to do the problem on a number line.

fractional part of the brownies had been sold? What fractional part is still for sale? Student correctly illustrates:

But then writes

i + i = ft·

3. Do not find a comm on denominator, but apply standard algorithm (Siegler et al., 2010). 4. Struggle to find common denominator (lack of fluency with basic facts and with common multiples).

• • Student is given ~ - 1 = _ and is unsure what to use for a denominator.

• • •

• •

5. Mixed numbers: Subtract the smaller traction from the larger (Petit et al., 2010; Siegler et al., 2010; Spangler, 2011).

When given a problem like 3~ - 1~. student subtracts the smaller fraction from the larger

6. Mixed numbers: Don't know what to do when one number Is a whole number.

Rlr a problem like 4 - ~- student places an 8 under the 4 to get ! - See Video Exampie 15.6 of Jacky's thinking.

(i - }).

Ask students to model the problem on a number line or with fraction strips. Go to NCTM's Illuminations •Making and Using Fraction Strips" (or other online fraction tool) to explore problems. Offer games and activities to reinforce multiplication facts. Do common multiple activities, such as Activity 15.6. Discuss that while finding a smaller common de nom~ nator (e.g., 12) has advantages, any denominator (e.g., 24) will work. Ask students to discuss strategies they use to find common denominators. Explore Interference Task, which uses common multiples to determine how often two orbiting satellites will cross paths (adapt to tasks with single-digit values).

For all of these mixed number challenges, strategies to help include: • Include more mixed numbers throughout addition of fraction units (rather than saving them for end of unit). • Use models and contexts, in particular the number line- think about how straight forward 4 - l is on the number line or ruler. • Make these errors part of public discussions, for example using worked examples with such errors.

k-

7. Mixed numbers: Focus only on When given a problem like 14~ the whole-numbers In the problem. student writes 11~ or 11~.

Mylab Education

Activity Page: Interference Task

Mylab Education

Video Example 15.5

3~.

Listen and watch how to address common errors in adding fractions.

Mylab Education

Video Example 15.6

Listen to Jacky's thinking as she tries to subtract a fraction from a whole number.

386

Chapter 15 Developing Fraction Operations

Activity 15.6

CCSS-M: 4 .0A.8 .4 ; 5 .NF.A.1

Use pairs of numbers between 2 and 12. Write the LCM on reverse.

Common Multiple Cards Use Common Multiple Cards with pairs of numbers t hat are potential denominators. Most STUDENTS should be less than 12 (see Figure 15.6). Or, with give students a deck of cards. Place students in SPECIAl NEEDS pairs. On a student's turn, he or she turns over a Common Multiple Card, or two cards from a deck of cards, and states a common multiple {e.g., for 6 and 8, a student m ight suggest 48). The partner gets a chance to suggest a smaller common multiple (e.g., 2 4). The student suggesting the least common multiple (LCM) gets to keep the card. Be sure to include pairs that are prime, such as 9 and 5; pairs in which one is a multiple of the other, such as 2 and 8; and pairs that have a common divisor, such as 8 and 12. For students with disabilities, start with the pairs wherein one number is a multiple of the other.

Mylab Education Activity Page: Common Mult iple Cards

m

Standards for Mathematical Practice MP1. Make sense of problems a nd persevere in solving them .

6,9 3,4

r-'\ 18

2,6

12

Greatest common divisor cards can be made the same way.

FIGURE 15.6

Least common multiple (LCM) flash cards.

Mylab Education

Self-Check 15.2

Multi pi ication

Can you think of a situation that requires using multiplication of fractions? Have you used this a multipl ication of fraction algorithm outside of mathematics classes? Often the answer to these questions is no. It is not that there are no situations involving multipl ication of fractions, but that because the meaning is not understood, the algorithm is never put into use. As you will see in the sections that follow, the foundational ideas of iterating (counting) fractional parts and partitioning are at the heart of understanding multipl ication of fractions. If students have not had sufficient experiences with these two ideas, you must find ways to engage them in iterating and partitioning so that they are ready to apply these ideas to multiplication. Multiplication of fractions is really about scaling. If you scale something by a factor of 2, you multiply it by 2. "W hen you scale by 1 (1 times the size), the amount is unchanged (identity property of multiplication). Similarly, multiplying by! means taking half of the original size, while mu ltiplying by 1! means the original size plus half the origi nal size. This scaling concept can enhance students' ability to decide whether their answers are reasonable. A possible progression of problem difficulty is developed in the sections that follow.

Contextual Examples and Models When working with whole numbers, we would say that 3 X 5 means "3 sets of 5" (equal sets) or "3 rows of 5" (a rea or array) or "5 three times" (number line). Notice the set, area, and linear models fitting with the multipl ication structures. Different models must be used and aligned with contexts so that students get a comprehensive understanding of multipl ication of fractions. The story problems that you use to pose multiplication tasks to students need not be elaborate, but it is important to think about the numbers and contexts that you use in the problems. A possible progression of problem difficulty is developed in the sections that follow.

Multiply a Fraction by a Whole Number. These problems look like this: 5 x

!.

6 X ~. 10 X i, and 3 X 2! and in the CCSS-M are introduced in grade 4. Look again at the wording of this subsection-because the word fraction preceded wbole numbe1·, there is sometimes confusion over what this means, but to multiply a fraction by a whole number means examples such as the ones provided here. Students' first experiences should be of th is type

Multiplicat ion

because conceptually it is a dose fit to multipl ication of whole numbers using the idea of equal sets. Consider the following two situations: Marvin ate 3 pounds of meat every day. How much m eat did Marvin eat in one week? Murphy ate ~ pounds of meat every day. How much meat did Murphy eat in one week?

-

387

Standards for

i i i Mathematical Practice MPS. Look for and express regularity in repeated reasoning.

WPause & Reflect What expressions represent each situation? What reasoning strategies would you use to solve each? •

For Man~n the expression is 7 (groups of) 3 pounds, or 7 X 3. You can solve this by skip counting 3 + 3 + 3 + 3 + 3 + 3 + 3 to get the answer of 21 pounds {but if you know your facts, you just multiplied). Murphy similarly ate 7 X ~ pounds, and it can similarly be solved by skip counting, this time by his portion size of thirds: ~ + ~ + ~ + ~ + ~ + ~ + t, wh ich in total is seven-thirds ( Notice the skip counting, called itemting, is the meaning behind a whole number times a frat.'tion.

0.

Activity 15.7

CCSS-M: 4.NF.B.4a, b

Hexagon Wholes Distribute a set of hexagons to students. To start, designate the yellow hexagon as the whole. Ask students the fractional value of the green, blue, and red pieces. Ask students to find how many wholes for different quantities: 5 blue pieces?

ENGLISH LEARNERS

10 green pieces?

Ask students to write a fraction equation to represent their problem. For the two examples here: 5 X

i

= 1 ~ or ~

10 X

g=

!j or 1 !

or 1 ~

Create a variety of tasks. Other wholes can be used-for example, using two hexagons as one whole to vary the types of fractions t hat ca n be used. Els may benefit from having the shapes labeled with their names and the fraction symbols written with fraction words {e.g., ~ is "one-sixth")

Circular Fraction Pieces can be used in addition to o r instead of pattern blocks to do Activity 15.7. Notice that in the answers given, the fraction (improper fraction) shows a pattern that can be generalized as a X = i, an important pattern for students to discover from having explored many examples that follow this pattern. Linear examples are also important to include. Use Jumps on the Ruler-Multiplication Acti,~ty Page to explore jumps of equal length (see A1tmtltwds and the CCSSM practices. Reston, VA: NCTM. Leinwand, S. (2007). Four teacher-friendly postulates for thriving in a sea of change. Mathm11Jtics Te11ch"; 100(9), 582-584. Ma, L. (1999). Knowing tmd teaching elnnmttnym11thmu1tics: Teachers' undentantling ofjimdnmnJtai mtJtbnuatics i11 ChintJ tmd th~ United States. Mahwah, NJ: Erlbaum . Maloney, E., G underson, E. Ramirez, G., Levine, S., & Beilock, S. (20 14). Teachrrs' math anxiety •·elates to girls' am1 boys' math achievemmt. Unpublished manuscript. Morris, A. K., & H iebert, J . (20 17). Effects of teacher preparation courses: Do graduates use what they learned to plan mathematics lessons' Americ11n Educational Rese11rch Joumtt!, 54(3), 524-567.

662

National Center for Education Statistics. (2014). Retrieved from "~vw. nationsreportcard.gov/reading_math_20 15/#'grade:4 NCTM (National Council of Teachers of Mathematics). ( 1989). Cu17·iruimuand roaiuntio11 Jttmdartls[err school 1J11Jtbrmlltics. Reston, VA: NCTM. NCTM (Nationa l Council of Teachers of Mat he matics) . (1995). Assemnmt stand11rds fo•· school 11111thematics. Resto n, VA: NCTM . NCTM (National Council of Teachers of Mathematics). (2000). P1·inciples tmtl standards for school mathm~t~tics. Reston, VA: NCTM. NCT M (National Council of Teachers of Mathematics). (2006). Cm~ •·imlum f'ocal poims for p•·ekim1n·g mtm tbrough gmtle 8 7111lthm1lltics: A lfiU'st jorcohennce. Reston, VA; NCTM. NCTM (National Council ofTcachers of Mathematics). (2014). Position J1tltrmrnt 011 11aess tmd equity in mathematics etlucntio11. Reston, VA: NCTM. NGA Center (National Governors Association Center for Best Practices) & CCSSO (Council of Chief State School Officers). (2010). C01711110ll core state >1tmdtmls. Retrieved from corestandards .org OECD . (2016) . Equations tmtl ineqtJIJiities: Making mathe111/Jtics 11mssible to all. Pari s: OECD Publ ishing. dx.doi .org/10.178/9789264258495-en . Palardy, G. J ., & Rumberger, R. W. (2008). Teacher effectiveness in the first grade: The importance of background qualifications, attitudes, and instructional practices for student learning. Educationlll Evaluation tmd Policy Analysis, 30, 111-140. Porter, A., McN1aken, J., Hwang, J ., & Yang, R. (201 1). Common core standards: The new U.S. intended curriculum. Educational Rmtmhn; 40(3), I 03-1 16. Rittle-Johnson, B., Schneider, M., & Star,J. R. (2015). Nota one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. EduciJtioniJI P>ychology Review, 27(4), 587-597. Rothwell, J . (2014). Still sell,.ching: Job vacancies tmtl STEM skills. Washington, DC: Brookings Institution. Steen, L. A (Ed.). (1997). Why numbers count: Qwmtit11tivelitm1cyjor trn11on·ow's Americ11. New York: The College Board. Stigler,]., & H iebert, J. (2009). C losing the teaching gap. Phi Delt11 1V1pptm, 91(3), 32-37. Tobias, S. ( 1995). Ove1·coming 111ath anxiety. New York: Norton. Toppo, G. (2015). Johnny can't always read, but slide in math scores puzzles. USA Todtq, October 28, 1. Walkowiak, T. A. (2015). Information is a common core dish best served first. Phi Delta 1V1ppan, 97(2), 62-67. Warshauer, H. (2015). Strategies to support productive struggle. M11thm1t1tics Teaching in the Middle School, 20(7), 39{}-393. Watts, T. W., Dun can, G.].. Siegler, R. S., & Davis-Kean, P. E. (20 14). W hat's past is pro logue: Relatio ns between early mathematics knowledge and high school achievement. Educ11tional Rmtmhe1; 43(7), 352-360. Wiggins, G. (2013,May 5). Letter to cl1e Editor. New Ym·k Times, 2.

References

Chapter 2 Anderson, L. \V., & Krnthwohl, D. R. (Eds.) (200 1). A tn.xon(J1ny

ftrr /taming, ttarbing, and asstmng: A rtVmo11 of Blo&m's Taxontrmy of tdurationnl objmivu: Complttt &lititnt. New York, NY: Addison \ \'esley, Longman. Ball, D. L. ( 1991). Mogic:~l hopes: .\bnipuloth·es and the refonn of moth edUc:ltion. Ammran £Jurator, 16(2), 1+-18, 46-47. Baroody, A. J ., Fcil, Y., & Johnson A. R. (2007). An altemati,·e reconcepruoliution of procedurol ond conceptual knowledge. Journal for Rrmmb in .llotbttnnlln £Jurario11, JS(2), 115-131. Boalcr, J. (1016) . •ltntbttnntunl mmdsm: U11lrasbnzg mulrnt pormtial tbro11gh trtiltit·~ math, msplring 11JtiSIIgi1Jg, and im1ovnrhY uaching. San Franci;co, CA: jossey Ba. \\ hitacre, l., Schoen, R. C., Chompagnc, Z., & Goddard, A. (20 16). Relational thinking: What's the difference? uarbing Childrrn ,\lothnnntus, 2J(S), 303-309. \\'ilburne, J. M., \\'ildmann, T., Morret, M., & Scipanm-ic, J. (20 H ). Classroom stntegies to make sense and perse\' ere. ·"fatbmrntia Ttnrhi11g i11 tbt .Hidtflt Srbool, 20(3), 14+-151.

Chapter 3 Aguirre, J. M., Mayfield-Ingram, K., &

Martin, D. (20 13). Tbr impart of idmtiry in K-8 Tlltllbrmatirs ltnmiug nud ttncbing: Rnhiuki11g equity-bam/ pmctim. Reston, VA: NcrM. AL\11$ (2005). Lookiug 01 Lint~: llltt'l"t'stiug objtrt.r lllltilinrarfimrtiolls. Fresno, CA: ALVIS Education Foundation. Barlow, A. T. (20 10). Bui lding word problems: W hat docs it toke? TrarbiugCbilr/r·w Mntlmnatia, / 7(3), 140- 148. Bay-Wil~ams, J. M., & Martinic, S. L. (2009). M111b tmd 11011-jirtion: C.·adrs 6-8. Sausalito, CA: Mari lyn Burns Books. Bay-Willioms,J. M., & Stokes Levine, A. (2017). The role of concepts and procedures in developing fluency. In D. Spangler &J. Wanko (Eds.), Enhnnri11grlassrom11 pra(lirtwitb rmnrrb brbintl PtA. Reston, VA:NCTM. Boalcr, J. (2008, April). Promoting "rclotionol equity" and high mathcm2tics ~chievcment through an innovotive mixed ability approach. Bntish EduratiDflnl RmarrbJoumnl, Jol, 167-194.

664

Refe rences

Boaler,j., (2016). Mathrmatic11i111indsets. San Francisco: Jossey-Bass. Boale r,)., & Scngupta-lrving, T. (2016). The many colors of algebra: The impact of equity focused teaching upon student learning and engagement. The Jomwttl ofMathnnatictJI Behavio1; 41, 179-190. Boalcr, J., & Selling, S. K. (2017). Psychological imprisonment or intellectual freedom? A longitudinal study of contrasting school mathematics approaches and their impact on adults' lives. Jotmwl for ResetJrch in Mllthmllftics &lucation, 48(1), 78-105. Bransford,)., Brown , A., & Cocking, R., Eds. (2000). Hrr\! ,\1ath Solutions Publictions. P6ly.J, G. (1945). Huw to solvt it. Princeton, NJ: Princeton University Press. Pugalec, D . K., Arbaugh, F., Bay-Wi llia ms, j . M., Farrell, A., Mathews, S., &. Royster, D. (2008). N1rvig11ting through matl!r"'"tical romlfctions in gnulrs 6-8. Reston, VA: NCTM. Rasmussen, C., Yackel, E., & IGng, K. (2003). Social and sociomathematical norms in the mathematics classroom. In I I. L. Schoen & R. I. C harles (Eds.), Teaching mathm111tia through problmt solving: Grtulrs 6-12 (pp. H3-154). Reston, \'A: NCTM. Rowling, J. K. (1998). HinT] Potttr tmd tbt sorrtrtr's sront. :-lew York, !'."Y: A. A. U..ine. Schneider, ,\I., Rittle-Johnson, B., & Sc:ar, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Dtvtlopmwtal Ps)'chology, 47, 1525-1538. Schroeder, T. L., & Lester, F. K.,Jr. ( 1989). Developing understanding in mathematics via problem solving. In P. R. Trafton (Ed.), Nrw dirwions for rln11rntmy scbooi1111Jtim11111ics (J1p. 3 1-42). Reston, VA:NCTM. Silver, E. A., & Stein, J\11. K. (1996). The QUASA R project: The "revolution of the possible" in mathematics instructional reform in urban middle schools. Urban Educat1o11, J0(4), 476-52 1. Smith, M. S., Bill, V:, & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Math"tructiotJ in the stnmltmis-basetl dtJs>l·o01II. (3rd Edition). Reston, VA: NCTM and New York, NY: Teachers College Press. Smith, M. S., Bill, V., & Hughes, E. K. (2008). T hinking through a lesson: Successfully implementing high-level tasks. MnthemtJtics Trttching in the Middle School, 14(3), 132-138. Smith,M. S., & Stein,M. K (2011). 5 Pnicticesf07·orchestrati11gproductive 11/ltthmiiitics dimm-ious. Thou~and Oaks, CA: Corwin Press. Star, J. R., & Verschaffel, L. (2016). Providing support for student sense making: Recommendations from cognitive science for the teaching of mathematics. In Jinfa Cai (Ed.), Compmditm1 for Resem-ch i11 Mtithmllltics &lucntiou. Reston, VA; NCTM. Stein, M. K., Remillard, J ., & Smith, M. S. (2007). How curriculum in fluences student learning. In F.J. Lester, J r. (Ed.), Seroud handbook of.-esel/rch 011 uutthemntirs tmchn1g and Imming (pp. 3 19-3 69). C harlotte, NC: In formation Age Publishing.

667 Tomlinson, C. A., & .\lcTighe, J (2006). lnttgratmg diffrrrntiattd lllstrttltlon. Alex:mdria, VA: ~otion for Supervision and Curriculum De,·elopment. \Vall3ce, A. II. (2007). Anticipating student responses to impro,-e problem solving. Mathn11atics T.arhing 111 tht Middlr School, 12(9), 50+-51 1. Walk, L., & Lassak, M. (2017). Making homework matter to students. M11tlm111ttics T.11ching in the Middle School, 22(9), 547-553. Whitenack,] . W., Cavey, L. 0., & I Jenney, C. (2015). /t's Elementary: A f>llrtnt'! Guidt to K-5 M11thmmtics. Reston, VA: NCTM. \Vieman, R., & Arbaugh, F. (20 14). Making homework more meaningful. Muthmrutia Ttucbing i11 tiN MitliUr Schoo~ 20(3), 160-165.

Chapter 5 Acc:lrdo, A. L., & Kuder, S.]. (20 17). .\ loniroring student learning in algebra. Mathl'mutiaT.arhing intbt MitkUt&bool, 22(6), 351- 359. Association of Mathematics Teacher &lucators, & Kational Council of Supervisors of Mathematics. (n.d.). lmprrr.:ing studmt achir.:e11/tllt in 1/tothnllatics tbrough formlllivt nsstStmellt iu imtruction. Joint position paper, amte.net/sitesldefJult/files/overvicw_arnte_ncsm_ position_paper_formative_assessmcnt.pdf. 13"rlow, A. T ., Gerstenschlager, N . E., & l larrnon, S. E. (2016). Learning from the un known student. 'TiHrrhing Chiklrm Mathelltntics, 21(2), 74-81. 13ny, W. S. (20 II). A collective ca.~c srudy of the influence of reachers' beliefs and knowledge on error-hondling practices during class discussion of mathematics. Journal for Rtmzrrh m .Uatbnnatics Eduttltion, 42(1), 2-38. Bra)',\V., & Sant:agata, R. (201-J). ,\lalcing mathcmatic:ol errors: Springbo:lrds for learning. ln K Karp & A. Roth-.\ lcDuffie (Eds.), Using rmarrb to imprut·• instnutit11l (pp. 139-HS). Reston, VA: KCD\1. Casa, T. M., Firmcnder, J. M., C•hill, J., Cardetti, F., Choppi n, J. M., Cohen, J., Zawodniak, R. (20 16). Typrt of mrd purposes Jf;r d~mouory math~mlltirnl uwiting: '"flukfortt ruommtndations.

Retrieved from mathwriting.cducat:ion.uconn.cdu. Danielson, C., & Luke, M. (2006). If I only had one question: Partner-quizzes in middle school matl>ematics. Mathrmatirs Teachillg ill thr Middlt School, /2(4), 206-2 13. Daro, P., ,\1osher, F., & Corcoran, T. (20 I I). ulll7ring trajmorirehmsihle fin· English Letmrm: The SlOP model (4th ed.). New York: Pearson. Flores, M., H inton, V., & Strozier, S. (2014). Teaching subtraction and multip liC11tion with regrouping using the concreterepresentational-abstract sequence and strategic instruction model. U11ming Disabilities Resem·ch tmd Pnrctice, 29(2), 7 5-88. Gagnon,] ., & Maccini, P. (200 1). Preparing student~ with disabilities fo r algebra. u11ching E.mptional Childnm, 34(1), 8-15. Garrison, L. (1997). Making the NCTM's Standards work fo r emergent English speakers. Ut/Chillg Chiltl•·m Mllthmmtics, 4(3), 132-138. Gavin, M . K., & Sheffield, L. J . (20 10). Using curriculum to develop mathematical promise in the middle grades. In M . Saul, S. Assouline, & L. J . Sheffield (Eds.), The J>mk ill the 111idtlle: Developillg mathematictii/y gifted studmts in the 111iddle gmdes (pp. 5 1-76). RestOn, VA: NCTM, National Association of Gifted C hildren, & National Middle School A~sociation . G6mez, C. L. (2010). Teaching with cognates. Utiching Childrm Mathemtitics, 16(8), 470-474. G riffin, C. C., J ossi, M. H., & van Garderen, D. (20 14). Effective mathematics instruction in in clusive schools. In J. McLeskey, N. L. Waldron, F. Spooner, & B. Algozzin e (Eds.), Handbook ofeffective inclusive schools: Resem·ch and Jmrctice (pp. 26 1-274). New York: Routledge. G utierrez, R. (2009). Embracing the inherent tensions in teaching mathematics from an equity stance. DniiM"liCJ and EductJtion 18(3), 9-16. G utierrez, R. (20 15). HOLA: H unt for opportunities-learn-act. Mathmwtics Ut/Che1·, 109(4), 270-277. Haas, E., & Gort, M. (2009). Demanding more: Legal stan dards and best practices for E nglish language learners. Bilingual Resmrch Joumttl, 32, 115-135. Heddens, J . (1964). Todtq~ mathnuatics: A guide to concepts tmd 111ethotls iu elemmtmy school 111athmmtics. Chicago, IL: Science Research 1

A~sociatcs.

!-lord, C., & Marita, S. (20 14). Students with learning disabilities tackle multistep problems. Mlllhntiiltics UtJching in the Middle School, 19(9), 548-555. Hunter, A. E., Bush, S. B. & Karp, K. (20 14). Systematic in terventions for teaching ratios. Mathem11tics Ullchillg iu the Middle School, 19(6), 360-367. J anzen, J . (2008). Teaching English language learners in the content areas. Rroirystem. Illuminations lesson at illuminations.nctrn.org/Lesson Detail .aspx'id=L245 Kersaint, G., Thompson, D. R., & Petkova, M . (2009). uachingmtithe111llrics to E11glish ilmguage letmren. New York, NY: Routledge. Khisty, L. L. (1997). Making mathematics accessible to Latino students: Rethinking instructional practice. In M. Kenney & J. T rentacosta (Eds.), Multicultural tmdgmdn· equity in them11thematics dns;To01u: The gift oftlivmity (pp. 92-101). Reston, VA: NCTM. Kisker, E. E., Lipka, J ., Adams, B. L., Rickard, A, Andrew-Ihrke, D., Yan ez, E. E., & M illard, A. (2012). The potential of a cul turally based supplemental mathematics curriculum to improve the mathematics performance of Alaska native and other students. JoumtTI for ResetiTCh nr MathmfiJtics Educatioll, 43(1 ), 75-1 13. Lewis, K. (2014). D ifference not deficit: Reconceptualizing mathematiC11l learning disabilities. Jounwl fo•· Resem·ch in MllthmfiJtics &lucatiou, 45(3), 351-396. Mack, N. K. (20 I I). Enriching number knowledge. UtJching Childm1 Mtithmurtics, 18(2), 101-109. Maldonado, L. A, T urner, E. E., Dominguez, H., & Empson, S. B. (2009). English language learning from, and contributing to, mathematiC11l discu.~ions. In D . Y. White &J. S. Spitzer (Eds.), Re>pondiug to diversity: Gnrdes f>Tl!-K-5 (pp. 7-22) Reston, VA: NCTM. Mand, D . B., Miller, S. P., & Kennedy, M. (2012). Using the concreterepresentational-abstract seque nce with integrated strategy instruction to teach subtraction with regrouping to students with learning disabilities. Lem7liug DistTbilities Rem,.ch tTnd Pmctice, 27(4),152-166. Mathis, S. (1975). The Hrmd1·ed Penny Box. New York Houghton M ifflin. Matthews, M. S., & Peters, S.]., (20 17). Methods to increase the identification rate of students from traditionally underrepresented populations for gifted services. InS. Pfeiffer, E. Shaunessy-Dctrick & M . N icpon (Eds.) Amn·iam Ps:ychologiml Axsociation htmdbook au giftedness tTnd talmt. Washington, DC: American PsychologiC11 l

A'isociation. Mazzocco, M. M. M., Devlin, K. T., & McKenney, S. J. (2008). l~ it a filet? Timed arithmetic performance of children with mathematiC11l learning disabilities (MLD) varies as a function of how MLD is defined. Develof>1tlfllflll Nmropsycbology, 33(3), 3 18-344. McMaster, K. L., & Fuchs, D . (2016). C lasswide intervention using peer-assisted learning strategies. In S. J imerson, M. Burns, & A. VanDerHeyden (Eds.), Htmdbook of •·espouse to iutervention (pp. 253-268). New York: Springer. M idobuche, E. (2001). Building cultural bridges between home and the mathematics classroom . UtJching Children Mllthnuatics, 7(9), 500-502. MiiJer, S. P., & Kaffar, B.J. (2011). Developing addition with regroupingcompctence among second graders with mathematics difficulties. broextigatiolls ill Mathe111atics Letmrillg, 4(1), 24-49. Moffett, G., Malzahn, K., & Driscoll, M. (20 14). Multimodal communication: Promoting and revealing students' mathematical thinking. In K. Karp & A Roth-Me D uffie (Eds.), Aunual perjpertives in mathematics etlucll'tion 2014: Using research to improve instructioll (pp. 129-139). Reston, VA; NCTM. Moschkovich,J. N. (1998). Supporting the participation of English language learners in mathematiC11l discussions. F01· the Lmming of Mtithmurtics, 19(1), 11- 19.

670

Refe rences

Murrey, D. L. (2008). D ifferentiating instruction in mathematics for the English language learner. Mathnnatirs Teaching in the Middle School, 14(3), 146-15 3. Natio nal Center for Educatio n Statistics. (2011). Retrieved from n ces. ed. govIna tions repo rtca rd/ i tm rl sx/sea rch. a spx > subject=mathematies National Research Council, Mathematics Learning Study Committee. Kilpatrick,)., Swafford,]., & Findell, B. (Eds.). (2001). Adding it up: Helping childrm lenm 111athematics. Washington, DC: T he National Academics Press. NCES (2017). NAEP data exp/(Jrer. Retrieved from nces.ed.gov/ nationsreportcard/naepdat:A/ NCTM (National Counc il of Teachers of Mathematics). (2007). Rmtmb brief' Effective strategies for tetJcbing >TIIdmts with dijjimlties inmllthnuntia. Retrieved from www.nctm.org/ncws/contcnt. aspx'id=8452 NCTM (National Council ofTeachers of Mathematics). (201 1,July). Position stiJtrmrnt on interuentians. Reston, VA: NCTJ\1. NCTM (National Council ofTeachers of Nlathematics). (2012). Position statnue11t on closing the oppo11'11nity g11p in 11/llthemntics edumtirm. Reston, VA: NCTM. NCTM (National Council ofTeachers of l\1athematics). (2014). Position stiJtrmrnt on tJcctsY tmd equity in mathnuaticr education. Reston, VA:NCTM. NCTM (National Council ofTeachers ofl\1athematics). (2016). Position J1lltrment on providing oppfrrttmities for students with exaptiomt! fromise. Reston, VA: NCTM. OECD (20 16). Equations tmd inequalities: Making mathematics accessible to 11/1, PISA. Paris, France: OECD Publ ishing. dx.doi.org/10. 1787/9789264258495-en Plucker, J. A., & Peters, S.). (2016). Excellence gaps in edttctJtion: Erptmding oppo1T1mitiesfor flllmted students. Cambridge, MA: Harvard Education Press. Pugalee, D. K., Harbaugh, A., & Quach, L. H. (2009). The human graph project: G iving students mathematical power through differentiated instruction. In M. W. Ellis (Ed.), Re>ponding to tlive.-sity: Gnules 6-8 (pp. 47-58). Reston, VA: NCTM. Read,). (2014). Who rises to the top' Peabody Riflectrn; 81(1), 19-22. Reis, S., & Renzulli,). S. (2005). Cun·imlum compacting: An easy >ttm to diffinmtiatingfor high potmtial mulents. Waco, TX: Prufrock Press. Remillard, J. T, Ebb)', C., Lim, V., Reinke, L., Hoe, N., & Magee, E. (20 14). Increasing access to mathematics through locally relevant curriculum. In K. Karp & A. Roth-McDuffic (Eds.), Anntllll perspectives in mathemntic.s education 2014: Using ,.rsfnrch to hnpruv;o imtruction (pp. 89-96). Reston, VA: NCTM. Renzulli, J. S., G ubbins, E. J., McMillen, K. S., Eckert, R. D., & Little, C. A. (Ed~.). (2009). Sptems &modelsfrn· tleveloping frog>·llms for the gifted & flllrnted (2 nd ed.). Mansfield Center, CT: C reative Learning Press. Ronau, R. , & Karp, K. (20 12). T he power over trash : Integrating mathematics, science and children's literature. Adapted from an article in M11thm11Jtics TrtJCbing in the Middle School, in H. Sherard (Ed.), Real Wo..td Mlfth. Reston, VA: NCTM. Rotigcl, J., & Fello, S. (2005). Mathematically gifted students: How can we meet their need~' Gifted Child Today, 27(4), 46-65. Sadler, P., & Tai, R. (2007). The two pillars supporting college science. Science, J/7(5837), 457-458. Saul, M., A~souline, S., & Sheffield, L.J. (Eds.). (2010). The pe11k in the mit/die: Developing11tathnnatimlly gifted studmts in the middle g>·atks. Reston, VA: NCTM, National Association of G ifted Children, & National Middle School Association .

Seidel, C., & McNamee,). (2005). Teacher to teacher: Phenomenally exciting joint mathematics-English vocabulary project. Mathe1/Jatics UlfcMng in the Middle School, 10(9), 461-463. Sheffield, L. J. (Ed.). (1999). Developingmttthmllltically pnrmi,-ing stttdmts. Reston, VA: NCTM. Simic-Muller, K. (2015). Social justice and proportional reasoning. Mttthmllltics TrtJChing in the Middle School, 2I(3), 162-168. Storcygard,J. (201 0). My kitls am: Making11tath accessible to all le~~mn>, K-5. PortSmouth, NH: Heinemann. Tomaz, V. S., & David, M. M. (2015). How students' everyday situations modify classroom mathematical activity: The case of water consumption.Jotmwl fo•· Research in Mlfthe111atics Education, 46(4), 455-496. Thompson, K. D. (20 17). W hat blocks the gate' Exploring current and former English learners' math course-taking in secondary school. Amnican Educational Rese~~nh ]oun111l, 54(4), 757-798. Torgcsen,J. K. (2002). The prevention of reading difficulties.Joumal ofScbool Ps;·chology, 40(1), 7-26. Turner, E. E., Ccled6n-Pattichis, S., Marshall, M., & Tennison, A. (2009). Fijense amorcitos, les voy a contra una historia: The power of story to support solving an d discussing mathematical problems among Latino an d Latina kindergarten students. In D. Y. White & ). $. Spitzer (Eds.), Responding to dive·rsity: Gr11des P•·e -K-5 (pp. 23-42). Reston, VA: NCTM. Turner, E., Sugimoto, A. T., Stoehr, K., & KurL, E. (2016). C reating Inequalities from Real-vVorld Experiences. Mttthmllltics Teaching in the Mit/die School, 22(4), 236-240. Wager, A. (2012). In corporating out-of-school mathematics: From cultural context to embedded practice. Jounwl of Mathematics UlfchN· Edumtion, 15(1), 9-2 3. Whiteford, T. (200912010). Is mathematics a universal language> Ulfching Cbildm1 MatbemtJtics, I 6(5), 276-283. Wigfield, A., & Cambria,). (2010). Expectancy-Value theory: Retrospective and prospective. In S. Karabenick & T. C. Urdan (Eds.), The tlrcntle tJhentl: Throrfticn/ prrspectives on motivation am/ nchiror1/lent (advances in 1/lOtivation and lfchieve111mt) (Vol. 16, pp. 35-70). Bradford, UK: Emerald G roup Publishing Limited. Woodward, ). (2006). Developing automaticity in multiplication facts: Integrating strategy instruction with timed practice drills. Lmmrng Dist1bility Qtumerly, 29(4), 269-289. Ysseldyke, J . (2002). Response to "Learning disabilities: Historical perspectives." In R. Bradley, L. Danielson, & D . Hallahan (Eds.), 1dmtification ofImming disabilities: ResetJrch to frttctice (pp. 89-98). Mahwah, NJ: Erlbaum. Zike, D. (2003). Dinah Zike's tellfhing mathe111atics with foldabh G len coe: McGraw-Hill.

Chapter 7 Baer, E. (1990). This is tbeu·ay we go to school. New York: Scholastic. Barood)', A. J . ( 1987). Children's11tatbnnatical thinking: A developmmtlll jhm1fW01·k for p•·eschool, primmy, tmd special education tet1chers. New York: Teachers College Press. Baroody, A J., Li, X., & Lai, M. L. (2008).Toddlers'spontaneous attention to number. Mlfthmiiitics Thinking tmd Leaming, I 0, 1-31. Betts, P. (20 15). Counting on using a number game. Trachrng Cbiltlrm Mllthmllltics, 21(7), 430-436. Brown, M. 'vV. (1947). Goodnight 11toon. New York, NY: Harper & Row.

C lements, D., Baroody, A., & Sarama,J. (2013). Backg>vund•·esem·cb on early 1/IIllhmllltics. Background Research for the National

References Governor~ Association (NGA) Center Project on Early Mathematics. Washington, DC: NGA. C lements, D., & Sarama,). (2014). Lmming 1mtl teaching mdy math: The Imming trajecturies appr011ch (2 nd ed.). New York, I'.'Y: Routledge. Dcan, J ., & Dean, K. (2016). Pete the cat 1mtl the missing mpc11kes. New York: HarperCollins. Ethridge, E. A., & King,J. R. (2005). Calendar math in preschool and primary classrooms: Questioning the curriculum. Em·ly Cbiltlbootl Education ]rmm11l, 32(5), 291-296. Fosnot, C. T., & Dolk, M . (200 1). Y11Ite sttmtltmls. Retrieved from corestandards.org Siegler, R. S. (2010). Playing numerical board games improves number sense in children from low-income backgrounds. In BJEP Monograph Series II, Nttmbn· 7-Untlen1tmtling tltlmbe-r tlroelopmmt tmtl tliffimlties (Vol. 15, No. 29, pp. 15-29). British Psychological Society. Slobodkin a, E. (1938). Caps fo.-stiie. New York, NY: Scholastic Books. Vout~ina, C. (20 16). Oral counting sequences: A theoretical discussion and analysis through the lens of representational redescription. Etltuationtii SttJtlies in Mtithmrtitics, 93, 175-193 . Watts, T. W., D uncan, G. ] ., Siegle r, R. S., & Davis-Kean, P. E. (20 16). W h at's past is prologue: Relations between early mathematics knowledge and high school achievement. Educiitional Remmber, 43(7), 352-360. Wright, R. ]., Stanger, G., Stafford, A. K, & Martland,]. (2006). Teaching n11mbn· in the classroom with 4-8 year olds. London, UK: Paul Chapman Publications/Sage.

Chapter 8 Baroody, A.]., Wilkins,]. L., & T iilikai nen, S. H. (2003). The development of children's understanding of additive commutativicy: From protoquantitive concept to general concept? In

671

A.]. Baroody & A. Dowker (Ed~.), The development ofaritbmttic concepts and skills (pp. 127-160). Mahwah , N]: Erlbaum. Blote, A., Licffering, L., & Ouwchand, K. (2006). The development of many-to-one counting in 4-ycar-old children. Cognitive Droelopmmt, 21(3), 332-348. Byrd, C., MCJ'-Jeil, N., C hesney, D., & J\1atthews, P. (2015). A specific misconception of the equal sign acts as a barrier to the learning of early algebra. Lettmi11g tmd lmlividtllll Dijfm?Jces, 38(2), 6 1--67. Caldwell, J., Kobett, B., & Karp, K. (2014). Essmtialundmttmdingof t1dtlitiou ami mbtmction in practice, K'·tJtfes K-2. Resron, VA: NCTM. C hampagne, Z. M., Schoen, R., & R;ddell, C. (2014). Variations in both-addends unknown problems. uacbing Chi/d,·m Mtithmllltics, 21(2), 114-12 1. C lark, F. B., & Kamii, C. ( 1996). Identification of multiplicative thinking in children in grades 1-5. ]oun111l for Resem·cb in Matbe71/lltics EduciJtiou, 27(1), 4 1-51. C lement, L., & Bernhard,). (2005). A problem-solving alternative to using key words. Mtithmllltics Utlcbing iu the Middle School, 10(7), 360-365. C respo, S., & N icol, C. (2006). C hallenging preservicc teachers' mathematical understanding: The case of division by zero. School Science and Mtithmllltics, 106(2), 84-97. C uyler, M. (20 I 0). Guinw pigs 11dtlup. New York, NY: Walker. De Corte, E., & Verschaffcl, L. (1985). Beginning first graders' initial representation of arithmetic word problcm~.Jotmw/ o[Mtithm111ticiii Bebavi01·, 4, 3-21. D ing, M. (2010, October 7). Email interaction. D ing, M. (20 16). Opportunities tO learn inverse relations in U.S. and C hinese textbooks. Mllthmmticiii Thinking and Lwming. 18(1), 45-68. D ing, M., & Auxter, A. (20 17). Children's strategies to solving additive inverse problems: A preliminary analysis. Mtithrmlltics Etfucntioual Rese~~,.ch Joumal, 29, 73-92. D ing, M., & Li, X. (2010). A comparative analysis of the distributive property in U.S. and Chinese elementary mathematics textbook~. Cognition 1md lmtmction, 28(2), 146--180. D ing, M., & Li, X. (20 14). Transition from concrete to abstract representations: The distributive property in a Chinese tcxthook series. Educational Studies in Mtithmllltics, 87, 103-121. D ing, M ., Li, X., Caprano, M. M., & Caprano, R. (20 12). S upporting meaningful initial learning of the associative pro perty: C ross-cultural differences in texthook presentations. lnte-mlltionlll Jotmwl for Studies in Mtithmllltics &lucation, 5(1), 114-130. D ixon, ). K., Nolan, E. C., Lott Adams, T., Tobias,]., & Barmoha, G. (20 16). Makiug smse o[111athm111tics[01· Wichiugf5'·11des 3-5. Bloomington, IN: Solution Tree Press. Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essentitJI tmdrnttmding ofnumbn· 11nd 71l11JU'I"1Jtion fur teaching 711nthe711lltics in prekindergmtm-grntfe 2. Reston, VA: NCTM. D rake,)., & Barlow, A. (2007). Assessing students' level of understanding multiplicatio n thro ugh problem writing. Ullching Childrm Mt~tbematics, 14(5), 272-277. D rescher, D . (2008). Wh11t~ biding in there? Edinburgh, Scotland: Floris Books. Fagnant, A., & Vlassis,). (20 13). Schematic representations in arithmetical problem solving: Analysis of their impact on grade 4 students. Etlucntiou Studies in Matbemtitics, 84, 149-168. Feldman, Z. (2014). Rethinking factors. Mtithmllltics Te11ching iu the Middle School, 20(4), 230-236. Fosnot, C. T., & Dolk, M. (2001). Yotmgmt~tbmulticitmsatwork: Constmcting111ultiplic1Ition anti divisiou. Port~mouth, NH: Heinemann .

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References

Gerofsky, S. (2009). Genre, simulacra, impossible exchange, •nd the real: I low posnnodem theory problematises "ord problems. In L. \ 'ersch•ffel, B. Greer,& w: \: Dooren (Eds.), Wordsnt~drrorlds: Mtxltlmg vn-bnl dmriptions of situations (p)>. 21 -3 8). Rotterdam: Sense Publishing. Goldstone, R. L., & Son,]. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. Tbr Jounwl ofthr Lenming Scimces, 14, 69-110. lleng, M.A., & Sudarshan, A. (2013). "Bigger number means you plus!" Teachers learning to use clinical interviews to understood students' mothemotirnl thinking. Educutiou Studies in M111hnnntirs, 83, 471-485. Ilolbert, S. & Borlow, A. (20 12n0 13). Eng:aging relucront problem soh-ers. Tturhmg Childrro Mathmtatia, 19(5), liG-315. llord, C. & .\brito, S. (2014). Students with learning dis:>bilities t~clde multistep problems. .\Lzthmtatta Tturhmg 111 thr Middlr School, /9(9), HS-555. I hunker, D. ( I 994, April). .\1uhi-strptrvmiprobk111s: A stratrgyforrtnptnrrring nudmrs. Presented at the Annual ,\1eeting of the Notional Cow>cil ofTcochers of i\Iathcmatics, Indianapolis, IN. Jong, C. & Ma&'I'Udcr, R. (20 14). Beyond cookies: Underst:Jndingvariousdivision models. Tet~chingChild•-rn M11thnnntia, 20(6), 367-373. Karp, K., Bush, S., & D ougherty, B. (20 14). 13 n iles that expire. "lftuhing Chiltlrm Mnthmwtirs, 21 (2), 18-2 5. Kinda, S. (2013). Generating scenarios of division as sharing •nd grouping: 1\ study ofjapanese elementory and university students. Matht'111Uttral Thinking and Lrarning, 15, 190-200. Knuth, E. & Stephens, A. (2016). The equal sign does motter. In E. Silver & P. Kenney (Eds.), .\Torr lrssons 1rumrd}w11 rarnrrh toolmnr 2 (pp. 135- 139). Reston, \'A: :'\"CDl ,\latney, G. T., & Daugherty, B.~- (2013). Seeing spots and de,..,loping multiplicative sense making. .\1atbrmatia Ttntbmg i11 thr .Uiddlr School, / 9(3), 148-155. McElligott, M. (2007). Bran tbirtrm. New York, NY: Putnam J u•·enile. Miller-Bennett, V. (20 17). Understanding the meaning of the cquol sign. Nrw 1!11gllllul Jourrwl ofMntbmwtics, 50(1), 18-25. Murata,/\. (2008). Mathematics teaching and lc:>rni ngas a mediating process: The c•se of rope diagrams. Mnthmullirnl Tbinking tmd Lt~tming, 10, 374-406. National RcS