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Elementary and Middle School Mathematics Teaching Developmentally FOURTH CANADIAN EDITION John A. Van de Walle Late of Virginia Commonwealth University Karen S. Karp University of Louisville Jennifer M. Bay-Williams University of Louisville Lynn M. McGarvey University of Alberta Sandra Folk University of Toronto With Contributions by Jonathan Wray Howard County Public Schools Toronto. Editor ...

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10 9 8 7 6 5 4 3 2 1 CKV

Library and Archives Canada Cataloguing in Publication

Van de Walle John A author

Elementary and middle school mathematics teaching developmentally John A Van de Walle Late of Virginia Commonwealth

University Karen S Karp University of Louisville Jennifer M Bay Williams University of Louisville Lynn M McGarvey

University of Alberta Sandra Folk University of Toronto Fourth Canadian edition

Revision of Elementary and middle school mathematics teaching

developmentally John A Van de Walle et al 3rd Canadian ed Toronto Pearson Allyn Bacon c2011

Includes bibliographical references and index ISBN 978 0 205 87195 7 pbk

1 Mathematics Study and teaching Elementary 2 Mathematics Study and teaching Middle school I Karp Karen S

author II Bay Williams Jennifer M author III McGarvey Lynn M author IV Folk Sandra author V Title

QA135 6 V36 2014 372 7 044 C2013 906895 3

ISBN 978 0 20 587195 7

About the Authors

John A Van de Walle was a professor emeritus att

Virginia Commonwealth University He was a

mathematics education consultant who regularly

gave professional development workshops for K 8 8

ited and taught in elemen

teachers in the United States and Canada He visited

tary school classrooms and worked with teachers to implement student

centered math lessons He co authored the Scott Foresman Addison Wesley

Mathematics K 6 series and contributed to the Pearson School mathematics

program enVisionMATH Additionally he wrote numerous chapters and

articles for the National Council of Teachers of Mathematics NCTM

books and journals and was very active in NCTM including serving on the

board of directors chairing the educational materials committee and speaking at national and

regional meetings

Karen S Karp is a professor of mathematics education at the University of

Louisville Kentucky Prior to entering the field of teacher education she was

an elementary school teacher in New York Karen is a co author of Feisty

Females Inspiring Girls to Think Mathematically which is aligned with her

research interests on teaching mathematics to diverse populations With

Jennifer Karen co edited Growing Professionally Readings from NCTM Publi

cations for Grades K 8 and co authored along with Janet Caldwell Developing

Essential Understanding of Addition and Subtraction for Teaching Mathematics

in Pre K Grade 2 She is a former member of the board of directors of the

National Council of Teachers of Mathematics NCTM and a former presi

dent of the Association of Mathematics Teacher Educators AMTE She con

tinues to work in classrooms with elementary and middle school teachers and with teachers at

all levels who work with students with disabilities

Jennifer M Bay Williams is a professor of mathematics education at the

University of Louisville Kentucky Jennifer has published many articles on

teaching and learning in NCTM journals She has also co authored numerous

books including Developing Essential Understanding of Addition and Subtraction

for Teaching Mathematics in Pre K Grade 2 Math and Literature Grades 6 8 Math

and Nonfiction Grades 6 8 and Navigating Through Connections in Grades 6 8

Jennifer taught elementary middle and high school in Missouri and in Peru

and continues to work in classrooms at all levels with students and with teach

ers Jennifer is on the board of directors for TODOS Equity for All is the

editor for the 2012 NCTM Yearbook and is a former president of the Associa

tion of Mathematics Teacher Educators AMTE

About the Contributor

Jonathan Wray is the technology contributor to Elementary and

Middle School Mathematics Teaching Developmentally He is the instruc

tional facilitator for secondary mathematics curricular programs in

the Howard County public school system He is the president elect

of the Association of Maryland Mathematics Teacher Educators

AMMTE and past president of the Maryland Council of Teachers of

Mathematics MCTM He has been recognized for his expertise in

infusing technology in mathematics teaching receiving the Out

standing Technology Leader in Education award for his school dis

trict from the Maryland Society for Educational Technology MSET

Jon is also actively engaged in NCTM serving on the editorial panels

of Teaching Children Mathematics and ON Math He has served as a primary and interme

diate grades classroom teacher gifted talented resource teacher elementary mathematics

specialist curriculum and assessment developer grant project manager and educational

consultant

About the Canadian Author

Lynn M McGarvey is an associate professor of mathematics educa

tion at the University of Alberta Lynn s research focuses on the

mathematical reasoning of young children She has written many ar

ticles for research and professional audiences particularly on the top

ics of algebraic thinking spatial reasoning and patterning in the early

years She is a long time member of NCTM and has served on a

number of task forces and committees including as editorial panel

chair for Teaching Children Mathematics She is a former junior high

school teacher who now spends a considerable amount of time work

ing with children and teachers in preschools and kindergartens Lynn

has taught mathematics curriculum and pedagogy courses to thou

sands of pre service and in service elementary teachers and has won

multiple teaching awards for this work

Brief Contents

Teaching Mathematics Foundations and Perspectives

Chapter 1 Teaching and Learning Mathematics Chapter 5 Blending Teaching and Assessment 64

in the Twenty First Century 1

Chapter 6 Teaching Mathematics

Chapter 2 Exploring What It Means for All Learners 84

to Know and Do Mathematics 11

Chapter 7 Tools for Learning Mathematics 103

Chapter 3 Mathematical Inquiry through

Rich Tasks and Classroom Discourse 28

Chapter 4 Preparing to Teach and Planning

for Mathematics Learning 45

SECTION II

Development of Mathematical Concepts and Procedures

Chapter 8 Developing Early Number Chapter 16 Developing Strategies

Concepts and Number Sense 119 for Fraction Computation 310

Chapter 9 Developing Meanings Chapter 17 Developing Concepts

for the Operations 139 of Decimals and Percents 333

Chapter 10 Helping Students Master Chapter 18 Proportional Reasoning 352

the Basic Facts 162

Chapter 19 Developing Measurement

Chapter 11 Developing Whole Number Concepts 369

Place Value Concepts 183

Chapter 20 Geometric Thinking and

Chapter 12 Developing Strategies Geometric Concepts 396

for Addition and Subtraction Computation 208

Chapter 21 Developing Concepts

Chapter 13 Developing Strategies of Data Analysis 428

for Multiplication and Division Computation 231

Chapter 22 Exploring Concepts

Chapter 14 Algebraic Thinking Generalizations of Probability 448

Patterns and Functions 253

Chapter 23 Developing Concepts

Chapter 15 Developing Fraction Concepts 284 of Exponents Integers and Real Numbers 466

APPENDIX Guide to Blackline Masters 485

REFERENCES 497

Preface xvii

Teaching Mathematics Foundations and Perspectives

The fundamental core of effective teaching of mathematics combines an understanding of how children learn how

to promote that learning by teaching through problem solving and how to plan for and assess that learning on a

daily basis Introductory chapters in this section provide perspectives on trends in mathematics education and the

process of doing mathematics These chapters develop the core ideas of learning teaching planning and assessment

Additional perspectives on mathematics for children with diverse backgrounds and the role of learning tools

e g manipulatives technology are also discussed

An Invitation to Do Mathematics 12

CHAPTER 1 Searching for Patterns 12

Teaching and Learning Mathematics Analyzing a Situation 14

in the Twenty First Century 1 Generalizing Relationships 14

Experimenting and Explaining 16

What Are Your Memories of School Mathematics 1 Engaging in Mathematical Inquiry 16

Mathematics Proficiency 17

Twenty First Century Knowledge and Skills 2

What Does It Mean to Learn

Diversity in Today s Classrooms 2

Mathematics 20

Mathematics Curriculum in Canada 3

Mechanical Metaphors for Learning 20

Goals for Student Learning 4

Learning Mathematics as Acquisition 21

Mathematics Content 4

Learning Mathematics along a Linear Path 21

Mathematical Processes 5

Ecological Metaphors for Learning 22

An Invitation to Learn and Grow 7

Learning Mathematics as Coherence 23

Becoming a Teacher of Mathematics 7

Learning Mathematics as Outward Growth 24

RESOURCES FOR CHAPTER 1 An Example of Mathematics Learning 24

Recommended Readings 9 Learning Theories and Implications

Online Resources 9 for Teaching 25

REFLECTIONS ON CHAPTER 1 Promoting Dispositions for

Doing Mathematics 26

Writing to Learn 10

For Discussion and Exploration 10 RESOURCES FOR CHAPTER 2

Recommended Readings 26

Online Resources 27

REFLECTIONS ON CHAPTER 2

Exploring What It Means to Know

Writing to Learn 27

and Do Mathematics 11 For Discussion and Exploration 27

What Does It Mean to Do Mathematics 11

A Classroom Environment for Doing Mathematics 12

viii Contents

Follow Up to Inquiry The After Phase of a Lesson 50

CHAPTER 3 Teacher Actions in the After Phase 50

Mathematical Inquiry through Rich Tasks Process for Preparing a Lesson 51

and Classroom Discourse 28 1 Curriculum Learning Expectations 52

2 Student Experiences 52

Why Mathematical Inquiry 28 3 Select Design or Adapt a Rich Task 53

4 Design the Lesson Assessments Tools 54

A Shift in the Role of Tasks

5 Plan the Before Phase of the Lesson 54

in the Classroom 29

6 Plan the During Phase of the Lesson 55

The Focus of Inquiry 30

7 Plan the After Phase of the Lesson 55

What Is a Rich Task 30 Final Revisions to Your Lesson Plan 56

Features of a Rich Task 31 Applying the Planning Process 56

Variable Entry and Exit Points 31 Applying the Three Phase Lesson Template

Thinking Mathematically 32 to Other Lesson Structures 56

High Levels of Cognitive Demand 34 Applying the Three Phase Model to Short Tasks 56

Relevant Contexts 34 Applying the Three Phase Model in Learning Centres 57

Evaluating and Adapting Tasks What Do You Need Homework and Parental Involvement 57

to Do to Make Them Rich er 36 Effective Homework 58

Task Selection Guide 36 Beyond Homework Families Doing Math 58

Developing Concepts and Procedures Resources for Families 59

through Tasks 36 Frequently Asked Questions 59

Concepts 36

RESOURCES FOR CHAPTER 4

Procedures 38

Recommended Readings 60

What about Drill or Practice 38 Online Resources 61

New Definitions of Drill and Practice 38

REFLECTIONS ON CHAPTER 4

What Practice Provides 39

What Drill Provides 39 Writing to Learn 61

When Is Drill Appropriate 39 For Discussion and Exploration 61

Drill and Student Errors 39 EXPANDED LESSON Fixed Area 62

Creating a Culture of Mathematical Inquiry 40

Classroom Discourse 40

Questioning Considerations 42

How Much to Tell and Not to Tell 42

Preparing to Teach 43 Blending Teaching and Assessment 64

RESOURCES FOR CHAPTER 3

Purposes for Assessing Learning

Recommended Readings 43 in Classrooms 64

Online Resources 44

What Should Be Assessed 66

REFLECTIONS ON CHAPTER 3

Mathematics Content 66

Writing to Learn 44 Mathematical Processes 67

For Discussion and Exploration 44

Mathematical Disposition 67

What Tools Can Be Used to Gather Evidence

of Mathematics Learning 67

CHAPTER 4 Performance Based Tasks 68

Questions and Conversations 69

Preparing to Teach and Planning

Interviews 69

for Mathematics Learning 45

Writing to Learn and Assess 70

Journals 72

A Three Phase Lesson Format 45

Admit and Exit Slips 73

Introduction to Inquiry The Before Phase of a Lesson 45

Teacher Actions in the Before Phase 46

Inventories 74

Development of Inquiry The During Phase of a Lesson 48 Summative Assessment Tools 74

Teacher Actions in the During Phase 48 Cumulative Projects 74

Contents ix

Teacher Made Tests 75 Students with Moderate Severe Disabilities 97

Provincial Assessment Programs 76 Students Who Are Mathematically Gifted and Talented 99

National and International Studies 76

Final Thoughts 101

The Pan Canadian Assessment Program PCAP 77 RESOURCES FOR CHAPTER 6

Trends in International Mathematics Recommended Readings 101

and Science Study TIMSS 77 Online Resources 101

Ways to Document and Communicate REFLECTIONS ON CHAPTER 6

Evidence of Learning 77

Writing to Learn 102

Documenting Formative Assessment 78 For Discussion and Exploration 102

Rubrics 79

What Gets Graded Gets Valued 81

From Assessment Tools to Grades 82

Using Assessments to Shape Instruction 82 Tools for Learning Mathematics 103

RESOURCES FOR CHAPTER 5

Tools for Learning 103

Recommended Readings 82

Online Resources 83 Types of Learning Tools 105

Concrete Materials and Physical Models 106

REFLECTIONS ON CHAPTER 5

Visual and Graphic Representations 107

Writing to Learn 83

Technology Based Tools 108

For Discussion and Exploration 83

Calculators in Mathematics Instruction 110

When to Use a Calculator 110

Benefits of Calculator Use 111

Graphing Calculators 112

Portable Data Collection Devices 113

Teaching Mathematics for All Learners 84 Digital Tools for Mathematics Learning 113

Instructional Applications 114

Differentiated Learning and Teaching 84

Concept Instruction 114

Tiered Tasks 84

Problem Solving 114

Learning Centres 86

Drill and Reinforcement 115

Choice Boards Menus and Think Tac Toe 86

Guidelines for Selecting and Using Digital Resources 115

Think Pair Share 86

Guidelines for Using Digital Content 115

Graphic Organizers 86

How to Select Appropriate Digital Content 115

Diversity in Our Classrooms 87

Final Thoughts 116

First Nations M tis and Inuit Students 87

Students Who Are Culturally and Linguistically Diverse 88 RESOURCES FOR CHAPTER 7

Culturally Diverse Students 88 Recommended Readings 116

Students Who Are English Language Learners ELLs 90 Online Resources 117

Providing for Students with Special Needs 93 REFLECTIONS ON CHAPTER 7

Prevention Models and Interventions for All Students 93 Writing to Learn 118

Students with Mild Disabilities 95 For Discussion and Exploration 118

x Contents

SECTION II

Development of Mathematical Concepts and Procedures

This section serves as the application of the core ideas of Section I Here you will find chapters on every major con

tent area in the pre K 8 mathematics curriculum Numerous inquiry based tasks and problems to engage students

are interwoven with a discussion of the mathematical content and how children develop their understanding of that

content At the outset of each chapter you will find a listing of Big Ideas the mathematical umbrella for the chapter

Also included are ideas for incorporating children s literature technology and assessment These chapters are de

signed to help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future

CHAPTER 8 CHAPTER 9

Developing Early Number Concepts Developing Meanings for the Operations 139

and Number Sense 119

Addition and Subtraction Problem Structures 140

Promoting Good Beginnings 119 Join and Separate Problems 140

Number Concepts Quantity Counting Part Part Whole Problems 141

and Knowing How Many 120 Compare Problems 141

Quantity and the Ability to Subitize 120 Problem Difficulty 142

Early Counting 121 Computational and Semantic Forms of Equations 142

Numeral Writing and Recognition 123 Teaching Addition and Subtraction 142

Counting On and Counting Back 124 Contextual Problems 142

Comparing Sets More Than Less Than MATH MAKES SENSE 144

and Equal To 125

Model Based Problems 145

Early Number Sense 126

Properties of Addition and Subtraction 147

Relationships for Numbers 1 through 10 127 Multiplication and Division Problem Structures 148

One and Two More One and Two Less 127 Equal Groups Problems 148

Anchoring Numbers to 5 and 10 128 Comparison Problems 149

Part Part Whole Relationships 130 Area and Array Problems 149

Dot Cards as a Model for Teaching Number Combination Problems 151

Relationships 133

Teaching Multiplication and Division 151

Relationships for Numbers 10 through 20 134

Contextual Problems 151

Early Place Value Concepts 134

Remainders 152

Extending More Than and Less Than

Model Based Problems 153

Relationships 134

Properties of Multiplication and Division 155

Number Sense in Their World 135

Strategies for Solving Contextual Problems 156

Estimation and Measurement 135

Analyzing Context Problems 156

Data Collection and Analysis 136

Two Step Problems 158

Classroom Routines 136

RESOURCES FOR CHAPTER 9

RESOURCES FOR CHAPTER 8

Literature Connections 159

Literature Connections 137

Recommended Readings 159

Recommended Readings 137

Online Resources 160

Online Resources 138

REFLECTIONS ON CHAPTER 9

REFLECTIONS ON CHAPTER 8

Writing to Learn 160

Writing to Learn 138

For Discussion and Exploration 160

For Discussion and Exploration 138

Contents xi

Basic Ideas of Place Value 184

CHAPTER 10 Integration of Base Ten Groupings with Counting by Ones 185

Helping Students Master the Basic Facts 162 Role of Counting 185

Integration of Groupings with Words 186

Developmental Nature of Basic Fact Mastery 162 Integration of Groupings with Place Value Notation 186

Approaches to Fact Mastery 163 Base Ten Models for Place Value 187

Guiding Strategy Development 164 Groupable Models 187

Reasoning Strategies for Addition Facts 165 Pregrouped or Trading Models 187

One More Than and Two More Than 166 Nonproportional Models 188

Adding Zero 167 Developing Base Ten Concepts 188

Using 5 as an Anchor 167 Grouping Activities 188

Make 10 167 The Strangeness of Ones Tens and Hundreds 189

Up Over 10 167 Grouping Tens to Make 100 190

Doubles 168 Equivalent Representations 191

Near Doubles 169 Oral and Written Names for Numbers 192

Reinforcing Reasoning Strategies 169 Two Digit Number Names 192

Reasoning Strategies for Subtraction Facts 170 Three Digit Number Names 193

Subtraction as Think Addition 170 Written Symbols 194

Down Over 10 171 Assessing Place Value Concepts 195

Take from the 10 171 Patterns and Relationships with Multidigit

Reasoning Strategies for Multiplication Numbers 195

and Division Facts 172 The Hundreds Chart 195

Doubles 172 Relationships with Benchmark Numbers 197

Fives 172 Connecting Place Value to Addition and Subtraction 198

Zeros and Ones 172 Connections to Real World Ideas 202

Nifty Nines 173

Numbers beyond 1000 203

Using Known Facts to Derive Other Facts 174

Extending the Place Value System 203

Division Facts 175

Conceptualizing Large Numbers 204

Mastering the Basic Facts 175

RESOURCES FOR CHAPTER 11

Effective Drill 175

Games to Support Basic Fact Mastery 176 Literature Connections 205

Recommended Readings 206

Fact Remediation 177 Online Resources 206

What to Do When Teaching Basic Facts 179

REFLECTIONS ON CHAPTER 11

What Not to Do When Teaching Basic Facts 179

Writing to Learn 206

RESOURCES FOR CHAPTER 10 For Discussion and Exploration 207

Literature Connections 180

Recommended Readings 180

Online Resources 181

REFLECTIONS ON CHAPTER 10

CHAPTER 12

Writing to Learn 181 Developing Strategies for Addition

For Discussion and Exploration 182 and Subtraction Computation 208

Toward Computational Fluency 209

Student Generated Personal Strategies 211

CHAPTER 11 Standard Algorithms 212

Developing Whole Number Development of Meaningful Strategies 214

Creating an Environment for Meaningful Strategies 214

Place Value Concepts 183

Models to Support Meaningful Strategies 214

Numeration System 183

Student Generated Strategies

for Addition and Subtraction 217

Pre Base Ten Understandings 184

Adding and Subtracting Single Digit Numbers 217

Counting by Ones 184 Adding Two Digit Numbers 218

xii Contents

Subtracting by Counting Up 219 Computational Estimation from

Take Away Subtraction 219 Student Generated Strategies 246

Extensions and Challenges 220 Stop before the Details 246

Algorithms for Addition and Subtraction 221 Use Related Problem Sets 246

Standard Algorithm for Addition 221 Computational Estimation Strategies 247

Alternative Algorithms for Addition 223 Front End Methods 247

Standard Algorithm for Subtraction 223 Rounding Methods 247

Alternative Algorithms for Subtraction 224 Compatible Numbers 248

Introducing Computational Estimation 225 Using Tens and Hundreds 248

Understanding Computational Estimation 225 Estimation Experiences 249

Suggestions for Teaching Computational Estimation 225 Calculator Activities 249

Computational Estimation Strategies 227 Using Whole Numbers to Estimate Rational Numbers 251

Front End Methods 227 RESOURCES FOR CHAPTER 13

Rounding Methods 227 Literature Connections 251

Compatible Numbers 228 Recommended Readings 252

RESOURCES FOR CHAPTER 12 Online Resources 252

Literature Connections 229 REFLECTIONS ON CHAPTER 13

Recommended Readings 229 Writing to Learn 252

Online Resources 230 For Discussion and Exploration 252

REFLECTIONS ON CHAPTER 12

Writing to Learn 230

For Discussion and Exploration 230 CHAPTER 14

Algebraic Thinking Generalizations

Patterns and Functions 253

CHAPTER 13

Developing Strategies for Multiplication Algebraic Thinking 254

and Division Computation 231 Generalization from Arithmetic 254

Generalization with Operations 254

Towards Computational Fluency Generalization in the Hundreds Chart 255

with Multiplication and Division 232 Generalization through Exploring a Pattern 256

Development of Meaningful Strategies Meaningful Use of Symbols 257

for Multiplication and Division 232 The Meaning of the Equal Sign 257

Models to Support Meaningful Strategies 232 The Meaning of Variables 262

Student Generated Strategies Making Structure in the Number System

for Multiplication 235 Explicit 265

Complete Number Strategies Including Doubling 236 Making Conjectures about Properties 265

Multiplication of Larger Numbers 236 Justifying Conjectures 267

Standard Algorithms for Multiplication 238 Odd and Even Relationships 267

One Digit Multipliers 238 Study of Patterns and Functions 267

Two Digit Multipliers 239 Repeating Patterns 268

Alternative Algorithms for Multiplication 240 Growing Patterns 270

Student Generated Strategies for Division 240 Linear Functions 274

Missing Factor Strategies 240 Mathematical Modelling 276

Cluster Problems 241 Teaching Considerations 277

Standard Algorithms for Division 242 Emphasize Appropriate Algebra Vocabulary 277

One Digit Divisors 242 Connecting Representations 278

Alternative Division Algorithms 244 Algebraic Thinking across the Curriculum 280

Computational Estimation in Multiplication RESOURCES FOR CHAPTER 14

and Division 245 Literature Connections 281

Understanding Computational Estimation 245 Recommended Readings 282

Suggestions for Teaching Computational Estimation 245 Online Resources 282

Contents xiii

REFLECTIONS ON CHAPTER 14 Addition and Subtraction 314

Writing to Learn 283 Contextual Examples and Student Generated Personal

For Discussion and Exploration 283 Strategies 314

Models 315

Developing the Algorithms 317

CHAPTER 15 Fractions Greater Than One 319

Addressing Errors and Misconceptions 319

Developing Fraction Concepts 284

Multiplication 321

Contextual Examples and Models 321

Meanings of Fractions 284

Fraction Constructs 285

Developing the Algorithms 324

Why Fractions Are So Difficult 285 Factors Greater Than One 325

Addressing Errors and Misconceptions 326

Models for Fractions 286

Area Models 286

Division 326

Length Models 288 Contextual Examples and Models 326

Set Models 289 Answers That Are Not Whole Numbers 328

Developing the Algorithms 329

Concept of Fractional Parts 289

Addressing Errors and Misconceptions 330

Fraction Size Is Relative 289

Fraction Language 290 RESOURCES FOR CHAPTER 16

Partitioning 290 Literature Connections 331

Sharing Tasks 292 Recommended Readings 331

Online Resources 331

Iterating 293

Fraction Notation 296 REFLECTIONS ON CHAPTER 16

Fractions Greater Than One 296 Writing to Learn 331

Estimating with Fractions 297 For Discussion and Exploration 332

Equivalent Fractions 298

Conceptual Focus on Equivalence 298

Equivalent Fraction Models 299 CHAPTER 17

Developing an Equivalent Fraction Algorithm 301

Developing Concepts of Decimals

Comparing Fractions 303

and Percents 333

Comparing Fractions Using Number Sense 303

Using Equivalent Fractions to Compare 305

Extending the Place Value System 334

Teaching Considerations for Fraction Concepts 305

A Two Way Relationship 334

PROBLEM BASED LESSON Equivalent Fraction Regrouping 334

Challenge 306 The Role of the Decimal Point 335

RESOURCES FOR CHAPTER 15 Connecting Fractions and Decimals 336

Base Ten Fractions 336

Literature Connections 308

Recommended Readings 308 Developing Decimal Number Sense 339

Online Resources 308 Familiar Fractions Connected to Decimals 339

REFLECTIONS ON CHAPTER 15 Computation with Decimals 343

Writing to Learn 309 The Role of Estimation 343

For Discussion and Exploration 309 Addition and Subtraction 344

Multiplication 344

Division 346

CHAPTER 16 Introducing Percents 346

Developing Strategies for Fraction Models and Terminology 347

Percent Problems in Context 348

Computation 310

Estimation 349

Understanding Fraction Operations 311 RESOURCES FOR CHAPTER 17

Conceptual Development Takes Time 311 Literature Connections 350

An Inquiry Based Number Sense Approach 311 Recommended Readings 351

Online Resources 351

Computational Estimation 312

xiv Contents

REFLECTIONS ON CHAPTER 17 Area 379

Writing to Learn 351 Comparison Activities 379

For Discussion and Exploration 351 Using Models of Area Units 380

The Relationship between Area and Perimeter 381

Developing Formulas for Area 382

CHAPTER 18 Student Errors and Misconceptions 382

Proportional Reasoning 352 Areas of Rectangles Parallelograms Triangles and Trapezoids 383

Circumference and Area of Circles 385

Ratios 352 Volume and Capacity 386

Types of Ratios 352 Comparison Activities 386

Ratios Compared to Fractions 353 Using Models of Volume and Capacity Units 386

Two Ways to Think about Ratios 353 Using Measuring Cups 387

Proportional Reasoning 354 Developing Formulas for Volumes of Common Solid Shapes 387

Connections between Formulas 388

Proportional and Nonproportional Situations 354

Additive and Multiplicative Comparisons in Problems 355 Mass and Weight 388

Covariation 356 Comparison Activities 389

Develop a Wide Variety of Strategies 360 Using Models of Mass Units 389

Mental Strategies 360 Angles 390

Ratio Tables 362 Comparison Activities 390

Using Models of Angular Measure Units 390

CONNECTED MATHEMATICS Grade 7 Comparing Using Protractors and Angle Rulers 390

and Scaling 363

Double Line Strip Comparison 364

Comparison Activities 391

Percents 365

Reading Clocks 392

Cross Products 365

Elapsed Time 392

Teaching Proportional Reasoning 366

RESOURCES FOR CHAPTER 18 Recognizing Coins and Identifying Their Values 393

Literature Connections 366 Counting Sets of Coins 393

Recommended Readings 367 Making Change 394

Online Resources 367

RESOURCES FOR CHAPTER 19

REFLECTIONS ON CHAPTER 18

Literature Connections 394

Writing to Learn 368 Recommended Readings 394

For Discussion and Exploration 368 Online Resources 394

REFLECTIONS ON CHAPTER 19

CHAPTER 19 Writing to Learn 395

For Discussion and Exploration 395

Developing Measurement Concepts 369

The Meaning and Process of Measuring 370

CHAPTER 20

Concepts and Skills 370

Introducing Nonstandard Units 372 Geometric Thinking

Developing Standard Units 372 and Geometric Concepts 396

Instructional Goals 372

A Brief History of the Metric System 373 Geometry Goals for Students 396

Important Standard Units and Relationships 373 Spatial Sense and Geometric Reasoning 397

The Role of Estimation and Approximation 374 Geometric Content 397

Strategies for Estimating Measurements 374

Developing Geometric Thinking 397

Tips for Teaching Estimation 375

The van Hiele Levels of Geometric Thought 397

Measurement Estimation Activities 375

Implications for Instruction 400

Length 376

Learning about Solids Plane Figures

Comparison Activities 376

and Their Properties 401

Using Models of Length Units 376

Solids Plane Figures and Properties for Level 0 Thinkers 401

Making and Using Rulers 378

Contents xv

Shapes and Properties for Level 1 Thinkers 405 RESOURCES FOR CHAPTER 21

Shapes and Properties for Level 2 Thinkers 410 Literature Connections 446

Learning about Transformations 413 Recommended Readings 446

Transformations for Level 0 Thinkers 413 Online Resources 446

Transformations for Level 1 Thinkers 415 REFLECTIONS ON CHAPTER 21

Transformations for Level 2 Thinkers 417 Writing to Learn 447

Learning about Location 417 For Discussion and Exploration 447

Location for Level 1 Thinkers 419

Location for Level 2 Thinkers 421

Learning about Visualization 421 CHAPTER 22

Visualization for Level 0 Thinkers 422

Visualization for Level 1 Thinkers 423

Exploring Concepts of Probability 448

Visualization for Level 2 Thinkers 424

Introducing Probability 448

RESOURCES FOR CHAPTER 20

Likely or Not Likely 449

Literature Connections 425 The Probability Continuum 450

Recommended Readings 426

Online Resources 426 Theoretical Probability and Experiments 453

Theoretical Probability 453

REFLECTIONS ON CHAPTER 20

Experiments 454

Writing to Learn 426 Why Use Experiments 456

For Discussion and Exploration 427

Use of Technology in Experiments 456

Sample Spaces and Probability of Two Events 457

Independent Events 457

CHAPTER 21 Area Models 459

Dependent Events 460

Developing Concepts of Data Analysis 428 Simulations 461

RESOURCES FOR CHAPTER 22

What Does It Mean to Do Statistics 429

Is It Statistics or Is It Mathematics 429 Literature Connections 463

Recommended Readings 463

The Shape of Data 429

Online Resources 464

The Process of Doing Statistics 430

REFLECTIONS ON CHAPTER 22

Formulating Questions 430

Classroom Questions 430 Writing to Learn 464

For Discussion and Exploration 464

Beyond One Classroom 431

Data Collection 431

Collecting Data 431

Using Existing Data Sources 432 CHAPTER 23

Data Analysis Classification 432 Developing Concepts of Exponents

Attribute Materials 433 Integers and Real Numbers 466

Data Analysis Graphical Representations 434

Bar Graphs and Tally Charts 435 Exponents 466

Circle Graphs 436 Exponents in Expressions and Equations 466

Numerical Data Graphs 437 Order of Operations 467

Scatter Plots 439 Negative Exponents 470

Data Analysis Measures of Centre Scientific Notation 470

and Variability 440 Integers 472

Averages 440 Contexts for Exploring Integers 472

Understanding the Mean Two Interpretations 441 Quantity Contexts 472

Variability 443 Linear Contexts 473

Box Plots 444 Meaning of Negative Numbers 474

Interpreting Results 445 Models for Teaching Integers 474