N i N T h E D i T i o N G L o B A L E D i T i o N 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 With Contributions by Jonathan Wray Howard County Public Schools Boston Columbus Hoboken Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Executive Acquisitions Editor: Meredith Fossel Series Editorial Assistant: Maria Feliberty Executive Development Editor: Linda Bishop Vice President, Director of Marketing: Margaret Waples Senior Marketing Manager: Christopher Barry Program Manager: Maren Beckman Project Manager: Christina Taylor Project Manager, Global Edition: Purnima Narayanan Senior Acquisitions Editor, Global Edition: Sandhya Ghoshal Senior Project Editor, Global Edition: Daniel Luiz Manager, Media Production, Global Edition: M. Vikram Kumar Senior Manufacturing Controller, Production, Global Edition: Trudy Kimber Editorial Production Service: MPS North America LLC Manufacturing Buyer: Deidra Skahill Electronic Composition: Jouve Interior Design: Diane Lorenzo Cover Image: © CristinaMuraca/Shutterstock Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text or on page C-1. Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Authorized adaptation from the United States edition, entitled Elementary and Middle School Mathematics: Teaching Developmentally, 9th edition, ISBN 978-0-13-376893-0, by John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams, published by Pearson Education © 2016. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, withouteither the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners.The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISBN 10: 1-292-09769-8 ISBN 13: 978-1-292-09769-5 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. 10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 Typeset in Janson Text LT Std Roman by MPS North America LLC. Printed and bound by R.R. Donnelly Kendalville in the United States of America. About the Authors John A. Van de Walle The late John A. Van de Walle was a professor emeritus at Virginia Commonwealth University. He was a leader in mathematics education who regularly offered professional develop- ment workshops for K–8 teachers in the United States and Canada focused on mathematics instruction that engaged students in mathematical reasoning and problem solving. He visited many 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 original Pearson School mathematics program enVisionMATH. Additionally, John was very active in the National Council of Teachers of Mathematics (NCTM), writing book chapters and jour- nal articles, serving on the board of directors, chairing the educational materials committee, and speaking at national and regional meetings. Karen S. Karp Karen S. Karp is a professor of mathematics education at the University of Louisville (Ken- tucky). Prior to entering the field of teacher education she was an elementary school teacher in New York. Karen is the volume editor of Annual Perspectives in Mathematics Education: Using Research to Improve Instruction and is the co-author of Developing Essential Understanding of Addition and Subtraction for Teaching Mathematics in Pre-K–Grade 2, Discovering Lessons for the Common Core State Standards in Grades K–5, and Putting Essential Understanding of Addition and Subtraction into Practice Pre-K–Grade 2. She is a former member of the board of directors for the National Council of Teachers of Mathematics (NCTM) and a former president of the Associ- ation of Mathematics Teacher Educators. She continues to work in classrooms with teachers of students with disabilities. Jennifer M. Bay-Williams Jennifer M. Bay-Williams is a mathematics educator at the University of Louisville (Kentucky). 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 teachers. Jennifer has published many articles on teaching and learning in NCTM journals. She has also authored and co-authored numerous books, including Developing Essential Understanding of Addition and Subtraction for Teach- ing Mathematics in Pre-K–Grade 2, Math and Literature: Grades 6–8, Math and Nonfiction: Grades 6–8, Navigating through Connections in Grades 6–8, and Mathematics Coaching: Resources and Tools for Coaches and Other Leaders. She is on the board of directors for the National Council of Teachers of Mathematics (NCTM) and previously served on the Board of Directors for TODOS: Equity for All and as secretary and president for the Association of Mathematics Teacher Educators (AMTE). 33 About the Contributor Jonathan Wray is the technology contributor to Elementary and Middle School Mathe- matics, Teaching Developmentally (6th–9th editions). He is the instructional facilitator for Sec- ondary Mathematics Curricular Programs in the Howard County Public School System. He is the president of the Association of Maryland Mathematics Teacher Educators (AMMTE) and past president of the Maryland Council of Teachers of Mathematics (MCTM) and serves as manager of the Elementary Mathematics Specialists and Teacher Leaders (ems&tl) Project. 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). Jon is also actively engaged in the National Council of Teachers of Mathematics (NCTM), serving on the Emerging Issues and Executive Commit- tees. He has served as a primary and intermediate grades classroom teacher, gifted/talented resource teacher, elementary mathematics specialist, curriculum and assessment developer, grant project manager, and educational consultant. 4 Brief Contents SeCtion i teaching Mathematics: Foundations and Perspectives ChAPter 1 teaching Mathematics in the 21st Century 25 ChAPter 2 exploring What it Means to Know and Do Mathematics 37 ChAPter 3 teaching through Problem Solving 57 ChAPter 4 Planning in the Problem-Based Classroom 81 ChAPter 5 Creating Assessments for Learning 108 ChAPter 6 teaching Mathematics equitably to All Children 128 ChAPter 7 Using technological tools to teach Mathematics 151 SeCtion ii Development of Mathematical Concepts and Procedures ChAPter 8 Developing early number Concepts and number Sense 166 ChAPter 9 Developing Meanings for the operations 191 ChAPter 10 Developing Basic Fact Fluency 218 ChAPter 11 Developing Whole-number Place-Value Concepts 246 ChAPter 12 Developing Strategies for Addition and Subtraction Computation 271 ChAPter 13 Developing Strategies for Multiplication and Division Computation 301 ChAPter 14 Algebraic thinking, equations, and Functions 323 ChAPter 15 Developing Fraction Concepts 363 ChAPter 16 Developing Fraction operations 395 ChAPter 17 Developing Concepts of Decimals and Percents 427 ChAPter 18 ratios, Proportions, and Proportional reasoning 453 ChAPter 19 Developing Measurement Concepts 477 ChAPter 20 Geometric thinking and Geometric Concepts 512 ChAPter 21 Developing Concepts of Data Analysis 550 ChAPter 22 exploring Concepts of Probability 582 ChAPter 23 Developing Concepts of exponents, integers, and real numbers 606 APPenDix A Standards for Mathematical Practice A-1 APPenDix B nCtM Mathematics teaching Practices: from Principles to Actions A-5 APPenDix C Guide to Blackline Masters A-7 APPenDix D Activities at a Glance A-13 55 This page intentionally left blank Contents Preface 15 SeCtion i teaching Mathematics: Foundations and Perspectives The fundamental core of effective teaching of mathematics combines an understanding of how students 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 students with diverse backgrounds and the role of technological tools are also emphasized. ChAPter 1 Mathematical Proficiency 47 teaching Mathematics How Do Students Learn Mathematics? 50 in the 21st Century 25 Constructivism 50 Sociocultural Theory 51 Becoming an Effective Teacher of Mathematics 25 Implications for Teaching Mathematics 51 A Changing World 26 Connecting the Dots 54 Factors to Consider 27 Reflections on Chapter 2 55 The Movement toward Shared Standards 28 Writing to Learn 55 Principles and Standards for School Mathematics 29 For Discussion and Exploration 55 Common Core State Standards 30 Resources for Chapter 2 56 Principles to Actions 33 Recommended Readings 56 An Invitation to Learn and Grow 34 Becoming a Teacher of Mathematics 34 Reflections on Chapter 1 36 ChAPter 3 Writing to Learn 36 teaching through Problem Solving 57 For Discussion and Exploration 36 Resources for Chapter 1 36 Problem Solving 57 Recommended Readings 36 Teaching for Problem Solving 58 Teaching about Problem Solving 58 Teaching through Problem Solving 61 ChAPter 2 Features of Worthwhile Tasks 61 exploring What it Means to Know High Levels of Cognitive Demand 62 and Do Mathematics 37 Multiple Entry and Exit Points 62 What Does It Mean to Do Mathematics? 37 Relevant Contexts 65 Evaluating and Adapting Tasks 67 Verbs of Doing Mathematics 38 Developing Concepts and Procedures through Tasks 68 An Invitation to Do Mathematics 39 Concepts 68 Searching for Patterns 39 Procedures 69 Analyzing a Situation 40 What about Drill and Practice? 71 Generalizing Relationships 41 Experimenting and Explaining 42 Orchestrating Classroom Discourse 73 Where Are the Answers? 44 Classroom Discussions 73 What Does It Mean to Be Mathematically Questioning Considerations 75 Proficient? 44 How Much to Tell and Not to Tell 76 Relational Understanding 45 Writing to Learn 77 77 8 Contents Problem Solving for All 78 Assessment Methods 111 Reflections on Chapter 3 80 Observations 111 Writing to Learn 80 Interviews 113 For Discussion and Exploration 80 Tasks 116 Resources for Chapter 3 80 Rubrics and Their Uses 119 Recommended Readings 80 Generic Rubrics 120 Task-Specific Rubrics 121 Writing as an Assessment Tool 122 Student Self-Assessment 123 ChAPter 4 Tests 124 Planning in the Problem-Based Classroom 81 Improving Performance on High-Stakes Tests 125 A Three-Phase Lesson Format 81 Communicating Grades and Shaping Instruction 125 The Before Phase of a Lesson 82 Reflections on Chapter 5 126 The During Phase of a Lesson 85 Writing to Learn 126 The After Phase of a Lesson 87 For Discussion and Exploration 126 Process for Preparing a Lesson 89 Resources for Chapter 5 127 Step 1: Determine the Learning Goals 90 Recommended Readings 127 Step 2: Consider Your Students’ Needs 90 Step 3: Select, Design, or Adapt a Worthwhile Task 91 Step 4: Design Lesson Assessments 91 ChAPter 6 teaching Mathematics equitably to All Step 5: Plan the Before Phase of the Lesson 92 Children 128 Step 6: Plan the During Phase of the Lesson 93 Step 7: Plan the After Phase of the Lesson 93 Mathematics for ALL Students 128 Step 8: Reflect and Refine 93 Providing for Students Who Struggle and Those with Special More Options for the Three-Phase Lesson 94 Needs 130 Short Tasks 94 Prevention Models 130 Learning Centers 95 Implementing Interventions 131 Differentiating Instruction 96 Teaching and Assessing Students with Learning Disabilities 135 Open Questions 96 Teaching Students with Moderate/Severe Disabilities 137 Tiered Lessons 97 Culturally and Linguistically Diverse Students 138 Parallel Tasks 99 Culturally Responsive Instruction 139 Flexible Grouping 99 Focus on Academic Vocabulary 140 Planning for Family Engagement 101 Facilitating Engagement during Instruction 143 Communicating Mathematics Goals 101 Implementing Strategies for English Language Learners 144 Family Math Nights 102 Providing for Students Who Are Mathematically Gifted 145 Homework Practices 104 Creating Gender-Friendly Mathematics Classrooms 147 Resources for Families 105 Gender Differences 147 Involving All Families 106 What Can You Try? 148 Reflections on Chapter 4 107 Reducing Resistance and Building Resilience 149 Writing to Learn 107 Reflections on Chapter 6 150 For Discussion and Exploration 107 Writing to Learn 150 Resources for Chapter 4 107 For Discussion and Exploration 150 Recommended Readings 107 Resources for Chapter 6 150 Recommended Readings 150 ChAPter 5 ChAPter 7 Creating Assessments for Learning 108 Using technological tools to teach Mathematics 151 Integrating Assessment into Instruction 108 What Is Assessment? 109 Tools and Technology 151 What Should Be Assessed? 110 Technology-Supported Learning Activities 152 Contents 9 Calculators in Mathematics Instruction 154 How to Select Appropriate Digital Content 160 When to Use a Calculator 155 Mathematics Resources on the Internet 162 Benefits of Calculator Use 155 How to Select Online Resources 162 Graphing Calculators 156 Emerging Technologies 162 Portable Data-Collection Devices 158 Reflections on Chapter 7 165 Appropriate and Strategic Use of Digital Tools 158 Writing to Learn 165 Concept Instruction 159 For Discussion and Exploration 165 Problem Solving 159 Resources for Chapter 7 165 Drill and Reinforcement 159 Recommended Readings 165 Guidelines for Selecting and Using Digital Resources for Mathematics 160 Guidelines for Using Digital Content 160 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 content area in the pre-K–8 mathematics curriculum. Numerous problem-based activities to engage students are interwoven with a discussion of the mathematical content and how students 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, integrations with the mathematical practices, and formative assessment notes. These chapters are designed to help you develop pedagogical strategies and to serve as a resource for your teaching now and in the future. ChAPter 8 Developing early number Concepts and number ChAPter 9 Sense 166 Developing Meanings for the operations 191 Promoting Good Beginnings 167 Teaching Operations through Contextual Problems 192 The Number Core: Quantity, Counting, and Knowing How Many 168 Addition and Subtraction Problem Structures 192 Quantity and the Ability to Subitize 168 Change Problems 193 Early Counting 169 Part-Part-Whole Problems 194 Numeral Writing and Recognition 172 Compare Problems 194 Counting On and Counting Back 173 Problem Difficulty 195 The Relations Core: More Than, Less Than, and Teaching Addition and Subtraction 196 Equal To 174 Contextual Problems 196 Developing Number Sense by Building Number Model-Based Problems 198 Relationships 176 Properties of Addition and Subtraction 201 Relationships between Numbers 1 through 10 176 Multiplication and Division Problem Structure 203 Relationships for Numbers 10 through 20 and Equal-Group Problems 203 Beyond 184 Comparison Problems 203 Number Sense in Their World 186 Area and Array Problems 205 Calendar Activities 186 Combination Problems 205 Estimation and Measurement 187 Teaching Multiplication and Division 205 Data Collection and Analysis 188 Contextual Problems 206 Reflections on Chapter 8 189 Remainders 207 Writing to Learn 189 Model-Based Problems 207 For Discussion and Exploration 189 Properties of Multiplication and Division 210 Resources for Chapter 8 189 Strategies for Solving Contextual Problems 212 Literature Connections 189 Analyzing Context Problems 212 Recommended Readings 189 Multistep Problems 214