Table of Contents

Introduction 1

Low-Floor, High-Ceiling Tasks 2

Youcubed Summer Camp 3

Memorization versus Conceptual Engagement 4

Mathematical Thinking, Reasoning, and Convincing 5

Big Ideas 9

Structure of the Book 9

Note on Materials 17

Manipulatives and Materials Used in This Book 18

Activities for Building Norms 21

Encouraging Good Group Work 21

How Many Do You See? Learning to Reason, Convince, and Pose Questions 23

Big Idea 1: Building with and Talking about Shapes 31

Visualize: Build(ings) 33

Play: Tangram Puzzles 40

Investigate: Tangram Squares 52

Big Idea 2: Tens and Ones Are Useful Ways to Organize 59

Visualize: Organizing and Counting a Collection 61

Play: How Many Dots? 68

Investigate: Boxes of Pencils 74

Big Idea 3: Representing and Modeling Joining and Separating Situations 81

Visualize: Showing the Story 83

Play: Playing with Problem Types 89

Investigate: Library Wonders 99

Big Idea 4: Using Data to Describe and Wonder about Our World 107

Visualize: Organizing the Natural World 109

Play: Displaying Data 115

Investigate: Our Trash, Ourselves 124

Big Idea 5: Equal Means the Same 133

Visualize: Hungry, Hungry Monsters! 136

Play: True or False? 155

Investigate: Exploring Relations 165

Big Idea 6: Building with Numbers within 20 175

Visualize: Seeing Spots 177

Play: Rolling Ten 186

Investigate: Addition Table Patterns 195

Big Idea 7: Finding Patterns in Numbers 203

Visualize: Patterns in the Hundred Chart 205

Play: Skipping across the Hundred Chart 213

Investigate: Weaving Patterns 220

Big Idea 8: Using Place Value to Add and Subtract 231

Visualize: Recess! 233

Play: Playing with Place Value across Problem Types 243

Investigate: Making a Dollar 251

Big Idea 9: Using Units to Measure Our World 259

Visualize: Which One is Longer? 261

Play: Measuring Our World in Cubes 267

Investigate: Measuring with Cuisenaire Rods 275

Big Idea 10: Partitioning Shapes into Equal Parts 281

Visualize: Half and Half-of-a-Half 283

Play: Analyzing Tangrams 293

Investigate: What’s in a Flag? 299

Appendix 313

1" Grid Paper 314

Hundred Chart 315

Grid Paper 316

About the Authors 317

Acknowledgments 319

Index 321

About the Author

JO BOALER is the Nominelli-Olivier professor at Stanford. Former roles have included being a maths teacher in London schools. She is author of 18 books, numerous articles and a White House presenter on women and girls. She co-founded www.youcubed.org, is a regular contributor to news and radio in the US and was named as one of the 8 educators "changing the face of education" by the BBC.

JEN MUNSON is an assistant professor of learning sciences at Northwestern University, a profession- al developer, and a former classroom teacher. She received her PhD in mathematics education from Stanford University. Her research focuses on responsive, equitable mathematics instruction.

CATHY WILLIAMS is the co-founder and the executive director of youcubed at Stanford University. Before working at youcubed, she was a high school math teacher and worked in mathematics curriculum and administration at the county and district levels in California.