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 10

Note on Materials 16

Manipulatives and Materials Used in This Book 17

Activities for Building Norms 19

Encouraging Good Group Work 19

Paper Folding: Learning to Reason, Convince, and Be Skeptical 21

Big Idea 1: Taking Apart Prisms and Polygons 24

Visualize: How Big Is the Footprint? 26

Play: Shards of a Shape 38

Investigate: Rising from the Footprint 46

Big Idea 2: Folding and Unfolding Objects 57

Visualize: Folding Cubes in Your Mind 60

Play: Folding Nets in Your Mind 70

Investigate: Filling Our Nets 85

Big Idea 3: Expanding the Number Line 92

Visualize: Folding around Zero 94

Play: Bouncing around the Number Line 101

Investigate: Going 2-D 107

Big Idea 4: Finding and Using Unit Rates 116

Visualize: Seeing Unit Rates 118

Play: Seeing the Best Deal 128

Investigate: How Fast Do You Walk? 136

Big Idea 5: Reasoning with Proportions 142

Visualize: Jump! Jump! 144

Play: Seeing Animals in a New Way 149

Investigate: A Seat at the Table 156

Big Idea 6: Visualizing the Center and Spread of Data 164

Visualize: The Shape of Data 167

Play: What Does Mean Mean? 178

Investigate: When Does Mean = Median = Mode = Range? 185

Big Idea 7: Using Symbols to Describe the World 192

Visualize: Cuisenaire Rod Equivalents 195

Play: Math Mobiles 204

Investigate: Radial Patterns 215

Big Idea 8: Generalizing 221

Visualize: Generalizing Number Patterns 223

Play: Generalizing Strategy 230

Investigate: Generalizing Visual Patterns 239

Appendix 246

Centimeter Grid Paper 247

Grid Paper 248

1" Grid Paper 249

Isometric Dot Paper 250

Dot Paper 251

About the Authors 252

Acknowledgments 254

Index 255

About the Author

JO BOALER is a professor of mathematics education at Stanford University and co-founder and faculty director of youcubed. She serves as an advisor to several Silicon Valley companies and is a White House presenter on girls and STEM (Science, Technology, Engineering, and Math). The author of seven books, including Mathematical Mindsets, and numerous research articles, she is a regular contributor to news and radio in the United States and England.

JEN MUNSON is a postdoctoral fellow at Northwestern University, a professional 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.