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

Paper Folding: Learning to Reason, Convince, and Be a Skeptic 22

Big Idea 1: Moving Shapes 27

Visualize: What Does It Mean to Be the Same? 29

Play: Pixel Puzzles 35

Investigate: Slide It, Flip It, Turn It 45

Big Idea 2: Zooming In and Out 53

Visualize: What is Similarity? 55

Play: Find the Fakes 60

Investigate: Growing on a Grid 65

Big Idea 3: Analyzing Proportional Relationships 71

Visualize: Dropping the Altitude 73

Play: Seeing Triangles under the Line 79

Investigate: Stairway to Eleven 85

Big Idea 4: Comparing Patterns of Growth 91

Visualize: Squared Squares 93

Play: Skip-Counting Arrays 100

Investigate: Stacking Pennies 107

Big Idea 5: The Ins and Outs of Functions 117

Visualize: Growing Functions 120

Play: Getting Warmer! 134

Investigate: The Functions of Near Squares 140

Big Idea 6: Finding Patterns in the Clouds 149

Visualize: What’s Going On in This Graph? 151

Play: What’s the Story Here? 158

Investigate: Creating Data Visualizations 166

Big Idea 7: Completing the Real Number System 177

Visualize: Square Sides 179

Play: Between 4 × 4 and 5 × 5 187

Investigate: The Hypotenuse Hypothesis 193

Big Idea 8: Discovering Pythagoras 201

Visualize: Unpacking Pythagoras 204

Play: Pythagorean Triples 209

Investigate: Approximating Square Roots 214

Big Idea 9: Exploring the Geometry of Ice Cream 221

Visualize: Building Intuition about Volume 223

Play: Comparing Cylinders 227

Investigate: Scooping Up Volume 233

Appendix 239

Centimeter Grid Paper 240

Grid Paper 241

Isometric Dot Paper 242

Dot Paper 243

About the Authors 245

Acknowledgments 247

Index 249

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

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.