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

Activities for Building Norms 17

Encouraging Good Group Work 17

Paper Folding: Learning to Reason, Convince, and Be a Skeptic 21

Big Idea 1: Thinking in Cubes 23

Visualize: Solids, Inside and Out 25

Play: City of Cubes 33

Investigate: A Box of Boxes 44

Big Idea 2: Estimating with Fractions 53

Visualize: Making Snowflakes 55

Play: Fraction Blizzard 61

Investigate: Wondering with Fractions 67

Big Idea 3: Using Fraction Equivalence 81

Visualize: Picking Paintings Apart 83

Play: Make a Fake 94

Investigate: Squares with a Difference 101

Big Idea 4: Exploring the Coordinate Plane 115

Visualize: Getting around the Plane 118

Play: Ship Shape 124

Investigate: Table Patterns 133

Big Idea 5: Seeing and Connecting Patterns across Representations 143

Visualize: Two-Pattern Tango 145

Play: Pattern Carnival 153

Investigate: Seeing Growth on a Graph 159

Big Idea 6: Understanding Fraction Multiplication Visually 169

Visualize: Fractions in a Pan 172

Play: Pieces and Parts 180

Investigate: The Sum of the Parts 187

Big Idea 7: What Does It Mean to Divide Fractions? 199

Visualize: Creating Cards 201

Play: Cuisenaire Trains 209

Investigate: Fraction Division Conundrum 217

Big Idea 8: Thinking in Powers of 10 223

Visualize: The Unit You 225

Play: Filling Small and Large 233

Investigate: Museum of the Very Large and Small 239

Big Idea 9: Using Numbers and Symbols Flexibly 247

Visualize: Seeing Expressions 250

Play: Inside Pascal’s Triangle 261

Investigate: The 1492 Problem 268

Appendix 279

Centimeter Dot Paper 280

Isometric Dot Paper 281

About the Authors 283

Acknowledgments 285

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 doctoral candidate at Stanford University, a professional developer, and a former classroom teacher. She works with teachers and school leaders across the United States to develop 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.