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 Index 287

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.