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: Connecting 2D and 3D Worlds 27 Visualize: Seeing Slices 29 Play: Playing with Clay 34 Investigate: The Area of Slices 40 Big Idea 2: Constructing Figures to Scale 47 Visualize: What is Scale? 50 Play: A Fishy Image 55 Investigate: Magnifying Solids 61 Big Idea 3: Seeing Proportional Relationships 69 Visualize: Growing Dough 72 Play: is It Proportional? 80 Investigate: Sizing Up Proportions 86 Big Idea 4: Understanding Percents in the World 93 Visualize: Painting Percents 95 Play: Building Benchmarks 103 Investigate: What's It Going to Cost? 109 Big Idea 5: Building Intuition about Probability 119 Visualize: is It Fair? 121 Play: Fair Sums? 131 Investigate: What's in the Bag? 136 Big Idea 6: Modeling Probabilities 145 Visualize: What's the Chance of That? 147 Play: Probability Carnival 155 Investigate: Go, Go Galton Board! 163 Big Idea 7: Sampling to Understand Populations 171 Visualize: Sampling Dots and Beads 173 Play: Catch and Release 180 Investigate: Wildlife Populations 186 Big Idea 8: Operating with Opposites 197 Visualize: Line Dancing 199 Play: Integer Tic-Tac-Toe 207 Investigate: Coordinating Multiplication 216 Big Idea 9: Using Algebra as a Problem-Solving Tool 225 Visualize: Case by Case 228 Play: (Non) Linear Pattern Puzzles 237 Investigate: Linear Letters 245 Big Idea 10: Exploring Circles 251 Visualize: Building Hunches about Circumference 253 Play: Dissecting Circle Area 260 Investigate: Unraveling Circles 267 Appendix 273 Centimeter Grid Paper 274 1" Grid Paper 275 Snap Cube Grid Paper 276 Isometric Dot Paper 277 Dot Paper 278 About the Authors 279 Acknowledgments 281 Index 283
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.