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 x 4 and 5 x 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
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.