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 Note on Materials 16 Manipulatives and Materials Used in This Book 17 Activities for Building Norms 19 Encouraging Good Group Work 19 Paper Folding: Learning to Reason, Convince, and Be Skeptical 21 Big Idea 1: Taking Apart Prisms and Polygons 24 Visualize: How Big Is the Footprint? 26 Play: Shards of a Shape 38 Investigate: Rising from the Footprint 46 Big Idea 2: Folding and Unfolding Objects 57 Visualize: Folding Cubes in Your Mind 60 Play: Folding Nets in Your Mind 70 Investigate: Filling Our Nets 85 Big Idea 3: Expanding the Number Line 92 Visualize: Folding around Zero 94 Play: Bouncing around the Number Line 101 Investigate: Going 2-D 107 Big Idea 4: Finding and Using Unit Rates 116 Visualize: Seeing Unit Rates 118 Play: Seeing the Best Deal 128 Investigate: How Fast Do You Walk? 136 Big Idea 5: Reasoning with Proportions 142 Visualize: Jump! Jump! 144 Play: Seeing Animals in a New Way 149 Investigate: A Seat at the Table 156 Big Idea 6: Visualizing the Center and Spread of Data 164 Visualize: The Shape of Data 167 Play: What Does Mean Mean? 178 Investigate: When Does Mean = Median = Mode = Range? 185 Big Idea 7: Using Symbols to Describe the World 192 Visualize: Cuisenaire Rod Equivalents 195 Play: Math Mobiles 204 Investigate: Radial Patterns 215 Big Idea 8: Generalizing 221 Visualize: Generalizing Number Patterns 223 Play: Generalizing Strategy 230 Investigate: Generalizing Visual Patterns 239 Appendix 246 Centimeter Grid Paper 247 Grid Paper 248 1" Grid Paper 249 Isometric Dot Paper 250 Dot Paper 251 About the Authors 252 Acknowledgments 254 Index 255
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 postdoctoral fellow at Northwestern University, a professional developer, and a former classroom teacher. She received her PhD in mathematics education from Stanford University. Her research focuses on 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.