The most challenging parts of teaching mathematics are engaging students and helping them understand the connections between mathematics concepts. In this volume, you'll find a collection of low floor, high ceiling tasks that will help you do just that, by looking at the big ideas at the seventh-grade level through visualization, play, and investigation.

During their work with tens of thousands of teachers, authors Jo Boaler, Jen Munson, and Cathy Williams heard the same messagethat they want to incorporate more brain science into their math instruction, but they need guidance in the techniques that work best to get across the concepts they needed to teach. So the authors designedMindset Mathematics around the principle of active student engagement, with tasks that reflect the latest brain science on learning. Open, creative, and visual math tasks have been shown to improve student test scores, and more importantly change their relationship with mathematics and start believing in their own potential. The tasks inMindset Mathematics reflect the lessons from brain science that:

With engaging questions, open-ended tasks, and four-color visuals that will help kids get excited about mathematics,Mindset Mathematics is organized around nine big ideas which emphasize the connections within the Common Core State Standards (CCSS) and can be used with any current curriculum.

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, includingMathematical 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.

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: Whats It Going to Cost? 109

Big Idea 5: Building Intuition about Probability 119

Visualize: is It Fair? 121

Play: Fair Sums? 131

Investigate: Whats in the Bag? 136

Big Idea 6: Modeling Probabilities 145

Visualize: Whats 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