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 × 4 and 5 × 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