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.