Acknowledgments xi About the Authors xiii Introduction xvii I. Why Problem Solving? Chapter 1: Rewards for Problem-Based Approach: Range, Rigor, and Resilience 5 Range Ignites Curiosity 5 Rigor Taps Critical Thinking 9 Resilience is Born Through Creativity 10 Chapter 2: Maximize Learning: Relevance, Authenticity, and Usefulness 13 Student Relevance 13 Mathematical Relevance 14 Mathematical Relevance: The Math Circle Example 16 Curriculum Relevance 18 Authenticity: The Cargo Cult Science Trap 21 Authenticity in Learning 22 Usefulness 25 Chapter 3: Creating a Math Learning Environment 27 Know Yourself: Ego and Grace 27 Know Your Students 30 Know Your Approach 35 Chapter 4: What is the Telos? 47 Autonomy to Solve Your Problems 47 Mastery Through Inquiry 48 Purpose with Competitions 50 Quadrants of Success 52 Chapter 5: Gains and Pains with a Problem-Based Curriculum 57 Teachers 58 Students 61 Parents 67 II. Teaching Problem Solving Chapter 6: Five Steps to Problem-Based Learning 75 Start with Meaningful Problems 75 Utilize Teacher Resources 79 Provide an Active Learning Environment 91 Understand the Value of Mistakes 97 Recognize That Everyone is Good at Math 99 Chapter 7: The Three Cs: Competitions, Collaboration, Community 103 Competitions 103 Collaboration 107 Community 117 Aspire to Inspire: Stories from Awesome Educators 121 Chapter 8: Mini-Units 147 Relate/Reflect/Revise Questions 147 Roman Numeral Problems 148 Cryptarithmetic 151 Squaring Numbers: Mental Mathematics 155 The Number of Elements of a Finite Set 157 Magic Squares 159 Toothpicks Math 163 Pick's Theorem 165 Equilateral versus Equiangular 168 Math and Chess 170 Area and Volume of a Sphere 172 III. Full Units Chapter 9: Angles and Triangles 177 Learning Objectives 177 Definitions 177 Angles and Parallel Lines 177 Summary 180 Chapter 10: Consecutive Numbers 185 Learning Objectives 185 Definitions 185 Chapter 11: Factorials! 191 Learning Objectives 191 Definitions 191 Chapter 12: Triangular Numbers 199 Learning Objectives 199 Definitions 199 Chapter 13: Polygonal Numbers 205 Learning Objectives 205 Definitions 205 Chapter 14: Pythagorean Theorem Revisited 213 Learning Objectives 213 Definitions 213 Pythagorean Theorem 214 Rectangular Boxes 214 Euler Bricks 216 Assessment Problems 219 Chapter 15: Sequences 221 Learning Objectives 221 Definitions 221 Introduce a Geometric Progression 222 Chapter 16: Pigeonhole Principle 227 Learning Objectives 227 Definitions 227 Chapter 17: Viviani's Theorems 235 Learning Objectives 235 Definition 235 Chapter 18: Dissection Time 239 Learning Objectives 239 Definitions 239 Chapter 19: Pascal's Triangle 245 Learning Objective 245 Summary 249 Chapter 20: Nice Numbers 255 Learning Objectives 255 Definitions 255 Index 259.