Math Wonders to Inspire Teachers and Students

Cover Page......Page 1
Title Page......Page 2
ISBN 0871207753......Page 3
CONTENTS......Page 6
Foreword......Page 10
Preface......Page 13
1 The Beauty in Numbers......Page 18
1.1 Surprising Number Patterns I......Page 19
1.2 Surprising Number Patterns II......Page 22
1.3 Surprising Number Patterns III......Page 23
1.4 Surprising Number Patterns IV......Page 24
1.5 Surprising Number Patterns V......Page 26
1.7 Amazing Power Relationships......Page 27
1.8 Beautiful Number Relationships......Page 29
1.9 Unusual Number Relationships......Page 30
1.10 Strange Equalities......Page 31
1.11 The Amazing Number 1,089......Page 32
1.12 The Irrepressible Number 1......Page 37
1.13 Perfect Numbers......Page 39
1.14 Friendly Numbers......Page 41
1.16 Palindromic Numbers......Page 43
1.17 Fun with Figurate Numbers......Page 46
1.18 The Fabulous Fibonacci Numbers......Page 49
1.19 Getting into an Endless Loop......Page 52
1.20 A Power Loop......Page 53
1.21 A Factorial Loop......Page 56
1.22 The Irrationality of √2......Page 58
1.23 Sums of Consecutive Integers......Page 61
2 Some Arithmetic Marvels......Page 64
2.1 Multiplying by 11......Page 65
2.2 When Is a Number Divisible by 11?......Page 66
2.3 When Is a Number Divisible by 3 or 9?......Page 68
2.4 Divisibility by Prime Numbers......Page 69
2.5 The Russian Peasant’s Method of Multiplication......Page 74
2.6 Speed Multiplying by 21, 31, 41......Page 76
2.7 Clever Addition......Page 77
2.8 Alphametics......Page 78
2.9 Howlers......Page 81
2.10 The Unusual Number 9......Page 86
2.11 Successive Percentages......Page 89
2.12 Are Averages Averages?......Page 91
2.13 The Rule of 72......Page 92
2.14 Extracting a Square Root......Page 94
3 Problems with Surprising Solutions......Page 96
3.1 Thoughtful Reasoning......Page 97
3.2 Surprising Solution......Page 98
3.3 A Juicy Problem......Page 99
3.4 Working Backward......Page 101
3.5 Logical Thinking......Page 102
3.6 It’s Just How You Organize the Data......Page 103
3.7 Focusing on the Right Information......Page 105
3.8 The Pigeonhole Principle......Page 106
3.9 The Flight of the Bumblebee......Page 107
3.10 Relating Concentric Circles......Page 109
3.11 Don’t Overlook the Obvious......Page 110
3.12 Deceptively Difficult (Easy)......Page 112
3.13 The Worst Case Scenario......Page 114
4 Algebraic Entertainments......Page 115
4.1 Using Algebra to Establish Arithmetic Shortcuts......Page 116
4.2 The Mysterious Number 22......Page 117
4.3Justifying an Oddity......Page 118
4.4 Using Algebra for Number Theory......Page 120
4.5 Finding Patterns Among Figurate Numbers......Page 121
4.6 Using a Pattern to Find the Sum of a Series......Page 125
4.7 Geometric View of Algebra......Page 126
4.8 Some Algebra of the Golden Section......Page 129
4.9 When Algebra Is Not Helpful......Page 132
4.10 Rationalizing a Denominator......Page 133
4.11 Pythagorean Triples......Page 134
5 Geometric Wonders......Page 140
5.1 Angle Sum of a Triangle......Page 141
5.2 Pentagram Angles......Page 143
5.3 Some Mind-Bogglers on......Page 148
5.4 The Ever-Present Parallelogram......Page 150
5.5 Comparing Areas and Perimeters......Page 154
5.6 How Eratosthenes Measured the Earth......Page 156
5.7 Surprising Rope Around the Earth......Page 158
5.8 Lunes and Triangles......Page 160
5.9 The Ever-Present Equilateral Triangle......Page 163
5.10 Napoleon’s Theorem......Page 166
5.11 The Golden Rectangle......Page 170
5.12 The Golden Section Constructed by Paper Folding......Page 175
5.13 The Regular Pentagon That Isn’t......Page 178
5.14 Pappus’s Invariant......Page 180
5.15 Pascal’s Invariant......Page 182
5.16 Brianchon’s Ingenius Extension of Pascal’s Idea......Page 185
5.17 A Simple Proof of the Pythagorean Theorem......Page 187
5.18 Folding the Pythagorean Theorem......Page 189
5.19 President Garfield’s Contribution to Mathematics......Page 191
5.20 What Is the Area of a Circle?......Page 193
5.21 A Unique Placement of Two Triangles......Page 195
5.22 A Point of Invariant Distance in an Equilateral Triangle......Page 197
5.23 The Nine-Point Circle......Page 200
5.24 Simson’s Invariant......Page 204
5.25 Ceva’s Very Helpful Relationship......Page 206
5.26 An Obvious Concurrency?......Page 210
5.27 Euler’s Polyhedra......Page 212
6 Mathematical Paradoxes......Page 215
6.1 Are All Numbers Equal?......Page 216
6.2 −1 Is Not Equal to +1......Page 217
6.3 Thou Shalt Not Divide by 0......Page 218
6.4 All Triangles Are Isosceles......Page 219
6.5 An Infinite-Series Fallacy......Page 223
6.6 The Deceptive Border......Page 225
6.7 Puzzling Paradoxes......Page 227
6.8 A Trigonometric Fallacy......Page 228
6.9 Limits with Understanding......Page 230
7 Counting and Probability......Page 232
7.1 Friday the 13th!......Page 233
7.2 ThinkBefor e Counting......Page 234
7.3 The Worthless Increase......Page 236
7.4 Birthday Matches......Page 237
7.5 Calendar Peculiarities......Page 240
7.6 The Monty Hall Problem......Page 241
7.7 Anticipating Heads and Tails......Page 245
8 Mathematical Potpourri......Page 246
8.1 Perfection in Mathematics......Page 247
8.2 The Beautiful Magic Square......Page 249
8.3 Unsolved Problems......Page 253
8.4 An Unexpected Result......Page 256
8.5 Mathematics in Nature......Page 258
8.6 The Hands of a Clock......Page 264
8.7 Where in the World Are You?......Page 268
8.8 Crossing the Bridges......Page 270
8.9 The Most Misunderstood Average......Page 273
8.10 The Pascal Triangle......Page 276
8.11 It’s All Relative......Page 280
8.12 Generalizations Require Proof......Page 281
8.13 A Beautiful Curve∗......Page 282
Epilogue......Page 285
Acknowledgments......Page 288
Index......Page 289
About the Author......Page 293
If you like this book, you’ll LOVE the membership!......Page 295