# Math Wonders to Inspire Teachers and Students by Alfred S. Posamentier

Rajan Sharma | 06-Feb-2016 |

#### Hi friends, here Rajan Sharma uploaded notes for Mathematics with title Math Wonders to Inspire Teachers and Students by Alfred S. Posamentier. You can download this lecture notes, ebook by clicking on the below file name or icon.

Chapter 1 The Beauty in Numbers . . . . . . . . . . . . . . . . . . 1
1.1 Surprising Number Patterns I . . . . . . . . . . . . . . . 2
1.2 Surprising Number Patterns II . . . . . . . . . . . . . . . 5
1.3 Surprising Number Patterns III . . . . . . . . . . . . . . 6
1.4 Surprising Number Patterns IV . . . . . . . . . . . . . . 7
1.5 Surprising Number Patterns V . . . . . . . . . . . . . . . 9
1.6 Surprising Number Patterns VI . . . . . . . . . . . . . . 10
1.7 Amazing Power Relationships . . . . . . . . . . . . . . . 10
1.8 Beautiful Number Relationships . . . . . . . . . . . . . . 12
1.9 Unusual Number Relationships . . . . . . . . . . . . . . 13
1.10 Strange Equalities . . . . . . . . . . . . . . . . . . . . . . 14
1.11 The Amazing Number 1,089 . . . . . . . . . . . . . . . 15
1.12 The Irrepressible Number 1 . . . . . . . . . . . . . . . . 20
1.13 Perfect Numbers . . . . . . . . . . . . . . . . . . . . . . 22
1.14 Friendly Numbers . . . . . . . . . . . . . . . . . . . . . . 24
1.15 Another Friendly Pair of Numbers . . . . . . . . . . . . 26
1.16 Palindromic Numbers . . . . . . . . . . . . . . . . . . . . 26
1.17 Fun with Figurate Numbers . . . . . . . . . . . . . . . . 29
1.18 The Fabulous Fibonacci Numbers . . . . . . . . . . . . . 32
1.19 Getting into an Endless Loop . . . . . . . . . . . . . . . 35
1.20 A Power Loop . . . . . . . . . . . . . . . . . . . . . . . . 36
1.21 A Factorial Loop . . . . . . . . . . . . . . . . . . . . . . 39
1.22 The Irrationality of √2 . . . . . . . . . . . . . . . . . . . 41
1.23 Sums of Consecutive Integers . . . . . . . . . . . . . . . 44
Chapter 2 Some Arithmetic Marvels . . . . . . . . . . . . . . . . . 47
2.1 Multiplying by 11 . . . . . . . . . . . . . . . . . . . . . . 48
2.2 When Is a Number Divisible by 11? . . . . . . . . . . . 49
2.3 When Is a Number Divisible by 3 or 9? . . . . . . . . . 51
2.4 Divisibility by Prime Numbers . . . . . . . . . . . . . . 52
2.5 The Russian Peasant’s Method of Multiplication . . . . 57
2.6 Speed Multiplying by 21, 31, and 41 . . . . . . . . . . . 59
2.7 Clever Addition . . . . . . . . . . . . . . . . . . . . . . . 60
2.8 Alphametics . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.9 Howlers . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.10 The Unusual Number 9 . . . . . . . . . . . . . . . . . . 69
2.11 Successive Percentages . . . . . . . . . . . . . . . . . . . 72
2.12 Are Averages Averages? . . . . . . . . . . . . . . . . . . 74
2.13 The Rule of 72 . . . . . . . . . . . . . . . . . . . . . . . 75
2.14 Extracting a Square Root . . . . . . . . . . . . . . . . . 77
Chapter 3 Problems with Surprising Solutions . . . . . . . . . . . . 79
3.1 Thoughtful Reasoning . . . . . . . . . . . . . . . . . . . 80
3.2 Surprising Solution . . . . . . . . . . . . . . . . . . . . . 81
3.3 A Juicy Problem . . . . . . . . . . . . . . . . . . . . . . 82
3.4 Working Backward . . . . . . . . . . . . . . . . . . . . . 84
3.5 Logical Thinking . . . . . . . . . . . . . . . . . . . . . . 85
3.6 It’s Just How You Organize the Data . . . . . . . . . . . 86
3.7 Focusing on the Right Information . . . . . . . . . . . . 88
3.8 The Pigeonhole Principle . . . . . . . . . . . . . . . . . 89
3.9 The Flight of the Bumblebee . . . . . . . . . . . . . . . 90
3.10 Relating Concentric Circles . . . . . . . . . . . . . . . . 92
3.11 Don’t Overlook the Obvious . . . . . . . . . . . . . . . . 93
3.12 Deceptively Difficult (Easy) . . . . . . . . . . . . . . . . 95
3.13 The Worst Case Scenario . . . . . . . . . . . . . . . . . 97
Chapter 4 Algebraic Entertainments . . . . . . . . . . . . . . . . . . 98
4.1 Using Algebra to Establish Arithmetic Shortcuts . . . . 99
4.2 The Mysterious Number 22 . . . . . . . . . . . . . . . . 100
4.3 Justifying an Oddity . . . . . . . . . . . . . . . . . . . . 101
4.4 Using Algebra for Number Theory . . . . . . . . . . . . 103
4.5 Finding Patterns Among Figurate Numbers . . . . . . . 104
4.6 Using a Pattern to Find the Sum of a Series . . . . . . 108
4.7 Geometric View of Algebra . . . . . . . . . . . . . . . . 109
4.8 Some Algebra of the Golden Section . . . . . . . . . . . 112
4.9 When Algebra Is Not Helpful . . . . . . . . . . . . . . . 115
4.10 Rationalizing a Denominator . . . . . . . . . . . . . . . 116
4.11 Pythagorean Triples . . . . . . . . . . . . . . . . . . . . . 117
Chapter 5 Geometric Wonders . . . . . . . . . . . . . . . . . . . . . 123
5.1 Angle Sum of a Triangle . . . . . . . . . . . . . . . . . . 124
5.2 Pentagram Angles . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Some Mind-Bogglers on . . . . . . . . . . . . . . . . 131
5.4 The Ever-Present Parallelogram . . . . . . . . . . . . . . 133
5.5 Comparing Areas and Perimeters . . . . . . . . . . . . . 137
5.6 How Eratosthenes Measured the Earth . . . . . . . . . . 139
5.7 Surprising Rope Around the Earth . . . . . . . . . . . . 141
5.8 Lunes and Triangles . . . . . . . . . . . . . . . . . . . . 143
5.9 The Ever-Present Equilateral Triangle . . . . . . . . . . 146
5.10 Napoleon’s Theorem . . . . . . . . . . . . . . . . . . . . 149
5.11 The Golden Rectangle . . . . . . . . . . . . . . . . . . . 153
5.12 The Golden Section Constructed by Paper Folding . . . 158
5.13 The Regular Pentagon That Isn’t . . . . . . . . . . . . . 161
5.14 Pappus’s Invariant . . . . . . . . . . . . . . . . . . . . . . 163
5.15 Pascal’s Invariant . . . . . . . . . . . . . . . . . . . . . . 165
5.16 Brianchon’s Ingenius Extension of Pascal’s Idea . . . . 168
5.17 A Simple Proof of the Pythagorean Theorem . . . . . . 170
5.18 Folding the Pythagorean Theorem . . . . . . . . . . . . 172
5.19 President Garfield’s Contribution to Mathematics . . . . 174
5.20 What Is the Area of a Circle? . . . . . . . . . . . . . . . 176
5.21 A Unique Placement of Two Triangles . . . . . . . . . . 178
5.22 A Point of Invariant Distance
in an Equilateral Triangle . . . . . . . . . . . . . . . . 180
5.23 The Nine-Point Circle . . . . . . . . . . . . . . . . . . . 183
5.24 Simson’s Invariant . . . . . . . . . . . . . . . . . . . . . 187
5.25 Ceva’s Very Helpful Relationship . . . . . . . . . . . . . 189
5.26 An Obvious Concurrency? . . . . . . . . . . . . . . . . . 193
5.27 Euler’s Polyhedra . . . . . . . . . . . . . . . . . . . . . . 195
Chapter 6 Mathematical Paradoxes . . . . . . . . . . . . . . . . . . 198
6.1 Are All Numbers Equal? . . . . . . . . . . . . . . . . . . 199
6.2 −1 Is Not Equal to +1 . . . . . . . . . . . . . . . . . . . 200
6.3 Thou Shalt Not Divide by 0 . . . . . . . . . . . . . . . . 201
6.4 All Triangles Are Isosceles . . . . . . . . . . . . . . . . 202
6.5 An Infinite-Series Fallacy . . . . . . . . . . . . . . . . . 206
6.6 The Deceptive Border . . . . . . . . . . . . . . . . . . . 208
6.7 Puzzling Paradoxes . . . . . . . . . . . . . . . . . . . . . 210
6.8 A Trigonometric Fallacy . . . . . . . . . . . . . . . . . . 211
6.9 Limits with Understanding . . . . . . . . . . . . . . . . . 213
Chapter 7 Counting and Probability . . . . . . . . . . . . . . . . . . 215
7.1 Friday the 13th! . . . . . . . . . . . . . . . . . . . . . . . 216
7.2 Think Before Counting . . . . . . . . . . . . . . . . . . . 217
7.3 The Worthless Increase . . . . . . . . . . . . . . . . . . . 219
7.4 Birthday Matches . . . . . . . . . . . . . . . . . . . . . . 220
7.5 Calendar Peculiarities . . . . . . . . . . . . . . . . . . . . 223
7.6 The Monty Hall Problem . . . . . . . . . . . . . . . . . 224
7.7 Anticipating Heads and Tails . . . . . . . . . . . . . . . 228
Chapter 8 Mathematical Potpourri . . . . . . . . . . . . . . . . . . . 229
8.1 Perfection in Mathematics . . . . . . . . . . . . . . . . . 230
8.2 The Beautiful Magic Square . . . . . . . . . . . . . . . . 232
8.3 Unsolved Problems . . . . . . . . . . . . . . . . . . . . . 236
8.4 An Unexpected Result . . . . . . . . . . . . . . . . . . . 239
8.5 Mathematics in Nature . . . . . . . . . . . . . . . . . . . 241
8.6 The Hands of a Clock . . . . . . . . . . . . . . . . . . . 247
8.7 Where in the World Are You? . . . . . . . . . . . . . . . 251
8.8 Crossing the Bridges . . . . . . . . . . . . . . . . . . . . 253
8.9 The Most Misunderstood Average . . . . . . . . . . . . 256
8.10 The Pascal Triangle . . . . . . . . . . . . . . . . . . . . . 259
8.11 It’s All Relative . . . . . . . . . . . . . . . . . . . . . . . 263
8.12 Generalizations Require Proof . . . . . . . . . . . . . . . 264
8.13 A Beautiful Curve . . . . . . . . . . . . . . . . . . . . . 265