Author(s): Shai Simmons
Publisher: The Mathematical Association of America
Year: 2011
Language: English
Pages: 240
cover
......Page 1
Copyright page
......Page 3
Title page
......Page 4
Acknowledgements......Page 8
Contents......Page 10
What is this Book About?......Page 14
Preliminaries......Page 15
How to Use this Book......Page 16
Introduction: How to Read Mathematics......Page 18
1 An Example of Mathematical Writing---The Birthday Paradox......Page 23
2 Our Reader Tackles the Birthday Paradox......Page 24
3 Challenges......Page 30
A Simple Lesson......Page 34
The Math Teacher as Conductor......Page 35
The Effective Teacher......Page 42
Challenges......Page 43
Don't Reach for Your Calculator (Yet)......Page 46
A Magic Trick......Page 51
Clever Calculations......Page 53
Pythagorean Triples......Page 55
Challenges......Page 59
Have Another Piece of Pie, Zeno?......Page 64
A Simpler Way......Page 67
Euclid, Proofs, and Writing Mathematics......Page 68
Challenges......Page 71
Thinking Like a Mathematician---Lessons from a Medieval Rabbi......Page 74
Rabbi Levi ben Gershon and his Sums......Page 77
The Number of Squares and Rectangles in a Grid......Page 85
The Triangle PuzzleA Model Mathematical Problem......Page 88
Challenges......Page 94
What is Mathematics Good For?......Page 96
Knock Hockey......Page 97
Basketball......Page 99
Rates of Growth......Page 102
Challenges......Page 105
Average I. The Arithmetic Average......Page 110
Average II. The Harmonic Average......Page 111
Average III. The Geometric Average......Page 112
Exploring Averages......Page 115
Challenges......Page 122
Algorithms---The Unexpected Role of Pure Mathematics......Page 126
A Solution to the Two Jug Puzzle......Page 127
Euclid's Algorithm......Page 129
The Efficiency (or Speed) of an Algorithm......Page 132
The Egyptian Multiplication Algorithm......Page 134
The Fast Modular Exponentiation Algorithm......Page 136
Greatest Common Divisors, Algorithms, and E-Commerce......Page 138
E-Commerce and Cryptography......Page 139
The Rest of the Story---The RSA Algorithm......Page 141
Challenges......Page 142
A Proof of Pythagoras' Theorem......Page 146
A Personal Experience......Page 148
The Height of a Pyramid......Page 153
Challenges......Page 157
Example 1: FOIL......Page 162
Example 2: Square Roots......Page 164
Example 3: The Quadratic Equation......Page 166
A ``Real Life" Quadratic Equation......Page 171
Challenges......Page 173
A Carnival Game......Page 176
Craps......Page 179
The Careless Casino......Page 184
Challenges......Page 185
Soccer Balls and Counting Tricks......Page 192
Euler's Formula......Page 193
Platonic Solids......Page 195
Counting......Page 196
The Puzzle of the Soccer Ball......Page 197
Challenges......Page 199
Pizza Pi and Area......Page 202
Pi in the Bible......Page 203
The Area of a Circle......Page 206
A Hard Puzzle Involving Areas......Page 208
Challenges......Page 212
Back to the Classroom......Page 218
A Classroom Experience......Page 219
Emphasizing Understanding......Page 221
What Next?......Page 226
Resources for Rediscovering Mathematics......Page 228
Further Reading......Page 232
Index......Page 234
About the Author......Page 240
