Rediscovering Mathematics: You Do The Math

Cover\nCopyright page\nTitle page\nAcknowledgements\nContents\nA Guide for the Reader\n	What is this Book About?\n	Is this Book for You?\n	Preliminaries\n	Why Bother?\n	How to Use this Book\nIntroduction: How to Read Mathematics\n	1 An Example of Mathematical Writing—The Birthday Paradox\n	2 Our Reader Tackles the Birthday Paradox\n	3 Challenges\nMathematical Discovery in the Classroom\n	1.1 A Simple Lesson\n	1.2 The Math Teacher as Conductor\n	1.3 The Effective Teacher\n	1.4 Challenges\nDon’t Reach for Your Calculator (Yet)\n	2.1 A Magic Trick\n	2.2 Clever Calculations\n	2.3 Pythagorean Triples\n	2.4 Challenges\nHave Another Piece of Pie, Zeno?\n	3.1 A Simpler Way\n	3.2 Euclid, Proofs, and Writing Mathematics\n	3.3 Challenges\nThinking Like a Mathematician—Lessons from a Medieval Rabbi\n	4.1 Rabbi Levi ben Gershon and his Sums\n	4.2 The Number of Squares and Rectangles in a Grid\n	4.3 The Triangle Puzzle—A Model Mathematical Problem\n	4.4 Challenges\nWhat is Mathematics Good For?\n	5.1 Knock Hockey\n	5.2 Basketball\n	5.3 Rates of Growth\n	5.4 Challenges\nThree Averages\n	6.1 Average I. The Arithmetic Average\n	6.2 Average II. The Harmonic Average\n	6.3 Average III. The Geometric Average\n	6.4 Exploring Averages\n	6.5 Challenges\nAlgorithms—The Unexpected Role of Pure Mathematics\n	7.1 A Solution to the Two Jug Puzzle\n	7.2 Euclid’s Algorithm\n	7.3 Other Methods for Calculating Greatest Common Divisors\n	7.4 The Efficiency (or Speed) of an Algorithm\n	7.5 The Egyptian Multiplication Algorithm\n	7.6 The Fast Modular Exponentiation Algorithm\n	7.7 Greatest Common Divisors, Algorithms, and E-Commerce\n	7.8 E-Commerce and Cryptography\n	7.9 The Rest of the Story—The RSA Algorithm\n	7.10 Challenges\nPythagoras’ Theorem and Math by Pictures\n	8.1 A Proof of Pythagoras’ Theorem\n	8.2 A Personal Experience\n	8.3 The Height of a Pyramid\n	8.4 Challenges\nMemorizing Versus Understanding\n	9.1 Example 1: FOIL\n	9.2 Example 2: Square Roots\n	9.3 Example 3: The Quadratic Equation\n	9.4 A ‘‘Real Life\" Quadratic Equation\n	9.5 Challenges\nGames and Gambling\n	10.1 A Carnival Game\n	10.2 Craps\n	10.3 The Careless Casino\n	10.4 Challenges\nSoccer Balls and Counting Tricks\n	11.1 Euler’s Formula\n	11.2 Platonic Solids\n	11.3 Counting\n	11.4 The Puzzle of the Soccer Ball\n	11.5 Challenges\nPizza Pi and Area\n	12.1 Pi in the Bible\n	12.2 The Area of a Circle\n	12.3 A Hard Puzzle Involving Areas\n	12.4 Challenges\nBack to the Classroom\n	13.1 A Classroom Experience\n	13.2 Emphasizing Understanding\n	13.3 What Next?\nResources for Rediscovering Mathematics\n	Organizations\n	Teaching-Focused Internet Sites\n	Content-Focused Internet Sites\nFurther Reading\nIndex\nAbout the Author