Math Wonders To Inspire Teachers And Students Posamentier

Are you "proud" to admit that you never liked math? Were never good in math? Are you struggling to pique your students' interest in math? Are you bored by the routine, mechanical aspects of teaching to the test in mathematics? This book offers a plethora of ideas to enrich your instruction and helps you to explore the intrinsic beauty of math. Through dozens of examples from arithmetic, algebra, geometry, and probability, Alfred S. Posamentier reveals the amazing symmetries, patterns, processes, paradoxes, and surprises that await students and teachers who look beyond the rote to discover wonders that have fascinated generations of great thinkers. Using the guided examples, help students explore the many marvels of math, including The Amazing Number 1,089 . Follow the instructions to reverse threedigit numbers, subtract them, and continue until everyone winds up with . . . 1,089! The Pigeonhole Principle . All students know that guesstimating works sometimes, but now they can use this strategy to solve problems. The Beautiful Magic Square . Challenge students to create their own magic squares and then discover the properties of Dürer's Magic Square. The author presents examples to entice students (and teachers) to study mathematicsto make mathematics a popular subject, not one to dread or avoid. TeX output 2003.03.17:2346 Cover Page 1 Title Page 2 ISBN 0871207753 3 CONTENTS 6 Foreword 10 Preface 13 1 The Beauty in Numbers 18 1.1 Surprising Number Patterns I 19 1.2 Surprising Number Patterns II 22 1.3 Surprising Number Patterns III 23 1.4 Surprising Number Patterns IV 24 1.5 Surprising Number Patterns V 26 1.6 Surprising Number Patterns VI 27 1.7 Amazing Power Relationships 27 1.8 Beautiful Number Relationships 29 1.9 Unusual Number Relationships 30 1.10 Strange Equalities 31 1.11 The Amazing Number 1,089 32 1.12 The Irrepressible Number 1 37 1.13 Perfect Numbers 39 1.14 Friendly Numbers 41 1.15 Another Friendly Pair of Numbers 43 1.16 Palindromic Numbers 43 1.17 Fun with Figurate Numbers 46 1.18 The Fabulous Fibonacci Numbers 49 1.19 Getting into an Endless Loop 52 1.20 A Power Loop 53 1.21 A Factorial Loop 56 1.22 The Irrationality of √2 58 1.23 Sums of Consecutive Integers 61 2 Some Arithmetic Marvels 64 2.1 Multiplying by 11 65 2.2 When Is a Number Divisible by 11? 66 2.3 When Is a Number Divisible by 3 or 9? 68 2.4 Divisibility by Prime Numbers 69 2.5 The Russian Peasant’s Method of Multiplication 74 2.6 Speed Multiplying by 21, 31, 41 76 2.7 Clever Addition 77 2.8 Alphametics 78 2.9 Howlers 81 2.10 The Unusual Number 9 86 2.11 Successive Percentages 89 2.12 Are Averages Averages? 91 2.13 The Rule of 72 92 2.14 Extracting a Square Root 94 3 Problems with Surprising Solutions 96 3.1 Thoughtful Reasoning 97 3.2 Surprising Solution 98 3.3 A Juicy Problem 99 3.4 Working Backward 101 3.5 Logical Thinking 102 3.6 It’s Just How You Organize the Data 103 3.7 Focusing on the Right Information 105 3.8 The Pigeonhole Principle 106 3.9 The Flight of the Bumblebee 107 3.10 Relating Concentric Circles 109 3.11 Don’t Overlook the Obvious 110 3.12 Deceptively Difficult (Easy) 112 3.13 The Worst Case Scenario 114 4 Algebraic Entertainments 115 4.1 Using Algebra to Establish Arithmetic Shortcuts 116 4.2 The Mysterious Number 22 117 4.3Justifying an Oddity 118 4.4 Using Algebra for Number Theory 120 4.5 Finding Patterns Among Figurate Numbers 121 4.6 Using a Pattern to Find the Sum of a Series 125 4.7 Geometric View of Algebra 126 4.8 Some Algebra of the Golden Section 129 4.9 When Algebra Is Not Helpful 132 4.10 Rationalizing a Denominator 133 4.11 Pythagorean Triples 134 5 Geometric Wonders 140 5.1 Angle Sum of a Triangle 141 5.2 Pentagram Angles 143 5.3 Some Mind-Bogglers on 148 5.4 The Ever-Present Parallelogram 150 5.5 Comparing Areas and Perimeters 154 5.6 How Eratosthenes Measured the Earth 156 5.7 Surprising Rope Around the Earth 158 5.8 Lunes and Triangles 160 5.9 The Ever-Present Equilateral Triangle 163 5.10 Napoleon’s Theorem 166 5.11 The Golden Rectangle 170 5.12 The Golden Section Constructed by Paper Folding 175 5.13 The Regular Pentagon That Isn’t 178 5.14 Pappus’s Invariant 180 5.15 Pascal’s Invariant 182 5.16 Brianchon’s Ingenius Extension of Pascal’s Idea 185 5.17 A Simple Proof of the Pythagorean Theorem 187 5.18 Folding the Pythagorean Theorem 189 5.19 President Garfield’s Contribution to Mathematics 191 5.20 What Is the Area of a Circle? 193 5.21 A Unique Placement of Two Triangles 195 5.22 A Point of Invariant Distance in an Equilateral Triangle 197 5.23 The Nine-Point Circle 200 5.24 Simson’s Invariant 204 5.25 Ceva’s Very Helpful Relationship 206 5.26 An Obvious Concurrency? 210 5.27 Euler’s Polyhedra 212 6 Mathematical Paradoxes 215 6.1 Are All Numbers Equal? 216 6.2 −1 Is Not Equal to +1 217 6.3 Thou Shalt Not Divide by 0 218 6.4 All Triangles Are Isosceles 219 6.5 An Infinite-Series Fallacy 223 6.6 The Deceptive Border 225 6.7 Puzzling Paradoxes 227 6.8 A Trigonometric Fallacy 228 6.9 Limits with Understanding 230 7 Counting and Probability 232 7.1 Friday the 13th! 233 7.2 ThinkBefor e Counting 234 7.3 The Worthless Increase 236 7.4 Birthday Matches 237 7.5 Calendar Peculiarities 240 7.6 The Monty Hall Problem 241 7.7 Anticipating Heads and Tails 245 8 Mathematical Potpourri 246 8.1 Perfection in Mathematics 247 8.2 The Beautiful Magic Square 249 8.3 Unsolved Problems 253 8.4 An Unexpected Result 256 8.5 Mathematics in Nature 258 8.6 The Hands of a Clock 264 8.7 Where in the World Are You? 268 8.8 Crossing the Bridges 270 8.9 The Most Misunderstood Average 273 8.10 The Pascal Triangle 276 8.11 It’s All Relative 280 8.12 Generalizations Require Proof 281 8.13 A Beautiful Curve∗ 282 Epilogue 285 Acknowledgments 288 Index 289 About the Author 293 If you like this book, you’ll LOVE the membership! 295

Are you "proud" to admit that you never liked math? Were never good in math? Are you struggling to pique your students' interest in math? Are you bored by the routine, mechanical aspects of teaching to the test in mathematics?

This book offers a plethora of ideas to enrich your instruction and helps you to explore the intrinsic beauty of math. Through dozens of examples from arithmetic, algebra, geometry, and probability, Alfred S. Posamentier reveals the amazing symmetries, patterns, processes, paradoxes, and surprises that await students and teachers who look beyond the rote to discover wonders that have fascinated generations of great thinkers. Using the guided examples, help students explore the many marvels of math, including

* The Amazing Number 1, 089. Follow the instructions to reverse three-digit numbers, subtract them, and continue until everyone winds up with... 1, 089!
* The Pigeonhole Principle. All students know that guesstimating works sometimes, but now they can use this strategy to solve problems.
* The Beautiful Magic Square. Challenge students to create their own magic squares and then discover the properties of Dürer's Magic Square.

The author presents examples to entice students (and teachers) to study mathematics--to make mathematics a popular subject, not one to dread or avoid.

Note: This product listing is for the Adobe Acrobat (PDF) version of the book.

SNAP Bronze Award Winner! Engaging students in math lessons and encouraging them to explore math principles is much easier-and lots more fun-when you use this book's amazing collection of quirks, illustrations, problems, and surprises. Dazzle your students with Peculiar number patterns and properties Clever shortcuts and tricks for avoiding cumbersome mathematic processes Curious problems that reveal problem-solving strategies Algebra problems that shed light on patterns in mathematics and the behavior of numbers Activities and demonstrations that uncover timeless geometric principles Used as attention-getters, motivators, or enrichment activities, these math wonders leave students questioning and exploring further, regardless of their grade level or ability. This book offers ideas to enrich instruction and help teachers explore the intrinsic beauty of math. Through dozens of examples from arithmetic, algebra, geometry, and probability, the symmetries, patterns, processes, paradoxes, and surprises that have facilitated generations of great thinkers are revealed. Activities include: (1) The Beauty in Numbers; (2) Some Arithmetic Marvels; (3) Problems with Surprising Solutions; (4) Algebraic Entertainments; (5) Geometric Wonders; (6) Mathematical Paradoxes; (7) Counting and Probability; and (8) Mathematical Potpourri. (KHR) A veteran teacher provides examples of the beauty and wonder of numbers and mathematics that are guaranteed to get students--and teachers--more interested in math. A veteran teacher provides examples of the beauty and wonder of numbers and mathematics that are guaranteed to get student sand teachers more interested in math