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Copyright, ii
About the Authors, iii
Preface, vii
1. The Challenge of Teaching, 1
Today’s Students, Mathematics, and Society’s Need, 2
Crisis in Math Education, 2
The Teacher Shortage, 3
Goals and Challenges for Secondary Math Teachers Today, 4
The Standards, 6
Professional Development: A Teacher’s Resource, 9
Resources for Teachers, 10
A Mathematics Debate, 11
2. Long-Range and Short-Range Planning, 13
Planning for Instruction, 14
Long-Range Planning of the Curriculum, 14
Scope and Sequence, 14
Unit Plans, 15
Unit Planning Model, 15
Interdisciplinary Planning, 17
Planning From a Fully Developed Curriculum, 17
Sources to Assist in Long-Range Planning, 17
Short-Range Planning, 19
Rationale for Planning Model, 19
Daily Lesson Planning, 20
Lesson Plan Template, 20
Assigning Homework, 26
Format of the Homework Assignment, 31
Reviewing the Homework Assignment, 32
Checking Homework Assignments, 33
Another Sample Lesson Plan, 34
Spiraling Content, 37
Reflecting on Your Lesson, 43
Differentiated Instruction, 44
Tiered Lessons, 44
Additional Differentiation Strategies, 46
Cooperative Learning, 47
What Is Cooperative Learning?, 47
How to Structure Small Learning Groups, 48
The Teacher’s Role in Managing Small Group Learning, 50
Mathematical Tasks, 51
Final Thoughts on Lesson Planning, 55
3. Teaching More Effective Lessons, 61
Motivational Techniques, 62
What Is Motivation?, 62
Motivating Students: Ten Techniques, 62
Classroom Questioning, 72
Classroom Questioning Features to Develop, 72
Some Precautionary Considerations for Improving Classroom Questioning, 73
Ten Types of Questions to Avoid, 75
Classroom Questioning as a Means to Generate Higher Order Thinking, 78
Strategies for Teaching More Effective Lessons, 81
Using Tree Diagrams, or Branching, 81
Using Paper Folding or Cutting, 82
A Picture Is Worth a Thousand Words, 83
Recognizing Patterns, 85
Using Mathematical Models and Manipulatives, 86
Extending Familiar Concepts, 91
Using a Graphing Calculator, 96
Literacy in Mathematics, 97
Language Acquisition, 97
Comprehension, 98
Semantic Mapping, 99
Guided Reading, 99
Rich Mathematical Literacy Environment, 100
Writing, 100
Criteria for Evaluating Student Writing Samples, 101
Benefits of Writing Activities in the Mathematics Classroom, 102
4. The Role of Problem Solving, 105
A Psychological View of Problem Solving, 108
Problem-Solving Preliminaries, 109
An Introduction to Problem-Solving Strategies, 111
The Ten Problem-Solving Strategies, 117
Creating Mathematical Problems, 127
Creativity in Problem Solving, 127
5. Using Technology to Enhance Mathematics Instruction, 130
Calculators, 131
Appropriate Use of Calculators in the Mathematics Classroom, 131
Calculators as an Aid to Problem Solving, 132
Examples of Calculator Activities, 132
Solving Equations Using Matrices, 138
Calculus Applications, 140
Computers, 141
Appropriate Use of Computer Technology in the Mathematics Classroom, 141
Some Examples of Using the Computer to Enhance Instruction, 142
Introduction to Dynamic Geometry and The Geometer’s Sketchpad, 145
Using The Geometer’s Sketchpad, 146
Dynamic Data Analysis and Fathom, 156
Computer Algebra Systems, 156
Mathematical Equation Editors, 156
Multimedia, 157
Course Management Systems, 157
6. Assessment, 159
Assessment for Monitoring Student Progress, 160
Assessment for Making Instructional Decisions, 160
Data-Driven Instruction, 161
Using the Data to Inform Instruction, 161
Final Thoughts on Data-Driven Instruction, 162
Evaluating Student Achievement, 162
Using Performance Assessment Tasks, 163
Using Rubrics to Evaluate Student Work, 163
Constructing a Rubric, 176
Constructing a Class Test, 176
Administering a Test, 184
Grading a Test, 185
Interpreting Test Results, 186
Multiple-Choice Tests, 187
Determining the Course Grade, 187
Evaluating Programs, 187
7. Enriching Mathematics Instruction, 189
Enriching Mathematics Instruction with a Historical Approach, 190
Enrichment Techniques for All Levels, 197
The Gifted Student, 199
Using Calculators to Enrich Instruction, 202
Models and Manipulatives That Enrich Instruction, 204
8. Extracurricular Activities in Mathematics, 209
The Mathematics Club, 210
Mathematics Teams, 210
Mathematics Contests, 212
Mathematics Projects, 212
The Mathematics Fair, 214
Cooperation with a University, 214
The School Mathematics Magazine, 214
The Mathematics Assembly Program, 214
Guest Speakers Program, 216
Class Trips of Mathematical Significance, 216
Peer Teaching Program, 217
The Computer, 217
The Bulletin Board, 217
Enrichment Units for the Secondary School Classroom, 223
Cross-Catalogue of Enrichment Units, 224
1. Constructing Odd-Order Magic Squares, 231
2. Constructing Even-Order Magic Squares, 233
3. Introduction to Alphametics, 235
4. A Checkerboard Calculator, 237
5. The Game of Nim, 239
6. The Tower of Hanoi, 241
7. What Day of the Week Was It?, 243
8. Palindromic Numbers, 248
9. The Fascinating Number Nine, 250
10. Unusual Number Properties, 252
11. Enrichment with a Handheld Calculator, 254
12. Symmetric Multiplication, 256
13. Variations on a Theme—Multiplication, 258
14. Ancient Egyptian Arithmetic, 260
15. Napier’s Rods, 263
16. Unit Pricing, 264
17. Successive Discounts and Increases, 265
18. Prime and Composite Factors of a Whole Number, 267
19. Prime Numeration System, 269
20. Repeating Decimal Expansions, 271
21. Peculiarities of Perfect Repeating Decimals, 273
22. Patterns in Mathematics, 274
23. Googol and Googolplex, 276
24. Mathematics of Life Insurance, 278
25. Geometric Dissections, 280
26. The Klein Bottle, 282
27. The Four-Color Map Problem, 283
28. Mathematics on a Bicycle, 285
29. Mathematics and Music, 288
30. Mathematics in Nature, 290
31. The Birthday Problem, 292
32. The Structure of the Number System, 293
33. Excursions in Number Bases, 295
34. Raising Interest, 297
35. Reflexive, Symmetric, and Transitive Relations, 299
36. Bypassing an Inaccessible Region, 301
37. The Inaccessible Angle, 303
38. Triangle Constructions, 304
39. The Criterion of Constructibility, 306
40. Constructing Radical Lengths, 308
41. Constructing a Pentagon, 310
42. Investigating the Isosceles Triangle Fallacy, 311
43. The Equiangular Point, 313
44. The Minimum-Distance Point of a Triangle, 315
45. The Isosceles Triangle Revisited, 317
46. Reflective Properties of the Plane, 319
47. Finding the Length of a Cevian of a Triangle, 321
48. A Surprising Challenge, 323
49. Making Discoveries in Mathematics, 324
50. Tessellations, 325
51. Introducing the Pythagorean Theorem, 327
52. Trisection Revisited, 330
53. Proving Lines Concurrent, 332
54. Squares, 334
55. Proving Points Collinear, 335
56. Angle Measurement with a Circle, 337
57. Trisecting a Circle, 339
58. Ptolemy’s Theorem, 341
59. Constructing π, 343
60. The Arbelos, 345
61. The Nine-Point Circle, 347
62. The Euler Line, 348
63. The Simson Line, 350
64. The Butterfly Problem, 352
65. Equicircles, 354
66. The Inscribed Circle and the Right Triangle, 356
67. The Golden Rectangle, 358
68. The Golden Triangle, 360
69. Geometric Fallacies, 363
70. Regular Polyhedra, 365
71. An Introduction to Topology, 367
72. Angles on a Clock, 369
73. Averaging Rates—The Harmonic Mean, 371
74. Howlers, 373
75. Digit Problems Revisited, 375
76. Algebraic Identities, 376
77. A Method for Factoring Trinomials of the Form ax² + bx + c, 378
78. Solving Quadratic Equations, 380
79. The Euclidean Algorithm, 382
80. Prime Numbers, 384
81. Algebraic Fallacies, 386
82. Sum Derivations With Arrays, 388
83. Pythagorean Triples, 390
84. Divisibility, 392
85. Fibonacci Sequence, 394
86. Diophantine Equations, 396
87. Continued Fractions and Diophantine Equations, 398
88. Simplifying Expressions Involving Infinity, 400
89. Continued Fraction Expansion of Irrational Numbers, 402
90. The Farey Sequence, 405
91. The Parabolic Envelope, 407
92. Application of Congruence to Divisibility, 409
93. Problem Solving—A Reverse Strategy, 411
94. Decimals and Fractions in Other Bases, 414
95. Polygonal Numbers, 415
96. Networks, 418
97. Angle Trisection—Possible or Impossible?, 420
98. Comparing Means, 421
99. Pascal’s Pyramid, 423
100. The Multinomial Theorem, 425
101. Algebraic Solution of Cubic Equations, 427
102. Solving Cubic Equations, 429
103. Calculating Sums of Finite Series, 432
104. A General Formula for the Sum of Series of the Form, 434
105. A Parabolic Calculator, 437
106. Constructing Ellipses, 439
107. Constructing the Parabola, 441
108. Using Higher Plane Curves to Trisect an Angle, 443
109. Constructing Hypocycloid and Epicycloid Circular Envelopes, 445
110. The Harmonic Sequence, 447
111. Transformations and Matrices, 449
112. The Method of Differences, 452
113. Probability Applied to Baseball, 454
114. Introduction to Geometric Transformations, 455
115. The Circle and the Cardioid, 458
116. Complex-Number Applications, 460
117. Hindu Arithmetic, 462
118. Proving Numbers Irrational, 464
119. How to Use a Computer Spreadsheet to Generate Solutions to Certain Mathematics Problems, 466
120. The Three Worlds of Geometry, 467
121. πie Mix, 470
122. Graphical Iteration, 471
123. The Feigenbaum Plot, 474
124. The Sierpinski Triangle, 476
125. Fractals, 478
Appendix: Additional Exercises, 481
References, 485
Index, 507

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