
The Challenge of Teaching 


1  (13) 

Today's Students, Mathematics, and Society's Need 


2  (12) 


2  (1) 


3  (1) 

Goals and Challenges for Secondary Math Teachers Today 


4  (3) 

The Standards: Twelve Years of Growth 


7  (3) 

Professional Development: A Teacher's Resource 


10  (2) 


12  (1) 


12  (2) 

LongRange and ShortRange Planning 


14  (49) 

LongRange Planning of the Curriculum 


15  (1) 


15  (1) 


16  (5) 


16  (2) 

Interdisciplinary Planning 


18  (1) 

Planning From a Fully Developed Curriculum 


18  (1) 

Sources to Assist in LongRange Planning 


18  (3) 


21  (26) 

Rationale for Planning Model 


21  (1) 


21  (1) 


21  (6) 


27  (6) 

Format of the Homework Assignment 


33  (1) 

Reviewing the Homework Assignment 


34  (2) 

Checking Homework Assignments 


36  (1) 

Another Sample Lesson Plan 


37  (3) 


40  (6) 

Reflecting on Your Lesson 


46  (1) 

Differentiated Instruction 


47  (1) 


48  (4) 

What Is Cooperative Learning? 


48  (1) 

How to Structure Small Learning Groups 


48  (3) 

The Teacher's Role in Managing Small Group Learning 


51  (1) 


52  (2) 

Final Thoughts on Lesson Planning 


54  (9) 

Teaching More Effective Lessons 


63  (46) 


64  (10) 


64  (1) 

Motivating Students: Ten Techniques 


64  (10) 


74  (10) 

Classroom Questioning Features to Develop 


74  (2) 

Some Precautionary Considerations for Improving Classroom Questioning 


76  (2) 

Ten Types of Questions to Avoid 


78  (3) 

Classroom Questioning as a Means to Generate Higher Order Thinking 


81  (3) 

Strategies for Teaching More Effective Lessons 


84  (17) 

Using Tree Diagrams, or Branching 


84  (1) 

Using Paper Folding or Cutting 


85  (1) 

A Picture Is Worth a Thousand Words 


86  (2) 


88  (2) 

Using Mathematical Models and Manipulatives 


90  (5) 

Extending Familiar Concepts 


95  (5) 

Using a Graphing Calculator 


100  (1) 


101  (3) 


101  (1) 


102  (1) 


103  (1) 


103  (1) 

Rich Mathematical Literacy Environment 


104  (1) 


104  (5) 

Criteria for Evaluating Student Writing Samples 


106  (1) 

Benefits of Writing Activities in the Mathematics Classroom 


107  (2) 

The Role of Problem Solving 


109  (26) 

A Psychological View of Problem Solving 


112  (1) 

ProblemSolving Preliminaries 


113  (2) 

An Introduction to ProblemSolving Strategies 


115  (6) 

The Ten ProblemSolving Strategies 


121  (11) 

Creating Mathematical Problems 


132  (1) 

Creativity in Problem Solving 


132  (3) 

Using Technology to Enhance Mathematics Instruction 


135  (31) 


136  (10) 

Appropriate Use of Calculators in the Mathematics Classroom 


136  (1) 

Calculators as an Aid to Problem Solving 


137  (1) 

Examples of Calculator Activities 


137  (6) 

Solving Equations Using Matrices 


143  (2) 


145  (1) 


146  (20) 

Appropriate Use of Computer Technology in the Mathematics Classroom 


146  (1) 

Some Examples of Using the Computer to Enhance Instruction 


147  (4) 

Introduction to Dynamic Geometry and The Geometer's Sketchpad 


151  (1) 

Using The Geometer's Sketchpad 


152  (11) 

Dynamic Data Analysis and Fathom 


163  (1) 


163  (1) 

Mathematical Equation Editors 


163  (1) 


163  (1) 

Course Management Systems 


163  (3) 


166  (31) 

Assessment for Monitoring Student Progress 


167  (1) 

Assessment for Making Instructional Decisions 


167  (1) 

Evaluating Student Achievement 


168  (29) 

Using Performance Assessment Tasks 


169  (1) 

Using Rubrics to Evaluate Student Work 


169  (13) 


182  (1) 

Constructing a Class Test 


182  (8) 


190  (1) 


191  (1) 

Interpreting Test Results 


192  (1) 


193  (1) 

Determining the Course Grade 


193  (2) 

Final Thoughts on Evaluating Student Achievement 


195  (2) 

Enriching Mathematics Instruction 


197  (20) 

Enriching Mathematics Instruction with a Historical Approach 


198  (8) 

Enrichment Techniques for All Levels 


206  (1) 


207  (3) 

Using Calculators to Enrich Instruction 


210  (2) 

Models and Manipulatives That Enrich Instruction 


212  (5) 

Extracurricular Activities in Mathematics 


217  (14) 


218  (1) 


218  (2) 


220  (1) 


220  (2) 


222  (1) 

Cooperation with a University 


222  (1) 

The School Mathematics Magazine 


222  (2) 

The Mathematics Assembly Program 


224  (1) 


224  (1) 

Class Trips of Mathematical Significance 


224  (1) 


225  (1) 


225  (1) 


226  (5) 

ENRICHMENT UNITS FOR THE SECONDARY SCHOOL CLASSROOM 


231  (260) 

CrossCatalogue of Enrichment Units 


232  (6) 

Constructing OddOrder Magic Squares 


238  (2) 

Constructing EvenOrder Magic Squares 


240  (2) 

Introduction to Alphametics 


242  (2) 

A Checkerboard Calculator 


244  (2) 


246  (2) 


248  (2) 

What Day of the Week Was It? 


250  (5) 


255  (2) 

The Fascinating Number Nine 


257  (2) 

Unusual Number Properties 


259  (2) 

Enrichment with a Handheld Calculator 


261  (2) 


263  (2) 

Variations on a ThemeMultiplication 


265  (2) 

Ancient Egyptian Arithmetic 


267  (2) 


269  (1) 


270  (1) 

Successive Discounts and Increases 


271  (2) 

Prime and Composite Factors of a Whole Number 


273  (2) 


275  (2) 

Repeating Decimal Expansions 


277  (2) 

Peculiarities of Perfect Repeating Decimals 


279  (1) 


280  (2) 


282  (2) 

Mathematics of Life Insurance 


284  (1) 


285  (2) 


287  (2) 

The FourColor Map Problem 


289  (2) 


291  (2) 


293  (2) 


295  (2) 


297  (2) 

The Structure of the Number System 


299  (2) 

Excursions in Number Bases 


301  (2) 


303  (2) 

Reflexive, Symmetric, and Transitive Relations 


305  (2) 

Bypassing an Inaccessible Region 


307  (2) 


309  (1) 


310  (2) 

The Criterion of Constructibility 


312  (2) 

Constructing Radical Lengths 


314  (2) 


316  (1) 

Investigating the Isosceles Triangle Fallacy 


317  (2) 


319  (2) 

The MinimumDistance Point of a Triangle 


321  (2) 

The Isosceles Triangle Revisited 


323  (2) 

Reflective Properties of the Plane 


325  (2) 

Finding the Length of a Cevian of a Triangle 


327  (2) 


329  (1) 

Making Discoveries in Mathematics 


330  (1) 


331  (3) 

Introducing the Pythagorean Theorem 


334  (3) 


337  (2) 


339  (2) 


341  (1) 


342  (2) 

Angle Measurement with a Circle 


344  (2) 


346  (2) 


348  (2) 


350  (2) 


352  (2) 


354  (1) 


355  (2) 


357  (2) 


359  (2) 


361  (2) 

The Inscribed Circle and the Right Triangle 


363  (2) 


365  (3) 


368  (2) 


370  (2) 


372  (2) 

An Introduction to Topology 


374  (2) 


376  (2) 

Averaging RatesThe Harmonic Mean 


378  (2) 


380  (2) 


382  (1) 


383  (2) 

A Method for Factoring Trinomials of the Form: ax2 + bx + c 


385  (2) 

Solving Quadratic Equations 


387  (2) 


389  (2) 


391  (2) 


393  (2) 

Sum Derivations With Arrays 


395  (2) 


397  (3) 


400  (2) 


402  (2) 


404  (2) 

Continued Fractions and Diophantine Equations 


406  (2) 

Simplifying Expressions Involving Infinity 


408  (2) 

Continued Fraction Expansion of Irrational Numbers 


410  (3) 


413  (1) 


414  (3) 

Application of Congruence to Divisibility 


417  (2) 

Problem SolvingA Reverse Strategy 


419  (3) 

Decimals and Fractions in Other Bases 


422  (1) 


423  (3) 


426  (2) 

Angle TrisectionPossible or Impossible? 


428  (2) 


430  (2) 


432  (2) 


434  (2) 

Algebraic Solution of Cubic Equations 


436  (2) 


438  (3) 

Calculating Sums of Finite Series 


441  (3) 

A General Formula for the Sum of Series of the Form Σ 


444  (3) 


447  (2) 


449  (2) 

Constructing the Parabola 


451  (2) 

Using Higher Plane Curves to Trisect an Angle 


453  (2) 

Constructing Hypocycloid and Epicycloid Circular Envelopes 


455  (2) 


457  (2) 

Transformations and Matrices 


459  (3) 

The Method of Differences 


462  (2) 

Probability Applied to Baseball 


464  (1) 

Introduction to Geometric Transformations 


465  (3) 

The Circle and the Cardioid 


468  (2) 

ComplexNumber Applications 


470  (2) 


472  (2) 

Proving Numbers Irrational 


474  (2) 

How to Use a Computer Spreadsheet to Generate Solutions to Certain Mathematics Problems 


476  (1) 

The Three Worlds of Geometry 


477  (3) 


480  (1) 


481  (3) 


484  (2) 


486  (2) 


488  (3) 
Appendix Additional Exercises 

491  (5) 
References 

496  (20) 
Index 

516  (4) 
About the Authors 

520  