Preface

Acknowledgments

About the Authors

Introductory Idea

Coming to Terms With Mathematical Terms

Algebra Ideas

1. Introducing the Product of Two Negatives

2. Multiplying Polynomials by Monomials (Introducing Algebra Tiles)

3. Multiplying Binomials (Using Algebra Tiles)

4. Factoring Trinomials (Using Algebra Tiles)

5. Multiplying Binomials (Geometrically)

6. Factoring Trinomials (Geometrically)

7. Trinomial Factoring

8. How Algebra Can Be Helpful

9. Automatic Factoring of a Trinomial

10. Reasoning Through Algebra

11. Pattern Recognition Cautions

12. Caution With Patterns

13. Using a Parabola as a Calculator

14. Introducing Literal Equations: Simple Algebra to Investigate an Arithmetic Phenomenon

15. Introducing Nonpositive Integer Exponents

16. Importance of Definitions in Mathematics (Algebra)

17. Introduction to Functions

18. When Algebra Explains Arithmetic

19. Sum of an Arithmetic Progression

20. Averaging Rates

21. Using Triangular Numbers to Generate Interesting Relationships

22. Introducing the Solution of Quadratic Equations Through Factoring

23. Rationalizing the Denominator

24. Paper Folding to Generate a Parabola

25. Paper Folding to Generate an Ellipse

26. Paper Folding to Generate a Hyperbola

27. Using Concentric Circles to Generate a Parabola

28. Using Concentric Circles to Generate an Ellipse

29. Using Concentric Circles to Generate a Hyperbola

30. Summing a Series of Powers

31. Sum of Limits

32. Linear Equations With Two Variables

33. Introducing Compound Interest Using the "Rule of 72”

34. Generating Pythagorean Triples

35. Finding Sums of Finite Series Geometry Ideas

Geometry Ideas

1. Sum of the Measures of the Angles of a Triangle

2. Introducing the Sum of the Measures of the Interior Angles of a Polygon

3. Sum of the Measures of the Exterior Angles of a Polygon: I

4. Sum of the Measures of the Exterior Angles of a Polygon: II

5. Triangle Inequality

6. Don’t Necessarily Trust Your Geometric Intuition

7. Importance of Definitions in Mathematics (Geometry)

8. Proving Quadrilaterals to Be Parallelograms

9. Demonstrating the Need to Consider All Information Given

10. Midlines of a Triangle

11. Length of the Median of a Trapezoid

12. Pythagorean Theorem

13. Simple Proofs of the Pythagorean Theorem

14. Angle Measurement With a Circle by Moving the Circle

15. Angle Measurement With a Circle

16. Introducing and Motivating the Measure of an Angle Formed by Two Chords

17. Using the Property of the Opposite Angles of an Inscribed Quadrilateral

18. Introducing the Concept of Slope

19. Introducing Concurrency Through Paper Folding

20. Introducing the Centroid of a Triangle

21. Introducing the Centroid of a Triangle Via a Property

22. Introducing Regular Polygons

23. Introducing Pi

24. The Lunes and the Triangle

25. The Area of a Circle

26. Comparing Areas of Similar Polygons

27. Relating Circles

28. Invariants in Geometry

29. Dynamic Geometry to Find an Optimum Situation

30. Construction-Restricted Circles

31. Avoiding Mistakes in Geometric Proofs

32. Systematic Order in Successive Geometric Moves: Patterns!

33. Introducing the Construction of a Regular Pentagon

34. Euclidean Constructions and the Parabola

35. Euclidean Constructions and the Ellipse

36. Euclidean Constructions and the Hyperbola

37. Constructing Tangents to a Parabola From an External Point P

38. Constructing Tangents to an Ellipse

39. Constructing Tangents to a Hyperbola

Trigonometry Ideas

1. Derivation of the Law of Sines: I

2. Derivation of the Law of Sines: II

3. Derivation of the Law of Sines: III

4. A Simple Derivation for the Sine of the Sum of Two Angles

5. Introductory Excursion to Enable an Alternate Approach to Trigonometry Relationships

6. Using Ptolemy’s Theorem to Develop Trigonometric Identities for Sums and Differences of Angles

7. Introducing the Law of Cosines: I (Using Ptolemy’s Theorem)

8. Introducing the Law of Cosines: II

9. Introducing the Law of Cosines: III

10. Alternate Approach to Introducing Trigonometric Identities

11. Converting to Sines and Cosines

12. Using the Double Angle Formula for the Sine Function

13. Making the Angle Sum Function Meaningful

14. Responding to the Angle-Trisection Question

Probability and Statistics Ideas

1. Introduction of a Sample Space

2. Using Sample Spaces to Solve Tricky Probability Problems

3. Introducing Probability Through Counting (or Probability as Relative Frequency)

4. In Probability You Cannot Always Rely on Your Intuition

5. When “Averages” Are Not Averages: Introducing Weighted Averages

6. The Monty Hall Problem: “Let’s Make a Deal”

7. Conditional Probability in Geometry

8. Introducing the Pascal Triangle

9. Comparing Means Algebraically

10. Comparing Means Geometrically

11. Gambling Can Be Deceptive

Other Topics Ideas

1. Asking the Right Questions

2. Making Arithmetic Means Meaningful

3. Using Place Value to Strengthen Reasoning Ability

4. Prime Numbers

5. Introducing the Concept of Relativity

6. Introduction to Number Theory

7. Extracting a Square Root

8. Introducing Indirect Proof

9. Keeping Differentiation Meaningful

10. Irrationality of the Square Root of an Integer That Is Not a Perfect Square

11. Introduction to the Factorial Function x!

12. Introduction to the Function x to the (n) Power

13. Introduction to the Two Binomial Theorems

14. Factorial Function Revisited

15. Extension of the Factorial Function r! to the Case Where r Is Rational

16. Prime Numbers Revisited

17. Perfect Numbers