**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**