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