Table Of Contents:

• 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