| Preface | |
| Acknowledgments | |
| About the Authors | |
| Introductory Idea | |
| Coming to Terms With Mathematical Terms | |
| Algebra Ideas | |
| Introducing the Product of Two Negatives | |
| Multiplying Polynomials by Monomials (Introducing Algebra Tiles) | |
| Multiplying Binomials (Using Algebra Tiles) | |
| Factoring Trinomials (Using Algebra Tiles) | |
| Multiplying Binomials (Geometrically) | |
| Factoring Trinomials (Geometrically) | |
| Trinomial Factoring | |
| How Algebra Can Be Helpful | |
| Automatic Factoring of a Trinomial | |
| Reasoning Through Algebra | |
| Pattern Recognition Cautions | |
| Caution With Patterns | |
| Using a Parabola as a Calculator | |
| Introducing Literal Equations: Simple Algebra to Investigate an Arithmetic Phenomenon | |
| Introducing Nonpositive Integer Exponents | |
| Importance of Definitions in Mathematics (Algebra) | |
| Introduction to Functions | |
| When Algebra Explains Arithmetic | |
| Sum of an Arithmetic Progression | |
| Averaging Rates | |
| Using Triangular Numbers to Generate Interesting Relationships | |
| Introducing the Solution of Quadratic Equations Through Factoring | |
| Rationalizing the Denominator | |
| Paper Folding to Generate a Parabola | |
| Paper Folding to Generate an Ellipse | |
| Paper Folding to Generate a Hyperbola | |
| Using Concentric Circles to Generate a Parabola | |
| Using Concentric Circles to Generate an Ellipse | |
| Using Concentric Circles to Generate a Hyperbola | |
| Summing a Series of Powers | |
| Sum of Limits | |
| Linear Equations With Two Variables | |
| Introducing Compound Interest Using the "Rule of 72" | |
| Generating Pythagorean Triples | |
| Finding Sums of Finite Series Geometry Ideas | |
| Geometry Ideas | |
| Sum of the Measures of the Angles of a Triangle | |
| Introducing the Sum of the Measures of the Interior Angles of a Polygon | |
| Sum of the Measures of the Exterior Angles of a Polygon: I | |
| Sum of the Measures of the Exterior Angles of a Polygon: II | |
| Triangle Inequality | |
| Don't Necessarily Trust Your Geometric Intuition | |
| Importance of Definitions in Mathematics (Geometry) | |
| Proving Quadrilaterals to Be Parallelograms | |
| Demonstrating the Need to Consider All Information Given | |
| Midlines of a Triangle | |
| Length of the Median of a Trapezoid | |
| Pythagorean Theorem | |
| Simple Proofs of the Pythagorean Theorem | |
| Angle Measurement With a Circle by Moving the Circle | |
| Angle Measurement With a Circle | |
| Introducing and Motivating the Measure of an Angle Formed by Two Chords | |
| Using the Property of the Opposite Angles of an Inscribed Quadrilateral | |
| Introducing the Concept of Slope | |
| Introducing Concurrency Through Paper Folding | |
| Introducing the Centroid of a Triangle | |
| Introducing the Centroid of a Triangle Via a Property | |
| Introducing Regular Polygons | |
| Introducing Pi | |
| The Lunes and the Triangle | |
| The Area of a Circle | |
| Comparing Areas of Similar Polygons | |
| Relating Circles | |
| Invariants in Geometry | |
| Dynamic Geometry to Find an Optimum Situation | |
| Construction-Restricted Circles | |
| Avoiding Mistakes in Geometric Proofs | |
| Systematic Order in Successive Geometric Moves: Patterns! | |
| Introducing the Construction of a Regular Pentagon | |
| Euclidean Constructions and the Parabola | |
| Euclidean Constructions and the Ellipse | |
| Euclidean Constructions and the Hyperbola | |
| Constructing Tangents to a Parabola From an External Point P | |
| Constructing Tangents to an Ellipse | |
| Constructing Tangents to a Hyperbola | |
| Trigonometry Ideas | |
| Derivation of the Law of Sines: I | |
| Derivation of the Law of Sines: II | |
| Derivation of the Law of Sines: III | |
| A Simple Derivation for the Sine of the Sum of Two Angles | |
| Introductory Excursion to Enable an Alternate Approach to Trigonometry Relationships | |
| Using Ptolemy's Theorem to Develop Trigonometric Identities for Sums and Differences of Angles | |
| Introducing the Law of Cosines: I (Using Ptolemy's Theorem) | |
| Introducing the Law of Cosines: II | |
| Introducing the Law of Cosines: III | |
| Alternate Approach to Introducing Trigonometric Identities | |
| Converting to Sines and Cosines | |
| Using the Double Angle Formula for the Sine Function | |
| Making the Angle Sum Function Meaningful | |
| Responding to the Angle-Trisection Question | |
| Probability and Statistics Ideas | |
| Introduction of a Sample Space | |
| Using Sample Spaces to Solve Tricky Probability Problems | |
| Introducing Probability Through Counting (or Probability as Relative Frequency) | |
| In Probability You Cannot Always Rely on Your Intuition | |
| When "Averages" Are Not Averages: Introducing Weighted Averages | |
| The Monty Hall Problem: "Let's Make a Deal" | |
| Conditional Probability in Geometry | |
| Introducing the Pascal Triangle | |
| Comparing Means Algebraically | |
| Comparing Means Geometrically | |
| Gambling Can Be Deceptive | |
| Other Topics Ideas | |
| Asking the Right Questions | |
| Making Arithmetic Means Meaningful | |
| Using Place Value to Strengthen Reasoning Ability | |
| Prime Numbers | |
| Introducing the Concept of Relativity | |
| Introduction to Number Theory | |
| Extracting a Square Root | |
| Introducing Indirect Proof | |
| Keeping Differentiation Meaningful | |
| Irrationality of the Square Root of an Integer That Is Not a Perfect Square | |
| Introduction to the Factorial Function x! | |
| Introduction to the Function x to the (n) Power | |
| Introduction to the Two Binomial Theorems | |
| Factorial Function Revisited | |
| Extension of the Factorial Function r! to the Case Where r Is Rational | |
| Prime Numbers Revisited | |
| Perfect Numbers | |
| Table of Contents provided by Ingram. All Rights Reserved. |