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