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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 | |