College Geometry: Using The Geometer's Sketchpad, Preliminary Edition
by Barbara E. Reynolds; William E. FentonEdition: CD
Format: Paperback
Pub. Date: 2008-06-01
Publisher(s): Wiley
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Summary
From two authors who embrace technology and value the role of collaborative learning comes College Geometry Using The Geometer's Sketchpad. The book's truly discovery-based approach guides readers to learn geometry through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries. In the process, readers hone their understanding of geometry and their ability to write rigorous mathematical proofs. Each copy of the book comes with a CD-ROM containing Sketchpad documents that relate directly to the material in the text. These multi-page documents help readers launch into the book's activities and provide dynamic, interactive versions of all figures in the text. Readers will need access to the Sketchpad(TM) program.
Table of Contents
| Preface | p. xiii |
| Our Motivation, Philosophy, and Pedagogy | p. xiii |
| Chapter Dependencies | p. xv |
| Supplements | p. xvi |
| Acknowledgments | p. xvi |
| To the Student | p. xix |
| Using The GeometerG+&s Sketchpad: Exploration and Conjecture | p. 1 |
| Discussion Part I: Getting Started with Sketchpad | p. 2 |
| Activities | p. 3 |
| Discussion Part II: Observation? Conjecture? Proof | p. 5 |
| Some Sketchpad Tips | p. 6 |
| Questions, Questions, Questions! | p. 7 |
| Language of Geometry | p. 8 |
| Euclid's Postulates | p. 12 |
| Congruence | p. 14 |
| Ideas About Betweenness | p. 15 |
| Constructions | p. 16 |
| Properties of Triangles | p. 18 |
| Properties of Quadrilaterals | p. 19 |
| Properties of Circles | p. 20 |
| Exploration and Conjecture: Inductive Reasoning | p. 20 |
| Exercises | p. 21 |
| Chapter Overview | p. 24 |
| Mathematical Arguments and Triangle Geometry | p. 29 |
| Activities | p. 30 |
| Discussion | p. 31 |
| Deductive Reasoning | p. 31 |
| Rules of Logic | p. 32 |
| Conditional Statements: Implication | p. 34 |
| Mathematical Arguments | p. 37 |
| Universal and Existential Quantifiers | p. 38 |
| Negating a Quantified Statement | p. 40 |
| Congruence Criteria for Triangles | p. 42 |
| Concurrence Properties for Triangles | p. 43 |
| Brief Excursion into Circle Geometry | p. 46 |
| The Circumcircle of ABC | p. 46 |
| The Nine-Point Circle: A First Pass | p. 47 |
| Ceva's Theorem and Its Converse | p. 47 |
| Menelaus' Theorem and Its Converse | p. 48 |
| Exercises | p. 49 |
| Chapter Overview | p. 53 |
| Circle Geometry, Robust Constructions, and Proofs | p. 57 |
| Activities | p. 58 |
| Discussion | p. 60 |
| Axiom Systems: Ancient and Modern Approaches | p. 60 |
| Robust Constructions: Developing a Visual Proof | p. 62 |
| Step-by-Step Proofs | p. 62 |
| Incircles and Excircles | p. 65 |
| The Pythagorean Theorem | p. 66 |
| Language of Circles | p. 67 |
| Some Interesting Families of Circles | p. 68 |
| Power of a Point | p. 70 |
| Inversion in a Circle | p. 71 |
| The Arbelos and the Salinon | p. 73 |
| The Nine-Point Circle: A Second Pass | p. 75 |
| Methods of Proof | p. 75 |
| Exercises | p. 78 |
| Chapter Overview | p. 82 |
| Analytic Geometry | p. 87 |
| Activities | p. 88 |
| Discussion | p. 90 |
| Points | p. 90 |
| Lines | p. 93 |
| Distance | p. 97 |
| Using Coordinates in Proofs | p. 100 |
| Polar Coordinates | p. 102 |
| The Nine-Point Circle, Revisited | p. 105 |
| Exercises | p. 110 |
| Chapter Overview | p. 113 |
| Taxicab Geometry | p. 117 |
| Activities | p. 118 |
| Discussion | p. 122 |
| An Axiom System for Metric Geometry | p. 123 |
| Circles | p. 125 |
| Ellipses | p. 126 |
| Measuring Distance from a Point to a Line | p. 127 |
| Parabolas | p. 128 |
| Hyperbolas | p. 130 |
| Axiom Systems | p. 130 |
| Exercises | p. 131 |
| Chapter Overview | p. 132 |
| Transformational Geometry | p. 135 |
| Activities | p. 136 |
| Discussion | p. 139 |
| Transformations | p. 139 |
| Isometries | p. 140 |
| Composition of Isometries | p. 144 |
| Inverse Isometries | p. 148 |
| Using Isometries in Proofs | p. 149 |
| Isometries in Space | p. 150 |
| Inversion in a Circle, Revisited | p. 151 |
| Exercises | p. 155 |
| Chapter Overview | p. 158 |
| Isometries and Matrices | p. 161 |
| Activities | p. 162 |
| Discussion | p. 164 |
| Using Vectors to Represent Translations | p. 164 |
| Using Matrices to Represent Rotations | p. 165 |
| Using Matrices to Represent Reflections | p. 166 |
| Composition of Isometries | p. 168 |
| The General Form of a Matrix Representation | p. 170 |
| Using Matrices in Proof | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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