College Geometry: Using The Geometer's Sketchpad, Preliminary Edition

by Barbara E. Reynolds; William E. Fenton

• ISBN13: 9780470412176
• ISBN10: 0470412178
• Edition: CD
• Format: Paperback
• Copyright: 2008-06-01
• Publisher: Wiley

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.