College Geometry : A Computer Applications Approach using the Geometer's Sketchpad, core Text

by Reynolds, Barbara E.; Fenton, William E.

• ISBN13: 9781931914543
• ISBN10: 1931914540
• Edition: CD
• Format: Paperback
• Copyright: 2005-09-01
• Publisher: Springer Verlag

Summary

College Geometry Using The Geometer's Sketchpad(R) is unique in its truly discovery based approach to learning geometry with complete integration of Sketchpad. Through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries, students hone their understanding of geometry and their ability to write rigorous mathematical proofs. It is an ideal text for students planning to teach at the middle or secondary level. Each chapter follows the same format to best promote active learning:- Activities open each chapter allowing students the opportunity to explore topics and make conjectures using Sketchpad- Discussions cover important theorems and concepts that solidify the conjectures made in the activities- Exercises reinforce the discussions and advance student expertise with Sketchpad- Each new copy of the book is packaged with a valuable CD-ROM with Sketchpad documents that relate directly to the material in the text- These multi-page documents provide a starting point for every activity and provide dynamic interactive versions of all the figures in the text

Table of Contents

Preface	xiii
Our Motivation, Philosophy, and Pedagogy	xiii
Chapter Dependencies	xv
Supplements	xvi
Acknowledgments	xvi
To the Student	xix
Using The Geometer's Sketchpad: Exploration and Conjecture	1	(28)
Discussion Part I: Getting Started with Sketchpad	2	(1)
Activities	3	(2)
Discussion Part II: Observation → Conjecture → Proof	5	(16)
Some Sketchpad Tips	6	(1)
Questions, Questions, Questions!	7	(1)
Language of Geometry	8	(4)
Euclid's Postulates	12	(2)
Congruence	14	(1)
Ideas About Betweenness	15	(1)
Constructions	16	(2)
Properties of Triangles	18	(1)
Properties of Quadrilaterals	19	(1)
Properties of Circles	20	(1)
Exploration and Conjecture: Inductive Reasoning	20	(1)
Exercises	21	(3)
Chapter Overview	24	(5)
Mathematical Arguments and Triangle Geometry	29	(28)
Activities	30	(1)
Discussion	31	(18)
Deductive Reasoning	31	(1)
Rules of Logic	32	(2)
Conditional Statements: Implication	34	(3)
Mathematical Arguments	37	(1)
Universal and Existential Quantifiers	38	(2)
Negating a Quantified Statement	40	(2)
Congruence Criteria for Triangles	42	(1)
Concurrence Properties for Triangles	43	(3)
Brief Excursion into Circle Geometry	46	(1)
The Circumcircle of ΔABC	46	(1)
The Nine-Point Circle: A First Pass	47	(1)
Ceva's Theorem and Its Converse	47	(1)
Menelaus' Theorem and Its Converse	48	(1)
Exercises	49	(4)
Chapter Overview	53	(4)
Circle Geometry, Robust Constructions, and Proofs	57	(30)
Activities	58	(2)
Discussion	60	(18)
Axiom Systems: Ancient and Modern Approaches	60	(2)
Robust Constructions: Developing a Visual Proof	62	(1)
Step-by-Step Proofs	62	(3)
Incircles and Excircles	65	(1)
The Pythagorean Theorem	66	(1)
Language of Circles	67	(1)
Some Interesting Families of Circles	68	(2)
Power of a Point	70	(1)
Inversion in a Circle	71	(2)
The Arbelos and the Salinon	73	(2)
The Nine-Point Circle: A Second Pass	75	(1)
Methods of Proof	75	(3)
Exercises	78	(4)
Chapter Overview	82	(5)
Analytic Geometry	87	(30)
Activities	88	(2)
Discussion	90	(20)
Points	90	(3)
Lines	93	(4)
Distance	97	(3)
Using Coordinates in Proofs	100	(2)
Polar Coordinates	102	(3)
The Nine-Point Circle, Revisited	105	(5)
Exercises	110	(3)
Chapter Overview	113	(4)
Taxicab Geometry	117	(18)
Activities	118	(4)
Discussion	122	(9)
An Axiom System for Metric Geometry	123	(2)
Circles	125	(1)
Ellipses	126	(1)
Measuring Distance from a Point to a Line	127	(1)
Parabolas	128	(2)
Hyperbolas	130	(1)
Axiom Systems	130	(1)
Exercises	131	(1)
Chapter Overview	132	(3)
Transformational Geometry	135	(26)
Activities	136	(3)
Discussion	139	(16)
Transformations	139	(1)
Isometries	140	(4)
Composition of Isometries	144	(4)
Inverse Isometries	148	(1)
Using Isometries in Proofs	149	(1)
Isometries in Space	150	(1)
Inversion in a Circle, Revisited	151	(4)
Exercises	155	(3)
Chapter Overview	158	(3)
Isometries and Matrices	161	(18)
Activities	162	(2)
Discussion	164	(11)
Using Vectors to Represent Translations	164	(1)
Using Matrices to Represent Rotations	165	(1)
Using Matrices to Represent Reflections	166	(2)
Composition of Isometries	168	(2)
The General Form of a Matrix Representation	170	(2)
Using Matrices in Proofs	172	(2)
Similarity Transformations	174	(1)
Exercises	175	(3)
Chapter Overview	178	(1)
Symmetry in the Plane	179	(24)
Activities	180	(3)
Discussion	183	(15)
Symmetries	183	(1)
Groups of Symmetries	184	(2)
Classifying Figures by Their Symmetries	186	(4)
Friezes and Symmetry	190	(3)
Wallpaper Symmetry	193	(1)
Tilings	194	(4)
Exercises	198	(2)
Chapter Overview	200	(3)
Hyperbolic Geometry	203	(26)
Part I: Exploring a New Universe	204	(1)
Activities	204	(3)
Discussion	207	(8)
Hyperbolic Lines and Segments	207	(1)
The Poincare Disk Model of the Hyperbolic Plane	207	(2)
Hyperbolic Triangles	209	(2)
Hyperbolic Circles	211	(1)
Measuring Distance in the Poincare Disk Model	211	(2)
Circumcircles and Incircles of Hyperbolic Triangles	213	(1)
Congruence of Triangles in the Hyperbolic Plane	214	(1)
Part II: The Parallel Postulate in the Poincare Disk	215	(1)
Activities	215	(2)
Discussion	217	(5)
The Hyperbolic and Elliptic Parallel Postulates	217	(3)
Parallel Lines in the Hyperbolic Plane	220	(1)
Quadrilaterals in the Hyperbolic Plane	221	(1)
Exercises	222	(3)
Chapter Overview	225	(4)
Projective Geometry	229	(32)
Activities	230	(2)
Discussion	232	(24)
An Axiom System	232	(3)
Models for the Projective Plane	235	(5)
Duality	240	(6)
Coordinates for Projective Geometry	246	(5)
Projective Transformations	251	(5)
Exercises	256	(3)
Chapter Overview	259	(2)
Appendix A Trigonometry	261	(12)
Activities	262	(2)
Discussion	264	(6)
Right Triangle Trigonometry	264	(1)
Unit Circle Trigonometry	265	(1)
Solving Trigonometric Equations	266	(1)
Double Angle Formulas	267	(1)
Angle Sum Formulas	267	(2)
Half-Angle Formulas	269	(1)
The Law of Sines and the Law of Cosines	269	(1)
Exercises	270	(3)
Appendix B Calculating with Matrices	273	(8)
Activities	274	(1)
Discussion	275	(5)
Linear Combinations of Vectors	275	(1)
Dot Product of Vectors	276	(1)
Multiplying a Matrix by a Vector	276	(1)
Multiplying Two Matrices	277	(1)
The Determinant of a Matrix	278	(2)
Exercises	280	(1)
Bibliography	281	(2)
Index	283