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E-raamat: College Geometry with GeoGebra

(Cardinal Stritch University), (Bellarmine University)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 01-Feb-2021
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119718147
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  • Formaat: PDF+DRM
  • Ilmumisaeg: 01-Feb-2021
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119718147
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From two authors who embrace technology in the classroom and value the role of collaborative learning comes College Geometry Using GeoGebra, a book that is ideal for geometry courses for both mathematics and math education majors. The book's discovery-based approach guides students to explore geometric worlds through computer-based activities, enabling students to make observations, develop conjectures, and write mathematical proofs. This unique textbook helps students understand the underlying concepts of geometry while learning to use GeoGebra software—constructing various geometric figures and investigating their properties, relationships, and interactions. The text allows students to gradually build upon their knowledge as they move from fundamental concepts of circle and triangle geometry to more advanced topics such as isometries and matrices, symmetry in the plane, and hyperbolic and projective geometry.

Emphasizing active collaborative learning, the text contains numerous fully-integrated computer lab activities that visualize difficult geometric concepts and facilitate both small-group and whole-class discussions. Each chapter begins with engaging activities that draw students into the subject matter, followed by detailed discussions that solidify the student conjectures made in the activities and exercises that test comprehension of the material. Written to support students and instructors in active-learning classrooms that incorporate computer technology, College Geometry with GeoGebra is an ideal resource for geometry courses for both mathematics and math education majors.

Preface xiii
Especially for Students xiii
Notes for Instructors xv
Our Motivation, Philosophy, and Pedagogy xv
Prerequisites and
Chapter Dependencies
xviii
Acknowledgments xix
One Using GeoGebra
1(21)
1.1 Activities: Getting Started with GeoGebra
2(7)
1.2 Discussion: Exploring and Conjecturing
9(8)
Some GeoGebra Tips
9(1)
Constructing → Exploring → Conjecturing: Inductive Reasoning
9(1)
Language of Geometry
10(1)
Explorations, Observations, Questions
11(2)
The Family of Quadrilaterals
13(2)
Angles Inscribed in Circles
15(1)
Rules of Logic
16(1)
1.3 Exercises
17(3)
1.4
Chapter Overview
20(2)
Two Constructing → Proving
22(31)
2.1 Activities
23(3)
2.2 Discussion: Euclid's Postulates and Constructions
26(18)
Euclid's Postulates
26(1)
Congruence and Similarity
27(3)
Constructions
30(2)
Geometric Language Revisited
32(1)
Conditional Statements: Implication
33(2)
Using Robust Constructions to Develop a Proof
35(1)
Angles and Measuring Angles
36(2)
Constructing Perpendicular and Parallel Lines
38(1)
Properties of Triangles
38(2)
Euclid's Parallel Postulate
40(1)
Euclid's Constructions in the Elements
41(2)
Ideas About Betweenness
43(1)
2.3 Exercises
44(4)
2.4
Chapter Overview
48(5)
Three Mathematical Arguments and Triangle Geometry
53(31)
3.1 Activities
54(2)
3.2 Discussion
56(19)
Deductive Reasoning
56(2)
Universal and Existential Quantifiers
58(1)
Negating a Quantified Statement
59(1)
Direct Proof and Disproof by Counterexample
60(2)
Step-By-Step Proofs
62(3)
Congruence Criteria for Triangles
65(2)
The Converse and the Contrapositive
67(2)
Concurrence Properties for Triangles
69(2)
Ceva's Theorem and its Converse
71(2)
Brief Excursion into Circle Geometry
73(1)
The Circumcircle of AABC
74(1)
The Nine-Point Circle: A First Pass
74(1)
Menelaus' Theorem and Its Converse
75(1)
3.3 Exercises
75(6)
3.4
Chapter Overview
81(3)
Four Circle Geometry and Proofs
84(26)
4.1 Activities
85(2)
4.2 Discussion
87(16)
Axiom Systems: Ancient and Modern Approaches
87(2)
Language of Circles
89(1)
Inscribed Angles
89(1)
Mathematical Arguments
90(1)
Additional Methods of Proof
91(2)
Cyclic Quadrilaterals
93(2)
Incircles and Excircles
95(1)
Some Interesting Families of Circles
96(1)
The Arbelos and the Salinon
97(2)
Power of a Point
99(1)
The Radical Axis
100(2)
The Nine-Point Circle: A Second Pass
102(1)
4.3 Exercises
103(4)
4.4
Chapter Overview
107(3)
Five Analytic Geometry
110(31)
5.1 Activities
111(3)
5.2 Discussion
114(22)
Points
114(3)
Lines
117(5)
Distance
122(3)
Using Coordinates in Proofs
125(2)
Another Look at the Radical Axis
127(1)
Polar Coordinates
128(3)
The Nine-Point Circle, Revisited
131(5)
5.3 Exercises
136(3)
5.4
Chapter Overview
139(2)
Six Taxicab Geometry
141(19)
6.1 Activities
142(5)
6.2 Discussion
147(8)
An Axiom System for Metric Geometry
147(3)
Circles
150(1)
Ellipses
151(1)
Measuring Distance from a Point to a Line
152(1)
Parabolas
153(1)
Hyperbolas
154(1)
Axiom Systems
155(1)
6.3 Exercises
155(3)
6.4
Chapter Overview
158(2)
Seven Finite Geometries
160(23)
7.1 Activities
161(2)
7.2 Discussion
163(17)
An Axiom System for an Affine Plane
163(4)
An Axiom System for a Projective Plane
167(3)
Duality
170(3)
Relating Affine Planes to Projective Planes
173(3)
Coordinates for Finite Geometries
176(4)
7.3 Exercises
180(2)
7.4
Chapter Overview
182(1)
Eight Transformational Geometry
183(23)
8.1 Activities
184(2)
8.2 Discussion
186(14)
Transformations
187(1)
Isometries
187(5)
Other Transformations
192(1)
Composition of Isometries
193(3)
Inverse Isometries
196(2)
Using Isometries in Proofs
198(1)
Isometries in Space
199(1)
8.3 Exercises
200(3)
8.4
Chapter Overview
203(3)
Nine Isometries and Matrices
206(18)
9.1 Activities
207(2)
9.2 Discussion
209(11)
Using Vectors to Represent Translations
209(1)
Using Matrices to Represent Rotations
210(1)
Using Matrices to Represent Reflections
211(2)
Composition of Isometries
213(2)
The General Form of a Matrix Representation
215(2)
Using Matrices in Proofs
217(2)
Similarity Transformations
219(1)
9.3 Exercises
220(3)
9.4
Chapter Overview
223(1)
Ten Symmetry in the Plane
224(24)
10.1 Activities
225(3)
10.2 Discussion
228(16)
Symmetries
229(1)
Groups of Symmetries
230(2)
Classifying Figures by Their Symmetries
232(4)
Friezes and Symmetry
236(3)
Wallpaper Symmetry
239(1)
Tilings
239(5)
10.3 Exercises
244(2)
10.4
Chapter Overview
246(2)
Eleven Hyperbolic Geometry
248(34)
Part I Exploring a New Universe
250(1)
11.1 Activities: Part I
250(3)
11.2 Discussion: Part I
253(10)
Hyperbolic Lines and Segments
253(1)
The Poincare Disk Model of the Hyperbolic Plane
254(2)
Measuring Distance in the Poincare Disk Model
256(2)
Hyperbolic Circles
258(1)
Hyperbolic Triangles
258(2)
Circumcircles and Incircles of Hyperbolic Triangles
260(1)
Congruence of Triangles in the Hyperbolic Plane
261(2)
Part II The Parallel Postulate in Hyperbolic Geometry
263(1)
11.3 Activities: Part II
263(2)
11.4 Discussion: Part II
265(10)
The Hyperbolic and Elliptic Parallel Postulates
265(2)
The Angle of Parallelism
267(2)
The Exterior Angle Theorem
269(2)
Quadrilaterals in the Hyperbolic Plane
271(2)
Another Look at Triangles in the Hyperbolic Plane
273(1)
Area in the Hyperbolic Plane
274(1)
11.5 Exercises
275(4)
11.6
Chapter Overview
279(3)
Twelve Projective Geometry
282(1)
12.1 Activities
283(2)
12.2 Discussion
285(1)
An Axiom System
285(1)
Models for the Projective Plane
286(5)
Duality
291(5)
Coordinates for Projective Geometry
296(6)
Projective Transformations
302(5)
12.3 Exercises
307(3)
12.4
Chapter Overview
310
APPENDIX A Trigonometry
1(1)
A.1 Activities
1(3)
A.2 Discussion
4(7)
Right Triangle Trigonometry
4(1)
Unit Circle Trigonometry
5(2)
Solving Trigonometric Equations
7(1)
Double-Angle Formulas
7(1)
Angle Sum Formulas
8(2)
Half-Angle Formulas
10(1)
The Law of Sines and the Law of Cosines
10(1)
A.3 Exercises
11(2)
APPENDIX B Calculating with Matrices
13(1)
B.1 Activities
13(2)
B.2 Discussion
15(4)
Linear Combinations of Vectors
15(1)
Dot Product of Vectors
16(1)
Multiplying a Matrix Times a Vector
16(1)
Multiplying Two Matrices
17(1)
The Determinant of a Matrix
18(1)
B.3 Exercises
19
Bibliography 1(1)
Index 1