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College Geometry with GeoGebra
Barbara E. Reynolds (Author), William E. Fenton (Author)
9781119718116, Wiley
Paperback / softback, published 14 April 2021
368 pages
25.2 x 20.1 x 1.5 cm, 0.658 kg
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 Especially for Students Notes for Instructors Our Motivation, Philosophy, and Pedagogy Prerequisites and Chapter Dependencies Acknowledgments ONEUsing GeoGebra 1.1 Activities: Getting Started with GeoGebra 1.2 Discussion: Exploring and Conjecturing Some GeoGebra Tips Constructing −→ Exploring −→ Conjecturing: Inductive Reasoning Language of Geometry Explorations, Observations, Questions The Family of Quadrilaterals Angles Inscribed in Circles Rules of Logic 1.3 Exercises 1.4 Chapter Overview TWO Constructing → Proving 2.1 Activities 2.2 Discussion: Euclid’s Postulates and Constructions Euclid’s Postulates Congruence and Similarity Constructions Geometric Language Revisited Conditional Statements: Implication Using Robust Constructions to Develop a Proof Angles and Measuring Angles Constructing Perpendicular and Parallel Lines Properties of Triangles Euclid’s Parallel Postulate Euclid’s Constructions in the Elements Ideas About Betweenness 2.3 Exercises 2.4 Chapter Overview THREE Mathematical Arguments and Triangle Geometry 3.1 Activities 3.2 Discussion Deductive Reasoning Universal and Existential Quantifiers Negating a Quantified Statement Direct Proof and Disproof by Counterexample Step-by-Step Proofs Congruence Criteria for Triangles The Converse and the Contrapositive Concurrence Properties for Triangles Ceva’s Theorem and Its Converse Brief Excursion into Circle Geometry The Circumcircle of ΔABC The Nine-Point Circle: A First Pass Menelaus’ Theorem and Its Converse 3.3 Exercises 3.4 Chapter Overview FOUR Circle Geometry and Proofs 4.1 Activities 4.2 Discussion Axiom Systems: Ancient and Modern Approaches Language of Circles Inscribed Angles Mathematical Arguments Additional Methods of Proof Cyclic Quadrilaterals Incircles and Excircles Some Interesting Families of Circles The Arbelos and the Salinon Power of a Point The Radical Axis The Nine-Point Circle: A Second Pass 4.3 Exercises 4.4 Chapter Overview FIVE Analytic Geometry 5.1 Activities 5.2 Discussion Points Lines Distance Using Coordinates in Proofs Another Look at the Radical Axis Polar Coordinates The Nine-Point Circle, Revisited 5.3 Exercises 5.4 Chapter Overview SIX Taxicab Geometry 6.1 Activities 6.2 Discussion An Axiom System for Metric Geometry Circles Ellipses Measuring Distance from a Point to a Line Parabolas Hyperbolas Axiom Systems 6.3 Exercises 6.4 Chapter Overview SEVEN Finite Geometries 7.1 Activities 7.2 Discussion An Axiom System for an Affine Plane An Axiom System for a Projective Plane Duality Relating Affine Planes to Projective Planes Coordinates for Finite Geometries 7.3 Exercises 7.4 Chapter Overview EIGHTTransformational Geometry 8.1 Activities 8.2 Discussion Transformations Isometries Other Transformations Composition of Isometries Inverse Isometries Using Isometries in Proofs Isometries in Space 8.3 Exercises 8.4 Chapter Overview NINE Isometries and Matrices 9.1 Activities 9.2 Discussion Using Vectors to Represent Translations Using Matrices to Represent Rotations Using Matrices to Represent Reflections Composition of Isometries The General Form of a Matrix Representation Using Matrices in Proofs Similarity Transformations 9.3 Exercises 9.4 Chapter Overview TENSymmetry in the Plane 10.1 Activities 10.2 Discussion Symmetries Groups of Symmetries Classifying Figures by Their Symmetries Friezes and Symmetry Wallpaper Symmetry Tilings 10.3 Exercises 10.4 Chapter Overview ELEVEN Hyperbolic Geometry Part I: Exploring a New Universe 11.1 Activities Part I 11.2 Discussion Part I Hyperbolic Lines and Segments The Poincaré Disk Model of the Hyperbolic Plane Measuring Distance in the Poincaré Disk Model Hyperbolic Circles Hyperbolic Triangles Circumcircles and Incircles of Hyperbolic Triangles Congruence of Triangles in the Hyperbolic Plane Part II: The Parallel Postulate in Hyperbolic Geometry 11.3 Activities Part II 11.4 Discussion Part II The Hyperbolic and Elliptic Parallel Postulates The Angle of Parallelism The Exterior Angle Theorem Quadrilaterals in the Hyperbolic Plane Another Look at Triangles in the Hyperbolic Plane Area in the Hyperbolic Plane 11.5 Exercises The Upper-Half-Plane Model 11.6 Chapter Overview TWELVE Projective Geometry 12.1 Activities 12.2 Discussion An Axiom System Models for the Projective Plane Duality Coordinates for Projective Geometry Projective Transformations 12.3 Exercises 12.4 Chapter Overview APPENDIX A Trigonometry A.1 Activities A.2 Discussion Right Triangle Trigonometry Unit Circle Trigonometry Solving Trigonometric Equations Double Angle Formulas Angle Sum Formulas Half-Angle Formulas The Law of Sines and the Law of Cosines A.3 Exercises APPENDIX B Calculating with Matrices B.1 Activities B.2 Discussion Linear Combinations of Vectors Dot Product of Vectors Multiplying a Matrix Times a Vector Multiplying Two Matrices The Determinant of a Matrix B.3 Exercises BIBLIOGRAPHY INDEX
Subject Areas: Education [JN]
