Foundations in Geometry
Revised: September 2020
Course Description
Axiomatic approach to the study and development of Euclidean and non-Euclidean geometry.
Prerequisite: Junior standing or permission of Department Head. Three semester hours
Objectives:
At the conclusion of this course, the successful student will:
- Demonstrate knowledge of the synthetic, investigative, and deductive approaches to
Euclidean plane geometry.
- Demonstrate knowledge of the historical and mathematical significance of the parallel
postulate, including an introduction to a non-Euclidean geometry.
- Show proficiency in the basic concepts of mathematical logic and their use in proofs.
- Perform basic Euclidean constructions using traditional tools, a MiraTM, and a graphing utility such as the Geometer's SketchPad or Geogebra
Text
Reynolds, Barbara E. & William E. Fenton, College Geometry Using the Geometer’s Sketchpad, Wiley, 2012.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion
of individual instructors, subject to general university policy.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to
general university policy.
Course Outline
- Introducing Geometry
- Basic constructions using Geogebra, Mira, classical tools
- Recalling polygons, using mathematical language, thinking about logic
- Constructions leading to Proof
- Using constructions to develop proofs about angles and their relationships in Triangles,
Circles, Parallel and Perpendicular Lines
- Investigating Triangles
- Writing proofs in geometry
- Pythagorean Theorem, Triangle Congruence, Concurrence properties of triangle
- Investigating Circles
- Robust constructions using Geogebra as a basis for proofs
- Language of circles, circles related to triangles, special circles
- Analytic Geometry
- Establishing a unit, describing lines, connecting Pythagorean thm and the distance
formula
- Transformational Geometry
- Transforming static shapes using translations, reflections, rotations, and glide reflections
- Non-Euclidean Geometries
- Introduction to one or more non-Euclidean geometries
- Finite Geometries (optional)
- Affine and Projective Geometries; Geometries with finite number of points and lines=
- Isometries and Matrices (optional)
- Describing transformations using matrices and matrix operations
- Symmetries (optional)
- Cyclic and Dihedral symmetry groups, frieze, wallpaper, tiling, Penrose Tiles