Download MATH 498E—Geometry for High School Teachers

Math 484 Geometry for High School Teachers (Summer 2012)
Teacher: Dr. Frances Gulick (office: MTH2101, email: [email protected], phone: 301 405 5154, office hours: right
after class for a reasonable amount of time)
Text: College Geometry Using the Geometer's Sketchpad, 1st Edition, Reynolds and Fenton, Wiley Publishing, 2012 or
College Geometry, 1st Edition, with Geometer’s Sketchpad v5 Set by Barbara Reynolds, Nov. 2011.
Dates: June 26-July 26, Tuesdays and Thursdays from 9 am to 1:30 pm.
Objective. The objective of this course is to provide a solid foundation for in-service teachers in neutral and Euclidean
geometry and given them an introduction to non-Euclidean geometry through the Lénart sphere and the Poincaré disk model
for hyperbolic geometry as well as through the theory. In addition, teachers will gain experience in working with Geometers
Sketchpad so that they can use it in the classroom both as a presentation tool and as an exploration tool for the students. The
expectations of this course are consistent with the expectations of 400-level mathematics courses.
The expectation is that we will cover Chapters 1-4, 6, 8, and 11 in College Geometry. There will be supplementary material,
including an exploration with the Lénart sphere.
Topics
Axioms for Plane Geometry and immediate consequences
Existence and incidence
Distance and angle measure
Plane separation
SAS congruence
Proof
Writing proofs
Indirect proof
What is needed in a definition?
Neutral geometry
Triangle Congruence theorems
Triangle inequalities
Alternate Interior Angle Theorem
Saccheri-Legendre Theorem
Euclidean Parallel Postulate and equivalent
statements
Quadrilaterals
Neutral constructions
Euclidean Geometry
Basic theorems with the parallel postulate
Parallel Projection Theorem
Similarity theorems
Dividing a line segment into n congruent pieces
Concurrent lines (medians, angle bisectors, altitudes,
perpendicular bisectors)
Ceva's Theorem, Menelaus' Theorem
Area
Postulates (Neutral and Euclidean)
Decomposition and finding formulas
Circles
Angles and circles
Inscribed and circumscribed polygons
Circles and triangles, Euler line
Lénart Sphere
Intro to spherical geometry
Sum of angles in triangle
Failure of Incidence axioms
No parallel lines
Transformational Geometry
Translations, rotations, reflections
Isometries and congruence
Dilations and similarity
Inversion in a circle
Analytic form
Hyperbolic Geometry
Some basic properties
Poincaré Disk Model
COURSE GRADE: The course grade will be based on homework and worksheets(100 points), Sketchpad activities (100
points) and two exams – a take-home midterm exam (100 points) and an in-class, closed-book final exam (150 points).
You may work with others on your homework but what you turn in should be in your own words and mathematics (not
copied from anyone else or any other source). Homework is due at the beginning of class. Sketchpad activities will be
submitted via email. You may turn in one late assignment with no penalty if it is received within a week of the due date.
Worksheet solutions are due on the same day they are done in class. This course operates under the Honor Code of the
University of Maryland.