MATH 498E—Geometry for High School Teachers



Math 484 Geometry for High School Teachers (Summer 2012)

Teacher: Dr. Frances Gulick (office: MTH2101, email: ffg@math.umd.edu, 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.

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