Math 277: Geometry for Elementary Teachers

Prepared for:NSF Site Visit June 8, 2005

Design Team Members Prof. Ric Ancel – Mathematical

Sciences

Dr. Hank Kepner – Curriculum & Instruction

Melissa Hedges – Teacher-in-Residence

Review of Standards National Standards (PSSM) Wisconsin Model Academic Standards for

4th and 8th grade Milwaukee Public Schools Learning

Targets MET Report

End result: Compilation of a comprehensive list of Geometry topics

Course Goalsand Anticipated Outcomes

Develop students’: ability to visualize problems familiarity and facility with a wide

range of geometry facts and problem solving techniques

understanding of logical structure of geometry – axioms, conjectures, theorems and counterexamples

Course Overview

Geometry as a measuring tool Geometry of the Earth Geometry as a logical system Rigid motions and symmetry

Topic 1: Geometry as a Measuring Tool

Pythagorean theorem Similar triangles Measurement of large scale

distances and heights Units and accuracy issues

Topic 2: Geometry of the Earth

Spheres, planes, lines, great circles, axes and antipodes

Latitude and longitude coordinates Rotation of Earth and seasons Eratosthenes and class

measurements of the Earth’s circumference

Topic 3: Geometry as a logical system

The axiomatic method Axioms for geometry Theorems and proofs Incomplete proofs of basic

geometry theorems Incomplete proofs of properties of

quadrilaterals and their diagonals

Topic 4: Rigid Motions and Symmetry

Patty paper constructions Translations, rotations, reflections

and glide reflections: definition, construction and identification

Group concepts for rigid motions: identity, composition and inverse

A typical day in class Introduction to the subject by the

teacher Small group exploration of subject Report by groups to whole class

and class-wide discussion Connect to sample activities from

K-8 curriculum Discuss and assign homework

Sample Problem 1 Which of the following types of sets can

occur as the intersection of a sphere of radius r and a plane in 3-dimensional space?

a) The empty set b) One point c) Two points d) A circle of radius r e) A circle of radius < r f) A circle of radius > r g) A non-circular ellipse Test your answers by slicing an orange.

Sample Problem 2 Rank the distances between the

following five pairs of points on the globe from smallest to largest.

a) 62ºS, 85ºE and 62ºS, 110ºEb) 70ºS, 140ºW and 80ºS, 40ºEc) 62ºN, 170ºE and 62ºN, 170ºWd) 12ºN, 115ºW and 37ºN, 115ºWe) 17ºS, 10ºE and 17ºS, 15ºW c < a < e < d < b

Classroom approach to a problem

Students discuss problem in small groups with occasional coaching from teachers.

Representatives of groups present their solutions to class.

Class discourse on student solutions facilitated by teachers

Accomplishments and Challenges

Accomplishments Design team

collaboration Class format

encourages student engagement and enthusiasm.

Daily lesson plans

Challenges Refine activities,

add topics Thought provoking

written homework Tension between

concepts needed to learn geometry vs. the large number of topics taught in school

Topics to be shoehorned into course

Trigonometry? Creation of proofs in incidence

geometry Volume and surface area of

cylinders, cones and spheres Use of dynamic geometry Symmetry of plane patterns: cyclic,

dihedral, frieze and wallpaper groups Tesselations of the plane