Math 277: Geometry for Elementary Teachers

1. Math 277: Geometry for Elementary Teachers Prepared for: NSF Site Visit June 8, 2005

2. Design Team Members • Prof. Ric Ancel – Mathematical Sciences • Dr. Hank Kepner – Curriculum & Instruction • Melissa Hedges – Teacher-in-Residence

3. 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

4. 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

5. Course Overview • Geometry as a measuring tool • Geometry of the Earth • Geometry as a logical system • Rigid motions and symmetry

6. Topic 1: Geometry as a Measuring Tool • Pythagorean theorem • Similar triangles • Measurement of large scale distances and heights • Units and accuracy issues

7. 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

8. 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

9. 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

10. 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

11. 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 • Two points • A circle of radius r • A circle of radius < r • A circle of radius > r • A non-circular ellipse • Test your answers by slicing an orange.

12. 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, 110ºE b) 70ºS, 140ºW and 80ºS, 40ºE c) 62ºN, 170ºE and 62ºN, 170ºW d) 12ºN, 115ºW and 37ºN, 115ºW e) 17ºS, 10ºE and 17ºS, 15ºW • c < a < e < d < b

13. 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

14. 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 Accomplishments and Challenges

15. 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