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