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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)

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math 277 geometry for elementary teachers

Math 277: Geometry for Elementary Teachers

Prepared for:

NSF Site Visit

June 8, 2005

design team members
Design Team Members
  • Prof. Ric Ancel – Mathematical Sciences
  • Dr. Hank Kepner – Curriculum & Instruction
  • Melissa Hedges – Teacher-in-Residence
review of standards
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 goals and anticipated outcomes
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
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
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
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
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
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
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
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.
sample problem 2
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
classroom approach to a problem
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

Accomplishments and Challenges
topics to be shoehorned into course
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