Math 277: Geometry for Elementary Teachers

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Math 277 Geometry for Elementary Teachers

• Prepared for
• NSF Site Visit
• June 8, 2005

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Design Team Members

• Prof. Ric Ancel Mathematical Sciences
• Dr. Hank Kepner Curriculum Instruction
• Melissa Hedges Teacher-in-Residence

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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

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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

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Course Overview

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

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Topic 1 Geometry as a Measuring Tool

• Pythagorean theorem
• Similar triangles
• Measurement of large scale distances and heights
• Units and accuracy issues

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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
  Earths circumference

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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

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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

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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

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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 lt r
  
• A circle of radius gt r
• A non-circular ellipse
• Test your answers by slicing an orange.

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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 lt a lt e lt d lt b

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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

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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

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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