Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 6
Akihiko Takahashi

Aki reviewed the results of the voting on Friday by first listing the reasons participants had given for voting for the winning entry. See Aki's PowerPoint presentation [download PowerPoint file] for a summary of the comments. 23% of the group voted for Table 4's poster Going the Distance.

Aki awards origami hats to Table 4, who won for the best open-ended question during last week's Iron Chef competition, using a geoboard for their competition involving number of distances that could be created.

Aki introduced us to the topic of Open Ended - problems with multiple solution methods. He showed us the Crooked Transversal problem. [See crookedtransversal.gsp] In this problem, lines l and m are given to be parallel. The angle to P from line l is 40° and the angle to P from line m is 30°. What is the measure of angle P? This is a typical textbook problem. The traditional solution is to draw a line parallel to lines l and m through point P and then use the alternate interior angles property to calculate the measure of angle P is 70°.

We were asked to work in groups to solve this problem using other lines. The groups worked on this for half an hour and then their solutions and process among each other.

A summary of the discussion follows:

What were your first impressions?

  • The first thought was to make triangles.
  • The problem was pretty straight forward.

After you started on the problem, in what ways did your first impressions change?

  • Feel bad that one particular method (the book method for solving these types of problems) was presented to solve the problem (the book used the parallel postulate).
  • Students have more experience with triangles than with parallel lines and the parallel postulate (triangles provide a good entry point).
  • All participants approached it differently because they have a good foundation in mathematics that students might not possess.
  • Students will typically try what they were taught last (impressions as to how students might approach this).
  • Students need an opportunity to pull everything together (impressions as to how this problem addressed so many content areas).
  • It was good to explore and to try to understand the other methods.

What content was addressed?

  • Constructions
  • Proof (some solution diagrams depended on the order that lines intersect. Proof was needed to show diagrams were correct)
  • Algebra and Geometry connection
  • Sums of interior angles of polygons
  • Meaning of parallel postulate
  • Exterior angles of a triangle
  • Parallel lines and transversal properties
  • Circles and arc measures

What are the advantages of problems with multiple solutions?

  • Multiple entry points
  • Everyone can participate
  • Can understand concepts/connections better
  • It slows down the "quick" kids (must think deeply)
  • Generates discussion
  • Forces deep mathematical thinking - reconciliation of solutions
  • Found "large" number of solutions only as a group
  • Some organized their solutions by cases. Others found an infinite number of solutions.

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This material is based upon work supported by the National Science Foundation under Grant No. 0314808 and Grant No. ESI-0554309. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.