Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 7
Akihiko Takahashi

Aki referred to some of the work that was done during our morning class. There we were asked to come up with a formula for the area of a parallelogram whose vertices were (0, 0), (a, b), (c, d) and (a + c, b + d).

Two participants demonstrated their solutions using The Geometer's Sketchpad®.

Aki: What have we learned from this morning? For finding the area of a parallelogram - there are two methods. Anastasia's method involved adding something to the parallelogram (inscribing it in a rectangle.) We discussed this method before. Can you see any relationship between her method and what we did last week in the 11 hour?

The second method - Anita's . We also discussed it. In this method the parallelogram is transformed to one with the same area whose base is on the x-axis. You try to transform the shape into ones that are easier to find the area of.

Probably most of you tried to think about this problem. In this case we cannot just grab this problem to give to your students. We have to actually think about it. What is the benefit of it? We only have today and tomorrow to work on our problem and to prepare the poster.

We worked in our groups for the hour - trying to come up with a problem which meets the four conditions:

  1. Determine if the problem is appropriate
    1. Is the problem rich in mathematical content and valuable mathematically?
    2. Is the mathematical level of the problem appropriate for the students?
    3. Does the problem include some mathematical features that lead to further mathematical development?
  2. Anticipate students' responses to design a lesson.
  3. Make the purpose of using the problem clear.
  4. Make the problem as attractive as possible.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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.