Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 11
Akihiko Takahashi

Aki: Yesterday we started with a simple puzzle by using two same-size origami paper to make a square and we had several different methods. Next we changed the size of the sides of the origami paper, in this case 1 and 2. One solution came from you [Aki showed solution from previous day]. So now how do you minimize the number of cuts? What do you think?

Table Participant A: Are we allowed to fold before we cut?

Aki: So yesterday we had cut this way [Aki shows a square with a segment from the top right vertex to a point on the bottom closer to the right vertex than left.] If we make one more cut, we should be able to do it. How about if we put this small one here? [Aki draws the smaller square on the left side of the existing square, adjacent to it.] So the question is, could you move this piece? [Aki shows a Geometer's Sketchpad document showing these pieces moving around to form a square]

So is this is a right angle, what do you think?

Table Participant: We know the two top angles are complimentary, so yes it is.

Aki: Now we are going to change our original problem a little bit. Right now the ratio between the two squares is 1 and 2. What happens if it becomes 1 and 3? I have another sheet, 1 and 3, so I will give three sheets to each table. [Groups began working.]

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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.