Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 10
Akihiko Takahashi

Aki: Thank you very much for all your posters. What do you think about them? I would like to ask you to share your experience working with your table coming up with the poster. What activity did you spend most of the time on? What was most difficult? Anyone?

Table Participant A: Along the way we stopped a lot and worked on our own.

Table Participant B: One thing I noticed that we were bouncing ideas off each other. I noticed lots of people came up with things they wouldn't have by themselves. As I listened to others I began to get some ideas, where I started as a blank slate. Everyone seems to come into this as they were writing this for their students, whoever that was... we were able to overcome that, but it was in issue we had to deal with.

Aki: We each bring different knowledge and share with each other. This experience makes it better. This kind of activity we really want your student experience, not that one student brings an idea and everyone agrees, but each of you contribute significantly. You need to appreciate this - sometimes you fight, sometimes you revise, but this process is very important. This mathematical discussion enriches this process. We want to have this kind of environment in your classroom. This is why we spend time here working on this.

And now, we want to talk about products. The results were very interesting. Two groups it was very close, just one point difference. Let me show you. [On Powerpoint, Aki showed the percentage of the vote won and a few peer comments from Table 4's and Table 11's posters.]

The winner is both Table 4 and Table 11, they both tied. Please stand up and come to the front. [Aki presented all winners with Origami hats.]

So process is important, but product is also important. Looking at all these tables I noticed you include quite a bit of open-ended approach. Some of you participated last year, we spent time on this, but those of you who are new may not clearly understand yet what is an open-ended approach.

[Aki shared a definition from Shimada and Becker, see Powerpoint]. The reason they did this research was to find out how we can promote higher-level thinking among students. They also study what a problem is in a traditional math classroom. Almost always they use a closed problem, only one correct answer and only one method. What kind of conversation can we have, then? Since there is no wrong answer in your posters, you can bring a lot of different ways to discussion. Even in the traditional problem you might have many answers but only one approach.

[The Powerpoint relayed the three types of pen-ended approaches as problems with multiple solution methods, multiple solutions, and links to other problems.] This original research was done in the 70's in Japan. At this time the curriculum was traditional. However, we can find many open-ended approaches in Japanese textbooks after thirty years. [Aki showed an example from a 5th grade textbook having to do with congruent triangles. The main problem asks for which quadrilaterals, when you draw a diagonal, will you create two congruent triangles?] What's going to happen? What do you think? For all of these you can start talking, using the open-ended approach. This took thirty years to do this.

Let's make an origami puzzle now. Do you know the Tangram? You cut a square up and rearrange them to make new shapes. We are going to do a similar kind of activity. [On Powerpoint, By using two origami papers, let's make a larger square. You can cut your origami paper(s) only by straight lines then rearrange these pieces to make a square. You must use all the pieces the origami papers without overlapping.] One of the answers to this problem is on the poster of Table 5. Each table will have three sets and scissors. [Groups start working.]

Very interesting discussion. I would like to ask one group to share what they found. [Aki draws two squares on the mimeoboard, with diagonals drawn from each top left vertex to bottom right vertex. The four pieces were then arranged so that each right angle of the four triangles became a corner of a larger square, with a larger square "hole" in the middle.]

Unfortunately this is not included in today's activities. We don't want to have any gap. But it is a good idea.

What other kind of methods did you come up with?

Table participant C: If you take the new big one you drew up there and think origami, and fold each corner in along the diagonals. [Aki drew arrows to show each triangular piece from "wrong" answer flipped inward, to create a square that would work for this activity.]

Table Participant D: If you slide the two squares together to make a rectangle. From the midpoint on the top to the midpoint on each side of the top one, same on the bottom, then rotate the corners up to the sides and there you go.

Table Participant E: We just kind of made triangles out of them. We cut each square along both diagonals, then rearranged them in different ways. [This answer used eight congruent triangles. Aki drew this method and all others on mimeoboard.]

Aki: Okay, so this one has multiple solutions. All the shapes you made are congruent. What do you think? Congruent or different?

Table Participant D: You have to preserve the area, so if the area is x-squared plus y-squared, it will always be that.

Aki: That means do you think that all sides are equal [of the ending square]? Oh, that one is interesting! I cannot draw it. [Aki takes solution from a table and puts it under the projector for everyone to see.]

This is one of the ways to use open-ended approach, If you have many solutions from students, they are very interested in other solutions or approaches.

[Table Participant F explained that the group measured very exactly the side length of one square, then used that to cut the other square so it would form a partial border.]

Aki: From problem to problem, we are going to mixed problem by changing this problem a little. We are going to change the size of the origami paper. Now this time the ratio between the two squares is 1 and 2. It is exactly the same but the difference is the ratio of the sides is different now, 1 to 2. Each group will have three sets of these. [Again, Aki showed new problem on Powerpoint and groups began working.]

I've only seen this method so far. Leave the smaller one as it is. Cut the larger one down the middle, then cut one diagonal of each to make four right triangles. Then make this one. [Aki draws a figure where the four right triangles are positioned around the smaller one so that each side of the square is adjacent to part of the longer side of the each right triangle.]

Table Participant G: Basically we know that if the middle square has an area of one, the bigger one has an area of four. If we combine them, the new square has to have an area of five, so each side has to have a length of square root of five.

Aki: So we know a triangle with 2 and 1 will have a third length of square root of five.

Table Participant H: I tried this on the first one and it didn't work, but now all of a sudden it did. I had seen this before for a proof of the Pythagorean theorem.

Aki: Any other method? Since we only have three more minutes, we will come back tomorrow morning and finish discussion. This is not the end of this problem. See you tomorrow.

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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.