Park City Mathematics Institute
Secondary School Teacher Program

Reflection on Practice Class: Day 6
Akihiko Takahashi

Aki: Today is Monday, tomorrow will be off and then we have Wednesday, Thursday and Friday, four days. This group you will work as a team for this entire week.

This morning I had various interesting conversations about your weekend. Some of you went hiking (Aki shows a picture of a trail in Park City.). Where did you go? Did you start from here and climb up or were you lazy so you climb up with the lift, then climb down? There are various ways you can do it.

(Aki shows a picture of an Aspen grove.) It's beautiful in this area. I live in Chicago and looking at mountains makes me much more happy. We have no mountains there.

(Aki shows a picture of a hot-air balloon in mid-air.) I know somebody did hot-air ballooning. How was it?
Participant: It was great. Most of us were okay up there, twelve in a balloon.

Aki: Because the inside of the balloon needs to be very hot, if you go out in the morning it is much better. Last year I did it and you can see wild animals on the ground, like foxes. If you use a helicopter or airplane, it is too loud and the animals will be scared.

The reason why I'm showing this is if you had different experiences over the weekend, naturally you will start sharing your experiences. This is a good way to start the discussion among us.

This morning I was in the in the International Seminar and was sharing with some of the participants. I talked to a man from Cameroon who said "To teach problems is very challenging, because the classrooms are overcrowded: 100 students in each." I also spoke to someone from Singapore, who said "Our teachers are very comfortable teaching problem-solving to 40 students at a time." Would you be comfortable with 40 students in your class? Different experiences bring different conversations. Different perspectives make us think. Is 25 the ideal size for a classroom? Is 40 impossible to teach?

Today, each group is going to work independently to create your own problem. Then we will have eleven different problems. At the end of the week we can start exchanging problems. This activity we call "Iron Chef."

Has anyone seen Iron Chef? This is a television show that started in Japan and is now very popular. Can someone explain it a little?

Participant: I think you have the Iron Chef who is the champion chef and two challengers. There is a key ingredient, like octopus, and each chef must create a meal with that ingredient. Then a panel of connoisseurs eats the meals and judges them.

Aki: At the end of this week we are going to vote on who makes the best problem. We need to have a judge but you are going to judge by yourself. We will set up a website and you can click on the problem you think is best. I want to emphasize you can not vote for your own table. You need to explain why you think the problem you chose is the best.

So now, what is your ingredient?
Your ingredient is Origami. You must use this ingredient to create a nice problem for your students. I am going to give a set of origami to each table. You don't have to use everything.

What kind of problem might you be able to create?
Example one, a very simple one: Can you make a right triangle from a sheet of origami paper? You may use only a pair of scissors and a pencil. You could do this with trial and error, but if you want the student to justify it as a right triangle or isosceles triangle or an equilateral triangle, it becomes a much better problem.

For an equilateral triangle, for example, you need to justify why it is correct.
(Aki folds a piece of square origami paper in half, which creates a perpendicular bisector.) We know one of the vertices of an equilateral triangle will be on the perpendicular bisector. You can fold up from each bottom corner to a point to create two equal sides.

Another example is to fold it like this. (Aki folds a paper in half along the diagonal, then twice more, halving the area of the paper each time. Aki then cuts the right angle of the new triangle off.) When I open this, what kind of shape will I get? Don't do it by yourself, can you think about what kind of shape it will be?

Participant: I think there will be a V-shape on each side. (Participant comes to the board and draws a cross-like shape that looks like a square with isosceles triangles removed from each of the four sides)

(Example 3)
Aki: Now this is the previous one, but fold one more time before you cut. What kind of shape will you have?
Participant: I can't give you an exact answer but I think you'll get some cut-outs in the middle.
(Aki unfolds the paper and reveals a square with fours square holes in each corner, a shape resembling a window pane.)

Aki: You can create many interesting shapes if you keep cutting. On the internet there is a problem-creating website. Another example is to use one straight cut to make two equal pieces.
(On screen: As you know, you can cut a piece of Origami paper into two equal-size parts (equal area) by a straight line.)
You can fold a sheet in many ways to create two equal pieces. If you have two pieces together [sharing an edge or part of an edge], can you still cut two equal pieces with one cut? Can you do that?
Participant: Cut a line through both square centers.

Aki: Is this the only way or not? If you now create a center of two rectangles and cut a line through these points, this is another way. A third way is to create the missing square and find its center, then cut through this point and the center of the other square. There are other ways, but I am not going to tell you because this is what you are going to work on today.

On the board, the following is projected:

  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.

Aki: You need to determine if your problem is appropriate and then anticipate student's responses to design a lesson. Make sure the purpose of using the problem is clear. You need to make sure the goal is clear. Finally, make sure the problem is as attractive as possible. With your group, you need to do all these things on a poster. By Thursday you need to post it so everyone can see.

Question: Is making the problem clear directed to teachers, to students, or to both?
Aki: To both.

Question: Basically, we can only use paper, pencil and origami sheets.

Question: Is there some sort of special size or ratio of paper size we should start with?
Aki: Origami comes in many sizes, but any one you use is fine as long as it is a square.

Question: The presentation needs to be on one piece of poster?
Aki: Yes.

Aki: Also, designing the poster is important, because if you put too much information, it is too much. During all three days I will post the criteria [for the task] on the wall.
Are you ready? Let's do it!

Groups work for 45 minutes.

Aki: Currently, most groups are working to explore what kind of problem we can make. If you are studying one good lesson, you need to study ten. Then you need to focus on one from the ten. Maybe not until Wednesday you will know which way to go. You can start designing the poster on Thursday.

I hope you enjoyed very much and I will see you Wednesday for more.

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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.