Friday, January 9, 2015

1/6 Winter Session Starts. Pythagorean Theorem.

Today was the first session coming back from break.  We first talked about our goals for the quarter:

1. Have Fun.
2. Try out new problems / learn some new concepts.
3. Emphasize proving why things work.
4. Go to more contests.

I then had the kids who went talk about how the KPMT competition went. That actually was fun and  I think I'll have the kids do this after every off site event.  Next we warmed up with some game of 24 cards.  Interestingly I asked how many kids had seen them before and only 1 or 2 of the group had.  I would have predicted beforehand that more of them would have used them for enrichment in regular class.  So I  spent more time talking about the rules and guiding kids to try the easy ones first to get the hang of how they work. I let this go on until everyone had solved at least one card or more and said they could take any of the sheets home that they wanted to think more about.  Overall, this was a success and I'm planning to try out kenken or sudoku sheets as a future warmup.

Next came my big experiment. Previously I had surveyed and found that most kids knew the Pythagorean Theorem but did not know why it worked. So I pre-printed a bunch of pictures of a right triangle surrounded by the 3 squares created by its edges and brought along some scissors. I then gave the kids 3 choices.

1. Try to find a free form way to cut the smaller 2 squares into shapes that would fit into the larger one.

2. Try to find two larger squares of the same size that you could make out of the cutout shapes. The hope here was to independently discover:

From there I planned to walk them through why these picture prove the theorem i.e. the area of the 2 little squares + 4 triangles == area of the big square + 4 triangles.

3. Try to figure out a proof based on the widow's chair diagram i.e. the classical Euclidean proof.

Overall I had several kids find free-form versions and one set find the 2 squares and make it to the realization why those proved the theorem. I also had one boy who from the start knew the algebraic proof using the 2nd square i.e. (a+b)^2 == c^2 + 2ab.  I liked the experimentation that went on while they played with the shapes but I had trouble bringing the whole room to a final aha moment. I think I'm going to review the findings at the beginning of next week in a lecture format. My goal is to spring board from here to some investigations about the square root of 2. 

Finally for homework I gave out a copy of the first 5 questions from the AMC 8 contest this year. Hopefully enough kids will try them that we can discuss in a group next week. 

Classroom Issues

I had one disruptive student today. Repeated commands to stop doing things did not calm him down and he tended to find new outlets while I was walking around the room with the other kids. I particularly didn't like that he started bothering one of the other girls. Besides talking with his parents, I'm going to try an assigned seat next week for him to see if that helps provide focus. While I  want to have everyone who wants to be here have a chance to do so that can't come at the cost of ruining the atmosphere for others.


  1. I once did a pythagorean theorem activity (for older kids) in which I presented different sheets with:
    (a) examples of pythagorean triples and pictures of the triangles
    (b) several different proofs (esp 2 geometric reorganization ones and the proportional reasoning one)
    (c) a proof of the converse (if the sides have lengths that fit the formula, then the triangle is a right triangle)

    I had them talk about which they found more convincing, which they thought were not real proofs, which they found more elegant, etc. Among other epiphanies: they found the collection of examples more convincing than the proofs!

  2. That's an interesting observation. Since this is one activity I definitely intend to repeat, I'll have to give the Pythagorean triple examples a try and see how it resonates. One extension I never did last year would be to find the formulas to derive the triples. Unfortunately, I was unable to see a way to simplify the algebra involved enough to make it accessible.

    1. Consider having them work on finding infinitely many relatively prime triples as a prelim/alternative to finding all triples. At least one family is quite accessible: (2m+1, 2m^2_2m, 2m^2+2m+1) which the kids will probably identify based on odd squares or sides with difference of 1.

      If they identify that family, you can also ask them if there is a family of relatively prime triples where the hypotenuse and a side differ by 2?