Wednesday, October 18, 2017

10/17 Pythagorean Triples

http://mymathclub.blogspot.com/2016/10/12-triangles-and-their-link-to.html The inspiration for this week was a puzzle from the recent Pythagorize Seattle event thrown by MoMath that my friend Dan recommended.



Link: https://drive.google.com/open?id=0B6oYedIeLTUKOFpYR1VZek80Y1RYeEJqLUN4ZmtVWHVqRTVN

Each of the triangles above is a Pythagorean triple with one edge given. To solve the puzzle, you need to find the length of the side marked with a question mark.  Then translate each number into its position in the alphabet. For example if you find a side of length 5 that becomes the letter "E". Altogether this forms a 8 letter word scramble.

With 12 kids, I printed and cut out 3 sets of these triangles so we could have 3 groups working at the same time. To start off I quickly reviewed the basic Pythagorean Theorem on the whiteboard. Because I knew we would talk about it more in the middle I deferred any review of why it works.

Then I let each group work for about 10 minutes. What I found was that a lot of kids took out calculators and started plugging numbers. Walking around, if kids looked stuck I suggested estimating the missing sides edges and just trying out numbers near the estimate.

At this point I paused everyone to watch the following video:


Some of the parts here are a bit advanced so I stopped in the middle a few times to go over ideas like the Complex plane. And at the end I emphasized the generator functions:


  • 2uw
  • u^2 + w^2
  • u^2 - w^2
I then sent everyone back to finish working on the puzzle. From here the groups fairly quickly finished although not at the same time. I gave out a followup Pythagorean Triple geometry problem for the last few minutes:

See the first proof here:

No one had fully solved this yet before we left for the day so I'm tempted to come back here. I'd also like to have the kids find some of the patterns in the triples i..e one of them is always a multiple of 3, 4, and 5. This might make a good bridge with a day on modular arithmetic as well.


P.O.T.W:
Continuing with the Pythagorean theme:


When handing this out I told everyone to be on the lookout for another hidden triple. I think a few kids found it already before they left.



Leveling:
I'm getting to know the kids a little better. This week I was hoping the initial puzzle would have enough entries for everyone. That was mostly the case but I ended up working with one student a bit to get him started.  Now that I've heard what class everyone is in, I realize I have a larger gap than was indicated in the entry forms from Math7 to Algebra II.  If I'm going to throw algebra into the mix I'm going to need either to run different activities at the same time or plan how everyone will be able to approach and work on the problem at the same time.

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