Friday. Last period of the day. My energetic and lovely grade 9s. And the topic of the day is the Pythagorean theorem, which they have done in grades 7 and 8, so it's not always, uh, exciting, shall we say. However, the majority of students only seem to remember "a squared plus b squared equals c squared" as if every right triangle would be labelled with a, b and c, the latter being the hypotenuse. (I have been know to quickly draw a triangle labelled with a, b and c where a is the hypotenuse - that confuses them.) So today I decided to do part of an activity that I stole from Alex Overwijk (26 Squares), World Freehand Circle Drawing Champion. (That is another story which you can also read on his blog... it all started when we taught together many years ago, but being somewhat shorter than Alex, I was not part of any of it!)
Back to today. Students walked in to find these instructions:
Each group received a package with squares of side length 1 to 26. With four students per group, it did not take long to cut them out. I also provided them with an envelope in which to place the squares as they cut them out. Still M. managed to lose to the 2 square. Sigh... Then they got to play but quickly converged on creating a triangle with the sides of the squares. Perfect. Next question:
I gave each group a whiteboard and let them argue about who was right. "Of course you can make a triangle with any 3 squares!" "No, you can't - try with 1, 2 and 26." Z tries... "Well, that's just a fluke." FYI - by fluke he meant "I was wrong." I prompted some groups to think about if they made the smaller squares progressively larger, at what point would it actually form a triangle. This took the form of "What if that 3 was a 4? Keep going with that." They came up with the limiting case when the sum of the two smaller sides equaled the longer side and wrote it like this:
Off to work they went, trying out lots of possibilities. My instructions we clearly not adequate for this part as they came up with combinations that looked like they could be right triangles, but in fact, were not. We talked about this, and next time I would have them place the three triangles on the grid side of their whiteboards to ensure that they truly had a 90 degree angle. If time allowed I would also get them to go back and check their work after we established what makes this "work".
We talked about a combination that worked (3, 4, 5) and I asked why they thought we were using squares, not just strips of length 1 to 26. We talked about area.
We consolidated by listing combinations that worked and they noticed a pattern between:
so we came up with more! I also eventually showed them a list of Pythagorean triples on-line. One of them said "But there would be an infinite number of them, right?". : )
The time flew by - gotta love a Friday afternoon when they aren't all looking at the clock!