If you've been following along, I've been preparing to do a day based on the Pythagorean Theorem. Last year I tried out a fairly inquiry based approach where I had the kids try to find equivalent squares using cut out shapes on the way to one of the basic proofs. Overall, I wanted to improve on that approach this year especially since the fourth graders have much less exposure to the basics of the theorem. This time around, I decided to go at it more directly and outline some proofs as a group and work instead on extension problems.
To start Math Club off, I gave out jelly beans to reward the group for finishing all the take home assignments. We then went over the solution to last week's problem: problem of the week This was a fun factoring/basic number theorem exercise and I was pleased that one of the girls outlined a really good proof why the equation was impossible on the whiteboard. In a nutshell:
let Y be the units digit and X be the other ones.
So the original number 10X + Y and the transposed version is 100Y + X.
if 2(10X + Y )= (100Y + X) then 19X = 98Y and since 19 is not a factor of 98 and Y is a single digit. There is no way to satisfy the equation.
[Note from the future: I would flip this idea now - don't waste focus on the puzzle and jump right in. Save cool downs for when the kids lose focus and try to maximize time on the main task. I'd also have the kids work on the whiteboard on the problem sets]
I also wanted to work about 30-35 minutes on the main topic to maintain focus. So I then had the kids do a warm up sudoku puzzle from http://www.websudoku.com/ for about 15-20 minutes. Now that spring is really upon us, the kids often start out highly energetic after the stress of a school day. Its easy to get the kids excited about games and puzzles and this made the next transition which required more focus a bit easier than jumping in directly. As a side note, I talked about my agenda for the day and it was interesting how several kids were already really interested in the topic.
At the midpoint I gathered everyone back to the whiteboard. For my presentation this year I started with some basics:
- A review of what a square number is and how it relates to a geometric square. I had read on the web that students often don't realize this connection and sure enough when asking for ideas from the room it was not immediately obvious. This made for good group brainstorming.
- A quick review of what a right triangle is and which side is the hypotenuse and how to find the area of a right triangle.
- Some background on the discovery of the theorem by Pythagoras, the Chinese et. al and then a discussion of the classic diagram from above emphasizing the idea that the theorem is about area.
- Then I broke into 2 of the more accessible visual proofs:
The first one I chose uses the 2 equivalent squares below:
I had the kids tell me the area of each square after having everyone agree they were both the same. Its really easy to "cancel" the 4 triangles and the theorem falls out from there.
The second one involved rotation:
After drawing the two smaller squares side by side you construct the original triangle by adding in point B and rotate ACB to AA'G and then rotate BB'F to A'B'C'. This one I did twice emphasizing the hinges of the rotation. I'm glad I did two completely different methods since it showed there are multiple ways to a solution.
Next, we briefly discussed Pythagorean triples and I had the kids name the few they knew included the 3-4-5. Since the first few pages went further into that topic I stopped at that point. With my presentation finished I then handed out some quick practice triangles to find the missing side to. Interestingly that sheet had some larger numbers and it showed that some of the kids don't know how to find a square root by narrowing down the range its in. I'll have to do a session on that and perhaps the old hand algorithm which while a novelty nowadays is fun to try out.
Finally we finished with the main packet I found online: http://agmath.com/media/DIR_12306/9$20Pythagorean.pdf which had a lot of good followup problems. By this point the kids only had about 10 minutes so they mostly worked on the first 2-3 pages. But from walking the room, it looked like the basics had gelled and they were able to make progress. I'm half tempted to continue on the packet again next week since the following pages looked fun.
I chose the following for this week's problem of the week:
This is a repeat from one of my pi day worksheets that I don't think most of the kids finished the first time around. I'm hoping to get good results from the repetition and the fact we've covered more basic ideas by now.
Overall I was fairly pleased. The session had fairly good flow and the kids seemed excited along the way. I'll have to mull things over some more since I still feel like there is a way to get build up to the proofs more organically and this was the closest I've come to teaching a subject like an actual math class.
For next year:
Find the length AB. (5-12-13 Pyth. Triple)