This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)
I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.
What examples can we come up with? (3-4-5, 5-12-13, ...)
6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)
Maybe writing a list of all the perfect squares up to 400 will help us find more.
What patterns do we see?
- Odd + Even = Odd
- Middle number is a multiple of 4
- c = b+1 (after which I added 8-15-17 to our list)
One person was new, so we reviewed our first week's work for him.
We explored the "family" of triples with c = b+1. a2 + b2 = (b+1)2 becomes a2 = (b+1)2 - b2
= b2 + 2b + 1 - b2 = 2b-1. If a2 = 2b-1, then b = (a2+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...
X noticed that in the triples
3-4-5the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:
Another new person came, so we summarized for her. Then we explored triples where c = b+2.
I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)2, and thought he might have to distribute.
We verified that we get all of the primitive Pythagorean triples with c=b+2 using:
Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.