## Friday, May 4, 2018

### 5/2 Following my passion

This week I ended up switching my focus on the fly. I had been originally been planning on doing a graph theory math circle  activity centered around chicken pecking hierarchies. (Yes really)  But I became so excited about thinking about polynomial deltas that I ended up asking myself the question "Why not go with what you're excited about?" All the kids have enough background with polynomials so we could just jump in which was an added bonus.

I stuck with one part of my planning process. I had already decided I wanted to start with some group white boarding focusing on some of the geometry puzzles from @solvemymaths.

So I picked 3 of them included the now infamous pink triangle.  In each the goal is to figure out which fraction of the shape is shaded pink and usually any polygons are regular.

I put each of these up on different sections of the whiteboards and let the kids circulate among them forming organic groups. (Occasionally I'll nudge kids to work together) They then spent about 15-20 minutes attempting to find solutions while I circulated and interacted with individual groupings.  My particular focus this time was to emphasize thinking about the problems and coming up with ideas.  I used the "What do you notice/wonder?" prompt quite a bit.  There was a lot of good thinking but I definitely still see room for encouraging more experimentation.  At the end of the process I had everyone regroup for a discussion of what various people had found. Interestingly, the first pink triangle solution was analytic i.e. the student setup to equations for the lines and found the intersection.  I think this reflects the emphasis the curriculum places on these type approaches over pure synthetic reasoning.  Students don't see similar triangles quite as quickly. As an aside I had to explain the expression "broke the internet" to one of the boys as in "this puzzle just broke the internet this week."

For the second half, I switched over to looking at the patterns within polynomial deltas. See: http://mymathclub.blogspot.com/2018/05/polynomial-deltas.html  for my motivation.

Example:

x^2 + x  + 1

1    3
4
2    7         2
6        0
3    13       2
8
4    21

The way I structured this section was to demonstrate calculating deltas on a sample polynomial and have the kids then come with up with their own polynomials and look for patterns on what was occurring.

1. How many levels of deltas before you hit you and why?
2. Is there some pattern to what the second to  last value was etc?
3. What does this mean? I had one kid talk about velocity/acceleration and I think I would draw this out more if I repeated as well.

When they started coming up with enough ideas, I then suggesting trying to investigate general classes of polynomial like Ax^2 + Bx + C.  This generated lots of good distributive law practice on the whiteboards.

Finally for my 3rd prompt I asked can you go backwards as well as forwards and why do you think it does or doesn't work?

I also put up the original problem from the last post as an extension at the end for a few kids who worked quickly enough to get there.

This structure worked pretty well the one element I would improve on was to add in graphing. I only have a single computer to use but it occurred to me after the fact that I could have done some group desmos activities in the beginning with various polynomials and graphically shown the deltas.

[Another interesting angle to pursue would be function inverses. What's the behavior of the deltas when a function has an inverse vs. when it doesn't can we make a test?]

To close, I mentioned this kind of investigation is closely related to another branch of mathematics and had everyone take guesses at what it was.  As I expected no one even came close to saying Calculus which shows how mysterious it is .