Exploring a neat problem from James Tanton

I didn’t have an specific project planned for today and was lucky enough to see a really neat problem posted by James Tanton:

P(a+b) =P(a) + P(b) + kab, P(1)=1
For k=2 get 1,4,9,…sq nmbrs
For k=1 get 1,3,6,…triang nmbrs
For k=0 get 1,2,3,..counting nmbrs
k= -1?

— James Tanton (@jamestanton) July 7, 2017

I didn’t show the tweet to the boys because I thought finding the patterns would be a good exercise for kids. We started with the k = 0 case. This case is also good for making sure that kids understand the basics of functions required to explore this problem:

Next we looked at the k = 1 case.

Next we looked at the k = 2 case and then my younger son made a really fun little conjecture 🙂

At the end of the last video my younger son thought that the k = 3 case might produce the pentagonal numbers. I had to look up those numbers ( 🙂 ) while the camera was off, but I found them and we checked:

We ended by looking at Tanton’s challenge problem -> what happens when k = -1? I had the boys take a guess and then we looked at the first few terms and the boys were, indeed, able to solve the problem!

The boys had a lot of fun playing around with this problem and I was really excited they found a different pattern than the one Tanton was asking for!