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!

One my time through F – E + V = 2

We did a fun project earlier in the week inspired by Dave Richeson’s book:

Didn't know about this book until today @divbyzero pic.twitter.com/v7RE1Ku7oB

— Mike Lawler (@mikeandallie) July 1, 2017

That project is here:

Looking at Dave Richeson’s “Euler’s Gem” book with kids

During the project the kids had a little trouble counting the verticies, edges, and faces of one of the complex shapes. We solved the problem with our Zometool set, but I wanted to try a different approach and printed the shapes again:

So, with these shapes I went through the project again. First a quick review:

Next, now that we have shapes that fit together, can we count the faces, verticies, and edges?

My younger son was still having a little bit of trouble seeing the number of edges, so we slowed down a bit:

Finally we did a quick recap of how the cube helped us. I was trying to get the boys to think about the shape without touching it, but wasn’t super successful.

This was a fun 2nd look at the F – E + V = 2 formula. We’ll be doing more projects based on Richeson’s book throughout the summer.