This morning we started looking at combinations and “choosing” numbers. I think this is one of those topics that seems much easier after you’ve learned it than it seemed when you were learning it. After a quick introduction to the idea of choosing number, the boys worked on an example problem that involved counting the number of games in a round robin tournament:
The were able to work through that problem, but I thought that thinking through it again would still be helpful, so we reviewed the problem from start to finish.
First – how did we think about this problem before we talked about choosing numbers?
After that quick review we began to talk about choosing numbers. I let the boys explain their ideas about these numbers. One fun thing that happened in this part of our project is the boys discovered that there is a little bit of symmetry in these numbers:
Now we looked at patterns that arise in the choosing numbers. There’s a little trick that I mistakenly thought we’d already talked about – that 0! = 1. After telling them that was simply a definition, we moved on to finding a fun pattern that comes up in the choosing numbers – Pascal’s triangle!
Finally, as a way of confirming the connection to Pascal’s triangle, we looked to see if the addition relationship between two rows of the triangle also shows up in the choosing numbers. This is one of the first examples of a combinatorial proof that the boys have seen!
So, a really fun project showing that counting has some surprising connections to other topics in math. It was fun to hear their ideas (and their surprise) when they found the connection to Pascal’s triangle. Showing them a basic combinatorial proof at the end was fun, too – those proofs can be absolutely amazing!