Approaching an elementary counting problem several different ways

Had sort of a busy day today with many hours driving. When we finally got home we planned on working through a few new counting problems, but everyone – me included – was pretty tired. Instead I had the boys work through a single problem from the new chapter in our Introduction to Counting and Probability book :

How many games are played in a round robin tournament with 8 different teams?

What caught my attention was that each kid’s approach was pretty different. We went through a few different approaches to the problem (with 5 people) for our project today:

First up, my younger son’s solution – he essentially lists out all of the games and is careful to avoid overlaps:

My older son’s approach was more geometric:

Next I showed them an approach where you do not avoid over counting:

Finally – to end with a bit of fun – is there a geometric way to understand this over counting idea:

So a fun and quick little project. One of the surprising things that comes up when you are learning to count is that all of these connections between geometry and numbers seem to come up all the time!