We started a new chapter in our *Introduction to Counting and Probability* book today. This chapter covers more applications of choosing numbers and the first example was counting paths in a grid.

The boys and I spent 20 to 30 minutes talking through this example and found an answer to the specific problem. The first video below is their explanation of that solution. In the following videos we talk about an alternative way to solve the problem that involves choosing numbers.

The solution that the boys described in the last video is a fairly standard solution to the problem, though the connection to choosing numbers didn’t really emerge. I wanted to help them see that connection, but didn’t want to give it away. So, talking through that connection takes up most of the next three videos. It was harder to get to that connection that I was expecting, but I think the roughly 15 minute conversation we had about that connection was really productive.

Here is that conversation:

So, definitely challenging, but a really fun project. The connection with the choosing numbers here is one of those ideas that is “obvious” after you learn it, but takes some work to see when you are encountering it for the first time. I think this is a great example of why learning math isn’t always a nice, easy straight line.