I few weeks (months?) ago I learned about Justin Solonynka’s train counting project from a Tracy Johnson Zager tweet:

It looked like a super fun project and we finally got around to trying it ourselves last night. This is one that I’d like to have a 2nd crack at because the first time around didn’t go quite as well as I was hoping – but that’s life I suppose.

Still . . . I was happy about the counting ideas we did get to discuss in the project, and I was happy that we got to the end of a pretty difficult counting problem. Hopefully Justin’s incredible video all by itself is enough proof of how great a project this is, but here’s how ours went:

I started by having them open the puzzle box and ask some mathematical questions:

So, the boys decided to count the total number of trains you could make, and we started counting. The first path we went down didn’t quite capture all of the arrangements, but the mistake in counting was actually pretty instructive.

At the end of the last video the kids thought there would be only two trains with one train car, but at the beginning of this video they notice that there are actually 20. This observation means that we’ll have to correct the counting from the last video and they work through this revised counting problem here.

There’s a little bit of confusion about how to revise our counting procedure, but working through that confusion is a really important part of learning to count.

Finally, I wanted to show them a slightly different way to count the trains by keeping the factorials from the second video around. I didn’t do nearly as good a job here as I’d have liked to do, but eventually they found the idea of choosing numbers.

So, we got a nice contrast from the last video where we just looked at permutations. For three train cars, for example, the contrast is that 10*9*8 arrangements is the same number of arrangements as choosing 3 cars out of 10 and looking at the 3! ways you can arrange those cars.

So, a nice counting project even if our walk through it wasn’t so great. I’m happy that Tracy shared Justin’s video – it is a great way for kids to learn about counting.