What learning math sometimes looks like – a distance and time problem

Last week the math club at my older son’s school got started. The first day the kids received a handout with a few hundred practice problems and we’ve been working through them slowly – usually 5 problems per day.

One of the problems today gave my son a tremendous amount of difficulty. I was surprised – not by the fact that it was difficult for him – but rather the extent of the difficulty. Reading through the questions ahead of time, I’d thought all 5 problems were roughly the same level. It turned out that 4 were about the same level and this one was essentially 4 times as hard.

Even when we went to talk about it on camera I thought it would take 5 minutes – it took 20. I present that conversation below without much comment. Maybe not my best work helping him out either, by the way, but the 20 minute conversation below is what learning math sometimes looks like. Not a straight line, but zig-zaggy struggle.

Here’s the introduction to the problem and the first steps toward the solution, as well as a discussion about a few things that confused him the first time through the problem:

 

At the end of the last video my son was close to writing down some equations that would help solve the problem. Here he does write down those equations, but, unluckily, with one little error that we’ll straighten out in a bit.

 

At the end of the last video my son noticed that there must be a mistake in his equations – here we search for that mistake. Part of the difficulty in finding the mistake comes from translating the ideas from the problem into math.

 

In this last part we get to the solution of the problem – and the best thing is that after our long talk here things do seem to start making sense to him. That was nice to see.

 

Using Laura Taalman’s 3D printed pentagons to talk math with kids

Saw this incredible tweet from Laura Taalman a few days ago:

Print all the pentagons! 3D print every known convex tessellating pentagon @Thingiverse: http://t.co/vOHqnv3Ukl pic.twitter.com/VtdyMnla6m

— Laura Taalman (@mathgrrl) September 11, 2015

She’s created 3D print versions of all of the known pentagons that can be used to tile the plane!! The 15th pentagon was discovered in 2015, and that discovery is discussed by Evelyn Lamb in this amazing article:

There’s Something about Pentagons by Evelyn Lamb

I thought that the pentagons and tiling patterns would be really interesting for kids to see, so I spent the day printing the “cookie cutter” version of all 15 of them. Once the printing was done, we started talking about pentagons:

 

I had each of my kids pick a pair of pentagons to talk about – my son picked #8 and #13 (as numbered on Taalman’s program linked above). Here’s what he noticed about these pentagons:

 

Next, my older son talked about what he noticed about pentagon #5 and pentagon #9 – I was happy to hear him speculate about the possible tiling patterns formed by the shapes.

 

When I was printing the pentagons I happened to notice that #5 and #6 looked pretty similar. It turns out that there are two different ways to tile the plane with these shapes, so I did print out these tiling patterns. Luckily my son had picked #5.

It was fun to see that the 2nd tiling patter wasn’t totally obvious to the kids, but they figured it out eventually.

 

We wrapped up by playing around with the interface Taalman made for the pentagons. It is amazing to be able to see and play with these patterns. The boys played with it for 20 minutes after we finished filming 🙂

 

So, a super fun project. The great thing about 3D printing (and the thing I can’t say thank you enough to Laura Taalman for teaching me) is that holding these shapes in your hands leads to great conversations!