Today we took a look at the last problem. The kids are in 4th and 6th grades so they are at pretty different levels in mathematical development. I decided to have them work individually on the problem so that each kid could use his own ideas to solve it.

Here’s the problem (from the link above):

First up was my younger son. It was a struggle for him to see the pattern that was repeating. The first part of the project was primarily him working to see if the pattern that he thought repeated actually did repeat.

When the first pattern didn’t quite work, we tried again – this time he found a pattern that, indeed, did repeat:

At the end of the last part we needed to figure out how many of our equilateral triangles were part of the repeating shape. His approach was to tile the hexagon with equilateral triangles – a different approach than my older son used later:

Next up (about an hour later, so better lighting!) was my older son. He also struggled to find the repeating pattern. He even wanted to go to the white board to try to work there, but I wanted him to see the pattern in the Zometool set. Eventually he was able to reason that there must be two triangles and one hexagon in the repeating block. It was interesting to me that he came to that conclusion via calculation rather than seeing the pattern.

Finally, having seen the pattern we still needed to calculate the ratio of area of the triangles to the area of the repeating pattern. Again he approached this problem via calculation rather than through geometric ideas:

So, an interesting problem. The calculation itself isn’t that interesting, but finding the repeating pattern in the tiling is an interesting exercise for kids – as is finding the number of triangles in the repeating pattern itself.

I’m missing something basic about the hexagon-plus-two-triangles tile. Each hexagon touches six triangles, and each triangle touches three hexagons, and I’d like to derive 2 = 6/3 from this. Is there an obvious missing step?
(Of course I can see how to tile the plane with a hexagon-plus-two-triangles; I’m just trying to remove the creativity from that step.)

I saw the step you are looking for the other way around.

If you use as the repeating shape the hexagon plus 6 triangles and check how this tiles the plane, you’ll count each hexagon once and each triangle 3 times. So, to count correctly you need to divide the number of triangles by 3.

## Comments

I’m missing something basic about the hexagon-plus-two-triangles tile. Each hexagon touches six triangles, and each triangle touches three hexagons, and I’d like to derive 2 = 6/3 from this. Is there an obvious missing step?

(Of course I can see how to tile the plane with a hexagon-plus-two-triangles; I’m just trying to remove the creativity from that step.)

I saw the step you are looking for the other way around.

If you use as the repeating shape the hexagon plus 6 triangles and check how this tiles the plane, you’ll count each hexagon once and each triangle 3 times. So, to count correctly you need to divide the number of triangles by 3.

Yeah, that’s the sort of thing I was looking for!