Disclaimer right at the beginning of this post – we made a mistake in the last video and I didn’t notice it until I was writing up our conversation. I decided to write up the project anyway because mistakes are a part of learning and talking about math.
So . . . yesterday we did a project inspired by Matt Parker’s Menger Sponge video. Here’s the project:
and here’s Parker’s video:
Yesterday we studied the 2d version – the Sierpinski carpet – and today I wanted to talk about the 3d shape. Since the 2d shape was fairly complicated, I thought it would be worthwhile to review yesterday’s confirmation before diving into the new 3D shape:
After that review, we talked about the 3d version. For the first part of the 3d discussion, we talked about removing the middle “thirds” of each part of a cube. How many cubes do we remove? How big are they?
The 3d geometry ideas here are great for kids to talk through. The conversation about the shape is also a nice fraction conversation for kids.
The next part is where we started to make a mistake – I’ll get to that in a second.
Here we looked at the next step in the modified Menger Sponge process. We began to discuss what happens when you chop up our remaining cubes into 5 pieces (on each side) rather than 3 pieces. The kids were able to see this step fairly easily on the square, but it was a little more confusing on the cube. Luckily we had one of the Menger sponges to aid in the visualization.
The mistake we made was in counting the cubes that get removed when we chop the sides into 5 pieces. We forgot we were chopping each side into 5 pieces and counted as if we were chopping each side into three pieces.
When you chop the sides into thirds and “punch out” each middle third, you remove 3 small cubes with each “punch” and have to put back 2 because you punched out the middle cube 3 times.
When you chop the sides into fifths, though, you punch out 5 cubes with each punch, so you really are punching out 5 cubes with each punch and having to put back 2 at the end. So, rather than taking away 7 cubes, you take way 13. In general, chopping the sides into (2n + 1) pieces and removing the middles results in taking way 3*(2n+1) – 2 cubes. Unluckily we always said that we took away 7 😦
For the last part we extended our ideas to higher levels of the modified Menger sponge. Our mistake of taking away 7 cubes every time came along for the ride . . . We’ll have to correct this mistake in a new project:
So, a neat project even with the mistake. I really like using the geometric ideas from Parker’s video to talk a little geometry and talk a little fractions with kids.