Using our Zometool set to help with a challenging AMC 10 problem

We returned from a short trip this afternoon and I was looking for a quick project to do with my older son. I didn’t have anything in particular in mind other than something to do with geometry (and, I guess, something that we could have a little fun with since we were both tired from the trip). We glanced over a few old AMC 10 problems and found problem #24 from the 2006 AMC 10a to be exactly what we were looking for:

Follow this link and then scroll down to problem #24

The problem is simple to state but not at all easy to solve: You form an octahedron by connecting up the centers of a given cube, what is the ratio of the volume of the octahedron to the volume of the cube? Working through this problem with the help of our Zometool set made for a really fun roughly 20 minute exercise.

First we just built a cube that would allow us to connect up the centers of each face. We talked about why this shape was helpful, and also discussed some simple geometric ideas that we could learn from it.

 

Next we discussed what my son thought the answer would be (and I should have put this at the end of the last video!) and then built the octahedron inside of the cube:

 

Having noticed that we can now reduce the problem to finding the ratio of the volume of one pyramid to the volume of one of our smaller cubes, we extracted the pyramid / small cube bit from our larger cube and studied it a little more carefully. My son spent a little bit of time thinking about the surface area, but this idea didn’t help us get to the volume. Eventually we looked for a little more symmetry:


Finally, we filled in the rest of the cube with the green zome edges. We now had a shape that had 4 pyramids, but do those four pyramids make up the whole volume of the cube? It turned out to be a little hard for my son to see that there was a bit in the middle that was not filled in by the pyramids. My guess is that this bit of geometry would have been next to impossible for him to see without the model in front of him, so score another win for the Zometool set! Having gotten a little bit of an understanding of the geometry, we can see that the answer to the question is that the octahedron is either 1/8 or 1/6 of the cube. We still don’t have enough information to be sure, but his best guess by the end was that we had 1/6 of the cube filled in by the octahedron.

 

So, as usual, it doesn’t take too much work to find a great project by looking through some old AMC problems. This was a particularly fun one since we were able to build an object that is sort of on the hard side to visualize. I assume that the folks writing the test were looking for kids to use the (1/3) base * height formula for the volume of a pyramid, but this topic has not yet come up in our geometry book. The Zometool set helps us get an idea of the answer without the formula, though. Hopefully this little exercise helps build up a little bit of 3D geometry intuition. I can say honestly that I didn’t realize how small the octahedron was in relation to the cube until I was holding it in my hand.

to be continued tomorrow . . . .

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