Looking at an old AMC 10 problem with our Zometool set

  

Saw a neat problem from the 2015 AMC 10b earlier in the week:

Problem #17 from the 2015 AMC 10b

I thought it would be a fun problem for a Family Math project, though I did misjudge how long it would take to building the shape out of our Zometool set. We started off looking at the problem and I asked the boys for different ideas on how to approach it.

Right now both kids really like diving into calculations immediately, so I was trying to slow down that process and get some ideas about the geometry. Eventually my older son suggested that we build the shape.

There’s also some great conversation about the shape itself here. It is always really interesting to hear how kids think about geometry.

Next we built the shape off camera. As I said in the introduction, building the shape took longer than I expected. BUT, the nice thing about the building process is that it gave the boys time to get to know the shape somewhat. One puzzle for them involved trying to understand why the green struts didn’t fit quite right.

There was also some really good conversation around the base of the pyramids.

The question of whether or not the base of our pyramid was a square wasn’t quite resolved in the last video, so we went to the whiteboard to explore that question more carefully.

This second conversation seemed to help resolve the issue.

Now that we found that our shape was a rhombus and not a square, we had to figure out how to calculate the area. How do you do that?

Now we headed back to the Zometool shape to calculate the volume of the octahedron described in the original problem:

Finally, to wrap up we went back to the original problem to see if our solution was correct:

So, a really fun project even with the extra time required to build the shape. One of the many great things about the Zometool set is that kids can build shapes like this one rather than having to think about them in their heads or view them on a computer screen. Being able to have that shape right in front of you lets you explore lots of fun problems like this one!