A neat problem from my son’s math team

I saw this problem on a sheet my son brought home from math team practice:

A couple of people point out on twitter than the problem isn’t worded so well. That’s my fault. The actual problem was a little longer but I was trying to capture the main point of the problem and did so in a slightly sloppy way.

Anyway, my son struggled with the problem at the math team tryouts, and the problem seemed so cool to me that I wanted to do a little more than just explain it to him.

Here’s our discussion. In the first part we just try to understand what’s going on – his first question was how you could get different shapes?


The second part of the discussion was about how to determine the largest difference in volume. This turned into a nice discussion about numbers and arithmetic. How could we tell which of the two numbers we had was bigger?


Fun little problem – and definitely a heck of a challenging problem for 6th graders!

Attending to precision

Working through two problems with my kids today reminded me of the “attending to precision” idea in math education.

The first problem was #10 from 1994 American Junior High math exam:


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You can find the problem (and the whole test here)

The 1994 AJHME on Art of Problem Solving’s website

This problem gave my younger son a bit of difficulty today. There are a couple of ideas in this problem that require you to be careful with your counting.



The next problem was Problem #19 from the 2015 AMC 10a which my old son had a little difficulty with today.

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This problem required you to keep track of a few different geometric ideas and a few algebra / arithmetic ideas in order to get to the solution. No one of those ideas alone is super tough, but all of them together really require you to work carefully.





I really liked how these two problems required the kids to work carefully. That careful work makes getting to the solution really satisfying!

The Pythagorean Theorem in our house

The boys had an early morning trip with their cousins today, so I was looking for a pretty light and easy morning math project. Since my younger son is studying square roots right now and my older son is studying geometry, I settled on a Pythagorean theorem project.

We’ve talked a little bit about the Pythagorean theorem previously, so the material isn’t totally new to either of the boys. Still, I thought a quick review of the theorem and right angles would be a good way to get going:

With the quick review out of the way we went looking for right angles and right triangles in our house. The first one we found was a window shaped like a right triangle. This window is in a place that made measuring it a little tricky, but we managed to get some crude measurements:

We weren’t able to find any more right triangles, but we did find a lot of rectangles. The first rectangle measurement we made were of the bricks on part of our floor:

Our last measurement we made was of some of the windows in our living room. These windows were a little too tall for my son to measure on his own, but they were a lot easier to measure than the first window.

To wrap up we returned to the table to work through the calculations. As we could have probably guessed from the measuring, the window that was difficult to measure gave the worst results, but the three measurements for the brick and the living room window were really close to satisfying the Pythagorean theorem. Hopefully this was a fun way for the boys to see the Pythagorean theorem in a non-theoretical way.