What learning math sometimes looks like -> averages

My older son struggled with a problem about averages from the 2000 AMC 8 this morning. Tonight we revisited it and it turned out to be an interesting example to work through with both kids.

Problem #23 from the 2000 AMC 8

Here’s the problem:

There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is 6 4/7, then the number common to both sets of four numbers is . . . . ?

My older son started with a nice picture of the situation, but then turns down a difficult path by assuming that the numbers that average to 5 are all 5’s and the numbers that average to 8 are all 8’s. After seeing that this approach is going to run into trouble he finds an different – and better – path to the solution.


After going through it with my older son, I thought that the problem would be accessible to my younger son, too, so we gave it a shot. He also started down the path of assuming all 5’s and all 8’s for those two parts of the problem. Although this approach is a tough way to tackle this problem, he stays with it until the end. There’s some great insights about arithmetic from him along the way.


So, a nice example of how a 4th and 6th grader approach a problem a bit differently. Hopefully a nice example of what learn math looks like sometimes, too.

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