What learning math sometimes looks like: repeating decimals and greatest common factor

My older son’s math team has some great review problems that I’m using to talk through topics we haven’t talked about in a while.

Tonight I used a problem about repeating decimals to have a really nice conversation with my older son, and then a different problem from my older son’s homework to talk with my younger son about greatest common divisors. Neither conversation went in a straight line from start to finish, but the winding path hopefully helped them understand the concepts a little better.

The conversation with my older started with a short problem involving repeating decimals. We’ve not discussed this topic recently and the problem gives him a little difficulty initially:

After we finished the problem, we returned to the repeating decimals to see if there was a different way to see why .242424 \ldots was 24 / 99. He has some interesting ideas:

Finally, we talked about a different way to understand 24 / 99 and saw how that idea applied to a few other repeating decimals.

With my younger son, we talked through a problem about the greatest common divisor. As with the talk with my older son, it has been a while since we talked about this topic. His initial idea is to find all of the factors of both of the numbers in the problem. That’s going to end up take a while, though, so we searched for a few other ideas.

(also, sorry for the door bell and dog barking!)

Unfortunately we weren’t finding other ideas, so we backed up and tried to find the greatest common factor of 16 and 24. After talking about these numbers for a bit, he saw the idea of finding the greatest common divisor by factoring the numbers into primes.

We wrapped up by finding the prime factors of462 and then finding the greatest common divisor we’d been looking for.

So, to fun conversations about topics we hadn’t talked about in a while. It was fun to hear their ideas and also to watch them slowly find a few ideas that helped them get a better understanding of the topic. Definitely not a straight line to the answers in either case, but that what learning math sometimes looks like.

A problem my high school math teacher would have loved

Earlier tonight a friend of mine sent me a link to some old problems from a middle school math contest in California.

Berkeley Math Tournament middle school problems

I skimmed through the 2014 contest for fun and ran across a problem that my high school math teacher – Mr. Waterman – would have loved:

#16: Consider the graph of f(x) = x^3 + x + 2014. A line intersects this cubic at three points, two of which have x-coordinates 20 and 14. Find the x-coordinate of the third intersection point.

I’m a little surprised (to say the least!) to see a problem like this on a middle school test, but it is a really cool problem. Properties of roots of equations was one of Mr. Waterman’s favorite subjects.