Not every day teaching the boys goes super well, but when things do go well it is such a awesome feeling. Today was one of the great days.

I started with my younger son today. The math topic of the day was what we’ve been studying all week: greatest common divisor. We started with a relatively straightforward problem to review the topic – find the greatest common divisor of 32 and 48. He’s still learning about factoring, so I’m always happy to start off the day with a problem that gets him practice on an older topic.

The next problem was to find the greatest common divisor of 99 and 100. The answer to this problem is “obvious” if you have experience with number theory, but if you are just learning it doesn’t seem any different than any other problem. He jumped in and found the factors of 99 and 100. The fact that 99 and 100 have no divisors in common led to some confusion, so I sent him back to the beginning – what are some things that we notice about the two numbers?

After thinking for a little while, he noticed that since the difference between the two numbers was 1, the greatest common divisor had to be 1 because no prime numbers could divide both 99 and 100. As I prepared to move on to the next problem, my son turned to me and said:

I think the difference between the two numbers tells you something about the greatest common divisor. I wonder if anyone has ever noticed that before.

Ha. I was sort of stunned for a moment, honestly, but then I told me that someone had indeed noticed that before. Unfortunately we were nearly out of time, so we’ll have to pick up this topic after Thanksgiving. Nonetheless, that little observation made my day.

Our video was about a GDC problem that involves slightly more complicated factoring:

The topic with my older son today was a little bit more about the quadratic formula. We discussed some interesting properties of solutions when the coefficients of the quadratic expression are all real and all rational. One of the questions went something like this: If a quadratic equation has all rational coefficients and you know one of the solutions is 1 + $\sqrt{2}$, what is the other solution? We had a nice discussion about this topic and it felt like some of the ideas about real, rational, and complex numbers were resonating with him.

As part of that discussion I drew a few pictures of parabolas. We’ve not really talked about graphing quadratic expressions yet – that’s the next chapter – but we have been talking about the geometry informally. The interesting question he asked me that completely changed the course of what I’d planned to talk through this morning was this:

Why do you always draw the parabolas going up? Why not sideways?

Why indeed! What a fun question – we discussed this question in our movie:

I love the questions, and I love when we get to talk about a topic in a way that is quite a bit different than I planned.

One of those days when you really get to experience the joy of teaching.