My son and I were walking back from the park, and he was trying to convince me that I had time to play Clash of Clans, and that I really should start playing it.

He said, “Daddy, if you played one third of one quarter of your time, that would not be very much. See look, if you had one minute, one quarter would be 15 seconds, and one third of that would be five seconds. And if you only had 1/3 of that time, that would be five seconds divided by three. Daddy, what is five divided by three?”

I asked him, “What do you think it is?”

He replied, “One and one third, I think.”

I asked him, “So let’s check. What would one and one third times three be?”

He said, “Let’s see. One times three is easy. That’s three. One third times three is one. Oh, that would be four total, not five. Maybe it’s one and one half?”

I asked him, “We can check that. What would one and one half times three be?”

He said, “Hrmm. Four and a half. That’s not right either. It has to be bigger. What about one and three-quarters? If I multiply that by three, that’s six and one quarter. That’s too big. So it has to be between one and one-half and one and three quarters. Is two-thirds between one half and three quarters? Let me check. Two thirds times three is the same as two thirds plus two thirds plus two thirds. That’s one and one third plus two thirds is two. So it works! Five divided by three is one and two thirds!”

I then pushed my luck and asked, “What’s half of one third?”

My son said, “One quarter! See,” while showing me his hands, “here’s one quarter and here’s half of one third. They are the same size.”

A woman passed by at this point and overheard our conversation and my question to my son. She stopped to listen.

“Okay, so how many halves of one third fit into a whole?” I responded to my son. The woman at this point started to speak, but I said to my son, “And remember the point of this is for you to figure this out yourself.” The woman looked at me and nodded and then stopped speaking.

After some thinking my son said, “Six, because three thirds is a whole. Oh, but four one-fourths fit into a whole, so they aren’t the same size.” My son then said, “One half of one third is a sixth, right? Because it fits six times into a whole.”

I push my luck again, “What about a half of a sixth?”

My son said, “One twelfth, then one twenty-fourth, then one forty-eighth. Oh those are the same numbers from Threes! Except instead of getting larger, they keep getting smaller.”

We stopped our conversation about fractions at this point as my son started talking about how in the games of Threes, the numbers aren’t actually multiples of three and the game is misnamed.

This is the first time I have heard my son talk about fractions other than thirds and halves. I saw my role here  was to find the right questions to ask at the moment that my son has an intellectual need for other fractions. These kinds of conversations are common-place between me and my son, but still every time I am amazed by how much thinking about this he must be doing when I’m not around.