Christopher Danielson often inspires me with the mathematical conversations he has with his children...like this one. I'm working on my son's number sense, but unfortunately, most of his encounters with math typically involve being a prop for my math instruction.

Lately I've been trying to instill in my son the concept of half. And I wanted him to get used to some different terminology, so I tried to get him to use half and fifty percent interchangeably. I don't necessarily want him to understand that fifty percent means fifty out of one hundred. That would be way beyond him. I just want him to acknowledge that 50 percent means one-half.

The other day, he and I were waiting for new tires to be put on our car. We decided to go to a local fast food establishment while we waited and ate breakfast. We haven't eaten there in forever, and there was a reason for that...it was disgusting. The food was really greasy. The only decent thing on the menu was the strawberry smoothie.

While we were drinking our smoothies, I noticed that his cup was about half-way full.

Me: How much of your cup is filled with smoothie?

Emmett: Half.

Me: Great! And what percent is that?

Emmett: 100!

Me: No. Remember, we talked about this the other day.

Emmett: 0!

Me: No, try again. (Not very good leading questions on my part. And we've only discussed three different percents, so the only other choice was inevitable...)

Emmett: 50!

Me: Right! Fifty percent! Now compare that with my cup.... What percent do you think I have?

He looked at both cups. The height of his smoothie was the same height of my smoothie, but my cup was taller, so it was definitely less than half.

Emmett: It's 50 percent!

Woah! Time out. Why did he say fifty percent? Because the smoothies were at the same height, and therefore, must be the same percent? Because he only knows three different percents, and anything between 0% and 100% defaults to 50%? Because he was just getting good at saying "fifty percent" when I asked what percent something was?

I showed him how my cup was actually less than halfway full and that this would mean there was less than fifty percent. But at this point, his mind was gone...content to drinking his smoothie and ignoring my lesson for the day.

After reflecting on this, I realized that there could be a pretty good middle school lesson here.

Was it accurate to say that Emmett's cup was 50% full? How does the shape of the cup affect this?

What percent of his cup is full if the height of the smoothie is half the height of the cup?

What percent of my cup was full (or empty for you pessimistic types) if it was at the same height as Emmett's?