# (not) Explaining “Half”

“If you can’t explain it to a 6-year-old, you don’t understand it yourself.” – Albert Einstein

“I’ve drank half my water”

My daughter isn’t a 6-year-old… yet. I get a couple more months before Dr. Einstein’s quote applies. Regardless, today I found myself at the breakfast table trying to make sense of one of those topics that exists in the intersection of math and common language.

It’s good to make mathematics common place, but the problem with making technically-specific terms commonplace is that it often leads to the usage of the word becoming a bit less technical. ELA teachers will tell you that this has happened with the word “literally.” (For more on “literally”, this post will literally blow your mind.)

In that same spirit, today’s breakfast was abuzz with the word “half”. My daughter is comfortable with the word “half.” She will often take a couple of gulps from her cup leaving ~5 ounces in a 6-ounce cup. Then she’ll say, “I drank half my water.” To her that means that she has consumed a some portion of her water, more than “just a little”, and less than “all” or even “a lot”. It seems the developing spectrum looks something like this:

Water Drinking Spectrum

It’s clear that spectra like this make clear the need for fractions, but that’s a discussion for another time.

Well, today, she asked about what “half” means. Which was a question that I should be qualified to answer, but you wouldn’t know it from how the conversation went.

Her: “Dad, what does ‘half’ mean?”

Me: “Well, it’s like broken into two pieces that are the same size.”

Her: “No, like, I drank half my water”

[See? My explanation didn’t work because I was thinking the two pieces were the water in the cup and the empty space. She only sees water. That’s only one piece.]

Me: “That means that the amount of water that you have already drank is the same as the amount of water you have left.”

Sage: “I don’t get it.”

Me: [holding up clear cylindrical glass that is “about half-full”] “See? This glass has water up to here [I point at the water line], and then the rest of the glass is empty. It’s like there’s the same amount of water and empty.”

Sage: “Um… Maybe we should talk about this when I’m older.” [Sage continues eating oatmeal and the topic changes]

Good thing she’s only five. I have a few months to prepare for Dr. Einstein’s test of whether or not I really understand the idea of “half.”

# Dripping Water Problem, Act I

So, what kind of math could we do with this?

Seems like it depends on whether you are in an Algebra class, a geometry class, a physics class, or a calculus class.

What are the questions that come into your mind when you watch this?