In Algebra II, our second semester begins with function operations and inverses. Since function composition is a very commonly used concept outside of the math classroom, I wanted to introduce the idea within a context.
So I opened the lesson with this picture:
The goal for my students was to determine how much I paid for gas the day before. I gave them some information:
My students wanted to know how much gas was and then proceeded to spend 45 seconds commenting on how inexpensive gas is at the moment (I figured this would happen given that my a lot of my students are starting to drive).
Then I told them how many miles I had driven:
Now that they knew the price and the fact I had driven 117.2 miles, I wanted them to make a prediction.
I got anything from a $15 prediction to a $40 prediction. I was surprised to see that when I probed them about why they predicted the price that they did, quite a few of my students used the fact that I was filling up about a half tank of gas. Questions and comments started flying about how many gallons of gas my car’s tank would hold. I didn’t have an answer for them, and I tried to question how that would be a useful piece of information. They said that if they knew how many gallons the tank was in total, they could figure out how many gallons half a tank is and use that to determine how much I paid.
A conversation about how they didn’t know exactly what proportion of my gas tank was being filled. They weren’t happy…they said it was about half of a tank…I said that using the word “about” makes that information unusable…one or two students still weren’t happy.
However, we did come to the conclusion that it would be nice to know how many gallons of gas my car needed…and I could help them with that:
Then, I sent them off in their groups to determine how much I paid. Five minutes later, I saw a lot of work that looked like this:
This was typical of the work I saw from my students. There were a lot of good conversations about keeping track of units and how the process worked. After discussing with students the answer:
We had a conversation about what the process entailed. They decided that they took a mileage and turned it into a number of gallons, then took those gallons and turned it into a price. I then defined this as a composition of functions. Then we decontextualized and practiced.