# Two Area Problems

Geometric solids week is coming! Prior to starting a unit on surface area and volume, I wanted to spend a couple days working on their concept of area. I pillaged a couple of text books for some ideas.

Problem One:

We ended up having a great conversation about the best way to answer the question. They immediately wanted to tell me what they knew. I showed them the results from the poll here on my blog (thanks to all of you who voted), and asked them to vote themselves:

Then I asked them to justify their votes. Some students started to go into defining what larger meant (a really important idea). They wanted to give me that they “knew” that the soccer field is larger, but did not have a mathematical justification. Finally they asked for some measurements:

Some calculations ensued:

They eventually decided the area of the soccer field was larger. A student made a great point that the football field is longer and that we had to make sure we were clear about how we defined “larger.”

Problem Two:

These deals were taken straight from Papa John’s website. They immediately wanted measurements.

I wanted my students to explore which deal was better, and sent this problem home with them to work on. The work you see on the slide did not come easily. They came back with thoughts that I wasn’t expecting. I felt like this question was tailor made for area, yet a few of my students didn’t go there at all. They wanted to make the argument that they get “more inches of pizza” with the larger pizza deal, so it was worth it. We took some time to explore if an extra inch of pizza is the same whether you start with a one inch pizza or a fifty inch pizza:

They decided that that an extra inch isn’t always the same in a circle. Eventually, with more probing than I expected, we got to the idea of a unit rate. I know they’ve seen unit rates and they’ve seen area, but I don’t know that they’d seen them together before. To everyone’s surprise (including mine) the unit rates for both deals works out to cost about \$0.06 per square inch of pizza. Therefore, there isn’t a better deal…only how much pizza you actually want!