What you see below is the original post from 2013. Since that time, this problem has taken on some alternate forms. One alternative was suggested by Jeff in Michigan and appeared in a post in March of 2014. The other alteration I made myself in Desmos in May of 2015. Feel free to read on and consider checking out the many different ways there are to approach this interesting (and delicious) mathematical situation.
We’ve started our 3-D unit.
Once we get into the volume and surface area measures for 3-D figures, the textbook leads us to shapes called “composite figures” that look like this.
This can be a tricky image for student to try to work with, mostly because they’ve never seen anything that looks like that before. But they’ve seen composite figures. They are everywhere. But, removing the context can be enough to take this very applicable, contextual concept and make it abstract enough to be confusing.
In reality, composite figures are wonderfully applicable. (I say again, they are everywhere!) So, here’s my question: Why do we insist on giving them an abstract picture to start with? Why not start them with one of the many composite figures that will draw the students into a real context.
I present exhibit A: The Wedding Cake.
Your basic wedding cake, like the one shown at the top is three cylinders of differing sizes stacked on top of each other. What I like about the Wedding Cake is that the measures of volume and surface area matter in real time and without too much more background than a bit of story-telling (which I love to do).
Now, let’s toss an additional cylinder into the mix.
Now, you’ve got yourself a math problem!
Question to the students: How much can the baker of the above cake expect to spend on the lemon frosting that is on the exterior of that cake?
… and see where they go with it.