*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.

source: Geometry, copyright Holt, Rhinehart, Winston, 2007, Pg 684 #8

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.

Photo Credit: Flickr user “kimberlykv” – used under Creative Commons

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.

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that is a cool idea, could they use google sketchup to create their own shapes?

I’m sure they could. I don’t have a lot of experience with Google Sketchup. Can you create functions? Could you see how volume and surface area change as you visually change the image?

Dude, This is great stuff. My geometry class has been near yours in pacing all year, and It has been helpful to see the struggles, and the questions you’ve experienced with your students. I can admit that I haven’t always found great ways to keep it in real world context. this is anopther great example.

Thank you. I agree that it can certainly be challenging to find authentic contexts and this one certainly has its limitations, but the students were willing to engage it and it led to some good discussion, which are victories unto themselves.

But, you can have this and anything else on here that seems useful.

I love this. Just came across it reading your more recent post on the pans…my next stop….

The original textbook image doesn’t even make sense since the radius is 4 and the width of the side is 8 yet the cylinder does not appear to touch the edge.