I am a doughnut.

I cannot eat a doughnut without thinking of Eddie Izzard. Language caution in this video clip.

Ich bin ein Berliner aside, I needed a calorie-laden lesson for my 6th graders to welcome them back into the classroom after two beautiful spring break weeks.

(We are also two short weeks from state testing and should be reviewing for the test. But who wants to do that when one can eat a doughnut instead?!)

I got the idea for this lesson from NCTM.

Estimating Volume

I held up one doughnut and a centimeter cube and asked each student to write down an estimate of the doughnut's volume in cubic centimeters. But first, I needed to tap the kids' prior volume knowledge.

Me: So, what is volume? What is the volume of this doughnut?

Julia: Length times width times height.

Me: Hmm... Where is the length of this doughnut? Or its width?

Julia: Oh...

Matt: It's the inside of something.

Sam: But that's area.

Rapha: It's the whole thing. The doughnut itself.

Matt: No, I mean the space inside.

Julia: Oh... it's how many of those cubes fit inside the doughnut!

I always dread hearing kids give me a formula for anything unless I specifically asked for a "formula," but this time I didn't break into hives.

I asked if it would be okay that I just counted the centimeter cubes in each of the solids below to find volume. The kids immediately pointed out the empty space inside and one said, You have to add like 5 to your answer to make up for the space. Another student said, More like add 10.

The class agreed that the cylinder held more cubes than the sphere and that the doughnut would be bigger than the cylinder. I told them there were 69 green cubes in the cylinder and allowed them to change their doughnut volume estimate if they wanted to.

When a centimeter cube was placed on the doughnut box, almost everyone said, Wow! (This made me happy because it showed they thought of volume as volume, not length-times-width-times-height.) They recorded their estimates for the box's volume.

Estimating Surface Area

When I asked the kids about SA, they said the outside, all-around, total area, all the surfaces, and doughnut skin represented SA. I added that the amount of glaze could also represent SA.

I told them this grid paper had 22 by 18 rows or 396 square centimeters and asked for their estimate of the doughnut's surface area.

They also recorded their estimate of the box's surface area.

Measuring the Doughnut

To manage this potentially messy lesson, the doughnut had to stay on construction paper, and all measurements had to be taken with a paper ruler.

Students took as many different measurements of the doughnut as they thought necessary to figure out volume and surface area.

They ate the doughnuts, and Cristian wanted me to take a picture of him.

We'll wrap up the lesson tomorrow and discuss approximating for volume.

No, none of my 6th graders brought up any of these -us words: torus, annulus, calculus. However, we did a lot of estimating and measuring with visual and concrete models. We each gained 200 calories from this. 

Updated 04/17/12

In my comment, I told @Peter_Price about making a cylinder similar to the doughnut; then, I found this roll of masking tape that was close in size!

My 6th graders are familiar with area, so surface area was easy to grasp. (I felt smart when I could just peel off the tape to show the side SA.) I showed them my set of vinyl nets to the geo solids also.

I needed the students to figure out how to calculate volume on their own. I used these two models because of both the CDs and the patty paper sheets are very thin.

Student A: Find the area of one sheet and multiply by how many sheets there are.

Me: There are 1,000 sheets here, so area times 1,000 gives me volume?

Student B: No, that just gives you the area of 1,000 sheets, kind of like their surface area, but you're still doing area, aren't you?

Me: It would seem so.

I told the class that finding the area of one sheet was correct. And they also knew that it was the same area as the base of the box. Then I poured out the stack of CDs and started filling its container back up one by one. (Easier than stuffing the patty sheets back in.) I said quietly. Volume is three-dimensional.

The class watched, and soon, the hands went up. Before I could call on anyone, a few students couldn't contain themselves and violated our classroom rule of waiting to be called on before talking. Two kids yelled out, The height! Then more chimed in to say, Multiply by the height!