Why Wait for Calculus?

(I just had my very best lesson yesterday, on a Friday, thank you. I feel almost brilliant right now. And I only feel like this once every 47 years, so please stay and read this post!)

My own kids tell me they will stock up their dorm rooms and apartments with junk food and soda when they move out to make up for these years of deprivation. (And this is supposed to make me feel bad.) So when I intentionally bring home a snack, like this bag of kettle corn, I usually find it empty within 24 hours. But seeing the empty bag made me think of a volume activity that I could do with my 6th graders with all these other bags of Orville Redenbacher's popcorn.

But the activity I had in mind — maximizing the volume of a box — is commonly done in a pre-calc or calculus class. These are my 6th grade babies. But didn't we do okay with approximating the volume of a torus via my doughnut lesson? So, why can't we do this too? I have to get rid of the popcorn.

I randomly assigned the kids in pairs, gave each pair two sheets of white copy paper. I told them to use one paper at a time to make a box — the goal is to make the box as big as possible so it'll hold the most popcorn. But the box must be made simply like this: cut off 4 corners from the the paper, then fold up the sides and tape them together. I used a half-sheet (so they couldn't duplicate mine) to demonstrate what I meant.

They quickly went to work. A few students were NOT cutting off square corners, so the top edges of the sides didn't line up. Two groups folded in their papers, in addition to cutting off corners, so they had to re-do.

Ryan and Annamaria wanted to make a shallow box. Ryan said, "... it doesn't matter how high it is."

Rapha and Cristian made the biggest corner cuts that I saw in the first round.

Mike's and Roy's first box was the shallowest in the class, but they changed their mind for their second box.

With 10 minutes left in our first hour together, I asked the kids to measure the box and find the volume. They had no trouble with this since we did the doughnut. They recorded the volume inside each box, and I tacked them on the board. (The butter seeped through in few of the boxes.)

Well, that was fun. I pointed out that two of the bigger boxes were over 1,000 cubic centimeters. The bell rang. I said, "We'll wrap up this afternoon."

I didn't know what I was going to do to "wrap up" the lesson. The microwave actually overheated — my room stank of greasy popcorn.

There was a confidence in me, however, that the kids would help me figure out how/where this lesson could go next. I began the afternoon hour by going over what they'd learned in the morning. They said:

The four corners must be of the same size. (I never told them this in my instruction.)

Each corner must be a square. (I didn't tell them this either. Not everyone was convinced of this, so I cut non-square corners to show them.)

There was a limit to how big the square could be. (I loved this! And this made me ask, "Is there a minimum to the size of the square?" Their eyes squinted, almost as if they were trying to "see" how small these corner squares could get. One kid said, "No. Technically, no.")

The volume numbers that people wrote down could be wrong.

By then they understood the different boxes and their volumes depended on the size of the corner squares that would get cut off. We focused on this. I asked them to draw a 10 x 12 rectangle in their math journal. We removed 1 x 1 corners from this rectangle and found the volume. I guided them through the next 2 x 2 corners. They continued on their own.

Then I gave each kid another white piece of copy paper. We measured the length and width of the paper and agreed that the paper was 28 cm x 21.5 cm. I asked them to build a systematic table like the one they just did in their journal. I said something like this, "Because you now know how to figure out the volume without actually cutting and making the boxes, see if you could figure out what size square the corners should be to maximize volume."

I saw kids high-fiving each other, "The corner has to be 4 by 4!" Rapha and Cristian beamed after congratulating each other, "That was one of the boxes we made!" We ended class with that. I swore I felt myself tearing up.

On Monday we'll play around with this applet.

And we'll ask Wolfram Alpha to take the first derivative of the volume for us. (I'm pretty sure the class could write this equation V = (28-2x)(21.5-2x)(x) for me to enter into WA.) Well, I actually just did it, and WA gives the side of our corner square as approximately 4.01965. My kids got 4 — pretty damn good for 6th grade brute-force math. Now that I'm writing this, however, I am really most proud of how well the kids had worked together. I randomly paired them up — a handful of the pairs were like the odd couples: high/low, shy/outgoing, squirrelly/quiet, jock/nerd, princess/cowboy. There was not a whisper of whine when their names were called to pair up. How did I get so lucky?

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 wonderful weeks of spring break.

(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 for the doughnut's volume in cubic centimeters. But first, I needed to tap the kids' prior knowledge of volume.

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. They also agreed 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 top of the doughnut box, almost everyone said, Wow! (This made me happy because it showed they were really thinking of volume as volume and not as length-times-width-times-height.) They recorded their estimates for the box's volume.

Estimating Surface Area

What the kids said when I asked them about SA: the outside, all around, total area, all the surfaces, doughnut skin. I added that the amount of glaze could also represent SA.

I told them there were 22 by 18 rows or 396 square centimeters on this grid paper 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 were done 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 talk more about approximating for volume.

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

Updated 04/17/12

I was telling @Peter_Price in my comment about making a cylinder similar to the doughnut, then I found this roll of masking tape that was really 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, kinda 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!