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 calculus class. These are my 6th grade babies. But didn't we do okay with approximating the volume of a torus via my donut 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 donut. 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. Or it was just my imagination. 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?