Geometry, arithmetic, optimization, logic, and question asking.
SEPTEMBER 18, 2012: “What would it take to fill this room with popcorn?”
- “a big popcorn machine”
- “the machine running a long time”
- “a lot of patience”
- “corn, heat, oil”
- “a small factory”
- “a lot of people”
At this point in this term’s first math circle, participants began to debate whether these things were truly necessary to answer the question. Wouldn’t it depend upon how we defined the question? The students then built up a new list: questions that need to be answered before any calculations could begin:
- How big is the machine?
- Are we using unpopped or popped corn?
- Can unpopped corn be legally considered popcorn?
- If we use popped corn, would it be bagged?
- Which variety of corn optimizes packing efficiency in terms of space and uniformity?
- How efficient is the machine?
- What is the point of entry into the room for the popcorn?
- Is it even possible to fill the room without drilling a hole in the ceiling?
- Are we allowed to drill a hole in the ceiling?
- What is the size of the room?
- Should we assume that there is none of this furniture in the room?
- Do we have enough people to do this?
- How efficient are the people?
- What’s the most environmentally-sensitive method?
- What’s the most cost-effective method?
- Do we care about cost?
- Do we care about the time?
- Do we care about the environmental impact of this project?
- Who are “we” (and therefore what can we assume about the answers to these questions)?
I acted as secretary, writing the questions on the board as the students discussed them. The initial debate about the implications of pre-popped versus unpopped kernels was substantial, as was the debate about who “we” are. The students went down the question list and marked with a check the questions we knew we could answer. It turns out that there was only one of those. Then we marked with an X those that required assumptions to answer. The debate about assumptions took so long that the group decided to vote on assumptions rather seek consensus. At this point - about 30 minutes into class - I had said very few words. Now I interjected a few times to nudge them oh-so-slightly toward a mathematical/theoretical approach to the question rather than an engineering/applied approach. (“Why are we here in a Math Circle?”) We talked about what kind of real-world people would be faced with such a problem. (“The people at SmartFoods must have to do this all the time.”)
The students decided to measure the room with a measuring tape. “But wait,” pointed out D, “we don’t need to measure the area of the floor, because we could just measure a single ceiling tile, and then count up those tiles.” P agreed, adding that “the floor and the ceiling are the same size.” R and G felt strongly that the measuring tape would be more accurate. So they split into two groups and measured both ways.
Another shortcut was suggested by A: “Most modern buildings have a standard ceiling height of 8 feet, so can’t we just assume that this one is 8 feet?” She was looking at me.
“I don’t know. Can we?” I lobbed this question back to the group, as I did with most every question directed at me.
Most people felt that if we’re going to all this trouble to measure things, let’s be exact with the ceiling and make sure. While S attempted to stretch the measuring tape to the ceiling, R demonstrated that we could stand on a chair and be sure.
The group was so actively engaged that I left the room for a few minutes. When I returned, the students came back into a circle and reported their results. It turned out that A’s assumption about ceiling height was correct. But, the calculated ceiling dimensions and measured floor dimensions were different from each other.
R pointed out that the ceiling calculation didn’t include the strips that separated each tile. Then C and G pointed out that those strips were not even uniform in size. The group therefore decided to use the dimensions obtained by the tape measure on the floor. “What should we do with these measurements?” I asked.
“Figure out the size of the room,” said N. N, P, G, and several others then suggested calculating the space that a small amount of popcorn takes up, and multiplying to get the answer.
We moved to the other board to begin calculating. After a discussion about whether to and how to calculate area or volume, the group decided to multiply 24 feet 4 1/2 inches by 13 feet 8 1/2 inches. Hmmm…. how to do it?
“Could we convert to decimals?” suggested R. We tried various methods, but none made sense. P suggested converting feet to inches. Both R and C multiplied 24 feet by 12 to get 288 inches, but not everyone was convinced that this number was correct. We estimated to determine that 288 was in the right ballpark. A wanted to get her calculator to check, but she was carried into the next calculation by the rest of the group’s sentiment that two hand calculations and an estimate were enough in this case, since no one’s life depended upon the calculation. C called out the answer to the next calculation, I put it on the board, and we moved on from there to end up with the dimension of the room in inches. Then we were out of time.
I talked about what we’d do next time, and everyone left. That is, everyone left but A. She stood there alone, staring silently at the calculations on the board as I cleaned up. Soon G came back and joined her, as did R. Soon, A had her calculator out, and they double checked the numbers that had been calculated by only one person. Aha: one of those numbers was incorrect!
All this talk about measuring reminds me of two more things I want to tell you. First, one of my mentors, Sue VanHattum, is featured in a short film about doing math with young children at the Richmond Math Salon. Second, Paul Lockhart, of A Mathematician’s Lament fame, has a new book out called Measurement. I haven’t read it yet, but I’ve heard that he presents a beautiful case for math as art.
And even though we were measuring today, that’s just a small part of what came up. Before class began, G and A asked me, “What’s the topic of this course?” I was reluctant to say, since mathematics, like most subjects, cannot really be segmented into discreet topics. Everything overlaps. To answer this question, I said vaguely, “data.” They were excited because they imagined doing probability. I was thinking we might do some statistics. Today’s question involved geometry, arithmetic, optimization, logic, and maybe most importantly, question asking. I wonder what we’ll unexpected touch upon next week? My plan is to revisit our discussion about double-checking calculations. Then, we’ll continue and possibly expand the popcorn problem as I talk about the interesting history behind these types of rich and compelling math questions.*
Looking forward to seeing you and your children then.
*Thanks to Amanda Sereveny for the popcorn question.