A Matter of Perspective (Eye of Horus 5)
(October 15, 2013)
Students arrived to see 7 arrays of blocks on the table, with the following numbers of blocks in the piles respectively: 1, 2, 4, 8, 16, 32, and 64. Well, that was the numbers from one side of the table. When we looked at the arrays from the other side (designated “Egypt”), the numbers in the piles respectively were these: 64, 32, 16, 8, 4, 2, 1. My old sign that said “Do not touch my number” now said “Do not touch my number???”
This block set up evoked a variety of emotional reactions in kids.
“Oh no,” said K vehemently. “I am NOT participating in that. You didn’t even tell me the answer to the other one.”
“Not this again,” R exclaimed with a dramatic flourish of her hand on her forehead.
V and S ran up to the blocks and started counting them excitedly.
“What is it?” asked M, R, and L curiously, noticing that the word “number” was crossed out on the sign. Of course I sent that volley straight back at them.
Soon everyone was making conjectures. On the “American” (1, 2, 4…) side of the table, people realized pretty quickly that each number was double the one to its left. Not many kids were willing to count the blocks in the final array, but R identified it as 64 by noticing that it was an 8x8 table. D verified this as he had “memorized the binary sequence.” The rest of the group looked at him as though he were speaking Martian, so it was refreshing for the kids to see multiple approaches to a calculation.
People got frustrated attempting to decipher what the view from the “Egyptian” (64, 32…) side meant. At this point, everyone did identify that the blocks represented a function as opposed to a number. Some were even able to extrapolate that the next number in the sequence* after 1 would be ½, but an explanation of the rule itself was elusive. Then someone asked, “What does this function have to do with Egypt anyway?”
Then I took out a brown piece of paper, called it a brownie, cut it in half, and gave one piece to a student. I did the same thing with the remaining piece, and repeated and repeated and repeated. Each subsequent student got a smaller piece of brownie (1/2, 1/4, 1/8…), until the twelfth student had a speck.
“What if the parents sitting outside wanted some too, and this was the best brownie in the world, so they wanted pieces of THIS one. Could we keep cutting it to feed all of them? To feed more people? To feed everyone in the world?”**
Students quickly fell into two camps: the theoretical and the applied. The former argued yes, while the latter argued no. We had a fascinating debate, followed by a discussion about theoretical versus applied mathematics. (This discussion would be interesting to continue with your kids at home.)
A more problematic debate ensued: whether to keep cutting and examining paper, or stick to the math questions at hand. People vehemently took sides. I handled this by doing neither. Instead, I told them an Egyptian mythology narrative about Horus’ eye. Then I gave everyone a paper Eye of Horus. “Is it the right eye or left eye?” I asked. More debate. Then I handed out another Eye of Horus – this one with numbers all over it. Students discussed what it might mean, which eye was right and which was left, and whether our own left and right eyes are identical. The take-home math word from this discussion would probably be symmetry.
I asked the kids what the numbers on the Eye of Horus might mean. “You don’t know?!” exclaimed J, as she picked up some of the brown paper and started tearing it into squares - a seeming act of protest against a seemingly-impossible question. I say “seeming” because I asked her whether she had some sort of instinct that the numbers in the Eye were related to the numbers we generated by paper cutting. She, and several others, nodded. (Seems that the truth does come out in jest.)
I turned the group’s attention to the board for two function machines: “America” (doubling) and “Egypt” (halving). The in and out numbers ended matching the numbers of blocks in the arrays on the table, and also some of the numbers on the Eye of Horus. Now the kids understood both functions represented by the arrays. Kids started getting antsy after nearly an hour, so we did not pursue the fractional numbers in the Eye at this point.
Instead, we turned our attention to a table with one block on each end. The kids wondered whether it was a number or a function or something else. (They’re asking more and more mathematical questions each week – a huge jump in the development of their mathematical thinking.) I had the group move to the different sides of the table and asked each time, “Which color of block is on the right? On the left? On the top? On the bottom?” Then I asked, “What concept in math are we exploring?”
“Perspective,” said several people. I introduced the word “relative” and started using it. We were then sadly out of time. Some students had questions at the end, but I was not able to stay for questions today, so bring those questions back next time.
FYI, next time a member of our older Math Circle will be running a bake sale to benefit Women Against Abuse as you arrive with your kids.
*A sequence is a type of function where the first “out” number is the second “in” number. It is more restricted than a general function since it must adhere to an order.
**Thanks to Dr. Maria Droujkova for the idea of cutting up “brownies” into infinity.