After the total silence that accompanies very hard thinking, the kids were thinking aloud and producing conjectures.
After we played with Polydrons and Schläfli symbols* for about 10 minutes, we created a function machine. The students dictated the machine’s parts (various polygons, a cupcake, and sprinkles) as I drew it. They specified that the “in” number should enter the right eye and the “out” number exit through the cupcake. I put in the first number: 70.
The students corrected me when I put the “out” number in the wrong eye. Then they twisted their faces in thought in response to my statement “When you put in 70, 70 comes out.” Then, 90 went in and 50 came out. 40 went in and 100 came out. After the total silence that accompanies very hard thinking, the kids were thinking aloud and producing conjectures:
“As the in numbers go up, the out numbers go down.”
“Maybe they are negative numbers.”
“What happens when you put in 1?” (139 comes out.)
“What about 100?” (40.)
I threw them a bone: if you put in 2, 138 comes out. A few faces lit up. M noticed that “each pair adds up to 140!” All agreed enthusiastically, confident that they had solved the puzzle of what the machine was doing.
But I didn’t let them off so easily. “What is the machine doing to the numbers?”
“I don’t know,” said a few aloud.
Then D said tentatively, “If you put in a number, it finds another number where if you add it to the number that you put in, you get 140.” I asked him to say that again, and he did, more confidently, as other heads nodded in agreement.
“I think you’re right, but it’s hard to keep track of all those words,” I asked. "Is there some math operation word that could describe this in fewer words?”
A few kids suggested addition, but were unable to say what exactly was being added. Finally M asked, “Could it be subtraction?”
“What would you subtract?”
Several voices excitedly joined M in declaring, “Subtract the number from 140!” The rest of the group audibly exhaled in relief. I wrote on the board “subtract from 140” as the rule, and then we proceeded to change the machine and give it a new rule.
The second machine had an elephant where the cupcake had been, and used the function 2x+1. Of course, these young students did not express the function (rule) algebraically once they figured it out – their wording was either “add it to itself…” or “double it…” Interestingly, the students had a much easier time with this two-step addition machine than with the one-step subtraction machine.
“Let’s play Euler,” I suggested at this point. We returned to the table, remembered a few facts about the life of Euler (particularly his blindness and subsequent increased mathematical productivity), put polyhedra in our hands, and closed our eyes. I explained that Euler liked to trace paths along the edges of polyhedra. We imagined him doing it blind, and did so ourselves. I suggested that, as Euler did, we tried to count edges, vertices, and faces. Some found this easy, some found it hard, and some, well, peeked. As each student had picked up a different shape, our attempt to compare numerical results was not productive. So we went back to constructing.
M moved over to collaborate with J in her continued attempt to construct a solid from pentagons and hexagons. The class discussed whether it would be possible, and all agreed that yes, it could be done. J is convinced that it can be done with random placement of shapes. M is sure that you need to alternate rows of shapes. All are curious. To be continued….
“Look, I made another net for a cube!” announced D. I challenged him to see how many different nets for a cube exist. He happily set to work on that.
As they constructed, I picked up where our storytelling had ended last and told the life story of self-trained astronomer/mathematician Caroline Hershel. It’s always fun to hear children’s unfiltered language when describing history; today’s nugget was hearing D describe England as “the boss of every place.”
X continued her attempt to discover the undiscovered polyhedron, and to name it. But we were out of shapes on the table. I suggested getting some from the box on the floor. (Unknown to the group, only regular polygons were on the table, while the box was full of irregular shapes.) “Oooh, rectangles!” said X.
“These triangles are different,” noted M about the new shapes. I asked her how they differed, and she said “It’s acute.” I asked her whether the triangles we had been using were acute too, and she had to admit they were.
“It’s pointier,” specified J. M agreed with her, they both studied the different triangles, and simultaneously realized “all the sides are the same!” on the triangles we had been using.
Meanwhile, D had created a net by connecting a tetrahedron net to two cube nets. I dubiously asked, “Can you fold it into a solid?” He was sure that he could, and he was right.
We ended Math Circle with a brief discussion on the difference between pure (i.e. some of Euler’s) versus applied (i.e. Herschel’s) mathematics. After the students left, my photographer R (a few years older than the kids in the group) commented, “How did they get the cupcake function machine so quickly? I never could have gotten that one myself. I guess you must have to do function machines collaboratively.”
*For those curious, here are the details of today’s early geometry play: “It’s a triansquare!” said A, about the polyhedron she created. All the kids got right to work without a word from me. They created “a diamond with a missing piece,” “my party hat,” “a house of invisible monsters,” a “person going like this,” a “net for a cube,” a “moving train,” and “triangular pyramids.” We remembered the mathematical terms for various polygons and polyhedra, and then I put forth a challenge: I listed about 6 Schläfli symbols on the board and asked what they represented. M and N explained these symbols to A and K, who were not here last week, and also to those who forgot. Some got to work on these new shapes, while others continued to enjoy inventing their own.