The green tape on the rug hinted at a quadrilateral shape, but was actually composed of 4 line segments that did not meet at corners. Is it safe to say that the lines would definitely meet and form corners were they to be extended? If so, can you call this shape a square or a rectangle? If so, how can you determine which of those it is?
These questions arose as we began this week’s Math Circle. Every week, our group asks more good questions – such an important skill to a mathematician. Regarding the tape on the floor, we finally agreed to assume 2 things: (1) that the lines met at corners and (2) that we could measure (with rulers or our fingers) to determine which shape it is, but that we didn’t want to. O lobbied for calling it a “whatchamacallit,” but the collective opinion of the group leaned more toward calling it a “squaretangle,” and so it was dubbed.
I dropped a pile of pencils into the squaretangle for a round of the Takeaway Game. The students quickly ascertained which one was gone, so I took away three pencils the next time. I expected the children to say that three pencils were gone, but most of the kids described in detail one pencil that was missing. P said that three were gone, but was unable to describe any of them. Since each child had initially focused on just one attribute (number or appearance), I was curious to see if they would focus on two attributes now that they were aware of the possibility. I put the pencils in a pile again and immediately M and A starting counting them before closing their eyes. I asked why they were counting, and V’s eye lit up as he said “so you can subtract afterwards and know how many are missing!” With the emergence of this new strategy, everyone started counting, but no one got the same number, so we all counted together. One person counted in Italian, one in Spanish, and yet another in Roman Numerals. Then they closed their eyes for the last time, I removed four pencils, and everyone happily announced with open eyes, “seven are left; you took four!”
And speaking of the number seven, since Halloween is coming up, I mentioned a spooky thought about the number seven: I’ve heard that to some people long ago, the number seven was considered a bad-luck number because it does not correspond to an obvious number of things in the human body. The children brainstormed various body parts and their associated numbers; then we returned to the squaretangle.
I dropped a pile of wooden geometric shapes into the squaretangle. While the children’s eyes were closed, I removed the sphere. When they opened their eyes, N declared that “the circle” was missing. All agreed. I showed them the sphere and asked, “What is different about this from a circle?” “Oh, a circle is flat. This is a ball.” I told the children that mathematicians had a special name for balls, and O called out “sphere!” We had fun identifying the other shapes, especially the hexagonal prism. That one was fun in three ways: another name for a curse is a hex; “prism” sounds like “prison” (we wondered whether we could lock someone in there?); and “hexagonal prism” is just an awful lot of fun to say aloud. People described the octahedron as a diamond and the cylinder as a can or a tube. O said that we should put some beans in it. We played the Takeaway Game with the solids once or twice, then moved the shapes and ourselves to the table, where a box of apples was waiting.
We used the apples for Maria Droujkova’s “Apple Math” activity, in which children predict how many knife slices are needed to cut an apple into a particular solid shape. N first asked to cut it into a sphere, and the children delightfully called out “zero!” They agreed that while the apple was not an exact sphere, that we could approximate and call it one. Each child took a turn suggesting a shape, and we used a show of hands to make conjectures on the number of slices it would take. Consensus regarding conjectures on number of cuts was infrequent. My 12-year-old helper (and photographer and editor) Rachel kept track of predictions and results on the board. We arrived at a prickly juncture when one child said we could cut off the stem first, and then cut the top off. Others said we didn’t need to use 2 separate cuts for this result. It turned out that we needed to clarify the question to specify that we wanted the minimum number of cuts.
For every shape, we made conjectures before checking them with actual cuts. We came up with definitions of “top” and “bottom” as they regard to apples. We arrived at another prickly point when M conjectured that the bottom of the apple did not need to be sliced off horizontally to make the base of a cube since the apple lay “flat” on the tabletop. However, we had already done so. As we were eating the cast-offs from our cube, M examined the bottom piece and refined her conjecture: the bottom was not flat, it was “flattish.” Definitions are, of course, important in mathematics, and we discussed the definition of a conjecture.
When we cut the apple into a cylinder and each child received a disc of apple to eat, some noticed that the seeds made a star shape. I asked what geometric shape you get when you connect the points of a star, and no one was able to visualize it. I put it on the board, connected the points, and talked about both shape and building name “pentagon.”
At this point, V asked, “Is this one of the games that the children at the zoo played to help with their concentration?” I was surprised to realize that I had forgotten about our story. Our continued narrative has added continuity, interest, and context to our Math Circle. I answered in the affirmative, and also told the children that the magical animals in the zoo required food that had been cut into these geometric solid shapes. We then returned to the ongoing dying unicorn situation.
We recalled that we ended last week with failure when the slowest pair of zookeeper children crossed the bridge first to try to save the unicorn. I reminded them that last week we made two plans: to use M’s suggestion that we see what happens when the fastest go first, and J’s suggestion to use our imaginations (instead of our bodies) to figure it out. P was not at all interested in returning to this problem, and I wasn’t sure whether A and N had lost interest, so I changed the wording of the question to insert some hope. I told the children that I had recently learned that another zoo of magical creatures had experienced this exact same problem and had solved it mathematically. “Now we know it can be done.” Our question is no longer, “Can it be done?” but “How can it be done?” Fortunately, our group remembered how to combine rates, and was able to do the arithmetic mentally. N quietly said the correct sum when several people added incorrectly.
Unfortunately, the unicorn still died when our characters Ginny and Ron crossed the bridge first (in our minds) and Ginny returned to give the flashlight to the slower walkers. I reminded the Math Circle children that thiscan be solved. Someone proposed that we use our power to go back in time and give the unicorn another chance. We decided to make another mathematical attempt to save the unicorn next week. Then Math Circle ended with each child excitedly taking a whole apple home as a present.