Function Machines, Etc. (Eye of Horus 3)
(October 1, 2013)
In order to imbue our Math Circle with some levity, and to get kids thinking deeply about arithmetic operations, we played Function Machines.* On the board, I had drawn a blob-like shape, of which everyone had a different interpretation – a blob, a wave, etc… I asked for suggested design elements, then drew on the suggested stripes. “Why do we have to put stripes on it?” asked S.
“That’s the hardest question anyone has asked me in weeks,” I replied. “Hmmm… what do you think?” I volleyed back to the group.
“You have to add the stripes to indicate that the machine is doing something to the numbers you put in,” explained D.
“And each machine has to look different so you can keep track of the different rules,” explained another participant.
Then the game began. “Put in 10,” said someone. We made the sound (“beep”) and I announced, “22 comes out.” We put in a few more numbers. Kids were begging to say the rule, but respectfully bit their tongues and instead predicted the out numbers until everyone realized the rule. That moment of realization was obvious as faces that had been scrunched in concentration suddenly lit up and exclamations of “oh!” dotted the room.
And speaking of obvious things, in the first few rounds several kids remarked “that’s obvious” or “that’s easy” when a first in/out pair went on the board. “No, that’s not necessarily obvious,” I said each time, and gave some not-so-obvious functions so everyone could enjoy both some success and some struggle. Those remarks quickly ceased.
“Add 3 hands with long fingernails,” instructed L for the next machine.
“And 3 beeps,” suggested E. I put hands on the blob, the kids put in the number 10, we all said “beep beep beep,” and out came 20. Comments about obviousness immediately evaporated when in went 36 and out came 72. We did a few more, including 5 in and 10 out. J was so engaged in mathematical struggle that she wandered up to the board, started pointing to numbers, and described what didn’t make sense to her.
“When you put in 5 and 10 comes out, that’s a difference of 5. When you in 10 and 20 comes out, that’s a difference of 10. How can this be adding a certain number if the differences change?” Others nodded in agreement.
“Oh, I see!” announced Z to the group. “You make a duplicate of the number and then add it.”
“That’s multiplying by 2,” explained V.
“They’re the same thing,” said someone else.
“No they’re not,” chimed in another. “Are they?” Attention moved from each other back to me.
“What do you think?” Kids debated a bit, then asked me again.
“Multiplication can be a shortcut for addition, but it isn’t always.” Here we could have jumped further into the meaning of multiplication, ** but didn’t.
I can’t remember what design element the kids added to the machine next. Eventually it had a face (S), eyelashes, poofiness to the eyelashes (V), skulls and crossbones all over it (J), squiggly lines over the stripes (L), a horn, a nose, a skyscraper on the nose, a chimney, and a bat (R).
For the first number into the third iteration of the machine, J suggested “three-thousand eight-hundred, and two million. How do you write that?” The group collectively interpreted the “and” as “plus” to come up with 2,003,800. The out number was 2,003,799. Next was 5 in, 4 out. Most people had a conjecture about the rule, but wanted to try a few more numbers, including negatives, just for fun. No one really understood how -2 could produce -3, but that’s a topic for another time. Everyone but D agreed that the function was subtracting one. D suggested that it could also be adding negative one. Many people in the group have no idea what a negative number even is, but those who do know asked whether those two functions are the same. I said yes.
Next, I shook things up with a machine that counted the number of digits that each number has. In return, R shook things up with a different, but also correct, interpretation of this function. Her rule involved ranges of numbers: “Numbers from 1-99 produce 1, numbers from 100-999 produce 2, and so on.” This is definitely a different function generating the same result. An interesting conversation to have at home would be how some seemingly identical functions are truly the same (just simplified versions) while others are not. If you want some dinner-table math debate, give a function in which the same number goes in and then out. Elicit some guesses, then tell people that your function was plus 2 then minus 2.
In our session, we were ready for the pièce de résistance. Into the function machine went the suggested 17, out come 289. I heard gasps of “huh?” and “what!?” among the “that’s-easy” crowd. Kids were scratching their heads and looking at each other inquisitively. So caught up were they in mystification that no one batted an eye as we collectively recited “O is awesome” (at O’s suggestion) at this function machine’s sound. The next pair (12 in, 144 out) generated a few pleasant “hmmmmms.” But when zero went in and zero came out, a detected a hint of outrage. I asked for suggestions of strategy for selecting “in” numbers. Only E had a strategy. That strategy helped himself, but not the others. I began dictating the “in” numbers: “Let’s try 1…, Let’s try 2…, Let’s try 3…” Now a few kids were bouncing with excited conjectures about the rule.
“It’s 3 squared!”
“It’s 3 times 3!”
“I don’t get it.”
I drew a 3-by-3 grid and explained that the rule I had in mind was “how many blocks it takes to make a square with that length of side.” Now everything understood (I think). And the stage is now fully set for Narcissistic Numbers.
Speaking of Narcissus, we finished his story. I had a lot of fun dramatizing his death. The kids were incredulous that Narcissus didn’t realize that he was loving his own reflection. “Didn’t he see his mouth moving? Didn’t he see his body moving?” The group wondered whether there was some explanatory gap in the thousand or two years between the original oral telling/retelling and Ovid’s written account. Or maybe Narcissus was just so in love with himself that he couldn’t even see clearly.
We had begun the session, as seems to our custom, with Tens Concentration. I had fortuitously forgotten to remove the face cards and jokers from the deck, so the kids got to make their own rules about how to handle these cards. It was a nice exercise in consensus-building.
We broke with another of our seeming customs – blocks – and used none today. Even the talk about blocks was more abstract, with a quick diagram on the board.
Gina Gruenberg, who is co-teaching our upcoming sacred geometry and henna course, sat in on class today. Thanks for your visit, Gina, and for participating in our function machines game. This game, by the way, was atypical in the Math Circle tradition in that it is more teacher-led than student-led. An aspect of the game that was typical, though, is the handling of unfamiliar, unexpected content. While we used negative numbers and zero in our machines, no one fully understood their properties conceptually. But who cares? Those properties are not the point of the game, and curiosity has been aroused. (Someday I would like to do a Math Circle on the properties of negatives and zero.)
We ended today’s circle when I suggested that kids try to stump the group with topics that seem to have no relevance to math. I told them to think about it and bring in ideas next week. “I know something right now that has no math in it: there is NO math in pants!” declared L. But alas, the kids quickly pointed out the involvement of measurement and symmetry, so I guess there is math in pants.
*In Math Circles, Function Machines make a sound when processing the “in” number, and then spit out an “out” number. The kids design the machine, suggest input numbers, are given resultant output numbers, and try to guess the rules. Just as with a cell phone, each time a new design element and sound is added, the function somehow changes.
**Multiplication can be seen as repeated addition, but also differently, as a scaling factor. A sixteen-ounce soda bottle is not the same as two of those “new” eight-ounce bottles.