Friday, January 22, 2016
Starting up math club
Math Club 1/15/16
A week ago I restarted an activity that my children and I really enjoyed in New York: an enrichment math club. I had planned to hold the club at a local park, but when I woke up that day it was wet and grey, so I changed the venue to my house. Given an enthusiastic response when I posted it on Facebook, I had been concerned that not everyone would fit, but as it turned out only 2 families were able to come. This was actually a good number to begin with – three 10-year-olds (including a set of twins) and one younger sibling, as well as my own children.
My idea for today’s lesson was to approach binary in a way that would be generally accessible, since I wasn’t sure how many younger children would be involved, and then to move to a more formal introduction. I was inspired by James Tanton’s “Exploding Dots” course (http://gdaymath.com/courses/exploding-dots/), and started with a physical activity based on the 1 <– 2 machine. Here’s how the activity went.
First, I placed a few sheets of paper on the floor in a line. I pointed out that a sheet of paper is fairly small; only one person can comfortably stand on it. So, if you ever have more than one person on a sheet of paper, the two EXPLODE! Only one “survives” the explosion, and that person is knocked to the next piece of paper in line. New people can only join the game by standing on the first sheet of paper in the line. So, after the first person joins the game, there is one person on the first piece of paper. Add in a second person, and they EXPLODE, and one of the two is left standing on the second piece, with the first piece empty. Add a third, and s/he can stand on the first piece without a problem: a person is on each of the first two places. Adding a fourth person is fun: the two on the first place EXPLODE, one joins the person on the second place, they EXPLODE, and one goes and stands on the third piece. If you’re familiar with binary, you can see where this is going; if not, try it out yourself (or click through to the Exploding Dots link). Some of the participants wanted to keep one person in the first place and explode the second person into the next place. I wish I’d thought of letting them play with that for a while – it would have been a base 1 system – but I just tried to re-explain that only one person can stay in the game after the explosion.
After we’d added in 5 or so people, I got out the whiteboard, and we started over again, writing down what happened if we dumped on a whole bunch of people at once. Someone suggested trying to put 9 people on the first sheet of paper, which required all 5 of mine and all 3 ten-year-olds, plus myself (holding E1), but it didn’t work too well because several people (all mine) wandered off before being exploded and were hard to find again (and to keep track of). At this point we moved to using the whiteboard exclusively.
I drew 4 boxes, and we began making a chart, using what the children didn’t yet recognize as standard binary notation: 1 is 1, 2 is 10 (a person in the second place as a result of the 2 who exploded in the first place), 3 is 11 (a person in each of the first 2 places), 4 is 100 (only one person left on the 3rd place, with the first 2 empty). I had them take turns drawing each subsequent number of dots in the first place, carrying out explosions and writing the resulting binary number in a somewhat organized chart. We went up to eleven or so (1011 in binary).
At that point, I got out the main manipulative I had prepared, a set of 5 cards containing 1, 2, 4, 8, and 16 dots respectively. I let them look at the cards and say what they noticed. “They’re all even!” “Except 1 isn’t.” “And 6 is missing.” “And 10.” Finally, “Oh, you have to double them.” Once that idea was mentioned, everyone agreed: it was a sequence of numbers you get by repeated doubling. Once they’d realized that, I asked them to make numbers using the available cards: 7 (1+2+4), 13 (1+4+8), 21 (1+4+16). Is there only one way of making each of these numbers, or could you find another way to do it? Well, all 4 children who were still engaged with the activity did it the same way, and no one could think of another way. “Unless we got into teams and had 2 of each card.” But with just one set of cards, we agreed that there was only one way to make each number.
The next step was to ask if there was any connection between the activity with the cards and the activity with the exploding dots. This puzzled them for a little while, so I set up a set of dot cards in order beneath one of the binary numbers we’d written on the board, with each card marking a place. They were able to see that, for example, 1010 was the same as 8 + 0 + 2 + 0.
Someone noticed that five in binary is 101, and ten is 1010. “What’s twenty?” They found it was 10100, which was an interesting pattern. I asked if anyone had any guesses about what forty might be, and there was general agreement that it should be 101000. We had to add another place, though, since our dot cards only went up to 16, but when we’d added the 32 place we noticed that 40 was indeed 32 + 8. I had thoughts at this point about doing some addition (with carrying) and subtraction (with borrowing), but I could tell that people had about reached the end of their interest with the activity, so we wound up and everyone left.
Things that went well: Letting them say what they noticed about the base 2 cards, noticing patterns of doubling and halving (5, 10, 20, 40 translates to 101, 1010, 10100, 101000), letting them hold the whiteboard markers, asking children to explain their reasoning whether the answer was right or wrong (which helped me see how my explanations were going awry).
Things that didn’t go as well: Whiteboard markers running out of ink (repeatedly), allowing some people to dominate the conversation and failing to include others, not explaining the rules of the game carefully enough for everyone to understand that one of the people in the explosion disappears completely, drawing 4 boxes on the whiteboard before we needed all of them (should have added boxes only as needed).
Plan for next time: I have no idea which of the same children and which new ones – of which ages – will come, but, weather permitting, I’m hoping to do it at the park. We’ll look at fractals – there’s a gorgeous, enormous tree there, so tree fractals will flow naturally. I’ll try giving each child a sheet protector with cardstock inside it, and a (new) whiteboard marker, so they can each explore without one child dominating. I’m particularly enamored with Sierpinski’s triangle and the Koch snowflake, so those will certainly feature. I’ll plan to post again with how it goes.