For my second session with Jenna and Den I used another of my long-time favourite number activities. It another one that’s very accessible but can make people think quite hard.
On each side of a card square write a number. You don’t actually have to use a different number on each side, and they don’t actually have to be whole numbers, but that’s what most people do. And on another piece of card do things similarly – again, you don’t have to use whole numbers, and they don’t actually have to be different from the ones on the first card, but that’s what usually happens.
Now toss the two cards as if they were coins, and add the two numbers you see. Do it again, and record the totals you see; do this until you’re satisfied you’re not going to get any new totals.
If you do this with a class some groups are likely to find they’ve made three different totals, and some will find they’ve got four. If they have two new pieces of card and number these, do they still get three (or four) totals? Can they discover how you ensure you always get three different totals, or four different totals?
With only a small number of children I may steer it in a different direction. Here’s the account I wrote up for school of what happened with Jenna and Den, including some false starts and blind alleys:
Today I asked them to devise two double-sided cards with different numbers on each face, so that the four possible totals they could display would give a set of consecutive numbers. Before long they found – not quite by accident, but not completely by design (Den had first decided that one card should be 0/1, and suggested 3/4 for the other) – the cards 0/1 and 2/4, which generate 0+2, 1+2, 0+4, 1+4 (i.e. 2,3,4,5).
I asked them to find a second set and Jenna offered 2/3 and 4/6, making 6,7,8,9.
I asked them to generalise from this and they suggested one card had to be even / odd and the other even / even, but it didn’t take long to find a counter-example, before Den came up with the correct suggestion that the numbers on one card should have a difference of 1, and on the other a difference of 2.
I asked them to explore the situation with three cards. They thought there would be six combinations, and Jenna suggested the cards would need differences of 1, 2, and 3. They used a logical process to derive each combination in turn, and both contributed equally. Having reached six they realised there would be eight possibilities, and they observed that each number appeared in four combinations and were able to use this to check they had a complete set. However, one of the totals in their set was repeated, and Jenna then suggested the cards needed to show differences of 1, 2, and 4 (rather than 1, 2, and 3).
They wanted to explore four cards, which Jenna suggested would need to display differences of 1, 2, 4, and 8. They quickly devised the set 2/3, 7/5, 4/8, and 9/1. Den thought there would be 12 combinations, but they again used their logical strategy for generating every combination, and so decided there would be 16. They found these with no slips, and found the 16 showed every total from 12 to 27, once each.
It was fun working with these two All I Can Throwers, but in some ways I prefer using Two Cards with students whose thinking isn’t so streamlined. Jenna and Den did offer a couple of suggestions which didn’t work out, but they immediately corrected them and got back on track, so they missed out on a lot of the red herrings that most people might experience. Incidentally, I was intrigued that in both the explorations they’d done so far neither of them had shown the slightest inclination to make notes or do any recording on paper.
Actually, this exploration was only half of what we did in the session, and I’ll tell you about the other activity in my next post.