Pick up the pieces
I take a class of seven-year-olds once a week. We had been playing a pegboard game called four-in-a-row and working out strategies for winning. I walked in with my box of pegs and boards as usual, and noticed a large, half-finished jigsaw puzzle on a table.
I picked up a couple of pieces and showed them to the class. We discussed how they were based on a square, with either a tongue or a slot in the middle of each side.
"How many different kinds of pieces are there?" I asked, putting my box aside.
They began designing different possibilities. I copied them on to the board and asked the children to try to find ones we had not already drawn. We collected 12, and I numbered them for easy reference.
We looked at what we had. They decided that Brian's odd-looking 12 did not obey the rules, so we removed it. Were the others all different?
No: 5 and l0 were the same. We removed 10. They thought 4 and 5 were the same.
"How can you make 4 look the same as 5?" I asked.
"You can turn it round," they said.
"How?" This needed some detail before I would admit to being clear about what was to happen. I needed to know how far to turn it and which way.
It needed some practice at angles and rotation. I asked them to stand up, face the front of the room and then do a half turn to the right. We had to check that everyone knew left from right. Then they turned quarter turns. Two half turns took them back to where they started and so did four quarter turns.
When we looked again at the jigsaw pieces on the board "turn to the right" was ambiguous, because either the top or the bottom could turn to the right. Someone thought of "clockwise". How far? "A quarter turn." We removed piece number 4.
In the same way we removed pieces 8 and 9 because they were quarter-turn rotations of 6, one in each direction, and half-turn rotations of each other, and we removed 7 because it was a counter-clockwise quarter-turn rotation of 1. This left us with six.
Was that the lot? We could not find any more, but that that did not mean that we necessarily had them all. How could we sort them out?
Caroline suggested that we could take any piece and change over the tongues and slots. This paired off some of the pieces: but left us with two pieces that were paired with themselves.
It was a nice idea, but it did not produce any new pieces, nor convince us that there were no more.
We decided that there were different numbers of tongues and we could put the pieces in order according to how many tongues they had, beginning with the piece that had none.
A lot of discussion took place before everyone was clear that only the case with two tongues gave rise to two possibilities instead of just one.
Next lesson we sorted out edge pieces and corner pieces.
Binary connections I presented the problem to my group of bright 11-year-olds, telling them what had happened with the seven-year-olds. They quickly sorted out all the different pieces. (It is nice to see that they have learned some mathematics since they were seven.) They had just been looking at numbers in other bases, and it seemed a good idea to connect the problem to the binary system. The edge pieces, some of which look like this, have three different positions in which to put a tongue or a slot. The binary numbers with three digits, with their base ten equivalents, are:
If we interpret 0 as a slot and 1 as a tongue then this corresponds exactly to the edge pieces, so we know there are these eight possibilities.
What about corner pieces?
These have just two positions for slots and tongues, so they correspond to the two-digit binary numbers: 00, 01, l0, 11.
What about the other pieces?
A first thought was that they had four positions, so they corresponded to the four-digit binary numbers, of which there were 16. However, we knew there were only six. Of course some were rotations of each other.
We looked at the four-digit binary numbers. The rotations could be recognised in the digits; for example, the numbers 3, 6, 12 and 9: 0011
all represented the piece with tongue-tongue-slot-slot.
There were some interesting connections between some of the other equivalent numbers as well.
Puzzles We now embarked on some possible puzzles we could set for the seven-year-olds.
1. There are four corner pieces. Can you put them together to complete a 2 by 2 jigsaw puzzle?
2. If we add the eight edge pieces, can we complete a 12-piece jigsaw?
3. We can use the edge and corner pieces to make a 4 by 4 square jigsaw, but we shall need four "middle" pieces. What are the possibilities?
4. How many 4 by 3 jigsaws are there?
5. What are the possibilities for triangular pieces . . .
6. . . . or hexagonal pieces?
It is always nice to take advantage of something of current interest and get some mathematics out of it. One should always be flexible, and ready to suspend an original lesson plan if an opportunity like this occurs. Some of the seven-year-olds looked with renewed interest at their jigsaw puzzle afterwards. However, one did ask at the end, "Can we carry on with the pegboard game next time?" It was also an added point of interest for the 11-year-olds that the problem was one their younger colleagues had worked on, and it certainly gave more point to their problem-posing that the puzzles were to be presented to someone. The binary system of numeration may be somewhat old hat these days, but it is still a useful notation with a wide variety of interesting applications.
The combinatorial type of problem, in which we need to know the number of possible ways something can occur, is an important general idea in mathematics, particularly in probability and statistics. This particular problem involved ideas about rotation, angles and fractions in a meaningful context appropriate for the seven-year-olds.
I believe that most mathematical ideas should be treated as they occur in this way, in interesting problem-solving situations, rather than presented in separate lessons in some arbitrary syllabus order.
* David Fielker is a freelance writer and lecturer and teaches part time at the American Community School in Egham