Corner to corner;Mathematics
Taking a class of seven-year-olds once a week does not remove the problem of finding something that will keep everyone in a very mixed group profitably occupied.
I spent some time discussing diagonals with the class. I showed the children various quadrilaterals made from geometric strips and asked them to imagine the lines joining opposite corners. They mostly agreed there were always two diagonals, although they were unsure of the concave kite (figure 1). Of course, a diagonal does not have to be inside the shape, but seven-year-olds need more than a definition - it has to feel like a diagonal. A crossed quadrilateral caused much amusement (figure 2).
No matter. There was time enough to discuss the finer points when the children had gained more experience. Anyway, there is a limit to the time they can spend in a class discussion.
The class drew a triangle and decided it had no diagonals. A convex quadrilateral had two, just like the ones I had showed them.
I asked them to draw a pentagon and put in all its diagonals, using a ruler. Drawing a pentagon is much harder than drawing a triangle or a quadrilateral. Some drew four sides and others six. Some made it five diagonals, some made it fewer. I drew one on the board and we checked together. I labelled the vertices A to E so the children could tell me which vertices had not been joined. We agreed there were five diagonals.
Next we worked on the hexagon. This was more difficult. Even when they managed to draw a shape with six sides, some children had trouble using a ruler, so the diagonals did not quite join the vertices. Some tried to draw the diagonals without a ruler, and occasionally managed to draw more than one diagonal between two vertices. After another concerted effort at the board we agreed on nine diagonals.
We made a list on a flip chart:
3 0 4 2 5 5 6 9 I suggested the children look at the heptagon during the week.
Caroline replied: "You add on 5 and you get 14."
"Oh," was all I had time to say before I went off to my next lesson, trying to be encouraging and non-committal at the same time.
By the following week, everyone had tried the heptagon, and had obtained a wide range of answers. We obviously needed to look at it again more carefully. I had duplicated some sheets with irregular polygons on them, with between 3 and 12 sides. I asked the children to work on the heptagon again.
I went around the class, helping those who were still having trouble with the ruler, or asking if they were sure they had drawn all the diagonals. Then I noticed Caroline. As the others had started drawing, Caroline sat looking at the table on the flip chart that was still there from last week. After a few minutes she turned her sheet of polygons over, and on the back, copied the table. Then she continued it: 7 14 8 20 9 27 10 35 11 44 I sat beside her and wrote out the results for odd numbers of sides: 5 5 7 14 9 27 11 44 "What do you notice?" I asked.
Caroline said: " 14 is 7 plus 7."
I waited. "44 is 11 times 4." I wrote this against the 44, and asked again about 14.
"That's 7 times 2," she said, so I wrote that against the 14, and asked about the 5. "Zero," said Caroline."No, one."
I added this result, so we had: 5 = 1x5 14 = 2x7 27 = 44 =4x11
Now it was easy enough for her to guess that 27 was 3x9, and verify it. But I had left gaps between these lines, so the table began: 5 5 = 1x5 6 9 = 7 14 =2x7
"How many sixes make 9?" I asked. Caroline used her ruler as a number line. 9 was one 6, leaving 3 more. And 3 was half of 6. So 9 was 11Z2 times 6.
She worked on the 8 and the 20 in a similar fashion. 3x8 was 24, but she had to take off half of 8, so 20 was 21Z2 times 8. Now she could work out the other gaps, and add them all to the table. Then she worked out the result for 12 sides.
5 5 = 1x5 6 9 = 11Z2x6 7 14 = 2x7 8 20 = 21Z2x8 9 27 = 3x9 10 35 = 31Z2x10 11 44 = 4x11 12 54 = 41Z2x12
The rest of the class now needed some attention. I drew a heptagon on the board, labelled the vertices, drew in some of the diagonals, and asked what was missing (figure 3). Some of the diagonals were found, but not all of them. How could we check?
How many diagonals were there from each corner? By now most had 4, but two of the children had only 3. "Join the other 2," said Will.
It was difficult to go further with the whole class, so they went back to their sheets, and to the various stages they had reached.
Some were still having trouble drawing the diagonals, but they were getting much better at using their rulers, allowing for the gap between ruler and pencil and getting the lines to go to the corners.
Some were having difficulty drawing all the diagonals each time. Some children needed careful observation, with a clue from our discussion about the same number of diagonals from each corner, which some appreciated more than others. Some were learning to organise the way they drew the diagonals, working systematically from corner to corner.
The biggest problem for many was counting the diagonals once they had drawn them. Counting is difficult, especially when what you are counting is not arranged neatly in some obvious order. But they were finding out the best ways of organising this, too.
Caroline was continuing her number pattern. She had not been taught much about fractions, and nothing about multiplying them. But she knew how to find a half of something, and that was all she needed.
In a few years' time, she will be able to explain her number pattern in terms of a formula, match it against the idea that each corner has the same number of diagonals, and explain why it works. So will the rest of the class. For the moment they are all gathering mathematical experiences at a variety of levels. Perhaps this is what is meant by "differentiation".
David Fielker is a freelance writer and lecturer, and teaches part-time at the American Community School, Egham, Surrey. His book Extending Mathematical Ability is published in the autumn by Hodder and Stoughton