Algebraic Justification

The following mini-investigations illustrate the process of algebraic justification.

Investigation 1

Write the integers 1 to 20 on the board. Ask a pupil to select a number – circle the number. Now ask pupils to add together the numbers either side of the circled number – what do they get? Repeat with a different number – what happens?

Pupils will notice that the sum of the numbers is always twice the circled number – WHY?

Explaining by example: This is usually the starting point for explanations e.g. I circled 8. The number before 8 is 7, the number after 8 is 9. 7 plus 9 is the same as 8 plus 8 (take one of the 9 and put it on the 7).

Accept these explanations but point out that pupils have only convinced you that the pattern works for these particular examples – you need convincing that it will always work……..

Justifying a generalisation in Words: Pupils need to move on from particular examples to the general case. Usually they find it easier to do this in words first. E.g. “Whatever number I circle, the number before is one less than the circled number, the number after is one more than the circled number. When I add the numbers together the –1 cancels out the +1 so I just get left with two times the circled number”

Encourage pupils to write down their justification. They will not find this easy and will be happy to go along with the idea of finding an easier way of recording their reasoning………..

Justifying Algebraically: Move pupils on to an algebraic justification like this:

If the circled number is n

Then the number before it is n-1

And the number after it is n+1

Adding the number before to the number after gives n-1 + n+1

This simplifies to 2n

Now get pupils go through the same processes with a similar investigation:

Write down a list of all the even numbers from 2 to 40. Circle one number then add the numbers before and after. What do you find? Why does this happen?

Investigation 2

This investigation is closer to the T-totals task and will help pupils appreciate how to represent the relationship between numbers on a grid. Put a number grid on the board. Draw a square around a block of 4 numbers. Add the numbers in one diagonal. Add the numbers in the other diagonal – Try this for different blocks of 4 numbers on the grid – what do you notice?

1 / 2 / 3 / 4 / 5
6 / 7 / 8 / 9 / 10
11 / 12 / 13 / 14 / 15
16 / 17 / 18 / 19 / 20
21 / 22 / 23 / 24 / 25