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Introduction

The investigation was corner to corner, which means that within a four sided shape on a table of numbers you had to multiply the top left corner with the bottom right corner and then the bottom left corner with the top right corner, you then had to find the difference between the two multiplication’s. ## C x B and then subtract A x D

 A B C D

E.g.

 2 3 12 13

(On 10 x 10 square)

2x13=26

3x12=36

36-26=10

If I move the 2

Middle

6643-6603=40

## Here is a table for the difference for a ten by ten grid:

 Length of Square Difference 2 10 3 40 4 90 5 160 6 250 7 360 8 490 9 640 10 810

The difference is calculated by the (length of the square – 1)² x10, that is how I was able to calculate the other differences of the different length of square.

(L-1)² x 10= difference [algebraic form]

L = length of the square

The limitations with this formula are that the shape has to be a square, the length of the grid has to be 10 and the constant difference between each number has to be 1.

(n + a (L-1)) x (n + (L-1)) – (n x (n + a (L-1) + (L-1)= difference

I then simplified the above equation to make:

a( L² -2L + 1) = difference

a is the number of how many the grid goes across

Conclusion

I have found that the formulae for the square and the rectangle are very similar but in the rectangle formula the height of the rectangle is included but for a square the height is the same as the length so the height value is not needed in the square formula. So a formula for four sided shapes on a grid of numbers with a constant difference of 1 is a (Lh – h – L + 1), the a is how much the grid goes across, the h is the height of the shape and the L is the length of the shape.

## Rather than simplifying the formulae step by step I have jumped from the starting formula to the ending one, in the future I shall simplify my formulae step by step.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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