Opposite Corners Maths investigation

Introduction

For this investigation I am going to try and develop a formula to work out the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 10 x 10 grid composing of 100 squares.

I shall use tables to present my findings; I shall be making predictions and proving my predictions right or wrong with examples. I will be using algebra to prove any of the rules I manage to create by analysing my results.

Method

To begin with I shall first find the difference between the products of the numbers in the opposite corners of 2x3 rectangles.

1. 34 x 26 = 884

24 x 36 = 864

                    20

Do all 2x3 rectangles have a difference of 20?

2. 58 x 50 = 2990

48 x 60 = 2880

                     20

It appears that they do. I predict that all 2x3 rectangles have a difference of 20.

3

6

20

2

4

8

30

2

5

10

40

2

6

12

50

The area increases by 2 each time.  This is because the length is always being multiplied by the height of 2.

The difference increases by 10 each time, also each line of the grid is 10 squares wide so the next square vertically straight down is ten more than the square above it.

I can now think of some possible formulas.

L=length H=Height D=Difference

L – 1 X 10

The length subtracted by one multiplied by ten

Or

L – 1 X 5H

The length subtracted by one multiplied by five multiplied by the height.

The lowest corner subtracted from the highest corner subtracted by one, but only when the rectangle is aligned so that the shortest sides are at the top and base.

I shall now proceed to test these

I shall try testing this rule, on rectangles with a length of four, to see if this rule truly does work on all rectangles.

4 x 5

1. 5 x 31 = 155

    1 x 35 =   35

                120

4 x 6

1. 6 x 31 = 186

    1 x 36 =   36

                   150

The difference is advancing by 30.  4 x 7 will have a difference of 180.

4 x 7

1. 7 x 31 = 217

    1 x 37 =   37

                   180

My prediction is right.

However does the rue actually work?                                

Evaluation

        I believe I have completed my aims and succeeded in attaining the goals I set myself.  I have found the rule to work out the difference for any rectangle of any size on a 10 x 10 grid.

I have succeeded in doing this through extensive investigation on the various sized of different rectangles and come to the conclusion that that the Difference between the products of the numbers in the opposite corners of any rectangle = L-1 x 10 x (H-1) formula works flawlessly.