Investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a 100 square.

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Tom Gowing 11A GCSE Maths Coursework

Opposite Corners

Investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a 100 square.

During the investigation my aim is to find a formula to work out the difference of these products for any size rectangle on any size grid.

I will start off by working out the difference on different shapes and looking for a basic pattern. I will try moving the shapes around the grid to see how this affects the difference.

I am going to use the following 5 shapes during this part of the investigation:

A 2x2 square

A 3x3 square

A 2x3 rectangle

A 2x5 rectangle

A 3x5 rectangle

A 2x2 Square

For the 2x2 I have decided to try 3 different positions along the horizontal,

3 along the vertical and 3 along the diagonal this should tell me if the difference alters depending on which way you move the shape around the grid. I have done 3 along each axis to provide accurate results by highlighting any anomalous ones. The only way to get perfectly accurate results would be to test every position the shape could be on the grid, but this is a waste of time when you can take 3 measurements from each axis and see any anomalous results. I will also do one test at a random site on the grid to make sure that results are as close to perfect as possible.

I predict that the difference will increase as the shape moves across or down the grid.


1 x 12 = 12

2 x 11 = 22

        22 – 12 = 10

        5 x 16 = 80

        5 x 15 = 90

        90 – 80 = 10

        9 x 20 = 180

        10 x 19 = 190

        190 – 180 = 10

From this I can see that for a 2x2 square on a 100 grid the difference is equal no matter where the shape is on the horizontal axis. This disproves my theory.

I will now try moving the square vertically to see if the difference alters.

1 x 12 = 12

2 x 11 = 22

        22 – 12 = 10

        41 x 52 = 2132

        21 x 52 = 2142

        2142 – 2132 = 10

        81 x 92 = 7452

        82 x 91 = 7462

        7462 – 7452 = 10

From this I can also see that no matter where the shape is vertically on the grid the difference also remains at 10. I will now try moving the shape along the diagonal axis.

1 x 12 = 12

2 x 11 = 22

        22 – 12 = 10

        45 x 56 = 2520        

        46 x 55 = 2530

        2530 – 2520 = 10

        89 x 100 = 8900

        90 x 99 = 8910

        8910 – 8900 = 10

I can now also see that no matter where the shape is on the diagonal axis the difference remains the same.

I will now try a random position on the grid to finalise my results so far.

        75 x 86 = 6450

        76 x 85 = 6460

        6460 – 6450 = 10

This has confirmed my results.

I can now prove that no matter where a 2x2 square is on a 10x10 grid the difference remains at 10.

I now predict that this works for any square on a 10 x 10 grid and I will test this theory by trying it out on a 3x3 square.

I think it would be a waste of time to try as many tests on this as on the 2x2 grid, so I for the 3x3 I will just try 2 shapes on the horizontal, 2 shapes on the vertical and 2 shapes on the diagonal, along with the random position to ensure accurate results.

 

For the 3x3 I predict that he difference will remain the same for all of the tested positions, as it was in the 2x2 tests.


        1 x 23 = 23

        3 x 21 = 63

        

        63 – 23 = 40

        8 x 30 = 240

        10 x 28 = 280

        280 – 240 = 40

I can now confirm that for any square on a 10 x 10 grid it doesn’t matter where it is horizontally positioned, the difference remains the same.

I will now try this out vertically.

1 x 23 = 23

        3 x 21 = 63

        

        63 – 23 = 40

        71 x 93 = 6603

        73 x 91 = 6643

        6643 – 6603 = 40

Join now!

I can now confirm that for any square on a 10 x 10 grid it doesn’t matter where it is vertically positioned, the difference remains the same.

I will now test this diagonally.

1 x 23 = 23

        3 x 21 = 63

        

        63 – 23 = 40

        45 x 67 = 3015

        47 x 65 = 3055

        3055 – 3015 = 40

I will now try a random position on the grid to finalise my results so far.

        77 x 99 = 7623

        79 x 97 = 7663

        7663 ...

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