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• Level: GCSE
• Subject: Maths
• Word count: 3344

# &quot;Multiply the figures in opposite corners of the square and find the difference between the two products. Try this for more 2 by 2 squares what do you notice?&quot;

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Introduction

Gurprit Singh Khela

Opposite Corners-coursework

“Multiply the figures in opposite corners of the square and find the difference between the two products.

Try this for more 2 by 2 squares what do you notice?”

Investigate!

In this investigation I will research various squares and rectangles within selected grids such as 10 by 10, 11 by 11 and so forth. I will find patterns between the differences and the squares and rectangles within the grids. By the end of this investigation my aim is to achieve a formula that will connect and link the shape within the grid whether it is a square, rectangle etc. to the size of the grid itself. If I have time I may possibly go into 3 Dimensions and investigate the addition of the next dimension and see how this will affect the 2D work so far and how it can be linked.

To start off with I will take a 2 by 2 square out of the top left hand corner of my grid which for now is 10 by 10 size.

 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

2 x 11 =22

1 x 12 =12

I have multiplied the opposite corners and now will subtract the smaller number from the larger one as explained in the question above.

22 – 12 = 10

By subtracting the 2 totals I have found the difference which is 10.I wonder if this difference of 10 will remain constant if I change the position of the 2 by 2 square on the grid. To test this I will move the square along a place and see what the difference is.

 2 3 12 13

3 x 12 = 36

Middle

33) = x2 + 33x

(x + 3)(x + 30) = x2 + 33x + 90

(x2 + 33x + 90) – (x2 + 33x) = 90

The difference is 90 and I have proven this with the use of algebra and numbers therefore my prediction was wrong and now I shall collate the differences so far and see what I can conclude on.

Here are my results so far:

 Square Size Difference First Difference Second Difference 2 x 2 10 3 x 3 40 +30 4 x 4 90 +50 +20 5 x 5 ???? ???? ?????

This is interesting as the first difference isn’t constant but the second difference is 20.Looking at the sequence I predict that the 20 will remain the same for the second difference and the main difference for the 5 by 5 square will be 160.This is because I will add the 20 to the already increasing first difference of 50 which will be 70.Then I will add the 70 to the 4 by 4 main difference of 90 which will make 160.Therefore a 5 by 5 square in my opinion should have a difference of 160.Now I will test my new prediction.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45

1 x 45 = 45

5 x 41 = 205

205 – 45 = 160

 x x + 1 x + 2 x + 3 x + 4 x + 10 x + 11 x + 12 x + 13 x + 14 x + 20 x + 21 x + 22 x + 23 x + 24 x + 30 x + 31 x + 32 x + 33 x + 34 x + 40 x + 41 x + 42 x + 43 x + 44

x(x + 44) = x2 + 44x

(x + 4)(x + 40) = x2 + 44x + 160

(x2 + 44x + 160) – (x2 + 44x) = 160

My theory is correct and the difference has been proven to be 160.I have now found a pattern and have enough results to come up with a formula that will give me the results for any size square on a 10 by 10 size grid.

Conclusion

n= one side of the square

I have expanded my formula because now it will give me the difference from any size square within any size grid. To test my formula I will choose a 3 by 3 square on a 12 by 12 firstly work it out with the formula then the full way and see if the answers match.

With the formula:

x(n – 1) 2

12(3 – 1)2

The difference in theory is 48

I will now test the difference to see if it remains the same:

 1 2 3 13 14 15 25 26 27

3 x 25 = 75

1 x 27 = 27

75 – 27 = 48

The difference is 48

This proves that my formula will work with any grid and any size square within it at any place on the grid. I have now reached the limit of how squares within a 2 dimensional grid can be investigated. To add another variable into my work I am going to investigate rectangles. This will add a further variable because as with squares both the length and width are represented by 1 number whereas rectangles can have 2 completely different numbers. Therefore I am expecting to work out a much different formula from the one above with the inclusion of another variable. This formula should provide me with the following:

• Difference to any size square or rectangle on a size grid in any location on the grid.

I will work out the difference to a 2 by 3 rectangle, all the work from this point will remain on a 10 by 10 grid until stated otherwise.

 1 2 3 11 12 13

1 x 13 = 13

3 x 11 = 33

Difference is 20

 x x + 1 x + 2 x +10 x + 11 x + 12

(x + 2)(x + 10) = x2 + 12x + 20

x(x + 12) = x2 + 12x

Difference is 20

2 x 4:

 1 2 3 4 11 12 13 14

4 x 11 = 44

1 x 14 = 14

Difference is 30

 x x + 1 x + 2 x + 3 x +10 x + 11 x + 12 x + 13

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