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

"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?"

Extracts from this document...

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.

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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.

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3

12

13

3 x 12 = 36

...read more.

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.

image01.png

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

????

????

?????

image02.pngimage03.png

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.

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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.

...read more.

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:

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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.

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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:

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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

Page

...read more.

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