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  • Level: GCSE
  • Subject: Maths
  • Word count: 7972

Investigation of diagonal difference.

Extracts from this document...

Introduction

Maths coursework

 Investigating diagonal difference

Aim:

For this course work my task is to investigate the diagonal difference of different size cutouts on different size grids.

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E.g. this is a 2x2 cutout on a 10x10 grid.

Finding the diagonal difference of a cutout is achieved by finding the product of the bottom left corner and the top right corner, then the product of the bottom right corner and the top left corner and finally calculating the difference between these two products.

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I will investigate the diagonal differences for cutouts anywhere on a grid and use algebra to prove any rules I may discover. I will log my results in a table to help me to find any formulas that may become apparent as I proceed. As I investigate this problem I will make appropriate predictions from rules I may discover from the table of results I produce, and from any patterns that can be seen.

Method:

I will start by investigating the diagonal difference of 2x2 cutouts on 10x10 grids.

2x2 cutouts

What is the diagonal difference of a 2x2 cutout on a 10x10 grid?

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< ...read more.

Middle

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I will now calculate the diagonal difference of a 4 x 4 cutout. Once I have looked at 4 x 4 cutouts I will compare all my examples of different size cutouts and look to see if there are any patterns or relationships between them. Hopefully by the end of this analysis I will have found a general formula for which I can work out the diagonal difference for any size, square cutout on any size, square grid.

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I notice that a 4 x 4 cutout is 1 more in length than a 3 x 3 cutout and 10 more in height. From this I can convert the cutout straight into its algebraic form using my previous knowledge of 3 x 3 and other cutouts.

I will now convert the cutout into its algebraic form

Converting the cutout into its algebraic form.

n

n + 3

n + 3g

n + 3g + 3

Step 1Step 2Step 3

n

N + 3

n + 30

n + 33

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 I have now found solutions for 2 x 2, 3 x 3 and 4 x 4 cutouts. I will now analyse these cutouts and see if I notice any patterns. My main aim will be to try and find a general solution/formula for any square, size cutout on any square, size grid.

image09.pngimage12.pngimage09.pngimage11.png

n

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n + 3

n + 3G

n + 3G + 3

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n

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n + G

n + G + 1

nimage18.png

n + 2image19.png

n + 2G

n + 2G + 2

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Since analysing the different size, square cutouts I have noticed a pattern. The value added to n in the top right corner is 1 less than the length of the cutout. For example a 4 x 4 cutouts top right corner value would be n + 3 as shown above. This value seems to be common in the bottom two corners of a square cutout. It seems to relate to the number of G’s added to n in both bottom corners and what number must be added to the n and G solutions bottom right corner.

...read more.

Conclusion

I will now calculate the diagonal difference of the algebraic cutout

Calculating the diagonal difference of the algebraic cutout.

I will first calculate the top right corner multiplied by the bottom left corner. I will then calculate the top left corner multiplied by the bottom right corner. Once I have done this I will takeaway both products and I hopefully find the final general formulae for any size cutout on any size grid.

image107.pngimage41.pngimage110.pngimage23.pngimage109.pngimage23.pngimage107.png

D = (X – 1)(Y – 1)G could be the general formula for working out the diagonal difference for any size cutout. To check this I will now calculate the diagonal difference of a 3 x 5 cutout on a 10 x 10 grid using the formula.

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      21 x 5 = 105

=> 25 x 1 = 25

=> 105 – 25 = 80

=> 80

The diagonal difference is 80

I will now work out the diagonal difference using my general formula

(X - 1)(Y – 1)G

=> 5 – 1 = 4

=> 3 – 1 = 2

=> 4 x 2 = 8

=> 8 x 10 = 80

=> 80

                                                                 The General formula works !      

Conclusion

The universal general formula (X – 1)(Y – 1)G gives the correct diagonal difference of any size cutout weather it be a square or a rectangle. When considering the height and length of a square, where X is equal to Y, the expression (X – 1)(Y – 1)G simplifies to my earlier formula (X – 1)² G. I think it’s safe to assume that a rectangle cutout would have no change in its diagonal difference on a rectangle grid, by looking at my previous investigations.

Evaluation

I believe I have reached the targets and aims I set myself in the beginning. Since starting this coursework I have found a general formula that works on any size cutout on any size grid through progressive investigation and through using my algebraic methods.

I believe I have taken this coursework to its limit in the time I have been given.  I am happy with the work that I have produced and cannot think of anything I could have carried out better or improved upon.

Maths Coursework

                                                  X

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    Yimage114.png

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INVESTIGATION OF DIAGONAL DIFFERENCE

Khalil Sayed –Hossen 9B

...read more.

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