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Introduction

Shahid Mahmood Syed

11 N1 Ms Hirts

Opposite Corners

Introduction:

For this piece of Mathematics GCSE coursework I am going to find out the difference between the products of the numbers in the opposite corners of any squares that can be drawn on a 10 x 10 grid composing of 100 squares.

I shall try to use tables to present my findings; I will make the 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:

I will find out the difference between products for squares of 2 values in 10 x 10 grid.  I will do this to find out the general case for this grid.

 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

Example 1:

 1 2 11 12

(11 x 2) – (1 x 12)

= 22 – 12

= 10

Example 2:

 25 26 35 36

(35 x 26) – (25 x 36)

= 910 – 900

= 10

Example 3:

 63 64 73 74

(73 x 64) – (63 – 74)

= 4672 – 4662

= 10

So, from the above examples I can see that the difference is 10, now I will find out the general case algebraically.

GENERAL CASE:

 n n + 1 n + 10 n + 11

(n + 10)(n + 1) – n(n + 11)

= (n2 + 11n + 10) – (n2 + 11)

= 10

Middle

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Example 1:

 2 3 4 12 13 14 22 23 24

(4 x 22) – (2 x 24)

= 88 – 48

= 40

This shows that the difference is 4, now I will do one another example and then I will find out the general case algebraically.

Example 2:

 56 57 58 66 67 68 76 77 78

(76 x 58) – (56 x 78)

= 4408 – 4368

= 40

Now I will find the general case of this pattern and on the basis of my examples I predict that the difference would be 40.

GENERAL CASE:

 n n + 2 n + 20 n + 22

(n + 20)(n + 2) – n(n + 22)

= (n2 + 22n + 40) – (n2 + 22n)

= 40

This shows that my prediction was right.

4 x 4 BOXES

 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

Example 1:

 12 13 14 15 22 23 24 25 32 33 34 35 42 43 44 45

(42 x 15) – (12 x 45)

= 630 – 540

= 90

Example 2:

 66 67 68 69 76 77 78 79 86 87 88 89 96 97 98 99

(96 x 69) – (66 x 99)

= 6624 – 6534

= 90

GENERAL CASE:

 n n + 3 n + 30 n + 33

(n + 30)(n + 3) – n(n+33)

= (n2 + 33n + 90) – (n2 + 33n)

= 90

RESULTS TABLE

 Grid Size Square Size Difference 10 2 x 2 20 10 3 x 3 40 10 4 x 4 90

Now I will try to find out the general formula for grids so I will be able to find out the difference of boxes for any type of boxes in 10 x 10 grid.

Now when I have noticed the pattern in my boxes I will try and find a general case so if I have any number of squared boxes then I will be able to find out the difference of opposite corners that by the formula.

 n n + q - 1 n + 10 (q – 1) n + 11 (q – 1)

Conclusion

However what if the rectangle is aligned differently on the grid, so the shorted sides are at the top and bottom? Will the difference for that still be 20?

 45 46 55 56 65 66

46 x 65 - 45 x 66 = 20

Therefore, I can say that the difference is 20 no matter which we around we make the rectangle of 2 x 3.

I will now try rectangles with the same height of two, but different lengths.

 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

20

2 x 4:

 12 13 14 15 22 23 24 25

22 x 15 - 12 x 25 = 30

2 x 5:

 46 47 48 49 50 56 57 58 59 60

56 x 50 – 46 x 60 = 40

It seems to me that whenever I increase the width by one, the difference increases by ten.

So considering the above 2 results. I predict that the difference for a 2 x 6 rectangle will be 50.

2 x 6:

 71 72 73 74 75 76 81 82 83 84 85 86

81 x 76 – 71 x 86 = 50

So my prediction was correct

In order to compare and find out the general formula I will now put all of my results in the table below:

 LENGTH HEIGHT AREA DIFFERANCE 2 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.

-  -

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3 star(s)

This is a basic opposite corners investigation. It uses good concrete examples to make and test predictions but is limited by the only basic use of algebra. Specific strengths and improvements have been suggested

Marked by teacher Cornelia Bruce 18/04/2013

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