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Introduction

Maths Coursework         1st DRAFT!!!!                                   Mr Gazey

## Introduction

I am going to investigate and observe what happens when I find the product of the top left and bottom right hand side of a square, and vice versa find t he product of the top right and the bottom left hand side of a square. I will then calculate the difference between the two squares and try to see if there is a genuine or pattern with the numbers.

I am first going to use a two by two square and see what happens.

 12 13 22 23

12 X 23 = 276          Here we can see that the difference is equal to ten.

22 X 13 = 286

 54 55 64 65

55 X 64 = 3520         The difference equals ten.

54 X 65 = 3510

 68 69 78 79

68 X 79 = 5372          The difference equals ten.

69 X 78 = 5382

I have observed that the difference is always ten between the two totals. To prove that my results were accurate I did another example.

 81 82 91 92

81 X 92 = 7452    I was correct as my concluding example also had a difference of

91 X 82 = 7462    ten.

To further my investigation I am now going to observe what happens with a three by three square.

 1 2 3 11 12 13 21 22 23

1 X 23 = 23           Here we can see that the difference is forty.

3 X 21 = 63

 35 36 37 45 46 47 55 56 57

35 X 57 = 1995      The difference is forty

37 X 55 = 2035

 71 72 73 81 82 83 91 92 93

71 X 93 = 6603        The difference is forty.

73 X 91 = 6643

This time the difference is always forty. To prove my results were correct I did another example.

 78 79 80 88 89 90 98 99 100

78 X 100 = 7800     My results were correct.

98 X 80 = 7840

I then decided to do the same thing with a four by four square.

 1 2 3 4 11 12 13 14 21 22 23 24 31 32 33 34

Middle

20

25

26

27

28

29

30

35

36

37

38

39

40

45

46

47

48

49

50

55

56

57

58

59

60

5 x 60 = 300              The difference is two hundred and fifty.

10 x 55 = 550

 45 46 47 48 49 50 55 56 57 58 59 60 65 66 67 68 69 70 75 76 77 78 79 80 85 86 87 88 89 90 95 96 97 98 99 100

50 x 95 = 4750           The difference is two hundred and fifty.

100 x 45 = 4500

 23 24 25 26 27 28 33 34 35 36 37 38 43 44 45 46 47 48 53 54 55 56 57 58 63 64 65 66 67 68 73 74 75 76 77 78

23 x 78 = 1794           The difference is two hundred and fifty.

28 x 73 = 2044

This is clearly what I predicted as the difference is always equal to two hundred and fifty. I attempted another example to back up my results.

 1 2 3 4 5 6 11 12 13 14 15 16 21 22 23 24 25 26 31 32 33 34 35 36 41 42 43 44 45 46 51 52 53 54 55 56

1 x 56 = 56                The difference again is always two hundred and fifty.

51 x 6 = 306

To make further investigations quicker. I worked out a formula from my table of results above. With the formula it will enable to see what the differences are for any square without having to work out the products of the top right and bottom left hand number, and the top left and bottom right hand numbers. I could see that the differences of each square is always the size of square penultimate. Therefore I worked out the formula to be this;

(n-1) ² x 10

To ensure that my formula worked I tested it on a 2 x 32 square of which the difference was 10.

(n-1) ² x 10   =     (2-1)² x 10   =

1² x 10   = 10

This shows me that my formula was correct, as it gave me the correct difference.

To further my investigation I am going to prove my results algebraically for a 2 x 2 square.

 x x+1 x+10 x+11

Therefore : x(x+11) =                   (x+1)(x+10) =

x² + 11x                      x² + 10x + x + 10 =

x² + 11x + 10

Here we can see that no matter what the value of x may be, the difference will always be ten.

I am now going to prove my results algebraically for a three by three square.

 x x+2 x+20 x+22

Therefore : x(x+20)                        (x+1)(x+10) =

x² + 20x                        x² + 20x + 2x + 40 =

x² + 22x + 40

Again this proves my results, as the difference will always be forty between the two numbers.

I am now going to prove my results algebraically for a four by four square.

 x x+3 x+30 x+33

Therefore : x(x+33)                       (x+3)(x+30) =

x² +33x                        x² + 30x + 3x + 90 =

x² + 33x + 90

Here the difference between the numbers is + 90, which proves and supports my results in the table.

To conclude I am going to prove my results algebraically for a five by five square.

 x x+4 x+40 x+54

Therefore : x(x+54) =                            (x+4)(x+40) =

x² + 54x                               x² + 40x + 4x + 160 =

x² + 44x + 160

Again the difference will always be +160. This example supports and proves my results.

To further my investigation, I am now going to find out what happens in a rectangle, starting with a three by two rectangle, and observe any patters or trends.

 1 2 3 11 12 13

1 X 13 = 13                      The difference between them equals twenty.

3 X 11 = 33

 4 5 6 14 15 16

4 X 16 = 64                      The difference between them equals twenty.

6 X 14 = 84

 88 89 90 98 99 100

90 X 98 = 8820                The difference between them equals twenty.

88 X 100 = 8800

I have noticed that the difference is always equal to twenty in a three by two rectangle. To prove my results are correct, I am going to conclude by doing another example.

 81 82 83 91 92 93

81 X 93 = 7533                The difference between them equals twenty

83 X 91 = 7553

I have now chosen to enlarge my rectangle to three by four.

 41 42 43 44 51 52 53 54 61 62 63 64

Conclusion

 5 6 7 8 9 10 15 16 17 18 19 20 25 26 27 28 29 30 35 36 37 38 39 40 45 46 47 48 49 50

5 X 50 = 250                   Again we see that the difference between the two numbers

10 X 45 = 450                 will always be two hundred.

I put my results in a table in order to enable me to discover a pattern or trend, and also to find a formula with the size of my rectangles.

 Size Of Rectangle Difference between the products 2 X 3 20 3 X 4 60 4 X 5 120 5 X 6 200

My results have shown me exactly the same pattern as to what happened with using squares. The size of the rectangle equals the next rectangle’s product after it. For example the 2 X 3 rectangle multiply together to give the product of the next rectangle 3 X 4, which is 200 (multiplied by ten). From my table of results I am going to work out a formula, making it easier and quicker to work out differences of any size rectangle. This is what I came up with.

(L-1)(W-1) X 10

To ensure that my formula was correct and worked I tested it with an example. I used a 5 X 4 rectangle of which the difference is one hundred and twenty.

(L-1)(W-1) X 10 =

(5-1)(4-1) X 10 =

4 X 3 = 12                 Therefore : 12 X 10 = 120

As you can see my formula clearly worked, and gave me the correct difference of one hundred and twenty.

I am now going to prove my results algebraically for a 2 X 3 rectangle.

 x x+2 x+10 x+12

Therefore : x(x+12) =                     (x+2)(x+10) =

x² + 12x                        x² + 20x + 2x +20 =

x² + 22x + 20

This clearly shows that no matter what the value of x maybe in a 2 X 3 rectangle the difference is always going to be twenty.

Moses Ilori 11P             Miss Lucas

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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