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Introduction

Simon Bevan

Maths Coursework

## Investigation

The aim of this investigation is to find out the difference between the products of numbers in the opposite corners of any rectangle that can be drawn on a 100 square. I shall start off researching the 2x3 rectangle and then working onto bigger ones.

 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

2x3 Rectangle

 7 8 9 17 18 19
 1 2 3 11 12 13 88 89 90 98 99 100

1x13=13                          88x100=8800                           7x19=133

11x3=33                          98x90=8820                             17x9=153

20                                          20                                          20

Prediction

I predict that when I multiply a 2x3 rectangle the opposite corners will have a difference of 20.

2x2 Rectangle (square)

 72 73 82 83
 4 5 14 15 33 34 43 44

4x15=60                        33x44=1452                     72x83=5976

14x5=70                         43x34=1462                    82x73=5986

10                                       10                                      10

Prediction

I predict that when I multiply a 2x2 rectangle (square)

Middle

74

75

76

71

72

73

81

82

83

91

92

93

54x76=4104                                         71x93=6603

74x56=4144                                         91x73=6643

1. 40
 10 11 12 20 21 22 30 31 32

Prediction
I predict that when I

10x32=320                                                   multiply a 3x3 30x12=360                                                   rectangle (square) the

Opposite corners will

Have a difference of 40.

3x4 Rectangle

 42 43 44 45 52 53 54 55 62 63 64 65 17 18 19 20 27 28 29 30 37 38 39 40

42x65=2730                                                 17x40=680

62x45=2790                                                 37x20=740

60                                                                60

 68 69 70 71 78 79 80 81 88 89 90 91

68x91=6188

88x91=6248

60

Again I have noticed a pattern occurring each time the width increases by 1 the difference increases by 20.

Prediction

Using the theory I predict that when I multiply a 3x4 rectangle the opposite corners will have a difference of 60.

3x5 Rectangle

Prediction

I predict that when I multiply a 3x5 rectangle the opposite corners will have a difference of 80.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 73 74 75 76 77 83 84 85 86 87 93 94 95 96 97

1x25=25                                                   73x97=7081

21x5=105                                                 93x77=7161

1. 80
 40 41 42 43 44 50 51 52 53 54 60 61 62 63 64

40x64=2560

60x44=2640

80

3x6 Rectangle

Prediction

I predict that when I multiply a 3x6 rectangle the opposite corners will have a difference of 100.

 18 19 20 21 22 23 28 29 30 31 32 33 38 39 40 41 42 43 61 62 63 64 65 66 71 72 73 74 75 76 81 82 83 84 85 86

18x43=774                                             61x86=5246

38x23=874                                             81x66=5346

1. 100
 8 9 10 11 12 13 18 19 20 21 22 23 28 29 30 31 32 33

8x33=264

28x13=364

100

Pattern  Difference

3x2    20 3x3    40

3x4    60

3x5    80

3x6    100

##### A Graph to Show the Difference within the Rectangle 3n 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

3x5

X(X+24)                            (X+4) X+20)

X² +24X                               X(X+20) +4(X+20)

X² +24X                               X² +20X+4X+80

X² +24X                               X² +24X+80

X² +24X

-X² +24X+80

80

I shall now move on and investigate 4x4 Rectangle (square) and more Rectangles in the same way.

4x4 Rectangle (square)

 56 57 58 59 66 67 68 69 76 77 78 79 86 87 88 89 19 20 21 22 29 30 31 32 39 40 41 42 49 50 51 52

56x89=4984                                        19x52=988

86x59=5074                                        49x22=1078

1. 90
 1 2 3 4 11 12 13 14 21 22 23 24 31 32 33 34

Conclusion

L-1(G (H-1)) = L-1 x (Grid Width) x (H-1) = Difference

Evaluation

I have completed this piece of coursework and succeeded in my aims I set out to do. I have found the rule to work out the difference for any Rectangle of any size on any grid. I have done this by progressive investigation and the use of algebraic methods.

I have taken the coursework as far as I can in the amount of time I had been allotted. I am happy with the amount of work that I have done on this bit of coursework and can’t think of anything I could of carried out better or improved upon.

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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3 star(s)

This is a fairly well structured report that covers a wide variety of experimental examples. To improve this the minor errors need to be revised and more emphasis placed on the algebraic analysis of the patterns. There are specific strengths and limitations suggested throughout.

Marked by teacher Cornelia Bruce 18/04/2013

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