# OPPOSITE CORNERS

Extracts from this document...

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

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

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98

99

100

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.

- -

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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## Here's what a teacher thought of this essay

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