# My course work in maths is going to consist of opposite corners and/or hidden faces.

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Introduction

My course work in maths is going to consist of opposite corners and/or hidden faces. For my first course in mathematics I have been given the task of investigating the difference between the products of the numbers in the opposite corners of my rectangle that can be drawn on a 10 by 10 square.

I will start off by showing some examples of different rectangles and what the sum of the corners will be. This will help me to determine whether or not there is a pattern.

## Example

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 |

1* 12=12

2* 11=22

22-12=10

This example shows us that between the squares 1,2,11,12 the sum will be 10.

We will now go on to see what a square between 1,2,21,22 will be.

Middle

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

3*14=42

4*13=52

52-42=10

For now it seems as though the pattern is working the same for the rest of the table. We need to justify our conclusion by drawing another two examples.

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 |

3*24=76

4*23=96

96-76 = 20

Again the same results seem to be occurring in this row but to be sure we will do one more example.

## Example

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 |

3*34= 102

4*33= 132

132 – 102 = 30

As is confirmed by my results any rectangle that is 2 along and 2 down will be 10, then 2 along 3 down will be 20 and so on and so forth.

Now that we have a small result

Conclusion

## Example

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 |

1*33=33

3*31=93

93-33=60

This confirms my prediction.

Now that I have gathered some information I can create I table of my results

and of further predictions.

2*2=10 | 3*2=20 | 4*2=30 | ||

2*3=20 | 3*3=40 | 4*3=60 | ||

2*4=30 | 3*4=60 | 4*4=90 | ||

2*5=40 | 3*5=80 | 4*5=120 | ||

2*6=50 | 3*6=100 | 4*6=150 | ||

2*7=60 | 3*7=120 | 4*7=180 | ||

2*8=70 | 3*8=140 | 4*8=210 | ||

2*9=80 | 3*9=160 | 4*9=240 | ||

2*10=90 | 3*10=180 | 4*10=270 |

I could carry on but I am sure that you can see from the table that a 5*2 would be 40 then a 5*3 would be 80 and so on.

Justification

The formula for the sum of the opposite corners minus the sum of the two opposite corners is :-

(10x – 10) * (y – 1)

X = the number of squares across

Y = the number of squares down

An example of this is a 3 * 3 square would equal

(10*3 – 10) * (3-1)

= 20 * 2

= 40

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 |

If we work this out by multiplying the opposite corners and subtracted the sums we get

3 * 21 – 1 *23

= 63 – 23

= 40

This proves my formula works.

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

The general pattern for a 10 x 10 grid is identified. To improve this investigation more algebraic manipulation is needed to verify the identified pattern. There should be multiplication of double brackets and the identification of an nth term. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/04/2013