# GCSE Mathematic Coursework T-totals Aim: to find a pattern that connects the T- number with the T- total

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Introduction

GCSE Mathematic Coursework

T-totals

Aim: to find a pattern that connects the T- number with the T- total.

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 |

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 | p | 41 | 42 | 43 | 44 | 45 |

46 | 47 | 48 | p+r | 50 | 51 | 52 | 53 | 54 |

55 | 56 | p+2r-1 | p+2r | p+2r+1 | 60 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

## T- number = T 40 p = T-number

T- total = 263 q = T-total

Grid size = 9 x 9 r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

Formula for T-shape: q = 5p + 7r

Justification: (5 x 40) + (7 x 9) = 263

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 | p | 42 | 43 | 44 | 45 |

46 | 47 | 48 | 49 | p+r | 51 | 52 | 53 | 54 |

55 | 56 | 57 | p+2r-1 | p+2r | p+2r+1 | 61 | 62 | 63 |

64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 |

73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 |

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 |

## T- number = T41 p = T-number

T- total = 268 q = T-total

Grid size = 9 x 9 r = grid size

Calculations:

Middle

41

42

43

44

45

46

47

p+r

49

50

51

52

53

54

55

p+2r-1

p+2r

p+2r+1

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

## T- number = T39 p = T-number

T- total = 258 q = T-total

Grid size = 9 x 9 r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

## Formula for T-shape: q = 5p + 7r

Justification: (5 x 39) + (7 x 9) = 258

The examples above give evidence to justify that the formula (q = 5p + 7r) works for all T- shapes that are rotated 1800.

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 |

I will now alter the grid sizes to try and justify as to whether my formula works on different grid sizes.

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 | p | 36 | 37 | 38 | 39 | 40 |

41 | 42 | p+r | 44 | 45 | 46 | 47 | 48 |

49 | p+2r-1 | p+2r | p+2r+2 | 53 | 54 | 55 | 56 |

57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |

## T- number = T35 p = T-number

T- total = 231 q = T-total

Grid size = 8 x 8 r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

## Formula for T-shape: q = 5p + 7r

Justification: (5 x 35) + (7 x 8) = 231

The example above justifies clearly that the formula (q = 5p + 7r) works for all T-shapes that are rotated 1800.

1 | 2 | 3 | 4 | 5 |

6 | 7 | p | 9 | 10 |

11 | 12 | p+r | 14 | 15 |

16 | p+2r-1 | p+2r | p+2r+1 | 20 |

21 | 22 | 23 | 24 | 25 |

Now I am going to change the grid size to a completely contrasting size, this time the grid size is 5 x 5 and a contrasting T-number will be used.

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 |

## T- number = T8 p = T-number

T- total = 75 q = T-total

Grid size = 5 x 5 r = grid size

Calculations:

p + p + p + p + p = 5p

r + 2r + 2r + 2r = 7r

-1 + 1 = 0

## Formula for T-shape: q = 5p + 7r

Justification: (5 x 8) + (7 x 5) = 75

Although this time the T-number I have used is different from the numbers used before the outcome is still exactly the same. This justifies that the formula (q = 5p + 7r) works for any T-shape that has a rotation of 1800 regardless of the T-number or the grid size.

Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for T-shapes that are rotated 1800 is (q = 5p + 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.

9 x 9 general pattern:

T40 | T41 | T42 | T43 |

263 | 268 | 273 | 278 |

As the T-numbers increase by 1, the T-total increases by 5.

8 x 8 general pattern:

T35 | T36 | T37 | T38 |

231 | 236 | 241 | 246 |

As the T-numbers increase by 1, the T-total increases by 5.

5 x 5 general pattern:

T8 | T9 | T10 | T11 |

75 | 80 | 85 | 90 |

As the T-numbers increase by 1, the T-total increases by 5.

My formula to calculate the T-total is the same for all of these grid sizes (when the T-shape is rotated 1800). This justifies my formula, (q = 5p + 7r).

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 | p-2r-1 | p-2r | p-2r+1 | 33 | 34 | 35 | 36 |

37 | 38 | 39 | p-r | 41 | 42 | 43 | 44 | 45 |

46 | 47 | 48 | p | 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 |

Conclusion

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 |

p-2r-1 | p-2r | p-2r+1 | 4 | 5 |

6 | p-r | 8 | 9 | 10 |

11 | p | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 |

## T- number = T12 p = T-number

T- total = 25 q = T-total

Grid size = 5 x 5 r = grid size

Calculations:

p + p + p + p + p = 5p

-r + -2r + -2r + -2r = -7r

-1 + 1 = 0

## Formula for T-shape: q = 5p – 7r

Justification: (5 x 12) - (7 x 5) = 25

Overall after using various T-numbers and different grid sizes I have come to the conclusion that the formula for standard T-shapes is (q = 5p - 7r). Using the formula I can now calculate different T-totals for different grid sizes and show a general pattern. Below are the patterns from the grid sizes I used.

9 x 9 general pattern:

T49 | T50 | T51 | T52 |

182 | 187 | 192 | 197 |

As the T-numbers increase by 1, the T-total increases by 5.

8 x 8 general pattern:

T38 | T39 | T40 | T41 |

134 | 139 | 144 | 149 |

As the T-numbers increase by 1, the T-total increases by 5.

5 x 5 general pattern:

T12 | T13 | T14 | T15 |

25 | 30 | 35 | 40 |

As the T-numbers increase by 1, the T-total increases by 5.

My formula to calculate the T-total is the same for all of these grid sizes (standard T-shape). This justifies my formula, (q = 5p - 7r).

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

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