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• Level: GCSE
• Subject: Maths
• Word count: 2179

# 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

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p+r

49

50

51

52

53

54

55

p+2r-1

p+2r

p+2r+1

59

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62

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## 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

) works for all standard T-shapes. Below is another example but this time the grid size is 5 x 5.
 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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