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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.

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p

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p+r

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p+2r-1

p+2r

p+2r+1

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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

image00.png

Example:

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p

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p+r

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p+2r-1

p+2r

p+2r+1

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T- number = T41                                                          p = T-number

T- total = 268                                                              q = T-total

Grid size = 9 x 9                                                         r = grid size

Calculations:

...read more.

Middle

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p+r

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p+2r-1

p+2r

p+2r+1

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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

image00.png

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

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

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p

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p+r

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p+2r-1

p+2r

p+2r+2

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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.

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p

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p+r

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p+2r-1

p+2r

p+2r+1

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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.

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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.

image00.png

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

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

8 x 8 general pattern:

T35

T36

T37

T38

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

5 x 5 general pattern:

T8

T9

T10

T11

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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).

image00.png

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p-2r-1

p-2r

p-2r+1

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p-r

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...read more.

Conclusion

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

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p-2r-1

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p-2r+1

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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

image00.png

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

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

8 x 8 general pattern:

T38

T39

T40

T41

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

5 x 5 general pattern:

T12

T13

T14

T15

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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).

...read more.

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