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  • Level: GCSE
  • Subject: Maths
  • Word count: 1138

To find the relationship between the T-Total and the T-Number I must investigate the relationship between the T-number and the other numbers in the T-shape.

Extracts from this document...

Introduction

T-Totals

Part 1image00.png

T-number = 23

T-Total = 23+14+5+4+6 = 52

To find the relationship between the T-Total and the T-Number I must investigate the relationship between the T-number and the other numbers in the T-shape.

To do this I will take all the numbers in the T away from the T-number to produce the formula.

image01.png

Formula for the relationship between the T-number and T-total in a 9 by 9 grid = T + (T – 9) + (T – 18) + (T- 19) + (T – 17) = 5T – 63

To test this formula I will use another T on the 9 by 9 grid.

image06.png

T-number = 62

T-Total = 5T – 63

       = (5 x 62) – 63

       = 247

    43 + 44 + 45 + 53 + 62= 247

The formula has been tested and found to be correct as 5T-63.


Part 2

I will now use grids of different sizes to investigate the relationship between the T-total, the T-numbers and the grid size.

I will start with a six by six gird and find the relationship between the T- total and the T-number.

image07.png

T-number = 21

T- total = 63

As before I will show the relationship between the T-number and the other numbers in the T.image08.png

Formula for the relationship between the T-number and T-total in a 6 by 6 grid = T + (T – 6) + (T – 12) + (T – 13) + (T – 11) = 5T – 42

...read more.

Middle

To prove this formula I will show the relationship between the T-total and the T-number but I will involve the grid size.

image09.png

Formula for the relationship between the T-number, T-total and grid size = T + (T – g) + (T – 2g) + (T – 2g + 1) + (T  - 2g – 1) = 5T - 7g

I will test this formula on a different grid.

image10.png

T- number = 67

T-total = (5 x 67) – (7 x 12) = 251

42 + 43 + 44 + 55 +67 = 251

The formula is correct.


I shall now investigate formulas showing the relationship between the T-number, the T- number and the grid size when they are rotated through 90° c/w, 180° and 90° a-c/w.

I will start with a 180° rotation.image11.png

T- number = 32

T- total = 32 + 41 + 49 + 50 +51 = 223

As before to show a formula I must investigate the relationship between the T-number, T- total and the grid size.

image12.png

Formula for the relationship between the T-number, T-total and grid size at a 180° rotation =

T + (T + g) + (T + 2g) + (T + 2g –1) + (T + 2g + 1) = 5T + 7g

I note that this formula is the opposite of the formula for the vertical rotation (5T – 7g).


I will now investigate a formula showing a relationship between the T-number the T-total and the grid size at a 90° c/w rotation.

image13.png

T-number = 72

T- total = 72 + 73 + 74 + 61 + 87 = 367image02.png

...read more.

Conclusion

image04.png

This T will be called T1.
T-number = 55

T-total = 390

I will transform the T to new a new position by moving it 3 along and

 2 up. (-3

             2)

This T will be called T2image05.png

T-number = 75

T-total = 305


To investigate the relationship between the T-total, the T-number, the grid size and the transformations I will need to show the effects of moving the T two up and 3 along and how this relates to the numbers in the T and the grid size.

  • If you move the T up or down one each number in the T increases by the grid size
  • If you move the T left or right one each number increases by one.

With the bullet points above I can construct a formula, which show the relationship between the two transformed T’s.

T-total of the original T – ((number moved up or down x amount of numbers in the shape x grid size) + (number moved left or right x amount of numbers in the shape)) = The T-total of the second, transformed T.

To test this formula I will use the T’s from above.

390 – ((2 x 5 x 10) + (-3 x 5)) = 305

305 is the T-total of T2 confirming the formula.

This formula will work for a transformation of a rotated T or a different shape.

...read more.

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