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  • Level: GCSE
  • Subject: Maths
  • Word count: 1950

T-Total and T-Number Coursework

Extracts from this document...

Introduction

Frances Duffy        T-Total and T-Number Coursework        10H1 Mrs Smith

T-Total and T-Number Coursework

Introduction

Part 1:

Investigate the relationship between the T-total and the T-number

T-Number

T-Total

20

37

21

42

22

47

23

52

24

57

The table above shows the difference between the consecutive T-Totals as the T-Number increases by one. On grid sheet 1, the T-Shapes can be seen being translated across the 9 x 9 grid by one square each time. There is a pattern between the T-Totals as the T-Shape is translated each time, as each time the T-Total increases by 5, as shown in the table above.

Each T-Total on the diagrams increases by 5 each time it is translated one square across. This is because each square in the T-Shape increases by one each time it is translated, and as there are 5 squares in the T-Shape, a total increase of 5 is calculated for the T-Total.

Already from this, I can begin to create a formula for working out the T-Total for any T-Shape on a 9 x 9 grid.

n-19

n-18

n-17

n-9

n

The formula shown in the T-Shape above should work out the T-Total for any T-Shape on a 9 x 9 grid. I now plan to test this theory, by taking a few random sample T-Numbers from the 9 x 9 grid and using the formula to work out the T-Total.

...read more.

Middle

11

The T-Number of the T-Shape is 11, therefore n = 11.

I will now use my formula to work out that T-Total for this T-Shape:

T = 5n – 28

                                                T = (5 x 11) – 28

                                                T = 27

I know that this is correct because:

T = 2 + 3 + 4 + 7 + 11

                                                T = 27

For 5 x 5 Grid Size:

This is the first of two T-Shapes that I will do for the 5 x 5 grid.

1

2

3

7

12

I will work out a formula for all T-Shapes on a 5 x 5 grid:

n-11

n-10

n-9

image03.png

n-5

n

The formula worked out is 5n - 35.

The T-Number of the T-Shape is 12, therefore n = 12. I will now work out the T-Total:                

T = 5n – 35

                                                T = (5 x 12) – 35

                                                T = 25

I know that this is correct because:

T = 1 + 2 + 3 + 7 + 12

                                                T = 25

This is the second grid I will now do from the 5 x 5 grid, to check that my formula is correct.

2

3

4

8

13

The T-Number of the T-Shape is 13, therefore n = 13.

I will now use my formula to work out that T-Total for this T-Shape:

T = 5n – 35

                                                T = (5 x 13) – 35

                                                T = 30

I know that this is correct because:

T = 2 + 3 + 4 + 8 + 13

                                                T = 30

For each grid, I have now worked out a formula that will find the T-Total of any T-Shape on that grid.

        Now, I am going to incorporate the grid size into every formula.

For 3 x 3 grid, the formula for the T-Total is 5n - 21. By dividing 21 by the grid size, 3, and substituting it into the formula, I came up with the following:

T = 5n - 21

                                                                     3

T = 5n – 7g

Where n is the T-Number and g is the grid size.

I will now test this formula on the 3 x 3 grid T-Shape.

1

2

3

5

8

The T-Number of the T-Shape is 8, therefore n = 8. I will now work out the T-Total using the new formula:

T = 5n – 7g

T = (5 x 8) – (7 x 3)

T = 40 – 21

T = 19

I already know that 19 is the T-Total for this T-Shape, so I know that my new formula is correct.

I then found that the same algebraic formula, 5n – 7g, could also be used for the 4 x 4 and 5 x 5 grids.

For the 4 x 4 grid, the formula for the T-Total is 5n – 28. I again put the grid size, 4, into the equation. This is what I did:

T = 5n - 28

                                                                     4

T = 5n – 7g

Where n is the T-Number and g is the grid size.

I will now test this formula on the 4 x 4 grid T-Shape:

1

2

3

6

10

...read more.

Conclusion

2

5

7

8

9

The T-Number for this shape from a 3x3 grid is 2, therefore n = 2. The grid size is 3, therefore g = 3. I will now substitute these into the inverse formula that I predicted would be correct for this rotation:                                                                                

T = 5n + 7g

T = (5 x 2) + (7 x 3)

T = 10 + 21

                                                T = 31

I know that this is correct because:

T = 2 + 5 + 7 + 8 + 9

                                                T = 31

This proves that my prediction was correct and the expression works.

Rotation 90° Anti-Clockwise:

image00.png

n-5

n-2image01.png

n-1

n

n+1

The expression is the opposite of the expression finding the T-total of a T-shape rotated 90º clockwise, which was 5n+7. This is anti-clockwise, therefore the formula is reversed and the sign changed.

Any Grid Size – Any T-Number – Anywhere on the Grid

        Now that I have found the formulae for T-shapes when rotated either 90° or 180°, I will go on to try and find some formulae to work out the T-Total of any T-Shape, with any T-Number, on any size grid. I will do this using n as the T-Number, and g as the grid size.

        Below is a T-shape from a 3x3 grid.

1

2

3

5

8

I will now substitute into this t-shape n and g, where n is the t-number and g is the grid size.

n-2g-1

n-2g

n-2g+1

n-g

n

The T-Number of the T-Shape is 8, therefore n = 8. The grid size is 3, therefore g = 3. I will substitute these into the formulae to check if it is correct.

        

...read more.

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