# T-Total and T-Number Coursework

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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.

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 |

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 |

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:

n-5 | ||

n-2 | 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.

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

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