# My aim is to see if theres a relation between T total and T number and I will then work out the algebraic expressions so that the 50th term would be found out using the formula.

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Introduction

MATHS

T – Totals COURSEWORK

MR.UDDIN

T – Totals

PART 1:

My aim is to see if there’s a relation between T – total and T – number and I will then work out the algebraic expressions so that the 50th term would be found out using the formula.

1 | 2 | 3 |

12 | ||

22 |

The total of the numbers inside the T – shape is called the T – total.

The bottom number of the T is called the T – number.

So in this case the T – number would be 22.

The T – total is 22 + 12 + 2 + 1 + 3 = 40

T – Total in a 10 by 10 grid:

2 | 3 | 4 |

13 | ||

23 |

T – Total = 2 + 3 + 4 + 13 + 23 = 45

T – Number = 23

Another try:

3 | 4 | 5 |

14 | ||

24 |

T – Total = 3 + 4 + 5 + 14 + 24 = 50

T – Number = 24

I’ll put it in a table to see the difference:

T – number | T – total | Difference |

22 | 40 | |

23 | 45 | 5 |

24 | 50 | 5 |

As you can see that when the T – number increases by 1 the T – total increases by 5. This is because there are 5 boxes in the T –shape and each number increases by 1 will add up to five extra at the end.

Now I predict that T – total is going to be 55 in a 10 by 10 grid, now to check if it is right from a 10 by 10 grid:

4 | 5 | 6 |

15 | ||

25 |

4 + 5 + 6 + 15 + 25 = 55.

This here shows that my prediction was correct.

For a 10 by 10 grid, this is how the T – shape will look:

T-21 | T-20 | T-19 |

T-10 | ||

T |

Middle

5 x 50 – 7 = 243

Now I am going to do a 6 by 6 90º anticlockwise:

This would be the same as this:

T + 4 | T-2 | T - 8 |

T-1 | ||

T |

T + 4 + T – 2 + T – 8 + T – 1 + T = 5T -7

16 | 10 | 4 |

11 | ||

12 |

I have to put my formula to the test:

5 x 12 – 7 = 53

I need to test if the formula is right:

16 + 10 + 4 + 11 + 12 = 53

So the 50th term would be the following:

5 x 50 – 7 = 243

Now I am going to put the results in the table and see if there is a difference in the T – Shapes when I changed the grid size and rotated the T – Shape 90º anticlockwise.

Grid Size | Formula | Difference |

9 by 9 | 5T – 7 | 0 |

8 by 8 | 5T - 7 | 0 |

7 by 7 | 5T - 7 | 0 |

As you can see that there is no difference on these formulae, this is because when you put it into a T – shape, the right hand box and the left hand box will decrease as the grid size decreases. However this wouldn’t be a problem as both numbers has decreased by 1.

Example:

7 by 7 grid:

T + 5 | T-2 | T - 9 |

T-1 | ||

T |

5 – 2 – 9 – 1 = -7

6 by 6 grid:

T + 4 | T-2 | T - 8 |

T-1 | ||

T |

4 – 2 – 8 – 1 = -7

Now I am going to rotate the T - shape 90º clockwise to investigate the difference between the T – number and T – total in different size grids.

Conclusion

CONCLUSION:

During this coursework I found out many things about the T – shapes, the first thing was that the algebraic expression change as the grid size change, however it’s answer was negative and always was in the 7 times table.

Example: 5T – 70 & 5T - 63

However when the T –shape is rotated 90° anticlockwise, the grid size didn’t make a difference to the algebraic expression, infact it didn’t even change, although the numbers in the algebraic expression was negative.

Example: 5T – 7

In addition when I rotated the grid size 90° clockwise, the algebraic expression didn’t change, however it was a positive number, which is an exact opposite of what I found out when I rotated the T – shape 90° anticlockwise.

Example: 5T + 7

However when I rotated the T – shape 180°, the grid size changed the algebraic expression, however as the first it was always in the 7 times but in this case the number was a positive, because it was an exact opposite of the first one.

Example: 5T + 70 & 5T + 63

Moreover, when I did vector translations in my T- shape I found that if there was a vector as so: , the number would decrease by 5, however if there was a vector as so: , the number would increase by 5.

Example:

= 5T – 7P + 5a

= 5T – 7p – 5bP

= 5T – 7P + 5a – 5bP

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

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