# The Relationship between the T-number and T-total

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Introduction

## Mathematics coursework by Nanjie Lu

The Relationship between the T-number and T-total

#### Question One

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

First of all I want use letter n for T-number, use letter T for T-total, use letter g for size of number-grid.

This question already told me that the size of number-grid is nine. So I am copied a size nine number-grid below:

In this table I chose a group of numbers to compare them and try to find out a relationship between one number and the total amount of this group, the numbers are below:

1 | 2 | 3 |

10 | 11 | 12 |

I chose the number ten for basic number, I am going to use letter X for this basic number. Also I am going to use letter Y for the total amount of this group.

So the result is showing below:

X-9 | X-8 | X-7 |

X | X+1 | X+2 |

Next step is work out the how much Y is (the total amount of this group).

Add them all together equals:

## So the relationship between the Y and X is:

Middle

The result of T-shape three:

- n = 78
- T = 59 + 60 + 61 + 69 + 78 = 327
- 327 = 5 × 78 – 63

These two T-shapes are both correct; I used same formula as T-shape one.

So this means this relationship can be used for anywhere of a size 9 by 9 number-grid.

Key point – the formula of the size 9 number-grid is showing below:

Question two:

- Use grids of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, T-number and the grid size.

I chose two different sizes of number-grid to see the relationships between the T-number, T-total and the grid size.

The first one is size of 6 number-grid.

First of all I chose three T-shapes are showing below:

19 | 20 | 21 |

26 | ||

32 | ||

1 | 2 | 3 |

8 | ||

14 |

22 | 23 | 24 |

29 | ||

35 |

I am still using letter n for T-number, letter T for T-number try to find the relationship between the T-number and T-total in size six number-grid.

The result is showing below:

n – 11 | n – 12 | n – 13 |

n – 6 | ||

n |

Now I am going to work out the T-total of this T-shape.

I found the relationship between the T-total and T-number is:

##### T = 5n – 42

The next thing is check the answer see if it is really can used for these T-shapes.

The results are showing below:

In T-shape one:

- n = 14
- T = 1 + 2 + 3 + 8 + 14 = 28
- 28 = 5 × 14 – 42

In T-shape two:

- n = 32
- T = 19 + 20 + 21 + 26 + 32 = 118
- 118 = 5 × 32 – 42

In T-shape three:

- n = 35
- T = 22 + 23 + 24 + 29 + 35 = 133
- 133 = 5 × 35 – 42

Conclusion

From the table above I found out that we could not use some numbers next to the edge of number-grid, the reason for this is showing below:

The diagram above shows some numbers that I chose from the size nine number-grid, all these numbers could not be T-numbers. That is because we cannot use these numbers to form a complete T-shape. For example, I chose number one as a T-number, the meaning of T-number is the number at the bottom of the T-shape. But the problem is the number one is the top row of the whole number-grid so we cannot find any line above it to form a T-shape. Next we try the number ten as a T-number, still could not get a whole T-shape out, if we use ten the shape just like this:

Same problem as all the number in the last right hand-side column we cannot make a T-shape out use this numbers.

Now I found out that all the number in the first two rows across, the first column and last column cannot be a T-number.

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

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