# The image of the T-shape shown below is drawn onto a 9 by 9 grid. The total of the numbers inside the T-shape is 1 + 2 + 3 + 11 + 20 = 37

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Introduction

T-TOTALS

The image of the T-shape shown below is drawn onto a 9 by 9 grid. The total of the numbers inside the T-shape is 1 + 2 + 3 + 11 + 20 = 37, therefore the T-total is 37. The number at the bottom of the T-shape is called the T-number. In this particular T-shape the T-number is 20.

The problem I have been set is to investigate how the relationship between the T-total and the T-number is affected when I translate the T-shape onto different positions on the grid.

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In addition I am also going to use grids of different sizes and translate the T-shape onto different positions on these grids. By doing this I will be able to investigate relationships between the T-total, the T-number and the grid size.

Furthermore I am again going to use grids of different sizes. On these grids i8 am going to try transformations and combinations of transformations. From doing this I am going to investigate the relationship between the T-total, T-numbers, the grid size and the transformations.

I will display and explain any patterns relationships and formula I find.

Part 1

In this section of my coursework I am going to be investigating the relationship between the T-total and the T-number.

Below there is a 9 by 9 number grid with 3 T-shapes drawn onto it, the T-shapes are highlighted in a red colour. As you know the number at the bottom of the T-shape is called the T-number and all of the other numbers that are highlighted, add together to make the T-total.

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My initial idea was to draw 3 T-shapes onto the 9 by 9 grid.

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As shown above I chose to draw 3 T-shapes onto the 10 by 10 grid. Again I placed the T-shapes in such positions so they will be overlapping one another. The following diagrams show what each individual T-shape would look like:

1st T-shape:

2nd T-shape:

3rd T-shape:

T-number | T-total | ||

1st T-shape | 22 + 12 + 1 +2 + 3 | 22 | 40 |

2nd T-shape | 23 + 13 + 2 + 3 + 4 | 23 | 45 |

3rd T-shape | 24 + 14 + 3 + 4 + 5 | 24 | 50 |

Working out the relationship between the T-total and the T-number

To work out the relationship between the T-total and the T-number on the 10 by 10 grid I followed the same process I used on the 9 by 9 grid.

I decide to work out the relationship by using the 1st T-shape.

T-number | T-total | ||

1st T-shape | 22 + 12 + 1 +2 + 3 | 22 | 40 |

Again, the T-total was abbreviated to T and the T-number to N.

T N

40 = 22 + 12 + 1 + 2 + 3

The first T-shape had a T-total (T) of 40. I then needed to work out the difference between the T-number (N) and the rest of the numbers in the T-shape.

I used the same method that was applied in the previous section. This is what I done:

The difference between the T-number (N) and the other numbers in the T-shape was:

22 – 12 = 10

22 – 1 = 21

22 – 2 = 20

22 – 3 = 19

Therefore:

T = N + (N-10) + (N-21) + (N-20) + (N-19)

I then substituted my formula with the actual numbers from the 1st T-shape; this is the answer I got:

40 = 22 + (22-10) + (22-21) + (22-20) + (22-19)

I worked out the answer for the formula I had found:

40 = 22 + 12 + 1 + 2 + 3

If my formula so far was correct then the numbers I got above had to match those that were in the 1st T-shape (as this was the shape I was using to work out the relationship).

The 1st T-shape was:

From this I realized that the numbers I got from using my formula were the same to those that were in the T-shape. Therefore my formula must be correct.

So the formula for the T-shape on the 10 by 10 grid was:

T = N + (N-10) + (N-21) + (N-20) + (N-19)

I then simplified the formula by collecting the like terms. This is the final formula I came up with:

T = 5N – 70

## Testing my formula

To test that the formula I had found for the 10 by 10 was correct I randomly chose N to equal 26, 33 and 49. The grid shows the T-shapes for each of these T-numbers:

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{Formula I found: T = 5N – 63}

For the T-shape that had N to equal 26, this is what I predict:

T = (5 x 26) – 70

= 130 – 70

= 60

To check my prediction was correct, I added up all of the numbers in the T-shape.

26 + 16 + 5 + 6 + 7 = 60

For the T-shape that had N to equal 26, this is what I predict:

T = (5 x 33) – 70

= 165 – 70

= 95

To check that my prediction was correct, I added up all of the numbers in the T-shape.

33 + 23 + 12 + 13 14 = 95

For the T-shape that had N to equal 49, this is what I predict:

T = (5 x 49) –70

= 245 – 70

= 175

To check that my prediction was correct, I added up all of the numbers in the T-shape.

49 + 39 + 28 + 29 30 = 175

For each of the T-shapes I tested my prediction was correct. In each case my prediction was made based on the formula I found. Each of the T-shapes I tested gave an answer that matched my prediction therefore my formula for a T-shape on a 10 by 10 grid is correct.

Formula for T-shape on a 10 by 10 grid: T = 5N - 70

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Below there is an 11 by 11 number grid with 3 T-shapes drawn onto it, the T-shapes are highlighted in a red colour. As you know the number at the bottom of the T-shape is called the T-number and all of the other numbers that are highlighted, add together to make the T-total.

Again drew 3 T-shapes onto the 11 by 11 grid. As I done previously, I placed the T-shapes in such positions so they will be overlapping. The following diagrams show what each individual T-shape would look like:

1st T-shape:

2nd T-shape:

3rd T-shape:

T-number | T-total | ||

1st T-shape | 24 + 13 + 1 + 2 +3 | 24 | 43 |

2nd T-shape | 25 + 14 + 2 + 3 + 4 | 25 | 48 |

3rd T-shape | 26 + 15 + 3 + 4 + 5 | 26 | 53 |

Working out the relationship between the T-total and the T-number

To work out the relationship between the T-total and the T-number I decided to work on the 1st T-shape I had drawn.

T-number | T-total | ||

1st T-shape | 24 + 13 + 1 + 2 +3 | 24 | 43 |

Conclusion

The 1st T-shape was:

From this I realized that the numbers I got from using my formula were the same to those that were in the T-shape. Therefore my formula must be correct.

So the formula for the T-shape, when rotated 180 degrees, on the 9 by 9 grid was:

T = N + (N+9) + (N+17) + (N+18) + (N+19)

I then simplified the formula by collecting the like terms. This is the final formula I came up with:

T = 5N + 63

When I tested my prediction for the T-shape, when rotated 180 degrees, on a 9 by 9 grid the answer I got was the same as what I predicted. Therefore my overall formula must be correct.

## Testing my formula

Although I am sure that my formula is correct I still decided to test it on one more T-shape (that had been rotated 180 degrees on a 9 by 9 grid).

To test my formula I chose N to equal 52. When N = 52, I predict:

T = (5 x 52) + 63

260 + 63

323

To check my prediction was correct, I added up all of the numbers in the T-shape.

52 + 61 + 69 + 70 + 71 = 323

From this I can say that my formula is definitely correct so the formula for the T-shape, when rotated 180 degrees, on a 9 by 9 grid is definitely:

T = 5N + 63

Just by this I predict that for the:

- 10 x 10 grid the formula when the T-shape is rotated 180 degrees is: T = 5N + 70
- 11 x 11 grid the formula when the T-shape is rotated 180 degrees is: T = 5N +77

To check my prediction for when the T-shape is rotated 180 degrees on a 10 by 10 grid I chose N to equal 24, 36 and 43.

For the T-shape that had N to equal 24 this is what I predict:

T = (5 x 24) + 70

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

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