# T-Totals. I will take steps to find formulae for changing the position of the T in many ways using methods such as translation and rotation.

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Introduction

T-Totals

This grid has a T-shape in the top left corner of it. The numbers inside of the T are 1, 2, 3, 11 and 20. To find the T total of the T I must add these 5 numbers together. By adding these numbers the T total is 37. The T number in each T drawn is the bottom of the T, whichever way the T faces. This T number is 20. From this information I will take steps to find formulae for changing the position of the T in many ways using methods such as translation and rotation.

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The stages I will take for my investigation are the following: -

- Firstly I will find a formula for finding the T total using the T number and including the grid size

- Once I have found this formula, I will extend it and use it to find formulae to explain and solve translating the T

- I will translate the T
- Using what I have found from my formulae, my next step will be to move the T to the right and down, to the right and up, to the left and down and finally to the left and up
- These are all the possible translations with this shape
- Once I have found those formulae I will move on to rotation
- I will be rotating the T shape 90 degrees, 180 degrees and 270 degrees
- I will start with 90 degrees, then 270 degrees and then I will work on 180 degrees
- Rotation will be my final part of my investigation
- All the way through my work I will be including explanations and diagrams
- As well as using explanations of what I am doing, I will explain why I am doing it and why I get the answers I do
- I will be stating all the variables and when I add a new variable I will clearly state what it is.

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WORKING OUT AND RESULTS

T number/T total(original) | Equation | T number/T total(translated) | Tested |

18/34 | (5 x 18)–56 +(5 x5)x 8 | 58/234 | 41+42+43+50+58 = 234 |

63/259 | (5 x 63)– 56- (5x5)x8 | 23/59 | 6+7+8+15+23 = 59 |

My formula works and I am correct.

Working out the movement of vertical and horizontal in one formula: To move the T shape vertically and horizontally in one formula you must combine the four formulae that have been worked out previously. The formulae that need to be focused on are translating the T shape up and down, and translating it right and left.

Moving down and to the right: the formula for moving down is 5x – y + 5zb and the formula for moving right is 5x – y + 5a. As there is already 5x – y in both of the formulae this part of the formula does not be to be repeated but obviously needs to be in the final formula. The 5zb part of the first formula calculates the movement moving down and the 5a calculates the movement of moving to the right. These are the two movements that are made in the process so are vital and must be in the formula. Therefore, when the three parts of each formula or combined the formula 5x – y +5zb + 5a is created. I will now test this formula on a 10 x 10 grid.

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WORKING OUT AND RESULTS

Formula: 5x – y + 5zb + 5a

Substituting in the relevant information: (5 x 32) – 70 + (5 x 10 x 5) + (5 x 5)

(I have added brackets into the equation to make it clear where numbers has substituted the variables)

Simplified: 160 – 70 + 125 + 25

Answer: 365

Checking: 66 + 67 + 68 + 77 + 87

Answer: 365

My formula is correct and can be used in any grid size as it has a grid size variable.

Moving down and to the left: moving down and to the left consists of the same method of combining the two formulae but the formula for moving to left is different to moving to the right. The slight difference is that as the numbers are decreasing, the +5a becomes –5a because of the direction. The 5zb part of the formula still remains the same as the T shape is still moving in the downward direction. The formula has now become 5x – y + 5zb – 5a. I will now test this formula out on a 9 x 9 grid to show my variation in grid sizes and to prove that the grid size does not effect my result as there is a grid width variable.

(My grid is on the next page)

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WORKING OUT AND RESULTS

Formula: 5x – y + 5zb – 5a

Substituting in the relevant information: (5 x 35) – 63 + (5 x 9 x 5) – (5 x 6)

Simplified: 175 – 63 + 225 – 30

Answer: 307

Checking: 55 + 56 + 57 + 65 + 74

Answer: 307

My formula has worked for this example.

Moving up and right: with this next stage I will be incorporating the same formula into this new formula as I did with the previous one. I will be using the formula 5x – y + 5a in my formula. The second part to the formula is the formula for moving the T shape up. This formula is 5x – y – 5zb. This seems the same as the formula for the downward movement, but there is a minus instead of an addition sign. This is because the numbers decrease as they move upward. The formula for this movement becomes 5x – y – 5zb + 5a when the two formulae are combined. I will now test this formula on an 8 x 8 grid.

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WORKING OUT AND RESULTS

Formula: 5x – y – 5zb + 5a

Substituting in the relevant information: (5 x 42) – 56 – (5 x 8 x 3) + (5 x 5)

Simplified: 210 – 56 – 120 + 25

Answer: 59

Checking: 6 + 7 + 8 + 15 + 23

Answer: 59

I am correct with my example.

Moving up and left: moving up and left is using the same formula as moving up and right but with one adjustment. As we are decreasing our T total by moving it to the left, the additions sign next to the 5a changes to a minus sign. The formula has now become 5x – y – 5zb – 5a. I will now test this on an 11 x 11 grid.

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Conclusion

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WORKING OUT AND RESULTS

Formula: 5x + 7z

Substituting the relevant information: 52 x 5 + 7 x 10

Answer: 330

Checking: 52+62+72+71+73

Answer: 330

This is correct

Substituting in the relevant information: 5 x 35 + 7 x 10

Answer: 245

Checking: 35+45+55+54+56

Answer: 245

I am correct

Substituting in the relevant information: 5 x 77 + 7 x 10

Answer: 455

Checking: 77+87+97+96+98

Answer: 455

I am correct and my formula seems to work.

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In conclusion of my investigation on T totals, I am happy with the results I was able to obtain and I am pleased with the ways I went about my investigation. If I had had more time along the way I would have been able to achieve more challenging aspects of this task. I would have attempted enlargement and reflection. From my investigation I gained the following information:

- How to find the T total with any T number in any size grid
- How to find the T total of a T by moving it up, down, right and left
- I have found how to move the T over a certain amount of squares and up and down a certain amount of squares and finding the T total.
- How to find the T total of a T shape when it is at 90 degrees, 180 degrees and 270 degrees.

The formulae I have found are the following:

- 5x – y
- 5x – y + 5a, 5x – y – 5a, 5x – y + 5zb and 5x – y – 5zb
- 5x – y + 5zb + 5a, 5x – y + 5zb – 5a, 5x – y – 5zb + 5a, and 5x – y – 5zb – 5a.

Of the work I have done and completed, I am pleased that I have understood it all and that I have been able to follow my plan and not become stuck on where to go next. I feel as if I have planned my investigation well and that it was very useful throughout the whole of the T totals assignment.

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

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