# T- total T -number coursework

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Introduction

Abdul Anwar

Maths Coursework

PART 1

I have a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below: -

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The number 20 at the bottom of the t-shape will be called the t-number. All the numbers highlighted will be called the t-total. In this section there is an investigation between the t-total and the t-number.

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## For this t-shape the

T-number is 20

And the

T-total is37

## For this t-shape the

T-number is 21

and the

T-total is 42

As you can see from this information is that every time the t-number goes up one the t-total goes up five.

Therefore the ratio between the t-number and the t-total is 1:5

This helps us because when we want to translate a t-shape to another position. Say we move it to here

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I all ready know the answer to the one in red. To work out the one in green all we have to do is work out the difference in the t-number and in this case it is 54. We then times the 54 by 5 because it rises 5 ever time the t- number goes up. Then we + the t-total from the original t-shape and I come out with the t-total for the green t-shape. This is another way to work out the t-total.

What I need now is a formula for the relationship between the t-total and the t-number. I have found a formula which is 5t-number-63 = t-total.

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T-total = 1+2+3+13+24 = 43

T-number = 24

The t-total and the t-number have risen even though the t-shape looks to be in the same place. The t-number has risen by four and the t-total has risen by six. If I use the same rules I made in the last section it works. Here is the longer method

Difference

24-1= 23

24-2 = 22

24-3 =21

24-13 =11

TOTAL =77

Or the shorter way

7* 11 (grid size) = 77

Try out the new formula

5tn – 77= t-total

5*24-77=43

The same formula works with only changing the last number in the formula. This will be tried on a smaller grid size to make sure it is not if the grid size is bigger.

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T-number = 10

T-total = 1+2+3+6+10= 22

7 * 4 (grid size) = 28

5tn- 28= t-total

5*10-28=22

This has proven to work on a smaller scale. I can see that by changing the grid size I have had to change the formula but still managing to keep to the rule of how I get the number to minus in the formula.

PART 3

In this next section there is change in the size of grid. Also there is transformations and combinations of transformations. The investigation of the relationship between the t-total, the t-numbers, the grid size and the transformations.

If I turned the t- shape around 180 degrees it would look like this. When I have done this I should realise if I reverse the t-shape I should have to reverse something in the formula.

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It is obvious that I will have to change the minus sign to a different sign. I should try the opposite of minus which is plus

5tn + 63=t-total

5 * 2 + 63 = 73

Check to see if the formula has worked

T-number = 2

T-total = 2+11+19+20+21 =73

The reverse in the minus sign has worked.

The next step is to move the shape on its side. Again we nearly keep the same formula as I had at the beginning. Again I change the minus number. I can work out the number to minus by working out the difference in the t-number to each number in the t-shape.

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Difference

12-1 =11

12-10= 2

12-19= -7

12-11 = 1

TOTAL = 7

Formula

5tn - 7 =t-total

5*12 - 7= 53

Check to see if the formula is right

T-number = 12

T-total = 1 +10 +19 +11 +12 = 53

This formula has worked. If I rotated the t-shape 180 degrees, the same will happen, as what happened when the t-shape was turned 180 degrees from it is first original position. This is proven below.

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5tn + 7 = t-total

5* 70 + 7 = 357

Check

T-number = 70

T-total = 70+71+72+63+81 = 357

If I were to put the t-shape diagonally on the grid I find that the same rule applies again apart from I can not use the 2nd rule were I times the grid size by seven.

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Conclusion

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Red t-shape

5tn- (7*G) + 133 = t-total

5* 41 –63+ 133= 275

Blue t-shape

5tn- (7*G) -7 = t-total

5*41-63-7 = 135

I now have a formula for seven different rotations. The number at the end of the formula I plus by or in one case minus buy again is divisible by seven. I could say that the magic number for this piece of coursework is seven.

If there are formulas for rotation then surly there is for reflection. Here I have simply only done one type of reflection just to prove that reflection actually works. Here is the formula 5tn+ (12gm) = t-total. How do I get this formula is what we need to know.

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The answer to this is that I need to think of what I’m doing to each of the numbers in the t-shape from the blue t-shapes t-number. For the number 29 I have a grid movement of one so we get (tn+gm). For the number 38 I have a grid movement of two so I get (tn+2gm). For the numbers 46, 47 and 48 I have a grid movement of three and a total of three numbers, se I get 3(tn+3gm). The total of all of them together is (5tn +12*gridsize) = t-total.

This formula should be tested. The t-total of the blue t-shape is 37 and the t-total of the red t-shape is 208.

Formula

5tn+(12*gridsize)= t-total

5*20+ 12* 9 = 208

The formula has worked.

#### CONCLUSION

In this project I have found out many ways in which to solve the problem I have with the t-shape being in various different positions with different sizes of grids. The way I have made the calculations less difficult is by creating a main formula that changes for all the different circumstances.

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

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