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

{Formula I found: T = 5N – 63}        

For the T-shape that had N to equal 40, this is what I predict:

T = (5 x 40) – 63

   = 200 – 63

   = 137

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

40 + 31 + 21 + 22 + 23 = 137

For the T-shape that had N to equal 43, this is what I predict:

T =  (5 x 43) – 63

   =  215 – 63

   = 152

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

43 + 34 + 24 + 25 + 26 = 152

For the T-shape that had N to equal 47, this is what I predict:

T = (5 x 47) – 63

   = 235 – 63

   = 172

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

 

47 + 38 + 28 + 29 + 30 = 172

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 9 by 9 grid is correct.

Formula for T-shape on a 9 by 9 grid: T = 5N - 63

Part 2

In this section of my coursework I am going to be translating the T-shape onto different positions using different grid sizes. I am then going to investigate the relationship between the T-total, T-number and grid size.

Below there is a 10 by 10 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.

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:

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.

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:

{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

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:

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.

The T-total was abbreviated to T and the T-number to N.

T       N

43 = 24 + 13 + 1 + 2 + 3

The first T-shape had a T-total (T) of 43. 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 calculations and this is what I done:

24 – 13 = 11

24 – 1 = 23

24 – 2 = 22

24 – 3 = 21

Therefore:

T = N + (N-11) + (N-23) + (N-22) + (N-21)

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

43 = 24 + (24-11) + (24-23) + (24-22) + (24-21)

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

43 = 24 + 13 + 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 11 by 11 grid was:

T = N + (N-11) + (N-23) + (N-22) + (N-21)

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

T = 5N – 77

Testing my formula

To test that the formula I had found for the 11 by 11 was correct I randomly chose N to equal 28, 52 and 68. The grid shows the T-shapes for each of these T-numbers:

{Formula I found: T = 5N – 77}        

For the T-shape that had N to equal 28, this is what I predict:

T = (5 x 28) – 77

   = 170 – 77

   = 63

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

28 + 17 + 5 + 6 + 7 = 63

For the T-shape that had N to equal 52, this is what I predict:

T = (5 x 52) – 77

   = 260 – 77

   = 183

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

52 + 41 + 29 + 30 + 31 = 183

For the T-shape that had N equal to 68, this is what I predict:

T = (5 x 68) – 77

   = 340 – 77

   = 263

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

68 + 47 + 55 + 56 + 57 = 263

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 11 by 11 grid is correct.

Formula for T-shape on a 11 by 11 grid: T = 5N - 77

Working out the relationship between the T-total, the T-number and the Grid size.

As you already know T-total is T, T-number is N and I am going to abbreviate Grid size to G.

For the 9 by 9 grid            5N - 63

        +7

For the 10 by 10 grid            5N – 70        (difference of 7)

                                                               +7

For the 11 by 11 grid            5N - 77

9 by 9 grid:

T = 5N – (G x 7)

T = 5N – (9 x 7)        it is 9 x 7 because the grid size is 9

T = 5N – 63                same as the formula that I found for the 9 by 9 grid

10 by 10 grid:

T = 5N – (G x 7)
T = 5N – (10 x 7)        it is 10 x 7 because the grid size is 10

T = 5N –70                 same as the formula that I found for the 10 by 10 grid

11 by 11 grid:

T = 5N – (G x 7)

T = 5N – (11 x 7)        it is 11 x 7 because the grid size is 11

T = 5N –77                same as the formula that I found for the 11 by 11 grid

As each of the formulas shown above are the same as the ones I found earlier, the formula for any grid size given that the T-shape is straight and the same size as the one shown below-right, is:

T = 5N – 7G

Part 3

In this next section there is again a change in the grid size. Also there are transformations and combinations of transformations. I will be investigating the relationship between the t-total, the t-numbers, the grid size and the transformations.

I decided to rotate the T-shape 90 degrees, 180 degrees and 270 degrees.

I started by rotating the T-shape 180 degrees. The 9 by 9 grid below shows how the T-shapes looked like when I rotated it 180 degrees.

The following diagrams show what each individual T-shape would look like:

1st T-shape:

2nd T-shape:

3rd T-shape:

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

When I turned the T- shape around 180 degrees I realized that if I reversed the T-shape then I should also reverse something in the formula. It was obvious to me that I would have to change the minus sign in the previous formula to a plus sign (as a plus sign was the mathematical opposite of a minus sign).

So I predicted that the formula for the T-shape, when rotated 180 degrees, on a 9 by 9 grid, should be:

T = 5N + 63          (Where, as in previous formulas, T is the T-total and N is the T-number)

Nonetheless, to make sure my predicted formula was correct I decided to work on the first T-shape I had drawn.

As shown above I abbreviated T-total to T and T-number to N.

T      N

88 = 5 + 14 + 22 + 23 + 24

The first T-shape had a T-total (T) of 88. I then needed to work out the difference between the rest of the numbers in the T-shape and the T-number (N)

This is what I done:

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

14 – 5 = 9

22 – 5 = 17

23 – 5 = 18

24 – 5 = 19

Therefore:

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

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

88 = 5 + (5+9) + (5+17) + (5+18) + (N+19)

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

88 = 5 + 14 + 22 + 23 + 24

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, 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