My aim is to see if theres a relation between T total and T number and I will then work out the algebraic expressions so that the 50th term would be found out using the formula.

As you can see the 5 remains constant because there are five boxes in the T – Shape, however the numbers at the end follow the 7 times table.

Now I am going to do the P by P grid:

T – 2P + 1 + T – 2P + T – 2P – 1 + T – P + T = 5T – 7P

I’ll check it by a real life example:

12 by 12 grid:

5 x 38 – 7 x 12 = 190 – 84 = 106

To check if this is right I will do the following: 13 + 14 + 15 + 26 + 38 = 106

My formula is right.

So the 50th value would be:

5 x 50 – 7 x 12 = 160

PART 3:

Now I am going to rotate my T – shapes in different grid sizes to 90° anticlockwise, 180°, and 90° clockwise to see if I would get a different algebraic expression.

8 by 8 grid of 90º anticlockwise:

 Would be the same as:

T + 6 + T-2 + T - 10 + T -1 + T = 5T - 7

I’ll test this using a random position from the grid:

5 x 21 – 7 = 98

Check: 27 + 19 + 11 + 20 + 21 = 98

My formula is working. So the 50th term would be the following:

5 x 50 – 7 = 243

Now I’m going to do a 7 by 7 grid 90º anticlockwise:

Would be the same as:

T + 5 + T – 2 + T – 9 + T – 1 + T = 5T – 7

To check if it works:

5 X 14 – 7 = 63

To check if the formula is working right:

14 + 13 + 12 + 19 + 5 = 63

 

So the 50th term would be the following:

5 x 50 – 7 = 243

Now I am going to do a 6 by 6 90º anticlockwise:

This would be the same as this:

T + 4 + T – 2 + T – 8 + T – 1 + T = 5T -7

I have to put my formula to the test:

5 x 12 – 7 = 53

I need to test if the formula is right:

16 + 10 + 4 + 11 + 12 = 53

So the 50th term would be the following:

5 x 50 – 7 = 243

Now I am going to put the results in the table and see if there is a difference in the T – Shapes when I changed the grid size and rotated the T – Shape 90º anticlockwise.

As you can see that there is no difference on these formulae, this is because when you put it into a T – shape, the right hand box and the left hand box will decrease as the grid size decreases. However this wouldn’t be a problem as both numbers has decreased by 1.

Example:

7 by 7 grid:

 5 – 2 – 9 – 1 = -7

                 

6 by 6 grid:

                

4 – 2 – 8 – 1 = -7

Now I am going to rotate the T - shape 90º clockwise to investigate the difference between the T – number and T – total in different size grids. I think that the formulae will all be positive, as it is the exact opposite of a 90° anticlockwise direction.

8 by 8 grid of 90º clockwise:

 

Would be the same as:

T - 6 + T+ 2 + T + 10 + T + 1 + T = 5T + 7

I’ll test this using a random position from the grid:

5 x 41 + 7 = 212

Check: 35 + 43 + 51+ 42 + 41 = 212

My formula is working. So the 50th term would be the following:

5 x 50 + 7 = 257

I’m going to do a 7 by 7 grid 90º clockwise:

Would be the same as:

T + 5 + T – 2 + T – 9 + T – 1 + T = 5T – 7

Now to check if it works:

5 X 40 + 7 = 207

I need to check if the formula is working right:

40 + 41 + 42 + 35 + 49 = 207

 

So the 50th term would be the following:

5 x 50 + 7 = 257

Now I am going to do a 6 by 6 90º clockwise:

This would be the same as this:

T - 4 + T + 2 + T + 8 + T + 1 + T = 5T + 7

I need to put my formula to the test:

5 x 28 + 7 = 147

I need to test if the formula is right:

24 + 30 + 36 + 29 + 28 = 147

So the 50th term would be the following:

5 x 50 + 7 = 257

Now I am going to put the results in the table and see if there is a difference in the T – Shapes when I changed the grid size and rotated the T – Shape 90º clockwise.

As you can see that there is no difference on these formulae, this is because when you put it into a T – shape, the right hand box and the left hand box will decrease as the grid size decreases. However this wouldn’t be a problem as both these numbers have decreased by 1. On the other hand, this formula is opposite to the 90° anticlockwise rotation. This is because the shape has been reflected so the formulae are opposites as well.

Example:

7 by 7 grid:

 -5 + 2 + 9 + 1 = 7

                 

6 by 6 grid:

-4 + 2 + 8 + 1 = 7

I am going to rotate it 180º to investigate the difference between the T – number and T – total in different size grids.

8 by 8 grid of 180º:

Would be the same as:

T + 17 + T + 16 + T + 15 + T + 8 + T = 5T + 56

I’ll test this using a random position from the grid:

5 x 42 + 56 = 266

Check: 59 + 58 + 57 + 50 + 42 = 266

My formula is working. So the 50th term would be the following:

5 x 50 + 56 = 306

I’m going to do a 7 by 7 grid 180º:

Would be the same as:

T + 15 + T + 14 + T + 13 + T + 7 + T = 5T + 49

I need to check if it works:

5 X 30 + 49 = 199

I am going to check if the formula is working right:

45 + 44 + 43+ 37 + 30 = 199

 

So the 50th term would be the following:

5 x 50 + 49 = 299

Now I am going to do a 6 by 6 180º clockwise:

This would be the same as this:

T + 13 + T + 12 + T + 11 + T + 6 + T = 5T + 42

I need to put my formula to the test:

5 x 20 + 42 = 142

I need to make sure that my formula is right:

33 + 32 + 31 + 26 + 20 = 142

So the 50th term would be the following:

5 x 50 + 42 = 292

Now I am going to put the results in the table and see if there is a difference in the T – Shape when I changed the grid size and rotated it

180º.

As you can see there is differences of 7 each time the grid size go down, I expected this to happen as this is exactly what I have found earlier on in page 4.

Now I am going to investigate the relation of T – number and the T – total when translated in a vector. I will then translate it to an algebraic expression.

A vector will look like this:

The a will move the T – shape right and left, right would be a positive number, and left would be a negative number.

The b will moves the T – shape up and down, up would be a positive number; and down would be a negative number.

If a T – shape was translated into a vector of  this is how the T – Shape would look like:

T – 2P + 1 + T – 2P + T – 2P – 1 + T – P + T = 5T – 7P + 5a

To check if my algebraic expression is right, I’ll do the following example in a 9 by 9 grid.

Original:

After a vector translation of , the T –shape would be:

So the equation will work like the following: 5 x 74 – 7 x 9 + 5 x 1 = 312

Now to see if the formula worked: 56 + 57 + 58 + 66 + 75 = 312

My equation is working. This is because every time I translate the T – shape into a vector of, each square will move across adding 1 for each number, since there are 5 boxes, there will be 5 added at the end.

Now I am going to a vector translation of  and I will translate it into an algebraic expression.

If a T – shape was translated into a vector ofthis is how the T – Shape would look like:

T – 2P + 1 + bP + T – 2P + bP + T – 2P - 1 + bP + T – P + bP + T + bP =

5T – 7p – 5bP

To check if my algebraic expression is right, I’ll do the following example in a 9 by 9 grid.

Original:

After a vector translation of  it will be:

So the equation will work like the following: 5 x 74 – 7 x 9 – 5 x 1 x 9 =

370 – 63 – 45 = 262

Now to see if the formula worked: 56 + 57 + 58 + 66 + 75 = 262

My equation is working. This is because every time I translate the T – shape into a vector of, each square will move up taking away 1 for each number, since there are 5 boxes, there will be 5 taken away at the end.

Now I am going to do a vector translation of, this would be much easier as we just have to combine the two algebraic expressions together as shown below:

5T – 7P + 5a – 5bP

Now to check this formula I am going to do the following in a 9 by 9 grid:

Original:

After a vector translation of  it will be:

So the equation will work like the following: 5 x 74 – 7 x 9 + 5 x 1 – 5 x 1 x 9 =

370 – 63 + 5 – 45 = 267

Now to see if the formula worked: 47 + 48 + 49 + 57 + 66 = 267

My equation is working. This is because every time I translate the T – shape into a vector of, each square will move up taking away 1 from each number, on the other hand each square will move across adding 1 from each number, since there are 5 boxes, there will be 5 taken away and added back at the end.

CONCLUSION:

During this coursework I found out many things about the T – shapes, the first thing was that the algebraic expression change as the grid size change, however it’s answer was negative and always was in the 7 times table.

Example: 5T – 70 & 5T - 63

However when the T –shape is rotated 90° anticlockwise, the grid size didn’t make a difference to the algebraic expression, infact it didn’t even change, although the numbers in the algebraic expression was negative.

Example: 5T – 7

In addition when I rotated the grid size 90° clockwise, the algebraic expression didn’t change, however it was a positive number, which is an exact opposite of what I found out when I rotated the T – shape 90° anticlockwise.

Example: 5T + 7

However when I rotated the T – shape 180°, the grid size changed the algebraic expression, however as the first it was always in the 7 times but in this case the number was a positive, because it was an exact opposite of the first one.

Example: 5T + 70 & 5T + 63

 

Moreover, when I did vector translations in my T- shape I found that if there was a vector as so: , the number would decrease by 5, however if there was a vector as so: , the number would increase by 5.  

Example:

 = 5T – 7P + 5a

= 5T – 7p – 5bP

 = 5T – 7P + 5a – 5bP