T-Shape Investigation.

So my prediction for the Third position will be that it is increased by 5, which will equal 38.

3+4+5+10+16 = 38

This shows that my prediction is correct once again.

From my previous work on the nth term I have made a rule for the T-Total. This rule is 5n.

I have made this rule to make it easier to use once I have found a rule to predict the T- Total using just the T-number and the size of the grid it is drawn on.  

7 by 3 grid

I am now going to make a 7 by 3 grid to add results to my over all table which I am going to make.

The first T-shape (blue outline) T- Total is 1+2+3+9+16 = 31

The T- Number of this T-shape is 16

The second T-shape (red outline) T- Total is 2+3+4+10+17 = 36

The T- Number of this T-shape is 17

The third T-shape (brown outline) T- Total is 3+4+5+11+18 = 41

The T- Number of this T-shape is 18

I am now going to put my information in a table so I can look at more carefully.

I have not felt the need to make a prediction of what the brown T-shape’s T-Total was going to be because I have already explained that the T-Total will always increase by 5 when you move it to the right.

I am now going to look at the tables and the actually T-shapes to see if they have anything in common. I have also made tables to show the start of a rule I have made to predict the T- Total

5 by 3 grid

Using my 5n rule for the increase of the T- Total when you move it to the right, I am going to use it on the T- Number. And then using 7 multiply by the grid number and subtracting that answer from the 5n rule you should get the T- Total.  

12 is the T- Number of the first position

5 * 12 = 60  

7 * 5 = 35

If you subtract 35 from 60 you get 25 which is the T- Total of this T- Number

13 is the T- Number of the second position

5 * 13 = 65

7 * 5 = 35

If you subtract 35 from 65 you get 30 which is the T-Total of this T- Number

14 is the T- Number of the third position

5 * 14 = 70

7 * 5 = 35

If you subtract 35 from 70 you get 35 which is the T-Total of this T- Number

Here you can see that this rule works! I will prove that it works for the other grids that this rule works.

6 by 3 grid

16 is the T- Number of the third position

5 * 16 = 80

7 * 6 = 42

If you subtract 42 from 80 you get 37 which is the T-Total of this T- Number

 

Here my rule works again

7 by 3 grid

18is the T- Number of the third position

5 * 18 = 90

7 * 7 = 49

If you subtract 49 from 90 you get 41 which is the T-Total of this T- Number

Here you can see that this rule works for all of the grids.

I will now explain how it works.

 

Here is my T- Shape in blue. I have made this in algebraic expression in the next diagram.

T= the T- Number

G= the Grid size (number of columns)

T + (T-5) + (T-2G) + (T- 2G +1) + (T-2G -1) = 5T – 7G

Here you can see what my algebraic expression has showed. It has given me the rule to find the T-Total of any T-shape facing upward with just the knowledge of the T- Number and the size of the grid.

This works! You get the 5 multiplied by the T-Number by getting 5 numbers in the T- Shape.  By applying the 5n rule to this you get 5T.

You get the 7G because of the structure of the T-shape.

For the row with the T-number in it the row equals 0

The next row up equals –1G because you have to subtract the grid size to get the T-Number

The last row from the T-Number is -2G because this is what you have to get the T-Number from this point.  

When you add -1G and the three -2Gs you get -7G

I will prove that my rule works on a 10 by 10 grid.

5 * 65 – 7*10= 255 

I will add up the numbers in the T- Total to prove that my rule is correct.

44+45+46+55+65= 225

My rule is correct!

First Extension

Moving the T-Shape 90 degrees to the right

Here I have moved my T-Shape 90 degrees to the right. I am going to investigate this T-Shape in the same why I did the last.

5 by 3 grid

My first position is in blue. The T- Number is 6. The T- Numbers will be in this place. This is the same as the last T-Shapes.

 

The first T-shape (blue outline) T- Total is 6+7+8+3+13 = 37

The T- Number of this T-shape is 6

The second T-shape (red outline) T- Total is 7+8+9+4+14 = 42

The T- Number of this T-shape is 7

The third T-shape (brown outline) T- Total is 8+9+10+5+15 = 47

The T- Number of this T-shape is 8

I am now going to make a table to put these results in a table to make it easier to read.

Here again I see that the T- Total is going up in 5. This is for the same reason as I stated before.

There is 5 numbers in each T-Shape so if you move the T-shape over each individual number is in creased by 1.  Equaling 5 * 1    

E.g. 6 changes to 7

       7 changes to 8 etc

With the information that I poses from my last investigation I feel no need to write out the 6 by 3 grid and the 7 by 3 grid. I know what is going to happen.

So I am going find the rule for this T-Shape.

First I am going to write the algebraic expression of how to get all the numbers in the T-Shape from the T- Number.

 

This is my T-Shape.

T= T- Number

T+(T+1)+(T+2)+(T-3)+(T+7) = 5T+7   

T= T- Number

By adding together the method of obtaining the numbers in the T- Shape from the T- Number I found the method for finding the T- Total by just using the number of columns in the grid and the T- Number!

I will show that my rule works using a 10 by 10 grid.

5 * 33 + 7= 172

I will now add the numbers in the T- Shape to see if I am correct.

33+34+35+45+25=172

This method is correct!

By first multiplying 5 by the T-total it is making 5 equal numbers.

Then by adding the 7 it is putting 2 extra into 3 and then 1 extra into one.

You get the 5 multiplied by the T- Number by 5 numbers being used in the T- Shape.

5 * 33 = 165

165 / 5 = 33

         +1     +2    and then rearrange them to the original number in grid

This is how that method works.

Second Extension

Rotating the Original T- Shape 90 degrees to the left

Now I have rotated the T-shape 90 degrees to the left.  I am going to try to find a method to predict the T –Total by just knowing the T- Number and the grid size (the number of columns in the grid)

 

I will now draw a 5 by 3 grid with the T-shape translated 3 times to collect information to find the rule.

The T- number is the same has it has been in last investigations. It will be 8 in the first position of this 5 by 3 grid.

5 by 3 grid

The first T-shape (blue outline) T- Total is 8+7+6+1+11 = 33

The T- Number of this T-shape is 8

The second T-shape (red outline) T- Total is 9+8+7+2+12 = 38

The T- Number of this T-shape is 9

The third T-shape (brown outline) T- Total is 10+9+8+3+13 = 43

The T- Number of this T-shape is 10

I am now going to put this information into a table so that it is easier to read.

This table shows the same information as the tables I have made before. It shows that the T- Total always goes up in fives.

I will have already explained this. But I will once again explain why.

There is 5 numbers in each T-Shape so if you move the T-shape over each individual number is in creased by 1.  Equaling 5 * 1    

E.g. 8 changes to 9

       8 changes to 10 etc

I will not make any more tables of a 6 by 3 grid or a 7 by 3 grid because they show me the same information which I already know.

This is the T-Total will always go up in 5 if you move it to the right.

I am now going to try to find a method the same way I have done before.

I am going to break down the T- Shape in an algebraic expression to find the method.

Here is what it looks like as an algebraic expression

T = the T- Number

T+(T-1)+(T-2)+(T+3)+(T-7) = 5T-7   

T = the T- Number

By adding together the method of obtaining the numbers in the T- Shape from the T- Number I found the method for finding the T- Total by just using the number of columns in the grid and the T- Number!

I will now prove that this rule works using a 10 by 10 grid.

5*89-7 =438

I will add up the numbers in the T- Shape to make sure my answer is correct.

87+88+89+77+97= 438

This rule works!

I will now explain why it works.

By first multiplying 5 by the T-total it is making 5 equal numbers. But these numbers are 7 higher then the T- Total.

5*97= 485

Then my subtracting 7 it equals the T-Total.

This rule is the opposite of the last extension piece because it is in the opposite direction to that T- Shape.

Third Extension

Rotating the original T-Shape 180 degrees

Here I have moved my T-Shape 180 degrees. I am going to investigate this T-Shape in the same why I did the last.

5 by 3 grid

My first position is in blue. The T- Number is 6. The T- Numbers will be in this place. This is the same as the last T-Shapes.

 

The first T-shape (blue outline) T- Total is 11+12+13+7+2 = 45

The T- Number of this T-shape is 2

The second T-shape (red outline) T- Total is 12+13+14+8+3 = 50

The T- Number of this T-shape is 3

The third T-shape (brown outline) T- Total is 13+14+15+9+4 = 55

The T- Number of this T-shape is 4

I am now going to put this information into a table so that it is easier to read.

This table shows the same information as the tables I have made before. It shows that the T- Total always goes up in fives.

I will have already explained this. But I will once again explain why.

There is 5 numbers in each T-Shape so if you move the T-shape over each individual number is in creased by 1.  Equaling 5 * 1    

E.g. 11 changes to 12

       12 changes to 13 etc

I will not make any more tables of a 6 by 3 grid or a 7 by 3 grid because they show me the same information which I already know.

This is the T-Total will always go up in 5 if you move it to the right.

I am now going to try to find a method the same way I have done before.

I am going to break down the T- Shape in an algebraic expression to find the method.

Here is what it looks like as an algebraic expression

T = the T- Number

T+(T+G)+(T+2G)+(T+2G-1)+(T+2G+1) = 5T+7G   

T = the T- Number

By adding together the method of obtaining the numbers in the T- Shape from the T- Number I found the method for finding the T- Total by just using the number of columns in the grid and the T- Number!

I will now prove that this rule works using a 10 by 10 grid.

5*25+7*10= 195

I will add up the numbers in the T- Shape to make sure my answer is correct.

25+35+45+44+46= 195

This rule is correct.

This is how it works.

You get the 5 multiplied by the T-Number by getting 5 numbers in the T- Shape.  By applying the 5n rule to this you get 5T.

For the row with the T-number in it the row equals 0

The next row up equals +1G because you have to subtract the grid size to get the T-Number

The last row from the T-Number is +2G because this is what you have to get the T-Number from this point.  

When you add +1G and the three +2Gs you get +7G

This rule is the opposite of the first investigation because it is in the opposite direction to the original T- Shape.