22 47 position 3
I saw that the pattern between the T-no was 1 and the pattern between the T-total is 5 and so therefore 5t is the first part of my formula. I then multiplied the T-no by 5 and then took away the T-total from the answer. I found out that the difference is 63. We get 63 because it is the total of the difference between the T-no and the number in each of the other 4 squares.
This is my working out:
20 x 5 = 100 100 – 37 = 63
21 x 5 = 105 105 – 42 = 63
22 x 5 = 110 110 – 47 = 63
My formula is now 5T – 63.
I will now do the same on an 8by8 grid still using the given 3by2 T. I firstly placed my
3by2 T on the beginning of the grid.
I moved the T along 3 times in order to find a pattern. These are the 3 different positions in which I placed my 3by2T:
Position 1 Position 2
Position 3:
I put the T-no in column and the T-total in another column. I did this too find the pattern between the T-no’s and the T-totals.
T-no T-total
18 34 position 1
1 5
19 39 position 2
1 5
20 44 position 3
I saw that the difference between the T-no’s are going up in ones and the T-totals are going up in 5’s. The first part of my formula is 5T. Once again I multiplied the difference of the T-total which is 5 by the T-no. I then took away the T-total from my answer.
This is my working out:
18 x 5 = 90 90 – 34 =56
19 x 5 =95 95-39= 56
20 x 5 =100 100- 44= 56
My formula is now 5T – 56.
I got 56 because that is the total differences between the T-no and the numbers in each of the other 4 squares.
After finding out the formulas of a 3by2 T on a 10, 9 and 8 grid I can now generate the general rule for the given 3by2 T on any size grid. I put my results in a table which will help me generate the general rule.
I noticed that in all the formulas begin with 5T and the endings of the formulas are linked with the number 7. This is because they all have the difference of 7. I also noticed that whatever grid size it is multiplied with 7 and it would give me the last part of the formula.
The general rule for a 3by2 T on any size grid at a 0° rotation is: 5T – 7G
To prove that I am correct I will test it out with a 5by5 grid. This is my working out:
(5 x 12) – (7 x 5) = 25
60 - 35 = 25
I added all the numbers inside the T to check if my answer is correct= 1 + 2 + 3 +7 + 12 = 25
10 by 10
I can now put expressions inside the T instead of numbers to prove that my formula is correct.
I noticed that the centre column of the T is going up in 10’s because of the grid size; if the grid size is 10 then it would go up in 10s. With the T set out like this I can check that the formula for a 3by2 T on a 10by10 grid is correct. I added up all the T term expressions inside the T, this is done in the working below:
T-total= T-21 + T-20 + T-19 + T-10 + T
= 5tT – 70
I will now check to see if this formula is correct.
If T = 35 T-total = 5 x 35 – 70 = 105
14 + 15 + 16 + 25 + 35 = 105.
9 by 9
I can now put expressions inside the T instead of numbers to see if my formula is correct.
T = the T-no
I noticed that the centre column of the T is going up in 9’s because of the grid size. With T set out like this I can now check that the formula for a 3by2 T on the 9by9 grid is correct. I added all the expressions from inside the T, this done in the working below:
T-Total = T -19 + T -18 + T – 17 + T-9 + T
= 5T – 63
I will now check to see if this formula is correct.
So if T = 42 T-total = 5 x 42 – 63 = 147
210 - 63 = 147
23 + 24 + 25 + 33 + 42 = 147
I now know that my formula is correct.
8 by 8
I can now put expressions inside the T to check if my formula is correct.
The centre column is going up in 8’s because of the grid size. With my T set out like this I can now check that the formula of a 3by2 T on an 8by8 grid is correct. I added up all the expressions inside the T, this is done in the working below:
T-total = T -17 + T – 16 + T – 15 + T - 8 + T
= 5T – 56
I will now check to see if this formula is correct.
T = 62 T-total = 5 x 62 – 56 = 254
310 – 56 = 254
45 + 46 + 47 + 54 + 62 = 254
My formula is correct.
10 by 10
I can now explain this formula using algebra and so therefore this will help me find the rule for a 3by2 T on any size grid.
If G = the grid size – in this case G would be 10 because I am using a 10by10 grid.
T = Is the T-no.
I then added up the G terms together to get 7G which is the final expression.
(2G +1) + (2G) + (2G +-1) + (G) = 7G
So therefore T = 5T – 7G
I will now test this formula on a 10by10 grid to see it works:
Checking:
(5 x 22) – (7 x 10) = 40
110 - 70 = 40
From 5t come 5 which is multiplied by the T-no from my first position which is 22.
7 comes from 7G and is multiplied by the grid size which is 10.
By checking out this formula I can see that it works on a 10by10 grid.
9 by 9
I can now explain the expression using algebra.
G = 9
I added up all the G terms from inside the T to get 7G the final expression.
(2G+1) + (2G) + (2g-1) + (G) = 7G
T = 5T – 7G
I will now test out this formula to see if it works on a 9by9 grid.
Checking:
(5 x 20) – (7 x 9) = 37
100 - 63 = 37
20 + 11 + 3 +2 + 1 = 37
By checking this formula on 9by9 grid I can see that it works.
8 by 8
I can now explain the expressions using algebra.
G = 8
T = T-no
I added up all the G terms from inside the T and I got 7g as the final expression.
(2G – 1) + (2G) + (2G +1) + (G) = 7G
So the formula for T is: T = 5T – 7G
I will now test this formula out to see if it works on an 8by8 grid.
(5 x 18) – (7 x 8) = 34
90 - 56 = 34
1 + 2 + 3 + 10 + 18 = 34 This formula is correct.
This formula works on an 8by8 grid.
ROTATION
First I rotated the 3by2 T by 90o on a 10by10 grid
Position 1 Position 2
Position 3 Position 4
I drew a table to write drown my results
Formula: 5N+7
Test The Formula:
I picked any number from the 10by10 grid e.g. 34 and used the formula
5 x 34 +7 = 177
For me to make sure it was right I counted all the numbers in the T to check if it totaled to 177.
I added: 34+35+36+26+46=177
And it was right so this meant that the formula works
I rotated the 3by2 T by 90o on a 9 by 9 grid
Position 1 Position 2
Position 3 Position 4
Formula: 5N+7
Test The Formula:
I picked any number from the 9 by 9 grid e.g. 30 and used the formula
5 x 30 +7 = 157
For me to make sure it was right I counted all the numbers in the T to check if it totaled to 157.
I added: 30+31+32+23+41= 157
And it was right so this meant that the formula works
I rotated the 3by2 T by 90o on a 8 by 8 grid
Position 1 Position 2
Position 3 Position 4
Formula: 5N+7
Test The Formula:
I picked any number from the 8 by 8 grid e.g. 27 and used the formula
5 x 27 +7 = 142
For me to make sure it was right I counted all the numbers in the T to check if it totaled to 142.
I added: 27+28+29+21+37= 142
And it was right so this meant that the formula works
The main similarity that I have spotted is that when the T is turned 90o the formula is the same for any grid size for insistence a 9by 9 grids formula is 5N+7 as well as an 8 by 8 grids formula which is also 5N+7
TRANSFORMATION
I first of all changed the size of my T to a 5by2 T. By changing the size of my T it will eventually help me find the rule for any size T on any size grid.
5
I first of all placed my 5by2 T on the beginning of a 10by10 grid.
I moved my 5by2 T 3 times along the 10by10 grid. These are the 3 positions in which I put my 5by2 T:
Position 1:
T-total: 1+2+3+4+5+13+23 = 51
Position 2:
T-total: 2+3+4+5+6+14+15 = 58
Position 3:
T-total: 3+4+5+6+7+15+25 = 65
The T-no is at the bottom of the T. I added up the numbers inside each T to get the T-total. I then put the T-no in one column and the T-total in the other column in order to find a pattern.
T-no T-total
23 51 position 1
1 7
24 58 position 2
1 7
25 65 position 3
I noticed that the difference between the T-no is 1 and that the T-total numbers were going up in 7’s and so therefore the first part of my formula for a 5by2 T on a 10by10 grid is 7T. I then multiplied 7 by the T-no. I then took away the T-total from my answer. I found out that the difference is 110. The answer is 110 because that is the total of the differences between the T-no and the numbers in each of other 4 squares. This is what I did:
23 x 7 = 161 161 – 51 = 110
24 x 7 = 168 168 – 58 = 110
25 x 7 = 175 175 – 65 = 110
My formula for a 7by2 T on 10by10 grid is now 7T – 110
I can now put expressions inside the T instead of numbers to show that my formula is right.
G = grid size which is 10
T = T-no
I noticed that the centre column of the T is going up in 10’s because I am using the 10by10 grid. With the T set out like this I can see if my formula is correct. I added all the expressions inside the T; this is done in the working below:
T-total = T – 22 + T – 21 + T – 20 + T – 19 + T – 18 + T – 10 + T
= 7T – 110
I will now check to see if this formula works.
T = 57 T-total = 7 x 57 – 110 = 289
57 + 47 + 37 + 36 + 38 + 39 + 35 = 389 This formula is correct.
I can now explain the expressions using algebra.
I added up all the G terms inside the T and I got 11G as the last part of my formula. This is what I did:
(2G-2) + (2G-1) + (2G) + (2G+1) + (2G+2) + (G) = 11G
So T = 7T – 11G
I will now check to see if my formula for a 5by2 T on a 10by10 grid is correct.
(7 x 23) – (11 x 10) = 51
161 - 110 = 51
1 + 2 + 3 + 4 + 5 + 13 + 23 = 51
I found out that the formula for a 5by2 on a 10by10 grid is correct.
I will now do the same on a 9by9 grid. I placed the T on the first part of the 9by9 grid.
I moved the T along the grid 3 times. The 3 position in which I put my T on:
Position 1:
T-total: 1+2+3+4+5+12+21 = 48
Position 2:
T-total: 2+3+4+5+6+13+22 = 55
Position 3:
T-total: 3+4+5+6+7+14+23 = 62
I put the T-no in one column and the T-total in the other.
T-no T-total
21 48 position 1
1 7
22 55 position 2
1 7
23 62 position 3
I saw that the T-no is going up in 1 and the T-total is going up in 7. Therefore 7 is the first part of my formula. Once again I multiplied the difference of the T-total which is 7 by the T-no. I then took away the T-total from my answer.
This is my working out:
21 x 7 = 147 147 – 48 = 99
22 x 7 = 154 154 – 55 = 99
23 x 7 = 161 161- 62 = 99
My formula for a 5by2 T on a 9by9 grid is now 7T – 99.
I can put expressions inside the T to check if my formula is correct.
I saw that the centre column is going up in 9’s because of the grid size. I added all the expressions from inside the T. This is what I did:
T-total = T-20 + T – 19 + T – 18 + T – 18 + T – 16 + T – 9 + T
= 7T– 99
I will now test out this formula to find out if it is correct.
T = 51 T-total = 7 x 51 – 99 = 258
31 + 32 + 33 + 34 + 35 + 42 + 51 = 258 This formula is correct.
I can now use algebra to explain the expressions.
G = 9 the grid size
I added up all the G terms inside the T and I got 11G as the last part of my formula.
(2G -2) + (2G-1) + (2G) + (2G+1) + (2g+2) + (G) = 11G
So T = 7T – 11G
I will now check to see if this formula works on a 9by9 grid.
(7 x 21) – (11 x 9) = 48
147 - 99 = 48
1 + 2 + 3 + 4 + 5 + 12 + 21 = 48 This formula is correct.
This formula is correct and it works on a 9by9 grid.
I now know the general formula for a 3by2 T on any size grid and a 5by2 T on any size grid.
