We can test this formula by finding the T-Total, when we know the T-Number. The T-Number is 82, so the T-Total must be:
5n- 49 = x
(5*82) - 49= x
410 - 49= x
361 = x
We can also work out the T-Total by using the above grid; (67+68+69+75+82) = 361, however this tends to be time consuming, which is why we use a formula.
The next step of my investigation is to draw a six and four number grid, with 4 adjacent T-Shapes on it. These are illustrated below.
Six Number Grid
Table of Results (Six Number Grid)
We can test this formula by finding the T-Total, when we know the T-Number. The T-Number is 52, so the T-Total must be:
5n- 42 = x
(5*52) - 42= x
260 - 42= x
218 = x
We can also work out the T-Total by using the above grid; (52+46+40+39+41) = 218. This reassures us, we know that the formula is correct.
Four Number Grid
Four Number Grid
From the three investigations above we can already see a general pattern, which applies to all three of the experiments. I have tabulated the three formulas into a table, to makes it easier to stop the relationship.
The general pattern is 5n- 7g. This rule applies for all number grids, which have not been transformed. This is because the difference between adjacent T-Totals is 5, which gives us the 5n, while the second number is obtained by multiplying the grid length by 7. We can also deduce this formula through algebraic equation. All of the above algebraic equations have had one thing in common; this is shown below.
- g- is the length of grid.
I predict that the formula for the five number grids is 5n- 35 because we know the 5n remains the same, while the grid length is multiplied by 7 (5*7). This means the formula will be 5n- 35. We can see whether my prediction is correct by investigating the five number grid. This is done below:
Five Number Grid
Table of Results (Five Number Grid)
The above results confirm that my prediction was correct because I obtained the same formula by using the 5n- 7g rule.
Rotation
To investigate further I will also find out whether the 5n- 7g rule applies when it comes to rotation. The yellow highlighted T-Shape below is rotated through 90°, 180° and 270° clockwise in a seven, six and five number grid.
Seven Number Grid
We can work out the formulae for each rotated shape by replacing the number with algebraic terms. This has been done below:
Rotation Through 90° Rotation Through 180°
Rotation Through 270°
The above results have been tabulated which makes it easier to see a relationship or pattern.
As we can see from the tabulated result there is a general pattern. We already know the formula for the original T-Shape; 5n- 49, similarly when the shape is rotated through 180°, the formula change, form 5n- 49 to 5n+ 49. This is because the T-shape is in the opposite position, it is a reflection and the signs are reversed with the reversed position.
Another pattern which can be seen form the results above is the rotation through 90° and 270°. The formulae are roughly the same; apart from the opposite sign. The formula for a T-Shape rotated through 90° is 5n+ 7 while the formula for a T-Shape rotated through 270° is 5n- 7.
The most obvious link is that, of the length of grid and when the shape is rotated through 90° and 270°. The length of the gird is 7; similarly when the T- Shape is rotated through 90° and 270°, the formulae are 5n+ 7 and 5n- 7 respectively.
Six Number Grid
As you can see left, the original yellow highlighted T-Shape is rotated through 90°, 180° and 270°. By rotating the T-Shapes we can find out a formula for each rotation.
Rotation through 90° Rotation through 180°
Rotation through 270°
The above results have been tabulated below:
As we have seen from the two above table; there is a general pattern starting to form. When the T-Shape is rotated through 90° the formula is 5n+7 whereas when the formula is rotated through 270° the formula is 5n-7. When can find the formula for a T-Shape which is rotated through 180° by using the equation 5n+7g.
Five Number Grid
The yellow highlighted T-Shape on the left is rotated through 90°, 180°, and 270° clockwise.
Rotation through 90° Rotation through 180°
Rotation through 270°
Once again we can see the pattern that forms. When the T-Shape is rotated through 90 and 270 degrees, the formula is always 5n+7 and 5n-7; respectively. On the other hand when the T-Shape is rotated through 180 degrees, the formula always is 5n+7g. We can prove that these equations apply to any rotations by using algebraic terms.
We can see from the above that:
- When the T is at 0° the formulae will be 5n-7g
- When the T is at 90° the formulae will be 5n+7
- When the T is at 180° the formulae will be 5n+7g
- When the T is at 270° the formulae will be 5n-7
I will now test my formulae to see if they are correct by using the formulae I found and seeing if I get the same results as I would do, using the grids. Using the T-Number 49 and the 9 grid, we can test whether the formula for 90° is right or not.
5n+7
(5*49)+7
245+7= 252
We can check to see if it’s right or not by doing the working out manually (49+50+51+42+60) = 252. This confirms that our first formula is correct.
I will check the formula for the 180° rotation by using the T-Number 26 and the 3 grid.
5n+7g
(5*26)+ (7*3)
130+21=151
Once again, we work out the answer manually to check whether our formula is correct or not, (26+29+32+33+31) = 151.
Our final formula is 5n-7, which is used for rotations of 270°. To check this answer we are going to use the 5 grid and T-Number 83.
5n-7
(5*83) -7
415-7 = 408
We then check our answer manually, (83+82+81+76+86) = 408. I got the same results manually as I did by using my formulae, which proves that my formulae for rotation are correct.
Translation
The second part to furthering my investigation is using translations. My aim is to establish a relationship between the T-Number, T-Total and translations, using the vectors ( ) for each translation. I will use the seven, six and five grid for my translations. By then using algebraic terms, I will try to work out a general formula for translation.
Firstly I will start by translating the T-Shape x- across, to look for a pattern. I will then translate the T-shape y- down, to establish a relationship. After working out a link I will then translate the T-Shape using the vectors ( ).
Seven Grid
The yellow highlighted T-Shape (left) is translated using the vectors ( ). The translated shape is highlighted blue.
To work out a formula for these translations we are going to have to replace the terms with algebraic expressions. This is done below:
Using the original T-Shape, I will work out a general pattern for the shape which has been translated 2 to the right.
The T-Number is 16 so this will be used as the base for the general T-Shape. This is shown below:
T-Number: 16
T-Total: 31
Original T-Shape ( ) Translated T-Shape
Below I have tabulated the results because it is easier to analyse them, when they are in a table.
From the results above and looking at the number grid we can see that all the numbers in the original T-Shape (highlighted yellow), have increased by 2; which is the horizontal column vector. Similarly, the T-Total increases by 10 because of (horizontal vector*difference between adjacent T-totals); we can work this out by (2*5).
As a result of this, I believe that if the horizontal vector is 3, each number in the original T-Shape would increase by 3, whereas the T-Total would increase by 15.
To prove that my prediction is correct and check whether the formula is correct I am going to try a different horizontal translation on the six number grid below:
Six Number Grid
The yellow highlighted T-Shape has been translated 3 to the left (blue highlighted T-Shape). I have replaced the numbers with algebraic terms below to find a formula:
Original T-Shape ( ) Translated T-Shape
The above results have been tabulated below because it’s easier to stop the pattern.
We can see from the results above that my prediction was correct because each number in the original T-shape increased by 3, due to its column vector. The T-total also increased by 15 because of (horizontal vector*difference between adjacent T-totals).
We can see that there is a general pattern which occurs, when a T-Shape is translated. This is shown below:
General formula for any T-Shape horizontal translation
We can use the above formula to work out the T-Total of the translated shape when already know the vectors of translation, T-Number and the grid length. I have randomly chosen to use the nine grid, T-Number 75 and a horizontal vector of 5.
N+x+((N-g)+x)+((N-2g)+x)+((N-2g)+x+1)+((N-2g)+x-1)
= 75+5+((75-9)+5)+((75-2*9)+5)+((75-2*9)+5+1)+((75-2*9)+5-1)
= 80+71+62+63+61
= 337
To check our answer we do it manually by adding up the number in the T-Shape which has been translated by a horizontal vector of 5; (80+71+63+62+61) = 337
The second part of translation is working out a general formula for a T-Shape, which has been translated y-down. This is shown below in a seven grid:
Seven Grid
The yellow highlighted T-Shape (left) has been translated using the vectors ( ). The new translated T-Shape is highlighted blue.
Below I have worked out the formula for both the original T-Shape and the translated shape.
Original T-Shape Translated T-Shape
We can work out a general formula any translation with vertical vectors. The below formula works for any translation:
General formula for any vertical translation
As I have worked out the general formula for any vertical or horizontal translation, I will now work out a general formula using the vectors ( ). Below is a seven number grid with a yellow highlighted T-shape on it; the shape has been translated to the pink highlighted T-Shape using the vectors ( ).
Seven Number Grid
The yellow highlighted T-Shape (left) has been translated using the vectors ( ), to the pink highlighted T-Shape. The blue T-Shape shows the horizontal translation ( ). By replacing the number in the T-shapes with algebraic terms, I will work out a general formula for translation.
Original T-Shape ( ) Translated T-Shape ( ) Translated T-Shape
We can work out a general formula for the T-shape which has been translated using vectors ( ). This has been done below:
General formula for translation
The above formula can be used to work out any translation. We can prove this by using a random number grid and different column vectors. To check my formula I will be using the nine number grid and the column vectors ( ).
We can check whether our formula is right or wrong by replacing the N and the column vectors in the formula.
The T-Number in our original T-Shape (highlighted yellow) is 47, so the new translated T-Number should be, N+a-bg. N=47 a=4 b=2 g=9
N+a-bg
47+4- (9*2)
51-18= 33
N+a-bg-g
47+4- (9*2)-9
51-18-9=24
N+a-bg-2g
47+4- (2*9) - (2*9)
51-18-18=15
N+a-bg-2g-1
47+4- (2*9) – (2*9) – 1
51-18-18-1= 14
N+a-bg-2g+1
47+4- (2*9) – (2*9) +1
51-18-18+1=16
From the above we can see that our formula is correct because we managed to work out the correct numbers in the translated T-Shape, by using the different formulae.
Double Transformation
The final part of my coursework is investigating the relationship between T-Number, rotation and translation. I have used the nine number grid to perform a translation using the vectors ( ) and then rotated 90° clockwise. I will then use algebraic T-Shapes to work out a relationship or general formula.
The original T-Shape (highlighted yellow) has firstly been translated using the column vectors ( ), (highlighted blue), then rotated 90° clockwise (highlighted red). Below I have worked out algebraic expression of each T-Shape.
Original T-Shape 90° Rotation ( ) Translation
90°Rotation and Translation
After looking at the above T-Shapes and rotation formulae, I started to find a pattern. The pattern I found is the general T-shape formula above. To make sure that this formula is correct I will check it, by performing another double transformation on a different grid size.
Seven Number Grid
The original T-Shape (highlighted yellow) in this number grid has been translated using the column vectors ( ), then rotated 90° clockwise.
General Formula for Translation and 90°Rotation
To make sure this rule applies to any 90° clockwise rotation and translation combined, I will use the original T-Number, and work out each number in the double transformed (highlighted red) shape. N=44 a=1 b=3 g=7
N+a-bg
44+1- (3*7)
45-21=24
N+a-bg +1
44+1- (3*7) +1
45-21+1=25
N+a-bg+2
44+1- (3*7) +2
45-21+2=26
N+a-bg- (g-2)
44+1- (7*3) - (7-2)
45-21-5=19
N+a-bg+ (g+2)
44+1- (7*3) + (7+2)
45-21+9=33
As you can see above, by using the formula I worked out each number in the double transformed (highlighted red) T-Shape. This proves that my formula is correct and works for any T-Shape which has been translated and rotated 90° clockwise. The following formula, 5N+5a-5bg+3, works for any 90°clockwise rotation followed by a translation.
Because my formula only works for T-Shape which has been translated and rotated 90°, I have to find separate formulae for translation and rotations of 180° and 270°. In the seven number grid below, the T-Shape has been translated using the column vectors ( ) and then rotated 180°.
Seven Number Grid
The original T-Shape (highlighted yellow) has been translated using the vectors ( ), then rotated through 180° (highlighted red).
Original T-Shape 180° Rotation ( ) Translation
180° Rotation and Translation
By looking at the general formula for translation, seven number grid and 180° rotation, I was able to come up with a general formula for 180° rotation and translation combined. Working out the formula was easy, because I used the T-Number (N), and the general formula for translation to start of with. I replaced the N=38, a=3, b=3 and g=7 into the general T-Shape for translation and then analysed my answers. Below is an example:
N+a-bg-g=27
38+3-(7*3)-g
41-21-7=13
N+a-bg+g=27
38+3-(7*3) +7
41-21+7=21
The general formula for 180° rotation, followed by any translation can be worked out by adding up all the algebraic expression in the T-Shape.
The last part of my coursework is to establish a link between, T-Number, 270° rotation and translation. In the six number grid below, the original T-Shape (highlighted yellow) has been translated using the column vectors ( ), then rotated 270°clockwise.
Six Number Grid
The original T-Shape (highlighted yellow), has been translated using the column vectors ( ), (highlighted blue). It has been rotated 270° clockwise (highlighted red.
Number Grid Formula Translation Formula 270° Rotation
By combining the three formulae above I was able to come up with a general for 270° clockwise rotation, followed by translation. I used the formula for 90° clockwise rotation, followed by translation to help me work out this. This is shown below:
By replacing the algebraic terms, N=32, a=2, b=3 and g=6 into the below formula I was able to work out a general formula for 270° clockwise rotation, followed by translation.
N+a-bg+1=15
32+2-(6*3) +1
34-18+1= 17
N+a-bg-1=15
32+2- (6*3)-1
34-18-1=15
I found out that the formula for 270° rotation, followed by translation is slightly different from the 90° rotation, followed by translation. The slight difference is shown below:
90° Rotation and Translation 270° Rotation and Translation
As we can see the formula are basically the same with the exception of a few signs changing from positive to negative.
By adding all the algebraic terms we can work out a general formula for a 270° rotation, followed by a translation.
270° Rotation and Translation
I have tabulated the formulae of each double transformation below:
As we can see from the results above, the formulae for 90° and 180° are identical, apart form one being +3 the other -3.
The aim of this coursework was to establish a link between the T-Number and the T-Total. I managed to do this by drawing 3 adjacent T-Shapes on each different sized grid and then working out the difference between each T-Total. I repeated this method a few times, to make sure the formula obtained was correct. The formula was then checked and put into use, for a number of other grids. The formula I come up with for the relationship between T-Number and T-Shape is 5n-7g.
Consequently, I had to further my investigations after working out the link between T-Number and T-Shape. I chose to find a link between T-Number, T-Total and rotation; however there were different formulae for each different rotation. Unlike the simple formula for the link between T-Number and T-Shape; working out formulae for rotations was a bit more complicated. I chose to use the T-Number as the center of rotation, for each different clockwise rotation. After rotating the T-Shape through 90°, 180° and 270°, I used algebraic terms based on the T-Number, to work out formulae for all the different rotations. When I obtained the three different formulae, I checked the answers by drawing another number grid and rotating a T-Shape; to make sure that grid size didn’t affect the link between T-Number, T-Shape and rotation. The formulae I obtained are tabulated below.
The next stage of my coursework was to find a relationship between the T-Number, T-Total and translation. Establishing a pattern for translation was very hard because the pattern wasn’t straightforward and harder to spot. Due to this, I resulted to using algebraic terms based on the T-Number. Firstly, I worked out the formula for any horizontal translation, using the column vector (a) and then for any vertical translation using the column vector (b). After working out formulae for both, I used algebraic terms to get a formula for translation; adding all the algebraic expressions together, I arrived with the general formula of 5N+5a-5bg-7g.
Finally, I finished the coursework by establishing the link between T-Number, T-Total and double transformation. When carrying out the double transformation, I chose to translate the original T-Shape before rotating it. This was because it was easier to stop the pattern once the shape had been translated, rather than rotated. By looking at the algebraic terms for translation and rotation; I realized that the formula for double transformation was slightly different than that of translation. This meant that I only needed to alter the formula of translation slightly. After working out formulae for each double transformation, I repeated my step on different number grids to check and make sure the formulae obtained were correct. The formulae I obtained are tabulated below:
Overall, I think that my coursework went very well because I achieved what I set out to do. The aim of the coursework was to find a link between T-Number and T-Total; I managed to do this and further investigations. If I could improve my coursework, I would write it out because it would be clearer to understand and much easier to draw tables and shapes.
Title Page Page Number
Introduction……………………………………………………………2-4
Main Part………………………………………………………………5-8
Rotation………………………………………………………………9-12
Translation………………………………………………………..13-18
Double Transformation………………………………………..19-24
Conclusion………………………………………………………..25-26