i.e.
T-total
T-total=37 T-total= 42
The T-numbers are increased by one. 42-37= 5. The T-total increases by five. This works for all my results, other examples:
T-total=57 T-total=62
The T-numbers are increased by one. 62-57= 5 . The T-total increases by five.
This obviously has a reason why, and as the T-number increases, the other components of the T increase 1, making a total sum of 5 more.
Therefore, n+1= t+5
- Algebraic formula connecting T numbers and totals.
In my hypothesis I stated that I believed there is a formula connecting the T-number and the T-total, and I investigated this.
First, I substituted the T-number for the letter n. Then, I managed to substitute the rest of the T shape using this n.
I used this general T and checked it worked with other T’s. It worked for all T’s.
i.e.
i.e
As it worked for all the T’s, I put the formula together creating a simple formula, which is used in form of substitution for achieving the T-total from the T-number.
- In this task I will different size grids, to investigate the correlation between the T-number, T-total and the translation of the T-shape in any grid size.
- First I will use a grid size 12 by 12, creating ten T’s for this investigation. I will do the same with a 5x5 grid. I will try to see the algebraic formula as found for the 9 by 11 grid.
- With the different T-totals and T-numbers I will create a result table to be able to see any observations.
- I will right down any observations noticed.
- I will go deeper into the investigation to try to find any algebra correlation between T-total and T-numbers of different grid sizes.
The 12 by 12 grid, and the ten T-shapes:
Results=
The 5 by 5 grid and the “T’s”:
Results=
From the different tables of result, we can make the following statements:
- In the 12 by 12 grid the increase of 1 in the T-number means also an increase of five.
- This also works for the 5 by 5 grid.
-
The odd T-numbers have a T-total that ends in 1, and in the even T-numbers the T-total ends in a 6. Which means that there is a sum of 5 each time. 1 + 5 = 6, 6 + 5 = 11
-
This also works for the 5 by 5 grid. But the odd numbers finish in 0, and the even T-numbers end in 5. 0 + 5 = 5, 5 + 5 = 10
- Representing my different observations.
Before I had found a equation for the T-total for the 9 by 9, but this formula doesn’t work for the rest of the grids.
i.e. 5n – 63 = T-total
123 x 5 – 63 = 552
T-number= 123, T-total = 531
As this formula didn’t work I found another formula for the 12 by 12 grid, the same way I investigated the formula for the 9x9 grid.
This formula isn’t valid for the size five grid, as the numbers after the 5n, change, the formula for a size grid five would be:
As I said before in both grids 12x12 and 5x5 have an increase of five in the T-total when the T-number is increased by one, and this also works with the 9 by 9 grid. So we can say that the rule n+1= t + 5 is true for all grids, where n is the T-number, and t is the T-total. The difference between the outcomes is notices by the starting number from where to add the 5 to. These are the lowest incomes for “T’s”
i.e. In 5x5 grid.
T-number= 12, T-total= 25
T-number= 13, T-total= 25 + 5 = 30.
The starting number to add, is 2
In 12x12 grid.
T-number= 26, T-total= 46
T-number= 27, T-total= 46 + 5 = 51
The starting number to add is 1.
In 9x11 grid.
T-number= 20, T-total= 37
T-number= 21, T-total= 37 + 5 = 42
The starting number to add is 1.
I believe, that there is an equation to link the different translations and I managed to find an equation for any translation, horizontal and/or vertical, that includes the “magic” number five. The result of this equation gives the T-total of the translated shape in this way:
Where v is the middle number of the T and g is the grid size. “a” and “b” are the translations. “a” is the vertical translation and “b” is the horizontal translation.
To prove this equation is correct I use a grid size of 10 , and an initial T, which was translated +2 vertically, and -2 horizontally.
Initial T:
V= 52
Translation: Translated T:
A= 2
T-total= 150
B= 1
With formula:
=(5x52-2x10)-(2(5x10)-5 x-2
= 240 – 100 + 10
= 150
This proves my equation is correct as; the T-total of the translated shape is equal to the mathematically translated shape, using the equation. I can state;
Any combination of translations (vertically and horizontally), can be found by using the equation
Were v is the middle number of the initial T, g is the grid width, a is the number by which to translate vertically and b is the number by which to translate horizontally. It is also apparent that this equation can be used for any singular vertical or horizontal translation, as the other value (vertical or horizontal) will be equal to 0. therefore it will cancel out that part of the equation as it would equal 0.
- Algebraic formula connecting T numbers and totals, with the different grid sizes.
I have made the substitution of any “T” with “n’s” (T-numbers) and “G’s”, the grid size.
In total this is= n-2G-1 + n-26 + n-2G+1 + n-G + n
= 5n - 7G
I have to check if this works for all grids.
T-number= 12
T-total= 25
Total= 25
This method works for a 5x5 grid. I will try with other grid sizes.
T-number= 26
T-total= 46
Total= 46
This method also works for this grid. So, I have come to the conclusion that there is a general rule for the relationship of T-numbers and T-totals in different size grids.
Method: Task II
- In this task I plan to investigate the relationship between rotations of the “T-shape” and the T-total and T-number.
- I will first use my 9 by 11 grid, to rotate the T 90º clockwise, and state any formula’s found.
- After I will rotate the T in 270º clockwise, (from initial shape) and state my observations and investigations.
- Use different grids again to find the relationship between the grid size and the rotation, with the T-numbers and T-totals.
Rotated T’s, 90º clockwise.
The initial T, is now rotated in 90º clockwise to form a
T-shape.
Some examples of this T, in the 9 by 11 grid are:
T-number= 10
T-total= 72
T-number= 41
T-total= 212
For this kind of rotation I have found a formula, with the substitution of the T-number by n. This formula is found by=
Which sums up to a total of=
n + n+1 + n+2 + n+11 + n-7=
5n + 7
i.e.
T-number= 76
T-total= 465
Total= 465
This works for all 90º rotated T-shapes from a 9 by x grid. This formula isn’t correct with other grids.
i.e.
T-number= 39
T-total= 202
Total= 188
The T-total and the formula total are different, showing that the formula doesn’t work for all grid type.
Rotated T’s, 270º clockwise.
The initial T, is now rotated in 90º clockwise to form a
T-shape.
For this kind of rotation I have found a formula, with the substitution of the T-number by n. This formula is found by=
Which sums up to a total of=
n-11 + n-2 + n+7 + n-1 + n
= 5n - 7
i.e.
T-number= 50
T-total= 243
With formula= 5 x 50 – 7
= 243
This formula simply works with the T shapes that have been rotated 270º clockwise in a 9 by x grid.
I have observed that all T have the formula 5n-/+x. 5n as there is five components in the T, but the rest of the equation depends on the rotation of the T. Opposite T’s have the same amount of x, but one is a negative x and one is a positive x.
i.e. Formula for = 5n- 63.
The opposite of this T = The formula = 5n+63
The formula for = 5n – 7
And the opposite of this
= 5n + 7
But all of these formula’s vary depending on the grid size, as the number after the 5n, changes depending on the difference between the components of the T and the rotation.
- I have observed that the number after the 5n in my equation is related to the 7 timetable, relating the grid size to 7.
For original T’s or 180º rotated T’s, the number after the 5n in the equation is found by 7 times the grid size, so this means there is a general rule between rotations and different grid sizes.
The formula for 180º rotation was:
From this I expected the formula for any gird size for 180º rotation would be: 5n+7G
Ex.
This task consists in the investigation between the enlargement of the T-shape in different grid sizes.
- First I will investigate the stretch of the T-top in a grid size 10, stating any formula found.
- I will check this formula with other grids. State a general formula for any T-top stretch of the T-shape in any grid size.
I investigated some T-stretches of 2, and here is a results table:
From this results table I can observe that each time the T-number is increased by one the T-total is increased by 7, two more than in the usual T. I can then perhaps state that, having s as the stretch, there is a formula in relation with the T-number, the T-total and the stretch.
To see if this equation works with any stretch I will try with a stretch of 4.
T-number= 24
T-total= 66
T-total= 75
24 + 1 = 66 + 5 + 4
= 75
I can see that this formula works for stretches that come in pairs, as the stretch has to be equal in both sides of the T. I can state this due to the fact that depending on which side the T stretches the number differs in a higher or lower manner, and for this formula to function the stretch has to be equal on both sides. I will now try this formula with a different grid size, such as a size grid 9.
T-number= 21
T-total= 48
Stretch= 2
The T-number is increased by one so:
21 + 1 = 48 + 5 + 2
= 55
From my investigations I can then state that the formula
Is applicable for all T-translations that are increased by one in any grid size with any even stretch.
Throughout this investigation I have found different relations between T-numbers and T-total in a T-shape. There is a series of formula’s that I have proved worked, that I now state:
- With a 9 by 11 grid the equation is correct anywhere in the grid.
-
Changing the vertical size of this grid makes no change, but if the width of the grid is changed the equation is changed.
- The equation for a 12 size grid is and this works anywhere on this grid size.
- The equation for the five size grid is. This works anywhere in this sort of width.
- The relationship between all the different grid sizes is the 5n, and the number that this is taken away from, as they are all multiples of seven.
- There is two equations relating any translation in any grid size. These are:
-
When the T-shape is rotated there are different rules for each individual rotation to find out the T-total. I have investigated these first with a size 9 grid and they are:
= 5n- 63 =5n+ 63
= 5n – 7 = 5n + 7
- I investigated these formula’s even further, and developed them for them to work with a T-shape rotation in any size grid. These formula’s were:
= 5n- 7G =5n+ 7G
= 5n – 7 = 5n + 7
