T-Total
Part 1
The aim of the investigation is to find out the relationship between the t-number and t-total. The t-number is the number in the t-shape, which is at the base of the T. The t-total is the sum of all numbers inside the t-shape.
I will start my investigation by looking at t-shapes on a 9 by 9 grid.
To solve the problem of finding the relationship between the t-number and t-total I will look at the information algebraically.
I will firstly assign a letter to the t-number of the shape, this letter will be T. I will then express the rest of the numbers in the t-shape with the letter assigned to the t-number in the t-shape. Therefore it will give me a standard expression to apply to all the t-shapes. Where the expression is equal to the t-total.
Examples
`
The expression works for all t-shapes in a 9 by 9 grid. I will now simplify the expression into a simple formula.
T + (T-9) + (T-17) + (T-18) + (T-19) = T-total
5T – 63 = T-total
I will see if this new formula still works.
5 × 20 – 63 = T- total
100 – 63 =T – total
37 = 37
5 × 21 - 63 = T- total
105 – 63 = T – total
42 = 42
Part 2
I will now as part of my investigation use different grid sizes, transformations of the t-shape and investigate the relationship between both. Then I will see how the t-number and the t-total relate to the new factors.
The smallest grid size can only be a 3 by 3 grid because that is the smallest grid size a t-shape can fit on. However this size grid will not allow me to translate the t-shape therefore I will start the investigation by using a 4 by 4 grid. Also I will be keeping the grid ...
This is a preview of the whole essay
42 = 42
Part 2
I will now as part of my investigation use different grid sizes, transformations of the t-shape and investigate the relationship between both. Then I will see how the t-number and the t-total relate to the new factors.
The smallest grid size can only be a 3 by 3 grid because that is the smallest grid size a t-shape can fit on. However this size grid will not allow me to translate the t-shape therefore I will start the investigation by using a 4 by 4 grid. Also I will be keeping the grid sizes square, so that the length of the grid is the same as its width. The largest grid size I can use is infinite.
I will now look at a 5 by 5 grid.
I will now look at a 6 by 6 grid.
I will now look at the data algebraically that I have collected from each of the grids. I will be using the letter T as before to represent the t-number and then express the rest of the numbers in the t-shape with the letter T. This should give me a formula as before which works for each of the t-shapes from each of the grids.
Data from 4 by 4 grid
10 + 6 + 3 + 2 + 1 = 22
T + (T-4) + (T-7) + (T-8) + (T-9) = T-total
Does this formula work for work for other t-shapes in a 4 by 4 grid.
6 + 7 + 8 + 11 + 15 = 47
T + (T-4) + (T-7) + (T-8) + (T-9) = T-total
Yes the formula does I will now simplify the formula.
T + (T-4) + (T-7) + (T-8) + (T-9) = T-total
5T – 28 = T-total
Data from 5 by 5 grid
1 + 2 + 3 + 7 + 12 = 25
T + (T-5) + (T-9) + (T-10) + (T-11) = T-total
Does this formula work for work for other t-shapes in a 5 by 5 grid.
3 + 4 + 5 + 9 + 14 = 35
T + (T-5) + (T-9) + (T-10) + (T-11) = T-total
Yes the formula does I will now simplify the formula.
T + (T-5) + (T-9) + (T-10) + (T-11) = T-total
5T – 35 = T-total
Data from 6 by 6 grid
1 + 2 + 3 + 8 + 14 = 28
T + (T-6) + (T-11) + (T-12) + (T-13) = T-total
Does this formula work for work for other t-shapes in a 6 by 6 grid.
2 + 3 + 4 + 9 + 15 = 33
T + (T-6) + (T-11) + (T-12) + (T-13) = T-total
Yes the formula does I will now simplify the formula.
T + (T-6) + (T-11) + (T-12) + (T-13) = T-total
5T – 42 = T-total
I will now look at all the formulas from each of the grids and see how they relate to each other and see if I can work out another formula, which will allow me to calculate the t-total whatever grid size, it may be.
The number, which you minus from the formula in all of the formulas are a multiple of seven. If you divide this figure by seven you get the grid size.
28 ÷ 7 = 4 42 ÷ 7 = 6
35 ÷ 7 = 5 63 ÷ 7 = 9
If we input this information into the formula we end up with an overall formula to work out the t-total by only knowing the t-number.
T-total = 5T – (7 ×Grid size)
T-total = 5T – 7G
I will now see if this formula works for any grid size.
5T – 7G = T-total
5 × 43 – 7 × 9 = T-total → 215 – 63 = T-total → 152 = T-total → 152 = 152
5T – 7G = T-total
5 × 22 – 7 × 10 = T-total → 110 – 70 = T-total → 40 = T-total → 40 = 40
5T – 7G = T-total
5 × 33 – 7 × 6 = T-total → 165 – 42 = T-total → 123 = T-total → 123 = 123
5T – 7G = T-total
5 × 16 – 7 × 7 = T-total → 80 – 49 = T-total → 31 = T-total → 31 = 31
I have tested the formula successfully and found that my formula works. The new formula only needs a one-figure grid size and the t-number and the formula will work.
Part 3
I will now investigate further into the relationship between the t-number and the t-total. To do this I will use different grid sizes and try other transformations as well as combination of the t-shape. This will allow me to see what affect it causes to the relationship between t-number and t-total.
I will start the investigation by using a 4 by 4 grid because it allows me to do the largest amount of transformations and combination of transformation on the smallest grid possible.
There are four main types of transformations; these are 1) translation 2) enlargement 3) rotation and 4) reflection. I will be using two types of transformations firstly; these will be translation, which I have already been using, and rotation. I will also be using these transformations in combination.
I will firstly start off by rotating the t-shape clockwise 90° and translating it on various grids.
Data from 4 by 4 grid
Data from 5 by 5 grid
Data from 6 by 6 grid
I will now look at the data I have collected algebraically.
Data from 4 by 4 grid
4 + 8 + 12 + 7 + 6 = 37
T + (T-2) + (T+2) + (T+6) + (T+1) = T-total
5T + 7 = T-total
Does this formula work for other t-shapes in the grid.
5T + 7 = T-total
5 × 9 + 7 = T-total
52 = 52
Data from 5 by 5 grid
5 + 10 + 15 + 9 + 8 = 47
T + (T-3) + (T+2) + (T+7) + (T+1) = T-total
5T + 7 = T-total
Does this formula work for other t-shapes in the grid.
5T + 7 = T-total
5 × 6 + 7 = T-total
37 = 37
Data from 6 by 6 grid
6 + 12 + 18 + 11 + 10 = 57
T + (T-4) + (T+2) + (T+8) + (T+1) = T-total
5T + 7 = T-total
Does this formula work for other t-shapes in the grid.
5T + 7 = T-total
5 × 26 + 7 = T-total
137 = 137
The formula 5T + 7 works for all t-shapes regardless of what grid size the t-shape has come from.
I will now rotate the t-shape from its original position 90° anti-clockwise and translating it on various grids. I will then collect data and assess this algebraically.
Data from 4 by 4 grid
Data from 5 by 5 grid
Data from 6 by 6 grid
The formula like the formula for the t-shape rotated at 90° clockwise does not differ from grid size to grid size.
I will see if this formula works for other t-shapes on other grid sizes.
T-shape from 10 by 10 grid
T-shape from 8 by 8 grid
The formula works for other grid sizes as well.
I will now rotate the t-shape from its original position 180° clockwise and translating it on various grids. I will then collect data and assess this algebraically.
Data from 4 by 4 grid
Data from 5 by 5 grid
Data from 6 by 6 grid
As you can see the formula varies from grid size to grid size. I will try to find a formula that will work for any of the t-shapes at 180° on any grid size.
As you can see from this table the formulas here are very similar to the formulas of the grid sizes with t-shape without any rotation to it. The difference between the two is that the value subtracted for the original t-shape is subtracted and the value for the t-shape rotated at 180° is added. Also the values added and subtracted are the same for each of the formulas of the same grid size.
Also the is a similarity with the value added to the formulas The formulas like the original t-shape formulas differ form grid size to grid size and again the values added are the same as the original t-shape formulas. So the values added are a multiple of seven when dividing the values added to the formula you get the size of the grid.
28 ÷ 7 = 4 42 ÷ 7 = 6
35 ÷ 7 = 5 63 ÷ 7 = 9
If we input this information into the formula we end up with an overall formula to work out the t-total by only knowing the t-number.
T-total = 5T + (7 ×Grid size)
T-total = 5T + 7G
I will now see if this formula works for any grid size.
5T + 7G = T-total
5 × 15 + 7 × 9 = T-total → 60 + 63 = T-total → 138 = T-total → 183 = 138
5T + 7G = T-total
5 × 6 + 7 × 8 = T-total → 30 + 56 = T-total → 86 = T-total → 86 = 86
5T + 7G = T-total
5 × 22 + 7 × 10 = T-total → 110 + 70 = T-total → 180 = T-total → 180 = 180
The formula works for other grid sizes as well.
5T – 7G = T-total ⇒ Overall formula for t-shape without any rotation.
5T – 7 = T-total ⇒ Overall formula for t-shape rotated 90°anti-clockwise.
5T + 7 = T-total ⇒ Overall formula for t-shape rotated 90°clockwise.
5T + 7G = T-total ⇒ Overall formula for t-shape rotated 180°.
The formulas with a negative rotation and no rotation at all have a value, which is subtracted in the formula. All the formulas with a positive rotation and a rotation of 180° have a value in the formula, which is added.
Each of the formulas have 7 and 5 as fixed values, only two of the formulas have the grid size involved in their formulas. This is because the t-totals differ greatly from grid size to grid size for t-totals, which come from t-shapes with a rotation of 180° and t-shapes without any rotation.
