# Maths coursework

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Introduction

Maths coursework

During my investigation I will be investigating whether there is a relationship between the T-number and the T-total.

The T-shape will look like: (in this example I will be using the numbers 1, 2, 3, 11, 20)

The T-total is the number at the bottom of the T-shape. The T-total is the sum of all the numbers inside the T-shape.

Throughout my investigation I will use a key to refer to the T-total, T-number and grid size.

For the first part of my investigation I will be investigating whether there is a relationship between T and N for numbers in a G9. On the first grid I have shaded places where N must not go; the reason that N cannot go in these places is because if there was a case where N was in these places then there would not be five numbers in the T-shape. Whilst trying to find the relationship I will move the T-shape systematically through each grid.

To find the relationship I could use:

- The sequence method
- Simultaneous equations
- Graphical methods

However, I will only use two of these methods. But for every relationship I will test whether the formula I conclude is correct, I will do this by randomly putting a T-shape into the grid and apply the formula into the numbers inside the T-shape. Also just to make sure that my conclusions are accurate I will use an algebraic approach; I use this approach because it shows a proof to an outcome.

Part 1, finding the formula relating T and N

The first thing that I notice when looking at T is that the values consistently ascend in 5’s when N ascend in 1’s. This states that there is a linear relationship.

Finding the formula.

Middle

G (grid size) | T (t-total) |

G9 G8 G7 | T = 5N - 63 T = 5N - 56 T = 5N – 49 |

I have noticed that in all of the formulas it is consistent that 5N is in the formula. Also the second term in the formula is the sum of G × -7.

Therefore I predict that the general formula is T = 5N – 7G

I will now test my prediction by using G10 and grid 11. I will use N = 25

T = (5 × 25) – (7 × 10)

T = 125 – 70

T = 55

Now I will use addition to see if T is the same as when I used the formula

T = 25 + 15 + 4 + 5 + 6 = 55

I can now say that the forula that I predicted is correct, this is because when I used the predicted formula the answer I got to was 55 and when I used addition the answer I got was the same of 55.

However if you look at G10 in grid 11 when N = 25 there is a relationship between T, N and G. this is:

I will now add up all of that is in the t-shape and put it into its simplest form:

T = N + (N – G) + (N – 2G) + (N – 2G + 1) + (N – 2G – 1)

T = 5N – 7G

Therefore this also correlates with the formula that I previously found, therefore the formula of T = 5N – 7G is correct.

Part 3

Here I will investigate the effect of a translation (x/y) on t-total.

Whilst doing this investigation I will use T2 as the new t-total.

Horizontal translation (x/0) for all grid sizes:

Firstly I will use G8 to find the effect of (1/0).

I can say that the formula for the t-shape in grid 3 is 5N – 7G, this is because I proved it in part 2. Now to find the formula in grid 4 compared to 5N – 7G

T in grid 3 = 34

T in grid 4 = 39

Here it shows that (1/0) is 5 more than (0/0). Therefore the formula here is

T2

Conclusion

T = 5N + 5x – 5Gy + 7 – 5d – 5dG

If these are then combined for a translation (c/d) then:

T = 5N +5x – 5Gy + 7 +5c – 5d – 5Gc – 5Gd.

However this is only my prediction, therefore I will now test this by using the formula first to find out the t-total of the rotated and translated shape, and then I will manually add up the five terms inside the rotated and translated shape and if the results both comply then the formula must be correct. I will test this on G10 on grid 10, the original t-shape will have N54 then this will be translated (-2/-1) then this shape will be rotated from the point (2/-1):

T = (5×54) + (5×-2) – (5×10×-1) + 7 + (5×2) – (5×-1) – (5×10×2) – (5×10×-1)

T = 282

Now I will add the five terms inside the rotated shape from the translated shape, and if the sum of this equals 282, then the formula works:

T = 55 + 56 + 57 + 47 + 67

T = 282

This means that the overall formula for a translation (x/y), followed by a rotation of 90º clockwise (c/d) from the new t-total is T = 5N + 5x – 5Gy + 7 + 5c – 5d – 5Gc – 5Gd.

Evaluation:

Therefore overall from my investigation, I have found that:

- For a translation (0/y) the general formula is T = 5N – 7G – 5Gy
- For a translation (x/0) the general formula is T = 5N – 7G + 5x
- For a translation (x/y) the general formula is T = 5N – 7G – 5Gy + 5x
- For a rotation 90º clockwise about point (c/0) T = 5N + 7 + 5c – 5cG
- For a rotation 90º clockwise about point (0/d) T = 5N + 7 – 5dG – 5d
- For a rotation 90º clockwise about point (c/d) T = 5N + 7 – 5dG – 5d – 5Gy + 5x
- For a translation (x/y),followed by a rotation of 90º clockwise from point (c/d) from the translated t-shape, T = 5N + 5x – 5Gy + 7 + 5c – 5d – 5Gc – 5Gd

However, due to time restriction I could only find the effects of a 90º clockwise rotation, but if time was not of the essence, then I could find the effect of a 180º rotation and 270º rotation and see if the is a connection between them all.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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