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• Level: GCSE
• Subject: Maths
• Word count: 4327

# T - ToTaLz WkD CwK- ChEk it OuT !!!!

Extracts from this document...

Introduction

## GCSE Maths Coursework

1) Investigate the relationship between the T-total and the T-number

2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the                      T- number and the grid size.

3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.

Plan

For my GCSE Maths coursework I am going to look and analyse at a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the T-shape.

This is highlighted in the colour red. This is shown below: -

The total number of the numbers on the inside of the T-shape is called the T-total.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

The total of the numbers inside the T-shape is 1+2+3+11+20=37                                                                                        This is called the T-total.

The number at the bottom of the T-shape is called the T-number.                                                                                         The T-number for this T-shape is 20.

By investigating the relationship I will translate the T-shape to different position, first  I will move the T-shape one square to the right every time, but when I get to the end of the line I will move onto the next line, I will be investigating the relationship between the T-total and the T-number.

Afterwards I will use different grid sizes to find each formula and then finally I will try to find out a general formula that will work out any        T-total on any grid size.

Last of all I will try to find out different formulas according to their transformations (e.g. 90º clockwise, 90º anti-clockwise and 180º)

Middle

48

49

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T = T-Number

 T-17 T-16 T-15 T-8 T 1 2 3 10 18

=

=

The T-total is: T + T – 8 + T – 15 + T – 16 + T – 17.                                                      If I put those numbers together this is what I would get = 5T – 56

If I translate the T-shape into different position:

T = T-Number

 T-17 T-16 T-15 T-8 T 442 43 44 51 59

=

=

The T-total is: T – 17 + T – 16 + T – 15 + T – 8 + T. This is the same one as

I have got above. If I add them together again, I will get the same

answer again: 5T – 56.

The solutions above shows that it doesn’t matter how I translate the

T-shape in the 8 x 8 grid into different positions. I would still get the

I will now test this method to see if this work                                                             5n - 56   (5 x 46) - 56 = 174                                                                   therefore T46= 29+30+31+38+46 = 174

The formula for a 8 by 8 square grid is 5n – 56

## 7

Again I am trying to find out another formula for 7x7

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
 T-Number T-Total 16 31 33 116 44 171

Using the idea of how the formula was found in all the grids before, but I have set it out a different way, to make the formula more understandable.

16 + [16 – (16 – 1)] + [16 – (16 – 2)] + [16 – (16 – 3)] + [16 – (16 – 9)]

= 16 + (16 – 15) + (16 – 14) + (16 – 13) + (16 – 7)

= 16 + 1 + 2 + 3 + 9

= 31

31 is also the T-total.

Using “n” instead of the T-number,

n + [n – (16 – 1)] + [n – (16 – 2)] + [n – (16 – 3)] + [n – (16 – 9)]

= n + (n – 15) + (n – 14) + (n – 13) + (n – 7)

= n + n – 15 + n – 14 + n – 13 + n – 7

= 5n – 49

Checking the formula:

5n – 49 = 5(33) – 49

= 165 – 49

= 116

116 is the T-total.

The formula for a 7 by 7 square grid is 5n – 49.

8

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

The formula for the 5 by 5 square grid

 T-Number T-Total 12 25 19 60 23 75

The difference between the T-totals and the T-number comes one after

each other has a difference of 5. For example:

 T-Number T-Total 12 25 19 60 23 75

]

 T-Number T-Total 17 50 18 55 19 60

(5)

]

Using the idea from the original formula (5n – 63) I put an “n” beside

the 5 as always because that’s the difference. Afterward I found the difference between 5n and the T-total.

I have used the same method as the 7 x 7 grid; I find it clearer and more understanding to adjust to.

For example:

12 + [12 – (12 – 1)] + [12 – (12 – 2)] + [12 – (12 – 3)] + [12 – (12 – 7)]

= 12 + (12 – 11) + (12 – 10) + (12 – 9) + (12 – 5)

= 12 + 1 + 2 + 3 + 7

=25

25 is also the T-total.

I am doing the same thing again to make sure it really works,

19 + [19 – (19 – 8)] + [19 – (19 – 9)] + [19 – (19 – 10)] + [19 – (19 – 14)]

= 19 + (19 – 11) + (19 – 10) + (19 – 9) + (19 – 5)

= 19 + 8 + 9 + 10 + 14

=60

60 is also the T-total.

Using n to replace the T-number,

n + [n – (19 – 8)] + [n – (19 – 9)] + [n – (19 – 10)] + [n – (19 – 14)]

= n + (n – 11) + (n – 10) + (n – 9) + (n – 5)

= n + n – 11 + n – 10 + n – 9 + n – 5

= 5n – 35

The formula for a 5 by 5 grid is 5n – 35.

10

The formula for the 6 by 6 square grid

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

By using the nth term, I will find out the formula which relates the                 T-number and the T-shape.

T-Number (n) 14   15     16    17

T-total (t)        28_33_  38_  43

5      5     5

5n

28        33        38        43

[          [    -42    ]         ]

70        75        80        85                                        5n - 42

By doing different grid sizes to work out a formula for each one, I tried to vary my formulas methods. First I used sequences (nth term) and then after I used algebra inside the T-shape therefore sometimes I used the simplifying down method.

I will now test this method to see if this work                                                             5n - 42   (5 x 27) - 42 = 93                                                                   therefore T27= 27+21+14+15+16= 93

The formula for a 6 by 6 square grid is 5n – 42

I am going to use algebra inside the T-shape to see if I still get the same formula using two different methods.

 T-17 T-16 T-15 T-8 T

The T-total is:

T – 17 + T – 16 + T – 15 + T – 8 + T.

This is the same one as

I have got above. If I add them together

and simplify the formula I will get the same answer again: 5T – 42.

9

General Formula

From exploring various different size grids, I have noticed a pattern, for each formula as showed:

 Grid Sizes Formulas 9 by 9 5n - 63 10 by 10 5n - 70 8 by 8 5n – 56 7 by 7 5n - 49 6 by 6 5n - 42 5 by 5 5n – 35

In all formulas, they all begin with 5n, because that how many numbers inside the T. The endings of the formulas are linked together by the number 7. This is because they all have the difference of 7 and also whatever grid size its  ‘times’ by 7 gives me the last part of the formula.

So now I will try to find a formula that links all the grids sizes up, that is called General Formula.

Then I will put the general formula into a T-shape in algebra and then I will work out the general formulas.

 N-(2W-1) N–2W N-(2W+1) N-W N

W= Width number

N = T-Number

The formula for the Value of the T-total in any grid size is:                              5N-7W= T-total

I am now going to put this formula forward and see if it is correct.             (W = width number is 9)

Method test:         5 x 70 = 350   -     7 x 9 =63       =        350 -63= 286                                                                                                           5N                 7W        =        T - total

This is the same answer which I got before, when I was using the              9 by 9 grid formula.

11

Section Three

In this next section I will use different grid sizes. Then I will try other transformations and combinations of transformations.                                                                                                     I will use a 9 x 9 Grid to find various formulas when the T-shape is rotated into different angles.                                                                                                            If I turn the T-shape around 180 degrees it would look like this:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

Conclusion

For example the formula will not work if it’s at 90º, 180º and 270º; the reason being is that the number patterns have changed.

Reflection formula – 5n + (12 x grid size)

16

Conclusion

In this project I have found out many ways in which to solve the problem that I have with the T-shape being in various different positions and with different sizes grids. The way I have made the formulas more understanding I used a general formula.

The size of the T-shape calculates the number before N in the formula and the grid size calculates the values of W. The number before W is calculated by looking at the rows and finding how many rows away from the T-number. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.

I will look at all the possible formulas for each grid that I have investigated.

 Grid 9 x 9 Formula Description 5n - 63 0º 5n + 63 180º 5n + 7 90º 5n - 7 270 Grid 10 x 10 Formula Description 5n - 70 0º 5n + 70 180º 5n + 7 90º 5n - 7 270
 Grid 8 x 8 Formula Description 5n - 56 0º 5n + 56 180º 5n + 7 90º 5n - 7 270

17

 Grid 7 x 7 Formula Description 5n - 49 0º 5n + 49 180º 5n + 7 90º 5n - 7 270
 Grid 6 x 6 Formula Description 5n - 42 0º 5n + 42 180º 5n + 7 90º 5n - 7 270
 Grid 5 x 5 Formula Description 5n – 35 0º 5n + 35 180º 5n + 7 90º 5n – 7 270
 General Formulas Formula Description 5n – (7 x grid size) 0º 5n + (7 x grid size) 180º 5n + 7 90º 5n – 7 270 5n + (12 x grid size) Reflection

The formula for any value of the T-Total in any grid size is

5n – 7w (grid size)

Another way of finding the general formula for the three different rotations are is to times grid size by seven and then you get the last part of the formula. The amount of numbers in the T decides on the number before ‘n’ in my formula.

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

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