• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
19. 19
19
20. 20
20
21. 21
21
22. 22
22
23. 23
23
24. 24
24
25. 25
25
• Level: GCSE
• Subject: Maths
• Word count: 3347

# This project consists of the investigation between patterns between T-numbers and T-totals.

Extracts from this document...

Introduction

• Investigation
• ## Introduction

This project consists of the investigation between patterns between T-numbers and T-totals. My investigation consists of different aims, but all of these aims have to do with this mathematical “T”. I believe that there is a direct link between these two, as the T-total increases depending on  the T-number. I think that there is nth sequence relating the T-number and the T-total, and therefore I will try and investigate this.

First, I will use a 9 by 11 grid and find the relationship between the T-number, and the T-total in this grid. I will draw other girds and also find out the correlation between the translation of T-numbers, T-totals and the gird size.

After this, I will go back to my initial 9 by 11 grid, and rotate the “T” and try to find the possible relationship of T-numbers and T-total of the rotated “T”. I believe that there is a formula for the T-total of T-shapes in any grid and in any rotation.

Subsequent to this investigation, I will use and enlargement of my initial “T” and use different examples to show the relationship between the enlargement and the T-number and T-total. From all my different investigations I will draw my conclusions giving explications for these.

1. I will draw a 9 by 11 grid,

Middle

i.e.

 79 80 81 89 98 98-19 98-18 98-17 98-9 98

i.e

 48-19 48-18 48-17 48-9 48 29 30 31 39 48

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.

1. 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.
2. 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.
3. With the different T-totals and T-numbers I will create a result table to be able to see any observations.
4. I will right down any observations noticed.
5. 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:

 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144

Results=

 T-number T-total T-number T-total 26 46 76 296 40 116 82 326 54 186 123 531 62 226 126 546 68 256 129 561

The 5 by 5 grid and the “T’s”:

 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

Results=

 T-number T-total 12 25 18 55 19 60 22 75
• Observations

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

 98 99 100 110 111 112 122 123 124

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.

 n-25 n-24 n-23 n-12 n

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:

 n-11 n-10 n-9 n- 5 n

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.

 41 42 43 51 52 53 61 62 63

Initial T:

V= 52

Translation:                       Translated T:

A= 2

 23 24 25 33 34 35 43 44 45

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.

 n-2G-1 n-2G n-2G+1 n-G n

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.

• 5 x 5 grid.

Conclusion

 1 2 3 4 5 10 11 12 13 14 19 20 21 22 23

T-number= 21

T-total= 48

Stretch= 2

The T-number is increased by one so:

 2 3 4 5 6 11 12 13 14 15 20 21 22 23 24

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.

• Conclusion

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:

1. With a 9 by 11 grid the equation is correct anywhere in the grid.
2. Changing the vertical size of this grid makes no change, but if the width of the grid is changed the equation is changed.
3. The equation for a 12 size grid is and this works anywhere on this grid size.
4. The equation for the five size grid is. This works anywhere in this sort of width.
5. 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.
6. There is two equations relating any translation in any grid size. These are:
7. 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

1. 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

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE T-Total essays

1. ## T-Totals Investigation.

51 52 53 54 55 56 57 58 59 60 61 62 63 64 T-number 18 19 20 21 22 23 T-total 34 39 44 49 55 59 +5 +5 +5 +5 +5 There is an addition of 5 again-(same as in the horizontal T-shapes of the 9 by 9 grid.

2. ## T-Shape investigation.

I first put the T in the top left side of an 8 by 8 grid. 1 2 3 4 9 10 11 12 17 18 19 20 25 26 27 28 But from my earlier mistakes, I will move the t shape along a number of times to identify a pattern on an 8 x 8 grid.

1. ## T-totals. I am going to investigate the relationship between the t-total, T, and ...

- 7 Using the validation of the formula for a rotation about an external point and the formula for a translation as evidence, it follows, from the algebraic argument above, that the formula for the combined transformation is correct. We need not therefore test it further.

2. ## Objectives Investigate the relationship between ...

T-total + (x*40), where x is the number of times you are translating in a downward direction, so if I translate down once it would be new T-total = current T-total + (1*40), T-total = T-total + 40, if I translate 5 times in a downwards direction it would be new T-total = current T-total + (5*40), T-total +200.

1. ## In this section there is an investigation between the t-total and the t-number.

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 It is obvious that we will have to change the minus sign to a different sign.

2. ## For my investigation, I will be investigating if there is a relationship between t-total ...

68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 T-total = (N-23)+(N-22)+(N-21)+(N-11)+(N)

1. ## For my investigation, I will be investigating if there is a relationship between t-total ...

86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 T-total = (N-21)+ (N-20) + (N-19) + (N-10)+( N) I then can simplify this to: T-total = 5N-70 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

2. ## T-shapes. In this project we have found out many ways in which to ...

This is proven below. 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

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to