• 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
• Level: GCSE
• Subject: Maths
• Word count: 1733

# T - Total. Looking at the 9-9 grid below and the T-shape drawn on it, The total number of the numbers on the inside of the T-shape is called the T-total

Extracts from this document...

Introduction

## Investigation into T shapes

Looking at the 9-9 grid below and the T-shape drawn on it,

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

82 83 84 85 86 87 88 89 90

The t-total for this T-shape is:

1+2+3+11+20=37

So 37 = T-total

The number at the bottom is the T-number, so the T-number for this shape is 20

Aims:

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, the T-numbers and the grid size.

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

Aim 1- the solution

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

T22=3+4+5+13+22

=47

T69= 50+51+52+60+69

=282

In the diagram below it shows the difference between the T-number and the other numbers. First is the T-shape in question:

1 2 3

11

20

This is the T-shape and here is the Difference T-shape:

N-19 N-18 N-17

N-9

N

This shows the difference N= T-number

Simplified this can be written as 5N - 63

In the T

Middle

Aim3- Transformations, (stretches) and there effects on the formula

I’ll do this with a 12 by 12 first, as this will give me enough accuracy to start with.

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

Stretch A will be called ST64 as it starts at 64, it’s a stretch of 2 in both directions.

St64=26+27+28+29+30+40+52+64

=296

Stretch A will be called ST64 as it starts at 64, it’s a stretch of 2 in both directions.

St64=26+27+28+29+30+40+52+64

=296

I think I can work out the formula using my previous method so:

12+24+36+(4*36)=216

216 / 12 = 18

This means the formula is:

8N-18W=T-total

8N= number of integers in the T-shape

18W=difference number calculated

Conclusion:

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

Below is a sheet with more simple things on it satisfying the same aim roughly explaining the above.

Conclusion

79

-

32

167

135

56

32

209

177

42

32

153

121

- 56

36

124

88

-

36

187

151

63

36

236

200

49

36

173

137

- 63

As you can see from the results the third difference of the difference is an inverse of the first difference of the difference.

So the formula for a T that has been rotated by 180 degrees should be an inverse of the formula used to work out a T in its upright position. Therefore

5N + 7W

It’s a plus sign as it’s the inverse of a minus sign.

Proof …

8 by 8 grid

1        2        3        4        5        6        7        8 (5 * 2) + (7 * 8) = 66

9        10        11        12        1314        15        16                2 + 10 +17 +18 +19 = 66

17        18        19        20        21        22        2324        It works !

25        26        27        28        29        30        31        32

However the formula will only work for T’s rotated by 180 degrees.

In terms of the T – number the 270 degree rotation is equal to :

5N – 7

I found this out by accident I found out that it worked for one and then I tried it on other size grids and it worked.

5 * 21 – 7 = 98

21 + 22 + 23 + 13 + 29 = 98

see it works. I then tested it on other places and it still worked. It wasn’t just a coincidence.

I took this further and used the same method as before in working out the inverse of this. To find out the T – total for a T that has been rotated by 90 degrees it will be the inverse of the 270 degree one.

5 * 14 + 7 = 77

14 + 15 + 16 + 24 + 8 =77   This also works.

5N + 7

5N - 7

5N – 7W

5N + 7W

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 Total and T Number Coursework

8x8 and 7x7 it should enable me to build up an accurate picture of any patterns that develop. I do not however expect that I will find just one general formula for a rotation of any degrees, there should be three separate formulas for 90 degrees, -90 degrees and finally 180 degrees.

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

To find this algebraic formula, I will find out a way to find the individual values in the T-shape: Let's refer to the T-number as 'n' * T34 25 26 27 34 35 36 43 44 45 * Tn 25 26 n-7 n n+1 n+2 43 44 n+11 The image

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

The next step is to move the shape on its side. Again we nearly keep the same formula as we had at the beginning. Again we change the minus number. We can work out the number to minus by working out the difference in the t-number to each number in the t-shape.

2. ## T-Total. I will take steps to find formulae for changing the position of the ...

51 52 53 54 55 56 57 58 59 60 61 62 63 64 WORKING OUT AND RESULTS Formula: 5x - y - 5zb + 5a Substituting in the relevant information: (5 x 42) - 56 - (5 x 8 x 3)

1. ## T totals. In this investigation I aim to find out relationships between grid sizes ...

grid width of 9), also adding a column for the difference between the number in that column and the once below. Middle number (v) T-Total (t) Equation used Difference 41 187 t = (5 x 41) + ( 2 x 9 )

2. ## T-Totals. We have a grid nine by nine with the numbers starting from 1 ...

32-23= 9 TOTAL= 63 This will happen to all the shapes this way up. To prove this I will do another. 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

1. ## T- total T -number coursework

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 5*57-63=t-total 5*57-63= 222 Check T-total = 38+39+40+48+57=222 This formula has proven to work.

2. ## My aim is to see if theres a relation between T total and ...

T - 15 + T - 8 + T = 5T - 56 To check this formula: 1 2 3 10 18 5 x 18 - 56 = 34 To check: 1 + 2 + 3 + 10 + 18 = 34 My formula is correct.

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