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• Level: GCSE
• Subject: Maths
• Word count: 8064

# T-Total.I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find,

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Introduction

## Introduction

In this investigation I aim to find out relationships between grid sizes and T shapes within the relative grids, and state and explain all generalizations I can find, using the T-Number (x) (the number at the bottom of the T-Shape), the grid size (g) to find the T-Total (t) (Total of all number added together in the T-Shape), with different grid sizes, translations, rotations, enlargements and combinations of all of the stated.

## Relations ships between T-number (x) and T-Total (t) on a 9x9 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 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

From this we can see that the first T shape has a T number of 50 (highlighted), and the T-total (t) adds up to 187 (50 + 41 + 31 + 32 + 33). With the second T shape with a T number of 80, the T-total adds up to 337, straight away a trend can be seen of the larger the T number the larger the 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

From these Extra T Shapes we can plot a table of results.

 T-Number (x) T-Total (t) 20 37 26 67 49 182 50 187 52 197 80 337

From this table the first major generalization can be made,

### The larger the T-Number the larger the T-Total

The table proves this, as the T-Numbers are arranged in order (smallest first) and the T-Totals gradually get larger with the T-Number.

From this we are able to make a formula to relate T-Number (x) and T-Total (t) on a 9x9 grid. Taking the T-number of 20 as an example we can say that the T-Total is gained by:

t = 20 + 20 – 9 + 20 – 19 + 20 – 18 + 20 – 17 = 37

The numbers we take from 20 are found, as they are in relation to it on the grid, as the T-Shape spreads upwards all numbers must be less by a certain amount, these are found by the following method;

Middle

36

37

38

39

40

41

42

43

44

45

As we can see we have a vertical translation of the first T-Shape (where v =23) by +3. Where v = 23, t = 105, and where v = 8, t = 30 (both found by using t = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 23 - 3).

 Middle number (v) T-Total (t) Equation used Difference 23 105 t = (5 x 23) + ( 2 x 5 ) 25 (105 – 80) 18 80 t = (5 x 18) + ( 2 x 5 ) 25 (80 – 18) 13 55 t = (5 x 13) + ( 2 x 5 ) 25 (55 - 30) 8 30 t = (5 x 8) + ( 2 x 5 ) N/a

From this we can see that 25 is the “magic” number for vertical translations by +1 on a grid width of 5, from this I can see a link with the “magic” numbers, as for as grid width of 5 it is 25, which is 5 x 5, and for a grid width of 9 it is 45 which is 9 x 5. We can also see that translations larger than 1 can be found by a(25) (were a is the figure which you want to translate by), i.e. if you wanted to translate the T-Shape vertically by +3 , the “magic” number would be found by 5x25.

To verify this we can see what the “magic” number is on a grid width of 10, we can predict it will be 50 (5 x 10).

 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 34 35 36 37 38 39 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

If we note the same form of table we have used before, we can find the “magic number”, the above graph shows a vertical translation of the T-Shape by +3, were v=46, t =210 which translates to, v=16, t=60.

 Middle number (v) T-Total (t) Equation used Difference 46 210 t = (5 x 46) + ( 2 x 10 ) 50 (210 – 160) 36 160 t = (5 x 36) + ( 2 x 10 ) 50 (160 – 110) 26 110 t = (5 x 26) + ( 2 x 10 ) 50 (110 - 60) 16 60 t = (5 x 16) + ( 2 x 10 ) N/a

As I predicted the “magic” number was 50, therefore I can generalize and state that;

Any vertical translation can be found by t=(5v-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

grid width.

In terms of the T-Number (x) instead of v,

Any vertical translation can be found by t=(5(x+g)-2g)-a(5g), were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

grid width.

Horizontal

Again, we shall use our standard gird size and position to establish our basic starting point;

 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

Here we can see our basic staring point (v = 41 therefore t = 187), with a vertical translation to the second shape (v = 44 therefore t = 202). Straight away, we can generalize that,

When a T-Shape is translated horizontally by a positive figure its T-Total is less than the

original T-Total

If we table these results along with all the horizontal translation results from 41 to 44 (for v), we should easily see a pattern (on a 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 44 202 t = (5 x 44) + ( 2 x 9 ) 5 (202 – 197) 43 197 t = (5 x 43) + ( 2 x 9 ) 5 (197 – 192) 42 192 t = (5 x 42) + ( 2 x 9 ) 5 (192 – 187) 41 187 t = (5 x 41) + ( 2 x 9 ) N/a

We can see an obvious relationship, that as the T-Shape is translated by +1 in a vertical direction the T-Total is larger by +45 than the previous T-Total. Therefore, we can state that,

As a T-Shape is translated horizontally by +1 on a grid width (g) of 9 the T-Total (t) is

+5 larger than the previous T-Total (t) (the origin)

It is obvious we can also state that:

As a T-Shape is translated horizontally by –1 on a grid (g) width of 9 the T-Total (t) is

-5 smaller than the previous T-Total (t) (the origin)

As when v = 32, t = 142 and with a translation horizontally by –1 v = 31, t = 137, and 137 – 142 = -5, therefore the above statement it correct. We now need to use the same method with different grid size to make a universal equation. I have chosen a grid width of 11 as that has a central number as does a grid width of 9, but make it vertically shorter as we do not need the lower number as they will  not be used. We know with will not make a difference to the final answer as proved in question 2.

 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

As we can see, we have a horizontal translation of the first T-Shape (where v =17) by +4. Where v = 17, t = 63, and where v = 21, t = 83 (both found by using t = 5v – 2g), if be draw up a table in the same format as the one we used for the 9x9 grid, we should be able to find some relationships (from 21 to 17).

 Middle number (v) T-Total (t) Equation used Difference 21 83 t = (5 x 21) + ( 2 x 11 ) 5 (83 – 78) 20 78 t = (5 x 20) + ( 2 x 11 ) 5 (78 – 73) 19 73 t = (5 x 19) + ( 2 x 11 ) 5 (73 – 68) 18 68 t = (5 x 18) + ( 2 x 11 ) 5 (68 – 63) 17 63 t = (5 x 17) + ( 2 x 11 ) N/a

From this we can see that  5 is the “magic” number again as for a grid width of 9 for horizontal translations. From this an obvious relation ship can bee seen that for all grid sizes, a horizontal translation of a T-Shape by +1, makes the T-Total +5 larger, but this is only a prediction. To verify this we can see what the “magic” number is on a grid width of 10.

 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

If we note the same form of table we have used before, we can find the “magic number”, the above table shows a vertical translation of the T-Shape by +4, were v=45, t =205 which translates to, v=49, t=225.

 Middle number (v) T-Total (t) Equation used Difference 49 225 t = (5 x 49) + ( 2 x 10 ) 5 (225 – 220) 48 220 t = (5 x 48) + ( 2 x 10 ) 5 (220 – 215) 47 215 t = (5 x 47) + ( 2 x 10 ) 5 (215 – 210) 46 210 t = (5 x 46) + ( 2 x 10 ) 5 (210 – 205) 45 205 t = (5 x 45) + ( 2 x 10 ) N/a

As I predicted the “magic” number was 5, therefore I can generalize and state that;

Any horizontal translation can be found by t=(5v-2g)+5a, were v is the middle number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

grid width.

In terms of the T-Number (x) instead of v,

Any horizontal translation can be found by t=(5(x+g)-2g)+5a, were x is the T-Number,

a is the figure by which the T-Shape is translated (e.g. +3 or –2) and g is the

grid width.

Combinations (diagonal)

For diagonal translation across a grid a combination of horizontal and vertical translations are used, therefore I predict that if I combine my 2 found equations for horizontal and vertical equations, I can generate a general formula for diagonal translations, which is a prediction I need to change. One simple change needs to be made to my horizontal translation equation, that as “a” was also used for the figure by which to translate (the same as the vertical translation), we have to substitute “a” with “b” in the horizontal equation otherwise we can only move the T-Shape, is fixed diagonal positions. We need now to combine the two equations, only take one instance of 5v-2g as only one T-Shape is being translated;

t=(5v-2g)-(a(5g))-5b

To prove this equation we need to again start with our stand grid and position and try it on a combination translation.

 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 above translation a combination of a horizontal equation of +3 and a vertical translation of +3 also, the origin T-Shape has a T-Total of 187, and the translated T-Shape has a T-Total of 67, using the equation we will try and generate the Second T-Total to prove our theory correct,

t=((5x41)-(2x9))-(3(5x9)-5x3)

t=(205-18)-(135-15)

t=187-120

t=67

My equation has been proved correct using this translation we must now try it on another grid size with another type of a combination translation, to verify that it is correct, I have chosen a grid width of 5, extended vertically to accommodate the combination translation:

 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

Here we can see a translation of +3 vertically, and +1 horizontally, with the original T-Shape having a T-Total of 105, and the translated T-Shape having a T-Total of 35, using our generated formula we can see if it correct, if it can translate with a v total of 23 to the T-Total of translated shape of 35;

T=((5x23)-(2x5))-(3(5x5)-5x1)

T=(115-10)-(75-5)

T=105-70

T=35

This proves my equation correct as; the correct translated T-Total is generated, thus we can state;

Any combination translation (vertically and horizontally), can be found by

using the equation of t=(5v-2g)-(a(5g))-5b were v is the middle number,

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 canceling out that part of the equation as it would equal 0. For example if we wanted to translate a T-Shape with a v number of 41 on a grid width of 9, by –3 horizontally, the equation would look like;

T=(5x41-2x9)-0(5x9)-(5x-3)

T=(205-18)+15

Conclusion

If we try this formula out;

T=5(58+1x9+2)+2

T=5(68)+2

T=347

Thus proving this formula works we now need to try it again on a different grid size with a different centre of rotation.

 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

Here we can see a rotation of 90 Clockwise with the centre of rotation being 36, the original T-Total is 170 and the rotated T-Total is 282. Using the formula we get;

T=5(36+2x10+0)+2

T=5(56)+2

T=282

Thus proving this formula works and it is obvious that it will work in the same fashion as my static (middle number as the centre) rotations, as they both find the position as the new V number then generate the t-total based on that number, therefore I can state;

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of v from the relative centre of rotation , is the grid width, and b is the vertical difference of v from the relative central number.

y is to be substituted by the ending required  by the type of rotation,

these are :

 Rotation (degrees) Direction Ending (y) 0 Clockwise - 2g 90 Clockwise + 2 180 Clockwise + 2g 270 Clockwise – 2 0 Anti-Clockwise – 2g 90 Anti-Clockwise - 2 180 Anti-Clockwise + 2g 270 Anti-Clockwise + 2

In terms of x (T-Number);

The T-Total of any rotation of a T-Shape with any centre of rotation on any grid size

can be found by using the formula of t=5(c+d(g)+b)+y were t is the T-Total, c is the

centre of rotation (grid value) d is the horizontal difference of x from the relative centre of rotation,

g is the grid width, and b is the vertical difference of x from the relative centre of rotation.

y is to be substituted by the ending required  by the type of rotation,

these are :

 Rotation (degrees) Direction Ending (y) 0 Clockwise - 7g 90 Clockwise + 7 180 Clockwise + 7g 270 Clockwise – 7 0 Anti-Clockwise – 7g 90 Anti-Clockwise - 7 180 Anti-Clockwise + 7g 270 Anti-Clockwise + 7

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