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• Unlimited access from just £6.99 per month   # Objectives Investigate the relationship between the t-totals and t-numbers. To translate the t-shape to different parts of the grid.

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Introduction    Objectives

• Investigate the relationship between the t-totals and t-numbers.
• To translate the t-shape to different parts of the grid.

Description

I am going to look at the relationship between the T-number and the T-totals I will translate the t-shape into different positions on different grids, I will be making 3 different grids,  8x8, 9x9, 10x10, I will rotate the

t-shape, translate it horizontally and vertically and work out a formula to all the transformations, I will also rotate the T-shapes and work out algebraic formulas for finding the T-totals of rotated T-shapes I will then write an analysis of my results, and lastly a conclusion.

Grids

 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

Translate Horizontally

I will start by calculating the T-total of the T-shapes, so I will be able to work out the difference between the three of them. I will be translating to the right.

The T-totals for these 3 T-shapes are as follows:

• T 18
 1 2 3 10 18

1+2+3+10+18=34

• T19
 2 3 4 11 19

2+3+4+11+19=39

• T20
 3 4 5 12 20

3+4+5+12+20=44

Table of results

 T-shape T-total Increment T18 34 +5 T19 39 +5 T20 44 +5

T-totals of the 3 T-shapes are: 34, 39 and 44

As you can see they increase by the integer ‘+5’ each time, they are translated to the right. Thus if translated to the left we would ‘-5’

A formula for finding the new t-total could be,  current T-total + 5, this would be right but it is not extensive enough a better formula would be: new T-total = T-total + (x*5), where x is the number of times you are moving to the right, so if I translate to the right once it would be new T-total = current T-total + (1x5), T-total = T-total + 5, if I translate 5 times to the right it would be new T-total = current T-total + (5*5), T-total +25.

Proof

At T-18, if I translated twice to the right,

Formula: ‘Current T-total + (x*5)’ x is the number of times you translate to a certain direction, in this case the right.

Middle

30

31

32

38

39

40

• Tn
 22 n 24 30 n+8 32 n+15 n+16 n+17

The image above shows exactly what needs to be done to n, the T-number to find the individual numbers in the T-shape.

n+n+8+n+15+n+16+n+17

=n+n+n+n+n+8+15+16+17=5n+56

Check

5n+56

Substitute the T-number into ‘n’

5x23+56 = 171

As you can see this formula will find the T-total of any T-shape rotated by180°

So let’s have a recap of all the algebraic formulas we have uncovered so far, and what they solve:

 Formula(8x8) what does it solve 5n-56 Works out the T-total of any T-shape translated horizontally or vertically, (left/right/up/down) 5n+7 Works out the T-total of any T-shape translated in a 90° angle 5n+56 Works out the T-total of any T-shape translated in a 180° angle

Rotating 270°

I will now find out a formula for finding the T-total of any T-shape rotated by 270°

 1 2 3 4 9 10 11 12 17 18 19 20 25 26 27 28
• T19
 2 3 4 11 19

2+3+4+11+19=39

• T19 270° Rotation
 9 10 11 17 18 19 25 26 27

9+17+25+18+19=88

 T-shape T-total Increment T19 39 T19(270°) 88 +49

As you can see the increment was ‘+49’ this allows me to build a simple formula.

x = current T-total + 49 where ‘x’ is equal to the new T-total

Therefore: x = 39 +49

x = 88

New T-total = 88

Although a good formula an algebraic formula is better as it doesn’t require an established T-total to be worked out, all I need is the T-number of the T-shape and I will be able to work it out,

• T19
 9 10 11 17 18 19 25 26 27
• Tn
 n-10 10 11 n-2 n-1 n n+6 26 27

The image above shows exactly what needs to be done to n, the T-number to find the individual numbers in the T-shape.

n+n-1+n-2+n-10+n+6

=n+n+n+n+n+6-1-2-10=5n-7

My formula for finding the T-total of any 180 rotated T-shape is therefore ‘5n-7’

Let’s check to see if it works. I will use the T-shape, T19 to test if this works.

5x19 – 7 = 88

The formula works, just as expected.

I will now show ALL the formulas that can be used in an 8x8 grid to calculate the T-total:

 Formula(8x8) what does it solve 5n-56 Works out the T-total of any T-shape translated horizontally or vertically, (left/right/up/down) 5n+7 Works out the T-total of any T-shape rotated in a 90° angle 5n+56 Works out the T-total of any T-shape rotated in a 180° angle 5n-7 Works out the T-total of any T-shape rotated in a 270° angle

Grid Change

I will move on to a 9x9 grid size

 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

Translating Horizontally

I will be exploring the relationship between the T-number and the T-totals depending on the translation of the T-shape. In this 9x9 grid, I will be translating to the right, and working out the formula for doing so, I will also work out an algebraic formula for working out the T-total of any T-shape in a 9x9 grid.

To begin with, I will first find the T-totals of these 3 T-shapes.

• T20
 1 2 3 11 20

1+2+3+11+20 = 37

• T21
 2 3 4 12 21

2+3+4+12+21 = 42

• T22
 3 4 5 13 22

3+4+5+13+22 = 47

Table of results

 T-shape T-total Increment T20 37 +5 T21 42 +5 T22 47 +5

As you can see the T-totals of the 3 T-shapes are: 37, 42 and 47

As you can see they increase by the integer ‘+5’ each time, they are translated to the right. Thus if translated to the left we would ‘-5’

A formula for finding the new t-total could be,  current T-total + 5, this would be right but it is not extensive enough a better formula would be: new T-total = T-total + (x*5), where x is the number of times you are moving to the right, so if I translate to the right once it would be new T-total = current T-total + (1*5), T-total = T-total + 5, if I translate 5 times to the right it would be new T-total = current T-total + (5*5), T-total +25.

Proof

At T-20, if I translated twice to the right,

Formula: ‘Current T-total + (x*5)’ x is the number of times you translate to a certain direction, in this case the right.

New T-total = 37 + (2*5) = 37 + 10 = 47

As you can see the T-total of T22 is ‘47’

Algebraic

I will now find the algebraic formula for this 9x9 grid, using the T-shape T20.

I will find this formula using an equation, to do so I will find the nth term

My prediction is ‘5*n - x = T-total’ where n = T-number

If I substitute my values it will get:

5*20 - x = 47

100 - x = 47

100 - 47 = x

63 = x

Therefore ‘5n – 63 = T-Total’ is my formula for finding the T-total of any t-shape.

I have used an equation method to find my formula; I could have used the algebraic difference method, to find it.

T20

 1 2 3 11 20

Tn

 n-19 n-18 n-17 n-9 n

As you can see from the above T-shape, we now know how to find all the individual values of the T-shape.

So the algebraic formula would be

n-19+n-18+n-17+n-9+n = ‘5n-63’

As you can see I have arrived at exactly the same formula of ‘5n-63’

Now I will test this formula, on T20

5x20-63 = 37

This formula will find the T-total of any T-shape on a 9x9 grid, as long as it is translated horizontally or vertically

Translating Vertically

Having found out an algebraic formula for finding the T-total of all horizontally/vertically translated T-shapes on a 9x9 grid, I will now find the simple formula for translating vertically, as I have already found the algebraic formula for a 9x9 grid.

To start of I will find the T-total of the following T-shapes:

T20 and T29

• T20
 1 2 3 11 20

1+2+3+11+20 = 37

• T29
 10 11 12 20 29

10+11+12+20+29 = 82

 T-shape T-total Increment T20 37 T29 82 +45

As you can see the T-total increased by ‘+45’, therefore an easy way to work out the T-total of a T-shape translated vertically downwards is:

New T-total = current T-total + 45

Although this is a perfectly valid formula, it is however not extensive enough, a better formula would be.

New T-total = current T-total + (x*45) where ‘x’ is the number of times the shape is translated downwards

For example if the T-shape T20 was translated once downwards to T29 it would be: new T-total = 37 + (1x45) = 37 + 45 = 82

If translated twice, it would be 37 + (2x45) = 37 + 90 = 127. This is the T-total of T38

I will now use the formula I found earlier to work out the T-total of this same T-shape (T29), just to prove that it works on all shapes as long as they are translated vertically or horizontally.

The formula is ‘5n-63’

So by substituting the values I get:

5x29 – 63 = 82

As you can see the formula works. I will now proceed unto rotation of the T-shapes in a 9x9 grid.

Rotating 90°

I will rotate the T-shape in a 90° direction, and see if I can find a formula to find the T-total of rotated T-shapes.

 1 2 3 4 10 11 12 13 19 20 21 22 28 29 30 31

I will be rotating T20 in a 90° direction; the T-number remains the same, ‘20’, but the T-total will change.

• T20 No rotation
 1 2 3 11 20

1+2+3+11+20 = 37

• T20 90° rotation
 11 12 13 20 21 22 29 30 31

13+22+31+21+20=107

 T-shape T-total Increment T20 37 T20(90°) 107 +70

As you can see from the table above, the 90° rotated T19 has a T-total of 102, a ‘+70’ increment.

I will now try this on another T-shape to see if I will get the same ‘+70’ increment.

I will use the T-shape of T34

 15 16 17 18 24 25 26 27 33 34 35 36 42 43 44 45
• T34 No rotation
 15 16 17 24 25 26 33 34 35

SUM method: 15+16+17+25+34=107

Algebraic Formula (5n-63): 5x34-63 = 107

• T34 90° Rotation
 25 26 27 34 35 36 43 44 45

27+36+45+35+34=177

 T-shape T-total Increment T34 107 T34(90°) 177 +70

We also have an increment of ‘+70’, therefore we know that, to find the T-total of a 90° rotated T-shape, we would be able to do so by simply adding ‘70’ to the current T-total.

x = current T-total + 70 (where ‘x’ is the new T-total to be found...)

Now I will find an algebraic formula for finding the T-total of any 90° rotated T-shape in a 9x9 grid. 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 above shows exactly what needs to be done to n, the T-number to find the individual numbers in the T-shape.

n+n+1+n+2+n-7+n+11

=n+n+n+n+n+1+2+11-7 (gather like terms) =‘5n+7’

Therefore my algebraic formula for finding the T-total of any 90° rotated t-shape is ‘5n+7’

I will now test this formula to see if it works, i will test it using the T34 T-shape

Substituting the T-number in place of ‘n’ I get:

5x34 + 7 = 177

The formula works!

Rotating 180°

I will now rotate the T-shape, T23 in 180.

 4 5 8 13 14 15 22 23 24 31 32 33 40 41 42 49 50 51
• T23 No Rotation
 4 5 6 13 14 15 22 23 24

To find the T-total of this T-shape, I will use the algebraic formula 5n-63

Where ‘n’ is the T-number of the T-shape.

5x23-63=52

Just to prove that this is correct: 23+14+5+4+6=52

• T23 180° Rotation
 22 23 24 31 32 33 40 41 42

23+32+40+41+42=178

 T-shape T-total Increment T23 52 T23(180°) 178 +126

You can clearly see from the above table of results, the T-total increases by an increment of ‘+126’ every time.

I will now find an algebraic formula for finding the T-total of any 180° rotated T-shape.

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’

• T23
 22 23 24 31 32 33 40 41 42
• Tn
 22 n 24 31 N+9 33 N+17 N+18 N+19

The image above shows exactly what needs to be done to n, the T-number to find the individual numbers in the T-shape.

n+n+9+n+17+n+18+n+19

=n+n+n+n+n+9+17+18+19=5n+63

Check

5n+63

Substitute the T-number into ‘n’

5x23+63= 178

As you can see this formula will find the T-total of any T-shape rotated by180°

So let’s have a recap of all the algebraic formulas we have uncovered so far, and what they solve:

 Formula(9x9) what does it solve 5n-63 Works out the T-total of any T-shape translated horizontally or vertically, (left/right/up/down) 5n+7 Works out the T-total of any T-shape translated in a 90° angle 5n+63 Works out the T-total of any T-shape translated in a 180° angle

Rotating 270°

I will now find out a formula for finding the T-total of any T-shape rotated by 270°

 1 2 3 4 10 11 12 13 19 20 21 22 28 29 30 31
• T21
 2 3 4 11 12 13 20 21 22

2+3+4+12+21=42

• T19 270° Rotation
 10 11 12 19 20 21 28 29 30

10+19+28+20+21=98

 T-shape T-total Increment T19 42 T19(270°) 98 +56

As you can see the increment was ‘+56’ this allows me to build a simple formula.

x = current T-total + 56 where ‘x’ is equal to the new T-total

Therefore: x = 42 +56

x = 98

New T-total = 98

Although a good formula an algebraic formula is better as it doesn’t require an established T-total to be worked out, all I need is the T-number of the T-shape and I will be able to work it out.

• T21

10

11

12

19

20

21

28

Conclusion

5n-56(8x8), 5n-63(9x9)

5n-56 + - 7

5n-56 – 7 = 5n-63

Looking at the table above, I can see that the formula for calculating the 90° rotations stays the same… ‘5n+7’

This means if you want to find the 90° rotation of any T-shape on any grid, simply using this formula: ‘5n+7’ will work out its T-total.

I can also see that the formula for calculating the 180° rotations increase by an increment of ‘+7’ as we increased the grid size…which shows yet again that the number is always a multiple of ‘7’

5n+56(8x8), 5n+63(9x9)

5n+56 + 7 = 5n+63

Looking at the table above, I can see that the formula for calculating the 270° rotations stays the same… ‘5n-7’

This means if you want to find the 270° rotation of any T-shape on any grid, simply using this formula: ‘5n-7’ will work out its T-total.

I have noticed all the numbers used in the formulas are multiple of ‘7s’

7, 56, 63, 70…

I have also noticed that all the formulas begin with ‘5n’; this is because they are always 5 numbers in a T-shape.

I have also noticed a clear relationship between the formula and the Grid number.

Tn where ‘n’ = the T-number

And where‘d’ = the grid number (8x8, dxd)

 n-2d-1 n-2d n-2d+1 n-d n

This is an algebraic formula showing the relationship between the T-total formula and the grid size number.

n+n-d+n-2d+n-2d-1+n-2d+1 = n+n+n+n+n+-d+-2d+-2d-1+-2d+1

= 5n-7d

The relationship between the T-total and the grid size is 5 * the T-number of the T-shape – 7 * the grid size.

Proof

Let’s use T23 in the 10x10 grid to prove this relationship

• T23
 2 3 4 12 13 14 22 23 24

2+3+4+13+23=45

5n-7d where ‘n’ = T-number and‘d’ = grid size number

By substituting the values I get:

5x23 – 7x10

115 – 70 = 45 as you can see there is a relationship.

This concludes my T-total project.

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

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