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  • Level: GCSE
  • Subject: Maths
  • Word count: 7061

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

Extracts from this document...

Introduction

image00.png

image01.png

image02.pngimage03.png

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

I will start with an 8x8 grid size

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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

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  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.

...read more.

Middle

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38

39

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  • Tn

22

n

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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°

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  • T19

  2

  3

  4

  11

  19

2+3+4+11+19=39

  • T19 270° Rotation

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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

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11

17

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19

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26

27

  • Tn

n-10

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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

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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

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2

3

11

20

1+2+3+11+20 = 37

  • T21

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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

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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

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3

11

20

1+2+3+11+20 = 37

  • T29

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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.

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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

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3

  11

  20

1+2+3+11+20 = 37

  • T20 90° rotation

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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

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  • T34 No rotation

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

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

  • T34 90° Rotation

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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

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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.

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  • T23 No Rotation

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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

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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

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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°

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3

4

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31

  • T21

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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

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21

28

...read more.

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

image04.png

...read more.

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