# Translate the T-shape to different positions on the grid Investigate the relationship between the T-total and the T-number

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Introduction

Sophie Melling

Maths Coursework – T-Totals

Coursework Question

Looking at this T-shape coloured on a 9 by 6 grid.

The total of the numbers inside the T-shape is

1+2+3+11+20 = 37

This is called the t-total

The number at the bottom of the T-shape is the T-number. The T-number for this T-shape is 20.

- Translate the T-shape to different positions on the grid
- Investigate the relationship between the T-total and the T-number

My Answer

- Yellow T-shape.

T-total = 1+2+3+11+20 =37

T-number = 20

Difference = 37 – 20 = 17

- Red T-shape.

T-total = 4+5+6+14+23 = 52

T-number = 23

+ 12

Difference = 52 – 23 = 29

- Jade T-shape.

T-total = 7+8+9+17+26 = 67 + 12

T-number = 26

Difference = 67-26 = 41

- Pink T-shape.

T-total = 28+29+30+38+47 = 172

T-number = 47

Difference = 172-47 = 125

- Green T-shape.

T-total = 31+32+33+41+50 = 187 + 12

T-number = 50

Difference = 149

- Blue T-shape.

T-total = 34+35+36+44+53 = 202 + 12

T-number = 53

Difference = 202-53 = 149

- The difference between the differences along the top row ( 1, 2, 3 ) is 12.
- The difference between the differences along the bottom row ( 4, 5, 6 ) is also 12.

T = (n – 19) + (n – 18) + (n – 17) + (n – 9) + n = 5n – 63

T- total = 5n – 63

T = (n – 19) + (n – 18) + (n – 17) + (n – 9) + n = 5n – 63

T- total = 5n – 63

I can use these equations to find the t-totals just from knowing the T-number. I predict that if the T-number is 50 then (5x50) – 63 = T-total.

250 – 63 = 187. This is correct as I have already worked out the T-total in 5) if you look back.

Middle

T-total = 3+4+5+13+22 = 47

When the T-shape moves across one there is an increase of 5 in the T-total

a = number moved across `

T = (n-2x-1+a) + (n-2x+a) + (n-2x+1+a) + (n-x +a) +(n+a) = 5n – 7x + 5a

T – total when the T-shape is moved across is :

5n – 7x = 5a

Vectors Moving Down

= 20 = T-number

T- total = 1+2+3+11+20 = 37

+ 45

= 29 = T-number

T-total = 10+11+12+20+29 = 82

+45

= 38 = T-number

T-total = 19+20+21+29+38 = 127

When the T-shape moves down there is an increase of 45 in the t-total. Depending on the grid size it will vary.

eg. This grid size is 9

45 ÷ 9 = 5

There are 5 squares in a T-shape

- each square = 9

q = number moved down

T = (n+9q) + (n-x+9q) + (n-2x-1+9q) + (n-2x+9q) + (n-2x+1+9q) = 5n – 7x + 45q

T-total when the t-shape is moved down is:

5n – 7x + 45q

Vectors on a different grid size – going down

I have already generalized vectors for going across the grid, but I have not generalized vectors for going down. So to find an equation I will try some relationships on a 12x12 grid.

= 26 = T-number

T-total = 1+2+3+14+26 = 46

= 38 = T-number + 60

T-total = 13+14+15+26+38 = 106

= 50 = T-number + 60

T-total = 25+26+27+38+50 = 166

= 62 = T-number + 60

T-total = 37+38+39+50+62 = 226

When the T-shape moves down 1

Conclusion

T-total when the shape is turned 270° clockwise is:

5n-7

Enlargements on a 9x6 Grid

I will now enlarge the T-shape and have a look at the t-number and t-total on this new size.

- T-number = 39

T-total = 1+2+3+4+5+12+21+30+39 = 117

- T-number = 51

T-total = 13+14+15+16+17+24+33+42+51 = 225

T = (n-4x-2) + (n-4x-1) + (n-4x) + (n-4x+1) + (n-4x+2) + (n-4x) + (n-2x) + (n-x) + n = 9n -234

= 234

9 (width of grid) = 26

t-total = 9n-26x

Enlargements 12x8 Grid

I will now enlarge the T-shape on a 12x8 Grid.

- T-number = 51

T-total = 1+2+3+4+5+15+27+39+51 = 147

- T-number = 79

T-total = 29+30+31+32+33+43+55+67+79 = 399

- T-number = 94

T-total = 44+45+46+47+48+58+70+82+94 = 534

I will now show you that my formula works.

Take the 2nd T-shape for example:

Using this formula:

n = 79

n- 12 = 67

n- (2x12) = 55

n- (3x12) = 43

n-(4x12) = 31

n- (4x12) -2 = 29

n- (4x12) -1 = 30

n- (4x12) +1 = 32

n – (4x12) +2 = 33

Therefore using only 79 as my figure, and using this formula,with x = the width of grid, I was able to work out the rest of the T-shape.

Transformations, on the enlarged T-shape

1) T-shape turned 90°

Tnumber = 19

T-total = 19 +20+21+22+23+5+14+32+41 = 197

2) T-shape turned 90°

T-number = 31

T-total = 31+32+33+34+35+26+17+44+53 = 305

T = (n-14) + (n-5) + (n+4) + (n+13) + (n+22) + (n+3) + (n+2) + (n+1) + n = 9n+26

T-total = 9n+26

Rotated at 180°

- T-number = 3

T-total = 3+12+21+30+39+37+38+40+41 = 237

- T-number = 16

T-total = 16+ 25+34+43+52+51+50+53+54 = 378

T = (n+34) + (n+35) + (n+36) + (n+37) + (n+38) + (n+27) + (n+18) + (n+9) + n

= 9n+ 237 237 ÷ 9 (grid size) = 26

T-total = 9n+26x

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

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