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• Level: GCSE
• Subject: Maths
• Word count: 1971

# To investigate the relationship between a number in a &quot;T&quot; shape placed on a grid of numbers, and the total of all the numbers in that &quot;T&quot; shape.

Extracts from this document...

Introduction

Maths Coursework

Objective: To investigate the relationship between a number in a “T” shape placed on a grid of numbers, and the total of all the numbers in that “T” shape.

Introduction: During this investigation I will focus mainly on the relationship between the size of the “T” against the size of the grid, and I will use this as a basis to find out any link between “T” total and “T” number.

Hypothesis: I believe there will be a link between the size of the “T” and the grid size.

I think there will also be a link between the size of the T and the grid size. These will either be linear or quadratic.

Methodology:

1. To begin, I will keep the “T” size constant and will change the side length of the square grid I place it on by one square each time. Then from these results I will investigate the link.
2. Then I will change the “T” size and investigate how that links to the size of the grid.

Middle

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N= 20     21     22     23     24     25     26

T=  37     42     47     52     57     62     67

5       5       5       5       5       5

Equations:

7 by 7 = 5n - 49

8 by 8 = 5n – 56

9 by 9 = 5n – 63

10 by 10 = 5n – 70

Each of these begins with 5n and ends with a multiple of 7. The exact multiple is the same as the length times 7. This makes the link between each total 5n – 7L, where L is length of one grid side This enables us to work out the “T” total and “T” number for that specific sized “T” on any grid size. The number 5 is the amount of numbers inside the “T”.

Second “T” size (6 numbers)

The first part of  the equation for this set of results will be 6n.

4 by 4 grid. 6 numbers in the “T”

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N= 14     15

T=  36     42

6

5 by 5 grid. 6 numbers in the “T”

 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

N= 17     18     19

T=  42     48     54

1. 6

6 by 6 =

N= 20     21     22     23

T=  48     54     60     66

6       6       6

Equations:

4 by 4 = 6n – 48

5 by 5 = 6n - 60

6 by 6 = 6n - 72

7 by 7 = 6n - 84

8 by 8 = 6n - 96

9 by 9 = 6n - 108

10 by 10 = 6n – 120

The link between each of these is the equation 6n – 12L

As before L equals the length of one of the grid sides, and the 6 is the amount of numbers in the T.

There is a pattern: an – bL

“a” is always the amount of numbers in the T.

Third “T” size (4 numbers)

4 by 4 grid. 4 numbers in the “T”

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

N= 6     7

T= 12   16

4

5 by 5: 4n – 15

6 by 6: 4n - 18

7 by 7: 4n - 21

8 by 8: 4n - 24

9 by 9: 4n - 27

10by 10: 4n – 30

The link between these is the equation: T = 4n – 3L

 N-5 N-4 N-3 N

Equations for the sizes of “T”s so far

4 numbers: T = 4n – 3L

4

5 numbers: T = 5n – 7L                1

5

6 numbers: T = 6n – 12L                1

6

7 numbers: T = 7n – 18L                1

7

8 numbers: T = 8n – 25L                1

8

9 numbers: T = 9n – 33L                1

9

10 numbers: T = 10n – 42L

0      1     2     3     4

3    3     7    12   18

4     5      6

0.5     1      1

3 = 0.5 x 12 + b x 1 + 0

3 = 0.5 + b

3 – 0.5 = 2.5

a = 0.5

b = 2.5

c = 0

Changing the grid size:

0.5n2 + 2.5n + 0

This links with    this number   in the equations

T = 4n –3L

Formula for working out the “T” total on any grid size and any sized “T” (end solution)

T = 4n – (0.5n2 +2.5n) x L

1     1     2     3     4

3     4     5     6     7

1     1     1     1

n + 3

T = (X + 3) x n – (0.5n2 + 2.5n) x L

Y = x2            5x

+

2              2

T = 4n – 3L

T = ? n – (   x2            5x   )

+

2              2

T = (y + X) x n – ((0.5n2 + 2.5n) x L)

Third Variable.

Width of the T.

1st T width

width of 5

grid size 6 by 6

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Conclusion

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N:     17     18     19

T:      42     49     56

7       7

The link here is therefore 7n – 77

I predict that on an 8 by 8 grid the outcome will be 7n – 88 and also that the link with this sized T will be 7n – 11g

 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

N:     19     20     21     22

T:      45    52     59     66

7       7       7

Prediction #1 correct: 7n – 88

6 by   6: 7n -   66

7 by   7: 7n -   77

8 by   8: 7n -   88

9 by   9: 7n -   99

10 by 10: 7n - 110

Second Prediction correct: 7n – 11g

2nd T size

9 numbers in the T

 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

N:     20     21

T:     60      69

9

Prediction: The link for a 9 by 9 grid will be 9n –135

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N:     22     23     24

T:     63     72      81

1. 9

Prediction correct: 9n – 135

The link for all this sized T is 9n-15g

5n – 7g

4

7n – 11g

4

9n – 15g

An - ?g     A being amount of numbers in the t. N being the t number therefore the

Unknown. G being the length of the grid.

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