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• Level: GCSE
• Subject: Maths
• Word count: 1565

# T-Totals.Plan: Part 1 I will investigate the relationship between the T-total and the T-number on a 9x9 grid. Part 2 I will investigate the relationship between the T-total, T-number and grid size.

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Introduction

T - Totals Investigation

T-Totals

Introduction:

This is an investigation on T-totals. By starting with a 9x9 grid, and numbering it, so the top left corner numbered with one and work downwards. An example of the grid is shown below.

 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 investigation will be done with a standard T-shape which is 3 along and from the centre 2 more down. As shown below.

 2 3 4 12 21

This is the t-number for a T-shape.

The T-total is all the numbers in the T-shape added together. The T-total for this specific example is: 2+3+4+12+21=42.

Plan:

Part 1 – I will investigate the relationship between the T-total and the  T-number on a 9x9 grid.

Part 2 – I will investigate the relationship between the T-total, T-number and grid size.

Part 1:

Diagrams:

 20 21 22 30 39 52 53 54 62 71

20+21+22+30+39=13252+53+54+62+71=292

T-total = 132                                    T-total = 292

 4 5 6 13 14 15 22 23 24

4+5+6+14+23=52

T-total = 52

From now on:

Let T-total be T!

Let T-number be n!

Now I have shown some examples, a table is required so the formula can be solved between the relationship of the T and n for a 9x9 grid.

n                                T

1                                X

X

19                                X

20                                37

21                                42

22                                47

23                                52

24                                57

25                                62

26                                67

27                                X

28                                X

29                                82

38                                127

80                                337

Middle

5n      100    105     110     115     120    125

T         37        42        47        52        57        62

5n to T    - 63    - 63     - 63     - 63    - 63     - 63

Therefore the table above shows the formula will have – 63 in, because – 63 is a constant difference.

The formula comes out to be 5n – 63 = T for a 9x9 grid only.

As a test if n = 38

With the formula: (5 x 38) – 63 = T

1. – 63 = T

127 = T

This is correct; the table shows that the formula works.

I predict that if the T-number is 40, with the formula of 5n – 63 = T for a 9x9 grid, T will equal: 137

T = 5n – 63

= (5 x 40) – 63

= 200 – 63

T = 137

To prove that the T-total for the T-number of 40 is 137:

 21 22 23 31 40

40 + 31 + 22 + 23 + 21= T

137 = T

So the formula is correct as the predicted T-total with the formula was correct, but only for a 9x9 grid.

I think that the formula will also work with incomplete T-shapes too, if 19 is used as n, I can use the formula obtained to prove it:

T = (5 x 19) – 63

= 95 – 63

T = 32

To show that this is correct:

 1 2 10 19

T=19+10+1+2

T= 32

It is correct the formula also got 32, showing the formula works for T-shapes even if they are not complete.

Part 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 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

Conclusion

grid size!

I think that I have found the relationship between grid size, T-number and T-total. Because of the table at the top of the page, shows how found that there is always 5n in the formula and then the grid size is multiplied by -7 to give T. So I think the formula is:

5n – 7G = T

To test the new formula of T = 5n -7G, I will have the grid size of 10 and n being 24.

The formula gets: T = (5 x 24) – (7 x 10)

T =    120    –     70

T = 50

To check this:

 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

T= 3+4+5+14+24= 50

Excellent, the formula obtained which links the T-total, T-number and grid size is:

T = 5n – 7G

Whilst doing this investigation a number of questions sprung to mind, what would happen if the T-shape was turned round, through 90o, 180o and 270o? I also wondered, what would happen to the T-total once the T-shape had been rotated around a number in the grid, how this would relate to the new T-total, the old T-number and the number the T-shape is rotated around? How would the T-total change when the T-shape is reflected? How would the T-total change when it is enlarged or elongated? I also thought that what would happen to the T-total if the T-shape is translated about the grid?

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

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