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• Level: GCSE
• Subject: Maths
• Word count: 2267

# t shape t toal

Extracts from this document...

Introduction

Maths Coursework

T-Total

1.

Here is a 9 by 9 grid.

 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 total of the numbers within the T shape is 1+2+3+11+20=37

I will call this the T-total.

The 20 at the bottom of the T shape will be called the T-Number.

 1 2 3 11 20

Here is the T shape by itself. As you can see the middle column is going down in 9’s because the grid size is 9 by 9.

 T-19 T-18 T-17 T-9 T

Here is a converted version of the original. This is so I can work out a formula to find the T-total on a 9 by 9 grid.

T-Total=T-19+T-18+T-17+T-9+T

T-Total=5T-63

Time for me to check if this formula works:

T means T-Number.

T=20

T-Total=5x20-63

T-Total=100-63

T-Total=37

To make sure it is not a fluke, I will do 2 more checks on the same grid size.

 16 17 18 26 35

T-Total=112

T-Total=5x35-63

T-Total=175-63

T-Total=112

 48 49 50 58 67

T-Total=272

T-Total=5x67-63

T-Total=335-63

T-Total=272

So now you know the formula for the 9 by 9 grid is T-Total=5T-63

2.

I will now find the formula for a 5 by 5 grid.

Here is the 5 by 5 grid.

 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

The total of all the numbers inside the T-shape is 1+2+3+7+12=25

As you can see, the middle column is going down in 5’s because of the grid size.

 1 2 3 7 12

T-Total=25

 T-11 T-10 T-9 T-5 T

Here is a converted version of the original. This is so I can work out a formula to find the T-total on a 5 by 5 grid.

Middle

The 3 at the top of the T-Shape will be called the T-Number.

 3 8 12 13 14

Here is the T-Shape by itself. As you can see the numbers in the middle is going up in 5’s because the grid width is 5.

Here is the converted version so I can work out the formula for the T-Total.

 T T+5 T+9 T+10 T+11

T-Total=(T)+(T+5)+(T+10)+(T+9)+(T+11)

T-Total=5T+35

Now to check if the formula works:

 3 8 12 13 14

T-Total=50

T-Total=5x3+35

T-Total=15+35

T-Total=50

Here is another check to make sure.

 14 19 23 24 25

T-Total=105

T-Total=5x14+35

T-Total=70+35

T-Total=105

Now I will see another formula for another grid size, which is 4 by 6.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

That will be the T-Shape I will be finding the T-Total formula.

Firstly here is the formula I need to use.

 T T+4 T+7 T+8 T+9

That is the converted version of the 4 by 6 grid size.

T-Total=(T)+(T+4)+(T+8)+(T+7)+(T+9)

T-Total=5T+28

Now to check if the formula works:

 7 11 14 15 16

T-Total=63

T-Total=5x7+28

T-Total=35+28

T-Total=63

Here is another example for that formula:

 14 18 21 22 23

T-Total=98

T-Total=5x14+28

T-Total=70+28

T-Total=98

Now I will figure out a formula to find the T-Total for an inverted T-Shape on any size grid.

 T T+N T+2N-1 T+2n T+2N+1

N=Grid Width

Here is the full formula to work when solving out the formula for an inverted T-Shape on any size grid.

T-Total=(T)+(T+n)+(T+2n)+(T+2n+1)+(T+2n+1)

T-Total=5T+7N

Now lets try that formula on the 5 by 5 grid.

 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

Conclusion

 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

This will be the T-Shape I will be using to find out if my formula works. I will firstly need to figure out a formula.

 T-6 T-2 T-1 T T+2

T-Total=(T)+(T-1)+(T-2)+(T-6)+(T+2)

T-Total=5T-7

Let me try this formula on my T-Shape:

 10 14 15 16 18

T-Total=73

T-Total=5x16-7

T-Total=80-7

T-Total=73

Now let’s try this on another T-Shape in the same grid size:

 17 21 22 23 25

T-Total=108

T-Total=5x23-7

T-Total=115-7

T-Total=108

Now this has been completed I will now try to find a formula to all T-Shapes rotated 270° to the right on any size grid. Although I should solve the full working out, I found that the formula 5T-7 could be worked for all grid sizes. Here is an example on a 9 by 8 grid:

 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

T-Total of T-Shape=158

T-Total=5x33-7

T-Total=165-7

T-Total=158

Here is another example:

 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

T-Total of T-Shape=123

T-Total=5x26-7

T-Total=130-7

T-Total=123

So now you understand that for the first T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5T-7n

Now you understand that for the inverted T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5T+7N

Now you understand that for the first sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5T+7

Now you understand that for the second sideways T-Shape the formula to work out a formula to solve any T-Total on any size table is –

T-Total=5T-7

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

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