Investigate the relationship between the T-total and the T-number.
GCSE Maths Coursework
Aneil Patel
Tasks
) Investigate the relationship between the T-total and the T-number
2) Use the grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total and the T- number and the grid size.
3) Use grids of different sizes again, try other transformations and combinations of transformations. Investigate relationships between the T-total and the T-number and the grid size and the transformations.
Plan
For my GCSE Maths coursework I am going to look and analyse at a grid nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the T-shape.
This is highlighted in the colour red. This is shown below: -
The total number of the numbers on the inside of the T-shape is called the T-total.
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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 called the T-number. The T-number for this T-shape is 20.
By investigating the relationship I will translate the T-shape to different position, first I will move the T-shape one square to the right every time, but when I get to the end of the line I will move onto the next line, I will be investigating the relationship between the T-total and the T-number.
Afterwards I will use different grid sizes to find each formula and then finally I will try to find out a general formula that will work out any T-total on any grid size.
Last of all I will try to find out different formulas according to their transformations (e.g. 90º clockwise, 90º anti-clockwise and 180º) and I will also use different grid sizes, therefore I will predict that I get (about three) formulas for each grid.
2
Section one
T-Number
T-Total
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31
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42
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47
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52
24
57
25
62
26
67
------------------
-------------------
29
82
30
87
31
92
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97
33
02
34
07
35
12
From this I have found out the T-shape and the T-number, by moving the T one square to the right. When I moved out the next like I noticed that I missed out two T-numbers (26, 27), this is why I have separated the table into two. The reason being was that the T- shape does not fit onto the grid. As you can see from this information is that every time the T-number goes up one the T-total goes up fives. Therefore the ratio between the T-number and the T-total is 1:5.
By using the nth term, I will find out the formula which relates the T-number and the T-shape.
T-Number (n) 20 21 22 23 24
T-total (t) 37_ 42_ 47_ 52_ 57
5 5 5 5
5n
37 42 47 52 57
[ [ -63 ] ]
100 105 110 115 120 5n - 63
3
This is the formula (5n + 63) this works for any T-Number and you use it to find out the T-Shape. However this formula will only be successful on a 9 x 9 grid.
My Method test
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) 5n - 63 (5 x 70) - 63 = 287 therefore T70= 51+52+53+61+70 = 287
2) 5n - 63 (5 x 74) - 63 = 307 therefore T74= 74+65+55+56+57= 307
From carrying out these methods I have found the correct formula and I have tested them to see if they work, as seen above. In both cases I got the same answer.
Alternatively I found out another way of finding out the formula as shown below:
This is the T-shape and here is the difference between the T-number. This shows the difference T= T-number.
T-19
T-18
T-17
T-9
T
4
I have notice that the centre column of the T-Shape is going up in 9's because of the table size which is 9 by 9. With the table set out like this a formula can be worked out to find any T-Total on this size grid. This is done in the working below:-
T-total = T-19+T-18+T-17+T-9+T = 5T-63
To see if this method works the both way, I will do an example T-total = 41-19+41-18+41-17+41-9+41 = 142 5T - 63 = 142
This is the same formula which I got before, by doing the nth term, so I now know there are two different ways of which working out the same formula.
I now have found the right formula, I will have to carry this information forward, to investigate the relationship between the T-total, the T-Number and different grid sizes.
5
Section Two
I know this works for the ...
This is a preview of the whole essay
To see if this method works the both way, I will do an example T-total = 41-19+41-18+41-17+41-9+41 = 142 5T - 63 = 142
This is the same formula which I got before, by doing the nth term, so I now know there are two different ways of which working out the same formula.
I now have found the right formula, I will have to carry this information forward, to investigate the relationship between the T-total, the T-Number and different grid sizes.
5
Section Two
I know this works for the grid 9 by 9 but I'm not sure if it'll work for any other grids because there are more intervals in between the first number of each line. I will use larger and smaller grid sizes to discover the different formulas.
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Here is a test for a 10 by 10 grid:
T22=1+2+3+12+22
=40 I notice this is 3 more than 9 by 9 I predict this is from the each extra number in each row T74=53+54+55+64+74=300
=300 I notice no similarities
As there are 10 in each row it's obvious that the row above will be 10 less than the row below. So 68 are 10 less than the T-number 78. If you calculate the whole T you realise that row 2 is 10 less than row 1 and row 3 is 20 less than row 1, but there are three relevant numbers in row 3 which are 19 less and 21 less than the T-number. These cancel out to form 20 each. This is the T-shape and here is the difference between the T-number. This shows the difference T= T-number.
T-21
T-20
T-19
T-10
T
T-total = T-21+T-18+T-17+T-9+T = The formula for a 10 by 10 square grid is 5n - 70
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I am now going to find a formula for the 8 by 8 grid: (a smaller grid size)
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T = T-Number
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=
=
The T-total is: T + T - 8 + T - 15 + T - 16 + T - 17. If I put those numbers together this is what I would get = 5T - 56
If I translate the T-shape into different position:
T = T-Number
T-17
T-16
T-15
T-8
T
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=
=
The T-total is: T - 17 + T - 16 + T - 15 + T - 8 + T. This is the same one as
I have got above. If I add them together again, I will get the same
answer again: 5T - 56.
The solutions above shows that it doesn't matter how I translate the
T-shape in the 8 x 8 grid into different positions. I would still get the
same answer:
I will now test this method to see if this work 5n - 56 (5 x 46) - 56 = 174 therefore T46= 29+30+31+38+46 = 174
The formula for a 8 by 8 square grid is 5n - 56
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Again I am trying to find out another formula for 7x7
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T-Total
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Using the idea of how the formula was found in all the grids before, but I have set it out a different way, to make the formula more understandable.
6 + [16 - (16 - 1)] + [16 - (16 - 2)] + [16 - (16 - 3)] + [16 - (16 - 9)]
= 16 + (16 - 15) + (16 - 14) + (16 - 13) + (16 - 7)
= 16 + 1 + 2 + 3 + 9
= 31
31 is also the T-total.
Using "n" instead of the T-number,
n + [n - (16 - 1)] + [n - (16 - 2)] + [n - (16 - 3)] + [n - (16 - 9)]
= n + (n - 15) + (n - 14) + (n - 13) + (n - 7)
= n + n - 15 + n - 14 + n - 13 + n - 7
= 5n - 49
Checking the formula:
5n - 49 = 5(33) - 49
= 165 - 49
= 116
16 is the T-total.
The formula for a 7 by 7 square grid is 5n - 49.
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The formula for the 5 by 5 square grid
T-Number
T-Total
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9
60
23
75
The difference between the T-totals and the T-number comes one after
each other has a difference of 5. For example:
T-Number
T-Total
2
25
9
60
23
75
]
T-Number
T-Total
7
50
8
55
9
60
(5)
]
Using the idea from the original formula (5n - 63) I put an "n" beside
the 5 as always because that's the difference. Afterward I found the difference between 5n and the T-total.
I have used the same method as the 7 x 7 grid; I find it clearer and more understanding to adjust to.
For example:
2 + [12 - (12 - 1)] + [12 - (12 - 2)] + [12 - (12 - 3)] + [12 - (12 - 7)]
= 12 + (12 - 11) + (12 - 10) + (12 - 9) + (12 - 5)
= 12 + 1 + 2 + 3 + 7
= 25
25 is also the T-total.
I am doing the same thing again to make sure it really works,
9 + [19 - (19 - 8)] + [19 - (19 - 9)] + [19 - (19 - 10)] + [19 - (19 - 14)]
= 19 + (19 - 11) + (19 - 10) + (19 - 9) + (19 - 5)
= 19 + 8 + 9 + 10 + 14
=60
60 is also the T-total.
Using n to replace the T-number,
n + [n - (19 - 8)] + [n - (19 - 9)] + [n - (19 - 10)] + [n - (19 - 14)]
= n + (n - 11) + (n - 10) + (n - 9) + (n - 5)
= n + n - 11 + n - 10 + n - 9 + n - 5
= 5n - 35
The formula for a 5 by 5 grid is 5n - 35.
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The formula for the 6 by 6 square grid
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By using the nth term, I will find out the formula which relates the T-number and the T-shape.
T-Number (n) 14 15 16 17
T-total (t) 28_33_ 38_ 43
5 5 5
5n
28 33 38 43
[ [ -42 ] ]
70 75 80 85 5n - 42
By doing different grid sizes to work out a formula for each one, I tried to vary my formulas methods. First I used sequences (nth term) and then after I used algebra inside the T-shape therefore sometimes I used the simplifying down method.
I will now test this method to see if this work 5n - 42 (5 x 27) - 42 = 93 therefore T27= 27+21+14+15+16= 93
The formula for a 6 by 6 square grid is 5n - 42
I am going to use algebra inside the T-shape to see if I still get the same formula using two different methods.
T-17
T-16
T-15
T-8
T
The T-total is:
T - 17 + T - 16 + T - 15 + T - 8 + T.
This is the same one as
I have got above. If I add them together
and simplify the formula I will get the same answer again: 5T - 42.
9
General Formula
From exploring various different size grids, I have noticed a pattern, for each formula as showed:
Grid Sizes
Formulas
9 by 9
5n - 63
0 by 10
5n - 70
8 by 8
5n - 56
7 by 7
5n - 49
6 by 6
5n - 42
5 by 5
5n - 35
In all formulas, they all begin with 5n, because that how many numbers inside the T. The endings of the formulas are linked together by the number 7. This is because they all have the difference of 7 and also whatever grid size its 'times' by 7 gives me the last part of the formula.
So now I will try to find a formula that links all the grids sizes up, that is called General Formula.
Then I will put the general formula into a T-shape in algebra and then I will work out the general formulas.
N-(2W-1)
N-2W
N-(2W+1)
N-W
N
W= Width number
N = T-Number
The formula for the Value of the T-total in any grid size is: 5N-7W= T-total
I am now going to put this formula forward and see if it is correct. (W = width number is 9)
Method test: 5 x 70 = 350 - 7 x 9 =63 = 350 -63= 286 5N 7W = T - total
This is the same answer which I got before, when I was using the 9 by 9 grid formula.
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Section Three
In this next section I will use different grid sizes. Then I will try other transformations and combinations of transformations. I will use a 9 x 9 Grid to find various formulas when the T-shape is rotated into different angles. If I turn the T-shape around 180 degrees it would look like this:
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When I done this I realise if I reverse the T-shape I should have to reverse something in the formula. It is obvious that I will have to change the minus sign to a different sign. I will try the opposite of the minus which is plus. 5n + 63 = T-total 5 x 2 + 63 = 73 I will check to see if the formula has worked: T-Number = 2 T-Total = 2+11+19+20+21 = 73 ( the reverse in the minus sign has worked)
My next step is to move the shape on its side at 90 degrees. To do this I must put the numbers inside the T-shape and work the formula out by using algebra. To do this I will have to find the difference in the T-number to each number in the T-shape.
n = T-Number
1n
n
-n
2n
-7n
63-70=-7
70-70=0
71-70=1
72-70 =2
81-70=11
=
=
2
So therefore (n) + (1-n) + (2n) + (11n) + (-7n) = 5n + 7 = T-total
Method test: 5n + 7 = T-total 5 x 70 + 7 = 357 I will Check to see if this formula is right. T-Number = 70 (T-total = 70+71+72+81+61 = 357)
I will now try finding the formula for the 270º turn, which I believe will be similar to the 90º, because I will just have to flip the T-shape over. Again I might have to change the plus sign, because I am just reflecting, the formula onto the other side, therefore I will have to reverse part of the formula.
I will work out the difference in between the T-number and the rest of the numbers in the T-shape.
68-57=11 68-66=2 68-75=-7 68-67=1 Total= -7
Method Test
5n - 7 = T-total T-Number = 68 5 x 68 - 7 = 333 Check: T-total = 68 + 57 + 66 + 75 + 67 = 333
Here are all the formulas I have come up with and these formulas only apply to the 9 by 9 grids.
Formula
Description
5n - 63
0º
5n + 63
80º
5n + 7
90º
5n - 7
270
I have now found all the formulas for 9 x 9 grid for different transformations and combination of transformation. My next task is to discover the rest of the formulas for an 8x8 grid, 7x7 and last of all 6x6. Also I will see if I can calculate a different general formula for all the various transformation that I will discover.
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I am now going to use a 7 by 7 grid, and I will try to find various formulas with different transformation.
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First I will find the formula, when the T-shape is rotated 270º as seen in green.
The difference between the numbers inside the T and the T-Number
25-16
25-23
25-24
25
25-30
6
23
24
25
30
=
=
I will use 'n' instead of T-number.
n + [n - (25 - 24)] + [n - (25 - 23)] + [n - (25 - 16)] + [n - (25 - 30)]
= n + (n -1) + (n + 2) + (n + 9) + (n - 5)
= n + n - 1 + n + 2 + n + 9 + n - 5 = 1 + 2 + 9 = 12 so then 5 - 12 = -7
= 5n - 7 (I have noticed that this formula is the same as the 9 x 9)
So therefore 5 x 25 - 7 = 118 and T25 = 16+23+30+24+25= 118
So I will assume that formula for rotating the T-shape 90º is 5n + 7
Method test = 5n + 7 = T-total
5 x 40 + 7 = 207
I will check to see if this is right:
T-shape 40 = 40+41+42+49+35= 207
Again I will predict that the formula will be 5n + 49 this is because I am reflecting (180º) the T-shape and then I will have to reverse the formula, as I did in the 9 x 9 grid.
Method test = 5n + 49 = T-total
5 x 6 + 49 = 79
I will check to see if this is right:
T-shape 6 =6+13+19+20+21 = 79
4
From already investigating two different size grids, I have found different formulas according to there rotation. I have noticed a pattern as the formulas have similarities because when rotating the T-shape 180º, you just have to reverse the sign, from its original formula. Also by rotating the T-shape 90ºand 270º you just have to change the sign and then put 7 at the end to obtain the T-Total.
To see if the pattern is followed I will just do one last grid to see if my predation is right.
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The formula for the 5 by 5 square grid
Already from carrying out similar investigations using different grids, I believe that I can predict all the formals and then I will carry out a method test to see if I am correct.
Method Test 270º
5n + 7 = T-total T-Number = 13 5 x 13 + 7 = 72 Check: T-total = 13+14+15+10+20 =72
Method test 180º = 5n + 35 = T-total
5 x 12 + 35 = 95
I will check to see if this is right:
T-shape 12 = 12+17+22+21+23= 95
Method Test 90º
5n - 7 = T-total (5 x 8) - 7 = 33
Therefore T8 = 8+7+6+11+1 = 33
I have predicted all the formulas and they are all correct, so therefore there is an obvious pattern between the grid sizes and the formulas.
5
I have now found three formulas for three different rotations. The number at the end of the formula is plus or minus and then seven or its seven times the width of the grid. The two main numbers used in this piece of coursework is seven and five.
If there are formulas for rotation then surly there is one for reflection. Below I will show out my working to get the formula.
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Light green is the original T-shape and then the blue T-shape is the reflection of the Green T-shape.
For the number 29 I have a grid movement (gm) of one, so I get (n + gm). For the number 38 I have a grid movement of two, so I get (n + 2gm).
For the numbers 46,47 and 48 I have a grid movement of three and also three numbers, so I get 3(n + 3gm). If I add them all together (n + gm) + (n + 2gm) + 3(n + 3gm) = (5n + 12 x grid size) = T-total
I will check this formula to see if I am correct:
5n + (12 x grid size)
5 x 20 + 12 x 9 = 208
So T-total = 29 + 38 + 46 + 47 + 48 = 208
The formula has worked
However there is one problem, the T-shape can only be reflected from it original position and from no other position that has been rotated.
For example the formula will not work if it's at 90º, 180º and 270º; the reason being is that the number patterns have changed.
Reflection formula - 5n + (12 x grid size)
6
Conclusion
In this project I have found out many ways in which to solve the problem that I have with the T-shape being in various different positions and with different sizes grids. The way I have made the formulas more understanding I used a general formula.
The size of the T-shape calculates the number before N in the formula and the grid size calculates the values of W. The number before W is calculated by looking at the rows and finding how many rows away from the T-number. If the T is regular then the W number is negative but if the T is flipped upside down the W number is positive.
I will look at all the possible formulas for each grid that I have investigated.
Grid
9 x 9
Formula
Description
5n - 63
0º
5n + 63
80º
5n + 7
90º
5n - 7
270
Grid
0 x 10
Formula
Description
5n - 70
0º
5n + 70
80º
5n + 7
90º
5n - 7
270
Grid
8 x 8
Formula
Description
5n - 56
0º
5n + 56
80º
5n + 7
90º
5n - 7
270
7
Grid
7 x 7
Formula
Description
5n - 49
0º
5n + 49
80º
5n + 7
90º
5n - 7
270
Grid
6 x 6
Formula
Description
5n - 42
0º
5n + 42
80º
5n + 7
90º
5n - 7
270
Grid
5 x 5
Formula
Description
5n - 35
0º
5n + 35
80º
5n + 7
90º
5n - 7
270
General Formulas
Formula
Description
5n - (7 x grid size)
0º
5n + (7 x grid size)
80º
5n + 7
90º
5n - 7
270
5n + (12 x grid size)
Reflection
The formula for any value of the T-Total in any grid size is
5n - 7w (grid size)
Another way of finding the general formula for the three different rotations are is to times grid size by seven and then you get the last part of the formula. The amount of numbers in the T decides on the number before 'n' in my formula.
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