Mathematics Coursework - T-totals
GCSE MATHEMATICS COURSEWORK
T-totals
0/3/'02
Jonathan Briggs
INTRODUCTION
Introduction
The question I have been asked has been set in three parts. The question is about T-shapes on different grids. The bottom number in the T is called the T-number the T-number is the largest number in the T-shape. All the numbers in the T-shape added together are called the T-total. In the whole of the coursework you are not allowed to expand the T-shape but you are allowed to turn it around.
In part 1of my coursework I am asked to investigate the relationship between T-totals and T-numbers. At the moment I do not have any notion how the T-total could be calculated in a standard way or of the relationship between the T-number and the other numbers in the T-shape. I did know that they had to be related because they were all numbers with in a sequence. I also knew that the T-total would depend on: ~
* The position of the T-shape on the grid;
* The size of the T-shape (for the purpose of this exercise the T-shape is fixed with only five numbers);
* The orientation of the T-shape on the grid.
These points are fairly obvious as any change to the T-shape will change the numbers within it.
From this knowledge I initially experimented practically by putting T-shapes in various positions on a 9 x 9 grid. At this stage (part 1 of this paper) the T-shape was fixed in a normal T position. Part 1 shows my approach and findings from experimenting with a standard T-shape on a 9 x 9 grid
In part 2 I did further experiments with grids of different dimensions attempting to repeat my findings from part 1 with these grids and attempting to develop a general formula for grids of all sizes
In part 3 I did more experiments, this time with the T-shape in different orientations on the grid; that is on its side in two positions and upside down. I also did different vectors. This section shows new formulas and new grids.
Below are some definitions which are necessary for consistency
Definitions
: T-shape: - This is a T shape inside a grid which only contains five numbers in this case.
2: T-total: - This is all five numbers inside the T-shape added together
3: T-number: - The number at the bottom of the T-shape.
4: Other numbers: - The other numbers in the T-shape excluding the T-number
5: Number of numbers in the T-shape: - the number of numbers inside the T-shape including the T-number
PART 1
Investigating the relationship between the T-total and the T-number
Part 1
Constraints
. The T- shape cannot be expanded.
2. The T-shape cannot be moved from the standard vertical T alignment.
3. In Parts 1 - 2 the T-shape will only be able to move left and right and down and up.
4. In part 3 you will be able to twist the T-shape.
9 x 9 Grid size
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My first step to establishing the relationship between the T-total and the T-number was to draw a series of six T-shapes on the 9 × 9 grid. I then added up the numbers in the T-shape to get the T-totals. In order to find the relationship between the T-total and the T-number I divided the T-number by the T-total for each T-shape. This showed that the T-number diminished as a proportion of the T-total as the T-shape moved to the right or down the grid. I also observed that the T-total went up by five for every one position moved to the right by the T-shape on the 9 × 9 grid.
Then I found out that five (that is the number of numbers in a T-shape) times the T-number minus the sum of each of the other number in the T-shape subtracted from the T-number gave you the T-total. From this I could derive a formula.
Formula
Equation 1:~
Where ~ N = T-total
T = T-number
A = the numbers inside the T-shape taken away from the T-number
N = 5T - ((T - A) + (T - B) + (T - C) + (T - D))
: N = 5T - (4T - (A + B + C + D))
Equation 2 is a generalisation of equation 1:~
Where ~ N = T-total
T = T-number
A - N = other numbers in the T-shape
X = number of numbers in the T-shape
N = TX - (T - 1) (X) - (sum of A to N)
Equation 3: ~
Where~ N = T-total
X = number of numbers in the T-shape
G = grid size
T-total = (N) + (N - 9) + (N - 2G)
+ (N - 2G - 1) + (N - 2G +1)
T-total = 5N - 7G
7 is the relationship of the T-total and the other numbers in the T-shape. This equation will work for different grid sizes. The only thing that will change is G, because G is the grid size.
The ratio between the T-number and the T-total reduces by four for every movement that the T-shape makes to the right.
T-total
37
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67
T-number
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Change in T-number
0
Change ...
This is a preview of the whole essay
7 is the relationship of the T-total and the other numbers in the T-shape. This equation will work for different grid sizes. The only thing that will change is G, because G is the grid size.
The ratio between the T-number and the T-total reduces by four for every movement that the T-shape makes to the right.
T-total
37
42
47
52
57
62
67
T-number
20
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26
Change in T-number
0
Change in T-total
0
5
5
5
5
5
5
Difference in T-number and T-total
7
21
25
29
33
37
41
Ratio of T-number to T-total
.85
2
2.14
2.26
2.38
2.48
2.58
Change of the ratio T-number to T-total
0
0.15
0.14
0.12
0.12
0.1
0.1
The first row shows the T-totals as you go across the grid. The second row shows the T-numbers as you go across the grid left to right, one column at a time. The third row shows the change in the T-numbers as you move left to right and the fourth row shows the change in T-totals. The difference in T-numbers and T-totals are shown in the fifth row. The sixth row shows the ratio of each T-number to each T-total and the seventh row shows the change of the ratio between the T-totals and the T-numbers.
The fifth row shows that the difference between the T-numbers and T-totals is increasing proportionally as the T-shape is moved to the right. This is going to happen as the T-number is changing by one and the T-total by five. The increasing difference is reflected by the increase in the ratio shown in row six as the T-shape moves to the right. Row seven shows that as you move the T-shape right, although the difference in the ratio between the number and T-shape is changing, the rate of change of this ratio is decreasing slightly.
PART 2
Use grids of different sizes. Translate the T-shape to different positions. Investigate relationships between the T-total, the T-number and the grid size.
Part 2
Constraints
As part 1 plus-
. The T- shape can not be expanded.
2. In this Part the T-shape will only be able to move left and right and down and up.
0 ×10
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As Part 1
I took a sample of six T-shapes on the 10 × 10 grid. I then added up the numbers in the T-shape to get the T-totals. In order to find the relationship between the T-total and the T-number I divided the T-number by the T-total for each T-shape. As with the 9 × 9 grid, this showed that the T-number diminished as a proportion of the T-total as the T-shape moved to the right or down the grid. I observed that as with the 9 × 9 grid the T-total went up by five for every one position moved to the right by the T-shape
I applied the same formula as I applied to the 9 × 9 grid as follows
Equation 1:~
Where ~ N = T-total
T = T-number
A = the numbers inside the T-shape taken away from the T-number
N = 5T - ((T - A) + (T - B) + (T - C) + (T - D))
: N = 5T - (4T - (A + B + C + D))
This formula works for the 10 x 10 grid as for the 9 x 9 grid
I then did the same analysis of my six sample T - Shapes as shown in the matrix below -
T-total
40
45
50
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60
65
70
T-number
22
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28
Change in T-numbers
0
Change in T-totals
0
5
5
5
5
5
5
Difference in T-number and T-total
8
22
26
30
34
38
42
Ratio of T-number to T-total
.81
.96
2.08
2.2
2.31
2.41
2.5
Change of the ratio T-number to T-total
0
0.15
0.12
0.12
.05
0.1
0.09
As in Part 1, this shows that the relationship between the T -Number and the T - Total remains the same as in the 9 × 9 grid.
1 × 11
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T-total
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48
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73
T-number
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30
Change in T-number
0
Change in T-total
0
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5
Difference in T-number and T-total
9
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31
35
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43
Ratio of T-number to T-total
.79
.92
2.04
2.15
2.25
2.34
2.43
Change of the ratio T-number to T-total
0
0.13
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0.11
0.1
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2 × 12
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T-total
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76
T-number
26
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31
32
Change in T-number
0
Change in T-total
0
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5
5
Difference in T-number and T-total
20
24
28
32
36
40
44
Ratio of T-number to T-total
.77
.89
2
2.10
2.2
2.29
2.38
Change of the ratio T-number to T-total
0
0.12
0.11
0.1
0.1
0.09
0.09
The relationship between the grid size and T-Total
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Whilst doing the work above on the various grid sizes and trying to find the formula to work out the T-total I realized that any number in the T-shape could be expressed/ described in terms of its relationship to the T-number or any other number.
The first example I did of this I took as A the top left hand number in the T-shape. I realized that B the next number across could be expressed as A + 1 and that C the final number across on the T bar could be expressed as A + 2. D the number above the T-number could be expressed as A + 10 and finally that E the T-number could be expressed as A + 19. I worked out the amounts you had to add to A to get D and E by counting across from C to D the number of spaces on the grid to get D, and by counting across from D to E to get E. By doing this I was able to develop the equation below.
Equation to work out the T total based on expressing numbers in the T-shape relative to the first number (9 × 9 grid)
On the diagram above A = 13, B = 14, C = 15, D = 23 and E = 32
A = A
B = A + 1
C = A + 2
D = A + 10
E = A + 19
T-total = 5A + 32
After doing this I found there was an alternative way of expressing numbers relative to each other and the grid size as follows
If N is the T number the number above N can be expressed as N - G where G is the width of the grid i.e.: nine in this case
The number above N - G can be expressed as N - 2G
The numbers each side of N - 2G can be expressed as N - 2G - 1 and N - 2G + 1 correspondingly. This gives the formula
Equation: ~
Where~ N = T-total
X = number of numbers in the T-shape
G = grid size
T-total = (N) + (N - 9) + (N - 2G)
+ (N - 2G - 1) + (N - 2G +1)
This simplifies to: ~ T-total = 5N - 7G
7 is the relationship of the T-total and the other numbers in the T-shape.
PART 3
Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-total, the T-number, the grid size and the transformations
Part 3
Constraints
. As in Part 2 except that it is possible to move the alignment of the T-shape in relation to the grid.
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This is with the T-shape on its side with the T-number facing right.
T-total
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T-number
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Change in T-numbers
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Change in T-totals
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Difference in T-number and T-total
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65
Ratio of T-number to T-total
4.42
4.46
4.5
4.53
4.56
4.59
4. 61
Change of the ratio T-number to T-total
0
0.04
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0.03
0.03
0.03
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The formulae for the T-total is: (N) + (N - 1) + (N - 2) + (N - 2 + G)
+ (N -2 + G)
T-total = (5N) - G
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This is with the T-shape on its side with the T-number pointing left
T-total
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T-number
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Change in T-numbers
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Change in T-totals
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Difference in T-number and T-total
47
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Ratio of T-number to T-total
5.7
5.64
5.58
5.54
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5.47
5.44
Change of the ratio T-number to T-total
0
-0.06
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-0.04
-0.04
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-0.03
The formulae for the T-total here is: (N) + (N + 1) + (N + 2) + (N + 2 - G) +
(N +2 +G)
T-total =5N + 7
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This is with the T-shape upside down
T-total
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T-number
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Change in T-numbers
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Change in T-totals
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Difference in T-number and T-total
71
75
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83
87
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95
Ratio of T-number to T-total
36.5
26.0
20.75
7.6
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4.0
2.87
Change of the ratio T-number to T-total
0
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5.25
3.15
2.1
.5
.13
The formula for the T-total here is: (N) + (N + G) + (N + 2G) +
(N + 2G - 1) + (N + 2G + 1)
T-total = 5N + 7G
I established all three of these equations in the same way as I established equation three in Part 1. These equations that I have applied will work on different grid sizes; the only thing that will change is G which is the grid size itself.
Transformations: vectors
A vector is a quantity which has magnitude (size) and direction. Vectors can be used to represent displacement, velocity, force, momentum and acceleration. A scalar is a quantity that has magnitude (size) but no direction. You can multiply a vector by a scalar to get another vector.
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By using Pythagoras theorem you can work out the size of the vectors.
A 4 = 4 + 4 = 16 + 16 = 32 = 5.66
4
B " " + " " + " = " "
C " " + " " + " = " "
D " " + " " + " = " "
E " " + " " + " = " "
F " " + " " + " = " "
G " " + " " + " = " "
H " " + " " + " = " "
I " " + " " + " = " "
J " " + " " + " = " "
The vectors stay equal throughout the grid.
I first tried my vectors on a 9 × 9 grid. I used a 4 × 4 vector, a 5× 5 vector and a 6 × 6 vector. My results were as follows:
The 4 × 4 vector came to a change of 32 in the T-number as above;
The 5 × 5 vector came to a change of 40 by the T-number;
The 6 × 6 vector came to a change of 48 by the T-number.
There is a clear link between the vector size and the rate of increase of the T-number and by extension the T-total.
I then tried using a 10 × 10 grid. I used a 4 × 4 vector, a 5× 5 vector and a 6 × 6 vector. The 4 × 4 vector came to a change of 36 by the T-number;
The 5 × 5 vector came to a change of 45 by the T-number;
The 6 × 6 vector came to a change of 54 by the T-number.
This also showed that the vectors are increasing, but this time the vectors were increasing by nine each time.
I then predicted that the 4 × 4 vector, a 5× 5 vector and a 6 × 6 vector on 11 × 11 grid would come to a change of 40 on a 4 × 4 vector;
A change of 50 on a 5 × 5 vector;
A change of 60 on a 6 × 6 vector.
These predictions were proved correct when I tried them. I worked them out by looking at the differences in the vectors on a 9 × 9 grid and a 10 × 10 grid.
I could also predict what the 7 × 7 vector and 8 × 8 vector would be on each grid by using the sequence increase, between 9× 9 grid, 10 × 10 grid and 11 × 11 grid which also proved to work when put to the test.
For a 4 × 5 vector on any grid the sequence would change.
More vectors
0 × 10 Grid size
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The vector dimensions on this Grid are 4 - and 3. It is possible to solve these vectors with the following equation;
Equation 1
Where: ~ G = grid size
TN = T-number
T-total = (TN + 4 + 3G + (TN - G + 4 + 3G) + (TN - 2G + 4 + 3G)
+ (TN - 2G + 1 + 4 + 3G + (TN - 2G - 1 +4 + 3G)
T-total = 5TN + 8G +20
T-total = 5TN + 8 × 10 + 20
T-total = 5TN + 100
Generalisation of equation 1
Where: ~ G = grid size
TN = T-number
A = 4 - (dimensions of vector)
B = 3| (dimensions of vector)
T-total = (TN + A + BG) + (TN - B +A + BG) + (TN - 2B + A + BG)
+ (TN - 2B + 1 + A + BG) + (TN - 2B - 1 +A + BG)
T-total = 5TN - 7B + 5A + 5BG
T-total = 5(TN - 1 G + A + BG)
Rotating vectors
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A vector with the T-shape on its side with the T-number facing right
Equation
Where: ~ G = grid size
TN = T-number
A = 4 - (dimensions of vector)
B = 3| (dimensions of vector)
T-total = (TN + 4 + 3G) + (TN - 1 + 4 +3G) + (TN - 2 + 4 + 3G)
+ (TN - 2 + G + 4 + 3G) + (TN - 2 - G + 4 +3G)
T-total = (TN + A + BG) + (TN - 1 + A + BG) + (TN - 2 + A + BG)
+ (TN - 2 + G + A + BG) + (TN - 2 - G + A + BG)
T-total = 5TN + 13 +15G
T-total = 5TN + 5A + 5B + 15G - 7
5 (TN + A + B + 3G - 1 )
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A vector with the T-shape on its side with the T-number facing left
Equation
Where: ~ G = grid size
TN = T-number
A = 4 - (dimensions of vector)
B = 3| (dimensions of vector)
T-total = (TN + 4 + 3G) + (TN + 1 + 4 + 3G) + (TN + 2 + 4 3G)
+ (TN + 2 - G + 4 + 3G) + (TN + 2 + G + 4 +3G)
T-total = (TN + A + BG) + (TN + 1 + A + BG) + (TN + 2 + A + BG)
+ (TN + 2 - G + A + BG) + (TN + 2 + G +A + BG)
T-total = 5TN + 5A + 5BG + 7
5 (TN + A + BG +1 )
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A vector with the T-shape upside down.
Equation
Where: ~ G = grid size
TN = T-number
A = 4 - (dimensions of vector)
B = 3| (dimensions of vector)
T-total = (TN + 4 + 3G) + (TN + G + 4 + 3G) + (TN + 2G + 4 + 3G)
+ (TN + 2G + 1 + 4 + 3G) + (TN + 2G - 1 + 4 + 3G)
T-total = (TN + A + BG) + (TN + G + A + BG) + (TN + 2G + A + BG)
+ (TN + 2G + 1 + A + BG) + (TN + 2G - 1 + A + BG)
T-total = 5TN + 5A + 5BG + 7G
5 (TN + A +BG + 1 G )
CONCLUSION
Conclusion
I tested my rules by experimenting with a variety of examples. I found the rule by experimenting with different grid sizes and different numbers inside my various grids.
I have found out that there is a definite relationship between the T-number and the T-total. I found that the T-total can be found by using several different formulae, depending on the size of the grid which the T-shape is in and the orientation of the T-shape.
I have made a variety of tables to test the formulae and to show the different orientations and the variations between the different sizes of grid.
The main equation that I have used for parts one and two is
T-total = 5N - 7G. This equation shows the relationship between the T-totals, the T-number and the grid size.
I have also used different transformations and orientations. It was also possible to create different equation for different orientations of the T-shape.
Then I tried using vectors which was more difficult. There was a formula for the vectors which I could find, I could find a link between vectors size and the extent to which T-number and T-total changed with a transformation.
