• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
17. 17
17
18. 18
18
• Level: GCSE
• Subject: Maths
• Word count: 4075

# T -Totals. From this 9*9 grid I will collect a number of T Shapes with T Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

Extracts from this document...

Introduction

Sohaib Mohammad 11H -- Maths Coursework -- T Totals

T –Totals

For this coursework I have been given a task which is to investigate the relationship between the T-total and the T-number. To start my investigation I will do this by working on a 9*9 grid. From this 9*9 grid I will collect a number of T –Shapes with T –Numbers after collecting these I will put them into a table and will investigate to find any relationship between them.

 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

Definition of the T –Total and T –Numbers

Consider the numbers in the T –Shaped figure shown in the grid by thick lines, redrawn underneath separately for clarity. The integer at the bottom of the T –Shape is 20 this I known as the T –number for the T –Shape:

 1 2 3 11 20

The T –Total is defined as the sum of all the integers shown in the above T –Shaped figure. For example the T –Total for the T –Shape above = 1+2+3+11+20 = 37

I firstly begin the investigation by starting of with a T shape, the following shows 4 T –Shapes collected from the first row:

 2 3 4 12 21

T –Number = 21

T –Shape = 2+3+4+12+21 = T –Total = 42

 3 4 5 13 22

T –Number = 22

T –Shape = 3+4+5+13+22 = T –Total = 47

 4 5 6 14 23

T –Number = 23

T –Total = 4+5+6+14+23 = 52

The next part shows 4 more T –Shapes collected from a different position on the grid, these T –Shapes are from the second row:

 10 11 12 20 29

T –Number = 29

T –Total = 82

 11 12 13 21 30

T –Number = 30

T –Total = 87

 12 13 14 22 31

T –Number = 31

T –Total = 92

 13 14 15 23 32

T –Number = 32

T –Total = 97

Middle

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

I’ve decided to take my first four T-Shapes from the 4th column and the 3rd row so I will start from the T –Number 18:

 4 5 6 12 19
 3 4 5 11 18

T –Number = 18                                            T –Number = 19

T –Total = 3+4+5+11+18 = 41                     T –Total = 4+5+6+12+19 = 46

 5 6 7 13 20 6 7 8 14 21

T –Number = 20                                      T –Number = 21

T –Total = 5+6+7+13+20 = 51               T –Total = 6+7+8+14+21 = 56

My next selection of four T –Shapes is from the 2nd column and 6th row so the starting T –Number would be 37:

 22 23 24 30 37 23 24 25 31 38

T –Number = 37                                                    T –Number = 38

T –Total = 22+23+24+30+37 = 136                     T –Total = 23+24+25+31+38 = 141

 24 25 26 32 39 25 26 27 33 40

T –Number = 39                                                    T –Number = 40

T –Total = 24+25+26+32+39 = 146                       T –Total = 25+26+27+33+40 = 151

The next step is to put the results into a table in order to see if there is any pattern amongst the results jus like in the 8*8 grid:

 T -Number T -Total 18 41 19 46 20 51 21 56
 T -Number T -Total 37 136 38 141 39 146 40 151

From both these tables I can see a pattern and that is that the T –Total increases by 5 every time the T –Number increases.

18,  19, 20, 21                   37,  38, 39, 40

5    5    5                            5    5    5

From this I am able to calculate a formula for this grid jus like for the 9*9 and 8*8 grid, to do this once again I will represent the numbers in one of the T-Shapes as “n”:

 n-15 n-14 n-13 n-7 n 23 24 25 31 38

Now to find out the formula I simply just add the T-Shape up:

T –Total = n+n-7+n-14+n-15+n-13 = 5n – 49 in order to check my equation I will once again pick a T –Number out of this 7*7 grid and then will substitute the T –Number into the equation as “n”:

 25 26 27 33 40

T –Number = 40

T –Total = 25+26+27+33+40 = 151

The next step is to substitute 40 into 5n – 49 which equals T-Total = 5(40) – 49

= 200 – 49

= 151

I was correct and the equation for the 7*7 grid is 5n – 49. I will investigate on one more grid which is the 6*6 after doing so it will help me enable to find a relationship between the grids so doing a 6*6 grid will only help me get a more accurate answer to the relationship between the grid sizes.

 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

I’ve decided to start from the T –Number 21 which is in the 3rd column and 4th row:

 8 9 10 15 21 9 10 11 16 22

T –Number = 21                                                              T –Number = 22

T –Total = 8+9+10+15+21 = 63                                      T –Total = 9+10+11+16+22 = 68

 10 11 12 17 23 11 12 13 18 24

T –Number = 23                                                      T –Number = 24

T –Total = 10+11+12+17+23 = 73                          T –Total = 11+12+13+18+24 = 78

The next four T –Shapes will be collected from the T –Number 34 which is at the 2nd column and 6th row:

 19 20 21 26 32 20 21 22 27 33

T –Number = 32                                                    T –Number = 33

T –Total = 19+20+21+26+32 = 118                      T –Total = 20+21+22+27+33 = 123

 21 22 23 28 34 22 23 24 29 35

T –Number = 34                                               T –Number = 35

T –Total = 21+22+23+28+34 = 128                 T –Total = 22+23+24+29+35 = 133

The next step is to put the results into tables:

 T -Number T -Total 21 63 22 68 23 73 24 78
 T -Number T -Total 32 118 33 123 34 128 35 133

From both these tables I can see a pattern and that is that the T –Total increases by 5 every time the T –Number increases.

21,  22, 23 ,24                        32,  33, 34, 35

5    5    5                                  5    5    5

From this I am able to calculate a formula for this grid jus like for the 9*9, 8*8 and 7*7 grid, to do this once again I will represent the numbers in one of the T-Shapes as “n”:

 n-13 n-12 n-11 n-6 n 19 20 21 26 32

Now to find out the formula I simply just add the T-Shape up:

T –Total = n+n-6+n-12+n-11+n-12  = 5n – 42 in order to check if this equation is right I will jus use the T –Number from the T- Shape above and will substitute it into the equation 5n – 42.

 19 20 21 26 32

T –Number = 32

T –Total = 19+20+21+26+32 = 118

The next step is substituting 32 into 5n – 42 equals = T –Total = 5(32) - 42

= 160 - 42

= 118

The equation is correct as I have managed to find the T –Total using the equation.

After collecting the equations for the four grid sizes the next step is to find and equation using the formulae of the grid sizes to achieve and equation which will enable us to find any T –Total from any grid size.

To do this I will begin by creating a table which will include all the equations from the four grid sizes:

 Order of Grid T -Number Equations 6 N 5N - 42 7 N 5N - 49 8 N 5N - 56 9 N 5N - 63

Looking at the table above, I can see that as the grid size increases by one the equation increases by 7 so therefore the grid size is in proportion to 7 which then helps to find an equation that will find any T –Total on any grid size:

So the equation is 5n – 7w, where w represents the grid size. In order to check if the equation is correct I will substitute the T –Shape below:

 19 20 21 26 32

Conclusion

In a 9*9 grid the equation equals 5n -63 and the rotated equation equals 5n + 7 I can notice the difference between the equations which is 70. In an 8*8 the equation equals 5n -56 and the rotated equation is 5n + 7 the difference here is of 63 and in a 7*7 grid the equation is 5n – 49 and the difference is 56:

By using the difference between the equations and then comparing them with each other I am able to formulate an equation for the T –Total of a rotated T –Shape of a grid size

Grid Size:   7           8          9

56         63       70

1. 7

From the above we can make the 7 a factor of the grid representing it as 7 as it is in proportion with the grid size because as the grid size increases by one the difference of seven also increases by seven so there fore leaving with an equation of 7(w +1) where w represents the grid size, this equation when added with the equation of a grid size helps to find the rotated  T –Total of a T –Shape for example on a 9*9 grid I would like to find the rotated T –Total of the T –Number 23 to do this I simply substitute the T –Number into the equation below along with the grid size:

5n – 63 + 7 (w+1) therefore Rotated T –Total = 5(23) – 63 + 7(9+1)

= 115 – 63 + 70

= 52 + 70

= 122

 16 23 24 25 34
 4 5 6 14 23

T –Number = 23

T –Total = 4+5+6+14+23 = 52

T –Number = 23

T –Total = 23+24+25+16+34 = 122

As you can see the calculation is correct for finding a rotated T –Total.

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

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE T-Total essays

1. ## T-Total Maths coursework

any grid size be it a 9 by 9 grid or a 65 by 65 grid. I also worked out that you always multiply the T-number 5 and to get the number which you have to subtract all you do is multiply the grid size by 9.

2. ## Urban Settlements have much greater accessibility than rural settlements. Is this so?

up from the residents of 'South Downs' - An OAP's residential area. This explains why there are hardly any pedestrians in most areas of South Darenth. This also explains the lack of traffic. South Downs has its own shop and has its own entertainment.

1. ## T-Shapes Coursework

In fact, if we take one constant Middle Number, 15, from each of the above tables, we get the following: Middle Number Tail Length (l) Sum of Wing Sum of Tail Total Sum (Wing + Tail) 15 1 45 25 70 15 2 45 60 105 15 3 45 105

2. ## T-Total. I will take steps to find formulae for changing the position of the ...

This is because I still want to find out the T total. Moving to the right: As each T total increases by 5 as it is moved along, I will have to times 5 by the amount of squares I will be moving along.

1. ## T-Shapes Coursework

By moving the T-Shape on the vertical, each number increases by 9 each time, because it is a 9x9 grid. This means that in the horizontal if the T-number went up by 1 each time, in the vertical T-Shapes, the T-Number would go up by 1 x 9 = 9

2. ## T totals. In this investigation I aim to find out relationships between grid sizes ...

are found by the following method; As there are 5 numbers in the T-Shape we need 5 lots of 20, the number adjacent to 20 is 11 which is 9 less than 20, the other numbers in the T-Shape are 1,2 & 3 which are 19, 18 & 17 less than 20.

1. ## T totals - translations and rotations

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

2. ## T-Totals. We have a grid nine by nine with the numbers starting from 1 ...

Here we are doing what we did in the last section but finding out more about the grid size and what it is capable of doing. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to