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Introduction

Zakwan Ahmed

10 A

Maths Coursework

Mr Uddin

T-total Investigation

Aim: Find a rule connecting the T-total and the T- number.

Investigation- Part one:

(A,B,C)

Investigation- Part Two:

Investigation- Part Three:

3

 1 2 3 12 2 22

I placed the 3by2 T on the beginning of the 10by10 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

I then moved the T along the 10by10 grid 3 times in order to find a pattern. These are the 3 positions in which I put my 3by2 T on:

Position 1:                                          Position 2:

 1 2 3 12 T-no 22 2 3 4 13 T-no 23

Position 3:

 3 4 5 14 T-no 24

The bottom number of the T is always the T-number. I added all the numbers inside the T- shape for each position and this gave me the T- total for e.g. for the first position 1 + 2+ 3+12+22 =40 and this is the T-total for the first position. I then put the T-no of each position in one column and the T-total of each position in another column to see the difference between the T- no’s and the T-totals.

T-no                        T-total

22                             40       position 1 +1                              + 5

23                             45       position 2 +1                              + 5

24                             50         position 3 I saw that the difference between the T-no’s is 1 and the difference between the T-totals is 5. The first part of my formula is therefore 5T. It is 5T because that is how many numbers inside the T. I multiplied the difference between the T-total which is 5 by the T-no. Underneath is my working out:

1. x  5 =110
2.  x  5=115
3. x  5=120

I then took away the T-total numbers so that I could find the difference between the T-no and the numbers in each of the 4 squares inside the T.

110- 40 =70

115- 45 =70

120- 50 = 70

Middle

T-G

T

I then added up the G terms together to get 7G which is the final expression.

(2G +1) + (2G) + (2G +-1) + (G) = 7G

So therefore T = 5T – 7G

I will now test this formula on a 10by10 grid to see it works:

Checking:

(5 x 22) – (7 x 10) = 40

110    -     70     = 40

From 5t come 5 which is multiplied by the T-no from my first position which is 22.

7 comes from 7G and is multiplied by the grid size which is 10.

By checking out this formula I can see that it works on a 10by10 grid.

9 by 9

I can now explain the expression using algebra.

G = 9

 T-2G+1 T – 2G T- 2G-1 T- G T

I added up all the G terms from inside the T to get 7G the final expression.

(2G+1) + (2G) + (2g-1) + (G) = 7G

T = 5T – 7G

I will now test out this formula to see if it works on a 9by9 grid.

Checking:

(5 x 20) – (7 x 9) = 37

100    -     63    = 37

20 + 11 + 3 +2 + 1 = 37

By checking this formula on 9by9 grid I can see that it works.

8 by 8

I can now explain the expressions using algebra.

G = 8

T = T-no

 T-2G-1 T-2G T-2G+1 T-G T

I added up all the G terms from inside the T and I got 7g as the final expression.

(2G – 1) + (2G) + (2G +1) + (G) = 7G

So the formula for T is: T = 5T – 7G

I will now test this formula out to see if it works on an 8by8 grid.

(5 x 18) – (7 x 8) = 34

90    -      56 = 34

1 + 2 + 3 + 10 + 18 = 34          This formula is correct. This formula works on an 8by8 grid.

ROTATION

First I rotated the 3by2 T  by 90o on a 10by10 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Position 1                Position 2

 3 11 12 13 23 4 12 13 14 24

Position 3                 Position 4

 5 13 14 15 25 6 14 15 16 26

I drew a table to write drown my results

 T-NUMBER T-TOTAL 11 62 12 67 13 72 14 77
 N-8 N N+1 N+2 N+12

Formula: 5N+7

Test The Formula:

I picked any number from the 10by10 grid e.g. 34 and  used the formula

5 x 34 +7 = 177

For me to make sure it was right I counted all the numbers in the T to check if it totaled to 177.

 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

And it was right so this meant that the formula works

I rotated the 3by2 T  by 90o on 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

Position 1                Position 2

 4 11 12 13 22 3 10 11 12 21

Position 3                 Position 4

 6 13 14 15 24 5 12 13 14 23
 T-NUMBER T-TOTAL 10 57 11 62 12 67 13 72 N-7 N N+1 N+2 N+11

Formula: 5N+7

Test The Formula:

I picked any number from the 9 by 9 grid e.g. 30 and used the formula

5 x 30 +7 = 157

For me to make sure it was right I counted all the numbers in the T to check if it totaled to 157.

 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

Conclusion T-no                          T-total

21                               48       position 1 1                                  7

22                               55       position 2 1                                  7

23                               62       position 3 I saw that the T-no is going up in 1 and the T-total is going up in 7. Therefore 7 is the first part of my formula. Once again I multiplied the difference of the T-total which is 7 by the T-no. I then took away the T-total from my answer.

This is my working out:

21 x 7 = 147         147 – 48 = 99

22 x 7 = 154         154 – 55 = 99

23 x 7 = 161         161- 62 = 99

My formula for a 5by2 T on a 9by9 grid is now 7T – 99.

I can put expressions inside the T to check if my formula is correct.

 T - 20 T - 19 T - 18 T-17 T -16 T – 9 T

I saw that the centre column is going up in 9’s because of the grid size. I added all the expressions from inside the T. This is what I did:

T-total = T-20 + T – 19 + T – 18 + T – 18 + T – 16 + T – 9 + T

= 7T– 99

I will now test out this formula to find out if it is correct.

T = 51     T-total = 7 x 51 – 99 = 258

31 + 32 + 33 + 34 + 35 + 42 + 51 = 258            This formula is correct. I can now use algebra to explain the expressions.

G = 9 the grid size

 T-2G-2 T-2G-1 T – 2G T-2G+1 T -2G+2 T – G T

I added up all the G terms inside the T and I got 11G as the last part of my formula.

(2G -2) + (2G-1) + (2G) + (2G+1) + (2g+2) + (G) = 11G

So T = 7T – 11G

I will now check to see if this formula works on a 9by9 grid.

(7 x 21) – (11 x 9) = 48

147    -      99     = 48

1 + 2 + 3 + 4 + 5 + 12 + 21 = 48           This formula is correct. This formula is correct and it works on a 9by9 grid.

I now know the general formula for a 3by2 T on any size grid and a 5by2 T on any size 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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