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  • Level: GCSE
  • Subject: Maths
  • Word count: 4292

T-total Investigation

Extracts from this document...

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

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 12

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 22

I placed the 3by2 T on the beginning of the 10by10 grid.

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 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:

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 T-no                                      image00.png

 22

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T-no image00.png

 23

Position 3:

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T-no image00.png

 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 1image00.png

+1                              + 5                        

       23                             45       position 2image00.png

+1                              + 5

      24                             50         position 3image00.png

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

...read more.

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.image01.png

This formula works on an 8by8 grid.

ROTATION

First I rotated the 3by2 T  by 90o on a 10by10 grid

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Position 1                Position 2

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Position 3                 Position 4

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I drew a table to write drown my results

T-NUMBER

T-TOTAL

11

62

12

67

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72

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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.

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I added: 34+35+36+26+46=177

And it was right so this meant that the formula works

I rotated the 3by2 T  by 90o on a 9 by 9 grid

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Position 1                Position 2

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Position 3                 Position 4

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T-NUMBER

T-TOTAL

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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.

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...read more.

Conclusion

image06.png

T-no                          T-total

             21                               48       position 1image00.png

1                                  7    

             22                               55       position 2 image00.png

1                                  7

             23                               62       position 3image00.png

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.image01.png

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.image01.png

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.

...read more.

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