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• Level: GCSE
• Subject: Maths
• Word count: 2195

# For this piece of GCSE maths coursework, I am aiming to find out the relationship between the T-Number and T-Total. To do this I am first going to use the 9 by 9 grid. After that, I am going to try out my rule by using different grids, 8 by 8, 6 by 6 etc

Extracts from this document...

Introduction

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For this piece of GCSE maths coursework, I am aiming to find out the relationship between the T-Number and T-Total. To do this I am first going to use the 9 by 9 grid. After that, I am going to try out my rule by using different grids, 8 by 8, 6 by 6 etc

Now I am going to do my first investigation, which is the 9 by 9 grid, and for this,             I am trying to find out the T-number and T-total.

 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
 47 48 49 57 66 46 47 48 56 65

N=65                                                         N=66

T=46+47+48+56+65=262                        T=47+48+49+57+66=267

 48 49 50 58 20 49 50 51 59 68

N=67N=68

T=49+50+51+59+68=277                                           T=48+49+50+58+67=272

 51 52 53 61 70
 50 51 52 60 69

N=69                                                            N=70                          T=50+51+52+60+69=282                       T=51+52+53+61+70=287

Key: - N= T-Number

T= T-Total

Now I am going to see the pattern that I made with the T-Number and the T-Total

 T-Number T-Total Pattern 65 262 5 66 267 5 67 272 5 68 277 5 69 282 5 70 287 5

I have found out that every time you move to the right, you add 5 and every time you move to the left, you subtract 5. I know this just by looking at the pattern grid above.

Now I am going to put my result into tables because I want to find the rule and the relation between the T-Total and the T-Numbers. In addition, my work would be neat, easier to read and it gives a good presentation of my work. It is also easy for me to conclude the results.

 T-Number 65 66 67 68 69 70 T-Total 262 267 272 277 282 287

I am going up by one

I am going up by five

Middle

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Rule in Algebra:-

 N-19 N-18 N-17 N-9 N 74-55 74-56 74-57 74-65 74=N

My aim was to find the T-Number for each of the numbers above. The first thing that I did was to find out the T-Number, which is eventually at the bottom of the T-shape. Then I subtracted the T-Number by the other numbers inside the T-Shape, for example, I subtracted 65 from 74, which gave the answers N-9+N-18+N-17+N-9+N, and this is how I found the T-Number for the entire number grid. I have noticed that the middle column is going up in nine and I think this is because of the size of the grid. I think that in every other grids which I would do the T-Number is going to gain on the grid size and go up in multiples e.g. 9 by 9 grid, the T-Number is going to go up in nine’s. I concentrated on the number 74 because it was the T-Number.                                                                                                                                       This is the formula for the T-Total= N-19+N-18+N-17+N-9+N= 5N-63

Now I am going to test my theory to see if it works with other T-Numbers.

 1 2 3 11 20 16 17 18 26 35

Key: - N= T-Number

T= T-Total

T =1+2+3+11+20=37                                          T=16+17+18+26+35=112                                                                   T-Total= N-19+N-18+N-17+N-9+N=5N-63      T-Total=N-19+N-18+N-17+N-9+N=5N-63 T=5x20-63=37                                                    T= 5x35-63=112

I have found out the formula 5N-63 works anywhere in the grid, I know this because I tested the formula two times for different T-Numbers and the formula works.                                                                                                                             The full formula for the nine by nine grid is: - T=5N-63

Now I am going to find out by using different grids the rule in algebra, I am doing this because I want to find out does the grid size matter and if it does how. I am going to do an eight by eight grid to start.

 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
 18 19 20 27 35 N-17 N-16 N-15 N-8 N

The formula for the T-Total is:-

T-Total = N-17+N-16+N-15+N-8+N=5N-56

T= 5x35-56=119

Now I am going to test this formula to see if it works

Key: - N= T-Number

T= T-Total

 9 10 11 18 26 30 31 32 39 47

T =9+10+11+18+26=74                                       T=30+31+32+39+47=179                                                                   T-Total= N-17+N-16+N-15+N-8+N=5N-56       T-Total= N-17+N-16+N-15+N-8+N=5N-56                                                          T= 5x26-56=74                                                    T= 5x47-56=179

I have found out that the formula 5N-56 works anywhere in 8 by 8 grid, I have tested the formula two times and this proves that my result does not have in faults in it.                                                                                                                                  The formula for the 8 by 8 grid is: - T=5N-56

Now I am going to do a seven by seven grid to find out the Algebraic rule

 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
 N-15 N-14 N-13 N-7 N 1 2 3 9 16

The formula for the T-Total is:-

T-Total = N-15+N-14+N-13+N-7+N=5N-49

T= 5x16-49=31

Now I am going to test this formula to see if it works

Key: - N= T-Number

T= T-Total

 11 12 13 19 26 17 18 19 25 32

Conclusion

5N-7x6

5N-42

7 by 7

5N- 7x7

5N-49

8 by 8

5N- 7x8

5N-56

9 by 9

5N- 7x9

5N-63

10 by 10

5N- 7x10

5N-70

Overall, the general rule for the T-Total is 5N-7G, where the N is T-Number and the G is grid number.

I recognised that if I multiply the number of grids by 7 I would get the T-Total.

General Rule: 5N-7G

Now I am going to test my general rule with two examples and then prove that I was right:-

 N-2G-1 N-2G N-2G+1 N-G N

General rule in algebraic form

now I will add the numbers to find the General rule, N-2G-1+N-2G+N-2G+1+N-G+N                                     I got the N-G because the middle square is the same as the grid number.

Using my rule that I found out I predict that if the grid size is 5 by 5 the rule is: 5N-35                  Now I am going to prove it 5N- 7G is 5N-7x5, so the rule is 5N-35.

Now I am going to predict and prove it for a 4 by 4 grid just to make sure that my general rule is correct. Therefore, the rule for a 4 by 4 grid is going to be 5N-28. Now I am going to prove it; 5N-7G is 5N-7x4, so the rule is 5N-28.

I have predicted and proved it twice to see if my general rule is 5N-7G and by my prediction, I have proved that my rule is correct.

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