# 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

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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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