# The object of this coursework is to find the relationship between the total value and the positioning of a T-shape on a number grid. I will investigate the following: What happens as the T-shape moves down the grid.

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Introduction

Introduction

The object of this coursework is to find the relationship between the total value and the positioning of a T-shape on a number grid. I will investigate the following:

- What happens as the T-shape moves down the grid.
- What happens as the T-shape moves across the grid.
- What is the smallest possible value for the T-number.
- What is the largest possible value for the T-number.
- What happens when the T-shape is rotated 90°, 180° and 270°

Starting on a 10x10 grid, I will draw my T-shape and investigate the 5 tasks stated above. Then I will move on to do the same on a 9x9 and 8x8 grid to see if I can find a relationship. On each grid, once the T-shape is rotated, I will move the T across and down for each angle it’s rotated at, all the results shall be listed into tables. After this I will try to find the smallest and largest possible value for the T-shape. Finally I will compare my results and see if there are any patterns which will hopefully supply me with an algebraic formula that can be used on an ‘N’ sized grid.

Here is my 10x10 grid, I will draw my T-shape and rotate it 90°, 180° and 270°, each time it is rotated I will move the shape down and across the grid 3 times and record the results into a table.

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 |

1 | 2 | 3 |

12 | ||

22 | ||

N-21 | N-20 | N-19 |

N-10 | ||

N |

T = N + N-10 + N-20 + N-21 + N-19

T = 5N = 70

This is a T-Shape on a 10 by 10 number grid. The total of the numbers inside the grid all add up to produce a T-Total. In this grid you add the numbers 1 + 2 + 3 + 12 + 13 which = 40. The N number is 22.

1 | 2 | 3 |

12 | ||

N 22 |

T = Total number in the shape

N = The number at the bottom

of the T (as shown)

Plan

1)

Middle

The total value of this T-Shape is 55 (4 + 5 + 6 + 15 + 25 = 55). I will now analyse the results of the T-Shape.

T-Number | T-Total |

22 | 40 |

23 | 45 |

24 | 50 |

25 | 55 |

You can see there is a pattern by looking at this table. When N goes up 1 the T goes up by 5. I will do one more T-Shape and test a prediction. I predict that when the N number goes up by 1 (so it will be 26,) the T-Total will go up by 5 (and be a total of 60.)

The total value of this T-Shape is 60 (5 + 6 + 7 + 16 + 26 = 60) and the N number is 26. This proves my prediction and the pattern correct.

Rule

I am now going to try and find a rule for a 10 by 10 grid to get from N to T (in terms of N.) This should give me a rule for which the only variable is N.

Value of T = 1 + 2 + 3 + 12 + 22 = 40 Value of N = 22

Value of T = N + N-10 + N-19 + N-20 + N-21 so

RULE: Value of T = 5N-70

To prove my rule I am now going to test it on the T-Shape above (where N = 22.)

5 x 22 - 70 = 40

I have now proved the relationship between the T and N numbers is N x 5 - 70. I am now going to prove the formula works by testing it on a T-Shape I have not yet used and check if the first result wasn’t just a guess.

Value of T = 7 + 8 + 9 + 18 + 28 = 70 Value of N = 28

5 x 28 - 70 = 70

These two graphs have proved that the formula works, and the formula is the same as the one for moving down the 10 by 10 grid. This means that if we know the number of the N, then we can work the out the total of the T on a 10 by 10 grid.

Conclusion

I will now use the rule 2G + 2 to see if the rule works. I will use the G (Grid) as 9 and N as 20.

So to get from G = 9 to N = 20 using the rule you times 9 by 2 to get 18, then add on 2 which gives 20. The rule works, however, I will still try the rule again in the 8 by 8 grid to prove it correct.

8 by 8 grid

I will first start with my T-Shape being stationary like the first T-Shape on the 10 by 10 and 9 by 9 grid.

The total of this T-Shape is 34 (1 + 2 + 3 + 10 + 18 = 34) and the N number is 18.

I will now rotate the T-Shape by 90° clockwise:

The total of this T-Shape is 52 (3 + 9 + 10 + 11 + 19 = 52) and the N number is 9.

I will rotate the T-Shape another 90° clockwise, and the T-Shape will have rotated 180° clockwise altogether.

The total of this T-Shape is 66 (2 + 10 + 17 + 18 + 19 = ) and the N number is 2.

I will finally rotate the T-Shape one more time at 90°, totalling a rotation of 270°.

The total of this T-Shape is 48 (1 + 9 + 10 + 11 + 17 = 48) and the N number is 11.

1 | 2 | 3 |

9 | 10 | 11 |

17 | 18 | 19 |

I now know that the smallest value on a 8 by 8 grid is 34, where the N number is 18 and the T-Shape is stationary.

I have now proved my prediction correct in which the T-Shape will be the smallest value when it is stationary on a 8 by 8 grid. This is the same for both the 10 by 10 and 9 by 9 grid.

I will now try and prove the rule for the last time on the 8 by 8 grid. I will use the G (Grid) as 8 and the N as 18.

So to get from G = 8 to N = 18 you times 8 by 2 which equals 16, and then add 2 which equals 18. This proves the rule is correct for the smallest value on a T-Shape.

Question 5) Largest Value of the T-Shape in the Number Grid

10 by 10 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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