# T-Shapes Coursework

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Introduction

T-Shapes Coursework

This coursework is based upon a grid of numbers in which a ‘T’ shape is set out with 3 numbers at the top row and 3 down the middle like this:

It is a shape with 5 boxes.

We can call the highlighted number the, t-number. We can also call the sum of all the numbers inside the t-shape the, t-total. This coursework investigates the relationships between the t-number and t-total, including different variables like various grid sizes and t-shape rotations.

Now, to begin with, I will have a 9x9 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 |

I have taken the t-shape, which consists of 40, 41, 42, 50 and 59. Adding these up I get 232.

If I move the t-shape one space left I get 39, 40, 41, 49 and 58. The t-total is 227. Notice the t-total has decreased by five.

If I move the t-shape two spaces right I get 42, 43, 44, 52 and 61. The t-total is 242, increasing the original by ten.

I moved the t one step right from the original, and predicted that the t-total from 232 would increase by five. 60+51+41+42+43=237. My prediction was correct

If I have a t-total, from that I can work out another t-total if the new t-shape is directly to the left or right of the first. I know that moving a t-shape left or right I increase or decrease the first t-total by 5 or more. This number could help me later.

Middle

T-total = 323

To move this one step right I would have a t-shape with the numbers 53, 62, 70, 71 and 72. I think the t-total is 323 + 5 = 328.

53 + 62 + 70 + 71 + 72 = 328. I was right.

To move this step one step left, I would have a t-shape consisting of 51, 60, 68, 69, and 70. I believe this t-total will be 323 – 5 = 318.

51 + 60 + 68 + 69 + 70 = 318.

I am sure that the rule of moving any rotated t-shape forward or backwards you just increase or decrease the t-total by 5 or more and it effectively works.

For the next section of working out the formula, I deducted the numbers, one by one, in the t-shape from the t-number.

4 + 11 + 12 + 13 + 22 = 62

This is the t-shape with the calculations shown in it to represent which box represents which calculation.

11 – 12 = -1

11 – 4 = 7

11 – 13 = -2

11 – 22 = -11

This shows the

t-shape with the

Results in it.

On the first t-shape I added the answers in the box up. The original answers were all minuses but on this t-shape we are left with pluses and minuses. I will still add these up I know that the answer will not be the same as before.

1 + 2 + -7 + 11 = 7

I remember that when I did turn the t-shape around and moved it left and right it remained its original rule of increasing or decreasing by five. I am going to still keep the 5 in the original equation but substitute the old –63 for the new 7.

n x 5 + 7 = t

Conclusion

I worked that out by using the information above me.

I thought for a while and came up with an idea to incorporate the grid size into the formula. It is:

(n x 5) -(+) (7 x g) = t

The g stands for grid and for grid you would look at whatever grid it is e.g. 3x3

and use the 3 as the g. Note: Begin as normal through the formula but when you encounter the brackets work them out separately and the answer from the brackets will tell you what to add or minus it by.

The 4 general formulas are:

(t x 5) - (7 x g) = t

(t x 5) + (7 x g) = t

t x 5 + 7 = t

t x 5 - 7 = t

Here they are demonstrated with a random grid. I

I randomly chose a 4x4 grid.

1 | 2 | 3 | 4 |

5 | 6 | 7 | 8 |

9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 |

(10 x 5) - (7 x 4) = 22

First I multiplied 10 by 5 which gave me 50,

then separately I worked out 7 x 4, that is

28. Finally I took 28 away from 50 which

gave me the answer.

1 + 2 + 3 + 6 + 10 = 22

(15 x 5) - (7 x 4) = 75 - 28 = 47

6 + 7 + 8 + 11 + 15 = 47

(2 x 5) + (7 x 4) = 10 + 28 = 38

6 + 10 + 11 + 9 + 2 = 28

(6 x 5) + (7 x 4) = 30 + 28 = 58

6 + 10 + 13 + 14 + 15 = 58

6 x 5 + 7 = 37

6 + 7 + 8 + 4 + 12 = 37

9 x 5 + 7 = 52

9 + 10 + 11 + 7 + 15 = 52

7 x 5 - 7 = 28

1 + 5 + 6 + 7 + 9 = 28

12 x 5 - 7 = 53

6 + 10 + 11 + 12 + 14 = 53

I have gained all possible formulas that can link you from a t-number to a t-total in any 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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