# T-Shape Investigation.

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Introduction

Nathaniel Cummings 10A Math Coursework Mrs. Young

## T-Shape Investigation

## Introduction

I was told to make a graph which was 6 by 3 grids like this:

1 | 2 | 3 | 4 | 5 | 6 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

## Then I had to put in a T-shape

1 | 2 | 3 | 4 | 5 | 6 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

From using the T-shapes I have to translate (slide) the T-shape to different positions on the grid. I then have to investigate the relationship between the t-number and T-total. By translating the T-shape I should be able to find a rule to predict the T-total.

The T-total is the numbers in the T-shape added up. In this case it is 1+2+3+8+14=28

A T-Shape is the shape which is made out the numbers in the grid.

The T-Number is at the bottom of each T-Shape. In this case the T-Number is 8.

To try to find a rule I am going to “Break down this problem”. I am going to draw a 5 by 3 grid the 6 by 3 and then a 7 by 3 grid. This will help me to see if there are any patterns which are the same between the grids if there are changed. For each grid I make I will translate the T-Shape to also see if there is any patterns.

5 by 3 grid

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

My first grid is a 5 by 3 grid. I have placed two T-shapes inside this grid. I am going to see I can find any patterns which are the same in these T-shapes.

I am going to see if I can see any patterns which are the same from looking at the T-Totals and the T-Numbers

The first T-shape (blue outline) T- Total is 1+2+3+7+12 = 25

The T- Number of this T-shape is 12

Middle

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 |

5 * 65 – 7*10= 255

I will add up the numbers in the T- Total to prove that my rule is correct.

44+45+46+55+65= 225

My rule is correct!

First Extension

Moving the T-Shape 90 degrees to the right

Here I have moved my T-Shape 90 degrees to the right. I am going to investigate this T-Shape in the same why I did the last.

5 by 3 grid

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

My first position is in blue. The T- Number is 6. The T- Numbers will be in this place. This is the same as the last T-Shapes.

The first T-shape (blue outline) T- Total is 6+7+8+3+13 = 37

The T- Number of this T-shape is 6

The second T-shape (red outline) T- Total is 7+8+9+4+14 = 42

The T- Number of this T-shape is 7

The third T-shape (brown outline) T- Total is 8+9+10+5+15 = 47

The T- Number of this T-shape is 8

I am now going to make a table to put these results in a table to make it easier to read.

T- Number | T- Total | |

First position | 6 | 37 |

Second position | 7 | 42 |

Third position | 8 | 47 |

Here again I see that the T- Total is going up in 5. This is for the same reason as I stated before.

There is 5 numbers in each T-Shape so if you move the T-shape over each individual number is in creased by 1. Equaling 5 * 1

E.g. 6 changes to 7

7 changes to 8 etc

With the information that I poses from my last investigation I feel no need to write out the 6 by 3 grid and the 7 by 3 grid. I know what is going to happen.

So I am going find the rule for this T-Shape.

Conclusion

This is the T-Total will always go up in 5 if you move it to the right.

I am now going to try to find a method the same way I have done before.

I am going to break down the T- Shape in an algebraic expression to find the method.

1 | 2 | 3 |

7 | 8 | 9 |

13 | 14 | 15 |

Here is what it looks like as an algebraic expression

1 | T | 3 |

7 | T+G | 9 |

T+2G - 1 | T+2G | T-2G + 1 |

T = the T- Number

T+(T+G)+(T+2G)+(T+2G-1)+(T+2G+1) = 5T+7G

T = the T- Number

By adding together the method of obtaining the numbers in the T- Shape from the T- Number I found the method for finding the T- Total by just using the number of columns in the grid and the T- Number!

I will now prove that this rule works using a 10 by 10 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 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

5*25+7*10= 195

I will add up the numbers in the T- Shape to make sure my answer is correct.

25+35+45+44+46= 195

This rule is correct.

This is how it works.

You get the 5 multiplied by the T-Number by getting 5 numbers in the T- Shape. By applying the 5n rule to this you get 5T.

For the row with the T-number in it the row equals 0

The next row up equals +1G because you have to subtract the grid size to get the T-Number

The last row from the T-Number is +2G because this is what you have to get the T-Number from this point.

When you add +1G and the three +2Gs you get +7G

This rule is the opposite of the first investigation because it is in the opposite direction to the original T- Shape.

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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