# Investigation to find out a formula for rectangles on different grid sizes.

Extracts from this document...

Introduction

Mariela Mezquita 10ª

Math, Mr Bhoja

30 April 2008

COURSEWORK

Aim: To find out a formula for rectangles on different grid sizes.

That is done by multiplying the corners of the rectangle and then we subtract them.

Hypothesis:

Eg. (on a 10x10 grid size with a 2x3 rectangle)

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 | ||

72 | 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 | ||

3 x 11= 33

1 x 13=13

33-13 = 20

Then, move on the rectangle.

A 2 x 3 cube…

1 | 2 | 3 |

11 | 12 | 13 |

2 | 3 | 4 |

12 | 13 | 14 |

3 | 4 | 5 |

13 | 14 | 15 |

As we can see, when we have a 2 x 3 rectangle, the final result will always be 20.

Explanation:

If we substitute the

Middle

1 | 2 | 3 | 4 |

11 | 12 | 13 | 14 |

21 | 22 | 23 | 24 |

2 | 3 | 4 | 5 |

12 | 13 | 14 | 15 |

22 | 23 | 24 | 25 |

It will always be 60..

Explanation:

x | x+3 | ||||

x+20 | x+23 | ||||

x(x+23)

x+3(x+20)

expand

x(x+23) = x²+23x

x+3(x+20) = x²+20x+3x+60

x²+23x

x²+20x+3x+60

=60

x² and 23x cancel each other out, leaving us with a result of 60.

If we change the

Conclusion

So:

3x4 = 6n

FORMULA ABOVE ONLY APLLIES FOR A 4 X 5 RECTANGLE ON ANY GRID SIZE

So:

4x5= 12n

FORMULA ABOVE ONLY APLLIES FOR A 5 X 6 RECTANGLE ON ANY GRID SIZE

So:

5x6= 20n

FORMULA ABOVE ONLY APLLIES FOR A 6 X 7 RECTANGLE ON ANY GRID SIZE

So:

6x7= 30n

Overall Formula:

N(H-1)(L-1)

Extension:

In my extension I will draw crosses instead of rectangles. Multiply the opposite sizes and try to find out a formula.

Hypothesis: I think it will be similar to the rectangle formula but a little bit more complicated.

Example:

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 | ||

72 | 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 | ||

This student written piece of work is one of many that can be found in our GCSE Open Box Problem section.

## Found what you're looking for?

- Start learning 29% faster today
- 150,000+ documents available
- Just £6.99 a month