# Maths Investigation- Grids

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Introduction

Karen Ng

10H

Maths Investigation- Grids

The first part of my task was to investigate the number of squares, which can be drawn on grids made from 11 lines.

I used 11 lines to make different grids on paper, alternating the use of lines. Starting with 1 horizontal and 10 vertical lines I work up (or down) so my grid then is 2 horizontal and 9 vertical. I counted the number of squares by outlining the shape of the square; this is shown in diagram 3.

Here is my table of results using 11 lines only.

Lines Down | Lines Across | 1x1 square | 2x2 Square | 3x3 square | 4x4 square | Total Squares |

0 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 1 | 0 | 0 | 0 | 0 | 0 |

9 | 2 | 8 | 0 | 0 | 0 | 8 |

8 | 3 | 14 | 6 | 0 | 0 | 20 |

7 | 4 | 18 | 10 | 4 | 3 | 35 |

6 | 5 | 20 | 12 | 6 | 2 | 40 |

5 | 6 | 20 | 12 | 6 | 2 | 40 |

4 | 7 | 18 | 10 | 4 | 3 | 35 |

3 | 8 | 14 | 6 | 0 | 0 | 20 |

2 | 9 | 8 | 0 | 0 | 0 | 8 |

1 | 10 | 0 | 0 | 0 | 0 | 0 |

0 | 0 | 0 | 0 | 0 | 0 | 0 |

As you can see, after the 6x5 result comes up, the rest of the results are exactly the

Middle

0

2

Total number of squares: 2

Best Combination: 3x2

This table of results is for 6 lines only

Lines Down | Lines across | 1x1 squares | 2x2 squares | 3x3 squares | 4x4 squares | Total Squares |

0 | 0 | 0 | 0 | 0 | 0 | 0 |

5 | 1 | 0 | 0 | 0 | 0 | 0 |

4 | 2 | 3 | 0 | 0 | 0 | 3 |

3 | 3 | 4 | 1 | 0 | 0 | 5 |

Total number of squares: 8

Best Combination 3x3

This table of results is for 7 lines only

Lines Down | Lines across | 1x1 squares | 2x2 squares | 3x3 squares | 4x4 squares | Total Squares |

0 | 0 | 0 | 0 | 0 | 0 | 0 |

6 | 1 | 0 | 0 | 0 | 0 | 0 |

5 | 2 | 4 | 0 | 0 | 0 | 4 |

4 | 3 | 6 | 2 | 0 | 0 | 8 |

Total number of squares: 12

Best Combination: 4x3

This table of results is for 8 lines only

Lines Down | Lines across | 1x1 squares | 2x2 squares | 3x3 squares | 4x4 squares | Total Squares |

0 | 0 | 0 | 0 | 0 | 0 | 0 |

7 | 1 | 0 | 0 | 0 | 0 | 0 |

6 | 2 | 5 | 0 | 0 | 0 | 5 |

5 | 3 | 8 | 3 | 0 | 0 | 11 |

4 | 4 | 9 | 4 | 1 | 0 | 14 |

Total number of squares: 30

Conclusion

## Conclusion

## The more lines you have, the more squares you will get. The rules I found out are; if you were trying to find out how many 1x1 squares were in a 6x5 grid, you would do 6-1=5 and 5-1=4, then, you times the answer ( 5 and 4 ) together, and that number will be the amount of 1x1 squares you will get in the grid. To find out how many 2x2 squares are in a grid, you do exactly the same thing, except you subtract 2 instead of 1 because now there are 2 boxes missing from each side.

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

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