# This piece of coursework is called 'Opposite Corners' and is about taking squares of numbers from different sized number grids

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Introduction

Algebra Coursework

Introduction

This piece of coursework is called ‘Opposite Corners’ and is about taking squares of numbers from different sized number grids. I will be multiplying the opposite corners together and subtracting to find the difference.

I will make a prediction for each grid and use a few examples to find a formula to prove my prediction right.

Method

To calculate the difference of the squares drawn on the number grids, I will be multiplying each of the diagonal numbers. Then I will subtract the smaller number away from the larger number to find the difference of the square.

To do this I will use algebraic equations, I will use ‘N’ to represent the smallest number and ‘g’ to represent the grid size. I will then compare my results to ‘N’, then to complete the equation by expanding, simplifying and cancelling the brackets to get my final result.

Prediction

I predict that a 2x2 square from a 5 wide grid, will have a final difference of 5.

Proof:

1 | 2 | 3 | 4 | 5 |

6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 |

16 | 17 | 18 | 19 | 20 |

Comparison:

N N+1

x

N+5 N+6

N (N+6) = N² + 6N

(N+1) (N+5) = N² + 5N + N + 5

= N² + 6N + 5

Difference:

(N² + 6N + 5) – (N² + 6N)

= 5

What I have noticed:

Middle

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24

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28

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36

Comparison:

N N+2

x

N+12 N+14

N (N+14) = N² + 14N

(N+2) (N+12) = N² + 12N + 2N + 24

= N² + 14N + 24

Difference:

(N² + 14N + 24) – (N² + 14N)

= 24

What I have noticed:

I have noticed that when a square that is 3x3, is taken from a 6x6 grid, the difference is always 24.

Prediction

I predict that a 4x4 square from a 6 wide grid, it will have a final difference of 54.

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 |

Comparison:

N N+3

x

N+18 N+21

N (N+21) = N² + 21N

(N+3) (N+18) = N² + 18N + 3N + 54

= N² + 21N + 54

Difference:

(N² + 21N + 54) – (N² + 21N)

= 54

What I have noticed:

I have noticed that when a square that is 4x4, is taken from a 6x6 grid, the difference is always 54.

Prediction

I predict that a 5x5 square from a 6 wide grid, it will have a final difference of 96.

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 |

Comparison:

N N+4

x

N+24 N+28

N (N+28) = N² + 28N

(N+4) (N+24) = N² + 24N + 4N + 96

= N² + 28N + 96

Difference:

(N² + 28N + 96) – (N² + 28N)

= 96

What I have noticed:

I have noticed that when a square that is 5x5, is taken from a 6x6 grid, the difference is always 96.

I have found out the difference for a six wid grid is g(x-1)².

I will now see if the same eqyuation works for a seven wide grid.

I will now carry out the same investigation but on a 7 wide grid

Prediction

I predict that a 2x2 square from a 7 wide grid, it will have a final difference of 7.

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 |

Conclusion

I have found out the difference for a seven wid grid is g(x-1)².

I now have found a formula that will work for and size number grid over five wide and square of 2x2. this formula is g(x-1)

Table of results:

5 wide grids

Square size | , Difference | Gap in Difference | Gap in Gap of Difference |

2x2 | 5 | +5 | n/a |

3x3 | 20 | +15 | +10 |

4x4 | 45 | +25 | +10 |

5x5 | 80 | +35 | +10 |

6 wide grids

Square size | , Difference | Gap in Difference | Gap in Gap of Difference |

2x2 | 6 | +6 | n/a |

3x3 | 24 | +18 | +12 |

4x4 | 54 | +13 | +12 |

5x5 | 96 | +42 | +12 |

7 wide grids

Square size | , Difference | Gap in Difference | Gap in Gap of Difference |

2x2 | 7 | +7 | n/a |

3x3 | 28 | +21 | +14 |

4x4 | 63 | +35 | +14 |

5x5 | 112 | +49 | +14 |

Conclusion

During my investigation, I have cancelled out, expanded and simplified brackets, and compared my results to ‘N’. I did this to find a final equation to be able to calculate the difference of any square of any size drawn on a number grid. I found this to be g (x-1) ². This formula works for any grid size over 5 and square size over 2x2.

If I had more time I would want to find out if the same formula works with a rectangle on a number grid and experiment with other quadrilaterals and see if the same equation works.

I would also see if I could find any other equation that would work.

Overall I think my investigation went well and I have found the final equation.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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