# opposite corners

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Introduction

GCSE Maths Coursework

Opposite Corners

I have been given the task to investigate the differences of the products of the diagonal opposite corners of a square on a 10x10 Grid with the numbers 1 to 100 to start with.

I will start with a 2 x 2 square on a 10 x 10 grid and discover the rule for it, then I will progress onto a 3 x 3 square on the same grid. I will then keep on going until I eventually find the rule for any sized square on a 10 x 10 grid.

2x2 Square

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 |

(2 x 11) – (1 x 12) = 10

(14 x 25) – (15 x 24) = 10

(8 x 17) – (7 x 18) = 10

(20 x 29) – (19 x 30) = 10

I have discovered that the answer is always 10 I will now use algebra to see if the answer is once again 10.

n | n+1 |

n+10 | n+11 |

(n+1)(n+10) – n(n+11)

(n2+11n+10) – (n2+11n)

10

As the algebraic equation also gives the answer of 10 I know it must be right. As I believe I can keep on learning throughout the investigation I will now move onto a 3x3 square on the same grid. I predict that once again all answers will be the same.

3 X 3 Square

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 |

(3 x 21) – (1 x 23) = 40

(6 x 24) – (4 x 26) = 40

(10 x 28) – (8 x 30) = 40

I believe the answer will always be 40 for a 3 x 3 square on this grid. So I will now use algebra to see if I am correct.

n | n+2 |

n+20 | n+22 |

(n+2)(n+20) – n(n+22)

(n2+20n+2n+40) – (n2+22n)

40

This proves that the answer is always 40 when a 3 x 3 square is placed on a 10 x 10 grid.

I am now going to use a 4 x 4 square on a 10 x 10 grid. The only difference will be that I will only use algebra as numbers are very time consuming.

4 x 4 square

n | n+3 |

n+30 | n+33 |

Middle

5 x 5 Square

n | n+4 |

n+40 | n+44 |

(n+4)(n+40) – n(n+44)

(n2+40n+4n+160) – (n2+44n)

160

I am now going to create a table displaying my results so far

Square | Opposite Corners |

2 x 2 | 10 |

3 x 3 | 40 |

4 x 4 | 90 |

5 x 5 | 160 |

6 x 6 | ??? |

I have noticed a pattern: the opposite corners are all square numbers of the square before multiplied by 10. Hence forth I predict that the product form the 6 x 6 square will have a difference of 250, I will now try it and see if my prediction is correct.

n | n+5 |

n+50 | n+55 |

(n+5)(n+50) – n(n+55)

(n2+50n+5n+250) – (n2+55n)

250

I was correct. I will now attempt to convert my verbal rule into algebraic terms for any square on a 10 x 10 grid.

n | n+(m-1) |

N+10(m-1) | N+11(m-1) |

(n+(m-1))(n+10(m-1)) – n(n+11(m-1))

(n2+10n(m-1)+n(m-1)+10(m-1) 2)-(n2+11n(m-1))

10(m-1) 2

I have now discovered the rule for any square on a 10 x 10 grid I am now going to try and find a general rule which will allow me to work out on any square on any grid. As the rule for any square on a 10 x 10 grid is 10(m-1) 2 I predict that if were to use a 7 x 7 grid the rule would be 7(m-1) 2 . I will now see if I am correct.

Conclusion

size.

H: the height of the rectangle. This is a non-restrictive measurement if it is required to be

as extra rows can be added easily to the bottom of the grid. Must be a whole positive

integer

G: The grid size which must be at least 2 x 2 upwards so that it has opposing corners. It

must also be a whole, positive integer.

S: The gap between the two numbers can be any number; positive, negative, decimals,

Fractions etc.

This is what my grid would look like

n | n+S | n+2S | …… | n(g-1)S |

n+GS | ||||

n+2GS | ||||

…… | …… | …… | …… | …… |

This is how I will work out the rule for any rectangle on any sized grid with any gap size between the numbers.

n | n+S(L-1) |

n+GS(H-1) | n+S(L-1)+GS(H-1) |

(n+S(L-1))(n+GS(H-1) – n(n+s(L-1)+GS(H-1))

(n2+nGS(H-1)+ns(L-1)+S2G(L-1)(H-1)) – (n2+ns(L-1)+nGS(H-1)

S2G(L-1)(H-1)

For Example

So if:

S= -0.1

G= 5

L=2

H=3

S2G(L-1)(H-1)

0.01 x 5 x 1 x 2 = 0.1

For Example

22 | 21.9 | 21.8 | 21.7 | 21.6 |

21.5 | 21.4 | 21.3 | 21.2 | 21.1 |

21.0 | 20.9 |

(21.9 x 21) – (22 x 20.9)

459.9 – 459.8

0.01

In conclusion I feel I have gone as far as possible with this investigation as I have covered all aspects of it as I can possibly think off. I am very happy with my results and have enjoyed doing this investigation.

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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## Here's what a teacher thought of this essay

This is a very well structured investigation. The relationships are clearly described through written, numeric and algebraic explanation. There are specific strengths and improvements suggested throughout.

Marked by teacher Cornelia Bruce 18/04/2013