# Opposite Corners.

Extracts from this document...

Introduction

Opposite Corners

Investigation:

Given a 100 Square, I am to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on the 100 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 |

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 |

2x3

A rectangle has been highlighted in the 100 square 1 and 13/11 and 3 are the numbers in the opposite corners product of the number in these opposite corners are

1x13=13

11x3=33

The difference between these products is 33-13= 20

A rectangle has been highlighted in the 100 square 4 and 16/14 and 6 are the numbers in the opposite corners product of the number in these opposite corners are

4x16=64

14x6=84

The difference between these products is 84-64= 20

A rectangle has been highlighted in the 100 square 7 and 19/17 and 9 are the numbers in the opposite corners product of the number in these opposite corners are

7x19=133

17x9=153

The difference between these products is 153-133=20

Conclusion:

Middle

5x18=90

15x8=120

The difference between these products is 120-90=30

A rectangle has been highlighted in the 100 square 21 and 34/24 and 31 are the numbers in the opposite corners product of the number in these opposite corners are

21x34=714

24x31=744

The difference between these products is 744-714=30

Conclusion:

I have concluded that my prediction was correct and the investigation told me that a 2 by 4 rectangle will give a difference of 30 in the opposite corners. I now know that a 2 by 2 rectangle will have a difference of 10, and 2 by 3 will have a difference of 20 and a 2 by 4 rectangle will have a difference of 30.

2x5

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 |

A rectangle has been highlighted in the 100 square 1 and 15/11 and 5 are the numbers in the opposite corners product of the number in these opposite corners are

1x15=15

11x5=55

The difference between these products is 55-15=40

A rectangle has been

Conclusion

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- 73 x 91 = 6643 _

71 x 93 = 6603

Diff = 40

- 25 x 43 = 1075 _

23 x 45 = 1035

#### Diff = 40

- 10 x 28 = 280 _

8 x 30 = 240

Diff = 40

Prediction:

I predict that the next 3 by 3 Rectangle will have a difference of 40.

Workings:

77 x 95 = 7315 _

75 x 97 = 7275

#### Diff = 40

Conclusion:

I have concluded that a 3 by 3 rectangle has a difference of 40 and I now know that a 3 by 2 rectangle has a difference of 20 and a 3 by 3 rectangle has a difference of 40.

3 by 4 Rectangles:

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 |

1. 50 x 67 = 3350 _

47 x 70 = 3290

Diff = 60

2. 6 x 23 = 138 _

3 x 26 = 78

Diff = 60

3. 54 x 71 = 3834 _

51 x 74 = 3774

Diff = 60

Prediction:

I that the next 3 by 4 rectangle will have a of 60.

Workings:

68 x 85 = 7580 _

65 x 88 = 7520

#### Diff = 60

Conclusion:

I have concluded that a 3 by 4 rectangle has a difference of 60.

3 by 5 Rectangles:

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 |

1. 6 x 22 = 132 _

2 x 26 = 52

Diff = 80

2. 50 x 66 = 3300 _

46 x 70 = 3220

#### Diff = 80

3. 66 x 82 = 5412 _

62 x 86 = 4332

Diff = 80

Prediction:

I predict that the next rectangle will have adifference of 80.

Workings:

35 x 51 = 1785 _

31 x 55 = 1705

Diff = 80

Conclusion:

I have concluded that a 3 by 5 rectangle has a difference of 80.

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 brief investigation grounded only in the experimental examples. In order to improve this investigation it needs to move from the literal into the formulaic. Algebraic expressions need to be formed and simplified to verify any patterns found.

Marked by teacher Cornelia Bruce 18/04/2013