# I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid

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Introduction

Opposite Corners

Opposite Corners

* Throughout this piece of work, the horizontal side will be referred to as the length, whilst the vertical side will be referred to as width.

Introduction/Aim

I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid. I am going to investigate different rectangles, of different areas, lengths, widths and positioning on a grid. After finding a pattern, I will try and prove that it will work for other cases by using algebra and making an algebraic formula. As I go along, I will have to record any ideas I have or patterns I see. Afterwards, I will go on to investigate how this rule may differs on a different sized grid.

2x2 Square, 10x10 Grid

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 |

This is a 10x10 grid. On it (outlines in red) is a 2x2 square.

Firstly, I’m going to see what the difference between the products of the corners is (D):

55x64=3520

54x65= 3510

3520-3510=10, D=10

Now: What if the same sized rectangle was placed in a different area of the grid?

28 | 29 |

38 | 39 |

29x38=1102

28x39=1092

1102-1092=10, D=10

The difference is still the same; I’m going to do one more example to confirm this:

1 | 2 |

11 | 12 |

2x11=22

1x12=12

22-12=10, D=10

From this I am assuming that the difference of the corners product (D) is the same wherever the square/rectangle is place.

Middle

24

25

33

34

35

One possible rule is taking one away from the length and multiplying by 10 will give the difference. This, written algebraically is: 10(L-1). However, this may not work with a width that isn’t 2 [I’ll look into this further on].

One formula that takes into account the width is: 5W(L-1). This can be written in words as the Length minus one, times by five times the width.

Checking Formulae

10(L-1), when L = 3: 10x(3-1) = 10x(2)=20? YES

5W(L-1), when L=3 and W=2: (5x2)x(3-1)= (10)x(2)=20? YES

3x‘n’ Rectangles, 10x10 Grid

I’m now investigating rectangles with heights of 3 and see whether they follow the same or similar pattern to those with heights of 2. After this I will see whether the formulae I came up with previously apply for these rectangles, and if not, how I can adapt them to.

3x2

1 | 2 |

11 | 12 |

21 | 22 |

2x21=42

1x22=22

42-22=20, D=20

3x3

61 | 62 | 63 |

71 | 72 | 73 |

81 | 82 | 83 |

63x81=5103

61x83=5063

5103-5063=40, D=40

3x4

26 | 27 | 28 | 29 |

36 | 37 | 38 | 39 |

46 | 47 | 48 | 49 |

29x46=1334

26x49=1274

1334-1274=60

3x5

16 | 17 | 18 | 19 | 20 |

26 | 27 | 28 | 29 | 30 |

36 | 37 | 38 | 39 | 40 |

20x36=720

16x40=640

720-640=80, D=80

The differences are still multiples of 10, however, they are going up in increments of 20.

Table of Results 2(and further predictions)

Area | Difference |

3x2 | 20 |

3x3 | 40 |

3x4 | 60 |

3x5 | 80 |

3x6 | 100 |

3x7 | 120 |

} +20

} +20

} +20

} +20 (we can assume this, looking at the pattern of the difference increasing in increments of 20)

} +20

Checking Previous Formulae

10(L-1), when L = 5: 10x(5-1)= 10x(4)= 40? NO, it should be double that (80).

5W(L-1), when L=5 and W=3: (5x3)x(5-1)= (15)x(4)=60? NO

Adapting The Formulae

I have decided to continue with the formulae 10(L-1) as the other formula doesn’t seem to work for other widths, whereas this one appears to be adaptable. For 10(L-1), the result is half the expected number, so to give the correct answer it can be multiplied by 2: 2x10(L-1) or 20(L-1). E.g. If L=4: 20x(4-1) = 20x(3) = 60? YES

4x‘n’ Rectangles, 10x10 Grid

To help me find a pattern, I’m going to do rectangles with a width of 4:

4x2

1 | 2 |

11 | 12 |

21 | 22 |

31 | 32 |

2x31=62

1x32=32

62-32=30, D=30

4x3

7 | 8 | 9 |

17 | 18 | 19 |

27 | 28 | 29 |

37 | 38 | 39 |

9x37=333

7x39=273

333-273=60, D=60

4x4

57 | 58 | 59 | 60 |

67 | 68 | 69 | 70 |

77 | 78 | 79 | 80 |

87 | 88 | 89 | 90 |

60x87=5220

57x90=5130

5220-5130=90, D=90

4x5

6 | 7 | 8 | 9 | 10 |

16 | 17 | 18 | 19 | 20 |

26 | 27 | 28 | 29 | 30 |

36 | 37 | 38 | 39 | 40 |

Conclusion

270

Like with rotating 90 degrees, the width is just swapped with the length vice versa. So to find out the new difference, you do:

360

Rotating a shape by 360 degrees just returns it to its original orientation and position. Obviously, this means the difference would have still stayed the same.

Evaluation

I think that I have completed the aims of this assessment and have reached the goals. I have found multiple patterns in the difference of opposites corners and used this to find individual rules. I then linked together the patterns and individual rules to make a general rule, which can work for any grid, width or length. I also went on to look at how rotating a shape would affect the difference, which was made easier by my previous work in creating a general rule. I also went very detailed in my explanations and proof of everything.

In the future, it would be interesting to see how this would work in 3D, for example, a cuboid inside a cube. This would be interesting as I’d have to take into account a whole new dimension which increases the work and calculations needed to come to any conclusions. It would also be interesting to see how it worked with different shapes like triangles and hexagons, although a specially designed grid would be necessary.

I am pleased with the way that I handled this task and, at the moment, cannot think of any improvements without looking at a more in-depth success criteria or the standard of other pieces of work.

Alexander Teh Jia-Hao

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 well thought out and constructed algebraic investigation. There is a good use of a general formula to generalise the relationships discovered. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/07/2013