# To find a relationship between the opposite corners in various shapes and sizes.

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Introduction

Gemma McCormick

## Opposite Corners

Aim: - To find a relationship between the opposite corners in various shapes and sizes.

First I am going to draw a square that will be 10 squares in depth and 10 squares in width. I am the going to number the squares 1 to 100, 1 in the top corner and 100 in the bottom square. I will then draw a series of two by two squares inside my main grid. The aim being to find out a relationship between the opposite corners inside the small boxes.

Ten by Ten (2x2)

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 |

I am now going to multiply the opposite corners in each box (Note the boxes are highlighted in red)

Box 1

14 x 25 = 350

15 x 24 = 360

I am now going to look at the difference in each corner by subtracting my two answers.

350 – 360 = 10

The difference between the opposite corners was ten; I am now going to see if this is repeated. To do this I will repeat the process until my results show a constant pattern.

56 x 67 = 3752

57 x 66 = 3752

3762 – 3752 = 10

The difference again is 10, this would lead me to believe that there is a pattern, but I am going to repeat this process again to see if still get the same result.

81 x 92 = 7452

82 x 91 = 7462

7462 – 7452 = 10

Again the difference is 10.

I am can now definitely see a pattern, so I am going to try and work at a relationship sing algebraic equations.

X | X+1 |

X+10 | X+11 |

X = 14

= 14 + 1 = 15

= 14 + 10 = 24

= 14 + 11 = 25

Middle

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

21 | 21 | 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 |

I have had to use numerous colours to show the squares that I am using as they are so close together and could not be seen clearly. I will now carry on with my investigation following the same instructions as before.

1 x 45 = 45

5 x 41 = 205

45 – 205 = 160

The answer to this square is 160 so I predict that the answer to the next square will be 160.

51 x 95 = 4845

55 x 91 = 5005

4845 –5005 = 160

My prediction was correct so I will try again with this square to see if I my results where accurate. I still can’t see a constant pattern.

56 x 100 = 5600

60 x 96 = 5760

5600 – 5760 = 160

I still can’t see a common pattern but maybe something will arise in the nine by nine grid.

Nine by Nine (2x2)

I am now going to start my nine by grid. I will use exactly the same processes as in the ten by ten 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 |

Now I will try and find a relationship between a 2x2 squares on a nine by nine grid.

5 x 15 = 75

6 x 14 = 84

75 – 84 = 9

I can now see a pattern between my grids but I will have to repeat this process to see if anything changes or if the pattern is consistent.

28 x 38 = 1064

29 x 37 = 1073

1064 – 1073 = 9

There is definitive pattern between the result here and the result in grid 10 by 10-square 2x2. I can make prediction about an eight by eight grid when investigate the 2x2 square I predict that the answer would be 8.

49 x 59 = 2891

50 x 58 = 2900

2891 – 2900 = 9

Conclusion

19 x 37 = 703

21 x 35 = 735

703 – 735 = 32

46 x 64 = 2944

48 x 62 = 2976

2944 – 2976 = 32

This part is correct maybe I can find more relationships and final formula if I carry on. I will now look at an algebraic equation for this section of work.

My equation: (X+18)

X = 6

= (6 + 18) =24

My equation works.

Eight by Eight (4x4)

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 |

I have to use multiple colours so that my squares can be clearly seen

I will now see if the third part of my prediction is correct.

5 x 32 = 160

8 x 29 = 232

160 – 232 = 72

The third part of my prediction was correct. I am going to find more evidence to prove this.

33 x 60 = 1980

36 x 57 = 2052

1980 – 2052 = 72

37 x 64 = 2368

40 x 61 = 2440

2368 – 2440 = 72

The third part of my prediction was definitely correct so I think that the fourth part of this section of investigation will be correct. I am now going to produce an algebraic equation for this section of work.

My equation:

(X+27)

I will now test my equation:

X= 33

= (33+27)= 60

My equation works.

Eight by Eight (5x5)

This is the final part of my numeric work and hopefully I can find an equation that works of all opposite corners.

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 |

For the end part of my investigation I am only going to take two examples from my grid. As I think that it will be too hard to recognise what squares I have used.

1 x 37= 37

5 x 33 = 165

37 – 165 = 128

27 x 63 = 1701

31 x 59 = 1829

1701 – 1829 = 128

My predictions where all correct as I’ve just completed the numeric section of work and it was I predicted. Now I am going to analysise all work to see if there is an overall section of work.

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

**

The general pattern for a 10 x 10 grid is identified. There is some extension of the investigation. However this is limited. To improve this investigation more algebraic manipulation is needed to verify the identified pattern. There should be multiplication of double brackets and the identification of an nth term. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/07/2013