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)

  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 = 14

    = 14 + 1 = 15

    = 14 + 10 = 24

    = 14 + 11 = 25

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

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

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)

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.

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.