To do this I am going to be drawing different size grids, including 2,3,4,5,6,7, and 8 grids

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Opposite Corners

Introduction

I have been set a piece of G.C.S.E. coursework called opposite corners. The aim of this coursework is to find the difference in the products of the opposite corners for any size square in any size grid. To do this I am going to be drawing different size grids, including 2,3,4,5,6,7, and 8 grids. This is what a 4 grid looks like,

2 3 4

5 6 7 8

9 10 11 12

3 14 15 16

It is called a 4 grid because it has got 4 colums. I will be drawing 2by2 squares in these grids, and then I will multiply the opposite corners to find the difference in the products. I will be drawing diagrams, tables, and graphs for my results, and I will be looking for any patterns. After I have finished the 2by2 squares in different size grids, I will be using different size squares and grids to find out an overall formula to find out the difference in the products of opposite corners for any size square in any size grid.

An example of what I am going to be doing is a 2by2 square in a four grid

) First I will draw a 4 grid

1 2 3 4

5 6 7 8

9 10 11 12

3 14 15 16

2) Then I will draw a 2by2 square in it

2 3 4

5 6 7 8

9 10 11 12

3 14 15 16

3) After that multiply the opposite corners of the 2by2 square

x6= 6

2x5= 10

4) Now I can find the difference between the products of the opposite corners, which is

0-6= 4

I am going to start this investigation with a 2by2 square, in the smallest grid it will fit into. The grid is a 2 grid. At the start of this investigation I will use three 2by2 squares in different parts of the grid, this is to prove that no matter where abouts in the grid the square is place that the difference in the products of the opposite corners is the same.

2

3 4 1x4=4 6-4=2

5 6 2x3=6 difference=2

7 8

9 10 7x10=70 72-70=2

1 12 8x9=72 difference=2

3 14

5 16 13x16=208 210-208=2

14x15=210 difference=2

I am now going to change the size of the grid to a three grid and try three 2by2 squares in it.

2 3 2x6=12 15-12=3

4 5 6 3x5=15 difference=3

7 8 9

0 11 12 7x11=77 80-77=3

3 14 15 8x10=80 difference=3

6 17 18

14x18=252 255-252=3

15x17=255 difference=3

I will now try 2by2 squares in a 4 grid.

2 3 4 1x6=6 10-6=4

5 6 7 8 2x5=10 difference=4

9 10 11 12

3 14 15 16 11x16=176 180-176=4

7 18 19 20 12x15=180 difference=4

13x18=234 238-234=4

14x17=238 difference=4

I am now going to change the size of the grid to a 5 grid and try three 2by2 squares in it.

2 3 4 5 1x7=7 12-7=5

6 7 8 9 10 2x6=12 difference=5

1 12 13 14 15

6 17 18 19 20 14x20=280 285-280=5

21 22 23 24 25 15x19=285 difference=5

26 27 28 29 30

21x27=567 572-567=5

22x26=572 difference=5

The last grid size I am going to try is a 6 grid.

2 3 4 5 6 1x8=8 14-8=6

7 8 9 10 11 12 2x7=14 difference=6

3 14 15 16 17 18

9 20 21 22 23 24 5x12=60 66-60=6

25 26 27 28 29 30 6x11=66 difference=6

21x28=588 594-588=6

22x27=594 difference=6

The pattern that I can see from finding the difference in the products of opposite corners for 2by2 squares is,

2 3 4 5 6

You can see that the difference in the products of the opposite corners is the same as the size of the grids. From this I predict that the difference of a 2by2 square in a 7 grid will be 7. To prove this I am going to draw a 2by2 square in a 7 grid and find the difference of the products of the opposite corners.

2 3 4 5 6 7 3x11=33 40-33=7

8 9 10 11 12 13 14 4x10=40 difference=7

5 16 17 18 19 20 21

I am now going to draw a table for my results of 2by2 squares in different size grids.

size of grid difference of products

of the opposite corners

in 2by2 squares.

2 2

3 3

4 4

5 5

6 6

7 7

I am now going to change the size of the squares to 3by3, I will be using different size grids. I cannot start with a 2 grid because a 3by3 square will not fit in it. So I will start with a 3 grid. From the first part of my investigation I proved that no matter where abouts in the grid I put the square, the difference in the products of the opposite corners was always the same, because of this I will only be using one 3by3 square in each grid.

2 3 1x9=9 21-9=12

4 5 6 3x7=21 difference=12

7 8 9

I will now try a 3by3 square in a 4 grid.

2 3 4 2x12=24 40-24=16

5 6 7 8 4x10=40 difference=16

9 10 11 12

3 14 15 16

I am now going to change the size of the grid to a 5 gridand draw a 3by3 square in it.

2 3 4 5 1x13=13 33-13=20

6 7 8 9 10 3x11=33 difference=20

1 12 13 14 15

6 17 18 19 20

I will now try a 3by3 square in a 6 grid.

2 3 4 5 6 3x17=51 75-51=24

7 8 9 10 11 12 5x15=75 difference=24

3 14 15 16 17 18

9 20 21 22 23 24

The pattern I can see from finding the difference in the products of the opposite corners for 3by3 squares is,

12 16 20 24

From my results I can see that the difference in the products of the opposite corners for 3by3 is the grid size muliplyed by 4. From this I can predict that for a 3by3 square in a 7 grid, the difference in the products of the opposite corners will be 28. I will now prove this prediction.

2 3 4 5 6 7

8 9 10 11 12 13 14 3x19=57 85-57=28

5 16 17 18 19 20 21 5x17=85 difference=28

22 23 24 25 26 27 28

I am now going to draw a table of my results.

size of grid difference of the products

of the opposite corners

of 3by3 squares

3 12

4 16

5 20
Join now!


6 24

7 28

I am now going to change the size of the squares to 4by4. I am going to start with a 4 grid because a 4by4 square will not fit into a 2 or 3 grid.

2 3 4

5 6 7 8 1x16=16 52-16=36

9 10 11 12 4x13=52 difference=36

3 14 15 16

I am now going to try a 4by4 square in a 5 grid

2 3 4 5

6 7 8 9 10 2x20=40 85-40=45

1 12 ...

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