Investigate the differences between products in a controlled sized grid.

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Aim

I am going to investigate the differences between products in a controlled sized grid.

Method

I am going to keep the grid size the same. Keeping the number of rows and columns the same. I am going to change the position of the box and the size of the size of the box altering the number of rows and columns.

Investigation on 2by 2 boxes in a 10 by 10 grid.

I am now going to multiply opposite numbers together in the box. This is to show the differences.

1*12=12

11*2=22

DIFFERENCE=10

                

9*20=180

19*100=190

DIFFERENCE=10

                

36*47=1692

37*46=1702

DIFFERENCE=10

Table of results

A        B A        B        

C        D C        D        

        

A        B

C        D

I have called these numbers A, B, C, D so that it will be easier to see the results in the table. It will also be easier to see which numbers I am going to multiply together.

By doing this I have found a pattern between all the examples of 2 by 2 boxes in a 10 by 10 grid. The difference is 10.

I am now going to try and find an algebraic equation to show the difference in a 2 by 2 box in a 10 by 10 grid.

I am going to call the top left hand number x; this is to form an algebraic equation. I am then going to represent the other numbers in relation to x.

x        x+1

x+10         x+11

I am now going to multiply these together as I did with the numbers to form an algebraic equation.

x (x+11)                        (x+1) (x+10)

=x²+11x                        = x²+11x+10

                

A, B, C and D represent numbers in the box. This can be represented in terms of x.

I am now going to subtract the two away from each other as I did when I did it numerically.

 x²+11x

 x²+11x+10

=10

I am now going to pick a 2 by 2 box at random to prove that this algebraic equation does work for 2 by 2 boxes in 10 by 10 grids.

I am going to substitute 4 for x in the equation. 4 represents the top left hand number in the box. This is to prove the equation is correct and that the difference is 10.

x²+11x                x²+11x+10

4²+(11*4)        4²+(11*4)+10        

=60        =70

70-60= 10

This proves that the equation does work for and 2 by 2 box in a 10 by 10 grid.

I am now going to investigate a 3 by 3 box in a 10 by 10 grid.

Investigation on 3 by 3 boxes in a 10 by 10 grid

I am now going to multiply opposite numbers together in the box. This is to show the differences.

4*26=104

6*24=144

DIFFERENCE= 40

41*63=2583

43*61=2623

DIFFERENCE=10

Table of results

A        B                                

C        D

A        B

C        D

By doing this I have found a pattern between all the examples of 3 by3 boxes in a 10 by 10 grid. The difference is 40.

I am now going to try and find an algebraic equation to show the difference in a 3 by 3 box in a 10 by 10 grid.

A, B, C and D represent numbers in the box. This can be represented in terms of x.

I am going to call the top left hand number x, this is to form an algebraic equation. I am then going to represent the other numbers in relation to x.

x        x+2

        

x+20        x+22

I am now going to multiply these together as I did with the numbers to form an algebraic equation.

x (x+22)                        (x+2) (x+20)

=x²+22x                        = x²+22x+40

I am now going to subtract the two away from each other as I did when I did it numerically.

x²+22x

 x²+22x+40

=40

I am now going to pick a 3 by 3 box at random to prove that this algebraic equation does work for 3 by 3 boxes in 10 by 10 grids.

I am going to substitute 8 for x in the equation. 4 represents the top left hand number in the box. This is to prove the equation is correct and that the difference is 40.

Join now!

x²+22x                x²+22x+40

8²+(22*8)        8²+(22*8)+40        

=240        =280

280-240= 40

This proves that the equation does work for and 3 by 3 box in a 10 by 10 grid.

I am now going to investigate a 4 by 4 box in a 10 by 10 grid.

Investigation on 4 by 4 boxes in a 10 by 10 grid

I am now going to multiply opposite numbers together in the box. This is to show the differences.

1*34=34

4*31=124

DIFFERENCE=90

57*90=5130

60*87=5220

DIFFERENCE=90

Table of results

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