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
A B
C D
A B
C D
By doing this I have found a pattern between all the examples of 4 by 4 boxes in a 10 by 10 grid. The difference is 90.
I am now going to try and find an algebraic equation to show the difference in a 4 by 4 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+3
x+30 x+33
I am now going to multiply these together as I did with the numbers to form an algebraic equation.
x (x+33) (x+3) (x+30)
=x²+33x = x²+33x+90
I am now going to subtract the two away from each other as I did when I did it numerically.
x²+33x
x²+33x+90
=90
I am now going to pick a 4 by 4 box at random to prove that this algebraic equation does work for 4 by 4 boxes in 10 by 10 grids.
I am going to substitute 5 for x in the equation. 5 represents the top left hand number in the box. This is to prove the equation is correct and that the difference is 90.
x²+33x x²+33x+90
5²+(33*5) 5²+(33*5)+90
=190 =280
280-290= 90
This proves that the equation does work for and 4 by 4 box in a 10 by 10 grid.
I am now going to investigate a 5 by 5 box in a 10 by 10 grid.
Investigation on 5 by 5 boxes in a 10 by 10 grid
1*45=45
5*41=205
DIFFERENCE=160
56*100=5600
60*96=5760
DIFFERENCE=160
Table of results
A B
C D
A B
C D
By doing this I have found a pattern between all the examples of 5 by 5 boxes in a 10 by 10 grid. The difference is 160.
I am now going to try and find an algebraic equation to show the difference in a 5 by 5 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+4
x+40 x+44
I am now going to multiply these together as I did with the numbers to form an algebraic equation.
x (x+44) (x+4) (x+40)
=x²+44x = x²+44x+160
I am now going to subtract the two away from each other as I did when I did it numerically.
x²+44x
x²+44x+160
=160
I am now going to pick a 5 by 5 box at random to prove that this algebraic equation does work for 5 by 5 boxes in 10 by 10 grids.
I am going to substitute 6 for x in the equation. 6 represents the top left hand number in the box. This is to prove the equation is correct and that the difference is 160.
x²+44x x²+44x+160
6²+(44*6) 6²+(44*6)+160
=300 =460
460-300= 160
This proves that the equation does work for and 5 by 5 box in a 10 by 10 grid.
Results table of all sized boxes investigated in a 10 by 10 grid.
I am now going to look back to the size f the box and see if there is any relation between the size of the box and the square numbers in the last column in the table. I think by doing this it will help me form a general equation for any size e square box in a 10 by 10 grid.
I am now going to try and form an algebraic equation using the information in the table.
I am going to give the size of the box a letter. It will be n.
I am going to give the size of the grid a letter as well. This will be x.
By doing this I will be able to form an algebraic equation.
(n-1) ²*x
= x(n-1)
x(n-1) ²=d
d is the difference.
This is the formula I have found from investigating different size boxes in a 10 by 10 grid. I think that by using this formula you will be able to find out the difference in any size box square box in any size square grid. I am going to investigate further to see if my conclusion is correct.
I am now going to investigate different size boxes in a 9 by 9 grid. I am going to use my formula and see if it is correct.
Investigation on a 2 by 2 box in a 9 by 9 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 2 by 2 box.
9(2-1) ²
=9
I am now going to show this numerically without using the equation to make sure it is correct.
1*11=11
2*10=20
DIFFERENCE= 9
Investigation on a 3 by 3 box in a 9 by 9 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 3 by 3 box.
9(3-1) ²
=36
I am now going to show this numerically without using the equation to make sure it is correct.
5*25=125
7*23=161
DIFFERENCE=36
Investigation on a 4 by 4 box in a 9 by 9 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 4 by 4 box.
9(4-1) ²
=81
I am now going to show this numerically without using the equation to make sure it is correct.
37*67=2479
40*64=2560
DIFFERNCE=81
From this it proves that the equation does work. I want to double check so I am going to do the same with different size boxes in an 8 by 8 grid.
Investigation on 2 by 2 boxes in an 8 by 8 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 2 by 2 box.
8(2-1) ²
=8
I am now going to show this numerically without using the equation to make sure it is correct.
1*10=10
2*9=18
DIFFERENCE=8
Investigation on 3 by 3 boxes in an 8 by 8 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 3 by 3 box.
8(3-1) ²
=32
I am now going to show this numerically without using the equation to make sure it is correct.
1*19=19
3*17=51
DIFFERENCE=32
Investigation on 4 by 4 boxes in an 8 by 8 grid.
x(n-1) ² =d
x represents the size of the grid
n represents the size of the box.
I am now going to substitute the letters for numbers in a 4 by 4 box.
8(4-1) ²
=72
I am now going to show this numerically without using the equation to make sure it is correct.
1*28=28
4*25=100
DIFFRENCE=72
This proves that the equation does show the difference for any size square box in any size square grid.
Therefore the general equation for any size square box in any size square grid is:
x(n-1) ² =d
I am now going to see if this general equation works if I change the shape. I am going to change it to a rectangle. I am going to keep the grid a square. I think that this equation will work for rectangles as well as squares.
I am now going to multiply these together as I did with the numbers to form an algebraic equation.
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. I am going to change the size of the rectangles and the position of the rectangles.
Investigation on 2 by 3 rectangles in a 10 by 10 grid
2 is the number of rows and 3 is the number of columns.
I am now going to apply the formula I found for squares and see if it worked with rectangles. I made a prediction and thought it would.
x(n-1) ² =d
I am now going to substitute the letters for numbers in a 2 by 3 box.
From looking and this I can see it will not work. In this equation n represents the number of rows or columns. In a square the number of rows and columns is the same. However it a rectangle they are different so two different letters will need to be allocated to the number of Rows and the number of Columns.
I am now going to investigate further. I am going to try and find an equation to solve the difference between opposite numbers in rectangles.
1*22=22
2*21=42
DIFFRENCE=20
36*57=2052
37*56=2072
DIFFRENCE=20
9*30=270
10*29=290
DIFFERENCE=20
Table of results
A B
C D
A B
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 3 rectangles in a 10 by 10 grid. The difference is 20.
I am now going to try and find an algebraic equation to show the difference in a 2 by 3 rectangle 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+1
x+20 x+21
I am now going to multiply these together as I did with the numbers to form an algebraic equation.
x (x+21) (x+1) (x+20)
=x²+21x = x²+21x+20
x²+21x
x²+21x+20
=20
I am now going to pick a 2 by 3 rectangle at random to prove that this algebraic equation does work for 2 by 3 rectangles 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 20.
x²+21x x²+21x+20
4²+(21*4) 4²+(21*4)+20
=100 =120
120-100= 20
This proves that the equation does work for and 2 by 3 rectangle in a 10 by 10 grid.
I am now going to investigate 2 by 4 rectangles in a 10 by 10 grid.
Investigation on 2 by 4 rectangles in a 10 by 10 grid
2 is the number of rows and 4 is the number of columns.
1*32=32
2*31=62
DIFFRENCE=30
36*67=2412
37*66=2442
DIFFERENCE=30
Table of results
A B
C D
A B
C D
By doing this I have found a pattern between all the examples of 2 by 4 rectangles in a 10 by 10 grid. The difference is 30.
I am now going to try and find an algebraic equation to show the difference in a 2 by 4 rectangle 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+30 x+31
I am now going to multiply these together as I did with the numbers to form an algebraic equation.
x (x+31) (x+1) (x+30)
=x²+31x = x²+31x+30
x²+31x
x²+31x+30
=30
I am now going to pick a 2 by 3 rectangle at random to prove that this algebraic equation does work for 2 by 3 rectangles 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 30.
x²+31x x²+31x+30
4²+(31*4) 4²+(31*4)+30
=140 =170
170-140= 30
This proves that the equation does work for and 2 by 4 rectangle in a 10 by 10 grid.
I am now going to investigate a 3 by 4 rectangle.
Investigation on a 3 by 4 rectangle in a 10 by 10 grid
3 is the number of rows and 4 is the number of columns.
1*33=33
3*31=93
DIFFERENCE= 30
Results table of all sized rectangles investigated in a 10 by 10 grid.
I am now going to try and form an algebraic equation using the information in the table.
I am going to give the number of rows a letter a letter. It will be n.
I am going to give the number of columns a letter as well. This will be m.
I am also going to give the size of the grid a letter. This will be x
By doing this I will be able to form an algebraic equation.
(n-1) (m-1)*x
= x(n-1) (m-1)
x(n-1) (m-1)=d
d is the difference.
This is the formula I have found from investigating different size rectangles in a 10 by 10 grid. I think that by using this formula you will be able to find out the difference in any size rectangle in any size square grid. I think that this formula will also work to find out any size square in an any size square grid.
I am now going to try to apply this formula. I am going to pick a random rectangle in any size square grid. then I am going to pick any size box in any size square grid.
x(n-1) (m-1)=d
This was taken from a 10 by 10 grid
x represents the size of the grid
n represents the numbers of rows
m represents the number of columns.
10(4-1) (2-1)
=30
This shows that the difference is 30. I am going to now check this.
46*59=2714
49*56=2744
DIFFERENCE=30
This proves that this equation is correct. I am now going to see if it works for squares.
This is also taken from a 10 by 10 grid
x(n-1) (m-1)=d
x represents the size of the grid
n represents the numbers of rows
m represents the number of columns.
10(2-1) (2-1)
=10
This shows that the difference is 30. I am going to now check this
54*65=3510
55*64=3520
DIFFERENCE= 10
Summary of results
By carrying out this investigation I found patters in the differences between the same sized squares in controlled grids but different numbers.
I also changed the grid sizes as well as the size of the square. I found a general formula for the difference:
x(n-1) ²
I found that this equation worked for different grid sizes as well as different square sizes.
I further carried out my investigation changing the shape, this time I investigated rectangles, but at the same time the grid was controlled.
I initially thought that the general equation for the differences in squares would also work for rectangles. This was not true. Instead I found that rectangles had their own general formula, which differentiated between the number of rows and the number of columns. This rule can be represented in the following manner:
x(n-1) (m-1)
This equation works for all sized rectangles in any size square grid. I found that the formula for rectangles also works for squares.
Evaluation
The aim of the investigation was to investigate the differences between products in a controlled sized grid.
I believe that I have fulfilled this aim as I found a general rule that fits all sized squares in any size square grid.
I further investigated a different shape that is a rectangle and also found a general rule for them.
I found that the rule for rectangles also works for squares.
If I were to carry out the investigation again I would investigate different shapes. Maybe parallelograms, to see if they also follow pattern.
I would also see if the general rule I found for rectangles is the general rule that governs all quadrilaterals. I would also change the grid size so that it was not square but instead vary the number of rows and columns. For example a 3 by 7 grid.
All in all I believe that my investigation was successful.