I predicted that the 5x5 square within a 5x5 grid will have a difference of 80 by following the pattern of the square numbers multiplied by the size of the grid, in this case 5x5. Also because all the other results were half of the 10x10 grid results I divided the result, 160 (4²x10) by 2 which is 80.
1x25= 25
5x21= 105
Difference= 105 – 25= 80
My prediction was correct; by using this method I am able to predict the results for the other 5x5 square for different sized grids.
9x9 Grid:
I am moving on to my third grid size, 9x9 grids. Within this grid I will work out the opposite corners with 2x2, 3x3 and 4x4 squares. The results from the opposite corners will help me find a pattern within all the different square sizes within the different sized grids.
2x2 Squares:
This is the smallest size of square I will be using.
21x31= 651
22x30= 660
Difference= 660 – 651= 9
7x17= 119
8x16= 128
Difference= 128 – 119= 9
60x70= 4200
61x69= 4209
Difference= 4209 – 4200= 9
So far, all my results are the same; this shows that again, like with all my results from the previous grids, all the results have been the same. I can start to see a pattern already. 9 is the answer for the 2x2 square in the 9x9 grid and the answer for the 2x2 square in the 10x10 grid is 10 and the 9 is one less than 10 and the 9x9 grid is one size less that the 10x10 grid.
3x3 Squares:
I will know use 3x3 squares within my grid to see whether the same or another pattern is present.
1x21= 21
3x19= 57
Difference= 57 – 21= 36
47x67= 3149
49x65= 3185
Difference= 3185 – 3149= 36
51x71= 3621
53x69= 3657
Difference= 3657 – 3621= 36
Like my results for the 2x2 square I have noticed that the 3x3 square all has results which are the same. This proves that there is definitely a pattern present.
4x4 Squares:
This is the largest square that I will be working out the opposite corners with. Hopefully the results I am left with will coincide with the results I obtained with the other 2 sizes of squares.
1x31= 31
4x28= 112
Difference= 112 – 31= 81
37x67= 2479
40x64= 2560
Difference= 2560 – 2479= 81
23x53= 1219
26x50= 1300
Difference= 1300 – 1219= 81
Again I have a set of results which are all the same, this must mean that like the other grids all the opposite corners of the squares link together to create a pattern.
Table of Results:
This table of results will help me to compare the results and find a pattern. Also it will help me see if the previous grids link in with this size grid.
[Prediction]
I have found that the pattern with the 9x9 grid is that again there is a repetition of the numbers 1,4,9 appearing in all the differences. In this case these numbers have been multiplied by 9 to come up with these results. The pattern here is that the differences of the opposite corners are the square numbers multiplied by the size of the grid. In this case all the numbers are being multiplied by 9.
Prediction:
I have predicted that the 5x5 squares opposite corners will have a difference of 144. I have done this by using the patterns I have obtained from the other size of the squares. There is a pattern that all the square numbers are being multiplied by the size of the grid, so what I did was use the next square number up from 9, 16, and multiplied it by 9.
I will now prove that my theory is correct.
23x63= 1449
27x59= 1593
Difference= 1593 – 1449= 144
My prediction was correct. By following the now familiar pattern I was able to predict a correct answer.
I am starting to see a pattern with all of the grids, the differences of all the opposite corners have been 1, for a 2x2 square, multiplied by the size of the grid, 4, for a 3x3 square, multiplied by the size of the grid and 9, for a 4x4 grid multiplied by the size of the grid.
12x12 Grid:
This is the largest size of grid I am using, and my using this I may be able to back up my theory that the differences of the opposite corners multiplied will always be the square numbers multiplied by the size of the grid.
I will find out the differences of the opposite corners of the 2x2, 3x3 and 4x4 squares within the 12x12 grid.
2x2 Squares:
34x47= 1598
35x46= 1610
Difference= 1610 – 1598= 12
106x119= 12614
107x118= 12626
Difference= 12626 – 12614= 12
52x62= 3380
53x64= 3392
Difference= 3392 – 3380= 12
All my results are 12, this coincides with my idea that the 2x2 square within a grid will always be 1² multiplied by the size of the grid. The answer is 1²x12, so therefore my idea is correct.
3x3 Squares:
4x30= 120
6x28= 168
Difference=168 – 120= 48
99x125= 12375
101x123= 12423
Difference= 12423 – 12375= 48
56x82= 4592
58x80= 4640
Difference= 4640 – 4592= 48
I have found that my theory for all 3x3 squares having a difference of 2² multiplied by the grid size is correct as all my results are the same and are in fact 2²(4) x 12.
4x4 Squares:
18x57= 1026
21x54= 1134
Difference= 1026 – 1134= 108
81x120= 9720
84x117= 9828
Difference= 9828 – 9720= 108
86x125= 10750
89x122= 10858
Difference= 10858 – 10750= 108
Again, all the results are the same and they coincide with the pattern 3² x size of the grid, 9 multiplied by 12 equals 108, the answer I got with every difference.
Table of Results:
[Prediction]
The pattern is the same as before, the same square numbers (1, 4 and 9...) are being multiplied by the size of the grid, in this case 12. If I were to do a 6x6 grid the differences would be half of the results I acquired here.
Prediction:
I predicted that the 5x5 square would have a difference of 192 by multiplying 16 (the next square number up from 9) by 12, the size of the grid. I did this because it follows on from my pattern before, the square numbers multiplied by the size of the grid.
I will now do an example to prove that my prediction of 192 is correct.
19x71= 1349
23x67= 1541
Difference= 1541 – 1349= 192
My prediction was correct. I knew the answer would be 192 as I followed the same pattern with all of the squares inside the grids, difference= the next square number x grid size.
Algebra:
Although I have found a pattern with all of the square sizes and I am sure it is correct, I need to use algebra to prove that it is correct. I will still be using the same size squares within the grid, 2x2, 3x3, and 4x4, and I will still be multiplying the top left by the bottom right and doing the same with the top right and bottom left and finding the difference. The only difference with what I will be doing is I am using letters instead of numbers to prove that our working is right.
10x10 Grids:
If I take a portion of a 10x10 grid:
And I draw a 4x4 square within this grid and work out the opposite corners.
36x69= 2484
39x66= 2574
Difference= 2574 – 2484= 90
I found 90 in the previous results but to prove that the theory is correct I can use algebra.
X x X+33= X(X+33)= X²+33X
X+3 x X+30= (X+3)(X+30)= X²+33X+90
F= XxX= X²
O= Xx30= 30X
I= 3xX= 3X
L= 3x30= 90
[30x+3X= 33X]
Difference= X²+33X – X²+33X+90= 90
This proves that the difference of the opposite corners multiplied of4x4 square in a 10x10 grid is definitely 90.
To check the accuracy of my theory I will do two more examples using a 2x2 square and a 3x3 square,
2x2 Squares:
6x17= 102
16x7= 112
Difference= 112 – 102= 10
XxX+11= X(X+11) = X²+11X
X+1xX+10= (X+10)(X+1)= X²+11X+10
F= XxX= X²
O= Xx1= 1X
I= 10xX= 10X
L= 1x10= 10
[1X+10X= 11X]
Difference= X²+11X – X²+11X+10= 10
This proves that in a 10x10 grid, a 2x2 square will all have opposite corners with differences of 10 when multiplied together.
3x3 Squares:
46x68= 3128
48x66= 3168
Difference= 3168 – 3128= 40
XxX+22= X(X+22)= X²+22X
X+2xX+20= (X+2)(X+20)= X²+22X+40
F= XxX= X²
O= Xx20= 20X
I= 2xX= 2X
L= 2x20= 40
[20X+2X= 22X]
Difference= X²+22X – X²+22X+40= 40
I have got the same answer as my number answer. My method of using algebra has made sure that the working I did previously is correct. So this means that a 3x3 square within a 10x10 grid will always have a difference of 40.
Table of Results:
This table of results will help me to compare the results from each square side and help me formulate a pattern.
My method of using algebra has made sure that the working I did previously is correct. So this means that a 3x3 square within a 10x10 grid will always have a difference of 40.
5x5 Grids:
I will now do the same with the 5x5 grid. If my theory is correct( 2x2=d=5, 3x3=d=20, 4x4=d=45) I will be left with differences of 5, 20 and 45.
2x2 Squares:
1x7= 7
2x6= 12
Difference= 12 – 7= 5
XxX+6= X(X+6)= X²+6X
X+1xX+5= (X+1)(X+5)= X²+6X-X²+6X+5
F= XxX= X²
O= Xx5= 5X
I= Xx1= 1X
L= 1x5= 5
[5X+1X=6X]
Difference= X²+6X – X²6X5= 5
The answer is the same as the answer I obtained by multiplying the opposite corners with numbers. I have used algebra to prove that the differences of a 2x2 square are 1²x5.
3x3 Squares:
13x25= 325
15x23= 345
Difference= 345 – 325= 20
XxX+12= X(X+12)= X²+12X
X+2xX+10= (X+2)(X+10)
F= XxX= X²
O= Xx10= 10X
I= 2xX= 2X
L= 2x10= 20
[10X+2X= 12X]
Difference= X²+12X – X²+12X+20= 20
This answer the same as the answer using numbers therefore proving that 20 is definitely the answer for the opposite corners of a 3x3 square, when multiplied, differences.
4x4 Squares:
6x24= 144
9x21= 189
Difference= 189 – 144= 45
XxX+18X(X+18)= X²+18X
X+3xX+15= (X+3)(X+15)= X²+18X+45
F= XxX= X²
O= Xx15= 15X
I= 3xX= 3X
L= 3x15= 45
[15X+3X= 18X]
Difference= X²+18X – X²+18X+45= 45
The difference is that same as the answers with number. The algebra method proves that the differences of the opposite corners multiplied of 4x4 square in a 5x5 grid will always be 45 (3²x5).
Table of Results:
This table of results is acting as an aid to help me find a pattern between the differences from the different grid sizes.
My working has shown the number 1, 4 and 9 again, just like in the previous results. Even with algebra these numbers are appearing showing that the answer to the opposite corners in the square size will always be a square number multiplied by the grid size. To help determine which square number it would be the number lower than the square size multiplied by the grid size, for example 4x4 square will have a difference of 3²xgrid size, 3 is one number lower than 4.
9x9 Grid:
I will be working out the opposite corners multiplied differences like I did before but I will then be working out the differences using algebra.
2x2 Squares:
31x41= 1271
32x40= 1280
Difference= 1280 – 1271= 9
XxX+10= X(X+10)= X²10X
X+1xX+9= (X+1)(X+9)= X²+10X+9
F= XxX= X²
O= Xx9= 9X
I= 1xX= 1x
L= 1x9= 9
[9X+1X= 10X]
Difference= X²+10X – X²+10X+9= 9
I have got the same result as my number result which shows that a 2x2 square within a 9x9 grid will always have a difference of 9.
3x3 Squares:
50x70= 3500
52x68= 3536
Difference= 3536 – 3500= 36
XxX+20= X(X+20)= X²+20X
X+2xX+18= (X+2)(X+18)= X²+20X+36
F= XxX= X²
O= Xx18= 18X
I= 2xX= 2X
L= 2x18=36
[18X+2X= 20X]
Difference= X²+20X – X²+20X+36= 36
The answer is the same, proving that no matter what the difference of a 3x3 square within a 9x9 grid will always be 36.
4x4 Squares:
11x41= 451
14x38= 532
Difference= 532 – 451= 81
XxX+30= X(X+30)= X²+30X
X+3xX+27= (X+3)(X+27)= X²+30X+81
F= XxX= X²
O= Xx27= 27X
I= 3Xx= 3X
L= 3x27= 81
[27X+3X= 30X]
Difference= X²+30X – X²+30+81= 81
Again my results have turned out to be all the same. This proves that the answer for 4x4 squares is 3²x9= 81.
Table of Results:
This table of results will help me compare the results and find a pattern and begin to find a formula relating the square size and grid size.
These results just back up my theory. Now I nearly have enough working to find a formula to work out the differences of the squares within a certain grid size.
12x12 Grids:
I will be using this grid size to check the accuracy of my working on the 12x12 grid. I will do a number calculation then go onto an algebraic method to prove that my answer is correct.
2x2 Squares:
70x83= 5810
71x82= 5822
Difference= 5822 – 5810= 12
XxX+13= X(X+13)= X²+13X
X+1xX+12= (X+1)(X+12)= X²+13X+12
F= XxX= X²
O= Xx12= 12X
I= 1xX= 1X
L= 1x12= 12
[12X+1X= 13X]
Difference= X²+13X – X²+13X+12= 12
I have got exactly the same results. This shows that a 2x2 square within a 12x12 grid will have a difference of 12.
3x3 Squares:
20x46= 920
22x44= 968
Difference= 968 – 920= 48
XxX+26= X(X+26)= X²+26X
X+2xX+24= (X+2)(X+24)= X²+26X+48
F= XxX= X²
O= Xx24= 24X
I= 2xX= 2X
O= 2x24= 48
[24X+2X=26X]
Difference= X²+26X – X²+26X+48= 48
My answers again have all turned out to be the same. This shows that there is a certain pattern and it proves that my previous working was correct.
4x4 Squares:
101x140= 14140
104x137= 14248
Difference= 14248 – 14140= 108
XxX+39= X(X+39)= X²+39X
X+3xX+36= (X+3)(X+36)= X²+39X+108
F= XxX=X²
O= Xx36=36X
I= 3xX= 3X
L= 3x36= 108
[36X+3X= 39X]
Difference= X²+39X – X²+39X+108= 108
All of my results are the same. This proves my previous working was correct, 3²x12= 108, which is the answer we were left with just now.
Table of Results:
This table shows all my results together, making it easier to compare. From my results now I am able to come up with a formula and confirm that all my work is correct.
Graph:
I am going to draw graphs to represent the differences I have found from the 2x2, 3x3, 4x4 and 5x5 squares inside of the 10x10, 12x12, 9x9 and 5x5 grids. The graph will be used to determine any differences, similarities and most importantly any patterns.
10x10 Grid:
5x5 Grid:
9x9 Grid:
12x12 Grid:
The graph I have drawn is the line y=X².
The numbers highlighted have appeared frequently throughout the investigation.
The line representing the 12x12 differences is the highest line and the line representing the 5x5 grid is the lowest line. The differences for 5x5 grid is exactly half of the 10x10 grid differences.
Formula:
Whilst using algebra a formula has been formulated. Throughout my work I constantly stated that the difference is the number lower than the grid sized is squared than multiplied by the grid size.
Difference= (square size – 1)² x grid size.
I will prove this formula by working out a difference I already know, 4x4 square within a 9x9 grid, using the formula.
D= s-1x grid size
= 4- 1
= 3²x9= 9x9
= 81
I know this is the right answer as I have worked it out before:
11x41= 451
14x38= 532
Difference= 532 – 451= 81
XxX+30= X(X+30)= X²+30X
X+3xX+27= (X+3)(X+27)= X²+30X+81
F= XxX= X²
O= Xx27= 27X
I= 3Xx= 3X
L= 3x27= 81
[27X+3X= 30X]
Difference= X²+30X – X²+30+81= 81
There is another, simpler formula for an easier way to work out the difference:
difference= (length-1)(width-1)x grid size.
I will work out another difference I already know using this formula.
2x2 square in a 10x10 grid:
d= (l-1)(w-1)x10
d= (2-1)(2-1)x10= 4-4+1= 1x10= 10
F= 2x2= 4
O= 2x-1= -2
I= -1x2= -2
L= -1x-1= 1
[-2-2= -4]
I know this is correct because I have worked it out before using numbers and algebra:
6x17= 102
16x7= 112
Difference= 112 – 102= 10
XxX+11= X(X+11) = X²+11X
X+1xX+10= (X+10)(X+1)= X²+11X+10
F= XxX= X²
O= Xx1= 1X
I= 10xX= 10X
L= 1x10= 10
[1X+10X= 11X]
Difference= X²+11X – X²+11X+10= 10
The formula has proven the differences without drawing the grids. This proves that there is a pattern and the pattern will carry on, the next square size will have a difference of the next square number multiplied by the grid size.
Conclusion:
In this investigation of opposite corners I have found that:
- A 2x2 square within an any sized grid will always have the difference of 1xgrid size / 1²xgrid size.
- A 3x3 square within an any sized grid will always have the difference of 4xgrid size / 2²xgrid size
- A 4x4 square within an any sized grid will always have the difference of 9xgrid size / 3²xgrid size.
And by using the method of prediction I have found that a 5x5 square within an any sized grid will always have a difference of 16xgrid size / 4²xgrid size.
To back up the results and to justify our theory, algebra has been used. We replaced numbers with algebra and a 2x2 square in a 10x10 grid would look like:
A 3x3 square in a 5x5 grid would look like:
A 4x4 square in a 9x9 grid would look like:
So basically the numbers have been replaced with an algebraic formula to see whether the results are right. I found out, using algebra, that:
- A 2x2 square within an any sized grid will always have the difference of 1xgrid size / 1²xgrid size.
- A 3x3 square within an any sized grid will always have the difference of 4xgrid size / 2²xgrid size
- A 4x4 square within an any sized grid will always have the difference of 9xgrid size / 3²xgrid size.
These results are the same as the number ones and using algebra has proved them correct.
There is a pattern with all the differences, 1 4 and 9, these are all square numbers and the line in the graph shows a line y=X². 1 is i², 4 is 2² and 9 is 3². The next sized grid sized grid will be the next square number and so on.
The formula that has been found has proven that the differences will always be square numbers multiplied by the grid size.
Extension- Rectangles and their opposite corners.
Introduction:
I have already investigated the opposite corners of squares in a grid and found that the differences are square numbers multiplied by square numbers. Now I am moving onto rectangles within a grid. I will try different sized grids and rectangles within the grid and work out the opposite corners, this is worked out by multiplying the top left by bottom right and finding the product of the top right by the bottom left and finding the difference.
10x10 Grid:
I will start by using a 10x10 grid, like with squares. If the pattern applies with differences of 10, 40 and 90.
I will draw a 3x2 and a 4x3 rectangle within this grid.
3x2 Rectangles:
33x45= 1485
35x43= 1505
Difference= 1505 – 1485= 20
88x100= 8800
90x98= 8820
Difference= 8820 – 8800= 20
I did more than one example to prove that my working out is accurate. Like with the squares all the answers are the same proving that it is correct and there may be a pattern.
4x3
12x35= 420
15x32= 480
Difference= 480 – 420= 60
63x86= 5418
66x83= 5478
Difference= 5478 – 5418= 60
Again I have got a set of results which are the same. This shows that again like the squares there may be a pattern.
5x4 Rectangles:
15x49= 735
19x45= 855
Difference= 855 – 735= 120
52x86= 4472
56x82= 4592
Difference= 4592 – 4472= 120
Table of Results:
I am going to compare the results and I will be finding out the pattern.
[Prediction]
With this table the pattern I have found is that as the number of rows in the rectangle increases the multiplier increases accordingly. I assume the multiplier increases accordingly with different sized grids.
Prediction:
I predicted that the 6x5 grid would have a difference of 200 by looking at the differences I was left with and how much difference they have from one another. I found that 20 went to 60 which is a difference of 40, then 60 goes to 120 which is 40, and then I predicted that the next number in the scale would be 80 as it coincides with the other numbers, as it is basically the 20 times table. I added 80 onto 120 and I was left with 200.
Also another way of working it out was I worked out what the difference divided into 20 by and I used that as an equation, for example 4x3 has a difference of 60, 60 equals 3x20. I found what 200 divided into 20 by and it was 10, none of the numbers link together although If you minus one of the multipliers by the multiplier in front of I, there seems to be a pattern:
1
3-1= 2
6-3= 3
[10-6= 4]
The difference goes up by one each time, one times table, so another pattern is that the difference will be the next number up.
I will now work out if my prediction is correct or not:
A portion of a 10x10 grid:
6x5 Rectangle:
32x77= 2464
37x72= 2664
Difference= 2664 – 2464= 200
My prediction was correct. This theory will help me with my algebra working.
Algebra:
10x10 grid:
I will be using the same method as before except I will be adding algebra into it, as I did before with the squares. The reason for using algebra is to check the accuracy of my work.
Again I will use the 3x2, 4x3 and 5x4 rectangles like before and I will work out the opposite corners and their differences using numbers and algebra.
3x2 Rectangles:
33x45= 1485
35x43= 1505
Difference= 1505 – 1485= 20
XxX+12= X(X+12)= X²+12X
X+2xX+10= (X+2)(X+10)= X²+12X+20X
F= XxX= X²
O= Xx10=10X
I= 2xX= 2X
L= 2x10= 20X
[10X+2X=12X]
Difference= X²+12X – X²+12X+20= 20
My results for both algebra and number are the same, this means that my previous working is correct and a 3x2 rectangle= 1x20.
4x3 Rectangles:
12x35= 420
15x32= 480
Difference= 480 – 420= 60
XxX+23= X(X+23)= X²+23X
X+3xX+20= (X+3)(X+20)= X²+23X+60
F= XxX= X²
O= Xx20= 20X
I= 3xX= 3X
L= 3x20= 60
[20X+3X=23X]
Difference= X²+23X – X²+23X+60= 60
I have got the same results again, proving that my calculations were correct in the first place.
5x4 Rectangles:
52x86= 4472
56x82= 4592
Difference= 4592 – 4472= 120
XxX+34= X(X+34)= X²+34X
X+4xX+30= (X+4)(X+30)= X²+34X+120
F= XxX= X²
O= Xx30=30X
I= 4xX= 4X
L= 30x4= 120
[30X+4X+34X]
Difference= X²+34X – X²+34X+120= 120
I have found that my results have turned out to be the same, showing that my working was correct but using algebra has made it so that all my working is accurate.
Conclusion:
Using rectangles in a 10x10 grid I have found that:
- A 3x2 rectangle will always have a difference of 20, 1x20
- A 4x3 rectangle will always have a difference of 60, 3x20
- A 5x4 rectangle will always have a difference of 120, 6x20
And by using the method of prediction I have found that a 6x5 rectangle will always have a difference of 200, 10x20, in a 10x10 grid.
After I worked that out, I then went on to using algebra, where I found that a 3x2 rectangle in a 10x10 grid would become this:
And still have a difference of 20 when the opposite corners were multiplied, like so:
XxX+12= X(X+12)= X²+12X
X+2xX+10= (X+2)(X+10)= X²+12X+20X
F= XxX= X²
O= Xx10=10X
I= 2xX= 2X
L= 2x10= 20X
[10X+2X=12X]
Difference= X²+12X – X²+12X+20= 20
And a 4x3 rectangle would look like this and still have a difference of 60:
XxX+23= X(X+23)= X²+23X
X+3xX+20= (X+3)(X+20)= X²+23X+60
F= XxX= X²
O= Xx20= 20X
I= 3xX= 3X
L= 3x20= 60
[20X+3X=23X]
Difference= X²+23X – X²+23X+60= 60
And a 5x4 rectangle could not contain the numbers, but algebra and the opposite corners will still have a difference of 120 when multiplied together:
XxX+34= X(X+34)= X²+34X
X+4xX+30= (X+4)(X+30)= X²+34X+120
F= XxX= X²
O= Xx30=30X
I= 4xX= 4X
L= 30x4= 120
[30X+4X+34X]
Difference= X²+34X – X²+34X+120= 120
I also managed to find a pattern, during this miniature investigation. Although the differences were not as concise as the differences for the squares, I managed to find a pattern using the rectangles in a 10x10 grid.
The pattern I have found is that as the number of rows in the rectangle increases the multiplier increases accordingly. I assume the multiplier increases accordingly with different sized grids.
Also I found that as the multiplier increases you can find the pattern by taking away the lower multiplier from the higher one and you are left with a pattern. So…
- As the first multiplier of 20 is one you are unable to subtract that from anything and the next multiplier is 3 so take 1 from that you are left with 2. The next multiplier up is 6, subtract 3 from that and you are left with 3. A row of consecutive numbers like so:
1
3-1= 2
6-3= 3
