I am going to investigate taking a square of numbers from a grid, multiplying the opposite corners and finding the difference of these two results. To start I used a 5x5 grid:
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Introduction
Maths Coursework
Number Grids
I am going to investigate taking a square of numbers from a grid, multiplying the opposite corners and finding the difference of these two results.
To start I used a 5x5 grid:
1  2  3  4  5 
6  7  8  9  10 
11  12  13  14  15 
16  17  18  19  20 
21  22  23  24  25 
Then I took 2x2 squares out of this grid and multiplied the opposite corners, to find the difference:
1  2 
6  7 
1 x 7 = 7
2 x 6 = 12
12 – 7 = 5
So the difference between the answers is 5. Next I took another 2x2 square from the same grid.
19  20 
24  25 
19 x 25 = 475
20 x 24 = 480
480 – 475 = 5
The difference is 5 again. I thought this could mean that all 2x2 squares in a 5x5 grid would come out with a difference of 5. To check this, I took another 2x2 square out of the grid to check.
12  13 
17  18 
12 x 18 = 216
13 x 17 = 221
221 – 216 = 5
So this shows that my prediction was right and every 2x2 square in a 5x5 grid should come out with a difference of 5.
Any 2x2 square in a 5x5 grid = Difference of 5
Now I am going to start taking 2x2 squares out of a 6x6 grid.
1  2 
7  8 
1 x 8 = 8
2 x 7 = 14
14 – 8 = 6
The difference is 6.
21  22 
27  28 
21 x 28 = 588
22 x 27 = 594
594 – 588 = 6
Again the difference is 6. So I can see like in the 5x5 grid there is a pattern. If I am right every 2x2 square in a 6x6 grid should have a difference of 6. To check if I am right I will take one more square out of the grid.
16  17 
22  23 
16 x 23 = 368
17 x 22 = 374
374 – 368 = 6
This shows that my hypothesis is right and every 2x2 square in a 6x6 grid will have a difference of 6.
Any 2x2 square in a 6x6 grid = Difference of 6
Middle








In this 3x3 square the only numbers in the grid we need to use are the corner numbers, so when drawing the 3x3 squares, with expressions in, from now on I will leave those middle numbers out.
x   
   
To find the top right corner I will look at a 3x3 square taken from a 5x5 grid.
1  2  3 
6  7  8 
11  12  13 
From this I have seen that the top right corner, is two more than the top left corner. This is the same in any other 3x3 square in any grid. So using the top left corner, x, we can say that the top right corner is x + 2.
x  x+2 
   
Now I’ll look at some 3x3 squares in different grids to find the expression for the bottom left corner.
3x3 square in a
5x5 grid:
1  2  3 
6  7  8 
11  12  13 
Looking at this example you can see that the bottom left corner is, 2 times the grid size more, than the top left corner. So it is 2g more than the top left corner. Meaning we can give the bottom left corner the expression, x + 2g.
x  x+2 
x+2g   
Then to find the last one like in the 2x2 square we add the same increment to the bottom left as we did to the top left to get the top right.
So the bottom right expression will be x + 2g + 2.
x  x+2 
x+2g  x+2g+2 
Then like the normal squares we just multiply the corners.
x(x+2g+2) = x²+x2g+2x
(x+2)(x+2g) = x²+x2g+2x+4g
Then take the answers away from each other to get the expression for the difference.
(x²+x2g+2x+4g)  (x²+x2g+2x) = 4g
So this means for a 3x3 square in any grid, the difference can be worked out by using this algebraic expression:
d = 4g
Now I have this I am going to try it on one of the 3x3 squares just to make sure it is right. I will use the equation to calculate the difference of a 3x3 square in a 9x9 grid, then I will actually use a real 3x3 square from a 9x9 grid to check if the equation is correct.
1  2  3 
10  11  12 
19  20  21 
d = 4x9 = 36
So if the equation is right the difference will be 36.
1 x 21 = 21
3 x 19 = 57
57 – 21 = 36
So the difference is 36 and the equation worked it out correctly. Now I know that I can use this equation for any 3x3 square in any square grid.
Now I am going to look at 4x4 squares.
Again we can call the first corner ‘x’.
x   
   
If this is the top line of a 4x4 square then we can just work out how much the top right corner will be more than x.
x       
So the top right corner is just x+3
x  x+3 
   
To find the bottom left corners expression I am going to have to look at some examples.
1  2  3  4 
6  7  8  9 
11  12  13  14 
16  17  18  19 
4x4 square in a:
5x5 grid:
1  2  3  4 
7  8  9  10 
13  14  15  16 
19  20  21  22 
Looking at this example you can see that the bottom left corner is, 3 times the grid size more, than the top left corner. So it is 3g more than the top left corner. Meaning we can give the bottom left corner the expression, x + 3g.
x  x+3 
x+3g   
Then the last corners expression is just, x + 3g + 3.
x  x+3 
x+3g  x+3g+3 
Conclusion
d = g(n1)²
This is the same as:
d = g(n1)(n1)
I am going to work out the difference for a 4x2 rectangle in a 6x6 square grid. First I will work it out using the above equation then work it out by actually taking the numbers and multiplying them out then getting the difference.
D = 6(41)(21)
D = 6 x 3
D = 18
So if the equation is right then the difference of the 4x2 rectangle in a 6x6 square grid should be 18.
Now I will take a 4x2 rectangle out of a 6x6 square grid.
8  9  10  11 
14  15  16  17 
8 x 17 = 136
11 x 14 = 154
154 – 136 = 18
The difference was 18. So the equation worked it out correctly.
To check if the equation is really correct I will take one more rectangle and test the equation with it.
A 6x3 rectangle from the above 6x6 square grid.
D = 6(61)(31)
D = 6 x 10
D = 60
1  2  3  4  5  6 
7  8  9  10  11  12 
13  14  15  16  17  18 
1 x 18 = 18
6 x 13 = 78
78 – 18 = 60
So the difference is 60 like the equation worked out.
Now I have double checked it, I am quite sure that the equation is correct. So to work out the difference of any rectangle in any square grid you just need to use this equation:
d = g (11)(w1)
Just to check this equation I will take a 4x3 rectangle from an 8x8 grid, work out its difference using the equation, then checking it by doing it properly.
12  13  14  15 
20  21  22  23 
28  29  30  31 
d = 8(41)(31) = 8 x 3 x 2 = 48
So if the equation is correct then the difference will be 48.
12 x 31 = 372
15 x 28 = 420
420 – 372 = 48
The difference is 48 so the equation is correct and we can use it to work out the difference for any rectangle in any square grid.
Mark Kirkwood
This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.
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Here's what a teacher thought of this essay
This is a solid investigation. There are patterns identified and developed with the use of algebra. This algebra however is kept to a fairly basic level. The algebra is clearly limited to experimental examples but this linkage should be more extensive. Specific strengths and improvements have been suggested throughout.
Marked by teacher Cornelia Bruce 18/04/2013