Number Grid Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results
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Introduction
Amanda Gaber
March 2006
Mathematics Coursework
Number Grid
Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results. I will start off by working this out on a 2x2 square in a 10 x 10 grid and then will investigate varying the widths and lengths of squares and rectangles.
Method: In order to simplify the process, the investigation has been divided into sections according to the size of squares in the grids.
2 x 2 Squares
To begin with, I used a 10 x 10 grid, looking at 2 x 2 squares:
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 | 26 | 27 | 28 | 29 | 30 |
31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Then I took 2 x 2 squares from this grid and multiplied the opposing corners to calculate the difference between the two products.
1 | 2 |
11 | 12 |
1 x 12 = 12
2 x 11 = 22
∴ 22 – 12 = 10
So the difference between the answers is 10. I then took another 2 x 2 box from the above 10 x 10 grid:
47 | 48 |
57 | 58 |
47 x 58 = 2726
48 x 57 = 2736
∴ 2736- 2726 = 10
The difference is 10 again. Perhaps this means that because it is a 10 x 10 grid, that all the differences would be 10. I would still like to further investigate this theory.
The grid below is once again a 2 x 2 box derived from the original 10 x 10 grid.
89 | 90 |
99 | 100 |
89 x 100 = 8900
90 x 99 = 8910
∴ 8910 – 8900 = 10
This once again confirms what I stated; that the difference between the products of cross-multiplied boxes will always equal 10 in a 10 x 10 grid.
I would like to determine if this is definitely correct, so I am going to do it again twice.
27 | 28 |
37 | 38 |
27 x 38 = 1026
28 x 37 = 1036
∴ 1036 – 1026 = 10.
43 | 44 |
53 | 54 |
43 x 54 = 2322
44 x 53 = 2332
∴ 2332 – 2322 = 10.
Middle
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3620
As before, the highlighted boxes are the ones I am going to calculate, to see if the difference between the product of multiplying the opposite corners, equals the size of the grid. In this case, I am looking for that number to be 19.
- 1 x 21 = 21
2 x 20 = 40
∴ 40 – 21 = 19
- 32 x 52 = 1664
33 x 51 = 1683
∴1683 – 1664 = 19
- 177 x 197 = 34869
178 x 196 = 34888
∴ 34888 – 34869 = 19
- 302 x 322 = 97244
303 x 321 = 97263
∴ 97263 – 97244 = 19
- 305 x 325 = 99125
306x 324 = 99144
∴ 91444 – 99125 = 19
Once again, I can conclude that the difference between the cross multiplied products is the size of the grid; 19.
To confirm this, the number 19 has been inserted into the formula to prove that this is correct.
n | n + 1 |
n + g | n + g + 1 |
n(n + g + 1) = n2 + ng + n
n + 1(n + g) = n2 + ng + n + g
∴ [n2 + ng + n +g] – [n2 + ng + n ] = g
where g = 19;
n(n + 19 + 1) = n2 + 19n + n
n + 1(n + 19) = n2 + 19n + n + 19
∴ [n2 + 19n + n +19] – [n2 + 19n + n] = 19
My prediction of what the difference would be was correct. So for any 2x2 square taken from any size grid, the difference will be the number of the grid size.
I can now work out the difference of any 2x2 square as long as I know the grid size.
5 x 5 grid = Difference of 5
19 x 19 grid = Difference of 19
100 x 100 grid = Difference of 100
31 x 31 grid = Difference of 31
58 x 58 grid = Difference of 58
1000x1000 grid = Difference of 1000
3 X 3 Squares
Having studied 2 x 2 boxes within different sized grids, I would like to study 3 x3 boxes within grids of varying sizes to see if there is an emerging pattern.
I have decided to look at an 8 x 8 grid first to see if there is a pattern.
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 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |
33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |
49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 |
57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 |
I am going to leave out all of the numbers in the middle of the 3 x 3 box and will concentrate on the four corners of this box.
Firstly, I need to work out the difference of the products for the highlighted 3 x 3 box.
26 x 44 = 1144
28 x 42 =1176
∴ 1176 – 1144 =32
From this, I can see that the difference is not the size of the grid (g). However, I will formulate an expression to predict the size of the difference for any 3 x 3 box within an 8 x8 grid,
The first top left hand box is shown as ‘n’ and the size of the grid is still ‘g’. The top right hand corner of the box is 2 more than the top left hand corner, so this is n+ 2. The bottom left hand corner of the box is two rows down exactly, so this n + 2g and finally, the bottom right hand corner of the box is n + 2g + 2.
Therefore;
n | n + 2 |
n + 2g | n + 2g+ 2 |
After multiplying the diagonal corners of the box, the algebraic formula is as follows:
(n )( n + 2g + 2) = n2 + 2ng + 2n
(n + 2)(n + 2g) = n2 + 2ng +2n + 4g
Then subtract either sides of the equation from each other to confirm what the difference is.
(n2 + 2ng +2n + 4g) – (n2 + 2ng + 2n) = 4g
This is accurate because the 8 x8 grid had a difference between the products of 32, which is equal to 4 x 8. Therefore, I can confirm that, according to my calculations, the difference between the products of the diagonal corners is 4g.
I will now use a different 3 x 3 box within the same 8 x8 grid to see if this rule remains the same;
6 | 7 | 8 |
14 | 15 | 16 |
22 | 23 | 24 |
Here, I will also multiply the corners only so:
6 x 24 = 144
8 x 22 = 176
∴ 176 – 144 = 32.
This proves that my prediction is correct.
I will now investigate if this formula for 3 x 3 boxes works in a different sized 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 | 26 | 27 | 28 |
29 | 30 | 31 | 32 | 33 | 34 | 35 |
36 | 37 | 38 | 39 | 40 | 41 | 42 |
43 | 44 | 45 | 46 | 47 | 48 | 49 |
The box I have highlighted will be used to work out the product of the corners and the difference between them.
1 x 17 = 17
3 x 15 = 45
∴45 – 17 = 28.
This fits the pattern, as 4 x 7 is equal to 28, where 7 is the grid size.
4 X 4 Square
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 | 26 | 27 | 28 | 29 | 30 |
31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
- 4 x 31 = 124
1 x 34 = 34
∴124 – 34 = 90
- 24 x 57 = 1368
27 x 54 = 1458
∴ 1458 – 1368 = 90
- 61 x 94 = 5734
64 x 91 = 5824
∴ 5824 – 5734 = 90
- 67 x 100 = 6700
70 x 97 = 6790
∴ 6790 – 6700 = 90
In a 10 x 10 grid, the difference between the diagonal corners of a 4 x 4 square is always 90.
2 x 3 Rectangle
As yet, I am unable to determine a general formula to predict the difference between multiplying the diagonals of a square or a rectangle in any sized grid. I will therefore investigate the difference between the products of the diagonal corners of a rectangle.
I am going to begin with a 2 x 3 rectangle in a 10 x 10 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 | 26 | 27 | 28 | 29 | 30 |
31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
Conclusion
If the number in the top left hand corner is n then the following squares will be…
For a 2 x 3 rectangle when the first square is n the difference is:
n | n+2 |
n+G | n+2+G |
(n+2) (n+G) = n2 + nG + 2n + 2G
n (n+G+2) = n2 + nG + 2n
∴ (n+2) (n+G) - n (n+G+2) = 2G
The algebraic expression for the difference of a 2 x 3 rectangle is 2G, the width of the grid multiplied by two.
For a 2x4 rectangle when the first square is n the difference is:
n | n+3 |
n+G | n+3+G |
(n+3) (n+G) = n2 + nG + 3n + 3G
n (n+G+3) = n2 + nG + 3n
∴ (n+3) (n+G) - n (n+G+3) = 3G
The algebraic expression for the difference for this size of rectangle is 3G, the width of the grid multiplied by three.
From this information, I have been able to tabulate my results:
L | H | nG |
2 | 3 | 2 |
2 | 4 | 3 |
3 | 5 | 6 |
From the table above, it is apparent that between 2 x 3 and 2 x 4 rectangles, nincreases by 1. This could be due to the height increasing by 1. I think that the height is probably related in some way to the difference value.
Evaluation:
I have formulated the algebraic formula that can predict the difference for any sized rectangle or square for any sized grid. Having investigated this observation, I can only ascertain that this is correct and that;
L – 1 x G (H -1)
Perhaps further investigations on different grid sizes would be suitable. Also, given more time, I would have liked to look at different shapes in different sized grids and to determine a general formula for them. For example, I would like to know if the same formula fits a rhombus on a 10 x 10 grid.
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 well written piece of work with only a couple of minor errors. This piece of work shows a good application of some algebraic techniques. There are specific strengths and improvements suggested throughout.
Marked by teacher Cornelia Bruce 18/04/2013