5 x 5 Box
1 x 45 = 45 56 x 100 = 5600
> 160 > 160
5 x 41 = 205 60 x 96 = 5760
Page 3
Michelle Solley
I am going to put this information into a table to show my findings.
Column A shows the difference between the answers to each sum.
Column B shows the increase from the answers to the sums in column A.
As you can see I have predicted that the 6 x 6 box in a 10 x 10 grid has a difference of 250 between each answer. The increase in column B rises by 20 each time.
For the purposes of writing a rule in algebra I am calling the box B. For example:
This 2 x 2 box in a 10 x 10 grid will be called B.
This 3 x 3 box in a 10 x 10 grid will also be called B.
I am now going to find a rule for getting the difference without having to draw a 10 x 10 grid to check.
Using the Above table to refer to I have worked out the following:
2 x 2 Box
(2 – 1) x 10 = 10
3 x 3 Box
(3 – 1) x 10 = 40
4 x 4 Box
(4 – 1) x 10 = 90
Page 4
Michelle Solley
This rule works for all equal sided boxes in a 10 x 10 grid, the rule is:
(B-1) X 10
I am now going to try the same thing on a 9 x 9 grid to see if the rule will work for all equal sided boxes whatever the grid size.
2 x 2 Box
1 x 11 = 11 4 x 14 = 56
> 9 > 9
2 x 10 = 20 5 x 13 = 65
3 x 3 Box
7 x 27 = 189 55 x 75 = 4125
> 36 >
9 x 25 = 225 57 x 73 = 4161
4 x4 Box
19 x 49 = 931 42 x 72 = 3024
> 81 > 81
22 x 46 = 1012 45 x 69 = 3105
I will now continue by drawing a 5 x 5 box in a 9 x 9 grid and predict what the differences will be for a 6 x 6 box in a 9 x 9 grid.
Page 5
Michelle Solley
5 x 5 Box
1 x 41 = 41 41 x 81 = 3321
> 144 > 144
5 x 37 = 185 45 x 77 = 3465
As I have done before I will place the information I have gathered into a table, and as above Column A shows the difference between the answers to each sum. Column B shows the increase from the answers to the sums in column A.
As you can see I have predicted that the 6 x 6 box in a 9 x 9 grid has a difference of 225 between each answer. The increase in column B rises by 18 each time.
The pattern I found for the 9 x 9 grid is similar to the pattern I found for the 10 x 10 grid and therefore this proves that whatever the grid size the rule I found for the 10 x 10 grid will be the same.
Page 6
Michelle Solley
Once again, using the table on page 5 to refer to the change in the rule is as follows:
2 x 2 Box
(2 – 1) x 9 = 9
3 x 3 Box
(3 – 1) x 9 = 36
4 x 4 Box
(4 – 1) x 9 = 81
The rule I have found works for all equal sided boxes in any size grid.
For the purposes of writing the final rule in algebra I am calling all grid sizes (10 x 10, 9 x 9, 8 x 8 etc) ‘G’ the final rule for boxes in any size grid is:
(B-1) G
Now that I have found a rule for equal sided boxes, I am going to look at rectangles in the same way. I will start with a 10 x 10 grid once again.
Page 7
Michelle Solley
2 x 3 Rectangle
1 x 22 = 22 4 x 25 = 100
> 20 > 20
2 x 21 = 42 5 x 24 = 120
2 x 4 Rectangle
6 x 37 = 222 9 x 40 = 360
> 30 > 30
7 x 36 = 252 10 x 39 = 390
2 x 5 Rectangle
31 x 72 = 2232 34 x 75 = 2550
> 40 > 40
32 x 71 = 2272 35 x 74 = 2590
2 x 6 Rectangle
46 x 97 = 4462 49 x 100 = 4900
> 50 > 50
47 x 96 = 4512 50 x 99 = 4950
My investigation of rectangles in a 10 x 10 grid has shown me that the answers to the sums increase in 10’s. As with the boxes I will show my findings in a table, Column A shows the difference between the answers to each sum. Column B shows the increase from the answers to the sums in column A.
Page 8
Michelle Solley
As you can see on page 7 I have predicted that the 2 x 7 rectangle in a
10 x 10 grid has a difference of 60 between each answer. The increase in the answers to the sums is 10.
I am now going to find a rule for rectangles and getting the difference without having to draw a 10 x 10 grid to check. I believe that the evidence in the table on page 7 will show a new rule for rectangles. Using the table to refer to I have worked out the following:
(2 – 1) x (3 – 1) x 10 = 20
(2 – 1) x (4 – 1) x 10 = 30
(2 – 1) x (5 -1) x 10 = 40
To change this rule into algebra I am calling 2 ‘W’ for width and 3,4 or 5 ‘L’ for length.
(W – 1) (L - 1) x 10
Based on my findings for boxes I believe that the above rule for rectangles will be the same no matter what the grid size. My prediction for the final rule is as follows:
(W – 1) (L – 1) G
I will now prove this by investigating further. I am going to input several rectangles into a 9 x9 grid.
Page 9
Michelle Solley
2 x 3 Rectangle
1 x 20 = 20 5 x 24 = 120
> 18 > 18
2 x 19 = 38 6 x 23 = 138
2 x 4 Rectangle
3 x 31 = 93 7 x 35 = 245
> 27 > 27
4 x 30 = 120 8 x 34 = 272
2 x 5 Rectangle
39 x 76 = 2964 44 x 81 = 3564
> 36 > 36
40 x 75 = 3000 45 x 80 = 3600
2 x 6 Rectangle
28 x 74 = 2072 32 x 78 = 2496
> 45 > 45
29 x 73 = 2117 33 x 77 = 2541
This investigation of rectangles in a 9 x 9 grid has shown me that the answers to the sums increase in 9’s. As with the 10 x 10 grid I will show my findings in a table, Column A shows the difference between the answers to each sum. Column B shows the increase from the answers to the sums in column A.
Page 10
Michelle Solley
My findings have proved that my prediction for the rule for rectangles is correct. As I did with the 10 x 10 grid I will show my workings that prove the rule.
(2 – 1) (3 – 1) x 9 = 18
(2 – 1) (4 – 1) x 9 = 27
As you can see I have predicted the answer to the 2 x 7 rectangle
(2 – 1) (7 – 1) x 9 = 54
My investigation has proved that there is a rule for finding the answer to a sum created by any equal sided box in any size grid. Once again, the rule is:
(B – 1) G
The investigation also proved that there is a rule for finding the answer to a sum created by any rectangle. The rule is:
(W – 1) (L – 1) G
