5986 – 5876 = 10
The difference between the two answers is 10.
49 50 49 x 60 = 2940
59 60 50 x 59 = 2950
The difference between the two answers is 10.
12 13 14 12 x 34 = 408
22 23 24 14 x 32 = 448
32 33 34 448 – 408 = 40
The difference between the two answers is 40.
78 79 80 78 x 100 = 7800
88 89 90 80 x 98 = 7840
98 99 100 7840 – 7800 = 40
The difference between the two answers is 40.
44 45 46 44 x 66 = 2904
54 55 56 46 x 64 = 2944
64 65 66 2944 – 2904 = 40
The difference between the two answers is 40.
7 8 9 10 7 x 40 = 280
17 18 19 20 10 x 37 = 370
27 28 29 30 370 – 280 = 90
37 38 39 40
The difference between the two answers is 90.
61 62 63 64 61 x 94 = 5734
71 72 73 74 64 x 91 = 5824
81 82 83 84 5824 – 5734 = 90
91 92 93 94
The difference between the two answers is 90.
1 2 3 4 1 x 34 = 34
11 12 13 14 4 x 31 = 124
21 22 23 24 124 – 34 = 90
31 32 33 34
The difference between the two answers is 90.
67 68 69 70 67 x 100 = 6700
77 78 79 80 70 x 97 = 6790
87 88 89 90 6790 – 6700 = 90
97 98 99 100
The difference between the two answers is 90.
Now that I have my results from each of the different size squares I done, I will put them into the table.
I now need to look for a pattern which is within the results.
After looking at the results I notice that if I ignore the noughts on the results, for example 1, 4, 9 . . . . . these numbers are all square numbers.
For the nth term, squaring must be involved.
When I square the number in the first column, I get the answer in the second column but on the line below. So instead of squaring n, I need to square (n – 1).
To get the nought, the answer will need to be multiplied by 10.
So the formula is: (n – 1) x 10
But in better form the formula would look like this: 10(n – 1)
The pattern tells me that the next number will be 160. I am now going to test my prediction.
10 x (5 – 1) = 10 x 16 =160
This proves that my formula works
I am now going to go through all the previous steps but adapting it by using rectangles instead of squares.
2 3 2 x 13 = 26
12 13 3 x 12 = 36
36 – 26 = 10
The difference between the two answers is 10.
6 7 6 x 27 = 162
16 17 7 x 26 = 182
26 27 182 – 162 = 20
The difference between the two numbers is 20
69 70 69 x 100 = 6900
79 80 70 x 99 = 6930
89 90 6930 – 6900 = 30
99 100
The differnec between the two answers is 30
31 32
41 42 31 x 72 = 2232
51 52 32 x 71 = 2272
61 62 2272 – 2232 = 40
71 72
The difference between the two answers is 40
The pattern shows that all the differences in the answers go up in 10’s.
2 3 4 2 x 34 = 68
12 13 14 4 x 32 - 128
22 23 24 128 – 68 = 60
32 33 34
28 29 30
38 39 40 28 x 70 = 1960
48 49 50 30 x 68 = 2040
58 59 60 2040 – 1960 = 80
68 69 70
The pattern shows that all the differences in the answers go up in 20’s.
For the squares the dimensions are the same, for example.
10(n-1)(n-1)
But for rectangles the dimensions will be different so I have to introduce a new letter. For example.
10(n-1)(m-1)
To test this I will use a 2 x 6 rectangle:
26 27
36 37
46 47
56 57
66 67
76 77
Using my formula I think the difference will be
2 – 1 = 1 1 x 5 = 5
6 – 1 = 5
5 x 10 = 50 using the formula I predict the the difference will be 50.
10(2 -1)(6 -1) = 10 x 5 = 50
This proves my formula works.
I have a theory that the reason I have to times by 10 is because I am usining a 10 x 10 grid. So I am going to test this theory by changing the size of the grid to a 5 x 5 grid, to see if the number I have to multiply by is 5.
I am going to test this theory on a 2 x 3 rectangle.
7 8
12 13
17 18
By using my previous formula I am going to guess what the difference is going to be.
2 – 1 = 1
3 – 1 = 2
1 x 2 = 2
2 x 5 = 5. I predict the number 5 will be the difference.
7 x 18 = 126
8 x 17 = 136
136 – 126 = 10 this provesmy theory was correct.
I have noticed the number you times it by is the size of the grid, so therefore to work out any rectangle on any size grid the formula would be:
g(n-1)(m-1)