Multiplying Opposite Corners in a Grid

I have a10x10 grid. The task is to select various size squares and rectangles, I will then be multiplying the opposite corners together then subtracting the results to get the difference between the two answers. I hope to find patterns and rules in the results, if I can find patterns I will use algebra to describe them. My aim is to find the general rule for any size rectangle on any size grid.Plan- investigate different squares on a 10×10 grid.

- Find an algebraic rule.- Repeat for rectangles on a 10×10 grid.- Adapt the rule for squares to fit rectangles.- Change the size of the grid- Find an overall rule for any rectangle on any grid.25 26 25 x 36 = 90035 36 26 x 35 = 910910 – 900 = 10The difference between the two answers is 10.1 2 1 x 12 = 1211 12 2 x 11 = 2222 – 12 = 10The difference between the two answers is 10.72 73 72 x 83 = 597682 83 73 x 82 = 59865986 – 5876 = 10The difference between the two answers is 10.49 50 49 x 60 = 294059 60 50 x 59 = 2950The difference between the two answers is 10.12 13 14 12 x 34 = 40822 23 24 14 x 32 = 44832 33 34 448 – 408 = 40The difference between the two answers is 40.

78 79 80 78 x 100 = 780088 89 90 80 x 98 = 784098 99 100 7840 – 7800 = 40The difference between the two answers is 40.44 45 46 44 x 66 = 290454 55 56 46 x 64 = 294464 65 66 2944 – 2904 = 40The difference between the two answers is 40.7 8 9 10 7 x 40 = 28017 18 19 20 10 x 37 = 37027 28 29 30 370 – 280 = 9037 38 39 40The difference between the two answers is 90.61 62 63 64 61 x 94 = 573471 72 73 74 64 x 91 = 582481 82 83 84 5824 – 5734 = 9091 92 93 94The difference between the two answers is 90.1 2 3 4 1 x 34 = 3411 12 13 14 4 x 31 = 12421 22 23 24 124 – 34 = 9031 32 33 34The difference between the two answers is 90.67 68 69 70 67 x 100 = 670077 78 79 80 70 x 97 = 679087 88 89 90 6790 – 6700 = 9097 98 99 100The 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.Size of SquareThe difference2 x 23 x 34 x 4104090I now need to look for a pattern which is within the results.

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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 10But 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 =160This proves that my formula worksI am now going to go through all the previous steps but adapting it by using rectangles instead of squares.2 3 2 x 13 = 2612 13 3 x 12 = 3636 – 26 = 10The difference between the two answers is 10.6 7 6 x 27 = 16216 17 7 x 26 = 18226 27 182 – 162 = 20The difference between the two numbers is 2069 70 69 x 100 = 690079 80 70 x 99 = 693089 90 6930 – 6900 = 3099 100The differnec between the two answers is 3031 3241 42 31 x 72 = 223251 52 32 x 71 = 227261 62 2272 – 2232 = 4071 72The difference between the two answers is 40Size of SquareThe difference2 x 32 x 42 x 5203040The pattern shows that all the differences in the answers go up in 10’s.2 3 4 2 x 34 = 6812 13 14 4 x 32 – 12822 23 24 128 – 68 = 6032 33 3428 29 3038 39 40 28 x 70 = 196048 49 50 30 x 68 = 204058 59 60 2040 – 1960 = 8068 69 70Size of SquareThe difference3 x 43 x 56080The 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 2736 3746 4756 5766 6776 77Using my formula I think the difference will be2 – 1 = 1 1 x 5 = 56 – 1 = 55 x 10 = 50 using the formula I predict the the difference will be 50.10(2 -1)(6 -1) = 10 x 5 = 50This 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.7 812 1317 18By using my previous formula I am going to guess what the difference is going to be.2 – 1 = 13 – 1 = 21 x 2 = 22 x 5 = 5. I predict the number 5 will be the difference.7 x 18 = 1268 x 17 = 136136 – 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: