Number Grid
...ht of 2. The difference increases in increments of 10. Also, each line of the grid is 10 squares wide so the next square vertically straight down the is ten more than the square above it. Possible Formulas L=Lenght H=Height D=Difference L-1 x 10 or The length subtracted by one multiplied by ten. But from past experience I doubt this rule would work with other sized rectangles without 2 as a side length L-1 x 5H or The length subtracted by one multiplied by five multiplied by the height. The lowest corner subtracted from the highest corner subtracted by one, but only when the rectangle is aligned so that the shortest sides are at the top and base. Do These Rules Work With Other Rectangles? 3 x X I will now look at rectangles where one of the sides has a length of 3 squares and see if the same patterns apply and investigate whether the above rules are true for rectangles of any shape. I believe I shall require only one example of each and will proceed on this principle. 3 x 3 3 x 4 3 x 5 What Do I Notice? The “L-1 x 5H” does not apply. But L-1 x 20 does. I feel confidant that L-1 is correct but I still need to find how H can be used to calculate what to multiply L-1 by. For 2 x X L-1 x 10 For 3 x X L-1 x 20 Could the universal rule be: D = L-1 x 10 x (H-1)? Subtract one from the height, multiply by 10 and then multiply by the height subtracted by one? Testing the Rule 4 x X I will now look at 4 x X and see if my new found rule truly encompasses all rectangles. 4 x 4 4 x 5 The difference is advancing by 30. 4 x 6 will have a difference of 150. 4 x 6 (table calculated with a calculator) Could the x 10 in my rule be related to width of my number grid, which is ten squares wide? To test this idea I will try my rule on a different sized grid. The Answer is forty, my rule works for any rectangle, on any grid. L-1 x (Grid Width) x (H-1) or Subtract one from the height, multiply by the width of the grid and then multiply by the height subtracted by one. Using Algebra to Look Closer: Calculating the rectangle’s values in relation to “X” If the number in the top left hand corner is X, then the following squares will be.... (G = Grid width) For a 3 x 2 rectangle when the first square is X the difference is: difference for this size of rectangle is: 2G, the width of the grid multiplied by two. This is correct as from my previous working on a rectangle of these proportions I know that on a 10 x 10 grid the difference is 20 (10 x 2 !) For a 4 x 2 rectangle when the first square is X the difference is: difference for this size of rectangle is 3G, the width of the grid multiplied by three. This is correct as from my previous working I know that on a 10 x 10 grid the difference is 30 (10 x 3 !) How can I produce an equation from the side lengths of this rectangle to give me 3? Possible equations: or I will look at the ?G rule in other rectangles to see if any of my above rules also apply and are true for rectangles across the board. the difference for this size of rectangle is 6G, the width of the grid multiplied by six. This is correct as from my previous working I know that on a 10 x 10 grid the difference is 60 (10 x 6 !) None of my rules apply, but what I can now build a table of my results: This must be because the H has increased by 1 also. This tells me that the H must be connected to the difference. But How? How can I get 2 from 2 & 3, and 3 from 2 & 4 using the same formula? If I subtract 1 from each of 2 & 3 I get: L=1 H=2 1 x 2 = 2. When I subtract one from each 2 & 4 I get: L=1 H=3 1 x 3 = 3 The answers to both of these short sums give me the figure to multiply G with to get the difference. I predict that this will work for 3 & 4: 3 - 1 = 2 4 - 1 = 3 2 x 3 = 6 So L-1 x H-1 x G = Difference ...