Opposite Corners Essay Example

Opposite Corners Essay Wx LDifferenceIncreaseWx LDifferenceIncrease2 x 320106 x 3100502 x 430106 x 4150502 x 540106 x 5200502 x 650106 x 6250502 x 760106 x 7300502 x 870106 x 8350502 x 980106 x 9400502 x 1090106 x 1045050We are investigating the difference between the products of the numbers in the opposite corners of any rectangles that can be put on a 100 square.2 x 3 Rectangles123111213To keep things simple I have started with rectangles with a width of 2 squares. I kept the width to two squares and increased the length by one square. (see results table above). I discovered that the width increases by 10 every time the length increases by 1.The difference can be worked out for all rectangles with a width of 2 squares by using several formulas:1. (Length 1 x 10 = Z)3 1x 10 = 20 = ZThen(Width x Z ) Z = difference of opposite corners2 x 20 20 = 20OR2. L = Length, W = Width(L 1) (10 (W-1)) = difference of opposite cornersExample:(3 1) x (10 (2 1)) = 20OR3.123111213Using algebra and going on the theor y that the width increases by 10 when the length is increased by 1, I have calculated the value of the corners. This formula can also work out the difference.(y+10) (y + 2) = y+ 20 +10y+ 2y= y + 20 +12yy ( y + 12)= y + 12y(y + 20 +12y) (y + 12y) = difference between product.Extending the problemThe difference between the opposite corners will still be the same even if you make a billion square grid because the length will still increase by 1 and the width will increase by 10.6 x 3 Rectangles123456111213141516212223242526I have changed the size of the rectangle to see if my formulas will work for it. (The results are in the table at the top of the first page)1. 6 1 x 10 = 50Then3 x 50 50 = 1002. (6 1) x (10 (3 1) = 1003.123456111213141516212223242526(y+20) (y + 5) = y+ 100 +20y+ 5y= y + 100 +25yy ( y + 25)= y + 25y(y + 100 +25y) (y + 25y) = difference between product.= 100ConclusionI have come to the conclusion that my three formulas work for all types of rectangles and squares . There are several ways to achieve the end result for the difference of the opposite corners.