This problem solving activity has an algebra focus.
A square is transformed by increasing its length and decreasing its other side by the same percentage amount.
How does the area change?
Suppose a square is transformed by increasing its length by 10% and decreasing its adjacent side by the same amount.
What is the resultant change in area?
What if the length and adjacent side of the original square were increased and decreased by 20% respectively?
How might you quickly determine the change in area of a square given any percentage increase in length and a decrease by the same percentage in the adjacent side?
In this problem students work with area, percentages and algebra.
Parts a) and b) lay a foundation for solving the general problem.
A square is transformed by increasing its length and decreasing its other side by the same percentage amount. How does the area change?
Suppose a square is transformed by increasing its length by 10% and decreasing its adjacent side by the same amount. What is the resultant change in area?
What if the length and adjacent side of the original square were increased and decreased by 20% respectively?
How might you quickly determine the change in area of a square given any percentage increase in length and a decrease by the same percentage in the adjacent side?
What is the change in area if the length is increased by one percentage amount and the adjacent side decreased by a different percentage amount.
Let the side of the square be L. Then the area of the rectangle after transformation is given by (L + 0.1L)(L– 0.1L) = L2 – (0.1)2L2 = L2 – 0.01L2 = 99L2. Hence the change in area is a decrease of 1%.
Let the side of the square be L. Then the area of the resulting rectangle is given by (L + 0.1L)(L– 0.1L) = L2 – (0.1)2L2 = L2 – 0.01L2 = 99L2. Hence the change in area is a decrease of 1%.
Suppose that the increase in one side is by x% and the decrease in the other is by y%. Then the new area is (L + xL/100)(L – yL/100) = L2(1 + x/100 – y/100 – xy/100). So the percentage change is x – y – xy. (A bit of algebra should convince you that if y = 1, this is always a decrease, while if y ¹ 1, then for this to be an increase we will need x > y/(1 – y).)
Note that if x = y, then the percentage change is –x2, as in c) above.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/transforming-square at 8:58pm on the 26th February 2024