Students will learn how to solve linear systems in two variables by substitution. They will solve word problems and mathematical ones. Next, give your students the following challenge:
Five congruent rectangles form the large rectangle picture below. Find the area of each small rectangle.
Let \(w\) be the long side of each rectangle, its width, and let \(h\) be the short side, its height. First, we notice \(w + h = 5,\) and \(2w = 3h.\) Solving equation 2 for \(w\) gives us \(w = (3/2)h.\) Using this in equation 1, we find \((3/2)h + h = 5.\) Clear the fraction by doubling both sides. We now have \(3h + 2h = 10.\) Solving for \(h,\) we find \(h = 2.\) But we also know \(w + h = 5,\) and thus, \(w = 3.\) Therefore, the area of each rectangle is \(6.\)
Conclude by giving your students these puzzles. Specifically, those that require making a system of equations to relate, and thus solve, unknown lengths in the figure. An additional puzzle can be found here.
Next, give your students these challenges:
Conclude by leading this investigation:
Irreptiles (proportion, similar shapes)