Conclude by giving your students these challenges:

- Square inside a Semi-Circle by Hieu Huy Nguyen
- Secret Transmissions by NRICH
- Changing Places by NRICH
- Skipped Counts

Find the area of the green rectangle:

Solution: Here's the solution. My method differs, and is preferable. It deviates immediately after finding the equations

$$\begin{align} & ab = 12 \\ & 9b - a = 23 \end{align}$$Here's what I did to avoid guess-and-check: First, I find \(w\) in terms of \(h.\)

$$\begin{align} & ab = 12 \\[0.5em] & a = \dfrac{12}{b} \\[0.5em] \end{align}$$Then I plug this into the second equation, and do some algebra:

$$\begin{align} & 9b - \dfrac{12}{b} = 23 \\[0.5em] & 9b^2 - 12 = 23b \\ & 9b^2 - 23b - 12 = 0 \\ & (9b + 4)(b - 3) = 0 \end{align}$$This gives the solutions \(h = -4/9, 3.\) But \(b\) is a length, and thus can't be negative. So \(b = 3.\) From this, it's easy to see the area of the green rectangle is \(24.\)