My older son had an after school activity today, so I talked through this problem with my younger son first. He looked at it in two different ways.

First he looked at the entire rectangle and subtracted away the areas that were not shaded. The arithmetic gave him a little difficulty, but he was able to work out the area:

Next I challenged him to find an alternate approach. This time he thought about shifting the top triangle to the left one unit. This approach is a nice little challenge for a younger kid.

When my older son got home I’d planned on going through a problem from an old AMC 8 that gave him a little trouble this morning (partially because of time). I hadn’t looked at the problem, though, and when I did look at it I got a nice surprise – it was a problem we could approach “in your head” just like the two problem from Pershan.

Here’s the problem on Art of Problem Solving’s site:

and here’s my older son talking through the problem this afternoon:

Even if was just a coincidence, I’m happy to see how the ideas you use to talk about beginning problems are also really useful in problems that seem much more complicated. Math is like that – the basic ideas can be really (and surprisingly) powerful.