In my Year 9 textbook, the start of the chapter on area and volume starts with a "review" of areas from past years. And by that, I mean it says something along the line of "you should remember these formulas from Year 8," then proceeds to list the formulas for the areas of various shapes.
Uhhh, no. Wasn't going to cut it. I'm not doubting the importance of remembering the formulas, but the book makes a huge assumption that all students understood the formulas completely last year, and just needed a quick reminder before they jumped back into, I don't know, completing boring lists of questions from the book I guess.
Instead, I didn't just want my class to remember the formulas, I wanted them to explain how they work, where they come from. I broke the class into groups and assigned them different shapes. They had to produce a poster demonstrating how we can discover the areas of their shape using other shapes - ideally this would be something of an informal proof. Here's some of the results:
This group didn't end up showing the formula, but they did show the general idea behind the proof. With a bit more prompting, I'm sure they would have gotten to the formula.
This group showed two different methods for proving the area of a parallelogram, which is awesome. I especially like that the one using the rectangle didn't involve any words. I think my students are slowly getting around the idea of using mathematics instead of English to communicate.
I made this one. I thought the circle was a bit beyond my class, but I still wanted a poster for it.
Rather than trying to explain my poster myself, I showed the class this video from Minute Physics: