Sometimes they work individually and I pick people at random to talk to the class, somtimes they work in pairs or groups.
Then I'll 'go at little crazy'. We'll do an SDPQ with two 1cm lines. Or two lines with obviously different but unmarked lengths. It sounds bizarre but the conversations are wonderful. Structural insights are everywhere. The students are really thinking hard. By now they know I'm serious when I say this is all about the journey and that I'm only a tiny bit interested in the answers.
When they're ready we'll move on to algebra. We can use letters, constants, linear term, quadratic terms and so on. Remember learned tricks only get half marks. They need to be able to explain structurally what's going on. They will need to take those structures they've developed with numbers and transfer them to the algebra. It's powerful, it's challenging, it's rewarding and its deep. You should expect the unexpected, namely that when students describe their thinking you will be hearing things you have never heard before and will need to go away and think about. Don't worry - if you get stuck you'll probably find the student who came up with it has had deeper thoughts themselves after a day or two. Those who say something innovative often think about it a lot after the lesson.
How do the Chinese do it? Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6