As I become increasingly comfortable with student-led inquiries guiding our learning, our math lessons are becoming more dynamic, unpredictable, and productive. There is also less emphasis on term/procedure memorization in favor of higher level, cognitive investigations and problem solving (and problem *posing*!).

Yesterday, in an attempt to explore properties of lines and angles, I asked the students to simply construct lines or line segments in their notebooks. Shortly after, I told them to limit the number of lines to five. The ability to actually use a ruler and pencil to construct a line, without ripping through the paper or breaking the pencil or letting the ruler slip, was more challenging for some students than I anticipated. But eventually, we all had five lines. Then questions guided the learning. The first, from me, was intentionally open, as opposed to a specific, restrictive plan or set of questions designed to force kids to see what I want them to see.

*“Mathematicians, what do you notice?”*

Students observed shapes and angles and intersections, as expected. Then Seth said, *“I wonder if there’s a maximum number of shapes, just like that ‘how many squares on a chessboard‘ problem we had.”*Â Oooh…good one. Seth was obviously on to something, and I, of course, hadn’t thought of it. As he explained his thinking, several students were already counting and exploring. Seth’s question obviously intrigued his classmates, but it also inspired me to wonder about intersections, so I posed the following: *“If we draw five lines, is there a maximum number of intersections that will occur?”*

This one question led to endless conjectures, debates, tests, and hypotheses. At some point I interrupted the busy hum of mathematical thinking in the room and suggested students start gathering data. They were doing this anyway, but I don’t think anyone had a systematic, written approach. Students eventually created a table like this one (Brianna’s):

And you can guess what happened next. Patterns! Predictions! Tests! Proofs! None of which, most likely, would have happened had I imposed a traditional “This is exactly what you’re going to learn today” approach. The trick is coming up with those generative questions, the ones Seth and so many of the students are good at asking.

Audience, what questions do you have? Looking at the images in this post (more student work/process below), thinking about the original question, what else could we explore?