Tangents and normals - a peer scaffolding lesson

Yesterday we had a nothing-special lesson, with no great outbursts of creativity, which nevertheless went very well and is the kind of staple lesson-setup that requires very limited preparation..

The goal: students should understand the concepts of tangents and normals to a curve, and be able to calculate the equations of these lines.
It's not exactly brain surgery, but then I find students often get lost in questions about tangents and normals: they have a hard time connecting the many relevant concepts: derivative, gradient, equation of a line, constant term, perpendicular, negative reciprocal, etc. They start doing funny things, like plugging in values of x into the derivative function instead of the original function in order to find the corresponding value of y. They lose track of what they're doing.

So for this lesson (as for almost all topics in calculus) we used algebraic and visual representations throughout, in parallel. I find it really helps students understand and keep track of their work, and check whether their work seems reasonable.

I put a "do now" question on the board: what is the equation of the tangent to the function..." and gave them a simple cubic function. Five minutes to work, in pairs, and everyone had found the gradient of the tangent, and many had also found its y-intercept. Some got a minus sign wrong, and could quickly see on the graph (which stretched close to, but not including the y-intercepts) that they must be mistaken. Go-through together and everyone's on track.

Follow up: is there any other point on the graph of f that has the same gradient of tangent? Five minutes pair-work, and most students set up, and at least attempt to solve, the resulting quadratic equation. A few needed a hint, because they attempted to set the equation of the tangent equal to its own gradient... Sure sign they were having a hard time connecting the derivative function with the gradient of tangent. This will be solved once we do more work on using the derivative for graphing the function.
Go-through together, and we're fine.

My only act of "telling" during this lesson was when I introduced the concept of a normal as a line perpendicular to the tangent at a certain point on the graph. A few students recalled that perpendicular lines have gradients that multiply to -1, and we were ready to go. There was a bit of the "do we use the same point? where do I plug this in?" going on, but when I brought their attention back to the graph on the board, they answered their own questions easily.

So all in all - students helped each other learn about tangents and normals, they worked efficiently during the whole lesson, and seemed to understand and enjoy the topic.