I haven’t seen a curriculum that develops quadratics quite in this way, but I’m having trouble giving up on this approach and going with anything else I’ve found. What do you think? Here’s how the unit would go:
FIRST TYPE OF EQUATION:
Step One: Learning to solve (x + a)(x + b) = c equations by treating them as multiplication equations. An important idea is that these equations can have 1, 2 or 0 solutions.
Step Two: In particular, learn how to efficiently solve (x + a)(x + b) = 0 and other similar, non-quadratic equations.
Step Three: Study these types of equations as functions. Check out what the zeroes represent in y = (x + a)(x + b).
(I’ve already done these activities in class while experimenting during the week before spring break.)
Step Four: Make generalizations about the graphs of these equations — about where the line of symmetry is, whether is curves up or curves down.
SECOND TYPE OF EQUATION:
Step Five: Check out a new type of equation or . (I mean that is non-negative.) Learn to solve these equations by using what you already know about solving linear equations, with the new twist of taking roots of each side. And notice that sometimes these equations have 2, 1 or 0 solutions, and learn precisely what sorts of equations will have
Step Six: Graph these new equations, or $y = x^2 – b$ especially in the case when is square. All that stuff above about lines of symmetry, zeroes, etc., study that but for these equations.
Step Seven: Big idea time. There are two equivalent ways of expressing many of these quadratic equations. No factoring, no multiplying binomials yet. Just notice: some of these equations produce the same graphs as ! (Mostly when is a square.) Let’s give arguments for why this is true, arguments about the zeroes, the lines of symmetry, and that these two equations share a vertex.
THIRD TYPE OF PROBLEM: MULTIPLYING AND FACTORING QUADRATICS
Step Eight: Learn to multiply binomials, like , and become equipped with a new algebraic way of doing the work of recognizing equivalent quadratic functions. Here we’ll especially focus on a difference of squares, .
Step Nine: Teach the rest of your quadratics unit at this point — including whatever other factoring you need to teach — while frequently asking the question “Will these equations produce the same graph or nah?”
This all seems to me a nice way to gradually build quadratics knowledge. If pushed on my design principles, I’d say that (a) I’m trying to be sensitive to the fact that the different types of equations that fall under ‘quadratics’ are of widely varying complexity and (b) I’m trying to make sure not to teach a connection between two mathematical objects before students have a chance to really become familiar with the different mathematical objects. (In other words, students would see lots of equations in factored/standard form before trying to connect them via multiplying or factoring one into the other.)
Is there any curriculum that structures a unit in a way that even roughly resembles this? I can’t develop too much of my own curricular stuff given my teaching load (four different courses: 3rd, 4th, 8th and Geometry) but I would love to try teaching this upcoming quadratics unit in something like this way.
Any materials or approaches you’ve seen for quadratics that resemble this? Can anyone talk me out of this approach? Where would I run into trouble, if I went against better judgement and developed my own materials for this unit?