If you want to teach for meaning, quadratics are extra hard for a simple reason: the way quadratics are usually taught, students have to plow through weeks of difficult skill-building before becoming convinced that these new skills will make them more powerful. (Dan Meyer would say it’s hard to convince them of their intellectual need to learn quadratics).
I’d like to outline a quadratics unit that makes kids feel powerful from the beginning. It’s a conceptual approach, and for a lot of folks, that means higher-order thinking and “more rigor”. Sorry, but my goal’s a little different: a conceptual approach that will knit topics together so my weakest students feel a coherent narrative arc through the unit.
Here’s a brief outline of the whole thing. (In later posts, I’ll describe each lesson and link to materials I have).
- Not everything’s a line. For example, there’s these things called parabolas. They let you do fun stuff, and they’re not hard. (Lesson: an adapted version of Will it Hit the Hoop, by Desmos.)
- How do you make this parabola shape? Oh, it’s easy if you use the zeros of the function. Wait, does that mean you can make other cool shapes, too? What shapes could you not make with this skill? (Lesson: my own role-play drama I call, “You don’t play with Legos one block at a time.” It introduces zeros of a function.)
- But what if I need to stretch or shift my graph to make a cool shape? (Lesson: applying vertical transformations to functions written in factored form).
- Real, actual catapult target practice. It brings together zeros of a function, parabolas, vertical transformations, and launching things. (Lesson: an Algebra 1 adaptation of a brilliant precalc lesson going around the MTBoS).
- Practice. Gotta practice zeros of a function to mastery. This includes trickier problems like finding the zeros of (7x+9)(5x – 1) by setting each factor equal to zero.
- Random skill interlude: finding factors of one thing that add up to something else. Motivated by the impossible challenge of solving x2+10x+26=2 by getting x by itself.
- So this guy jumps off a crane with no parachute. Wait, what? Yes, that’s a thing. We can’t tell you his rate of falling, because he keeps speeding up. Ack! It breaks all our math to have a slope that keeps changing! RIP, y=mx+b. What function will work here? Oh, functions with x2. (Lesson: one of my own creations… I guess you could call it, “See you next fall.”)
- Wait, why do x2 graphs also make parabolas? I get that y=(x+2)(x-2) makes a parabola, because it clearly has two zeros, at ±2. But y=x2-4 makes the same parabola? If the equations are the same, how do you go from one to the other?
- Skill-building on factoring and multiplying binomials. (There are a million blog posts out there on factoring, but there’s one little thing I do that seems to make it better for weaker students).