Teaching Quadratics for Meaning

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. Notice that skill-building on multiplying binomials and factoring quadratics don’t appear till Lesson 8.  (In later posts, I’ll describe each lesson and link to materials I have). Update 11/18/17: still haven’t gotten around to writing most of the rest of the posts, so I’m editing the below to include more detail and brief links to the materials.

  • 1. Will it Hit the Hoop, adapted to focus on factored form. For reasons described in this blog post, I really think quadratics needs to start with factored form rather than vertex or standard form. (After all, factoring is the biggest obstacle to enjoying and understanding quadratics, right?) I rewrote Will it Hit the Hoop, by Desmos, to channel the awesomeness of that lesson to those ends. Check out the post linked above for details.
  • 2. 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?  Here’s the main thrust of the lesson:
    • Lesson hook: I try to fool my students into believing that I don’t know how Lego’s work. I make it seem like I don’t know that they click together…as if I only play with one piece at a time.  They’re usually surprised but quite patient in explaining to me how Lego’s work. (Kids are so nice).
    • Once we establish that Lego’s are meant to be assembled into larger creations, I say “Oh! Wait, that’s exactly what we’re learning in math class!” Huh? They’re puzzled.
    • “Well,” I say, “all year we’ve been studying equations like y=x-4 or y=x+1. They’re both lines, which is okay, but a little boring, like an individual Lego piece. How can you put them together to build something that’s cooler?”
    • We explore whether adding them together, as in y=x-4+x+1, would make something cooler or just another line. Why?
    • If they get stuck, I prompt them with the parabola concept. I display the graph of y=(x-4)(x+1) and ask, “What operation goes where the question mark is to make a parabola from y = (x – 4) ? (x + 1).  Why?”
    • Then we go through a sequence of match-my-graph activities: I display a parabola, and they write the function.
    • But it’s not just parabolas we can make. Wouldn’t it be weird if someone just learned that you can click two Lego’s together, but then they always stopped after just two Lego’s?  We play around with making cubic, quartic shapes. We approximate a sine curve using a lot of factors. Make it playful — let kids suggest shapes and challenge the class to write the equations (they may need to use decimals, like (x – 0.4) if you do this.
    • I usually include at least one graph that’s not a function, such as a sideways parabola. Someone always points out that, although all the other weird shapes could be written with various factors, this one can’t because it’s not a function.
  • 3. But what if I need to stretch or shift my graph to make a cool shape? Here we’re still playing match-my-graph, essentially, but introducing graphs that have vertical transformations.  We focus on first writing the un-stretched or un-shifted function in factored form, and then stretching or shifting it.  I’ve always taught this with lame notes, but this year I hope to have a snazzy Desmos activity for it.
  • 4. Real, actual catapult target practice that applies zeros and vertical shifts. I got this idea from Julie Reulbach, but whereas her activity is for Precalc, you can successfully adapt this for Algebra 1. The idea is to place a projectile launcher on the floor and notice the landing spot. If, say, it’s 70 cm away from launch, then y=ax(x-70) is a good approximation of the parabola, measuring distance from the launcher. You can eyeball the height of the vertex while it’s flying, adjust the value of a on a Desmos slider, and graph the actual parabola.  Which lets you predict: if you launch it from a tabletop instead (introducing a vertical shift like we studied in Lesson #3), where will the projectile land?  Here is a catapult design students can make — I tried a cheap store-bought one last year but wasn’t satisfied with the repeatability, so I’ll either buy an expensive electronic one for physics class this year or try the make-your own design linked above.
  • 5. 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.
  • 6. 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. My secret trick here is to include lots of contrasting cases where the problem can almost be done using either 2 negatives, or only 1 negative, but one of those methods gives the wrong sign for the product. It turns out this is the key difficulty to hammer home repeatedly, even ad naseum, until everyone understands. Getting kids over this hurdle addresses about 50% of their difficulty with factoring.  Examples:

  • 7. Parabolas and y=x2  are connected because they both model things that are accelerating. The lesson is adapted from one I’ve already written about here. The Desmos activity linked within the post is the key that brings it all together — kids are always successful discovering the correct way to rearrange the dots into a square to calculate the the total number without adding them up.
  • 8. 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).
  • 9. Traditional quadratics topics, e.g., the quadratic formula, graphing parabolas using the vertex formula, etc.

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