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. Instead, I am front-loading the big idea of factored form of a quadratic. After all, factoring trinomials feels useless and painfully tedious unless you’re already a believer in the power of factored form. And only after you see what factored form can do will you see expressions like (x+1)(x-2) as important enough to want to multiply them out.   Update 11/18/17: Originally, I’d intended this to be a series of posts about each lesson in the unit, but instead I’ve decided to describe each lesson below and link to the materials. Update 5/1/18: I keep improving this post as I think about it. I guess it’s kind of a living document.

  • 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. What equations make parabolas (and other nonlinear graphs)?  Now we apply factored form to an awesome Desmos marbleslide. The little stars are all zeros of the function, which focuses student attention on the critical features in a really fun way. At the end of the lesson, students even learn to stretch or reflect their graphs to avoid the purple obstacles. Here are a couple thumbnails of the marbleslide:

Marbleslide

  • 3. Practice and refinement. Consolidation lesson, with a slight extension:
    • Part 3.1 Drill practice on the factored form of a quadratic equation. This includes parabolas where the zeros are not integers, so you have to set the factors equal to zero. For example, in (+ 9)(5x – 1), you have to solve 5x-1=0 to find the second zero.
      • Khan Academy’s quadratics unit begins with factored form, so those exercises are a great resource if you’re into Khan Academy.
    • Part 3.2 We also need to learn to apply vertical shifts to parabolas. (This may seem out of place, but it’s necessary for the catapult-shooting that we’ll do in the next lesson).  One resource for this bit is the last 5 screens of my Desmos marbleslide. They explicitly introduce vertical shifts, and compare them to vertical stretches.
  • 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 think I’ll try to obtain the expensive model featured in tweet below:

*Note: I think you can do Julie’s projectile lab without the LabQuest interface, which saves serious $, but then you need to buy the $11 power supply for the projectile launcher.

  • 6. Random skill interlude: finding factors of one thing that add up to something else. This is a boring topic, but I have a couple of honestly excellent lessons on this. For students who don’t know their multiplication facts, this skill can be quite difficult without careful lesson design.
    • Here is my first lesson, using all positive numbers. And here is the student handout.
    • Here is my second lesson, introducing negative numbers, which are really the crux of students’ difficulty with finding factor pairs. Here is the student handout for this lesson. My secret trick here is to include lots of contrasting cases where the problem can almost be done in two different ways, using either two negatives or one negative and one positive, 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.  Here’s a screenshot of the lesson showing what I mean. Notice that the problems are almost the same: one product is -6 while the other is 6, but both problems add up to -5.

Factor Pairs

 

  • 7. 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. I teach factoring when a>1 first, with the box method, and then teach a=1 as a special case to students after they show they can handle the a>1 case. Making the transition in this direction seems to be much easier than in the reverse direction.
  • 8. Introduction to Standard Form: When you multiply binomials, the x2  represents the acceleration. 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.
    • Note: This used to be my lesson 7. But as I continue tinkering with this quadratics unit, I’ve realized the key content of the lesson is the different role that the quadratic and linear terms play in standard form. So I think this is best as the intro to Standard Form.
  • 9. Traditional quadratics topics, e.g., the quadratic formula, graphing parabolas using the vertex formula, etc.

4 thoughts on “Teaching Quadratics for Meaning”

  1. One thing I love is that vertex form just sort of pops out of factored form. y = (x + 2)(x + 2) only has one root, so it has to be just touching the x-axis. And then we can apply those same vertical shifts that you make explicit as well. Not as useful for your project, but very useful for students!

    A lot of the New Visions materials are adaptable for my quadratics unit, I don’t know if they’d be useful for your’s.

    1. I never thought about it that way before — that putting a parabola in vertex form means taking a parabola that’s barely touching the x-axis and shifting it around until it matches the parabola you are trying to model. As long as the coefficient “a” matches as well, the final parabolas will be the same. I always thought about transforming to vertex form in terms of completing the square. This way is so much more natural. Wow.

      I still have to make my way through the New Visions curriculum in detail this summer. I’ve skimmed already. Glad you’re liking it.

      1. Yeah! It’s my first time teaching quadratics deeply in a few years and I’ve really loved how my kids can use either vertex or factored form in a lot of situations.

        And, once you can see both perspectives, the equivalence between x^2 – 9 and (x+ 3)(x – 3) comes out as a graphical truth before it’s an algebraic one…

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