Solving quadratics – what’s most important?

I’m currently teaching MPM2D, Grade 10 Academic math, and I’ve been having conversations about what’s most important for students to master in the quadratics part of the course. We’ve learned about the different forms of quadratic equations (standard, factored, vertex) and how transformations apply to give us vertex form, and we’ve started factoring.

The expectations I’m looking at right now are summarized here:

  • expand and simplify second-degree polynomial expressions, using a variety of tools and strategies
  • factor polynomial expressions involving common factors, trinomials, and differences of squares using a variety of tools and strategies
  • determine,through investigation,and describe the connection between the factors of a quadratic expression and the x -intercepts (i.e.,the zeros) of the graph of the corresponding quadratic relation, expressed in the form y=a(x-r)(x-s)
  • interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x -intercepts of the corresponding relations
  • express y = ax^2 + bx + c in the form y=a(x-h)^2+k  by completing the square in situations involving no fractions, using a variety of tools
  • sketch or graph a quadratic relation whose equation is given in the form y = ax^2 + bx + c , using a variety of methods
  • explore the algebraic development of the quadratic formula
  • solve quadratic equations that have real roots, using a variety of methods

Of all that stuff, what’s most important? There is time pressure in this course, and I’m trying to focus on what’s truly essential moving forward. I also want to make sure that my evaluation reflects the importance of each understanding and skill.

Have you taught this course (or the sequels, MCR3U and MCF3M)? What do you think?

2 thoughts on “Solving quadratics – what’s most important?

  1. Disclaimer: I haven’t taught this course (though I taught similar skills to some of my students when I was in England). But here are my thoughts anyway! :D

    I’ll turn this around and ask: What expectations are least important? (It’s worth pointing out our Grade 9/10 curriculum is 10 years old at this point, so we are due to be talking about revisions.)

    I have never been a huge fan of completing the square. To this day I still have to look up how to do it before I teach it to people. It’s really nifty if you approach it from a geometric perspective. But you soon learn better techniques for interrogating quadratics.

    Similarly, let’s not make a big deal about deriving the quadratic formula. The quadratic formula itself is boring and ugly; I don’t even want students to have to memorize it. If they need it so badly they can look it up—or better yet, get a computer to find the roots of the quadratic equation for them. The formula is an ugly hack, and I’m glad that the Abel-Ruffini Theorem scuttled any possibility of doing this for higher-order polynomials. Not that I’ll be telling Grade 10s about that. (I did have a student try to derive the cubic formula from first principles once. He got really close.)

    In my opinion, the importance of quadratics in our curriculum comes from them not being straight lines. Quadratics give us an opportunity to introduce students to the more abstract world of algebra. Quadratics still behave nice and predictably, but they are slightly more interesting and complicated than straight lines. So with this in mind, when we teach quadratics, we should think about how we’re equipping students to go on to tackle functions beyond quadratics. Do they understand that other functions have roots as well, that other functions might model other situations? Do they recognize that they will be able to apply what they’ve learned about quadratics to more functions down the road?

    Expanding and factoring are huge for this reason. I see so many people in Grade 11 and 12 courses who struggle with factoring—often because they learn one of the algorithms for doing it with complex trinomials without ever understanding the inverse relationship between factoring and expanding. But you need factoring and expanding as you move into higher-order polynomials.

    To these skills I’ll add one more, albeit a broad one: understanding the relationship between an equation and a graph. Students need to understand that both representations are equally “real” or valid. I’m not a graphical person; I love equations. But I know other people are the opposite. And being comfortable going from one to the other will help students remain resilient as the algebra gets tougher.

    This second skill is pretty broad, and I know with time pressure you can never quite nail it the way you want to. It covers a few of the expectations you mentioned. But if you can get students looking at equations/graphs and interpreting what they mean, or discussing how changing an equation would change the graph (or vice versa), I think that’s a very successful and worthwhile outcome to aim for. Unlike solving quadratics, interpreting them is not something a CAS will often be able to do for you.

    • Hey Ben,

      Those are some awesome thoughts! It’s always nice when the comments are longer than the original post ;)

      The issue of all these technical procedures is what got me really thinking about the relative importance of the skills they’re developing. From talking with other teachers of grade 11 and 12 courses, you’re right: factoring and expanding are a major barrier to success for students, and they’re critical.

      I’m feeling like I should be downplaying factoring trinomials (especially complex ones) in favour of mastering common factoring. The connection between the equation and the graph for a relation is surprisingly difficult to make, you’re right; I’m consistently surprised by students who can (for example) tell me the roots from a quadratic equation but couldn’t graph that parabola unless I first ask them about the roots.

      Thanks a bunch – lots to think about here!

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