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 ┬áby 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?