Many texts I’ve seen ask students to solve quadratics by factoring, then by graphing, then memorising the quadratic formula, and then if there’s time left, they are introduced to the idea of completing the square. I’ve done it before like this myself, and I have seen students struggle mightily and miserably. For me, beginning with factoring is problematic. While factoring trinomials can be satisfying, especially for students who like the puzzle solving parts of math class, this technique generally only works with problems that have been contrived by math teachers or textbook writers. Most quadratics we come across don’t naturally factor to nice clean numbers like y=(x+4)(x-2).
I was looking for a better approach, and came across James Tanton’s take on quadratics. He emphasizes symmetry as the key to studying and understanding quadratics, and right from the start, teaches completing the square (Although he relentlessly resists formal vocabulary, and calls it the “box method”) as the way to solve for x. His course gives a sequence of problems, each of which adds one level of complexity until students can solve just about any quadratic thrown at them. As he says, “The box method will never let you down.”
Developing a solid conceptual understanding of completing the square leads naturally to moving back and forth between standard and vertex form, and to the derivation of the quadratic formula – and students know why it works. And the best part is, only one of my students from this year’s group uses the quadratic formula as her go-to method. They all go straight for completing the square because they understand why and how it works, and are totally comfortable with the techniques. Most of them can recite the formula, but they are worried about making arithmetic mistakes, and are not as confident in their results as they are when they use symmetry and completing the square.
I strayed from Tanton’s approach in some ways – we figured out imaginary solutions with the honors group (Tanton advocates to leave out imaginary solutions when studying quadratics with Algebra 1 or 2 classes) – because they asked, and because they were ready to expand their horizons. And I did eventually teach factoring as a method of solving quadratics. Tanton suggests that factoring might be better included when we’re working on a discreet study of polynomials – but honestly, it was easy for me to fit in as the last method I taught. We took less than one full class period to discuss and practice factoring, and by the time we got there, students understood what the factors meant, and how they related to the roots of the quadratic. They appreciated the quickness of the solutions when factoring worked, and understood that if it didn’t work easily, that they could fall back on methods they know.
Every one of my Algebra students, even those who really struggle with math have had success moving through this sequence. Although this is only one year, and there are always other variables, I am convinced that this order makes much better intuitive sense. I’ll report back next year after I’ve had the chance to try this again.
BTW, some of my favorite resources to go deeply into studying quadratics include
- Des-Man – inspired by Fawn Nguyen and highly motivating activity.
- The new Desmos Parabola Polygraph – great for building vocabulary.
- Dan Meyer’s Basketball series – great for solidifying understanding of what the a,b,c, h and k mean in standard or vertex form. Also a highly motivating activity.
- Don Steward’s quadratic problems – he manages to consistently keep the learning about thinking and to keep connections between algebra and geometry.
- …and of course NRICH has some excellent quadratics problems that present nice challenges!