So factoring is super useful, yes. But at the Exeter Conference, one of the keynote speakers was making an impassioned, clarion call for CAS in the classroom and threw up an image. It was of which quadratics are actually factorable, and which aren’t. I tried to make my own 15 minute version of that to show you below (where and are non-negative, just because I got lazy). Apologies if there are any mistakes.
This image struck me so hard I can’t even tell you. Because although we teach the quadratic formula, in reality, most of our assessments which come after the quadratics unit give factorable quadratics. But in one powerful image, we are reminded that most quadratics are not factorable (at least, over the rationals). And we all know why we give factorable quadratics all the time — and it’s nothing to be ashamed of. We don’t want to have students spend all their time using the quadratic formula (and possibly generating incorrect answers) when we’re trying to teach an unrelated skill.
Still, the implicit lesson we’ve taught our students, by always giving nice, factorable quadratics is that most things are factorable. I mean, how many times have you been asked “is there a mistake in this question?” when you’ve given students a non-factorable quadratic on a test not on the quadratic unit? I thought so.
So next year I vow to show my students this chart, and remind them that most things in this amazing universe are NOT factorable. Heck, most quadratics that come up in engineering won’t even have integer coefficients, I will say, while showing ’em a picture of a falling cow and the equation governing its vertical motion in metric units. And that I tend to give more factorable quadratics than unfactorable quadratics because I want to save them computing time, not for any other reason.