My favorite topic to teach is completing the square. Weird, right? I just love it. I start by having kids solve radical expressions. I say “what cha gonna do?” and then say “square both sides!” We then move to solving equations where we need to square root both sides and I catch them in the chant! “what cha gonna do?” “Square both sides.” Really? Then they think. And then they think some more. “Oooo,” they say. X squared = plus or minus the square root of x. Major deal of course. So now the chant becomes, “what cha gonna do?’ and they say, “square root both sides.” Many forget the plus and minus no matter how many we do it. I do this before solving quadratics, which may be unconventional, but it works.
We solve equations with a squared term, a linear term and a constant or two and we discover how to complete the square in order to be able to take the square root of both side after some massaging. It’s awesome. It’s magical. It has absolutely nothing to do with quadratics as far as students know. We are merely balancing equations.
Then we do go into quadratics. There is always some smart alec that wants to dive into the quadratic formula. I hate this, but soldier on. They may NOT use the quadratic formula until they can prove it. End of story. We go into vertex form of a quadratic. They dig the structure and can assemble the vertex form of a quadric if they know the vertex and the multiplier. We then look at discovering coming up with a quadratic equation with far less information. We know the structure of vertex form because that is taught in Math 1 (supposedly). Of course the structure is re-taught. But, now we have a reason for knowing how to complete the square. Yeah! It’s so similar to coming up with the slope intercept form of a linear equation. Plug in what you know (what you are given) and chug away. That there must be a perfect square trinomial to make the square root of both side s of the equation is the secret weapon. All must balance. No illegal moves. Taking the multiplier into consideration is a challenge, but we get there.
Because I introduce completing the square multiple times and early, it sticks by the time we prove the quadratic formula. It gets revisited when we do geometry and prove the Pythagorean theorem. The sparks and light bulbs that go off are so fun!
I tell my kids when we first complete the square that this is a skill that could make them a hero some day. When I first started teaching algebra 7 years ago, my daughter was in a college chemistry class. Her class had a problem that looked unsolvable until a student suggested that they complete the square. That student was a hero. I tell my students that some day they too can be a hero with completing the square. They are instructed to notify me the day this happens. Sadly, no notifications so far.
I think that success in teaching completing the square comes from students being exposed to it over a period of time, even before they really need it. If you wait until they need it, the mind is too clogged with other ways to solve quadratics. I tell my kids, too frequently solving a quadratic via the quadratic formula is like taking a barge down a river. Completing the square is a kayak. Have fun! Use it. Be a hero!