Completing the Square

New animation: a geometric representation of completing the square. In this post, I present one way to use it as part of an algebra curriculum.

Many secondary school teachers figure that the derivation of the quadratic formula by completing the square can be shown to students, but have little hope of any understanding. They do not worry too much about it, because as they see it, the main point is for students to learn the quadratic formula, and to know how to use it to solve quadratic equations. Understanding it is not a priority.

I disagree.

In fact, I believe the opposite: while I have no objection to students memorizing and being able to use the quadratic formula, this is much less important than it used to be. Nowadays, any quadratic equation can be solved by entering it in Wolfram Alpha, or in a CAS calculator, and pressing the Enter key. On the other hand, completing the square is a nice bit of algebra, with other uses for those students who will pursue more advanced work in mathematics. But even for students who will go no further than Algebra 2, it is interesting and accessible, and helps demystify the quadratic formula.

Well, it’s not accessible if taught to eighth or ninth graders strictly through the manipulation of algebraic symbols. I’ve done that, and to be honest, I was reaching perhaps 10% of my students. I was able to turn this around, and make the whole thing accessible to perhaps 90% of my students, by postponing this topic until Algebra 2 (10th / 11th grade), and by using a geometric approach based on the Lab Gear manipulatives.

Here is a summary of my strategy. The key is to break the long and complicated procedure into understandable chunks.

– Make a rectangle, then make a square using the same blocks:

Doing this several times reveals visually what goes on, and makes it far easier to understand the process of completing the square. The “completing the square” animation is helpful at this point to help students put the process into words, and to generalize to non-whole-number values of b.

– Equal squares

Looking at increasingly complicated examples, I make the point that if we have an “equal squares” equation, we can reduce it to two linear equations. For example, if x² = 9, then x = 3 or x = -3. If x² + 6x + 9 = 7, then x + 3 = √7  or x + 3 = – √7. I avoid the phrase “take the square root of both sides”, which only encourages misunderstandings.

– Solving any quadratic

At this point, it becomes possible to combine what was learned into an overall “completing the square” strategy to solve quadratic equations.

– Connection to graphing

For a nice wrap-up to this unit, see this connection to graphing.

–Henri

(Much of this approach is spelled out in Algebra: Themes, Tools, Concepts, the textbook I co-wrote with Anita Wah. For more quadratic-related materials, click here.)