My Favorite: Completing the Square

Grad School: Take 1

I essentially learned how to teach math by thinking about how to teach completing the square. In my high school methods class with Yolanda Rolle, I was paired up with two of my classmates (Mary Feeley and Amanda Miles) to prepare a lesson on completing the square which we would then teach in a mock lesson to the rest of the class. We did some research and started to realize that there were a lot of different ways to approach this. Our concept map is a nice picture of this complexity.

We thought hard and long about the sequencing of a quadratics unit. In particular, we were wondering if students would have already seen the quadratic formula (and completing the square would be used to make sense of it) or if the quadratic formula was coming later. We looked at textbooks and saw both approaches. We decided that for our lesson, our “students” – who were really our classmates – had already solved quadratics using square roots, factoring, and the quadratic formula, although the quadratic formula made little sense to them at the time.

Our lesson started with the following worksheet.

We hoped that there would be some variance in solution strategies and we could follow up with a discussion about the pros and cons of each strategy. Since this was a grad school class, there were even more strategies that we had hoped: one “student” (Kate) used difference of two squares to solve these equations. We concluded that the quadratic formula was annoying but always worked, and that both factoring and square roots were pretty easy … but did they always work? We tried to show that square roots always worked by starting a discussion about how to solve $x^2+4x=7$ . I was glad that our method lent itself to a discussion about a big idea (comparing different methods), but we didn’t really get into why we need to add $(\frac{b}{2})^2$ .

Grad School: Take 2

I had originally thought that completing the square and quadratics were purely algebraic skills. After some reflection and more planning, my group taught the “class” a second lesson on completing the square. This time, we picked up where we left off and tried to focus on why adding $(\frac{b}{2})^2$ created a perfect square. And we used Geometry to do it!

We led the class through folding a piece of paper whose area represented the left side of our equation: $x^2+4x=7$ . Therefore, the area of this paper was 7.

We asked the class: how can we make this paper into a square, so that we could then use the square root method. Someone suggested that we add $4x$ to the bottom, but we concluded that this would not be helpful because then the area would be $7+4x$ and we would be no closer to using square roots. We concluded that we could only add a real number. We eventually led our clever “students” to the discovery we wanted: split the $4x$ in half,

and move one half to the other side.

Which smoothly transitioned us to a new question – what do we need to add in order to complete the square? – and we had no trouble answering with 4. We later generalizing this approach with $x^2+bx$ . I loved the use of geometry, but there was something forced about the discovery, and the entire lesson relied pretty heavily on the background knowledge of our “students.”

Second Year of Teaching – Algebra 2

In my first year of teaching, I tried using that lesson in my Precalculus Honors class, and it fell flat. The “discovery” turned into a demonstration, and I realized that I had to go back to the drawing board.

By my second year of teaching, I had developed a new approach to quadratics. Everything started with vertex form: $y=a(x-h)^2+k$ . Vertex form is easy to graph (find the vertex) and solve (use square roots). When I introduced quadratics in standard form: $y=ax^2+bx+c$ , my students didn’t know what to do. So we came up with a plan: change the equations back into vertex form. But how do we do that? We have to complete the square and factor.

While I was still doing a lousy job of motivating the $(\frac{b}{2})^2$ , I had made an important shift: instead of just teaching my students how to solve equations, I was trying to get them to understanding the structure of a quadratic function. This algebraic trick revealed the vertex of a parabola and facilitated finding the x-intercepts.

Fourth Year of Teaching – Mathematical Foundations

This year, I teach 2 sections of an Algebra 2 support class. While my students have definitely been taught completing the square (probably twice), it hasn’t stuck. These classes have a lot of students with low English proficiency and/or learning disabilities, who hate math and/or see themselves as terrible at math. In this last unit, I was focused on teaching multiplication, first with integers and then with algebraic expressions. We almost exclusively used an area model to multiply. Then, we started to use algebra tiles and the area model to factor. In this context, completing the square was a problem-solving rich application of multiplying and factoring.

In this activity, I asked my students, how many little squares (i.e. whole numbers) do we need to add to make a perfect square out of $x^2+2x$ , $x^2+4x$ , $x^2+6x$ , $x^2+8x$ , $x^2+10x$ , $x^2+12x$ , $x^2+100x$ , $x^2+bx$ , and $x^2+5x$ . And their approach was fascinating.

When the b-value was a multiple of 4, most students had the same approach: spread the x-tiles (the long thin rectangles) evenly around the four sides of the big square (the $x^2$ tile).

But when the b-value was not a multiple of 4, students were pretty stuck. Many students came up with solutions that were not actually squares, which led to a pretty good teaching moment.

Some students came up with some really wacky and fun solutions.

And when a couple students came up with an elegant solution, they were so confident that: (a) they could calculate the number of little squares without drawing them, and (b) they could explain their approach to their peers.

In this class, completing the square is not about solving equations or the structure of quadratic functions; completing the square is about problem solving and how area connects algebra and geometry.

Completing the square has a special place in my heart. By trying to teach completing the square, I have learned so much about a topic that I thought I understood as a teenager. I have taught completing the square 7 times – twice in grad school, twice in Algebra 2, twice in Precalculus, and now once in Mathematical Foundations. I don’t think I’m anywhere close to finding the best way to teach this topic, and I love that challenge.