# Engaging students: Completing the square

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Haley Higginbotham. Her topic, from Algebra: completing the square.

A2. How could you as a teacher create an activity or project that involves your topic?

To start the activity, I think I would do some examples of how to complete the square and see if anybody notices a pattern in how it is done. If not, I would give them some hints and some time to think about it more deeply, and maybe give them a few more examples to do depending on time and number of previous examples. After they have figured out the pattern, I would ask them if they knew why it worked to add (b/2)^2, and why they need to both add and subtract it. Then, we would go into the second part of the activity, which would require manipulatives. They would get into partners and model different completing the square problems with algebra tiles, and explain both verbally and in writing why adding (and subtracting) (b/2)^2 works to complete the square. I would probably also ask if you could “complete the cube,” and have them justify their answer as an elaborate.

B1. How can this topic be used in your students’ future courses in mathematics?

Completing the square is a fairly nifty trick that pops up a decent bit in Calculus 2, particularly in taking integrals of trig functions. Since they need to be in the specific form of (x+a)^2, or some variation thereof. If a student didn’t know how to complete the square, they would get stuck on how to integrate that type of problem. In addition, completing the square is useful when you want to transform a quadratic equation into the vertex form of the equation. It also could have applications in partial fraction decomposition if you are trying to simplify before doing the partial fraction decomposition, and has applications in Laplace transforms through partial fraction decomposition. It is also helpful in solving quadratic equations if it’s not obviously factorable and the quadratic equation is useful but can be tedious to use, especially if you don’t remember how to simplify radicals.

B2. How does this topic extend what your students have learned in previous courses?

Students typically learn, or at least have heard of, the quadratic formula before they have learned completing the square. Completing the square can be used to derive the quadratic formula, so they get more of an idea of why it works as opposed to just memorizing the formula. Also, if a student is having trouble remembering what exactly the quadratic formula is, they can use completing the square to re-derive it fairly quickly. Also, it ties the concepts of what they are learning together more so they are more likely to remember what they learned and less likely to see the quadratic formula and completing the square as two random pieces of mathematical information. Depending on the grade level, completing the square can also extend the idea of rewriting equations. They might have been familiar with turning point-slope form into slope intercept form, but not turning what is sometimes the standard form (the quadratic form) into the vertex form of the equation.

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