# 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 Deborah Duddy. Her topic, from Algebra: completing the square. What interesting word problems using this topic can your students do now?

Applying what is learned in the class is very vital in fact it is a process TEKS that teachers need to use to maximize student’s understanding. “When are we going to use this in real life?” and “Why do we need to know this?” are questions that students ask on a daily basis. Connecting material to the real world helps engage students and develops critical thinking. Describing a path of a ball, how far an item can be tossed in the air and how to maximize profits for a company are just some examples of how quadratics can be used in the real world.

One important event happens during high school; students receive their driver’s license. In their written driver’s test, students must know the distance needed to stop a car at certain speed limits. Using an example like the one below will be interesting for the students and help connect lesson material and real life. How could you as a teacher create an activity or project that involves your topic?

To begin class and get students involved with their learning, the class will participate in an activity. Each pair of students will have two different cards such as (x+2)^2 and x^2+4x+4, and any variations of these problems. They can only look at the (x+2)^2 card. Students will work out the problem on paper. Students will be asked to remember how to find the area of a square and then set up a square with the dimensions matching the first card. From there, the pairs would use algebra tiles (after knowing what each tile stands for) and attempt to “complete the square”. This activity will be used as an engage and a beginning explore for the students. This activity will help students see completing a square geometrically. How does this topic extend what your students should have learned in previous courses?

Completing the square is another way of solving/factoring the equation. The process of completing the square is to turn a basic quadratic   equation of ax^2 + bx + c = 0 into a(x-h)^2 + k = 0 where (h,k) is  the vertex of the parabola. Therefore this process is very beneficial because it helps students graph the quadratic equation given. In order to find h and k, students should be able to factor, square a term, find the square root and manipulate the equation.

In solving the equation by completing the square is to subtract the constant off the left side and onto the right side. Then students take the coefficient off the x-term divide it then square it. Students then add this number to both sides of the equations. By simplifying the right side of the equation, students give the perfect square. Then solve the equation left by taking the square root of both sides and determining x.

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