In the previous lesson, students were introduced to a quadratic in standard form that did not easily convert to a perfect square. I like to start today's class by looking first at a quadratic that is a perfect square and then moving to one that is not. We are essentially starting with Question 9 in Building the Perfect Square and then moving on to Question 10.
Sometimes I have assigned Question 10 for homework and I start class by asking students to report out how they adjusted their new expression to match the old one. Sometimes a review of both questions is helpful to students. The main part of the opening is to remind students that sometimes a quadratic is already a perfect square and so the vertex form is easy to find, and sometimes we have to make adjustments to "k" to make it equivalent to the original.
Next students move on to Question 11 Building the Perfect Square, encountering a quadratic where "b" is an odd number. They should be able to follow the same steps they took in Question 10 and again, adjust their new expression by adding or subtracting a number to make the "c" term out correctly. I pause here before going on to Question 12, which is often too big a leap for my students. I add in some practice at this point with two more quadratics that have a equal to 1 and then challenge students with two that have a equal to negative 1. The Standard Form to Vertex Form Practice resource gives some examples of sample problems.
We have a whole group discussion about how to handle a negative "a." I ask students how we can get the quadratic to look like an expression we are familiar with and go from there. Finally, we take the leap to Question 12, where "a" is equal to 2. This work is usually challenging for my students and I try to give them a lot of opportunities to explain their thinking. It is also sometimes hard for them to keep track of the number that they factored out of the original expression and then remember to redistribute it to compare their new vertex form with the original standard form.
At this point in class, I usually show a Khan Academy video that shows a more formulaic way to complete the square in order to get vertex form. I like to give students the option to choose which method they like better and what makes more sense to them.
Source URL: https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/ex3-completing-the-square (Accessed May 28, 2014)
In my experience, some students need a lot of practice completing the square, while other students learn the procedure really quickly. Some students are ready to work with more complicated problems while others are still working with equations where "a" is primarily 1.
In my current school, almost all of my students need a fair amount of time to practice. At this point in the lesson, I assign students differentiated practice based on where they are. A worksheet like Vertex Form of Parabolas may work well. Depending on how long the Investigation section takes in class, some of these problems may be assigned for homework.
Have students complete an Exit Ticket related to Reflection. Ask them to complete the follow prompt on an index card:
Which method do you prefer to Complete the Square? Why?
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