In previous activities, students identified features of a trinomial that make it a perfect square trinomial. They build on this work and use patterns noticed previously as they identify the constant term needed to create a perfect square trinomial. 

The purpose of this discussion is for students to articulate a process for rewriting a perfect square trinomial as a squared binomial. Here are some questions to discuss:

• “How did you determine what the constant term needed to be?” (I took half of the coefficient of the middle term, and then I squared it.)
• Display the equality . "What value should go in the blank to make this equation true? How does the value in the blank relate to the 10 in the original expression? How is it related to the 25?" (5 goes in the blank. It is half of the linear coefficient and the square root of the constant term.)
• "For the last expression, was the process of calculating the missing value the same as for the other expressions? How could the last expression be rewritten as a squared binomial?" (It would look like or .)

In this activity, students are introduced to completing the square for an equation of a circle. As students look at a pre-written version of the first few steps and analyze what was done, they are making sense of completing the square (MP1). Then they finish the process using skills from previous activities and determine the center and radius of the circle.

Students may have seen a similar process or the term "completing the square" in previous courses. This activity builds from that work as students complete the square for two different trinomials within the same equation using two different variables. Because students are making sense of completing the square in this activity, it is not necessary to revisit earlier course work before approaching this task. 

Students may be familiar with technology such as the Desmos graphing tool (available under Math Tools) that can graph an equation of a circle. If the technology available in the classroom graphs equations only in the form “”, then it will not be a helpful tool for this activity.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

In this activity, students find the radius and center of a circle, then graph the circle in the coordinate plane. In order to identify the radius and center, students rewrite the equation to see its features. As students apply the patterns and processes from earlier activities to complete the square and rewrite equations, they are making use of the structure of equations for a circle (MP7).

Using graphing technology may reduce barriers for students who need support with fine motor skills and students who benefit from extra processing time.

If students wonder why the equation form given in the activity statement is used at all, tell them that circles are one particular example of a set of shapes called “conic sections.” All conic sections can be defined by equations similar in form to the one in the activity statement, so sometimes we use that format so we can easily compare and contrast several different equations. They’ll look at another conic section in an upcoming activity.