Lesson 6

As a step toward completing the square, students practice identifying the constant term needed to build a perfect square trinomial.

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

Ask students to compare and contrast the last expression with the others. Was the process of calculating the missing value the same? How could the last expression be rewritten as a squared binomial? (It would look like \(\left(x+\frac52 \right)^2\) or \((x+2.5)^2\).)

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

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 in the form “\(y=\)”, 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).

If students get stuck, suggest that they look at the form for the equation of a circle that’s on their reference chart. Ask them why that form is useful and how it relates to what they’ve done in recent activities.

Tell students that the process of adding a value to both sides of an equation in order to create a perfect square trinomial is called completing the square. They may have seen a similar process in previous courses.

Here are some questions for discussion:

• “The goal was to rewrite the equation in the form \((x-h)^2+(y-k)^2=r^2\). Why did Elena want to do this?” (If the equation is in this form, we can easily read the center and the radius directly from the equation.)
• “What did Elena do in Step 2, and why?” (She wanted to create perfect square trinomials, so she added the values 9 and 100. To produce an equation equivalent to the original, she also needed to add those values to the other side of the equation.)
• “What did Elena do in Step 1, and why?” (Moving the 105 to the right side makes it easier to create the perfect square trinomials—the 105 would be distracting if it stayed on the left side. Grouping the terms with an \(x\) and those with a \(y\) makes it easier to rewrite the trinomials as squared binomials.)

Students complete the square to find the radius and center of a circle with no scaffolding.

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

If students are stuck, suggest they look back to Elena’s process.

Display the image of the circle and two forms of its equation for all to see:

\((x-1)^2+(y+2)^2=3^2\)

\(x^2+y^2-2x+4y-4=0\)

Ask students how these three items relate to each other. (They all represent the same thing in different ways.) Ask students how the concept of distance relates to any of the representations. (The equation \((x-1)^2+(y+2)^2=3^2\) says that the distance between a point \((x,y)\) on the circle and the center \((1,\text-2)\) is 3 units. The graph is a picture of the set of all such points. It’s harder to directly relate the concept of distance to the first equation.)

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.