Unit 7, Lesson 2
Equation of a Circle
Integrated Math 2
2.1
Warm-up
Standards Alignment
Building On
Addressing
HSG-GPE.A.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Building Toward
HSG-GPE.B.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point lies on the circle centered at the origin and containing the point .
Materials
NoneActivity Narrative
The purpose of this Math Talk is for students to connect the radius of a circle with the Pythagorean Theorem. It encourages students to think about what they know about distances between points and rely on the symmetry found in the points on a circle to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students build a general equation for a circle. As students use symmetry to determine whether points with negative values lie on the circle, they look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneDecide mentally whether each point lies on a circle centered at the origin with a radius of 5:
Activity Synthesis
The goal of this discussion is to review students’ strategies for finding the distance between points in the coordinate plane.
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
2.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students build the general equation of a circle. As students build from the specific circle equations in earlier activities to a generalized equation of a circle, they are expressing regularity in repeated reasoning (MP8).
Look for students who write instead of in the first equation. Explain that either one is correct, but it is conventional to write the variable as the first term in a binomial.
2.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
This activity builds toward completing the square. Students rewrite an equation of the form , putting it in the form . Then, they’re presented with an equation for a circle in which the squared binomials have been expanded. Students rewrite two perfect square trinomials in factored form, then identify the center and radius of the circle.
Lesson Synthesis
Display these four equations of circles.
Ask students to describe how they would find the center and radius of each circle. Here are some questions to elicit student strategies:
- “How can we use the format of the first equation to help us know the center and radius?” (The equation is already arranged in a format so that is subtracted from and is subtracted from , so the center must be . The right side is a value squared, which matches the format , so the radius is 60.)
- “How could you tell what the center is for the second equation?” (There's no or in the equation, so that means the center is the point .)
- “What is the first thing you would try with the third equation?” (I would rewrite it into two squared binomials so we can see what the center must be.)
- “How is the last equation similar to and different from the third equation?” (I would try to rewrite the equation as two binomials squared, but it is not quite in the format needed to do that. I would need to figure out some of the constant terms in order to rewrite this equation.)
Invite students to find the center and radius for the circles represented by the first three equations, if they have not shared them during the discussion. Tell students that, in an upcoming activity, they’ll learn how to rearrange the final equation to be able to identify the corresponding circle’s center and radius.
Student Lesson Summary
We can use the Pythagorean Theorem to test whether a particular point lies on a particular circle. For example, we know the point lies on the circle centered at with radius 5 because . We can use this reasoning to test whether a general point lies on a circle with center and radius . This is the general equation for a circle: .
Similarly, if we know the equation for a circle, then we can use that to find the center and radius. For example, a circle with equation has a center at and radius of 12.
Equations for circles are sometimes written in different forms, but we can rearrange them to help find the center and radius of the circle. For example, suppose the equation of a circle is written like this:
We can’t immediately identify the center and radius of the circle. However, if we rewrite the two perfect square trinomials as squared binomials and rewrite the right side in the form , the center and radius will be easier to recognize.
The first 3 terms on the left side, , can be rewritten as . The remaining terms, , can be rewritten as . The right side, 225, can be rewritten as 152. Let’s put it all together.
Now we can see that the center of the circle is and the circle’s radius measures 15 units.