Unit 6, Lesson 5
Squares and Circles
Geometry
5.1
Warm-up
Standards Alignment
Building On
6.EE.A.3
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression to produce the equivalent expression ; apply the distributive property to the expression to produce the equivalent expression ; apply properties of operations to to produce the equivalent expression .
Materials
NoneActivity Narrative
This Math Talk focuses on strategies for distributing pairs of binomials. It encourages students to think about algebraic strategies they have used in earlier courses and to rely on patterns for multiplying, squaring, and finding differences to mentally solve problems. The strategies elicited here will help students develop fluency and will be helpful later in the lesson when students complete the square to find the center and radius of a circle.
As students recall area models or patterns for representing the distributive property, they need to look for and make use of structure (MP7).
Launch
Remind students that, in a previous activity, they found that a circle can be defined by an equation in the form , where is the center of the circle and is the length of the circle’s radius. Tell them that, in the next several activities, they’ll use algebra to work with circles in the coordinate plane.
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 Synthesis
The goal of this discussion is to review students’ strategies for representing the distributive property.
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
5.2
Activity
Instructional Routines
MLR3: Critique, Correct, ClarifyMaterials
NoneActivity Narrative
This activity revisits work students have done in other courses while students practice squaring binomials. As students look for patterns in their results to help identify and rewrite perfect square trinomials, they express regularity in repeated reasoning (MP8). This will prepare students for future activities in which they will rewrite expressions in order to complete the square and write equations for circles.
In this activity, students critique a statement or response that is intentionally unclear, incorrect, or incomplete and improve it by clarifying meaning, correcting errors, and adding details (MP3).
This activity uses the Critique, Correct, Clarify math language routine to advance representing and conversing as students critique and revise mathematical arguments.
Launch
Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. Use an area model to show that is not equivalent to .
Supports accessibility for: Conceptual Processing; Visual-Spatial Processing
Supports accessibility for: Conceptual Processing; Visual-Spatial Processing
5.3
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 it 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 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 them 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
Suppose we square several binomials, or expressions that contain 2 terms. We get trinomials, or expressions that contain 3 terms. Does any pattern emerge in the results?
Each of the expressions on the right is called a perfect square trinomial because it is the result of multiplying an expression by itself. There is a pattern in the results: When the coefficient of in a trinomial is 1, if the constant term is the square of half the coefficient of , then the expression is a perfect square trinomial.
For example, is a perfect square trinomial because the constant term, 49, can be rewritten as (-7)2, and half of -14 is -7. This expression can be rewritten as a squared binomial: .
Two squared binomials show up in the equation for circles: . 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.