# Lesson 5

Squares and Circles

## 5.1: Math Talk: Distribution (5 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for distributing pairs of binomials. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to complete the square to find the center and radius of a circle.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they recall area models or other ways to represent the distributive property.

### Launch

Remind students that in a previous activity, they found that a circle can be defined by an equation in the form \((x-h)^2+(y-k)^2=r^2\) where \((h,k)\) is the center of the circle and \(r\) 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.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

*Representation: Internalize Comprehension.*To support working memory, provide students with sticky notes or mini whiteboards.

*Supports accessibility for: Memory; Organization*

### Student Facing

Distribute each expression mentally.

\(5(x+3)\)

\(x(x-3)\)

\((x+4)(x+2)\)

\((x-5)(x-5)\)

### Student Response

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### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

- “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
- “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
- “Do you agree or disagree? Why?”

*Speaking: MLR8 Discussion Supports.*Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.

*Design Principle(s): Optimize output (for explanation)*

## 5.2: Perfectly Square (15 minutes)

### Activity

Students practice squaring binomials. Then, they look for patterns in their results to help identify and rewrite perfect square trinomials.

### Launch

*Representation: Internalize Comprehension.*Represent the same information through different modalities by using diagrams. Use an area model to show that \(x^2+3^2\) is not equivalent to \((x+3)^2\).

*Supports accessibility for: Conceptual processing; Visual-spatial processing*

### Student Facing

- Apply the distributive property to each expression.
- \((x-7)(x-7)\)
- \((x+4)^2\)
- \((x-10)^2\)
- \((x+1)^2\)

- Look at your results. Each of these expressions is called a
*perfect square trinomial*. Why? - Which of these expressions are perfect square trinomials? If you get stuck, look for patterns in your earlier work.
- \(x^2-6x+9\)
- \(x^2+10x+20\)
- \(x^2+18x+81\)
- \(x^2-2x+1\)
- \(x^2+4x+16\)

- Rewrite the perfect square trinomials you identified as squared binomials.

### Student Response

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### Anticipated Misconceptions

Students may distribute the exponent across the addition inside the parentheses in the first part of the task. Ask them what it means to “square” something. Suggest that they write out the two factors, for example, \((x+4)(x+4)\), and then apply the distributive property.

If students aren’t sure of the definition of a trinomial, remind them that a *binomial* is an expression that has 2 terms. How might that relate to a trinomial?

### Activity Synthesis

Ask students to share their strategies for determining which trinomials were perfect squares, and for rewriting those expressions. For example, students may say that if the constant term is the square of half the coefficient of \(x\), then the expression is a perfect square trinomial. For the expression \(x^2+10x+20\), ask students what could be changed in order for it to be a perfect square trinomial (for example, if the final term were 5^{2} or 25, it would be a perfect square trinomial).

Invite students to imagine they were looking at the work of a peer who incorrectly rewrote \(x^2-8x+16\) as \((x-8)^2\). How could this student check their work? (One method is to distribute the expression \((x-8)^2\) to see if it matches the original expression.)

*Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct.*Before students share their solutions to the first question, present an incorrect answer and explanation. For example, “If I apply the distributive property to the expression \((x+4)^2\), then I get \(x^2+16\), because each term inside the parentheses is squared.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify the meaning of the distributive property. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to explain why \((x+4)^2\) expands to \(x^2+8x+16\). This will help students evaluate and improve on the written mathematical arguments of others.

*Design Principle(s): Optimize output (for explanation); Maximize meta-awareness*

## 5.3: Back and Forth (15 minutes)

### Activity

This activity builds toward completing the square. Students rewrite an equation of the form \((x-h)^2 + (y-k)^2 = r^2\), putting it in the form \(x^2 + y^2 + ax +by + c = 0\). Then, they’re presented with an equation for a circle in which the squared binomials have been expanded. Students rewrite 2 perfect square trinomials in factored form, then identify the center and radius of the circle.

### Student Facing

- Here is the equation of a circle: \((x-2)^2+(y+7)^2=10^2\)
- What are the center and radius of the circle?
- Apply the distributive property to the squared binomials and rearrange the equation so that one side is 0. This is the form in which many circle equations are written.

- This equation looks different, but also represents a circle: \(x^2+6x+9+y^2-10y+25=64\)
- How can you rewrite this equation to find the center and radius of the circle?
- What are the center and radius of the circle?

### Student Response

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### Student Facing

#### Are you ready for more?

In three-dimensional space, there are 3 coordinate axes, called the \(x\)-axis, the \(y\)-axis, and the \(z\)-axis. Write an equation for a sphere with center \((a,b,c)\) and radius \(r\).

### Student Response

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### Anticipated Misconceptions

Students may struggle to decide whether the coordinates of the circles’ centers are positive or negative. Encourage them to rewrite the equation in the form \((x-h)^2+(y-k)^2=r^2\). Remind them that we *subtract* the coordinates of the center from the given point \((x,y)\) to get the distance between the center and the point.

### Activity Synthesis

Ask students to rearrange the circle equation from the second problem so that there is a 0 on one side of the equation: \(x^2+6x+y^2-10y-30=0\). Display these 3 forms of this equation for all to see, emphasizing that these are all equivalent equations and therefore represent the same circle:

\((x+3)^2+(y-5)^2=8^2\)

\(x^2+6x+9+y^2-10y+25=64\)

\(x^2+6x+y^2-10y-30=0\)

The purpose of the discussion is to make connections between different forms of the equation in preparation for completing the square. Ask students:

- “In which form is it easiest to find the center and radius of the circle?” (The first one.)
- “Compare and contrast the second and third forms.” (Each form contains the terms \(x^2\), \(6x\), \(y^2\), and \(\text-10y\), but the constant terms are different.)
- “How can you go from the first form to the second one?” (Distribute the two sets of squared binomials.)
- “How can you go from the second form to the third one?” (Combine like terms.)
- “How can you go from the second form to the first one?” (Rewrite the perfect square trinomials as squared binomials, and rewrite 64 as 8
^{2}.)

*Conversing, Representing: MLR8 Discussion Supports.*Use this routine to amplify the language students use to make connections between different forms of the equation. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson, such as distribute, binomial, factor, and perfect square. For example, ask students, “Can you say that again, using the terms 'factor’ and ‘perfect square'?” Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.

*Design Principle(s): Support sense-making*

## Lesson Synthesis

### Lesson Synthesis

Display these 4 equations of circles for all to see.

- \(x^2+y^2=1\)
- \(x^2-18x+81+y^2-16y+64=4\)
- \((x-23)^2+(y+18)^2=60^2\)
- \(x^2+20x+y^2-24y+243=0\)

Ask students to rank these equations in order of how easy it is to find the center and radius of the circle. Ask students if they agree or disagree with each other’s rankings. Some students may struggle to identify the center and radius of the circle represented by the first equation. Invite students to share their strategies for working with this particular equation.

Invite students to find the center and radius for the circles represented by the first 3 equations. 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.

## 5.4: Cool-down - Center and Radius (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

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?

\((x+6)^2=x^2+12x+36\)

\((x-8)^2=x^2-16x+64\)

\((x+5)^2=x^2+10x+25\)

Each of the expressions on the right are called perfect square trinomials because they are the result of multiplying an expression by itself. There is a pattern in the results: When the coefficient of \(x^2\) in a trinomial is 1, if the constant term is the square of half the coefficient of \(x\), then the expression is a perfect square trinomial.

For example, \(x^2-14x+49\) 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: \((x-7)^2\).

Two squared binomials show up in the equation for circles: \((x-h)^2+(y-k)^2=r^2\). 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:

\(x^2-22x+121+y^2+2y+1=225\)

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 \(r^2\), the center and radius will be easier to recognize.

The first 3 terms on the left side, \(x^2-22x+121\), can be rewritten as \((x-11)^2\). The remaining terms, \(y^2+2y+1\), can be rewritten as \((y+1)^2\). The right side, 225, can be rewritten as 15^{2}. Let’s put it all together.

\((x-11)^2+(y+1)^2=15^2\)

Now we can see that the center of the circle is \((11, \text-1)\) and the circle’s radius measures 15 units.