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Unit 5, Lesson 8

Rewriting Quadratic Expressions in Factored Form (Part 3)

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Integrated Math 2

8.1

Warm-up

10 mins

Math Talk: Products of Large-ish Numbers

Standards Alignment

Building On

Instructional Routines

Math Talk MLR8: Discussion Supports

Materials

None

Activity Narrative

This Math Talk focuses on recalling strategies for multiplying. It encourages students to use the structure of the expressions to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students look at quadratic expressions that can be written as .

To multiply the values, students need to 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.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Activity

None

Student Task Statement

Evaluate mentally.

Activity Synthesis

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. 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.
Advances: Speaking, Representing

8.2

Activity

15 mins

Can Products Be Written as Differences?

Instructional Routines

None

Materials

To Gather

Scientific calculators

Activity Narrative

In this activity, students multiply expressions of the forms and . Through repeated reasoning, they discover that the expanded product of such factors can be expressed as a difference of two square numbers: (MP8). Students use diagrams and the distributive property to make sense of their observations.

In the last question, they have an opportunity to notice that an expression in the form is not equivalent to , to discourage overgeneralizing. Later in the lesson, they will use their understanding of the structure relating the equivalent expressions to transform quadratic expressions in standard form into factored form, and vice versa (MP7).

To answer the first two questions, some students may simply evaluate the expressions rather than reasoning about their structure. For example, to see if is equivalent to , they may calculate and and see that both are 91. Ask students to show that this is not a coincidence. They could show, for example, that would also be equivalent to .

As students work, look for those who expand the factored expression using the distributive property and leave the partial products unevaluated in order to show the equivalence of the two expressions.

8.3

Activity

10 mins

What If There Is No Linear Term?

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Math Community Chart

Activity Narrative

In this activity, students use the insights from an earlier activity to write equivalent quadratic expressions in standard form and factored form. They see that when a quadratic expression in standard form is a difference of two squares (a squared variable, , and a squared constant, ) and has no linear term, the factored form is . Even if the constant term is not a perfect square, we can still find the factored form, but the numbers in the factors would not be rational numbers. For example, the expression can be written as , because . If we expand , there will be no linear term because .

Students also notice that when a quadratic expression is a sum (instead of a difference) of a squared variable and a squared constant, it cannot be written in factored form.

Lesson Synthesis

To help students consolidate and articulate their understanding of the relationship between and , ask them to reflect, in writing or by talking to a partner, on questions such as:

“The expression has a sum and a difference, and so does . When expanded into standard form, why does one have a linear term but the other does not?” (Multiplying out gives two linear terms that are opposites, and , which add up to 0, so the linear term disappears. Multiplying out also gives two linear terms, and . Because they are not opposites, their sum is not 0, so the linear term remains.)
“Can be seen as a difference of two squares? Can it be written in factored form? If so, what would it be?” (Yes. squared is , and is . The factored form is .)
“Think of another example of a quadratic expression in factored form that, when rewritten in standard form, is a difference of two squares and does not have a linear term. What is the expression in standard form?”

to access Cool-down.

Student Lesson Summary

Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?

Let’s take as an example. To help us write it in factored form, we can think of it as having a linear term with a coefficient of 0: .

We know that we need to find two numbers that multiply to make -9 and add up to 0. The numbers 3 and -3 meet both requirements, so the factored form is .

To check that this expression is indeed equivalent to , we can expand the factored expression by applying the distributive property: . Adding and gives 0, so the expanded expression is .

In general, a quadratic expression that is a difference of two squares and has the form can be rewritten as .

Here is a more complicated example: . This expression can be written as , so an equivalent expression in factored form is .

What about ? Can it be written in factored form?

Let’s think about this expression as . Can we find two numbers that multiply to make 9 and add up to 0? Here are factors of 9 and their sums:

9 and 1, sum: 10
-9 and -1, sum: -10
3 and 3, sum: 6
-3 and -3, sum: -6

For two numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9, because multiplying a negative number and a positive number always gives a negative product.

Because there are no numbers that multiply to make 9 and also add up to 0, it is not possible to write in factored form using the kinds of numbers that we know about.

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Launch

Provide access to calculators, if requested.

Activity

None

Student Task Statement

Clare claims that is equivalent to and is equivalent to . Do you agree? Show your reasoning.
1. Use your observations from the first question to evaluate . Show your reasoning.
2. Check your answer by computing .
Is equivalent to ? Support your answer:

With a diagram:







Without a diagram:
Is equivalent to ? Support your answer, either with or without a diagram.

Student Response

to access Student Response.

Building on Student Thinking

Some students may struggle to generalize the pattern after just a few examples. Before starting the Activity Synthesis, provide additional factored expressions for students to expand, for instance , , , and .

Are You Ready for More?

Activity Synthesis

Invite previously identified students to share their responses and reasoning.

To help students generalize their observations, display the expression and a blank diagram that can be used to visualize the expansion of the factors. Ask students what expressions go in each rectangle.

Illustrate that when the terms in each factor are multiplied out, the resulting expression has two squares, one with a positive coefficient and the other with a negative coefficient ( and ), and two linear terms that are opposites ( and ). Because the sum of and is 0, what remains is the difference of and , or . There is now no linear term.



		, or

Emphasize that knowing this structure allows us to rewrite into factored form any quadratic expression that has no linear term and that is a difference of a squared variable and a squared constant. For example, we can write as because we know that when the latter is expanded, the result is .

Use the last question to point out the importance of paying attention to the particulars of the structure of these expressions (the subtraction in the first expression, the presence of both addition and subtraction in the second). For example, we can't use any patterns observed in this activity to rewrite in factored form.

Launch

Math Community
Display the Math Community Chart for all to see. Give students a brief quiet think time to read the norms, or invite a student to read them out loud. Tell them that during this activity they are going to choose a norm to focus on and practice. This norm should be one that they think will help themselves and their group during the activity. At the end of the activity, students can share what norm they chose and how the norm did or did not support their group.

Consider arranging students in groups of 2 and asking them to think independently about the problems before discussing their responses with their partner.

Ask students to write as many equivalent expressions as time permits while aiming to complete at least the first six rows and the last row of the table.

Activity

None

Student Task Statement

Each row has a pair of equivalent expressions.

Complete the table.

If you get stuck, consider drawing a diagram. (Heads up: One of them is impossible.)

factored form	standard form

Student Response

to access Student Response.

Building on Student Thinking

Some students may struggle to see the numbers in the expressions in standard form as perfect squares. Prompt them to create a list or table of square numbers (, , , and so on) to have as a handy reference. Others may benefit by rewriting both terms as squares before writing the factored form. Demonstrate how to rewrite as and as .

Are You Ready for More?

Activity Synthesis

Consider displaying the incomplete table for all to see and asking students to record their responses. Give the class time to examine the responses and to bring up any disagreements or questions. Discuss with students:

“How can we check if the expression in factored form is indeed equivalent to the given expression in standard form?” (We can expand the factored expression by applying the distributive property and see if it gives the expression in standard form.)
“Some of the expressions show a squared variable subtracted from a number, instead of the other way around. Can we still write an equivalent expression in factored form?” (Yes. As long as the expression in standard form can be written as a difference of two squares, it can be written in factored form.)
“What if the number is not a perfect square—for example, ?” (We can still write it in factored form, by thinking about what number can be squared to get 5. Both and can be squared to get 5. Regardless of which number we use, the factored form is .)
“Why can be written in factored form but cannot?” (One possible approach is to rewrite the former as . We learned previously that, to write this expression in factored form, we would need to look for two numbers whose product is -100 and whose sum is 0. The numbers 10 and -10 meet this requirement. For , however, we need two numbers whose product is 100 and whose sum is 0. No such numbers exist. To have a sum of 0, one number has to be positive and the other negative, so their product can’t be positive 100.)

Math Community
Invite 2–3 students to share the norm they chose and how it supported the work of the group or a realization they had about a norm that would have worked better in this situation. Provide these sentence frames to help students organize their thoughts in a clear, precise way:

“I picked the norm ‘.’ It really helped me/my group because .”
“I picked the norm ‘.’ During the activity, I realized the norm ‘’ would be a better focus because .”

MLR8 Discussion Supports. During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking

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Lesson 8

Rewriting Quadratic Expressions in Factored Form (Part 3)

Warm-up

Math Talk: Products of Large-ish Numbers

Standards Alignment

Building On

6.EE.A.3

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Activity Synthesis

Activity

Can Products Be Written as Differences?

Instructional Routines

Materials

To Gather

Activity Narrative

Activity

What If There Is No Linear Term?

Instructional Routines

Materials

To Gather

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Student Response

Building on Student Thinking

Are You Ready for More?

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Addressing

Building Toward

Standards Alignment

Building On

Addressing

Building Toward

HSA-APR.A.1

HSA-SSE.A.2

Standards Alignment

Building On

Addressing

HSA-SSE.A.2

Building Toward

HSA-REI.B.4.b

HSA-SSE.B.3.a