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Unit 5, Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

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Integrated Math 2

7.1

Warm-up

5 mins

Sums and Products

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

None

Activity Narrative

This Warm-up serves two purposes. The first is to recall that if the product of two numbers is negative, then the two numbers must have opposite signs. The second is to review how to add two numbers with opposite signs.

Launch

None

Activity

None

Student Task Statement

The product of the integers 2 and -6 is -12. List all the other pairs of integers whose product is -12.
Of the pairs of factors you found, list all pairs that have a positive sum. Explain why they all have a positive sum.
Of the pairs of factors you found, list all pairs that have a negative sum. Explain why they all have a negative sum.

Student Response

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Building on Student Thinking

Activity Synthesis

Ask students to share their list of factors. Once all the pairs are listed, highlight that each pair has a positive number and a negative number because the product we are after is a negative number, and the product of a positive and a negative number is negative.

Then, invite students to share their responses for the next two questions. Consider displaying a number line for all to see and using arrows to visualize the additions of factors. Make sure students understand that when adding a positive number and a negative number, the result is the difference of the absolute values of the numbers, and that sum takes the sign of the number that is farther from zero.

7.2

Activity

15 mins

Negative Constant Terms

Instructional Routines

MLR8: Discussion Supports

Materials

7.3

Activity

15 mins

Factors of 100 and -100

Instructional Routines

None

Materials

None

Activity Narrative

This activity aims to solidify students’ observations about the structure connecting the standard form and factored form. Students find all factor pairs of a number that would lead to a positive sum, a negative sum, and a zero sum. They then look for patterns in the numbers and draw some general conclusions about what must be true about the numbers to produce a certain kind of sum. Along the way, students practice looking for regularity through repeated reasoning (MP8).

Lesson Synthesis

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “What is your process for rewriting the quadratic expression in factored form ?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.

If time allows, display these prompts for feedback:

“ makes sense, but what do you mean when you say . . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”

Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

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Student Lesson Summary

When we rewrite expressions in factored form, it is helpful to remember that:

Multiplying two positive numbers or two negative numbers results in a positive product.
Multiplying a positive number and a negative number results in a negative product.

This means that if we want to find two factors whose product is 10, the factors must both be positive or both be negative. If we want to find two factors whose product is -10, one of the factors must be positive and the other negative.

Suppose we wanted to rewrite in factored form. Recall that subtracting a number can be thought of as adding the opposite of that number, so that expression can also be written as . We are looking for two numbers that:

Have a product of 7. The candidates are 7 and 1, and -7 and -1.
Have a sum of -8. Only -7 and -1 from the list of candidates meet this condition.

The factored form of is therefore or, written another way, .

To write in factored form, we would need two numbers that:

Multiply to make -7. The candidates are 7 and -1, and -7 and 1.
Add up to 6. Only 7 and -1 from the list of candidates add up to 6.

The factored form of is .

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Launch

Arrange students in groups of 2. Give students a few minutes of quiet work time to attempt the first question. Pause for a class discussion before students complete the second question. Ask students, “How do the constant term and the coefficient of the linear term help when rewriting an expression from standard form to factored form?” (The numbers added to the ’s in factored form are factors of the constant term and have a sum equal to the coefficient of the linear term.)

Next, ask students to work individually on the second question before conferring with their partner.

MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “How is this similar or different from the ones we've already rewritten?” This gives both students an opportunity to produce language.
Advances: Conversing

Representation: Develop Language and Symbols. Maintain a visible display to record new vocabulary. Invite students to suggest details (words or pictures) that will help them remember the meaning of the words and phrases.
Supports accessibility for: Language, Memory

Activity

None

Student Task Statement

Each row of this table should have a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.

factored form standard form
Each row in this table should have a pair of equivalent expressions. Complete the table. If you get stuck, consider drawing a diagram.

factored form standard form
How are the two tables similar and different?

Student Response

to access Student Response.

Building on Student Thinking

Are You Ready for More?

Launch

Give students time to complete the first two questions and then pause for a class discussion. If time is limited, consider arranging students in groups of 2 and asking one partner to answer the first question and the other to answer the second question. Alternatively, consider offering one pair of factors as an example in each table.

Before students answer the last question, consider displaying the completed tables for all to see and inviting students to observe any patterns or structure in them. Discuss questions such as:

“How are the factor pairs of 100 like or unlike those of -100?” (The numbers are the same, but the signs are different.)
“In the first two tables, what do you notice about factor pairs that give positive values?” (They are both positive.) “What about factor pairs that give negative values?” (They are negative.)
“In the next two tables, what do you notice about factor pairs that give positive values?” (One factor is positive and the other is negative. The positive number has a greater absolute value.)
“What about factor pairs that give negative values?” (One factor is positive and the other is negative. The positive number has a greater absolute value.)
“What do you notice about the pair of factors that give a value of 0?” (They are opposites.)

Encourage students to use these insights to answer the last question.

Activity

None

Student Task Statement

Consider the expression .

Complete the first table with all factor pairs of 100 that would give positive values of , and the second table with factors that would give negative values of .

For each pair, state the value they produce. (Use as many rows as needed.)

positive value of

factor 1 factor 2 (positive)

negative value of

factor 1 factor 2 (negative)
Consider the expression .

Complete the first table with all factor pairs of -100 that would result in positive values of , the second table with factors that would result in negative values of , and the third table with factors that would result in a zero value of .

For each pair of factors, state the value they produce. (Use as many rows as there are pairs of factors. You may not need all the rows.)

positive value of

factor 1 factor 2 (positive)

negative value of

factor 1 factor 2 (negative)

zero value of

factor 1 factor 2 (zero)
Write each expression in factored form:

Student Response

to access Student Response.

Building on Student Thinking

When completing the tables to find , some students may multiply the factors rather than add them. Remind them that what we are looking for is the coefficient of the linear term.

Consider completing one row of the table and displaying a rectangle diagram to remind students how the value of is obtained when we rewrite an expression such as in standard form. Applying the distributive property gives , or . Point out that in standard form, the product of the factors, 100, is the constant term. If the coefficient of the linear term is what we are after, we need to find the sum of the factors.

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Lesson 7

Rewriting Quadratic Expressions in Factored Form (Part 2)

Warm-up

Sums and Products

Standards Alignment

7.NS.A.1

7.NS.A.2

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Activity Synthesis

Activity

Negative Constant Terms

Instructional Routines

Materials

Activity

Factors of 100 and -100

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Are You Ready for More?

Activity Narrative

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

HSA-SSE.A.2

Standards Alignment

HSA-SSE.A.2

HSA-REI.B.4.b

Standards Alignment

HSA-SSE.A.2

HSA-REI.B.4.b

HSA-SSE.B.3.a

HSA-SSE.B.3.a