Unit 5, Lesson 7
Rewriting Quadratic Expressions in Factored Form (Part 2)
Integrated Math 2
7.1
Warm-up
Standards Alignment
Building On
7.NS.A.1
Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Addressing
Building Toward
HSA-SSE.A.2
Use the structure of an expression to identify ways to rewrite it. For example, see as , thus recognizing it as a difference of squares that can be factored as .
Instructional Routines
NoneMaterials
NoneActivity 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
NoneActivity
None- 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
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
In this activity, students encounter quadratic expressions that are in standard form and that have a negative constant term. They notice that, when such expressions are rewritten in factored form, one of the factors is a sum, and the other is a difference. They connect this observation to the fact that the product of a positive number and a negative number is a negative number.
Students also recognize that the sum of the two factors of the constant term may be positive or negative, depending on which factor has a greater absolute value. This means that the sign of the coefficient of the linear term (which is the sum of the two factors) can reveal the signs of the factors.
Students use their observations about the structure of these expressions and of operations to help transform expressions in standard form into factored form (MP7).
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
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
Supports accessibility for: Language, Memory
7.3
Activity
Instructional Routines
NoneMaterials
NoneActivity 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).
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.
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.
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 .