Unit 8, Lesson 8
Rewriting Quadratic Expressions in Factored Form (Part 3)
Algebra 1
8.1
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 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
Supports accessibility for: Memory; Organization
Activity
NoneEvaluate 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
Advances: Speaking, Representing
8.2
Activity
Instructional Routines
NoneMaterials
To Gather
Scientific calculatorsActivity 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
Instructional Routines
MLR8: Discussion SupportsMaterials
To Gather
Math Community ChartActivity 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?”
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.