Unit 8, Lesson 12
Forms of Quadratic Equations
Extra Support Materials for Algebra 1
12.1
Warm-up
Standards Alignment
Building On
Addressing
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 .
Materials
NoneActivity Narrative
The purpose of this Math Talk is to elicit strategies and understandings students have for rewriting a quadratic expression in standard form. It encourages students to think about multiplying binomials and to rely on what they know about combining like terms to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will need to be able to factor and complete the square in the associated Algebra 1 lesson.
To rewrite the expressions in standard form, 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.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneCreate equivalent expressions in standard form mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for multiplying binomials to rewrite quadratic expressions in standard form.
To involve more students in the conversation, consider asking:
- “Who can restate ’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 ’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. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
12.2
Activity
Instructional Routines
Take TurnsMaterials
NoneActivity Narrative
In this activity, students practice connecting quadratic expressions in the form with associated functions in standard form. In the associated Algebra 1 lesson, students are beginning the process of completing the square. Fluency with recognizing quadratic expressions in standard form that are equivalent to perfect squares in factored form is important for students to begin completing the square.
In this partner activity, students take turns matching equivalent quadratic expressions in different forms. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Students will refer to these matches in the next activity.
12.3
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
NoneActivity Narrative
In this activity, students use the matches from the previous activity to identify parts of different forms of equivalent expressions. They then notice patterns and draw conclusions connecting the forms of quadratic expressions. By using each of the pairs students express regularity in repeated reasoning (MP8).
When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas
Launch
Display this first pair of equivalent expressions for all to see.
and
Consider completing the row in the table together while students answer the questions:
- “Which expression is in factored form?” ()
- “Which expression is in standard form?” ()
- “What does it mean to say that a term is linear?” (The term consists of a coefficient and or just .)
- “What is a coefficient?” (A number multiplied by the variables in a term. For example, 18 is the coefficient of in the linear term for this example.)