Unit 8, Lesson 13
Constants in Quadratic Equations
Extra Support Materials for Algebra 1
13.1
Warm-up
Standards Alignment
Building On
8.EE.A.1
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, .
Materials
NoneActivity Narrative
The purpose of this Math Talk is to elicit strategies and understandings students have for finding a value that is needed to complete the square. It encourages students to think about expressions and to rely on what they know about operations involving fractions and exponents to mentally solve problems. The understanding elicited here will be helpful later in the associated Algebra 1 lesson when students will need to be able to complete the square.
By completing these calculations, students are making use of a structure that is needed for completing the square (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
NoneFor each value of , find mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for calculating .
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
13.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students solve equations of the form . In the associated Algebra 1 lesson, students solve equations using completing the square. After they complete the square, students can use the methods from this activity to solve the equation. Students look for and make use of structure (MP7) when they use the structure of equations of the form to solve problems.
13.3
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
In this activity, students use the patterns they noticed about quadratic expressions in standard form to add a constant value to the quadratic and linear term to make it a perfect square. After adding the value, students rewrite the expression in factored form. In the associated Algebra 1 lesson, students work to complete the square to solve equations. The work of this activity helps students develop some fluency with adding the correct constant as part of the process for completing the square. Students must look for and express regularity in repeated reasoning (MP8) when they find appropriate values to add to make a perfect square.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Arrange students in groups of 2. Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to find a constant term and to rewrite the expressions. Display words and phrases, such as “constant,” “half,” “squared,” “factor,” and “coefficient.”