Unit 7, Lesson 16
Graphing from the Vertex Form
Extra Support Materials for Algebra 1
16.1
Warm-up
Standards Alignment
Building On
Addressing
7.NS.A.2.c
Apply properties of operations as strategies to multiply and divide rational numbers.
Building Toward
Materials
NoneActivity Narrative
The purpose of this Math Talk is to elicit strategies and understandings students have for adding and squaring signed numbers. It encourages students to think about evaluating an expression like for an input as a sequence of operations in a conventional order and to rely on what they know about signed numbers to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will need to be able to evaluate a quadratic expression for a negative value of the variable.
Students notice and make use of structure if they notice that they can substitute their previous answer for part of the next expression (MP7). For example, if they notice that is the negation of the previous expression.
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
NoneEvaluate each expression when is -7, mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for evaluating expressions.
Ask students to share their strategies for each problem. Record and display their responses for all to see. 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
16.2
Activity
Instructional Routines
MLR2: Collect and DisplayMaterials
NoneActivity Narrative
In this activity, students create a table consisting of the coordinates of the points around a graph’s vertex. This prepares them to reason using the structure of expressions in the associated Algebra 1 lesson, where they perform a similar analysis without being prompted to create a table first. As students generalize about quadratic equations they are expressing regularity in repeated reasoning (MP8).
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
Display the functions and for all to see.
Ask: “How are they alike? How are they different?”
This activity is meant to be done without a calculator, as performing the calculations will help students notice the structure of the outputs of the function for different inputs. However, if the calculations will prevent any students from accessing the task, provide access to a scientific calculator.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to reason about the graphs of quadratic equations. Display words and phrases, such as “vertex,” “maximum,” “minimum,” “square,” “negative,” and “zero.”16.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In order to match each definition with a function, students could use the structure of the vertex form noticed in this lesson and in earlier Algebra 1 lessons, or they could start by substituting different values into each equation and compare those to the tables. When students use the structure of vertex form equations they are looking for and making use of structure (MP7).
Launch
Point out that the first question is just asking for a rough sketch, rather than carefully scaled axes and plotted points. The goal is for the vertex to be located in the correct quadrant and labeled correctly, and for the graph to open up or down as appropriate.