Unit 6, Lesson 12
Connections between Graphs and Equations
Extra Support Materials for Algebra 1
12.1
Warm-up
Standards Alignment
Building On
Addressing
HSF-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on evaluating functions at a given value of the input. It encourages students to think about functions as equations and to rely on what they know about solving equations to mentally solve problems. The understanding elicited here will be helpful later in this lesson when students will construct a table of values for a given function.
To evaluate functions at a given input value, 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
NoneHere is a function:
Evaluate mentally:
Activity Synthesis
The goal of this discussion is to review students’ strategies for evaluating equations.
- “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?”
Consider sketching a blank graph for all to see and asking:
- “What shape do we expect this graph to be?” (A line, because the output is 100 when the input is 0, and then the output decreases by 5 every time the input increases by 1.)
- “What are some points on the graph?” ()
- “Where can you see the 100 and the -5 on the graph?” (100 is the vertical intercept, and -5 is the slope.)
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
NoneMaterials
To Gather
Graphing technologyActivity Narrative
In this activity, students make connections between equations, tables, graphs, and verbal descriptions. They are given equations representing bank accounts and use that information to create tables, describe the situations, and use technology to graph the equations. If students also use technology to create the tables, they are using appropriate tools strategically (MP5).
Launch
Provide access to graphing technology. Students may need some assistance adjusting their graphing window to see the relevant features of each graph. Either help them, before they start, set the window to a domain of and range of or wait until they have attempted to graph at least one function and remind them of what they have learned about changing the graphing window.
12.3
Activity
Instructional Routines
MLR1: Stronger and Clearer Each TimeMaterials
To Gather
Graphing technologyActivity Narrative
Students continue to work with the functions from the previous activity but strategically modify them so that they represent new situations. Then students select a graph that represents the new situation and use technology to see if their new equation is correct (by seeing whether its graph matches the one they chose). Students reason abstractly and quantitatively (MP2) when making decisions about how to adjust equations to represent altered situations.
Launch
Continue to provide access to graphing technology. Since this work is a continuation of the previous activity, students should be able to get started without much intervention.