Equations and Their Graphs
5 min
Narrative
This Warm-up prompts students to carefully analyze and compare features of graphs of linear equations. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and to learn how they talk about characteristics of graphs.
The work here prepares students to reason about solutions to equations by graphing, which is the focus of this lesson.
Launch
Arrange students in groups of 2–4. Display the graphs for all to see.
Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask students to share their reasoning as to why a particular graph does not belong, and together to find at least one reason that each item doesn't belong.
Student Task
Which three go together? Why do they go together?
A
B
C
D
Sample Response
Sample response:
- A, B, and C go together because they all have a positive slope.
- A, B, and D go together because they are continuous graphs (a solid line rather than plotted points).
- A, C, and D go together because they have hours on the x-axis and dollars on the y-axis.
- B, C, and D go together because they are linear relationships with a positive y-intercept.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology that they use, such as "y-intercept" or "negative slope," and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
Standards
- 8.EE.B·Understand the connections between proportional relationships, lines, and linear equations.
- 8.EE.B·Understand the connections between proportional relationships, lines, and linear equations.
- 8.F.5·Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
- 8.F.B.5·Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- HSF-IF.C.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
10 min
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Equations and Their Graphs
5 min
Narrative
This Warm-up prompts students to carefully analyze and compare features of graphs of linear equations. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students know and to learn how they talk about characteristics of graphs.
The work here prepares students to reason about solutions to equations by graphing, which is the focus of this lesson.
Launch
Arrange students in groups of 2–4. Display the graphs for all to see.
Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask students to share their reasoning as to why a particular graph does not belong, and together to find at least one reason that each item doesn't belong.
Student Task
Which three go together? Why do they go together?
A
B
C
D
Sample Response
Sample response:
- A, B, and C go together because they all have a positive slope.
- A, B, and D go together because they are continuous graphs (a solid line rather than plotted points).
- A, C, and D go together because they have hours on the x-axis and dollars on the y-axis.
- B, C, and D go together because they are linear relationships with a positive y-intercept.
Activity Synthesis (Teacher Notes)
Invite each group to share one reason why a particular set of three goes together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.
During the discussion, ask students to explain the meaning of any terminology that they use, such as "y-intercept" or "negative slope," and to clarify their reasoning as needed. Consider asking:
- “How do you know . . . ?”
- “What do you mean by . . . ?”
- “Can you say that in another way?”
Standards
- 8.EE.B·Understand the connections between proportional relationships, lines, and linear equations.
- 8.EE.B·Understand the connections between proportional relationships, lines, and linear equations.
- 8.F.5·Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
- 8.F.B.5·Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- F-IF.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.
- HSF-IF.C.7.a·Graph linear and quadratic functions and show intercepts, maxima, and minima.