Solving Problems with Systems of Linear Inequalities in Two Variables
5 min
Narrative
This Warm-up prompts students to carefully analyze and compare graphs that represent linear equations and inequalities. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology that students know and how they talk about characteristics of graphed systems.
Launch
Arrange students in groups of 2–4. Display the graphs for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell students to share their reasoning about why a particular item 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 responses:
A, B, and C go together because:
- They are graphs of systems.
- There is more than 1 equation or inequality graphed.
A, B, and D go together because:
- They include a line with a negative slope.
A, C, and D go together because:
- They have at least 1 point as a solution.
- They have solid lines.
B, C, and D go together because:
- They are graphs of inequalities.
- They have at least 1 region shaded.
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, prompt students to explain the meaning of any terminology they use, such as “region,” “boundary,” “slope,” or “solution,” 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
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
10 min
20 min
Knowledge Components
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Solving Problems with Systems of Linear Inequalities in Two Variables
5 min
Narrative
This Warm-up prompts students to carefully analyze and compare graphs that represent linear equations and inequalities. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology that students know and how they talk about characteristics of graphed systems.
Launch
Arrange students in groups of 2–4. Display the graphs for all to see. Ask students to indicate when they have noticed one that does not belong and can explain why. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, tell students to share their reasoning about why a particular item 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 responses:
A, B, and C go together because:
- They are graphs of systems.
- There is more than 1 equation or inequality graphed.
A, B, and D go together because:
- They include a line with a negative slope.
A, C, and D go together because:
- They have at least 1 point as a solution.
- They have solid lines.
B, C, and D go together because:
- They are graphs of inequalities.
- They have at least 1 region shaded.
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, prompt students to explain the meaning of any terminology they use, such as “region,” “boundary,” “slope,” or “solution,” 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
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- A-REI.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
- HSA-REI.D.12·Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.