Summary
This lesson is an extension of the "Pie > Everything" lesson series and adds a second variable to the equation. The goal of this lesson is for students to understand and find possible solutions for two-variable inequalities. Students will translate real-life problems into linear inequalities and make connections between the two. Students will also write and graph two-variable inequalities.
Essential Question(s)
How can two-variable inequalities be used to represent relationships?
Snapshot
Engage
Students watch a video and discuss questions to begin thinking about variables.
Explore
Students consider a real-world scenario and find multiple solutions to the problem presented in the scenario.
Explain
Students resolve misconceptions about inequalities to gain a deeper understanding.
Extend
Students elaborate on their knowledge of inequalities by participating in an Amplify Classroom polygraph activity.
Evaluate
Students write a linear inequality represented by a given graph.
Materials
Lesson Slides (attached)
Movie Snacks handout (attached; one per student; print two-sided)
Guided Notes handout (attached; one per student; print one-sided)
Student devices with internet access
Engage
5 Minute(s)
Introduce the lesson using the attached Lesson Slides. Briefly introduce the essential question on slide 3. Move to slide 4 to identify the lesson's learning objectives. Review each of these with students to the extent you feel necessary.
Move to slide 5 and show the short Let’s All Go to the Lobby video.
After showing the video, show slide 6 and pose the following questions:
Would you rather…
Have two boxes of candy and no drink, or a small popcorn and unlimited drinks?
Invite two more friends to watch a movie with you and have no snacks, or go with one friend and have money to buy snacks?
Ask students to consider whether they would answer the questions the same way every time, or if they would answer differently depending on the circumstances.
Explore
20 Minute(s)
Display slide 7 and have students find an Elbow Partner. Have partners read and analyze the following scenario on the slide:
Scenario 1: Your family goes to the movies. The snack bar is all out of large popcorn containers and large cups. The only size they have left is small. Your family has $30 to spend. How many orders of popcorn and drinks can your family buy if a small popcorn costs $6 and a small drink costs $5? Your family does not want any change.
Pass out one copy of the attached Movie Snacks handout to each student. Have students work with their Elbow Partners to determine how many different combinations of popcorn orders and drinks can be purchased for $30 without receiving change. Have them record their answers in the blank chart provided on “Part A” of the handout.
Once students think they have found all possible combinations, have them move to “Part B” to plot the points on the graph. After students plot their points, direct them to respond to the questions in “Part C,” which have them connect the dots with a line, identify the type of graph, and explain their thinking.
Go to slide 8 and direct students’ attention to “Scenario 2” on their handout:
Scenario 2: Your family decided it was too difficult to determine how many orders of popcorn and drinks to buy so that they wouldn’t get any change back. They decided that they do not mind if they get change back. How many different combinations of popcorn orders and drinks can your family buy?
Have students figure out the new combinations based on this scenario and record them in the second blank chart on the handout. Once they have figured out all the new combinations, ask them to plot the points on the graph and complete the handout.
Explain
20 Minute(s)
Pass out a copy of the Guided Notes handout to each student and transition through slides 9–10. Explain to students that graphing linear inequalities is much like graphing linear equations; the first step is the same for both processes, but when graphing linear inequalities, students must consider whether the line should be solid or dashed, and which area to shade on the graph.
Display slide 11 and explain the difference between solid and dashed lines and when each line is used. Have students record when to use solid lines and when to use dashed lines on their handout by writing the ≥ and ≤ symbols in the table under the graph of the solid line and writing the < and > symbols in the table under the graph of the dashed line.
Show slide 12 and define test point as any point that is not on the line, in the context of linear equations. Demonstrate how to use the test point by algebraically substituting the x- and y-values into the inequality to see if the test point makes the inequality true or false.
Move to slide 13 and explain that if the test point makes the inequality true, then they should shade toward the test point. If the test point makes the inequality false, they should shade away from the test point.
Transition through slides 14–15 and demonstrate to students how to shade toward the test point when it makes the inequality true and how to shade away from the test point when it makes the inequality false. Have students continue to complete their handouts.
Display slide 16 and guide students through the process of graphing the example equation on the slide, 2y < 3x – 2. Ask students the following prompting questions on the slide as you work:
Will this be a solid or dashed line?
What point will you use?
Check students' understanding using the challenge question on slide 17, which asks students to graph the equation 2y – 4x ≥ 6. Allow students to work in groups or pairs to graph the linear inequality.
Move to slide 18 and have students reflect on Scenario 2, “Part B” from the Movie Snacks handout. Have groups or pairs share out their thought processes for creating the equation in Scenario 2. As they share, guide students to make connections between the activity and the rules of inequalities by asking them the following questions on the slide:
Where are all the points located on the graph you created?
Could you shade a region that would contain all your points?
Display slide 19 and see if students recognize the inequality from the Movie Snacks scenarios. Ask students what the numbers in the inequality represent. Repeat this process with slide 20. Ask students to raise their hand if their graph looked like the graph on slide 19. Then, ask students to raise their hand if their graph looked like the graph on slide 20. Lead a class discussion about why both responses are correct.
Extend
30 Minute(s)
Display slide 21 and provide students with your session code. Then, have them go to student.amplify.com/join and enter the session code.
Once students enter the session code, explain how the polygraph activity works:
The first round is quick and intended to help them understand how the game works.
Students will be automatically assigned partners to complete the activity. If you have an odd number of students, consider logging in as a student to be someone’s partner.
One student is assigned the role of “guesser” and the other is assigned the role of “picker.” The picker selects one of the inequalities. Then, the guesser will ask the picker “yes” or “no” questions to try to determine which graph the picker selected.
Have students play the game two times and answer the prompted questions at the end of the game.
As students work, use the teacher dashboard to see what questions students are asking each other. Remind and encourage students to use the vocabulary they learned during the lesson.
Evaluate
5 Minute(s)
Use the Exit Ticket instructional strategy to individually assess what students have learned from the lesson. Display slide 22 and have students write the inequality that is represented by the given graph. Have students write their answers on an index card, sticky note, piece of paper, etc.
Use student responses to determine if they need additional practice with graphing or writing linear inequalities or if they are ready for the next topic.
Resources
Burrns Luciano. (2021, May 15). Let's all go to the lobby! Intermission bumper 4K update [Video]. YouTube. https://www.youtube.com/watch?v=lm6IU6V-dE8
Illichman. (n.d.). Polygraph: Linear inequalities [Interactive activity]. Desmos. https://teacher.desmos.com/activitybuilder/custom/560ad6907701c30306330601
K20 Center. (n.d.). Bell ringers and exit tickets. https://learn.k20center.ou.edu/strategy/125
K20 Center. (n.d.). Desmos classroom. Tech tools. https://learn.k20center.ou.edu/tech-tool/1081
K20 Center. (n.d.). Elbow partners. https://learn.k20center.ou.edu/strategy/116
