Authentic Lessons for 21st Century Learning

Popcorn > Raisinets

Inequalities With Two Variables

Brittany VanCleave, Amber Stokes, Keiana Cross, Michell Eike | Published: January 10th, 2023 by K20 Center

  • Grade Level Grade Level 9th
  • Subject Subject Mathematics
  • Course Course Algebra 1
  • Time Frame Time Frame 80-90 minutes
  • Duration More 2 class periods


This lesson is an extension of the Memes > GIFs lesson, adding a second variable to the equation. The goal is to understand and find possible solutions for two-variable inequalities. Students will translate real-life problems into linear inequalities and make the connection between the two. They will also write and graph two-variable inequalities.

Essential Question(s)

How can two-variable inequalities be used to represent relationships?



Students watch a video and discuss questions to get them thinking about variables.


Students consider a real-world scenario and find multiple solutions to the problem.


Students resolve misconceptions to gain a deeper knowledge about inequalities.


Students elaborate on their knowledge of inequalities by playing a Desmos Classroom polygraph activity.


Students write linear inequalities represented by the given graphs.


  • Lesson Slides (attached)

  • Movie Snacks handout (attached; one per student; printed front/back)

  • Guided Notes handout (attached; one per student; printed front only)

  • Student devices with internet access (tablets, laptops, or desktops recommended)


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, pose the following questions.

Would you rather…

  1. Have two boxes of candy and no drink, or a small popcorn and unlimited drinks?

  2. Invite two more friends to watch a movie with you and have no snacks, or go with one friend and have the money to buy snacks?

Ask students to reflect on whether they would answer the question the same way every time, or differently depending on the circumstances.


20 Minute(s)

Display slide 6 and have students find an Elbow Partner to analyze the following scenario:

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 they buy if a popcorn costs $6 and a drink costs $5? Your family does not want any change.

Pass out a copy of the attached Movie Snacks handout to each student. Have students work in pairs to figure out how many different combinations of popcorn orders and drinks can be bought for $30 without receiving change. Have them put their answers in the blank chart provided on Part A of the handout.

Once they think they have found all possible combinations, have them move on to Part B where they can plot the points on the graph. After students plot their points, direct them to answer the questions in Part C, connect their dots with a line, and explain their thinking.

Go to slide 7 and direct students’ attention to Scenario 2 on their handout: 

Scenario 2: Your family decided it was way too difficult to determine the number of orders of popcorn and drinks they need to buy to get back no change. Now, they do not mind if they get change back from their $30. 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.


20 Minute(s)

Pass out a copy of the Guided Notes handout to each student and transition through slides 8–9. Explain to students that graphing linear inequalities is much like graphing linear equations: the first step is the same but the difference is they need to consider solid or dashed lines and shading with inequalities.

Display slide 10 and explain the difference between solid and dashed lines and when they occur. Have students record when to use solid lines and when to use dashed lines on their handout by writing the ≥ to and ≤ symbols in the table under the graph of the solid line and writing < and > in the table under the graph of the dashed line.

Display slide 11, define “test point,” and demonstrate how to use them.

Transition through slides 12–14 and demonstrate to students how to shade towards the test point when it makes the inequality true and to shade away from the test point when it makes the inequality false. Remind students to continue completing their handout.

Display slide 15 and work through graphing the example with students while asking the guiding questions on the slide.

Check students' understanding using the challenge question on slide 16. Allow students to work in groups or pairs to graph the linear inequality.

Move to slide 17 and reflect on the activity from the Scenario 2, Part B section of the Movie Snacks handout completed earlier. Have groups or pairs share their thought process for creating the equation in Scenario 2. As they do, start making connections between the activity and the rules of inequalities by asking the students the following questions:

  1. Where are all the points located on the graph you created for Scenario 2?

  2. Could you shade a region that would contain all your points?

Display slide 18 and see if students recognize this inequality from the Explore scenarios. Ask students what the numbers in the inequality represent. Repeat this with slide 19. Ask students to raise their hand if their graph looked like what they saw on slide 18. Repeat this with slide 19. Then have a class discussion about why they are both correct.


30 Minute(s)

Display slide 20 and provide students with your session code. Then, have students go to and enter the session code.

Once students enter the session code, explain to students how the polygraph activity works: 

  1. The first round will be quick and is for students to understand how the game works.

  2. Students will then 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.

  3. One student will be assigned the role of “guesser” and the other will be 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.

  4. 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 that they learned during the lesson.


5 Minute(s)

Use the Exit Ticket strategy to individually assess what students have learned from the lesson. Display slide 21 and have students write the inequality that is represented by the given graph. Have students write their answer 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 are ready for the next topic.