Authentic Lessons for 21st Century Learning

Two Worlds Collide, Part 1

Systems of Linear Equations: Graphing

Brittany VanCleave, Teresa Lansford, Michell Eike | Published: July 13th, 2022 by K20 Center

  • Grade Level Grade Level 9th, 10th, 11th
  • Subject Subject Mathematics
  • Course Course Algebra 1, Algebra 2
  • Time Frame Time Frame 50-75 minutes
  • Duration More 1-2 class period(s)


This lesson focuses on how to analyze and solve systems of linear equations by using the graphing method. The goal is for students to use prior knowledge to expand their understanding of equations and how they connect to real-world scenarios. Students will be able to identify, solve, and write equations using graphing as their tool. This is the first lesson of three in the "Two Worlds Collide" lesson series.

Essential Question(s)

How can systems of equations be used to represent situations and solve problems?



Students evaluate a statement by using the Always, Sometimes, or Never True strategy.


Students discover intersection points through a Desmos activity.


Students formalize their understanding of different types of systems of equations: consistent and independent, consistent and dependent, and inconsistent.


Students solve a problem based on a real-world scenario.


Students reflect on their understanding of the lesson.


  • Lesson Slides (attached)

  • Note Catcher handout (attached; one per student; printed front only)

  • Skate Park handout (attached; one per student; printed front only)

  • Paper

  • Pencil

  • Laptop or tablet with internet access


5 Minute(s)

Introduce the lesson using the attached Lesson Slides. Display slide 3 to share the lesson’s essential question. Display slide 4 to go over the lesson’s learning objectives. Review these slides with students to the extent you feel necessary.

Go to slide 5. Using the Elbow Partners strategy, have students discuss the first Always, Sometimes, or Never True statement: Two lines cross at only one point.

Give student pairs a couple minutes to analyze the statement and choose their claim. Once students have had time to discuss their claims, conduct a whole-class discussion by asking different pairs to share their viewpoints on the statement.


10 Minute(s)

Display slide 6 and provide students with your session code. Then, have students go to and enter the session code.

Pass out the attached Note Catcher handout to each student. Invite students to take notes on the top portion of the handout as they work through the activity.


20 Minute(s)

Assign student pairs or have each student find a partner. Display slide 7 and give student pairs time to read the definitions for consistent, inconsistent, dependent, and independent. Remind students these definitions are also on the Note Catcher handout for later reference.

Go to slide 8 and have student pairs use the definitions to match each graph on the slide with what they think is the best description: consistent and independent, consistent and dependent, or inconsistent. Ask students to draw, in pencil, a rough sketch of each graph in the corresponding columns on the Note Catcher handout.

As students finish sketching the graphs, go to slide 9. On the handout, ask student pairs to describe each type of system of equations. The goal is for each student to have both visual and verbal representations of the different types of systems of equations on their handout.

As students finish writing their descriptions, bring the class together to go through slides 10–12. Have students use these slides to confirm that they put the graphs in the right columns. Encourage students to write down any information from the slides that they may be missing on their handouts.

Ask for volunteers to share anything they wrote that was not on the slides. Use students’ responses to check for misunderstandings.


10 Minute(s)

Display slide 13. To expand students’ knowledge of using graphs and equations to solve real-world problems, read aloud the following scenario: You and your friend want to go to a skate park this weekend. There are two parks in the area, Scissortail and Silverstone. Scissortail costs 3 dollars to enter and 1 dollar for every hour you stay. Silverstone costs 5 dollars to enter and 50 cents for every hour you stay. Which skate park will you and your friend attend? Explain your reasoning.

Pass out the attached Skate Park handout to each student. Have students work individually to create and graph their own equations based on the given scenario. Using their findings, have students determine which skate park they want to attend and explain why on their handouts.

After providing time for students to work through the problem, go to slide 14. Ask students to share with the class which park they decided to attend and why.


5 Minute(s)

To wrap up the lesson, have students use the How Am I Feeling? What Am I Thinking? strategy to reflect on their learning. Display slide 15 and ask students to recreate the table on the slide on the back of their Skate Park handouts.

Go to slide 16. In the "How Am I Feeling?" column of the table, ask students to draw or write a description of how they feel about the content they’ve explored. In the "What Am I Thinking?" column, have students write a sentence that explains what they understand or think about the content they’ve explored. This could be a question or a comment about their learning or a description of the experience itself.

Use students’ responses to give you an understanding of how well they comprehend the material and what you might have to tweak moving forward. Make sure students have a clear understanding of the visual representation of how a system of equations works before the next lesson: "Two Worlds Collide, Part 2."


K20 Center. (n.d.). Always, sometimes, or never true. Strategies.

K20 Center. (n.d.). Elbow partners. Strategies.

K20 Center. (n.d.). How am I feeling? What am I thinking? Strategies.

K20 Center. (n.d.). Desmos Classroom. Tech tools.

Zimmermann, M. (n.d.). Solutions to Systems of Linear Equations [Interactive activity]. Desmos.