# A Parallel Perspective

## Writing Equations of Parallel Lines

Daniel Schwarz, Teresa Lansford, Michell Eike, Brittany VanCleave | Published: February 3rd, 2023 by K20 Center

• Subject Mathematics
• Course Algebra 1
• Time Frame 65-80 minutes
• Duration 2 class periods

### Summary

The lesson is an extension of the slope and point-slope forms of linear equations, with a focus on parallel lines. The goal is for students to understand which characteristics make a line parallel and apply that knowledge to solve problems. After completing the lesson, students will be able to define, recognize, and create parallel lines on graphs and with equations.

### Essential Question(s)

How do you know that a line is parallel to another line?

### Snapshot

Engage

Students judge if they should trust their eyes to determine if lines are parallel.

Explore

Students look at one set of buildings drawn from two different perspectives and determine which lines are parallel.

Explain

Students complete guided notes with the class to confirm that parallel lines have the same slope and learn how to write the equation of a parallel line.

Extend

Students find the equation of a line parallel to their line through an I Have Who Has activity.

Evaluate

Students apply their understanding of parallel lines to find the missing vertex of a parallelogram.

### Materials

• Lesson Slides (attached)

• Parallel Processing handout (attached; one per student; printed front only)

• Perspective Drawings handout (attached; one per student; printed front only)

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

• Guided Notes (Model Notes) document (attached; for teacher use)

• I Have Who Has Cards handout (attached; one per class; printed front only)

• Exit Ticket handout (attached; one quarter page per student; printed front only)

• Rulers

• Paper

• Pencils

• Coloring utensils (highlighters, markers, etc.; optional)

### Engage

Introduce the lesson using the attached Lesson Slides. Slide 3 identifies the lesson’s essential question: “How do you know that a line is parallel to another line?” Slide 4 identifies the lesson’s primary learning objectives.

Go to slide 5 and explain to students that they are about to see three images. For each image, they need to determine whether any lines are parallel, and if so, which ones. Take a moment to remind students that parallel lines are lines that do not and will never intersect. Alternatively, ask for a volunteer to remind the class what it means for two lines to be parallel. This is not yet the time to discuss slopes of parallel lines; that will be covered during the Explain section of the lesson.

Display slide 6. Ask the class to quietly think about which lines they think are or are not parallel and why. Ensure that everyone has time to think about the lines in the image.

After a couple of minutes, ask for 1–2 volunteers to share their thoughts about which lines are or are not parallel for the first image.

Transition through slides 7–8 and repeat the process with the other two images.

Now, have students find a partner or assign partners and pass out the attached Parallel Processing handout to each student. Here, students have the three images printed and closer to them. Tell students that they have about 5 minutes to determine which lines are parallel, if any. Allow students to use whatever resources they think would help them make a decision. Some students may ask to use a ruler to measure the distance between the lines. Some students may fold their paper along a line to make sure that the line is straight. Students can use markings that they remember from Geometry to indicate the pairs/sets of parallel lines, coloring utensils, or they can use any labeling system of their choice.

As students work, circulate the room. If anyone seems to finish really quickly, ask them questions about their conclusions or ask if they were to look at the image from a different angle (such as rotating their paper 90°) changes their opinion.

After students have had a chance to really look at the images, ask if anyone has changed their minds about which lines are parallel on any of the images. Go back through slides 6–8 and facilitate a whole-group discussion about which lines are parallel and how we can be sure. Emphasize to students that what is really important right now is not that they are right or wrong, but rather their reasoning.

Now transition through slides 9-12 and answer the question, “Were we right?” for each image. These slides explain why each image is an optical illusion and why scientists think we perceive the image the way that we do.

### Explore

Ask students, “Should we trust our eyes to determine if lines are parallel?” Facilitate a short discussion and push the discussion towards how math or even graph paper might be helpful.

Show slide 13 and give each student a copy of the attached Perspective Drawings handout. Have students use coloring utensils or other labels to indicate sets of parallel lines. Here, students see the same set of three buildings from two different perspectives: an aerial perspective and a drafting perspective.

Encourage students to try a different approach than what they tried during the Engage portion of the lesson. Remind them that we should not trust our eyes and ask if they can think of a different way to determine if two lines are parallel. Direct students to write their justifications regarding which lines are parallel.

End this portion of the lesson by bringing the class together for a whole-class discussion about what they noticed. Did they notice that the roof and floors of a building should be parallel and that depending on which perspective they had determined if this was graphically true? Also, help students understand that being parallel in three dimensions did not guarantee being parallel in two dimensions, but that graphing in three dimensions is a topic for a later course. Again, emphasize the importance of not trusting our eyes and the importance of needing the precision that mathematics offers.

### Explain

Go to slide 16. And give each student a copy of the attached Guided Notes handout. Ask pairs to determine whether the lines on the graph are parallel. Then ask for volunteers to share their thoughts. Discuss the characteristics that make up parallel lines.

After students feel comfortable with the first example, show slide 17 and have students independently determine if the lines are parallel. Call on a student to explain their thinking for the second example.

Display slide 19 and direct students’ attention to the third example on their handout. Ask guiding questions and walk through the example of how to write the equation of a line that passes through a given point and is parallel to a given line. Walk through the steps together to solve the problem.

Once you feel as though students are comfortable with the concepts, go to slide 20, and have them work with a partner or individually to write the equation for the fourth example. Move around the room to make sure everyone is on the same page.

Write the answer and work for example 4 on the board for students to check their work or ask for a volunteer to explain their work depending on the needs of your class. Use the attached Guided Notes (Model Notes) as needed.

### Extend

Show slide 21 and give each student one card from the I Have Who Has Card deck. Introduce students to the I Have Who Has strategy and explain what is on the card and the procedure for the activity. Students’ cards have an equation at the top of their card: “I have [this equation],” which is the answer to someone else’s card. At the bottom of their card, there is the question: “Who has a line parallel to mine that goes through the point (x, y)?”

Students are done when they have assembled into five groups of six students, where each group is a circle of students whose cards chain-link together. For example, Card 29 says, “I have y + 3 = 2(x – 1). Who has a line parallel to mine that goes through (–1, 3)?” The person with the answer to that question is the person holding Card 1, which says, “I have y – 3 = 2(x + 1)....” This continues until the sixth person in the group, who needs Card 14, which says, “...Who has a line parallel to mine that goes through (–1, 3)?” The answer is the person with Card 29, completing the circle.

Use the hidden slide 22 to check students’ work. Consider unhiding that slide to let students self-check at the end of the activity.

### Evaluate

Display slide 23 and use the Exit Ticket strategy to assess what students have learned individually. Give each student the Exit Ticket handout. Students are asked to determine the fourth vertex of a parallelogram given three vertices.

Use student responses to determine if the class is ready for the next topic or if remediation is needed. Use the hidden slide 24 to check students’ work. Consider unhiding the slide and using it the following day as bellwork.