standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Warm-up
This warm-up prompts students to compare four lines. It invites students to explain their reasoning and hold mathematical conversations, and allows you to hear how they use terminology and talk about lines and their properties. To allow all students to access the activity, each figure has one obvious reason it does not belong. Encourage students to move past the obvious reasons (e.g., line
\(t\)
has a different color) and find reasons based on geometric properties (e.g., a slope triangle of line
\(u\)
is not similar to the slope triangles of the other three lines).
So far, we have only considered lines with positive slopes. The purpose of this warm-up is to suggest similarities (same vertical and horizontal lengths of slope triangles) and differences (since they are not parallel, there is something fundamentally different going on here) between lines whose slopes have the same absolute value but opposite signs.
As students share their responses, listen for important ideas and terminology that will be helpful in the work of this lesson. Students may:
Identify lines that have the same slope.
Identify points where lines intersect.
Distinguish between lines with positive and negative values for slope and be able to articulate the difference clearly.
Launch
Arrange students in groups of 2–4 and provide access to geometry toolkits. Display the image of the four lines for all to see. Ask students to indicate when they have noticed one line that does not belong and can explain why. Give students 2 minutes of quiet think time and then time to share their thinking with their group. After everyone has conferred in groups, ask each group to offer at least one reason a particular line doesn’t belong.
Student Facing
Which line doesn’t belong?
[IMG]
Student Response
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Activity Synthesis
After students have conferred in groups, invite each group to share one reason why a particular line might not belong. Record and display the responses for all to see. After each response, ask the rest of the class if they agree or disagree. Since there is no single correct answer to the question asking which shape does not belong, attend to students’ explanations and ensure the reasons given are correct.
During the discussion, prompt students to use mathematical terminology (parallel, intersect, slope) correctly. Also, press students on unsubstantiated claims. For example, a student may claim that
\(u\)
does not belong because it has a different slope. Ask how they know for sure that its slope is different from the other lines. Demonstrate drawing a slope triangle and computing slope.
Based on the work done up to this point in the unit, students are likely to assume that the slope of
\(v\)
is
\(\frac13\)
. In the discussion, solicit the idea that there is something fundamentally different about line
\(v\)
compared to the others. You could use informal language like “uphill” and “downhill,” or “tilt direction.” The expressions positive and negative slope do not need to be introduced at this time.


aligns: yes