Writing Equations for Lines
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this activity, students identify lines with different slopes and draw a line with a particular slope. This activity reinforces the idea that different slope triangles whose longest side lies on the same line give the same value for slope.
Monitor for students who use different strategies to construct the line for part F. Here are some strategies students may use:
- Draw slope triangles.
- Count off horizontal and vertical distances without drawing slope triangles.
Select students using each method to share during the discussion.
Launch
Provide access to rulers or straightedges. If necessary, refer to the classroom display defining slope. Give students 5 minutes of quiet work time followed by a whole-class discussion.
Student Task
Here are several lines.
- Match each line shown with a slope from this list: 31, 2, 53, 1, 0.25, 23.
- One of the given slopes does not have a line to match. Draw a line with this slope on the empty grid (F).
Sample Response
- A: 23, B: 31, C: 1, D: 2, E: 0.25
- Sample response: (A valid response may or may not include a slope triangle similar to the one shown, but all lines should have slope 53.)
Activity Synthesis (Teacher Notes)
The goal is for students to practice finding the slope of a given line on a grid and to understand how different slope triangles can be used to draw or determine the slope of the same line.
Ask previously selected students to share how they drew their lines with a slope of 53. Sequence the discussion so that students who use slope triangles present their work first and students who count horizontal and vertical displacement (without drawing a triangle) present second. Help students see that the second method is the same as the first except that the slope triangle connecting two points on the line is only “imagined” rather than drawn. If time allows, demonstrate that moving up 3 then right 5 results in a line with the same slope as moving right 10 and up 6. Encourage students to draw slope triangles if it helps them to see and understand the underlying structure.
Here are some questions for discussion:
-
“Given a line, how can you determine its slope?” (Draw a right triangle where the longest side is on the line and divide the vertical length by the horizontal length.)
-
“Given a slope, how can you draw a line with the slope?” (Draw a right triangle with vertical and horizontal lengths whose quotient matches the slope, and then extend the longest side of the triangle.)
Advances: Conversing, Reading
Supports accessibility for: Organization, Conceptual Processing
Anticipated Misconceptions
Some students may find it difficult to draw a slope triangle for a line when one is not given. Prompt them to examine two places where the line crosses an intersection of grid lines.
Standards
- 8.EE.6·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.
- 8.EE.B.6·Use similar triangles to explain why the slope <span class="math">\(m\)</span> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <span class="math">\(y = mx\)</span> for a line through the origin and the equation <span class="math">\(y = mx + b\)</span> for a line intercepting the vertical axis at <span class="math">\(b\)</span>.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Writing Equations for Lines
10 min
Teacher Prep
Setup
Required Preparation
Narrative
In this activity, students identify lines with different slopes and draw a line with a particular slope. This activity reinforces the idea that different slope triangles whose longest side lies on the same line give the same value for slope.
Monitor for students who use different strategies to construct the line for part F. Here are some strategies students may use:
- Draw slope triangles.
- Count off horizontal and vertical distances without drawing slope triangles.
Select students using each method to share during the discussion.
Launch
Provide access to rulers or straightedges. If necessary, refer to the classroom display defining slope. Give students 5 minutes of quiet work time followed by a whole-class discussion.
Student Task
Here are several lines.
- Match each line shown with a slope from this list: 31, 2, 53, 1, 0.25, 23.
- One of the given slopes does not have a line to match. Draw a line with this slope on the empty grid (F).
Sample Response
- A: 23, B: 31, C: 1, D: 2, E: 0.25
- Sample response: (A valid response may or may not include a slope triangle similar to the one shown, but all lines should have slope 53.)
Activity Synthesis (Teacher Notes)
The goal is for students to practice finding the slope of a given line on a grid and to understand how different slope triangles can be used to draw or determine the slope of the same line.
Ask previously selected students to share how they drew their lines with a slope of 53. Sequence the discussion so that students who use slope triangles present their work first and students who count horizontal and vertical displacement (without drawing a triangle) present second. Help students see that the second method is the same as the first except that the slope triangle connecting two points on the line is only “imagined” rather than drawn. If time allows, demonstrate that moving up 3 then right 5 results in a line with the same slope as moving right 10 and up 6. Encourage students to draw slope triangles if it helps them to see and understand the underlying structure.
Here are some questions for discussion:
-
“Given a line, how can you determine its slope?” (Draw a right triangle where the longest side is on the line and divide the vertical length by the horizontal length.)
-
“Given a slope, how can you draw a line with the slope?” (Draw a right triangle with vertical and horizontal lengths whose quotient matches the slope, and then extend the longest side of the triangle.)
Advances: Conversing, Reading
Supports accessibility for: Organization, Conceptual Processing
Anticipated Misconceptions
Some students may find it difficult to draw a slope triangle for a line when one is not given. Prompt them to examine two places where the line crosses an intersection of grid lines.
Standards
- 8.EE.6·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.
- 8.EE.B.6·Use similar triangles to explain why the slope <span class="math">\(m\)</span> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <span class="math">\(y = mx\)</span> for a line through the origin and the equation <span class="math">\(y = mx + b\)</span> for a line intercepting the vertical axis at <span class="math">\(b\)</span>.