Linear Relationships

Lesson 6

Math

Unit 5

8th Grade

Lesson 6 of 15

Objective


Define slope and determine slope from graphs.

Common Core Standards


Core Standards

  • 8.EE.B.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.

Foundational Standards

  • 8.G.A.4
  • 8.G.A.5
  • 7.NS.A.2.B

Criteria for Success


  1. Define the slope of a line as the measure of vertical change over the measure of horizontal change between any two points on the line. 
  2. Understand that the slope of a line is the same through any two points on the line, which can be explained using similar triangles.
  3. Understand that positive slopes rise toward the right on the coordinate plane and negative slopes rise toward the left on the coordinate plane.
  4. Understand that slopes with an absolute value measure less than 1 are more “shallow” and slopes with an absolute value measure greater than 1 are more “steep.”
  5. Find the slope of a line from a graph.

Tips for Teachers


  • This lesson may be extended over two days in order to develop the concepts in all three Anchor Problems and to allow students time to practice on their own. 
  • Students will need an understanding that  $${-{{p}\over{q}}}$$ is the same as $${-p\over{q}}$$ and $${p\over{-q}}$$ (standard 7.RP.2b) when they consider negative slope. It may be helpful to review this standard prior to this lesson. 
  • Students will also recall the transformations that can be used to show similarity between two figures (8.G.4). 
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Anchor Problems

25-30 minutes


Problem 1

Three staircases are shown below.

a.   Which staircase is the steepest? Without doing any calculations, order the staircases from least steep to most steep.

b.   Slope is a measure of steepness. The greater the slope, the greater the steepness. Use the measurements provided in the diagram to justify the order of steepness you determined in part A.

c.   Another staircase climbs 8 feet over a distance of 10 feet. Where does this staircase fall in the order of steepness? 

Guiding Questions

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Student Response

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References

Finding Ways Staircase and Steepness

Staircase and Steepness by Fawn Nguyen is made available on Finding Ways under the CC BY 4.0 license. Accessed Feb. 24, 2018, 4:53 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Find the slope of the line in each graph below. 

a.   

b.   

c.   

Guiding Questions

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Student Response

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Problem 3

A line is drawn through the points A, C, E, and G as shown in the graph below.

  1. What transformation takes triangle $$ABC$$ to triangle $$EFG$$? Are the triangles congruent, similar, or neither? Explain.
  2. What transformation takes triangle $$ABC$$ to triangle $$ADG$$? Are the triangles congruent, similar, or neither? Explain.
  3. If two triangles are similar, then what do you know about their corresponding side lengths? 
  4. Will the slope of the line between any two points on the line always be the same? Explain your reasoning using similar triangles. 

Guiding Questions

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Student Response

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Problem Set

15-20 minutes


Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

5-10 minutes


Problem 1

Find the slope of each line graphed below.

a.   

 

b.   

Student Response

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Problem 2

For the line in part B, use a different pair of coordinate points to find the slope to show that the slope is the same through any two points on the line.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Next

Determine slope from coordinate points. Find slope of horizontal and vertical lines.

Lesson 7
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Comparing Proportional Relationships

Topic B: Slope and Graphing Linear Equations

Topic C: Writing Linear Equations

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