Loading [Contrib]/a11y/accessibility-menu.js

 Lesson Plan: Types of Slope

 

Lesson Summary

In this lesson, students will explore the concept of slope and learn to differentiate among the four types: positive, negative, zero, and undefined. Through interactive activities, visual aids, and discussions, students will analyze how the slope affects the direction of a line on a graph. By the end of the lesson, students will be able to identify and describe each type of slope and understand its graphical representation.

Lesson Objectives

  • Differentiate between positive, negative, zero, and undefined slopes.
  • Analyze how the slope affects the direction of the line on a graph.

Standards

  • CCSS.MATH.CONTENT.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.

Prerequisite Skills

  • Familiarity with the coordinate plane and plotting points
  • Understanding of the concept of slope as the ratio of the vertical change (rise) to the horizontal change (run) between two points
  • Experience with using interactive graphing tools or online resources (a Desmos activity is provided in this lesson)

Key Vocabulary

  • Slope: A measure of the steepness or incline of a line, calculated as the ratio of the rise to the run between two points on the line.
  • Rise: The vertical change between two points on a line.
  • Run: The horizontal change between two points on a line.
  • Positive Slope: A slope where the line rises from left to right, indicating a positive rate of change.
  • Negative Slope: A slope where the line falls from left to right, indicating a negative rate of change.
  • Zero Slope: A slope of a horizontal line, indicating no change; the rise is zero.
  • Undefined Slope: A slope of a vertical line, where the run is zero, making the slope undefined.
  • Coordinate Plane: A two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis, used to graph points, lines, and curves.
  • Linear Relationship: A relationship between two variables that, when graphed, produces a straight line, indicating a constant rate of change.

Multimedia Resources

 

 

Warm Up Activities

Choose from one or more activities.

Activity 1: Image Analysis

  1. Use these math clip art pieces to show different types of slopes: https://www.media4math.com/library/75389/asset-preview
  2. Ask students to observe the directions of the lines and describe what they notice about the slopes.
  3. Encourage students to share their observations and discuss the differences in line directions.

Activity 2: Data Table 1 – Hours vs. Earnings

Students examine the table below, which represents a person’s earnings based on the number of hours worked. They calculate the common difference and identify the pattern as linear with a positive rate of change.

Hours Worked

Earnings ($)

115
230
345
460
575

Discussion Questions:

  • What is the common difference in earnings as the number of hours worked increases?
  • How would you describe this pattern in words?
  • How can we represent this pattern as an equation?

Activity 2: Data Table 2 – Savings Account Decrement

The table below represents a person withdrawing $50 from a savings account each week. Students determine the common difference and identify the pattern as linear with a negative rate of change.

Week

Savings Balance ($)

0500
1450
2400
3350
4300

Discussion Questions:

  • What is the common difference in savings as the weeks progress?
  • Why is this pattern considered linear?
  • How can we express this pattern using an equation?

 

 

Teach

Introduction

In this lesson, students will learn about the concept of slope and how it describes the steepness and direction of a line on a coordinate plane. They will explore the four types of slope—positive, negative, zero, and undefined—and learn how to identify them from graphs, equations, and data tables.

The lesson will cover the following key concepts:

  • Understanding Slope: Slope is a measure of how much a line rises or falls as it moves from left to right. It is calculated using the formula: slope (m) = rise/run.
  • Types of Slope: Students will analyze the four types of slope:
    • Positive Slope: The line rises from left to right.
    • Negative Slope: The line falls from left to right.
    • Zero Slope: The line is horizontal, showing no change.
    • Undefined Slope: The line is vertical, meaning the run is zero.
  • Graphing Slope: Students will practice identifying the slope of a line by examining its graph.
  • Interpreting Slope in Real-World Contexts: Students will see how slope applies to real-life scenarios, such as speed, earnings, and savings.
  1. Play the following video, which describes the different types of slopes: https://www.media4math.com/library/75390/asset-preview
  2. Review the following vocabulary terms:
  3. Have students use this Desmos activity: https://www.media4math.com/library/75388/asset-preview. Students can change the slope of the line in two ways:
    • Clicking and dragging on one of the points
    • Changing the values for a, b, c, and d to modify the coordinates.
  4. Have students use the Desmos activity to show examples of these types of slopes:
    • Positive Slope
    • Negative Slope
    • Zero Slope
    • Undefined Slope

Example 1: Positive Slope

Problem: Given the points (2,3) and (6,7), find the slope of the line passing through these points.

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{7 - 3}{6 - 2} = \frac{4}{4} = 1 \]
  3. The slope is 1, meaning the line rises one unit for every unit it moves to the right.

 

Slope

 

Example 2: Negative Slope

Problem: Find the slope of the line passing through the points (1,5) and (4,2).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{2 - 5}{4 - 1} = \frac{-3}{3} = -1 \]
  3. The slope is -1, meaning the line falls one unit for every unit it moves to the right.

 

Slope

 

Example 3: Zero Slope

Problem: Find the slope of a horizontal line passing through (2,4) and (5,4).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{4 - 4}{5 - 2} = \frac{0}{3} = 0 \]
  3. A zero slope means the line is perfectly horizontal.

 

Slope

 

Example 4: Undefined Slope

Problem: Find the slope of a vertical line passing through (3,2) and (3,6).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{6 - 2}{3 - 3} = \frac{4}{0} \]
  3. Since division by zero is undefined, the slope is undefined.

Example 5: Real-World Application – Road Incline

Problem: A road rises 10 feet for every 50 feet of horizontal distance. What is the slope?

  1. Use the slope formula: \( m = \frac{\text{rise}}{\text{run}} \).
  2. Substituting the values: \[ m = \frac{10}{50} = \frac{1}{5} \]
  3. This means the road rises 1 foot for every 5 feet traveled forward.

Art Specs: A simple side-view diagram of a road with a rise of 10 feet and a run of 50 feet.

Example 6: Real-World Application – Water Drainage

Problem: A drainage pipe slopes downward 3 inches for every 12 inches of horizontal distance. What is the slope?

  1. Use the slope formula: \( m = \frac{\text{rise}}{\text{run}} \).
  2. Substituting the values: \[ m = \frac{-3}{12} = \frac{-1}{4} \]
  3. The negative slope means the pipe slopes downward.

 

 

Review

Lesson Summary

In this lesson, students explored the concept of slope and how it describes the steepness and direction of a line. They learned about the four types of slope:

  • Positive Slope: The line rises from left to right.
  • Negative Slope: The line falls from left to right.
  • Zero Slope: The line is horizontal, showing no change.
  • Undefined Slope: The line is vertical, meaning the run is zero.

Students applied the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) to calculate slope using coordinate points and real-world scenarios. Note: The next lesson in this series goes into more detail on the slope formula. They also learned how to interpret slope values and recognize linear patterns in data tables and graphs.

Key Vocabulary Review

  • Slope: The ratio of vertical change (rise) to horizontal change (run) between two points on a line.
  • Positive Slope: A line that rises from left to right, indicating an increasing relationship.
  • Negative Slope: A line that falls from left to right, indicating a decreasing relationship.
  • Zero Slope: A horizontal line with no change in y-values.
  • Undefined Slope: A vertical line where the x-values do not change.
  • Rise: The vertical difference between two points on a line.
  • Run: The horizontal difference between two points on a line.
  • Linear Relationship: A relationship that produces a straight-line graph, showing a constant rate of change.
  1. Facilitate a class discussion by asking students to share their observations and findings from the previous activity.
  2. Address any misconceptions or clarify any confusions that may arise.
  3. Provide individual practice problems or a short quiz to reinforce the concepts of positive, negative, zero, and undefined slopes.

Example 1: Positive Slope

Problem: Find the slope of a line passing through the points (-2,-1) and (3,4).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{4 - (-1)}{3 - (-2)} = \frac{4 + 1}{3 + 2} = \frac{5}{5} = 1 \]
  3. The slope is 1, meaning the line rises one unit for every unit it moves to the right.

Example 2: Negative Slope

Problem: Find the slope of a line passing through the points (4,6) and (8,2).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{2 - 6}{8 - 4} = \frac{-4}{4} = -1 \]
  3. The slope is -1, meaning the line falls one unit for every unit it moves to the right.

Example 3: Zero Slope

Problem: Find the slope of a horizontal line passing through (-3,5) and (4,5).

  1. Use the slope formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
  2. Substituting the values: \[ m = \frac{5 - 5}{4 - (-3)} = \frac{0}{7} = 0 \]
  3. A zero slope means the line is perfectly horizontal.

 

 

Quiz

Answer the following questions.

  1. Find the slope of the line passing through (2,3) and (6,9).
  2. Find the slope of the line passing through (-5,8) and (3,4).
  3. Find the slope of the line passing through (1,7) and (1,-2).
  4. Find the slope of the line passing through (4,-2) and (10,-2).
  5. Find the slope of the line passing through (-3,-5) and (6,1).
  6. If a line has a slope of \( \frac{2}{3} \), does it rise or fall from left to right?
  7. If a line has a slope of \( -4 \), does it rise or fall from left to right?
  8. If a line has a slope of 0, what kind of line is it?
  9. If a line has an undefined slope, what kind of line is it?
  10. Find the slope of the line passing through (-6,9) and (0,3). What can you say about the behavior of this line?

Answer Key

  1. \( m = \frac{9 - 3}{6 - 2} = \frac{6}{4} = \frac{3}{2} \)
  2. \( m = \frac{4 - 8}{3 - (-5)} = \frac{-4}{8} = \frac{-1}{2} \)
  3. \( m = \frac{-2 - 7}{1 - 1} = \frac{-9}{0} \) (undefined slope, vertical line)
  4. \( m = \frac{-2 - (-2)}{10 - 4} = \frac{0}{6} = 0 \) (zero slope, horizontal line)
  5. \( m = \frac{1 - (-5)}{6 - (-3)} = \frac{6}{9} = \frac{2}{3} \)
  6. The line rises from left to right because the slope is positive.
  7. The line falls from left to right because the slope is negative.
  8. The line is horizontal.
  9. The line is vertical.

  10. \( m = \frac{3 - 9}{0 - (-6)} = \frac{-6}{6} = -1 \)

    The line falls at a 45-degree angle from left to right because the slope is -1.