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 Lesson Plan: Visualizing Slope on a Graph

 

Lesson Summary

In this lesson, students will deepen their understanding of slope by engaging in hands-on activities that involve plotting points on a coordinate plane and analyzing the resulting lines. They will learn to calculate the slope by determining the ratio of the vertical change (rise) to the horizontal change (run) between two points. Through interactive tools and real-world examples, students will visualize how slope represents the steepness and direction of a line, and how it applies to various contexts such as ramps, staircases, and other inclined surfaces.

Lesson Objectives

  • Plot points on a coordinate plane.
  • Understand how slope is represented on a graph.
  • Calculate the slope of a line given two points 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.
  • CCSS.MATH.CONTENT.8.F.B.4: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Prerequisite Skills

  • Understanding of the coordinate plane and how to plot points
  • Knowledge of the concept of slope (rise over run)
  • Familiarity with similar triangles and their properties

Materials

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.
  • Coordinate Plane: A two-dimensional plane formed by the intersection of a vertical y-axis and a horizontal x-axis, used for plotting points, lines, and curves.
  • Plot: To mark a point on a graph or coordinate plane at the intersection of specified x and y values.
  • Linear Relationship: A relationship between two variables that, when graphed, produces a straight line, indicating a constant rate of change.
  • Steepness: The degree of incline of a line; a greater slope value indicates a steeper 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 zero, indicating a horizontal line with no vertical change.
  • Undefined Slope: The slope of a vertical line, which is undefined because the run (horizontal change) is zero.

Multimedia Resources

 

 

Warm Up Activities

Choose from one or more activities.

Activity 1: Visualizing Slopes

Review the concept of slope with a quick real-world example, such as the slope of a ramp or a hiking trail. Ask students to share their experiences with slopes in everyday life. 

Show these examples of stairs to compare steepness: https://www.media4math.com/library/75324/asset-preview

Activity 2: Building Staircases Using Unit Cubes

In this activity, students will use small unit cubes to construct staircases with specific slopes. This hands-on approach helps students visualize and physically build the concept of rise and run.

  • Provide students with unit cubes and a set of slope values (e.g., \( \frac{1}{1} \), \( \frac{2}{1} \), \( \frac{3}{1} \), \( \frac{1}{2} \)).
  • Ask students to build staircases where the rise (vertical change) and run (horizontal change) correspond to the given slope values.
  • Encourage students to compare the staircases to see how different slopes affect the steepness.
  • Discuss how the ratio of rise to run determines the angle of incline.

 

Slope

 

Activity 3: Building Ramps Using Straight Pieces of Wood

This activity allows students to model different slopes by setting up ramps using wooden planks. They will see how changes in rise and run affect the steepness of a ramp.

  • Provide students with wooden planks and small objects (e.g., books or blocks) to adjust the height.
  • Assign each group a specific slope value:
    • \( \frac{1}{2} \)
    • \( \frac{1}{1} \)
    • \( \frac{2}{1} \)
    • \( \frac{3}{1} \)
  • Have students adjust the height of one end of the plank to match the required rise while keeping the run fixed.
  • Encourage students to test rolling objects down their ramps and discuss how the slope affects speed and movement.
  • Connect the activity to real-world applications, such as wheelchair ramps and road inclines.

 

Slope

 

 

Teach

Introduction

The concept of slope is fundamental in mathematics, particularly in algebra and geometry. It describes how steep a line is and provides a mathematical way to measure change. In real-world applications, slope is used in construction, engineering, and physics to describe inclines, rates of change, and proportional relationships.

During this lesson, students will explore the meaning of slope by engaging in interactive activities that help them visualize how rise and run define the steepness of a line. They will use graphs, equations, and physical models to develop a deeper understanding of positive, negative, zero, and undefined slopes. The lesson is structured into three key sections:

  • Explore: Students will experiment with visual representations of slope by graphing points, constructing staircases, and modeling ramps.
  • Explain: Students will formalize their understanding of slope by learning the mathematical definition, calculating slopes using coordinates, and applying formulas.
  • Elaborate: Students will extend their knowledge by solving real-world problems involving slope and analyzing different linear relationships.

By the end of the lesson, students should be able to confidently determine the slope of a line using different methods and recognize its significance in various mathematical and real-world contexts.

Explore

Distribute graph paper and pencils to students. Provide them with a set of coordinates and ask them to plot the points on the coordinate plane. Then, have them connect the points to form a line. Encourage students to observe the steepness or flatness of the line they have drawn.

Review the following definitions:

Explain

Demonstrate how to calculate the slope of a line given two points on a graph using the ratio of rise over run. Show this video to see how to do that.

https://www.media4math.com/library/75386/asset-preview

Use this Desmos activity to have students explore slope as the ratio of rise over run. Students click and drag on the two points and then use their understanding of right triangles to find the ratio of the rise over the run.

https://www.media4math.com/library/75385/asset-preview

Elaborate

Use the Desmos activity from the previous section and have students find the slope for the line connecting the following pairs of points:

  • (0, 0) and (4, 4)
  • (1, 1) and (7, 4)
  • (2, 2) and (5, 8)
  • (0, 4) and (4, 0)

Have students click and drag on the points to place them on the proper coordinate. Or, if this is being presented to the class you can do this for them.

To calculate the slope have students:

  • Count the number of vertical spaces from one point to the other: This is the Rise.
  • Count the number of horizontal spaces from one point to the other: This is the Run.

Remind students:

  • The rise and the run are part of a right triangle where the line connecting the points is the hypotenuse. Show additional examples with both positive and negative slopes.
  • Lines with a positive slope point upward in going from left to right.
  • Lines with a negative slope point downward in going from left to right.

Example 1: Calculating a Positive Slope

Given two points \( A(2,3) \) and \( B(5,9) \), determine the slope by counting the rise and run on a coordinate grid.

Solution:

  1. Plot points \( A(2,3) \) and \( B(5,9) \) on a coordinate grid.
  2. Count how many units you move up from \( A \) to \( B \) (rise): **6 units**.
  3. Count how many units you move to the right from \( A \) to \( B \) (run): **3 units**.
  4. The slope is the ratio of rise to run: \[ \frac{6}{3} = 2 \]
  5. The positive slope means the line rises from left to right.

 

Slope

 

Example 2: Calculating a Negative Slope

Given two points \( C(1,8) \) and \( D(5,-4) \), determine the slope by counting the rise and run on a coordinate grid.

Solution:

  1. Plot points \( C(1,8) \) and \( D(5,-4) \) on a coordinate grid.
  2. Count how many units you move down from \( C \) to \( D \) (rise): **12 units down**.
  3. Count how many units you move to the right from \( C \) to \( D \) (run): **4 units**.
  4. The slope is the ratio of rise to run: \[ \frac{-12}{4} = -3 \]
  5. The negative slope means the line falls from left to right.

 

Slope

 

Example 3: Establishing Collinearity with a Positive Slope

Given three points \( A(2,4) \), \( B(4,10) \), and \( C(6,16) \), determine if they are collinear by checking if the rise and run are proportional.

Solution:

  1. Plot points \( A(2,4) \), \( B(4,10) \), and \( C(6,16) \) on a coordinate grid.
  2. Find the rise and run from \( A \) to \( B \): \[ \text{Rise} = 6, \quad \text{Run} = 2 \]
  3. Find the rise and run from \( B \) to \( C \): \[ \text{Rise} = 6, \quad \text{Run} = 2 \]
  4. Since the ratios are the same, the points are collinear.

 

Slope

 

Example 4: Establishing Collinearity with a Negative Slope

Given three points \( D(3,12) \), \( E(6,3) \), and \( F(9,-6) \), determine if they are collinear by checking if the rise and run are proportional.

Solution:

  1. Plot points \( D(3,12) \), \( E(6,3) \), and \( F(9,-6) \) on a coordinate grid.
  2. Find the rise and run from \( D \) to \( E \): \[ \text{Rise} = -9, \quad \text{Run} = 3 \]
  3. Find the rise and run from \( E \) to \( F \): \[ \text{Rise} = -9, \quad \text{Run} = 3 \]
  4. Since the ratios are the same, the points are collinear.

Example 5: Real-World Applications

Scenario: A wheelchair ramp must rise 6 feet for every 18 feet of horizontal distance. What is the slope of the ramp?

Solution:

  1. Identify the rise and run: \[ \text{Rise} = 6, \quad \text{Run} = 18 \]
  2. Find the ratio of rise to run: \[ \frac{6}{18} = \frac{1}{3} \]
  3. The slope of the ramp is \( \frac{1}{3} \), meaning for every 3 feet traveled horizontally, the ramp rises 1 foot.

 

 

Review

Lesson Summary

In this lesson, students explored the concept of slope by examining the relationship between the vertical change (rise) and horizontal change (run) of a line on a coordinate grid. They learned how to determine slope by counting rise and run, understood the difference between positive and negative slopes, and established collinearity by comparing slopes between points. Through real-world applications, students connected slope to everyday contexts such as wheelchair ramps and staircases.

The key takeaways from the lesson include:

  • Slope measures how steep a line is and is calculated as the ratio of rise to run.
  • A positive slope means the line rises from left to right, while a negative slope means it falls from left to right.
  • Collinear points have the same slope between consecutive pairs.
  • Slope appears in real-world settings such as ramps, roofs, and roads.

Review of Key Vocabulary

  • Slope: The measure of the steepness of a line, found by the ratio of rise (vertical change) to run (horizontal change).
  • 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 moves upward from left to right.
  • Negative Slope: A slope where the line moves downward from left to right.
  • Zero Slope: A horizontal line with no vertical change.
  • Undefined Slope: A vertical line where the run is zero, making the slope undefined.
  • Collinear Points: Points that lie on the same straight line, meaning they have the same slope between consecutive pairs.

Additional Examples

Example 1: Finding a Positive Slope

Given two points \( A(1,3) \) and \( B(4,12) \), determine the slope by counting the rise and run on a coordinate grid.

Solution:

  1. Plot points \( A(1,3) \) and \( B(4,12) \) on a coordinate grid.
  2. Count how many units you move up from \( A \) to \( B \) (rise): 9 units.
  3. Count how many units you move to the right from \( A \) to \( B \) (run): 3 units.
  4. The slope is: \[ \frac{9}{3} = 3 \]
  5. The positive slope means the line rises from left to right.

Example 2: Finding a Negative Slope

Given two points \( C(2,10) \) and \( D(7,-5) \), determine the slope by counting the rise and run on a coordinate grid.

Solution:

  1. Plot points \( C(2,10) \) and \( D(7,-5) \) on a coordinate grid.
  2. Count how many units you move down from \( C \) to \( D \) (rise): 15 units down.
  3. Count how many units you move to the right from \( C \) to \( D \) (run): 5 units.
  4. The slope is: \[ \frac{-15}{5} = -3 \]
  5. The negative slope means the line falls from left to right.

Example 3: Real-World Application

Scenario: A roof has a rise of 8 feet and a run of 24 feet. What is the slope of the roof?

Solution:

  1. Identify the rise and run: \[ \text{Rise} = 8, \quad \text{Run} = 24 \]
  2. Find the ratio of rise to run: \[ \frac{8}{24} = \frac{1}{3} \]
  3. The slope of the roof is \( \frac{1}{3} \), meaning for every 3 feet traveled horizontally, the roof rises 1 foot.

 

 

Quiz

Answer the following questions.

  1. What does slope measure in a graph?
  2. How do you determine the slope of a line using a coordinate grid?
  3. Which of the following slopes is positive? 
    (a) \( \frac{-3}{4} \) 
    (b) \( \frac{2}{5} \) 
    (c) \( \frac{-5}{2} \) 
    (d) \( \frac{-1}{6} \)
  4. Which of the following represents a line with a negative slope? 
    (a) A line that moves upward from left to right 
    (b) A line that moves downward from left to right 
    (c) A horizontal line 
    (d) A vertical line
  5. Given two points \( A(3,2) \) and \( B(7,10) \), what is the slope of the line passing through them?
  6. Given two points \( C(2,8) \) and \( D(6,-4) \), what is the slope of the line passing through them?
  7. Three points \( P(1,3) \), \( Q(3,7) \), and \( R(5,11) \) are plotted on a coordinate grid. Are they collinear? Explain why.
  8. A wheelchair ramp has a rise of 4 feet and a run of 12 feet. What is the slope of the ramp?
  9. Which of the following best describes a line with an undefined slope? 
    (a) The line is perfectly horizontal 
    (b) The line is perfectly vertical 
    (c) The line has a positive slope 
    (d) The line has a negative slope
  10. Explain how slope can be used in real-world situations. Provide one example.

Answer Key

  1. Slope measures the steepness of a line, showing the ratio of rise (vertical change) to run (horizontal change).
  2. Slope is determined by counting the rise (how far up or down the line moves) and the run (how far left or right the line moves) between two points.
  3. (b) \( \frac{2}{5} \) is a positive slope because the numerator and denominator are both positive.
  4. (b) A line with a negative slope moves downward from left to right.
  5. Count the rise and run between \( A(3,2) \) and \( B(7,10) \): 
    Rise = \( 10 - 2 = 8 \) 
    Run = \( 7 - 3 = 4 \) 
    Slope = \( \frac{8}{4} = 2 \)
  6. Count the rise and run between \( C(2,8) \) and \( D(6,-4) \): 
    Rise = \( -4 - 8 = -12 \) 
    Run = \( 6 - 2 = 4 \) 
    Slope = \( \frac{-12}{4} = -3 \)
  7. To check for collinearity, find the slopes of \( PQ \) and \( QR \): 
    Rise from \( P(1,3) \) to \( Q(3,7) \) = \( 7 - 3 = 4 \) 
    Run from \( P(1,3) \) to \( Q(3,7) \) = \( 3 - 1 = 2 \) 
    Slope of \( PQ \) = \( \frac{4}{2} = 2 \) 
    Rise from \( Q(3,7) \) to \( R(5,11) \) = \( 11 - 7 = 4 \) 
    Run from \( Q(3,7) \) to \( R(5,11) \) = \( 5 - 3 = 2 \) 
    Slope of \( QR \) = \( \frac{4}{2} = 2 \) 
    Since both slopes are the same, the points are collinear.
  8. Rise = 4 feet 
    Run = 12 feet 
    Slope = \( \frac{4}{12} = \frac{1}{3} \)
  9. (b) A line with an undefined slope is perfectly vertical because there is no horizontal change (run = 0).
  10. Examples of real-world applications of slope: 
    - Slope is used in designing wheelchair ramps, stairs, and roads to ensure safe and accessible inclines. 
    - Engineers use slope in roof construction to determine drainage angles. 
    - In physics, slope represents speed in position-time graphs.