Lesson Plan: Graphing Linear Equations

Lesson Summary

This lesson plan focuses on teaching students how to graph linear equations using the slope-intercept form. Students will learn to plot points on the coordinate plane, identify the slope and y-intercept of linear equations, and interpret their meanings in real-life contexts. The lesson aligns with Common Core Standards 8.EE.B.5 and 8.EE.B.6, emphasizing the graphing of proportional relationships and the use of similar triangles to explain consistent slope.

Lesson Objectives

Plot points on the coordinate plane
Identify the slope and y-intercept of a linear equation
Graph linear equations using the slope-intercept form
Interpret the meaning of slope and y-intercept in real-life situations

Common Core Standards

8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph.
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

Plotting points on the coordinate plane
Understanding of slope and y-intercept concepts

Key Vocabulary

Coordinate Plane: A two-dimensional plane formed by the intersection of a horizontal line (x-axis) and a vertical line (y-axis).
Ordered Pair: A pair of numbers (x, y) that represent the coordinates of a point on the coordinate plane.
Slope: The measure of the steepness of a line, calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.
Y-Intercept: The point where a line crosses the y-axis, representing the value of y when x is zero.
Slope-Intercept Form: A linear equation written in the form $ y = mx + b $, where m represents the slope and b represents the y-intercept.

Multimedia Resources

Math definitions of terms related to linear equations and functions: https://www.media4math.com/Definitions--LinearFunctions
Video definitions on the topic of linear equations and functions: https://www.media4math.com/MathVideoCollection--LinearFunctionsDefinitions

Warm Up Activities

Choose from one or more activities.

Activity 1: Review of the Coordinate Plane

For students that need to review graphing points on the coordinate plane review the first example from this video:

https://www.media4math.com/library/39514/asset-preview

Engage students by asking them to plot the following points on a coordinate plane:

(1, 2)
(2, 4)
(3, 6)

Use this Desmos activity to graph the points and ask students to notice any patterns.

https://www.desmos.com/calculator/cq834lfyfp

Then ask them to find additional coordinates that continue the pattern.

Activity 1: Desmos Activity: Graphing Coordinates

In this activity, students will learn how to use Desmos to plot points and visualize linear equations step by step.

Go to Desmos Graphing Calculator.
Click on the "+" button and select "Table" to create a set of ordered pairs.
Enter the following coordinates in the table:
- (-2, -4)
- (-1, -2)
- (0, 0)
- (1, 2)
- (2, 4)
Observe how the points align to form a straight line.
To confirm the equation of the line, type y = 2x in the equation bar and check if it passes through all plotted points.
Adjust different values of the slope and y-intercept to explore how the graph changes.

This activity helps students connect tables, equations, and graphs in an interactive and engaging way.

Activity 3: Review of Slope: Line Orientations

Before graphing linear equations, students will review the different orientations of lines based on their slopes.

Positive Slope: A line that rises from left to right. Example: $ y = 2x + 1 $.
Negative Slope: A line that falls from left to right. Example: $ y = -3x + 4 $.
Zero Slope: A horizontal line with no vertical change. Example: $ y = 5 $.
No Slope (Undefined): A vertical line that does not have a defined slope. Example: $ x = -2 $.


Positive Slope	Negative Slope	Zero Slope	No Slope

Teach

Definitions

Use the following video definitions to define key terms:

The slope-intercept form of a linear equation: https://www.media4math.com/library/74604/asset-preview
The y-intercept: https://www.media4math.com/library/74608/asset-preview
The slope: https://www.media4math.com/library/74617/asset-preview

Introduction to Graphs of Linear Equations

A linear equation represents a straight line when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form:

\[ y = mx + b \]

where:

$ m $ is the slope, which determines the steepness and direction of the line.
$ b $ is the y-intercept, which is the point where the line crosses the y-axis.

To graph a linear equation, follow these steps:

Identify the slope ($ m $) and y-intercept ($ b $).
Plot the y-intercept ($ b $) on the y-axis.
Use the slope to determine the next points. Move up/down (rise) and left/right (run) according to $ m $.
Draw a straight line through the points.

Examples

Demonstrate how to identify the slope and y-intercept from a given linear equation.

Start by showing this video about the slope-intercept form:
https://www.media4math.com/library/39543/asset-preview
Provide examples of graphing linear equations using the slope-intercept form. Use this slide show, which focuses on given the slope and the y-intercept, graph the linear equation
https://www.media4math.com/library/slideshow/math-examples-slope-intercept-form
Have students use this Desmos activity to explore the slope-intercept form:
https://www.media4math.com/library/40088/asset-preview
Here is the corresponding worksheet for this Desmos activity:
https://www.media4math.com/library/40089/asset-preview

Below are six examples demonstrating different cases of linear equations.

Example 1: Positive Slope, Crosses the Origin

Equation: $ y = 2x $

The slope ($ m $) is $ 2 $, meaning the line rises 2 units for every 1 unit it moves to the right.
The y-intercept ($ b $) is $ 0 $, so the line passes through the origin (0,0).
From (0,0), plot the next point by moving up 2 and right 1.
Continue plotting more points and draw the line through them.

Example 2: Positive Slope, Positive Y-Intercept

Equation: $ y = 3x + 2 $

The slope ($ m $) is $ 3 $, meaning the line rises 3 units for every 1 unit it moves to the right.
The y-intercept ($ b $) is $ 2 $, so the line crosses the y-axis at (0,2).
From (0,2), plot the next point by moving up 3 and right 1.
Connect the points with a straight line.

Example 3: Positive Slope, Negative Y-Intercept

Equation: $ y = \frac{1}{2}x - 4 $

The slope ($ m $) is $ \frac{1}{2} $, meaning the line rises 1 unit for every 2 units it moves to the right.
The y-intercept ($ b $) is $ -4 $, so the line crosses the y-axis at (0,-4).
From (0,-4), plot the next point by moving up 1 and right 2.
Draw the line through the points.

Example 4: Negative Slope, Crosses the Origin

Equation: $ y = -x $

The slope ($ m $) is $ -1 $, meaning the line falls 1 unit for every 1 unit it moves to the right.
The y-intercept ($ b $) is $ 0 $, so the line passes through the origin (0,0).
From (0,0), plot the next point by moving down 1 and right 1.
Connect the points with a straight line.

Example 5: Negative Slope, Positive Y-Intercept

Equation: $ y = -2x + 3 $

The slope ($ m $) is $ -2 $, meaning the line falls 2 units for every 1 unit it moves to the right.
The y-intercept ($ b $) is $ 3 $, so the line crosses the y-axis at (0,3).
From (0,3), plot the next point by moving down 2 and right 1.
Draw the line through the points.

Example 6: Negative Slope, Negative Y-Intercept

Equation: $ y = -\frac{3}{4}x - 5 $

The slope ($ m $) is $ -\frac{3}{4} $, meaning the line falls 3 units for every 4 units it moves to the right.
The y-intercept ($ b $) is $ -5 $, so the line crosses the y-axis at (0,-5).
From (0,-5), plot the next point by moving down 3 and right 4.
Connect the points with a straight line.

These examples help students visualize and understand how changes in the slope and y-intercept affect the graph of a linear equation.

Review

Summary of the Lesson

In this lesson, students learned how to graph linear equations using the slope-intercept form:

\[ y = mx + b \]

where $ m $ represents the slope and $ b $ represents the y-intercept. Students practiced identifying the slope and y-intercept, plotting points on a coordinate plane, and drawing straight lines to represent the equations. The lesson also covered different types of slopes: positive, negative, zero, and undefined.

Key Vocabulary

Coordinate Plane: A two-dimensional plane formed by the intersection of a horizontal (x-axis) and vertical (y-axis) line.
Ordered Pair: A pair of values $(x, y)$ that represent a point on the coordinate plane.
Slope: The ratio of vertical change (rise) to horizontal change (run) between two points on a line.
Y-Intercept: The point where the line crosses the y-axis ($ x = 0 $).
Slope-Intercept Form: The equation of a line written as $ y = mx + b $.

Multimedia Resources

Try this drag-and-drop game, which focuses on linear equations:
https://www.media4math.com/library/4829/asset-preview
Encourage students to graph linear equations using the slope-intercept form.
Address any misconceptions or questions that arise during the review.

Additional Graphing Examples

Example 1: Positive Slope

Equation: $ y = \frac{2}{3}x + 1 $

The slope ($ m $) is $ \frac{2}{3} $, meaning the line rises 2 units for every 3 units it moves to the right.
The y-intercept ($ b $) is $ 1 $, so the line crosses the y-axis at (0,1).
From (0,1), plot the next point by moving up 2 and right 3.
Draw a straight line through the points.

Example 2: Negative Slope

Equation: $ y = -\frac{4}{5}x + 2 $

The slope ($ m $) is $ -\frac{4}{5} $, meaning the line falls 4 units for every 5 units it moves to the right.
The y-intercept ($ b $) is $ 2 $, so the line crosses the y-axis at (0,2).
From (0,2), plot the next point by moving down 4 and right 5.
Draw a straight line through the points.

Example 3: Zero Slope (Horizontal Line)

Equation: $ y = -3 $

The slope ($ m $) is $ 0 $, meaning the line does not rise or fall.
The y-intercept ($ b $) is $ -3 $, so the line crosses the y-axis at (0,-3).
Since the slope is zero, plot a horizontal line through $ y = -3 $.

Example 4: Undefined Slope (Vertical Line)

Equation: $ x = 4 $

This equation does not follow the slope-intercept form because it represents a vertical line.
The line passes through $ x = 4 $, meaning all points on the line have the same x-coordinate.
Plot a vertical line at $ x = 4 $, extending infinitely up and down.

Quiz

Answer the following questions.

Plot the points (1, 2), (-2, -1), and (3, 4) on the coordinate plane.
Identify the slope and y-intercept of the equation y = 2x + 3.
Graph the equation y = -1/2x + 4 on the coordinate plane.
If the slope of a linear equation is 3 and the y-intercept is -2, what is the equation?
Interpret the meaning of the slope and y-intercept in the equation y = 0.5x + 10. You save fifty cents a day in a piggy bank that already has an amount of money in it.
A line passes through the origin and through (4, 9). What is its equation?
Determine if the point (3, -1) lies on the line represented by the equation y = 2x - 5.
Graph the equation 3y = 6x - 9 on the coordinate plane.
Explain the relationship between the slope of a line and its steepness.
This equation represents a car slowing down every second at a constant speed (in miles per hour): y = -5x +50. What is the car's initial speed? What does the slope represent?

Answer Key

Slope = 2, y-intercept = 3
y = 3x - 2
The slope represents the 50 cents saved a day. The y-intercept is the amount of money initially in the piggy bank (\$10).
y = 9/4x
No, the point (3, -1) does not lie on the line.
The steeper the line, the greater the slope value (positive or negative).
Initial speed is 50 mph. The car slows down by 5 mph every second.