Lesson Plan: Introduction to Linear Equations

Lesson Summary

This lesson introduces middle school students to linear equations, focusing on their structure and graphical representation. Students will explore the concept of linear relationships, learn how to recognize and construct linear equations, and understand their significance in real-world contexts. Through guided examples and interactive activities, students will develop a foundational understanding of slope, y-intercept, and graphing on the coordinate plane. The lesson also sets the stage for deeper exploration of slope-intercept form, graphing techniques, and real-world applications in future lessons.

Lesson Objectives

Define linear equations and their components (variables, coefficients, constants)
Identify linear equations from given equations or word problems
Represent linear equations using tables, graphs, and algebraic expressions

Common Core Standards

8.EE.B.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph.
8.EE.C.7: Solve linear equations in one variable.

Prerequisite Skills

Understanding of variables and algebraic expressions
Familiarity with the Cartesian coordinate plane
Basic arithmetic operations (addition, subtraction, multiplication, division)

Key Vocabulary

Linear Equation – An equation that represents a straight line when graphed. It has the general form $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept.
Variable – A symbol (usually a letter) that represents an unknown value in an equation.
Slope ($m$) – A measure of the steepness of a line, calculated as the ratio of the change in $ y $ to the change in $ x $: \[ \frac{\Delta y}{\Delta x} \]
Y-Intercept ($b$) – The point where a line crosses the y-axis ($ x = 0 $).
X-Intercept – The point where a line crosses the x-axis ($ y = 0 $).
Coordinate Plane – A two-dimensional plane formed by the intersection of the x-axis and y-axis.
Ordered Pair – A pair of numbers $ (x, y) $ that represents a point on the coordinate plane.
Graph of an Equation – A visual representation of all the solutions to an equation plotted on the coordinate plane.
Function – A relationship between input (x-values) and output (y-values) where each input has exactly one output.
Proportional Relationship – A relationship where two quantities increase or decrease at the same rate, resulting in a straight line that passes through the origin: \[ y = mx \]

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 of these activities.

Activity 1: Applications of Linear Equations

Introduce the concept of linear equations by showing real-life examples. Use this slide show:

https://www.media4math.com/library/slideshow/applications-linear-equations

These slide shows provide a deeper dive into these applications of linear equations (click on the slide show links):

Cricket chirps vs. Temperature slide show: https://www.media4math.com/library/slideshow/application-linear-functions-cricket-chirps
The cost vs. time for renting equipment slide show: https://www.media4math.com/library/slideshow/application-linear-functions-cost-vs-time
Distance vs. time: https://www.media4math.com/library/slideshow/application-linear-functions-distance-vs-time

Activity 2: Linear vs. Non-Linear Equations

Use this slide show to compare and contrast linear and non-linear graphs:

https://www.media4math.com/library/slideshow/linear-vs-non-linear-graphs

Teach

Definition and Components of Linear Equations

Define linear equations as equations that form a straight line when graphed. Use this slide show that provides video definitions of linear equations and linear functions:

https://www.media4math.com/library/slideshow/linear-equations-and-functions-definitions

Explain the components: variables, coefficients, and constants.

Here are some additional definitions to review:

Slope: https://www.media4math.com/library/74617/asset-preview
y-intercept: https://www.media4math.com/library/74608/asset-preview
Coefficient: https://www.media4math.com/library/74631/asset-preview

Identifying Linear Equations

Introduce linear equations in standard form and show how this form relates to the slope-intercept form. Use this slide show:

https://www.media4math.com/library/slideshow/linear-equations-standard-and-slope-intercept-form

Representing Linear Equations

Introduce the three ways to represent linear equations: tables, graphs, and algebraic expressions. Use this slide show of examples of multiple representations of linear equations:

https://www.media4math.com/library/slideshow/multiple-representations-linear-equations

Demonstrate how to create a table of values and plot points on the coordinate plane. Use this Desmos activity to explore the three representations of linear equations:

https://www.desmos.com/calculator/tfoy9bwov2

Explain the concept of slope and y-intercept, and their relationship to the equation's form. Use this Desmos activity to explore slope-intercept form:

https://www.media4math.com/library/40088/asset-preview

Example 1: Data Table and Graph

Let's consider a real-world example: a car rental company charges a daily rental fee.

The cost of renting a car is $40 per day. The relationship between the number of rental days ($ x $) and the total cost ($ y $) is linear.

Data Table

Days Rented ($ x $)	Total Cost (\$) ($ y $)
1	40
2	80
3	120
4	160
5	200

The equation that represents this data is:

$ y = 40x $

Graphing this equation, we get a straight line passing through the origin.

Example 2: Simple Linear Equation of the Form $ y = kx $

Consider the equation:

$ y = 3x $

We calculate a few points:

When $ x = 0 $, $ y = 3(0) = 0 $
When $ x = 1 $, $ y = 3(1) = 3 $
When $ x = 2 $, $ y = 3(2) = 6 $
When $ x = 3 $, $ y = 3(3) = 9 $
When $ x = 4 $, $ y = 3(4) = 12 $

Graphing these points gives a straight line. The slope is 3, which is the coefficient of $ x $, meaning the line rises 3 units for every 1 unit it moves to the right.

Example 3: Modeling a Linear Equation of the Form $ Ax + By = C $

Suppose a car has a fuel tank capacity of 12 gallons, and the driver wants to track the fuel remaining after driving a certain distance. The car’s fuel efficiency is 30 miles per gallon.

The relationship between the number of miles driven ($ x $) and the gallons of fuel remaining ($ y $) can be modeled by:

\[ x + 30y = 360 \]

where:

$ x $ is the number of miles driven
$ y $ is the gallons of fuel remaining
The car starts with 12 gallons, so at $ x = 0 $, $ y = 12 $

Solving for $ y $:

\[ y = \frac{360 - x}{30} \]

Calculating a Few Points:

When $ x = 0 $, \[ y = \frac{360 - 0}{30} = 12 \] (Full tank)
When $ x = 90 $, \[ y = \frac{360 - 90}{30} = 9 \] (9 gallons left)
When $ x = 180 $, \[ y = \frac{360 - 180}{30} = 6 \] (6 gallons left)
When $ x = 360 $, \[ y = \frac{360 - 360}{30} = 0 \] (Empty tank)

Graphing this equation shows a straight line that decreases as more miles are driven, illustrating how fuel is consumed at a constant rate.

Example 4: Modeling a Linear Equation of the Form $ y = mx $

Suppose a worker earns $15 per hour.

The equation for total earnings ($ y $) based on hours worked ($ x $) is:

$ y = 15x $

Calculating a few points:

When $ x = 0 $, $ y = 15(0) = 0 $
When $ x = 4 $, $ y = 15(4) = 60 $
When $ x = 8 $, $ y = 15(8) = 120 $

Graphing this equation, we get a line passing through the origin with a slope of 15, indicating an increase of $15 for each additional hour worked.

Example 5: Modeling a Linear Equation of the Form $ y = mx + b $

Consider a cellphone plan with a monthly fee of $20 plus $5 per GB of data used.

The equation is:

\[ y = 5x + 20 \]

Calculating a few points:

When $ x = 0 $, \[ y = 5(0) + 20 = 20 \]
When $ x = 2 $, \[ y = 5(2) + 20 = 30 \]
When $ x = 4 $, \[ y = 5(4) + 20 = 40 \]

Graphing these points results in a straight line that starts at $ y = 20 $ and has a slope of 5, representing an increase of $5 for each additional GB used.

Review

Use this slide show to review linear equations and functions, along with an application of slope:

https://www.media4math.com/library/slideshow/linear-equation-review

You can also assign this worksheet, which reviews multiple representations of linear equations and functions:

https://www.media4math.com/library/34317/asset-preview

Review of Key Vocabulary

Linear Equation – An equation that represents a straight line when graphed. It has the general form $ y = mx + b $.
Variable – A symbol (usually a letter) that represents an unknown value in an equation.
Slope ($m$) – The rate of change of a line, calculated as: \[ \frac{\Delta y}{\Delta x} \]
Y-Intercept ($b$) – The point where a line crosses the y-axis ($ x = 0 $).
X-Intercept – The point where a line crosses the x-axis ($ y = 0 $).
Coordinate Plane – A two-dimensional plane formed by the x-axis and y-axis.
Ordered Pair – A pair of numbers $ (x, y) $ representing a point on the coordinate plane.
Function – A relationship where each input ($ x $) has exactly one output ($ y $).
Proportional Relationship – A linear relationship where the line passes through the origin ($ y = mx $).

Additional Review Examples

Example 1: Temperature Conversion (Linear Relationship)

The relationship between Fahrenheit ($ F $) and Celsius ($ C $) can be modeled by:

\[ F = \frac{9}{5} C + 32 \]

Calculating a few points:

When $ C = 0 $, \[ F = \frac{9}{5}(0) + 32 = 32 \]
When $ C = 10 $, \[ F = \frac{9}{5}(10) + 32 = 50 \]
When $ C = 20 $, \[ F = \frac{9}{5}(20) + 32 = 68 \]

Graphing this equation results in a straight line that models the conversion between Fahrenheit and Celsius.

Example 2: Taxi Fare Calculation

A taxi company charges a base fare of $3 plus $2 per mile driven. The total fare ($ y $) based on miles ($ x $) can be modeled as:

\[ y = 2x + 3 \]

Calculating a few points:

When $ x = 0 $, \[ y = 2(0) + 3 = 3 \] (Base fare)
When $ x = 2 $, \[ y = 2(2) + 3 = 7 \]
When $ x = 5 $, \[ y = 2(5) + 3 = 13 \]

This equation represents a real-world scenario where fares increase at a constant rate per mile.

Example 3: Savings Growth Over Time

Maria starts with $50 in her savings account and deposits $10 each week. The amount saved ($ y $) after $ x $ weeks is given by:

\[ y = 10x + 50 \]

Calculating a few points:

When $ x = 0 $, \[ y = 10(0) + 50 = 50 \] (Initial savings)
When $ x = 3 $, \[ y = 10(3) + 50 = 80 \]
When $ x = 6 $, \[ y = 10(6) + 50 = 110 \]

Graphing this equation shows how Maria's savings grow linearly over time.

Quiz

Answer the following questions.

Which of the following is a linear equation?
a) y = x² + 3
b) 2x + 5y = 10
c) x³ - y = 0
d) 3x - 2y + 4 = 0
Identify the coefficient of x in the equation: 4x + 2y = 8.
What is the y-intercept of the equation y = 2x + 3?
What is the general form of a linear equation in slope-intercept form?
What is the general form of a linear equation in standard form?
Determine if the equation 5x + 2y - 3 = 0 is linear or non-linear.
Which equation represents this situation: The cost of renting a car is \$25 plus $0.20 per mile.
a) y = 0.2x + 25
b) y = 25x + 0.2
Identify the variables, coefficients, and constant in the equation: y = -2x + 5.
Find the slope of the line represented by the equation y = $ \frac{1}{2} $x + 3.
Determine if the equation x² + y² = 25 is a linear equation or not.

Answer Key

b and d
4
3
y = mx + b
Ax + By = C
Linear
a
Variables: x, y; coefficients: 1, -2; constant: 5
$ \frac{1}{2} $
Not a linear equation

Lesson Plan: Introduction to Linear Equations

Lesson Summary

Lesson Objectives

Common Core Standards

Prerequisite Skills

Key Vocabulary

Multimedia Resources

Warm Up Activities

Activity 1: Applications of Linear Equations

Activity 2: Linear vs. Non-Linear Equations

Teach

Definition and Components of Linear Equations

Identifying Linear Equations

Representing Linear Equations

Example 1: Data Table and Graph

Data Table

Example 2: Simple Linear Equation of the Form \( y = kx \)

Example 3: Modeling a Linear Equation of the Form \( Ax + By = C \)

Calculating a Few Points:

Example 4: Modeling a Linear Equation of the Form \( y = mx \)

Example 5: Modeling a Linear Equation of the Form \( y = mx + b \)

Review

Review of Key Vocabulary

Additional Review Examples

Example 1: Temperature Conversion (Linear Relationship)

Example 2: Taxi Fare Calculation

Example 3: Savings Growth Over Time

Quiz

Answer Key