Linear Functions in Real Life: From Stories to Graphs

Materials Needed

• Paper or graph paper
• Pencil, ruler, and colored pens or highlighters
• Calculator or spreadsheet app
• Sticky notes or index cards
• Optional: Desmos or another graphing tool
• Optional real-world data, such as phone plans, rideshare fares, freelance rates, or utility costs

Learning Objectives

By the end of the lesson, the learner will be able to:

• Identify the slope and y-intercept of a linear function.
• Explain what slope and y-intercept mean in a real-world situation.
• Write a linear equation in the form y = mx + b from a table, graph, or written scenario.
• Graph a linear function accurately using a table or the slope-intercept form.
• Use a linear model to make predictions and explain whether the model makes sense.

Success Criteria

Successful work will include:

• Correctly identifying m as the slope and b as the y-intercept.
• Labeling axes, choosing a reasonable scale, and plotting points accurately.
• Explaining slope using units, such as “$25 per hour” or “3 fewer customers per day.”
• Writing an equation that matches the given graph, table, or situation.
• Using the equation to answer a practical question and justifying the answer.

Lesson Overview

Suggested time: 75–90 minutes

Big idea: A linear function describes a constant rate of change. Its graph helps us see how one quantity changes in relation to another.

Introduction: Hook and Purpose

Real-Life Hook: Which Option Is the Better Deal?

Imagine choosing between two freelance payment plans:

• Plan A: $100 upfront plus $30 per hour
• Plan B: No upfront payment, but $45 per hour

Ask:

• Which plan pays more for 2 hours of work?
• Which plan pays more for 10 hours?
• Is there a point where the two plans pay the same amount?

Invite the learner to make an estimate before calculating. Explain that linear functions give us a practical way to answer these questions and compare changing quantities.

Share the Learning Target

Say: “Today, you will learn how to translate real-life situations into linear equations and graphs. You will use slope to describe a rate of change, the y-intercept to describe a starting amount, and graphs to make predictions.”

Body: Instruction and Practice

Part 1: Explore the Structure of a Linear Function

I Do: Introduce y = mx + b

Explain that the standard slope-intercept form of a linear equation is:

y = mx + b

• y is the output or dependent variable.
• x is the input or independent variable.
• m is the slope, or constant rate of change.
• b is the y-intercept, or starting value when x = 0.

Use this example:

y = 30x + 100

Explain:

• The starting amount is $100, so b = 100.
• The amount increases by $30 for every hour, so m = 30.
• x represents hours worked.
• y represents total earnings.

Connect the equation to a table:

Hours, x	Earnings, y = 30x + 100
0	$100
1	$130
2	$160
3	$190

Emphasize that the output increases by the same amount each time. That constant change is what makes the relationship linear.

Quick Check

Ask the learner:

1. What does the 30 mean in this situation?
2. What does the 100 mean?
3. What would the model predict for 5 hours?

We Do: Interpret Different Equations

Work through the following equations together. For each one, identify the slope, y-intercept, and a possible real-world meaning.

1. y = 12x + 8
2. y = -4x + 50
3. y = 0.5x + 20

Possible interpretations:

• y = 12x + 8: A service begins with an $8 fee and adds $12 per unit.
• y = -4x + 50: A tank begins with 50 gallons and loses 4 gallons per minute.
• y = 0.5x + 20: An account begins with $20 and grows by $0.50 per day.

Think-Pair-Share alternative: If working with others, have learners explain one equation to a partner. If working independently, record a one-minute verbal explanation or write it in a complete sentence.

Part 2: Find Slope from a Table and Graph

I Do: Calculate Rate of Change

Use the slope formula:

m = (change in y) ÷ (change in x)

For two points, this is written as:

m = (y₂ − y₁) ÷ (x₂ − x₁)

Model with the points (2, 160) and (5, 250):

m = (250 − 160) ÷ (5 − 2) = 90 ÷ 3 = 30

The slope is 30, meaning the earnings increase by $30 per hour.

We Do: Analyze a Table

Examine this table:

Days, x	Account Balance, y
0	240
2	220
4	200
6	180

Work together to determine:

1. Whether the relationship is linear.
2. The slope.
3. The y-intercept.
4. The equation.
5. What the model predicts after 10 days.

Expected equation: y = -10x + 240

Discuss why the slope is negative and what the y-intercept means in context.

Formative Assessment: Error Analysis

Present this incorrect reasoning:

“The account drops from $240 to $220, so the slope is −20.”

Ask the learner to explain the error. The change is −$20 over 2 days, so the rate is:

−20 ÷ 2 = −10 dollars per day

Part 3: Graph a Linear Function

I Do: Graph Using the Intercept and Slope

Model the equation:

y = 2x + 3

1. Start with the y-intercept: plot (0, 3).
2. Read the slope as rise over run: 2/1.
3. From (0, 3), move up 2 and right 1 to reach (1, 5).
4. Repeat to find another point, such as (2, 7).
5. Draw a straight line through the points.
6. Label both axes and include units when appropriate.

Explain that a graph is not just a picture. It shows the starting value, the direction of change, and the rate of change.

We Do: Graph and Interpret

Graph:

y = -3x + 12

Work together to identify:

• The y-intercept: (0, 12)
• The slope: −3 = −3/1
• Two additional points, such as (1, 9) and (2, 6)
• What the line means if x represents time and y represents the amount remaining

Ask: “What does the line eventually predict? Is that prediction realistic for every value of x?” Guide the learner to consider the domain. For example, an amount remaining cannot usually be negative, so the model may only make sense until the quantity reaches zero.

You Do: Choose a Graphing Method

Choose one method:

• Graph on paper using the y-intercept and slope.
• Create a table of values and plot the points.
• Use a graphing calculator or Desmos, then annotate the graph by identifying the slope and y-intercept.

Graph two of the following:

1. y = 4x − 1
2. y = −0.5x + 6
3. y = 1.25x + 2

Part 4: Real-World Application Challenge

You Do: Build and Compare Two Linear Models

Return to the opening payment plans:

• Plan A: A(x) = 30x + 100
• Plan B: B(x) = 45x

Here, x represents the number of hours worked and the output represents total earnings.

Complete the following:

1. Create a table showing earnings for 0, 2, 4, 6, 8, and 10 hours.
2. Graph both plans on the same coordinate plane.
3. Identify the slope and y-intercept of each plan.
4. Determine the break-even point by solving:
   30x + 100 = 45x
5. Decide which plan is better for 2 hours, 10 hours, and 20 hours.
6. Write a recommendation for someone choosing between the plans.

Expected break-even calculation:

30x + 100 = 45x
100 = 15x
x = 6⅔ hours

The plans pay the same after approximately 6 hours and 40 minutes. Plan A is better below that point, while Plan B is better above it.

Creative Choice

Choose one of the following options:

• Model the cost of a subscription with a monthly fee and a per-use charge.
• Model earnings from a side business with a fixed startup cost or fee.
• Model travel distance at a constant speed, including a starting location.
• Model savings over time with a starting balance and regular deposits.
• Invent two competing plans and determine when one becomes better than the other.

Your model must include:

• A written scenario
• A table with at least five values
• A linear equation
• A labeled graph
• One prediction
• A short explanation of whether the model is realistic and what its limitations are

Formative Assessment Throughout the Lesson

• Vocabulary check: Explain slope and y-intercept in your own words.
• Equation check: Identify m and b in a given equation.
• Table check: Determine whether the rate of change is constant.
• Graph check: Point to the y-intercept and describe the direction of the line.
• Reasonableness check: Decide whether a prediction makes sense in the real-world context.
• Feedback pause: After each completed graph or equation, compare it with the success criteria and correct one detail if necessary.

Summative Assessment

Linear Model Demonstration

Submit or present the completed real-world application challenge. The learner should demonstrate the ability to:

Criteria	Evidence of Success
Equation	The equation correctly represents the situation.
Slope	The slope is correctly calculated and explained using units.
Y-intercept	The starting value is correctly identified and interpreted.
Graph	The graph has labeled axes, an appropriate scale, accurate points, and a clear line.
Prediction	The learner uses the equation or graph to make a correct prediction.
Communication	The learner explains the model clearly and identifies at least one limitation.

Differentiation and Adaptations

Additional Support

• Provide a reference card showing y = mx + b, the slope formula, and vocabulary definitions.
• Use whole-number slopes before introducing fractions and decimals.
• Give partially completed tables or graphs.
• Allow the learner to use a graphing tool to verify calculations.
• Use sentence frames such as: “The slope is ___, which means ___ for every ___.”
• Begin with concrete contexts involving money, distance, or time before moving to abstract equations.

Extension

• Compare two linear models and calculate their intersection algebraically and graphically.
• Investigate horizontal and vertical lines and explain why a vertical line is not a function of x.
• Use a data set with slight variation and decide whether a linear model is appropriate.
• Explore the difference between interpolation and extrapolation.
• Write a short critique explaining why a linear model may become unrealistic outside a particular domain.

Flexible Delivery Options

• Homeschool: Complete the investigation through discussion, written work, and a personal finance example.
• Classroom: Use pairs or small groups for the plan comparison and gallery walk.
• Training or adult learning: Replace the example with pricing, productivity, staffing, or budgeting data from the participants’ work contexts.
• Digital format: Use a shared spreadsheet or graphing tool and explain the model in a short audio or video recording.

Conclusion: Closure and Recap

Three-Minute Recap

Ask the learner to complete these statements:

• “The slope tells me…”
• “The y-intercept tells me…”
• “A linear relationship has…”
• “I can graph y = mx + b by…”
• “One real-world situation where a linear model is useful is…”

Exit Ticket

Given the function:

C(x) = 18x + 35

1. Identify the slope and y-intercept.
2. Describe what each represents in a realistic situation.
3. Calculate C(7).
4. Explain what the graph would look like.

Expected response:

• Slope: 18
• Y-intercept: 35
• C(7) = 18(7) + 35 = 161
• The graph is a straight line crossing the y-axis at 35 and rising 18 units for every 1-unit increase in x.

Final Reflection

Write one sentence answering:

“How can a graph help someone make a better real-world decision?”