Mission: Design a Roller-Coaster Ride with Linear Functions
Materials Needed
- Graph paper or a graphing tool such as Desmos
- Pencil, colored pencils, and ruler
- Calculator, optional
- Sticky notes or index cards
- Two different-colored markers
- Practice worksheet or notebook
- Optional: coins, toy cars, or small objects for a ramp investigation
Learning Objectives
By the end of the lesson, the learner will be able to:
- Explain what the slope and y-intercept mean in a real-world situation.
- Identify linear functions from tables, graphs, equations, and stories.
- Write a linear equation in the form y = mx + b.
- Graph a linear function using its slope and y-intercept.
- Use a linear model to make and explain predictions.
Success Criteria
You are successful if you can:
- Correctly identify m as the slope and b as the y-intercept.
- Explain slope as the rate of change, or how much y changes when x changes.
- Plot the y-intercept and use rise over run to graph a line.
- Write an equation that matches a table, graph, or real-life situation.
- Explain what your answer means using complete sentences and units.
Lesson Overview
Suggested time: 60–75 minutes
Big question: How can a straight line help us describe and predict what happens in the real world?
Introduction: Hook and Preview
Hook: The Streaming Subscription Challenge
Imagine that a music app charges a one-time sign-up fee of $5 and then $3 each month.
Ask:
- How much would the app cost after 1 month?
- How much would it cost after 4 months?
- Could you predict the cost after 10 months?
- Would a graph help you see the pattern?
Have the learner make a quick table:
| Months, x | Total Cost, y |
|---|---|
| 0 | $5 |
| 1 | $8 |
| 2 | $11 |
| 4 | $17 |
Explain that this is a linear relationship because the cost increases by the same amount each month.
Vocabulary
- Linear function
- A rule that creates a straight-line graph and has a constant rate of change.
- Rate of change
- How much one quantity changes compared with another quantity.
- Slope
- The rate of change of a line. It is often written as m.
- Y-intercept
- The point where a line crosses the y-axis. It is often written as b.
- Equation
- A mathematical rule that connects two quantities.
Body: I Do, We Do, You Do
Part 1: I Do — Understanding y = mx + b
Explain that many linear functions can be written in this form:
y = mx + b
- m is the slope, or rate of change.
- b is the y-intercept, or starting value.
- x is the input.
- y is the output.
Use the subscription example:
y = 3x + 5
Explain:
- The slope is 3: the cost increases by $3 each month.
- The y-intercept is 5: the starting fee is $5 when the number of months is 0.
- If x = 4, then y = 3(4) + 5 = 17.
Quick Check
Ask the learner:
- What does the 3 mean?
- What does the 5 mean?
- What would the cost be after 8 months?
Expected answer: y = 3(8) + 5 = 29, so the cost would be $29.
Part 2: I Do — Finding Slope
Introduce the slope formula:
m = rise ÷ run = change in y ÷ change in x
Use the points (1, 8) and (4, 17) from the subscription table:
- Change in y: 17 − 8 = 9
- Change in x: 4 − 1 = 3
- Slope: 9 ÷ 3 = 3
The slope is 3, which means the cost increases by $3 for every additional month.
Show the learner that slope can be:
- Positive: the line rises from left to right.
- Negative: the line falls from left to right.
- Zero: the line is horizontal.
Part 3: We Do — Graphing a Line Together
Graph the equation:
y = 2x + 1
Work through these steps together:
- Identify the y-intercept: b = 1. Plot the point (0, 1).
- Identify the slope: m = 2 = 2/1.
- From (0, 1), rise 2 units and run 1 unit to plot (1, 3).
- Repeat to plot another point, such as (2, 5).
- Use a ruler to draw a straight line through the points.
Think-Pair-Share
If working with a group, have learners discuss with a partner. For homeschool use, the learner can explain the answer aloud to an adult or record a short explanation.
Discuss:
- What would happen to the graph if the y-intercept changed from 1 to 4?
- What would happen if the slope changed from 2 to 1/2?
- How does the graph show that the function is increasing?
Part 4: We Do — Match Representations
Match each description to its equation. Explain how you know.
- A taxi charges a $4 starting fee and $2 per mile.
- A plant is 6 centimeters tall and grows 3 centimeters each week.
- A video game player starts with 20 points and loses 5 points each round.
Possible equations:
- y = 2x + 4
- y = 3x + 6
- y = −5x + 20
Ask the learner to underline the starting value and circle the rate of change in each equation.
Formative Check
Ask the learner to create a sentence for one equation. For example:
“The taxi starts at $4 and costs $2 for every mile traveled.”
Part 5: You Do — Linear Function Detective
Complete the following tasks independently. Show your work and explain your reasoning.
Task A: Complete a Table
Use the equation y = 4x − 2.
| x | y |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 |
Task B: Graph the Function
Graph y = 4x − 2 on graph paper or in a graphing tool.
- Plot the y-intercept.
- Use the slope to find at least two more points.
- Draw the line.
- Label the axes and include units if appropriate.
Task C: Interpret the Graph
Answer these questions:
- What is the slope?
- What is the y-intercept?
- Is the line increasing or decreasing?
- What is the value of y when x = 5?
- What does the y-intercept mean if x represents hours and y represents money earned?
Real-World Choice Project: Build Your Own Linear Story
Choose one scenario or create your own:
- Saving money each week
- Tracking distance during a bike ride
- Calculating the cost of a snack stand
- Measuring plant growth
- Tracking points in a game
- Planning the cost of a trip
Create the following:
- A short story describing the situation.
- A table with at least five input-output pairs.
- An equation in the form y = mx + b.
- A labeled graph.
- One prediction using your equation.
- A two- or three-sentence explanation of what the slope and y-intercept mean.
Project Example
“I save $7 each week and already have $10. My equation is y = 7x + 10. The slope, 7, means I save $7 per week. The y-intercept, 10, means I started with $10. After 6 weeks, I will have y = 7(6) + 10 = $52.”
Assessment
Formative Assessment During the Lesson
- Ask the learner to identify the slope and y-intercept in several equations.
- Have the learner explain what a point on a graph means.
- Check whether the learner uses rise over run correctly.
- Use the think-pair-share questions to listen for accurate explanations.
- Ask the learner to predict what happens when the slope or y-intercept changes.
Summative Assessment: Exit Challenge
A school club sells T-shirts. Each shirt costs $8, and there is a one-time design fee of $12.
- Write an equation for the total cost, y, of buying x shirts.
- Identify the slope and y-intercept.
- Find the cost of 6 shirts.
- Graph the equation.
- Explain what the slope and y-intercept mean in this situation.
Answer Key
- Equation: y = 8x + 12
- Slope: 8, or $8 per shirt
- Y-intercept: 12, or the $12 design fee
- Cost of 6 shirts: y = 8(6) + 12 = 60, so the cost is $60
Summative Rubric
| Skill | Excellent | Developing | Needs Practice |
|---|---|---|---|
| Equation | Correct equation with clear variables | Minor error but shows the correct structure | Equation is missing or does not match the situation |
| Graph | Accurate, labeled line with correct scale | Mostly accurate with one labeling or plotting error | Points or line do not match the equation |
| Interpretation | Clearly explains slope and y-intercept in context | Explains one correctly or needs more detail | Meanings are confused or missing |
| Prediction | Uses the equation correctly and includes units | Method is mostly correct but has a calculation error | Prediction is missing or unsupported |
Differentiation and Adaptations
Support for Learners Who Need More Help
- Use color coding: one color for the slope and another for the y-intercept.
- Provide the sentence frame: “The slope means ___ for every ___.”
- Begin with positive whole-number slopes before using fractions or negative slopes.
- Allow the learner to use a table before writing the equation.
- Use a physical ramp or toy car to demonstrate “rise” and “run.”
- Provide graph paper with the axes already labeled.
Extensions for Advanced Learners
- Graph and compare two linear functions. Decide where they intersect.
- Investigate lines with fractional or negative slopes.
- Write an equation from two points without being given the y-intercept.
- Explain why a relationship may look linear for a while but not continue forever.
- Use a graphing tool to create a design, logo, or picture made from several line segments.
Flexible Learning Formats
- Hands-on: Use a ramp, measuring tape, and toy car to record distance over time.
- Digital: Use a graphing calculator or Desmos to change m and b and observe the graph.
- Verbal: Explain an equation as a story to an adult, partner, or recording device.
- Written: Complete the table, equation, graph, and project in a notebook.
Conclusion: Closure and Recap
Return to the big question: How can a straight line help us describe and predict what happens in the real world?
Have the learner complete these statements:
- A linear function is...
- The slope tells me...
- The y-intercept tells me...
- To graph y = mx + b, I first...
- One real-world situation that can be modeled by a linear function is...
Final Reflection
Ask the learner to rate their confidence from 1 to 5 and answer:
- What part of linear functions was easiest?
- What part needs more practice?
- Where might you use a linear function outside of math class?
Key takeaway: A linear function describes a constant rate of change. In y = mx + b, m tells how quickly the output changes, while b tells the starting value.