Piecewise Functions in the Real World: “Which Price Plan Wins?”

Materials Needed

• Paper or whiteboard
• Pencil
• Calculator, optional
• Ruler, optional
• Colored pencils or highlighters, optional

Learning Objectives

By the end of this 20-minute lesson, you will be able to:

• Explain why a real-world situation can be modeled with a piecewise function.
• Write a piecewise function from a verbal description.
• Evaluate a piecewise function for a given input.
• Use a piecewise function to solve a real-world problem and explain what the answer means.

Success Criteria

You are successful when you can:

• Identify the different rules and the input conditions for each rule.
• Use the correct rule for a given input.
• Show your work and label your answer with appropriate units.
• Explain how the function represents the real-world situation.

Introduction: Hook and Objective — 2 Minutes

Hook: Imagine you are ordering a ride-share. The company charges a $4 starting fee, plus $2 per mile for the first 5 miles. After 5 miles, the price changes to $1.50 per mile for every mile beyond 5. How could one function describe both pricing rules?

Explain:

A piecewise function uses different rules for different parts, or “pieces,” of a situation. Today, you will use piecewise functions to solve real-world problems involving changing prices.

Quick prediction: Which ride would cost more: a 3-mile ride or an 8-mile ride? Why might the company use different pricing rules?

Body: I Do — Teacher/Parent Modeling — 5 Minutes

Example: Ride-Share Pricing

The ride-share company charges:

• $4 starting fee
• $2 per mile for the first 5 miles
• $1.50 for each mile beyond 5 miles

Let x represent the number of miles and C(x) represent the total cost.

Step 1: Identify the input conditions

There are two situations:

• For rides of 5 miles or less: 0 ≤ x ≤ 5
• For rides longer than 5 miles: x > 5

Step 2: Write the rule for each situation

For the first 5 miles:

C(x) = 4 + 2x

For a ride longer than 5 miles, the first 5 miles cost $10, and each additional mile costs $1.50:

C(x) = 4 + 2(5) + 1.50(x − 5)

So the piecewise function is:

Important idea: The second rule begins with the cost of the first 5 miles, $14 total, and then adds $1.50 for each extra mile.

Step 3: Evaluate the function

Find the cost of an 8-mile ride.

1. Since 8 > 5, use the second rule.
2. C(8) = 14 + 1.50(8 − 5)
3. C(8) = 14 + 1.50(3)
4. C(8) = 18.50

Answer: An 8-mile ride costs $18.50.

Quick check: Ask, “Why would it be incorrect to use the first rule for an 8-mile ride?”

Body: We Do — Guided Practice — 6 Minutes

Problem: Streaming Data Plan

A data plan costs $20 per month and includes up to 5 gigabytes of data. Each gigabyte over 5 costs $4.

Let x represent the number of gigabytes used and P(x) represent the monthly price.

Work Together

1. Identify the conditions:

   What happens when x ≤ 5? What happens when x > 5?

2. Write the piecewise function:

   P(x) =
   20, if 0 ≤ x ≤ 5
   20 + 4(x − 5), if x > 5

3. Evaluate the function:

   Find P(8).

   Because 8 > 5, use the second rule:

   P(8) = 20 + 4(8 − 5) = 20 + 12 = 32

Answer: Using 8 gigabytes costs $32.

Think-pair-share option: If working with a parent or partner, explain why the expression is 4(x − 5) instead of 4x. If working independently, explain your answer aloud or write one sentence.

Formative Check

Answer these questions:

1. What is P(4)?
2. Which rule should be used for P(6)?
3. What does the number 5 represent in this situation?

Answers: P(4) = $20; use the second rule for P(6); 5 is the number of gigabytes included in the monthly plan.

Body: You Do — Independent Real-World Challenge — 5 Minutes

Choose One Challenge

Option A: Arcade Game

An arcade charges $8 for the first hour of play and $3 for each additional hour. Let h be the number of hours played.

1. Write a piecewise function for the cost.
2. Find the cost of playing for 4 hours.
3. Explain what each piece of your function means.

Option B: Printing Service

A printing service charges $6 for up to 20 pages. Each page beyond 20 costs $0.15.

1. Write a piecewise function for the cost of printing p pages.
2. Find the cost of printing 50 pages.
3. Explain why the second rule uses p − 20.

Expected Work for Option A

C(h) =
8, if 0 < h ≤ 1
8 + 3(h − 1), if h > 1

C(4) = 8 + 3(4 − 1) = 17, so 4 hours costs $17.

Expected Work for Option B

C(p) =
6, if 0 ≤ p ≤ 20
6 + 0.15(p − 20), if p > 20

C(50) = 6 + 0.15(50 − 20) = 6 + 4.50 = 10.50, so 50 pages costs $10.50.

Conclusion: Closure and Recap — 2 Minutes

Complete this sentence:

A piecewise function is useful when...

Review the process:

1. Read the situation carefully.
2. Identify where the rule changes.
3. Write a rule for each interval.
4. Choose the correct rule for the given input.
5. Evaluate and label the answer.

Exit Ticket

A museum charges $12 for admission and includes the first 2 hours of parking. Each hour beyond 2 costs $3.

1. Write a piecewise function for the total cost of parking and admission.
2. Find the cost for 5 hours of parking.
3. In one sentence, explain what makes this a piecewise situation.

Exit Ticket Answer:

C(t) =
12, if 0 ≤ t ≤ 2
12 + 3(t − 2), if t > 2

C(5) = 12 + 3(5 − 2) = 21, so the total cost is $21. It is piecewise because the cost rule changes after 2 hours.

Differentiation and Adaptations

• Support: Provide a table with columns labeled “input condition,” “rule,” and “evaluation.” Let the learner use a number line to mark where the rule changes.
• Additional support: Begin with whole-number prices and use the phrase “base cost plus extra cost” before introducing symbolic notation.
• Extension: Ask the learner to create a piecewise pricing plan for a fictional business, graph it, and explain whether the graph has a jump or connects at the rule change.
• Digital option: Use a graphing calculator or spreadsheet to make a table of values and compare the two rules visually.
• Verbal option: Have the learner explain each step aloud rather than writing every explanation.

Assessment Summary

• Formative: Prediction, quick checks, guided practice, and think-pair-share explanation.
• Summative: Independent challenge and exit ticket.
• Mastery benchmark: Correctly identifies the rule conditions, writes both pieces, evaluates at least one input accurately, and explains the real-world meaning with appropriate units.