Function Notation Mission: Decode the Real World

Materials Needed

• Pencil and paper or a whiteboard
• Calculator, optional
• Timer, optional
• Function Mission Cards or the scenarios provided below

Standard

MA.912.F.1.2: Evaluate functions given in function notation and interpret outputs in real-world contexts.

Learning Objectives

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

• Evaluate a function such as f(3) by substituting the input for the variable.
• Explain what a function output means in a real-world situation.
• Use function notation to solve at least three real-world problems with 80% accuracy.

Success Criteria

I can:

• Replace the input value wherever the variable appears.
• Follow the order of operations accurately.
• State the answer with units and explain what it means in context.

Introduction: The Function Decoder Hook (2 minutes)

Hook: Say: “Imagine a vending machine. You put something in, and the machine gives something back. A function works in a similar way: it takes an input and produces exactly one output.”

Explain that in a function such as f(x) = 2x + 3, x is the input and f(x) is the output.

Tell the student: “Today you will learn how to decode function notation and use it to answer real-world questions.”

Body: I Do, We Do, You Do

I Do: Teacher Modeling (5 minutes)

Use the following function:

f(x) = 3x + 2

1. Identify the input. For f(4), the input is 4.
2. Substitute 4 for every x:
   f(4) = 3(4) + 2
3. Evaluate using the order of operations:
   f(4) = 12 + 2 = 14
4. State the output: f(4) = 14.

Key idea: The number inside the parentheses is the input. The answer after evaluating is the output.

Real-world example: Suppose c(d) = 3d + 2 represents the cost of renting a bicycle for d hours. The rental company charges $2 initially and $3 per hour.

c(4) = 3(4) + 2 = 14

Interpretation: Renting the bicycle for 4 hours costs $14.

We Do: Guided Practice (6 minutes)

Work through each problem together. Ask the student to explain each step before moving on.

Problem 1: Arcade Game

The function p(g) = 5g + 10 represents the number of points earned after completing g game levels.

Evaluate p(3).

Guiding questions:

• What number replaces g?
• What expression do we get after substitution?
• What does the answer represent?

Solution: p(3) = 5(3) + 10 = 25. The player earns 25 points after completing 3 levels.

Problem 2: Smoothie Shop

The function s(n) = 4n + 2 represents the cost of buying n smoothies, including a $2 delivery fee.

Evaluate s(5).

Solution: s(5) = 4(5) + 2 = 22. The cost of 5 smoothies is $22.

Quick Check: Ask the student: “What is the difference between the input and the output?”

Expected response: “The input is the value placed into the function. The output is the result produced by the function.”

You Do: Function Mission Challenge (5 minutes)

Tell the student: “You are now the Function Decoder. Solve each mission independently. Show substitution, calculation, and a sentence interpreting the answer.”

1. Streaming Service: The function c(m) = 8m + 5 represents the cost of a streaming service for m months, including a one-time $5 setup fee. Evaluate c(3).
2. Plant Growth: The function h(w) = 2w + 6 represents the height of a plant after w weeks. Evaluate h(7).
3. Ride Share: The function r(m) = 1.50m + 4 represents the cost of a ride that travels m miles. Evaluate r(6).

Answer Key

• c(3) = 8(3) + 5 = $29: Three months cost $29.
• h(7) = 2(7) + 6 = 20 feet: The plant is 20 feet tall after 7 weeks.
• r(6) = 1.50(6) + 4 = $13: A 6-mile ride costs $13.

Conclusion: Decode and Reflect (2 minutes)

Ask the student to complete these statements aloud or in writing:

• “When I see f(4), the 4 is the __________.”
• “To evaluate a function, I __________ the input for the variable and then simplify.”
• “In a real-world problem, the output tells me __________.”

Final recap: Function notation names the input and output. To evaluate a function, substitute the input for the variable, calculate carefully, and explain the answer using real-world units and meaning.

Assessment

Formative Assessment

• Listen for accurate use of the terms input and output.
• Check substitution during the “We Do” problems.
• Ask the student to explain why units such as dollars, miles, or feet are important.

Summative Exit Challenge

The function t(x) = 12 + 0.75x represents the total cost of a taxi ride that includes a $12 starting fee and costs $0.75 per mile.

Evaluate t(8) and interpret the result in a complete sentence.

Expected answer: t(8) = 12 + 0.75(8) = 18. An 8-mile taxi ride costs $18.

Differentiation and Choice

• Support: Provide the substitution frame: f(input) = formula with input substituted. Allow the student to use a calculator and highlight the variable in the function.
• Hands-on option: Use coins, blocks, or tokens to represent a starting fee and repeated cost.
• Verbal option: Let the student explain each solution aloud instead of writing every step.
• Extension: Have the student create a function for a personal scenario, such as snack costs, savings, game points, or travel distance. Then ask a family member to evaluate the function for a chosen input.
• Challenge: Ask the student to compare f(3) and f(5) and explain how changing the input changes the output.