Understanding Function Composition

---

Grade Level:

10th Grade

Curriculum Area and Standards:

Common Core State Standards (CCSS) – Mathematics

• CCSS.MATH.CONTENT.HSF.BF.A.1.C: Compose functions. Understand the process of function composition and how output values from one function become input values for another.
• CCSS.MATH.CONTENT.HSF.BF.A.2: Build new functions from existing functions.

---

Lesson Duration:

40 minutes

Lesson Focus:

Composition of functions, emphasizing teamwork and differential learning strategies.

---

Lesson Objectives:

By the end of the lesson, students will:

1. Understand the concept of function composition and its notation [e.g., (f ∘ g)(x)].
2. Be able to compute the composition of two functions analytically.
3. Work collaboratively in teams to solve problems with varying levels of challenge, catering to differential learning.
4. Strengthen their critical thinking and communication skills while solving real-world function application problems.

---

Materials Needed:

• Whiteboard and markers
• Printed handouts with differentiated practice problems (Easy, Moderate, Challenge)
• Small dry erase boards and markers for each team
• Calculators
• Slips of paper for team activity roles

---

Lesson Plan Breakdown:

1. Warm-Up Activity: Discovering Relationships (5 Minutes)

1. Begin by asking students: "Do you use input-output machines in real life? What kinds of processes take something, change it, and send it out?"
   • Quick examples to discuss: ATM withdrawals (input uses account info, output gives money) or vending machines.
   • Relate real-world situations to mathematical functions, establishing input and output relationships.
2. Introduce the idea of combining machines: "What if one machine’s output fed into another machine? How would the overall process change things?"

---

2. Direct Instruction: Function Composition (10 Minutes)

1. On the board: Write two functions:
   f(x) = 2x + 3 and g(x) = x².
   • Demonstrate composition notation: (f ∘ g)(x) and (g ∘ f)(x).
   • Walk through solving:
     • (f ∘ g)(x) = f(g(x)) = f(x²) = 2(x²) + 3 = 2x² + 3.
     • (g ∘ f)(x) = g(f(x)) = g(2x + 3) = (2x + 3)² = 4x² + 12x + 9.
2. Emphasize the importance of order: Composition is not commutative! (i.e., (f ∘ g)(x) ≠ (g ∘ f)(x).)
3. Tie back to the earlier analogy of input-output machines: the order of feeding the machines matters!

---

3. Teamwork Activity: Function Machine Challenge (20 Minutes)

Team Structure:

• Divide the class into 6 teams of 4–5 students each.
• Assign each student a role (e.g., Calculator, Recorder, Problem Solver, Presenter, Checker). Rotate as needed during the activity.

Steps:

1. Handout Distribution:

   • Hand out printed problems to each team with three difficulty levels (Easy, Moderate, Challenge).
     • Easy: Basic compositions (linear functions).
     • Moderate: Polynomial and rational functions, involving some fractions.
     • Challenge: Real-world scenarios requiring interpretation of output.

2. Teamwork Component:

   • Each team solves problems collaboratively on small dry erase boards.
   • The teacher circulates to prompt students with guiding questions like:
     • "Who is taking charge of this role?"
     • "How can we break this problem into smaller steps?"
     • "Why does the order of composition matter here?"

3. Incorporate Differential Learning:

   • Teams that finish "Easy" problems quickly can attempt the "Moderate" or "Challenge" set to deepen understanding.
   • Scaffold struggling teams with strategic hints without revealing answers.

Example Challenge Problem:

"A company offers a discount function d(x) = 0.8x to reduce a price by 20%, and a tax function t(x) = 1.1x to add 10% sales tax. If an item’s original price is $50, what will the final price be after applying the discount first and then the tax? What if the tax is applied before the discount?"

---

4. Review and Reflection (5 Minutes)

1. Discuss common misconceptions identified during teamwork:
   • Emphasize the significance of order in composition.
   • Call on teams to present their solutions to challenge problems.
2. Closure question:
   • "Can you think of a situation in real life where combining two processes in a specific order would make a difference?" (e.g., washing a car before drying it vs. drying first.)

---

5. Exit Ticket (Handed in as Students Leave)

• Create an Exit Ticket with two short problems:
  1. (f ∘ g)(x) = 3x – 2; g(x) = x². Find (f ∘ g)(2).
  2. State which order is correct and compute: A beverage costs $4. Tax is 5%, and a coupon reduces the price by $1.

---

Assessment and Differentiation:

1. Formative Assessment:
   • Monitor teamwork and facilitate struggling teams during the activity.
   • Check accuracy of dry erase board results during team presentations.
2. Differentiated Learning:
   • Easy, Medium, and Challenge problems accommodate diverse skill levels.
   • Scaffolding questions and strategic team assignments address equitable learning needs.
3. Exit Ticket: Analyze solutions to gauge individual learning.

---

Extension Idea for Early Finishers:

Develop their own real-world problem requiring function composition and exchange with another team to solve.

Homework Suggestion:

Provide a worksheet with additional composition problems, including real-life applications (e.g., inventory pricing, temperature conversions).

---

Classroom Setup:

• Arrange desks in clusters of 4–5 for team collaboration.
• Ensure all materials (boards, calculators, problem handouts) are ready before class begins.

---

Wrap-Up Notes for Teachers:

This lesson integrates mathematical content with interpersonal skill building. It leverages teamwork and differential learning strategies to deepen understanding of function composition, providing a robust and engaging experience for students.