Trigonometry in Action

Overview

This lesson focuses on 11th Grade students applying trigonometric functions to solve real-world problems based on US Common Core Standards - HSF.TF.8 (Building Functions and Applying Trigonometric Functions). The overarching goal is to show how trigonometry relates to practical applications, including architectural design, engineering problems, and navigation. The Marzano framework is applied to structure the lesson with clear learning goals, student engagement, practice, reflection, and feedback.

Lesson Duration: 56 minutes

Class Size: 20 students

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Learning Objective

Students will be able to:

1. Analyze real-world problems involving right triangles and periodic phenomena.
2. Apply sine, cosine, and tangent functions to model and solve these problems.
3. Demonstrate a conceptual understanding of trigonometric relationships and their practical applications.

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Materials Needed

• Graphing calculators
• Protractors
• Rulers
• Whiteboard/markers
• Printed problem sets (scenario-based challenges)
• A model of a Ferris wheel (miniature or diagram)
• Smartboard/projector if available

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Structure

Step 1: Setting Learning Goals (6 minutes)

• Hook (2 minutes): Show an image of a Ferris wheel, a suspension bridge, and a mountain slope. Ask, “What do these have in common?” Facilitate a short discussion to prompt curiosity.
• Learning Goals (4 minutes):
  • Write today's learning goals on the board:
    1. Interpret and model real-world problems using trigonometric functions.
    2. Use technology and tools to solve applied trigonometry challenges.

Step 2: Direct Instruction (10 minutes)

1. Real-World Connections (3 minutes):
   Explain how trigonometry is used in architecture, navigation, and physics. For example:

   • Architects use trigonometry to calculate roof slopes.
   • Engineers apply it when designing roller coasters to measure curves.
     Include visuals (e.g., Ferris wheel with height vs. time model) or stories to captivate interest.

2. Concept Review (3 minutes):

   • Quickly recap the foundational concepts: SOHCAHTOA and periodicity in sinusoidal functions.
   • Define the differences between sine, cosine, and tangent functions with a visual unit circle.

3. Worked Example (4 minutes):
   Solve a practical problem collaboratively on the board:
   Problem: A rescue team needs to calculate the height of a cliff using trigonometry. With an angle of elevation of 45° from a distance of 100 feet, find the height.
   Walk them through the use of the tangent function:
   tan(45°) = height/100 → height = 100 ft.

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Step 3: Group Activity (20 minutes)

Real-World Application Challenges

Break students into 5 groups of 4, assigning each group one challenge. Each group will:

• Analyze a scenario-based word problem.
• Model the situation using trigonometric functions.
• Solve using a step-by-step process and graph the results where applicable.

Challenges:

1. Ferris Wheel Problem: Given a Ferris wheel with a radius of 30 feet and center at 50 feet above the ground, model the height of a passenger over time using a sinusoidal function.
2. Navigation Challenge: A ship spots a lighthouse at an angle of elevation of 30° from 200 yards offshore. Calculate the lighthouse's height.
3. Road Construction: An engineer needs to find the incline angle of a 500 ft ramp that rises 40 ft vertically.
4. Satellite Problem: Model the height of a satellite using sinusoidal functions based on its positions and time.
5. Bridge Design: Estimate the length of a suspension cable in a bridge design based on angles and height of poles.

Instructions:

• Use tools like protractors, calculators, and graph paper provided.
• Draft clear steps for solving.
• Present the solution on poster paper or whiteboard space.

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Step 4: Guided Practice and Feedback (12 minutes)

1. Group Presentations (10 minutes):
   Each group presents their solution to the class for 2 minutes. While one group presents, others evaluate using a provided checklist:

   • Did the group correctly identify the trigonometric relationship?
   • Is their solution logical?
   • Are their calculations accurate?

2. Teacher Feedback (2 minutes):
   Provide constructive guidance and corrections as needed, praising effort and innovative approaches.

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Step 5: Closure and Reflection (6 minutes)

• Class Reflection (2 minutes):
  Ask, “Where else could we apply these concepts?” Build a quick think-pair-share activity to prompt students.
• Exit Ticket (4 minutes):
  Each student writes down:
  1. One real-world application of trigonometry they learned today.
  2. A problem-solving step they found challenging or easy to understand.

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Homework

Assign differentiated practice:

1. Solve 5 real-world trigonometry problems from the textbook.
2. Create a short real-world scenario that would require using trigonometric functions to solve, and explain how you'd solve it.

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Differentiation Strategies

1. For advanced learners: Encourage them to derive trigonometric identities or explore inverse trigonometric functions in the homework problem.
2. For struggling learners: Provide scaffolded problem sets with step-by-step guides.

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Assessment

1. Formative: Evaluate group work, exit tickets, and group presentations for understanding and application ability.
2. Summative: Use tomorrow's quiz to assess mastery of today's concepts.

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Marzano Elements Addressed

• Establishing Learning Goals: Goals are explicitly stated and reviewed.
• Engaging Students: Real-world applications and group work foster active engagement.
• Providing Feedback: Feedback during presentations supports deeper learning.
• Practicing and Deepening Understanding: Students apply knowledge through collaborative problem-solving.
• Reflecting on Learning: Reflection reinforces conceptual grasp and identifies areas for improvement.

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Notes for the Teacher

• Emphasize the relevance of trigonometry to careers, making the subject more meaningful.
• Foster collaboration throughout the group task to strengthen team-based problem-solving skills.
• Check for understanding often and adjust pacing if needed.