Noise-canceling headphones are amazing. They don’t just block the sounds from coming into your ears, but actively destroy them. They actually make more noise so that you end up hearing less noise. How does that work?

In this lesson, students use transformations of trigonometric functions to explore how sound waves can interfere with one another, and how noise-canceling headphones use incoming sounds to figure out how to produce that sweet, sweet silence.

Students will

Given the frequency of a pure tone, calculate the period of its sound wave

Sketch the graph of a chord composed of two tones and estimate its period and frequency

Describe any apparent relationship between a chord’s period and its perceived pleasantness

Sketch the graph of a pure tone’s noise-canceling counterpart and construct multiple functions to describe it

Develop general procedures for generating a noise-cancelling wave from a given pure tone

Examine some difficulties in describing noise-canceling waves for chords

Explain how noise-canceling headphones work and why they are not always completely effective

Before you begin

Students should be familiar with the general form of sinusoids (i.e. f(x) = Asin(Bx – C) + D or f(x) = Acos(Bx – C) + D) and how changes in the parameters affect their graphs.

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Building Functions (BF), Interpreting Functions (IF)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF)

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Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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Topic:
Building Functions (BF), Interpreting Functions (IF)

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Building Functions (BF), Functions (F), Interpreting Functions (IF)

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Building Functions (BF), Interpreting Functions (IF), Using Probability to Make Decisions (MD)

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Creating Equations (CED), Building Functions (BF), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

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Topic:
Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI)

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Building Functions (BF), Interpreting Functions (IF)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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Topic:
Building Functions (BF), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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Topic:
Building Functions (BF), Creating Equations (CED)

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Topic:
Building Functions (BF), Linear, Quadratic, and Exponential Models (LE)

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Building Functions (BF), Linear, Quadratic, and Exponential Models (LE), Seeing Structure in Expressions (SSE)

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Mathalicious lessons provide teachers with an opportunity to teach standards-based math through real-world topics that students care about.

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