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), Linear, Quadratic, and Exponential Models (LE)

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Building Functions (BF), Creating Equations (CED), Interpreting Functions (IF), Reasoning with Equations and Inequalities (REI), Similarity, Right Triangles, and Trigonometry (SRT)

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

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

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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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Building Functions (BF), Interpreting Functions (IF)

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Building Functions (BF), Interpreting Categorical and Quantitative Data (ID), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE)

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

When should NFL teams go for it on fourth down? Students use quadratic functions to develop a model of expected points. They then apply this model to determine when teams should punt the ball, and more importantly, when they shouldn’t.

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

How much should Nintendo charge for the Wii U? Students use linear functions to explore demand for the Wii U console and Nintendo's per-unit profit from each sale. They use those functions to create a quadratic model for Nintendo's total profit and determine the profit-maximizing price for the console.

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

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

How have video game console speeds changed over time? Students write an exponential function based on the Atari 2600 and Moore's Law, and see whether the model was correct for subsequent video game consoles.

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

Do social networks like Facebook make us more connected? Students create a quadratic function to model the number of possible connections as a network grows, and consider the consequences of relying on Facebook for news and information.

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