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)

How far away from the TV should you sit? Students use right triangle trigonometry and a rational function to explore the percent of your visual field that is occupied by the area of a television.

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

How can you make money in a pyramid scheme? Students learn about how pyramid schemes work (and how they fail), and use geometric sequences to model the exponential growth of a pyramid scheme over time.

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

How much do you really pay when you use a credit card? Students develop an exponential growth model to determine how much an item really ends up costing when purchased on credit.

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

Could Inspector Javert have survived the fall? Students use quadratic models to determine how high the bridge was in Les Misérables, and explore the maximum height from which someone can safely jump.

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

How much can you trust your memory? Students construct and compare linear and exponential models to explore how much a memory degrades each time it's remembered.

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

How much should you bid in an auction? Students use probability, expected value, and polynomial functions to develop a profit-maximizing bidding strategy.

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

How can we improve our calendar? Students examine some other ways to keep track of dates, and use number sense and function concepts to convert between different calendars.

Topic:
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.

Topic:
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.

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

Which size pizza should you order? Students apply the area of a circle formula to write linear and quadratic formulas that measure how much of a pizza is actually pizza, and how much is crust.

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

How have temperatures changed around the world? Students use trigonometric functions to model annual temperature changes at different locations around the globe and explore how the climate has changed in various cities over time.

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

How should pharmaceutical companies decide what to develop? In this lesson, students use linear and quadratic functions to explore how much pharmaceutical companies expect to make from different drugs, and discuss ways to incentivize companies to develop medications that are more valuable to society.

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

How has the human population changed over time? Students develop exponential models to analyze human population growth and explore the impact this growth will have in areas around the world.

Topic:
Building Functions (BF), 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)

Why are so many Americans dying from opiate overdoses? Students use exponential decay and rational functions to understand why addicted patients seek more and stronger opioids to alleviate their pain.

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

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

How have temperatures changed around the world? Students use periodic functions to compare long-term average monthly temperatures to recorded monthly temperatures, evaluate evidence of climate change, and discuss possible consequences.

Topic:
Functions (F), Interpreting Categorical and Quantitative Data (ID), Interpreting Functions (IF)

What time should school start in the morning? Students use periodic functions to compare the alertness levels of adults vs. teenagers over the course of the day and debate the merits of starting school later.

Is higher education a good investment? Students write and solve systems of linear equations to determine how long it would take to pay off various degrees and discuss the pros and cons of different educational paths.

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

How dangerous are heat and humidity? Students use polynomial functions to explore the heat index and discuss the life-and-death consequences that cities around the world will face in the coming years.

Topic:
Interpreting Functions (IF)

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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 do the rules of an election affect who wins? Students calculate (as a percent) how much of the electoral and popular vote different presidential candidates have received, and add with integers to explore elections under possible alternative voting systems.

Topic:
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