Because we use it every day, we tend to forget that our calendar is pretty strange. The year is broken up into months of different lengths, seemingly without much of a pattern. The same date can occur on different days of the week in different years. Is this really the best way to carve up a year?

In this lesson, students examine some other ways to keep track of dates, and use number sense and function concepts to convert among different calendars. What might a more reasonable calendar look like?

Students will

Convert dates among different proposed calendars using multiples and factors

Describe conversion rules in terms of functions, domain, and range

Use function composition and notation to describe two-step conversion rules

Reason about multiples and remainders to develop rules for converting dates into days of the week

Explain why functions can have inverses that fail to be functions themselves

Make recommendations for improvements to both the current and proposed calendars

Before you begin

Students should be able to reason about factors and multiples, and they should know how to obtain the remainder from a whole number division. Most of the questions revolve around function topics such as domain, range, inverses, and composition, so this lesson can serve as a way to discuss those topics in the context of calendars.

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Topic:
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)

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

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

What are some ways to encrypt secret messages? Students explore function concepts using ciphers to encrypt messages both graphically and algebraically; they try to decrypt some messages too. In the end, they’ll learn what makes for a useful cipher, and what makes a cipher impossible to decode.

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)

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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 do noise-canceling headphones 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.

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

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)

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

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

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

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

How have video game consoles changed over time? Students create exponential models to predict the speed of video game processors over time, compare their predictions to observed speeds, and consider the consequences as digital simulations become increasingly lifelike.

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