It can be incredibly frustrating to snap what you think is a great picture, only to have it come out blurry. Of course it helps to have a steady hand, but there’s electronic help available if you’re willing to cough up some extra cash for a lens with vibration reduction.

In this lesson, students interpret graphs and use right triangle trigonometry to explore the relationship between focal length, viewing angle, and blurriness. In the end, they figure out when vibration reduction might help them take clearer pictures, and when it might not be worth the cost.

Students will

Use right triangle trigonometry to calculate unknown distances from a diagram

Write general expressions to describe a lens’s vertical coverage of an object in terms of its viewing angle and the camera’s distance from the subject

Calculate absolute and relative error for a camera that moves while taking a picture

Interpret graphs describing how a lens’s viewing angle and relative error are related to its focal length

Develop questions to help someone decide under what circumstances it might make sense to purchase vibration reduction in a lens

Before you begin

Students should be familiar with the basic definitions of the trigonometric ratios, and be able to write expressions involving the relationships among the sides and angles of a right triangle.

When you buy a bigger TV, how much more do you really get? Students use the Pythagorean Theorem and proportional reasoning to investigate the relationship between the diagonal length, aspect ratio, and screen area of a TV.

Topic:
Geometry (G), Similarity, Right Triangles, and Trigonometry (SRT)

How fast is the Earth spinning? Students use rates, arc length, and trigonometric ratios to determine how fast the planet is spinning at different latitudes.

Topic:
Circles (C), Similarity, Right Triangles, and Trigonometry (SRT)

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 does two people's love for one another change over time? Students investigate the effect of coefficients on recursive functions, and explore whether or not romance can be modeled with mathematics.

Topic:
Interpreting Functions (IF), Seeing Structure in Expressions (SSE)

Does the same sound always sound the same? Students come up with equations in several variables to explore the Doppler Effect, which explains how sound from a moving object gets distorted.

How should pharmaceutical companies decide which drugs to develop? Students create linear and quadratic functions to explore how much pharmaceutical companies profit from different drugs and consider ways to incentivize companies to prioritize medications that are valuable to society.

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

How do municipal fines affect people with different incomes? Students write, solve, and graph systems of linear equations to determine how long it takes to pay off a ticket and debate the fairest ways for cities to raise revenues without harming their poorest residents.

Topic:
Creating Equations (CED), Expressions and Equations (EE), Reasoning with Equations and Inequalities (REI), Functions (F)

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)

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)

Should the government increase the minimum wage? Students use systems of linear equations to explore the relationship between wage and labor, analyze the economics of fast-food restaurants, and debate whether the federal government should increase the minimum wage.

Topic:
Creating Equations (CED), Interpreting Functions (IF), Linear, Quadratic, and Exponential Models (LE), Reasoning with Equations and Inequalities (REI)

How much should Nintendo charge for a video game console? Students use linear and quadratic models to analyze and discuss the relationship between the price of a Wii U console and profits for Nintendo.

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

How much does age matter in a relationship? Students use a system of linear inequalities to explore the popular dating rule-of-thumb, ‘half plus seven’, and debate how important age -- and other factors -- are in healthy relationships.

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

What’s the ideal size of a soda can? Students create rational functions to explore the relationship between volume, surface area, and cost to determine the optimal size of a soda can.

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)