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# Ferris Wheels and Trigonometry

Lesson 8 of 11

## Objective: SWBAT apply knowledge of right triangle trigonometry to find the exact height of a rider at any point on the Ferris Wheel.

## Big Idea: After many days of investigation, students will finally apply their previous knowledge to this new problem--and take the first steps to extend right triangle trigonometry to all points on the unit circle.

*85 minutes*

#### Warm-Up

*30 min*

Today's Warm-Up is a chance for students to use special right triangles to consider all the pieces of the Ferris wheel graphs. To complete the task, they need to actually create the complete data table based on the given information, and, then turn this into a graph.

Likely the biggest challenge for students on Problem 1 will be to figure out how to find the exact outputs using radicals and the Pythagorean Theorem. Also, it will not be easy for them to see how these numbers increase and decrease because they will be writing them in radical form, so I like to ask students to find radical form first, and then convert this to decimal form.

Problem 2 builds on yesterday's review of right triangle trigonometry. As students transition to this problem, I will ask them how they think the task relates to Ferris wheels. Considering this question prepares students for the big idea of the day.

Problem 3 is a real challenge and serves as an extension for students who have already mastered the other big ideas. *How can you get information about the Ferris wheel out of the data table? *This problem will be an on-going challenge for students who are ready for it. I plan to give my students lots of time to explore this task and I will offer them very little assistance.

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My goal for the Find Missing Entries in Ferris Wheel Data Tables tasks is to push students to use function rules and shortcuts to figure out what is going on. In each problem, there is different information missing, so students have to think in different ways (**MP1**). I want to keep them on their toes by giving them slightly different information each time. Students can use all sorts of strategies to figure these problems out: guessing and checking, entering the data into the computer and finding points that fit the curve, setting up systems of equations. Hopefully they will come up with lots of different strategies. I will try to partner students with peers who developed different strategies so that they can discuss their methods.

Some questions I will ask students as they work are:

*Do you think you have found the only possible solution here? How do you know?**Which parameters can you determine given the data? Which parameters are missing? Which parameters are given?**Can you set up equations to solve for the unknown parameters?*

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#### Closing

*10 min*

Since each section of the lesson today has been somewhat open ended, I leave time for students to debrief the lesson together. At the end of this lesson, they often end up teaching each other something, or arriving upon a good question together. During this brief time, I circulate quickly and if I notice any students are not talking with each other, I ask them why and they usually say, “We understand everything,” which is obviously not accurate, so I push them to discuss a specific aspect of the lesson, or to look ahead to the next day’s lesson using the prompt, “*Are there function rules that create these data tables*?”

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Introduction to the Ferris Wheel Problem
- LESSON 2: Ferris Wheel (Graph) Symmetries
- LESSON 3: Graphing Ferris Wheel Heights
- LESSON 4: Changing Ferris Wheels
- LESSON 5: Ferris Wheel Speeds
- LESSON 6: From Degrees to Radians, Ferris Wheel Style
- LESSON 7: Ferris Wheels and Triangles
- LESSON 8: Ferris Wheels and Trigonometry
- LESSON 9: Ferris Wheel Function Rules
- LESSON 10: Ferris Wheel Unit Review
- LESSON 11: Ferris Wheel Summative Assessment