# From Degrees to Radians, Ferris Wheel Style

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## Objective

SWBAT convert between degrees and radians and to describe the concept of radians using arc-length.

#### Big Idea

If you know the angular speed of a Ferris wheel, can you determine how far a rider travels in a given amount of time? Use angular speed to convert degrees to radians.

30 minutes

## Investigation

30 minutes

I will begin by stating clearly that I do not expect that I will need to show my students how to answer the questions asked on the Radians to Degrees Ferris Wheel worksheet. If some students are not sure how to start, I will give them a brief clue, "Use circumference." Since I am confident that my students can do the work, I will be patient. I am prepared to give them lots of time to think about the tasks, and if necessary what my hint means—how can we use this?

I would really like my students to generate the necessary algorithms on their own, even if they are not able to express these formulas algebraically. For this section of the class,my goal is gain confidence that they can develop their own methods for converting between angles and distance. In my experience, when it comes to working with units of measurement, the more confidence students possess, the easier it is for them to develop fluency with a concept like angle measure with radians.

## The Unit Circle

20 minutes

At long last, here it is Radians and Degrees Ferris Wheel. This worksheet is a relatively tradition representation of angle and degree measure on a unit circle. I pose this task simply and let my students have plenty of time to work on the task. I plan to give students up to 20 minutes to work on this, so that they can discover as many patterns and relationships within the degree and radian labels as possible. I think that the opportunity to apply, summarize and discover makes it easier for students to appreciate the benefits of memorizing the numerical values tha they will be recording on the worksheet.

I ask my students to fill out the sheet, either by creating their own formulas (MP4) or repeating the same calculations (MP8). One point of emphasis is to ask students to use fractions to represent angle measures in radians. Some students are likely to want to use decimals to write their radian measures, so it is important to encourage them from the start to use fractions, which will enable them to see the patterns better.

## Closing

5 minutes

The whole point of this lesson is for my students to anchor their understanding of the relationship between radian measure and degreee measure  in contexts and patterns that are really meaningful. This means that the closing is essential to attaining my goals for the lesson. I plan to ask the following questions:

• You just wrote two numbers next to each point on the circle. What do both of those numbers mean?
• How do the two numbers you wrote relate to speed (both angular speed and actual speed)?
• Explain why you need to know the radius of the circle to convert between angular and actual speed.

I will first ask students to briefly discuss these questions in groups. Then, I will facilitate a brief sharing of answers. Before finishing class, I will ask students to write their answer on a sheet of paper as today's check-out.

My hope is that even though students do not yet have a formal definition for radians, they will be able to explain the meaning of and relationship between two different ways to label each point on a unit circle.