SWBAT determine the actual speed of a rider on a Ferris wheel given the angular speed of the wheel and vice versa.

How fast is the rider actually moving as the wheel turns? How do changing the angular speed and the radius of the wheel affect the speed of the rider? Students make a generalization about how these changes affect the rider's speed.

30 minutes

To get today's lesson off to a good start I plan to give my students 30 minutes or so to work on Ferris Wheel Speeds Warm-Up. These three problems leverage multiple representations to take students deeper in their understanding of trig functions. Each problem gives the same information in a different form. A good guiding prompt for students is, "Try to figure out how to get as much information as you can out of each representation."

Problem 2 allows for a rich discussion because it is interesting for students to think about how many different Ferris wheels could have those same points. An interesting idea to think about is whether or not the max and min that are given are right “next to each other” or whether the graph has max and min points in between those. For advanced students, push them to really think about this. One way to push them is to say: “I can find more than one Ferris wheel that fits those requirements. How do you think I did that?” I will only say this after the students have had some time to think about the question already.

Problem 4 is set up to preview the day’s Investigation. After giving all students 30 minutes or so to work on the first 3 problems, I ask them to talk about the problem on the back. I don't want them to solve it directly.I want students to think about and discuss how they could arrive at an answer. Even if students answer to the problem, I will not tell them whether they are correct or not. Instead, I will ask them to try to convince another student that their answer is correct (**MP3**).

I want Problem 4 to generate conversation about what speed means. There is deliberately more information provided than students will need to answer the question. They are also missing a reminder about how to find the circumference of the wheel. I will give students 5+ minutes to talk about this problem. At the right moment, I will ask students to share ideas as a class, so that the problem provides a natural segue into the lesson.

40 minutes

10 minutes

The Ferris Wheel Speeds task gives students a chance to look at some relationships—the big picture here is to continue the theme of thinking about different functions that arise in real world situations, when we examine different variables. This task can be used as an extension, or as a stand-alone task to supplement any lesson where you feel that it could fit.

In today’s lesson, I use the task as a challenge to push student thinking. Because all of the students in my Precalculus class are concurrently enrolled in Chemistry, they are working with dimensional analysis, so some of them figure out how to apply that to this problem.

10 minutes

There are some really abstract ideas going on in this lesson, so the closing is a good time to talk about what it all really means. **What is speed?** We normally think of speed as distance over time, so angular speed is something different, but somehow there is a way to translate between these different ways of measuring speed.

I will ask students to respond to these questions:

- What is speed? What are some different ways to think about speed?
- What kinds of
*unit conversions*did you need to perform in this lesson? Why is it important to be able to convert between different units of measurement? - Do you get the same amount of information about a Ferris wheel if you know its angular speed or its actual speed? Explain. If you know one, can you find the other? Why or why not?

Students may not have good answers to these questions yet, but we are going to keep looking at these questions, because this is our entry point into understanding radians as an alternative angle measure.