SWBAT create trigonometric equations to match graphs. SWBAT make use of the equation to determine the amplitude and period of sinusoidal functions.

Practice makes perfect! Students gain confidence relating the graph to the equation by means of their key features.

15 minutes

In the previous lesson, we already began a discussion of the Ferris Wheel problem. Today, we begin class by focusing on the final modeling problem: a mass bouncing on a spring.

If one of the groups of students has worked out a (mostly) complete solution, I will ask them to share it with the rest of the class. A document camera is really useful for this. As they present their solution, I will encourage the rest of the class to ask questions. I will also ask my own questions to probe their understanding and to help synthesize some of the concepts for everyone. The conversation will be fairly open, with students raising their hands to chime in whenever they have a question or comment. (**MP 3**)

Some of the questions I will be sure to ask are:

- How did you decide between sine and cosine for this function?
- How can you tell when to use a negative coefficient?
- Do you have a strategy for making sense of frequency vs. period?
- How do you use inverse functions to solve for the independent variable?
- How can you make use of the symmetry and periodicity of the function to draw conclusions?
- How does the graph help with
*all*of this?

While there are certainly some *incorrect* responses to these questions, there are a number of correct responses and I hope to hear many different ones from around the room.

25 minutes

To help students build their confidence with both equations and graphs of sinusoidal functions, I will hand out Sine and Cosine Practice and provide ample time in class to complete it. At first, I'll ask the students to work independently, but once I'm satisfied that everyone's on the right track I'll allow them to work in small groups if they'd like. Meanwhile, I will circulate to check for understanding and to answer questions.

In this practice set, the students will first be writing sine and cosine equations to match given graphs. They'll have to pay attention to the amplitude and period of the function, and they'll have to decide between the sine or cosine functions. (To check their work, I will encourage them to make use of their calculators to test their equation.) After moving from graphs to equations, they'll complete a few problems that simply ask them to determine the amplitude and period from the equation alone. I expect the amplitude to be easy, but determining the period will require that they keep the target value of 2*pi in mind at all times. (**MP 7**; Take a look at the ** Making Use of Structure **reflection on this section for more thoughts on this subject.)

5 minutes

At the end of class, I will take about 5 minutes to run through the solutions to the Sine and Cosine Practice problems. From my observations, I'll already have a pretty good idea of how everyone is doing, and I'll have already corrected a number of mistakes, but I'd like the students to leave knowing where they stand.

I will ask students one-by-one to share their solutions to the problems, and we'll address any discrepancies as they arise. The goal is for students to leave class feeling confident in their understanding of the differences between the sine and cosine functions, as well as the relationship between the graph and equation.