SWBAT model situations that can be modeled by sinusoidal functions.

Students collect data, analyze the data, and construct models using their knowledge of trigonometric functions.

10 minutes

This project asks students to apply the ideas learned in the unit to real world situations. Students will collect and analyze data, develop a mathematical model, then describe the development of the model in a written report. My students have three days to complete the project.

Students can work in groups of 2 or 3 students. I have found that more than three students is unproductive. The project works best when students work in pairs. Today, I will introduce the project and students will have time to brainstorm topics for research.

When students come into class today the following question is on the board:

**What real world situations can be modeled by a sinusoidal curve? **

I ask students to share their ideas on the board, creating a list of possible project topics.

After a list of ideas is present I will ask, **"Of the ideas on the board, which ones could we research to find data to analyze?"** I'll place check marks next to the ones that students identify. We'll use this process to narrow down the list of possible topics. At the same time, it is often helpful to let students continue to add new topics at this point.

Here are some topics that students like to explore:

- moon phases
- sunrise
- sunset
- average daily temperature
- monthly average rainfall
- modeling the swing of a pendulum

20 minutes

Now that students have some ideas, I will provide more information about the requirements for the Sinusoidal Project, including the scoring guide. There are two options for the project.

- Students use the internet to find data on a real world phenomena such as sunrise. The students collect and organize data into tables and then graphs so they can write a mathematical model.
- Students design an experiment. The students collect data from the experiment than make a model for the experiment. An example of this would be to measure the time it takes for a pendulum to swing.

As the students consider the options, I often answer several questions about the project requirements.

I have made a spreadsheet for the students to use during the project. This spreadsheet allows students to input their data and then determine the parameters a, b, c, d for the equation y=a sinb(x-c)+d. When the parameters are inputted a graph of best is produced on the graph. Students can quickly change the parameters to improve the model. The students use this template so they can focus on analyzing the data collected.

After students have reviewed the options we look over the scoring guide. I show students some similar projects that could be improved with revision. The issues that I want to bring to my students attention are:

- Project quality suffers when the data analysis was not not critically and carefully
- Editing and revising the report is a necessary step
- Organization of space and presentation of content for display makes a big difference

I have found that it makes a big difference to students when I show them examples of student work so they understand what good work looks like, and, what incomplete work looks like.

10 minutes

During the final minutes of class, students will choose their partners and begin to select a topic for the project. As students talk about their ideas, I move around to listen to what my students are considering. Students often have great ideas, but sometimes they are too involved to complete within the time frame for this project. For example, students may want to model a predator vs. prey relationship. If so, I will ask students, "How do you plan to collect the data? What data will you collect?" I want to make sure that students have an idea about how to get themselves started.

**Important Note**: There are always groups that want to research situations that increase and decrease, but are not periodic. Examples of these are:

- gasoline prices
- airline ticket prices
- student misconduct by month
- number of people riding a bus

When students consider options like this I always ask students to explain why they think the situation is sinusoidal. Then, I ask students how they will collect the data. And importantly, I say, **"What will you do if you collect the data and determine it is not sinusoidal?"** For this project it is important for students to use situations that are sinusoidal, but I might be swayed to let students explore a function if I think that they can explain why a model didn't really work.

As class ends I explain that we will be working on this over the next 2 class periods. The sooner you have your topic the sooner you can get started and finish your work.