Writing Sinusoidal Models

To begin today's class I ask students to work together with their groups on a problem from our text (Larson p. 327 #83).  After 3 to 4 minutes, I will ask several students to share their equations with the class.  I will graph each of the equations so students can see and verify whether or not the equation is correct. If there are errors, we will work together to identify the key feature and correct the mistake.

Depending on how the students do, I may extend the Bell Work by choosing another example from page 328 of our text (Larson's Precalculus with Limits 2nd edition). 

In this section of the lesson we will work with trigonometric functions, using them to model problems in context. For many students this is easier than working on trig functions in a more abstract case. I use tables, graphs, and problems from two texts for examples:

• Lial's Trigonometry 8th ed.
• Stewart's Precalculus Mathematics for Calculus 5th ed.

I order the problems so that students begin with graphs and then finish with word problems. My students are more comfortable working with visual examples first, then progressing to problems in textual form.

For this lesson I also have more examples prepared than I will be able to use. This gives me a chance to adjust to students' needs.  I provide a handout so my students can easily take notes as they work.  Giving students the handout facilitates annotating and helps students work a little more quickly.

For most of these examples I do not tell the students which trigonometric function to use. For the first Example 1 some students want to use sine and others want to use cosine. A great discussion occurs when the students are asked to justify the function they want to use. Students will discuss how sine and cosine are just a horizontal shift of -pi/2. This is where we may put the graph on the board to verify the shift. Some will say discuss how the period of sine begins at the midline while cosine begins at the maximum. Interestingly, after discussing the functions, most students consider a sine curve to be the more efficient choice.  

For Example 2 it is important that students read the scale of the graph. Students will not notice how the axis are not at the origin. We discuss how graphs can be changed to confuse students.

As we move through the examples I ask students what the key features of the graph mean in real world terms.  Students need some questioning to understand that the midline is the average and the amplitude is the amount of fluctuation for the average.  I use local temperature data to help students understand the connections I ask these questions:

• If we graph the temperature over time (months) what would the average temperature be for the graph?
• How could we find the amplitude of the graph?
• What is the amplitude for the temperature?
• How would we find the horizontal shift of the graph?
• Is the average temperature always in January? About when is the average temperature.

As we move through the examples I have students work independently and then share their work. When students put different equations on the board I will ask:

• Do these equations represent the same equation?
• What is the same or almost the same for the equations?
• What will always be the same? different?

After working on several examples I want to gain some insight into the process students are using to to write equations for their models. I ask students to answer these questions. These questions allow me to assess students' fluency in determining models.  As students answer the questions, I will write up what they say or rephrase the comment. I also ask students to rephrase what is said to make sure students have a method.

After discussing the questions I give students the Modeling worksheet.  This assignment will be turned in and graded. Nonetheless, I encourage my students to work together. I'll say, "You can ask other students or myself questions." I want to know what my students are able to do using available support.

As they get started I will circulate and ask questions such as the ones below to guide students through their work:

• After reading the problem what can you tell me about the situation?
• What words let you know that this is the maximum(minimum, amplitude, period frequency)?
• How could you determine if your equation will give you the correct answer?