More Modeling with Periodic Functions

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SWBAT use sine and cosine functions to model periodic phenomena. SWBAT interpret key features of the mathematical model in context.

Big Idea

Periodic phenomena are all around us and mathematical models help us to understand them.

Getting Started

5 minutes

By now the students are familiar with the use of sine and cosine functions, so today's lesson doesn't require much introduction.  I will hand out Periodic Functions, and tell the class that I'd like them to take a couple of minutes to read the first problem and begin to make some headway (MP1).

The scenario of the bouncing spring should be familiar (bungee jumping?), so I expect everyone to be able to make a good beginning on their own.  I'll move around the room to make sure of this and to answer any questions.

(Just to be on the safe side, I'd suggest borrowing a spring, a brass weight, and a ring stand from the science department.  Then you can mesmerize your students with a real, live, bouncing mass!)

Working Together

20 minutes

As the students work together to complete the first modeling problem, I'll move around the room simply observing at first.  Initially, I'll encourage any struggling students to use the given information to identify the amplitude, period, and midline of the periodic function.  With these quantities known, they should use the general sine or cosine equation to create a mathematical model (MP4).

Typically, I allow my students to place themselves into groups with the understanding that there can never be any exclusion.  If self-selected groups aren't going in this lesson, then I would select heterogeneous groups of three or four students.

Once the students have an equation, they can begin putting it to use.  When asked to evaluate whether the mass is moving up or down at 5 seconds, students may use a graph, evaluate a second, nearby point, or divide 5 seconds by the known period.  In any case, they should be able to offer some explanation.

Finally, in a nod to some of the new expectations of the CCSS, I've included a couple of questions about the average speed of the mass.  These questions both look ahead to Calculus and look back to common problem types from middle school.  I will welcome discussions of the difference between speed and velocity, and I'll certainly demand that my students provide units for their answers.

Discussing the Solution

10 minutes

After 20 minutes, I expect most of the class to have completed the majority of the first problem, so this is a good time for a discussion of the solution.

Before beginning the conversation, I'll ask a student to write his or her equation on the board.  If students have used both sine and cosine functions, I'll be sure to have both on the board.  I'll also have GeoGebra handy to display a graph of the function for all to see.

To begin the conversation, I'll simply call on a student to explain the different parts of the equation.  "Why should we add 10.2 to the cosine function?  What does 10.2 correspond to in the context of the problem?"  Similarly, I'll call on students to explain the other constants.  Next, we'll confirm our answers to parts (b) and (c).

Once everyone agrees on the first time the mass is 10 cm above the table, I'll ask, "When will it next be that high?"  This should prompt a brief discussion of the periodicity of the function and the limitations of the inverse function.

Finally, I'll ask for a volunteer to explain how we can calculate average speed.  The main point to get across is that we use the slope between two nearby points.  Of course, "nearby" is a relative term, and it will be useful to discuss how the average speed would change depending on which two points are chosen.

Working Together...continued

15 minutes

At this point, I'd like all students to move on to the second problem (they can revisit the first one later, if necessary).  Before they leave class, I want to be sure that everyone has created a model and begun working with it to some extent.

Again, the key to beginning the modeling process is to recognize the periodicity of the function and use the given information to identify its period, amplitude, and midline.  In this case, students will have to choose a starting point for the function and call it t = 0.  This choice will determine or be determined by their use of sine or cosine. (MP 4)  

Note: Different groups may come up with slightly different models due to discrepancies in the given information.  The tidal data for Joggins Wharf was obtained from this site.  In my answer key, I've used a sine function based on a period of 25.4167 hours.

I expect some students to have trouble converting the times given in hours:minutes:seconds to a decimal number of hours.  This is a healthy challenge, however, that they should all be able to overcome with a little perseverance! (MP 1)

When students are asked to compute a rate of change this time, the time interval is not specified.  What's really asked for is an approximate instantaneous rate of change, so the students will need to choose two "nearby" points.  Of course, "nearby" is a relative term, so I'll expect slightly different rates from the different groups.

Wrapping Up

5 minutes

Students will continue working until the moment class ends.  During the last 5 minutes, I'll make a note of just how much progress each group has made.  This will serve as a starting point for the next lesson, since tonight's homework is to spend about 20 minutes making some progress individually.  It is a good idea to have a quick discussion with each group focused on these two questions: How far have you come today, and what goal should we set for your homework?  This will create some explicit accountability for all students.  At the beginning of the next lesson, I'll be asking the groups to reconvene to share their insights.