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# Modeling with Periodic Functions

Lesson 5 of 8

## Objective: SWBAT model periodic phenomena with trigonometric functions.

## Big Idea: The sine and cosine functions are flexible tools for modeling a wide array of periodic phenomena.

*47 minutes*

#### Jumping Right In

*2 min*

The last few lessons have seen me at the front of the classroom quite a bit more than I like. Today will be different.

As class begin I will simply hand out Sinusoidal Functions with the explanation that today they're on their own. They have all the tools they need - radian measure, the unit circle, the sine and cosine functions - and I'm confident that they can solve these problems.

As usual, they will begin by working on their own so that I can do some quick observations as a formative assessment.

#### Resources

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#### Working Individually

*10 min*

The class will be pretty quiet for the first 10 minutes as the students grapple with these problems on their own. I will move around the room checking for understanding and briefly stopping to work with individual students. In 10 minutes, I expect most students to complete the first problem and most of the second.

To understand the purpose of these problems, please see this short video.

#### Resources

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Once most students have completed (or nearly completed) the first two problems, I'll allow them to begin working together on Sinusoidal Functions in small groups. They should begin by checking their work up to this point and resolving any differences. I'll be available to help explain any mistakes that the students can't figure out on their own, and I'll be ready to point out any that they have not noticed (**MP3**).

Once they're confident that the solutions to the first two problems are correct, they should move on to solve the remaining three. Note that although students are not explicitly instructed to find the amplitude, period, and midline for the 3rd function, these three quantities will help them to construct the graph more efficiently.

For the modeling problems, the students will need to interpret the given situation to determine the amplitude, frequency, and midline. Then they will use these to create both an equation and a graph. Be careful how much "help" you provide here - it's easy to say too much and wind up doing all the thinking for the students! I will not move beyond the hint: "Can you tell from the given information what the amplitude, period, and midline are?" (**MP 2**).

The final question is the most challenging because students must use the inverse sine function to solve for multiple values of* t*. To make matters worse, they must use the symmetry of the function to identify other values of *t* that will yeild the same function value. A good one to practice perseverance on!

#### Resources

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#### Sharing Strategies

*5 min*

At the end of class, I will try to help student synthesize what they've learned by asking them to share some of their strategies for approaching these problems.

I might ask, "Have you found some strategies that might be helpful for your classmates? What part of the problem do you find it easier to do first? Which part is the most difficult?"

I'm never sure what they'll come up with, but I expect things like these:

- I like to draw horizontal lines for the midline, maximum, and minimum.
- I like to draw vertical lines showing where the period will begin & end.
- I like to graph the roots first, then the max/min points, then connect the dots.
- I find it helps to remember that (period)*(frequency) = 2*pi.

We'll end class with this conversation and the assurance that we'll discuss the solution to the more challenging final question tomorrow.

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*Responding to Jacob Nazeck*

Thanks for your quick response. I had not noticed that omega in addition of being called angular velocity, is also called angular frequency. It is more descriptive to the sine curve than just regular frequency. But I do think we need to make it clear to the students that it is angular frequency versus just frequency. I will make sure the students know the difference units of the two values.

Good job on the lessons and thanks for sharing them. The lessons fit very well with the common core. This is the first year of my school implementing common core curriculum fully in Algebra II. We usually can not get to trig unit. Now that we can skip the linear, systems and matrix, we will be here. I think I will borrow a lot of your ideas. I am worried about how much time I will spend on getting special angle trig value for all quadrants. I also teach PreCalc. It still take the class a lot of time to go over that. And still produce poor result on the test.

| 2 years ago | Reply

Jun,

Thanks for the feedback! I think that we're dealing with a difference of definitions here. In terms of ordinary frequency, you're absolutely right that period & frequency are reciprocals. In physics, for instance, we might say that a wave has a period of 3 seconds per cycle, and therefore a frequency of 1/3 cycles per second. Clearly, 3/1 * 1/3 = 1.

But suppose we are thinking only mathematically, and we consider the function y = sin(3x). This function completes one cycle in (2/3)pi radians, so its period is (2/3)pi. But what is it's frequency? What unit do we use?

My first instinct would be to ask, "How many cycles are completed in 2pi radians?" In this case, then, our function would complete (2pi) divided by (2/3)pi cycles, or exactly 3 cycles in 2pi radians. That would mean that in this case period*frequency = 2pi. It's my understanding that this is the ordinary way to consider frequency in mathematics (see this" target="_blank" >http://mathworld.wolfram.com/Sinusoid.html">this entry at Wolfram MathWorld for a reference).

On the other hand, you might reply that my unit is really *1* radian, not 2pi radians (which is certainly true). In this case, the frequency of my function would be 1 divided by (2/3)pi, or 3/(2pi). This is clearly the reciprocal of the period, so in this case period*frequency = 1.

So which is it? It looks like it can be both, if" target="_blank" >http://en.wikipedia.org/wiki/Angular_frequency">if Wikipedia can be trusted. In my classes, I'll continue to use 2pi as the baseline for calculating frequency. That said, I sure hope that one of my students asks me exactly the same question you did - what an awesome discussion that would be!

Thanks,

Jake

| 2 years ago | Reply

I like your lessons a lot. But period and frequency is reciprocal of each other. They time together is 1 not 2pi.

| 2 years ago | Reply*expand comments*

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
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- UNIT 10: End of the Year

- LESSON 1: What do Triangles have to do with Circles?
- LESSON 2: Introducing: Radians!
- LESSON 3: The Trigonometric Functions
- LESSON 4: Generalizing the Sine Function
- LESSON 5: Modeling with Periodic Functions
- LESSON 6: Practicing with Sine and Cosine
- LESSON 7: More Modeling with Periodic Functions
- LESSON 8: The Tangent Function