# Frequency versus Period

## Objective

SWBAT determine the difference between the period and the frequency of a periodic function.

#### Big Idea

I know the frequency how do I determine the period for a function?

## Bell Work

10 minutes

To start the day I display several graphs that appear to be periodic. I ask my students to determine if they are periodic, and if so, find the period.  I allow my students to discuss the graphs with each other and then share answers with the class.

Once we discuss the periodicity of the graphs I will ask some followup questions:

• Which of these functions are sinusoidal?  (I expect some students will say that the first one is not, but the other two are sinusoidal.)
• Isn't a sinusoidal graph a periodic?
• Can a function be sinusoidal and not be periodic?

Before moving on, I expect to spend some time talking about the graph on the right side of the display. Some of my students will argue that is not a sinusoidal.  If so, I will ask:

1. Could we use sine or cosine to model the equation?
2. Does a mathematical model have to match the data perfectly?

The responses to the second question will give me some important insights. Some students usually say it has to match exactly. This misconception often results from seeing too many data sets that have a really good fit. I plan to remind my students that:

• models are used to predict what will happen
• models can sometimes result in poor predictions
• models are a good fit as long as they are helpful (better than working without a model)

A great example is the weather. Many models are put together to determine what will happen each day. The weather prediction changes continually, and, it can sometimes be very wrong. In my community, students are aware of this because we often have a forecast of snow and nothing happens or vice-versa.

## Frequency

20 minutes

As we begin to move on from the Bell Work, I ask my students to get out Sinusoidal Definitions, the resource we have used in our previous lessons. As they do, I rewrite the definition of period that we used yesterday on the board. Then, I ask my students to discuss two questions:

• What does it mean to describe the frequency of a periodic function?
• What is the difference between period of a function and the frequency of a function?

At first, my students will struggle with the difference, with respect to describing a function.  Today, I want my students to think about the units that would be used to describe each quality of a function.

Frequency describes the #of cycles/unit of time

Period measures the unit of time/one complete cycle

This will help them determine a method for using the frequency to write models for functions. After students have discussed the differences with each other, I will ask groups to share their ideas with the class. As students share, I will question students until we see how the ratios for period and frequency are different.

I have prepared a Frequency worksheet to help students understand the difference between period and frequency: Students will use Desmos as they work to write a rule for finding the coefficient on of x in a periodic function. Students work on page 1 of the resource for about 15 minutes.  We then come back as a class to discuss the questions on page 1. I have students go to the board to share their results from question 1. We then spend time on question 2-5. Students explain their reasoning as we go through the questions. I will restate important ideas to make sure the students are understanding.

## Closure

10 minutes

During the final minutes of the lesson, I will make sure that all of my students are working on questions 6 and 7 from the Frequency worksheet. As students work I move around and check their student work. Then, with about five minutes left, I will give my students an equation. I'll ask my students to work in groups to determine the amplitude, vertical shift, horizontal shift, period, frequency and midline. I will collect one answer sheet from each group as an Exit Slip. The exit slips allow me to identify misunderstandings that need to be addressed.