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# Making Waves and Determining Mathematical Relationships

Lesson 4 of 10

## Objective: Students will be able to determine and apply the mathematical equation that relates speed, frequency and wavelength.

*70 minutes*

#### Making Waves Activity

*55 min*

The goal of this lesson is to help students develop the wave equations that relate frequency and wavelength of a wave and the period and wavelength of a wave (SP2). Students create waves and focus on measuring the different aspects of a transverse wave and then using that information to engage in computational thinking (SP5). Students collect and analyze the data (SP4) from the waves they draw in the activity to reinforce the relationships that they have already found in the simulation activity during the previous lesson and combine those relationships to create the wave equation. These activities all lead towards mastery of NGSS performance expectation HS-PS4-1.

To start out class, I provide each group with one copy of the Making Waves Activity. I only give one copy so students must work together in order to complete this activity. Once each group has a copy, I discuss the purpose and the materials that they will be using for this activity and then I ask students to look at the roles that are on the paper. I ask each group member to select a role and write their name next to it. The roles include Puller (pull the paper at constant speed), Wavemaker (move the marker back and forth on the paper at a constant speed), and Timer. If there are 4 people I ask them to add the role of Data Presenter (write the data on whiteboard) so that each person has a role. I ask the Data Presenter (or the timer if there is a group of 3) to read the procedure to their group. While the Data Presenter is reading, I tell the others to listen to what their job is for their specific role of Puller, Wavemaker or Timer so that they can summarize what they will be doing as part of this activity.

After each group is done reading, I ask for a volunteer Puller and Timer to participate in the demonstration of the activity. I act as the Wavemaker and I ask the other Wavemakers to tell me what to do. They respond with moving the marker in a back and forth motion on the paper at a constant speed. Then I ask what the Puller will do and the response is to pull the paper at a constant speed when the Timer says go. Finally I ask what the Timer will do and they respond with say go when they wavemaker is moving the marker at a constant speed and to stop the timer when the wavemaker has reached the end of the paper. I also add that the Timer or the Data Presenter should help to guide the paper so it goes as much in a straight line as possible. After the demonstration, I tell students that the Data Presenter is be in charge of taking the data from the summary table and putting it up on the board for a discussion.

To make sure each group member is working even after they make their high and low frequency waves, I ask that each group member take a different colored marker to complete the different data questions that are asked in the packet. When I collect the packets to look over, I look through to see that there are equal amounts of work done in each color to show that everyone is working.

Before students move to the back, I ask them to make predictions on three different parts of the lab as a group. As shown in the Making Waves Teacher Notes, I ask students to consider what they think will happen with the amplitude, speed and wavelength as they look at low frequency versus high frequency. As a table group, students must decide on a prediction based on the information that they have learned from previous activities in class and their own prior knowledge.

After these instructions, I allow students to go back to their lab tables and get started. I give them 30 minutes to work as a group to complete all the steps of the activity including data collection and calculations. Each lab table has a 2-meter pieces of butcher paper, 2 meter sticks, a stopwatch and a box of markers. As students work on their waves, I walk around to make sure that there are two distinct waves (high and low frequency). The students waves come out like in the picture below. I suggest that groups open up their paper so they can see both waves and have two people work on each wave to collect the data that is needed for the activity in the packet.

As students work through collecting the data of amplitude, wavelength, speed, frequency and period for each wave, I walk around to make sure they are all on task and working diligently. Some groups make it through the tasks quickly and break up the work efficiently, while others take longer to get through the data collection section of the activity. As groups finish up, the Data Recorder summarizes the data in the summary table at the bottom of the Making Waves Activity from the data collection and calculations they have done.

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#### Wave Equation Discussion

*15 min*

After students have finished the Making Waves Activity, I ask one member of the group to put their summary data table up on the board. We use the class data to discuss the different relationships that the groups found based on their waves.

To start the discussion, I have the students focus on the frequency column and notice that not every group got the same frequency, which is okay since we all had different waves.

Then I ask students to look at the period column and see what they notice about the period at a high frequency vs. a period at a low frequency. Students notice that there is an inverse relationship which is something that we have already talked about in this unit but it will help students to develop formulas later in the discussion.

Next, we look at the speed column and I ask students to see if they notice any patterns with speed and frequency. As we look through each group's speeds, about half have a higher speed at low frequency and the other half have a higher speed at high frequency. I talk to them about the fact that speed is typically something that we talk about as a constant. For example, the speed of sound is 3.0 x 10^8 m/s and the speed of sound in the air is 340 m/s. So when we talk about speed of waves it does not have a direct or inverse relationship to frequency.

Finally, we look at wavelength which is the most important relationship that we look at in this activity. Students notice that the high frequency wave had a shorter wavelength, which means that frequency and wavelength are inversely proportional.

After we have discussed the data, I ask students to turn to the Wave Calculation Example in their packets. This is where I ask students to help me to develop the equations, as shown in the video below.

After we develop the equations, I run through solving the example problem, as shown below. I make sure to point out that you can use either of the two speed equations to solve for speed since you have wavelength, frequency and period.

#### Resources

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After the example problem during the discussion, I ask students to complete problems #1-5 (the front page) of the Wave Equation WS. I want them to work on some of the problems on their own for homework to see how they do with the new equations. This worksheet is meant to be a good assortment of problems that require students to solve for all of the different quantities in at least one problem. Students complete the remaining problems in class with their groups in the next class.

#### Resources

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- LESSON 1: Period and Frequency of a Pendulum
- LESSON 2: Periodic Motion and Waves
- LESSON 3: Wave Relationships
- LESSON 4: Making Waves and Determining Mathematical Relationships
- LESSON 5: Using the Wave Equation to Solve Problems
- LESSON 6: Be the Expert: Wave Phenomena Jigsaw, Day 1
- LESSON 7: Be the Expert: Wave Phenomena Jigsaw, Day 2
- LESSON 8: Wave Lab Stations, Day 1
- LESSON 9: Wave Lab Stations, Day 2
- LESSON 10: Waves and Wave Phenomena Test