Tsunami+and+Sine+Waves

** Introduction: **
Students will use their knowledge of sine waves to explore connections to the world such as tsunami waves. Students will analyze the tsunami wave at different times as it approaches shore. They will use a program called Logger Pro to translate the wave to a sine equation.

Teacher Candidate:
Erica Carlson, Kathryn Clow, Megan McDermott, Nina Ratliff

Date & Time:
February 24, 2011 at 11:00 AM

School District & Cooperating Teachers:
Dr. Mark Oursland

Grade Level & Subject:
High School Junior, Pre- Calculus

Algebra 9-12
· Select and justify functions and equation s to model and solve problems. (EALR M2.1.A)

Geometry, Algebra 9-12
· Construct new functions using the transformations f(x ? h), f(x) + k, cf(x), and by adding and subtracting functions, and describe the effect on the original graph(s). (EALR M3.2.E)

· Understand the difference between sine and cosine waves
· Use correct math terminology to describe sine and cosine waves · Be able to graph and manipulate sine and cosine equations · Apply sine and cosine waves to real world situations

Assessment Strategies:

 * 1) Students will have a worksheet to fill out during lecture to help them in activity .
 * 2) Throughout the lesson, students may come up to the board to answer examples, or they can answer questions to check for understanding.
 * 3) Students will be handed an activity worksheet to guide them through the activity. At the end of class, groups or individuals will each explain different parts of their findings to the class on the document camera.

Grouping of Students for Instruction:
Students will be in groups of 3 or 4. They may also work individually depending on the number of computers.

Instructional Materials, Equipment, and Technology Needed:

 * Guided Notes worksheet
 * [[file:Guided Notes.docx]]
 * Computers with Logger Pro installed
 * Activity Worksheet for Logger Pro
 * <span style="font-family: 'Cambria','serif';">﻿﻿[[file:Tsunami and Sine Waves WS.docx]]
 * <span style="font-family: 'Cambria','serif';">Tsunami Wave Pictures
 * [|Wave_1.png]
 * [|Wave_2.png]
 * [|Wave_3.png]
 * [|Wave_4.png]
 * <span style="font-family: 'Cambria','serif';">Power Point Slide Show
 * [[file:Tsunamis and Sine Waves.pptx]]

Prior Knowledge and Skills:
<span style="font-family: 'Cambria','serif'; margin: 0in 0in 0pt;">Students should know:
 * <span style="font-family: 'Cambria','serif';">The basic knowledge of the shape of a sine and cosine wave
 * <span style="font-family: 'Cambria','serif';">The terms amplitude, period, crest, trough, phase shift, and vertical shift
 * <span style="font-family: 'Cambria','serif';">The equations for sine and cosine waves.
 * <span style="font-family: 'Cambria','serif';">How to translate from a sine graph to an equation and an equation to a sine graph

New Knowledge:
Students will learn about:
 * <span style="font-family: 'Cambria','serif';">Tsunami waves
 * <span style="font-family: 'Cambria','serif';">Connecting sine waves to real life situations

Learning Experiences :

 * Introduction:
 * Hello, we are going to be your teachers today. We are going to review sine and cosine waves as well as connect them to the real world. Today’s real life situation is tsunami waves. For those who do not know what a tsunami wave is, here is a video.
 * Play tsunami movie ( [] ) Click on Anatomy.
 * “Does anyone remember the 2004 tsunami in the Indian Ocean? It was one of the most destructive tsunamis in history. These pictures illustrate in the power point show the impacts of tsunamis. Since this deadly tsunami, mathematicians have been modeling tsunamis to make more accurate predictions on when they occur. This research’s goal is to save more lives by giving people more advanced warning to when they will strike."
 * “Researchers model tsunami waves by representing them as sine waves”
 * [sine wave on the board]
 * “The period is usually called the wavelength. And the height is twice the amplitude, or the distance from crest to trough”
 * [label on sine diagram]
 * “Does anyone know what are some differences between a normal wind-generated wave vs. Tsunami wave?”
 * [Let students try to answer, then show wind –generated waves diagram on slide 4]
 * “Speed is one difference; a normal wave travels about 10-20 mph whereas a Tsunami is 450-650 mph in deep ocean. They also differ in wavelength. Regular waves have wavelengths of 300-600 feet, but a Tsunami has an extremely long wavelength from 60-300 miles”
 * “As you can see in this diagram, there is a difference between tsunamis in the deep ocean and tsunamis as they approach the coast”
 * [show diagram on slide 5]
 * “It slows down as it approaches the shore, to about 30-200 mph”
 * “The wavelength (period) decreases, to less than 10,000 feet”
 * [draw graphs on the board of a tsunami in deep water vs. shallow water and label the wavelengths]
 * “On the other hand the height increases. In deep waters a tsunami is usually no more than 3 feet in height and is usually undetectable to ships. However, the height increases dramatically as it approaches the shore. Tsunamis can be hundreds of feet tall”
 * [show differences in height on graphs]
 * “Before we analyze tsunami waves on Logger Pro, let’s make sure we remember the ins and outs of a trigonometric wave.” Draw a sine wave up on the left board with an amplitude of 1 and a period of 2π. (This board will be our term board and be left up for the remainder of the class period so write neat and enough room for the rest of the terms.) “What kind of wave is this?”, then draw a cosine wave with a amplitude of 1 and a period of 2π and ask “Which wave is this?”. After getting the correct answer, say “ What is the difference between a sine and cosine graph?”.
 * “Since we have already studied these graphs before, can someone come up and show us where the amplitude is on the graph? So it the height from the top of the wave to the middle of the wave. Can someone else come up and show us where the period is? The period is the length of one interval. Can a third person show is where the crest of the wave is? How about the trough? Can someone show us the trough? Thank you for those who came up.”
 * “Now students, we are going to talk about phase and vertical shifts. Remember phase shifts are also called horizontal shifts. So if I draw a graph like this, (draw a graph on the right board that has a vertical shift of π) then what kind of shift did I do? What about this graph? (Draw a graph under the vertical shift graph with a phase shift of 1) Now what about this graph? (Draw a graph with a vertical shift of π/2 so they know the cosine graph is just the sine graph with a vertical shift) Can we combine shifts? (Draw a graph with a vertical shift of π and a phase shift of 1) So what would the equations of these graphs look like? (Write up the general formula of and go through what each variable means) The //a// is the amplitude, //b// is the variable that determines how long the wave interval is, //k// is the vertical shift, and //h// is the phase shift. The cosine equation has the same variables. Let’s write the equations for the four graphs we just drew. There is one more equation I want us to remember which is the equation of the period. If you remember correctly, the equation for the period is where b is the same as in the equation. For example, our graph has b=1 so our period is 2π (Show this calculation on left board with equation).
 * “Now we are going to show a simulation of tsunami waves in Logger Pro”
 * [open up Logger Pro]
 * Logger Pro
 * Show the full video of tsunami wave.
 * Show wave 1. Describe what the auto fit tells us about the graph. Notice the amplitude of wave.
 * Open wave 4. Plot the points on the picture. Use the curve fit application. Notice the difference between this wave’s amplitude and wave 1’s amplitude. Have students discuss why this is true.
 * Activity
 * Worksheet is attached.
 * Each student will have their own worksheet to turn in, but they may work in groups.
 * Conclusion
 * Go over answers
 * Students come to board to show answers.

Extensions:
One option for an extension would be to assign a project where the students form groups and create their own video of a tsunami or tidal wave. Since tsunami waves are more of a challenge to create a simulation for, the students may create waves using different objects such as string or a slinky.

Another option would have the students find different real world situations that can be represented by a sine wave. This would help expand the idea that math expands to outside the classroom.

If the students just need more practice with writing equations or graphing sine waves, then this worksheet provides more examples.

Teacher Reflections:
Introducing the lesson with the video of a tsunami was a good way to hook the students and get them interested in the lesson. In order to ensure the students were truly engaged in the lesson after the movie, we could have asked the students to give their theories on how tsunamis relate to mathematics instead of telling them. However, the video itself did a good job in gaining their interest and introducing the topic of the lesson.

Since this lesson requires computers with Logger Pro installed, it may not be possible to have students use Logger Pro. The lesson can still be effective and the activity can still be engaging by completing the activity as a class. More pictures of the tsunami wave could be used and the students could individually make predictions about each wave as the class is working through the worksheet together.

Before the lesson, each student received guided notes that they could fill out definitions and graphs as the lesson was being taught. This gave the students something they could look back on during and after the lesson. As a whole, this lesson is engaging for students and helps them make the connections between trigonometry and the real world. Having a real world application helps trigonometry come to life for high school students who are still struggling with the mathematical concepts of trigonometry.