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# Modeling the Speed of Tsunamis with Square Roots

Lesson 1 of 8

## Objective: Students will be able to write find the speed of a tsunami given a square root function.

*50 minutes*

#### Introduction to Tsunamis

*20 min*

This lesson was inspired by a wonderful lesson “Faster than Speeding Tsunamii” from the Science & Math Investigative Learning Experiences Program (SMILE) based out of Oregon State University. I am using this lesson as a one day modeling activity to provide some perspective on how square roots are useful in real life.

I begin by asking the students to describe what they know about tsunamis. I then share with them that I had a friend who was in Phuket, Thailand during the 2004 tsunami that killed as many as 230,000 people. No one knew if she was alright for several days. Thankfully, she had been up at her hotel which just happened to be on a hill when the tsunami hit and wasn't injured. I then show them a clip from that tsunami (not taken by my friend).

We then discuss how tsunamis are formed. The lesson plan from SMILE is an excellent resource for this information. Detailed presentation notes are located in the PowerPoint.

Setting the tone for problem based learning, I ask students to think about what information people might need to know when predicting or learning from tsunamis. I guide the discussion to where mathematical models are very important. This leads me to share that the speed a tsunami travels can be modeled by the function *S(d) = √9.8d*, where *d* is the depth of the water in meters, *S(d)* is the speed of the tsunami in meters per second, and *9.8* represents the acceleration of the water due to gravity in *m/s.*

At this point, I pass out the Tsuami Student Handout which contains a graph of the function. I have them describe to their partner what they can determine about tsunamis according to this graph. We then share as a class. I ask them to identify the speed of the tsunami at 2000 meters using the graph and then prove it solving algebraically. I then ask them to do the same thing for a speed of 100 meters per second.

Given the authenticity of a real-world application we can have an insightful class discussion about the function’s real-world domain and range. The deepest place in the ocean is the Mariana Trench at 10,911 meters deep. I don’t bring this up but wait for it to come up in the discussion.

SMILE has graciously given me permission to use their lesson.

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#### Making a Plan

*30 min*

Now I introduce the problem.

*Information has just come in via satellite that a tsunami has been located off Hilo, Hawaii at an ocean depth of 4,500 m. *

*National authorities are trying to establish how much preparation time these cities will have: *

*• San Diego, California *

*• Newport, Oregon *

*• Kodiak, Alaska *

I have my students team up in groups of four. Their goal is to make a plan to solve this problem. I have given them no information other than the assignment and a map. My aim is for them to make their own plan as well as decide what information they need (**Math Practices 1 and 5**). I have them determine:

*1. What I want to know. *

*2. My plan to make that happen. *

*3. Information needed. *

Once they have had an adequate time to talk, we share as a class. This activity and the subsequent discussion are the highlight of the lesson. Students need to be able to take a problem and solve it without being guided step by step. The discussion provides scaffolding for those groups that are less confident as well as helps all of the students identify any missing portions of their plan.

I have one group volunteer to share their plan and needed information which I record on the white board. I then ask if any other group has a different plan or portion that wasn't included by the first group. This should bring up any holes in the first plan. If something is not brought up, like converting meters into kilometers for example, then I will ask guiding questions of the entire class. For example, I may say " What would be the most useful way to present this information?"

Once we have a full plan, I address the needed information on the board. The only information regarding the distance between the cities is their longitude and latitude.

Kodiak, Alaska: 57.7931° N, 152.3942° W

Newport, Oregon: 44.6044° N, 124.0547° W

San Diego, California: 32.7150° N, 117.1625° W

Hilo, Hawaii: 19.7056° N, 155.0858° W

1^{o} latitude ≈ 110 kilometers

1^{o} longitude ≈ 70 kilometers

Any time remaining in class will be spent working on the assignment. Each student is asked to identify how long it will take the tsunami to reach each location and then write a complete explanation of each step they used to identify the solution (**Math Practice 3**). The remainder of the work is to be completed on their own time. I generally give several days before it is due.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Modeling the Speed of Tsunamis with Square Roots
- LESSON 2: Transformations on Quadratic Functions Day 1 of 2
- LESSON 3: Transformation of Quadratic Functions Day 2 of 2
- LESSON 4: Square Root Functions
- LESSON 5: Radical Equations Day 1
- LESSON 6: Radical Equations Day 2
- LESSON 7: Radical Functions and Equations Review
- LESSON 8: Radical Function Test