# Analyzing Formulas & How Variables Relate

## Speed versus Size

## Opening

# Speed versus Size

Vincent Strouhal, a physicist from Czechoslovakia, discovered a number that compares the speeds of flying animals with respect to their wing sizes. To calculate the Strouhal number, you multiply the wing flapping frequency by the distance between the tip of the wing at its highest and lowest points, and then divide the result by the animal’s cruising speed.

- How does the wing-flapping frequency of the hummingbird compare to that of the albatross?
- About how many wing flaps per second do you think the hummingbird and the albatross make when flying at cruising speed?

## Hummingbird and Albatross

## Opening

# Hummingbird and Albatross

The distance between the tip of the wing at its highest and lowest points is called the *amplitude*. The bird shown below has an amplitude of 2.5 meters.

- How do you think the amplitude of the hummingbird compares to the amplitude of the albatross?
- Do you think the Strouhal numbers for the hummingbird and the albatross are close to each other or far apart?

## Problem-Solving Process

## Opening

# Problem-Solving Process

During this unit, you will use the problem-solving process to solve the problems. Review the problem-solving process and discuss with your classmates.

Problem-Solving Process

- Understand the problem situation.
- What is the problem asking you to find out?
- What information is provided?
- What are the quantities that vary?
- How are the variables related?

- Represent the situation.
- Write a formula to show how the variables of the Strouhal number relate to each other.

- Answer the questions.
- Check that the mathematical answer makes sense.

## Math Mission

## Opening

Compare the relative speeds of the hummingbird and the albatross using Strouhal numbers.

## Find and Compare Strouhal Numbers

## Work Time

# Find and Compare Strouhal Numbers

To calculate the Strouhal number, you multiply the wing-flap frequency by the distance between the tip of the wing at its highest and lowest points, and then divide the result by the animal’s cruising speed.

Use the data in the table and the problem-solving process to answer these questions.

- What is the Strouhal number for the hummingbird?
- What is the Strouhal number for the albatross?
- How do these two numbers compare?

Begin by writing a formula for calculating the Strouhal number for any bird. To help you write this formula, ask yourself:

- What are the variables in the problem?
- How are the variables related?
- What information about Strouhal numbers is given? How can you use this information?
- What mathematical operations do you need to include in your formula?

## Prepare a Presentation

## Work Time

# Prepare a Presentation

Prepare a presentation on your findings for Strouhal numbers. Explain how the Strouhal numbers compare to each other. Support your answers with calculations, diagrams, and formulas to explain why your answers make sense.

# Challenge Problem

The Strouhal number for the Eurasian Kestrel is 0.24. The bird has an amplitude of 0.345 meters and a wing-flap frequency of 5.7 flaps per second.

- What is its cruising speed?

## Make Connections

## Performance Task

# Ways of Thinking: Make Connections

Take notes about classmates’ calculations and comparisons of the Strouhal numbers, and other students’ use of the problem-solving process.

As your classmates present, ask questions such as:

- What is the relationship between the variables in the problem? How did you use this relationship to help you write your formula?
- As the wing-flap frequency of a bird increases, what happens to the amplitude? Why?
- Do you think the Strouhal numbers for all birds compare in the same way that the numbers for the hummingbird and the albatross compare? Why or why not?
- Which steps of the problem-solving process were the most helpful when answering the questions?

## Reflect on Your Work

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Something I learned today that surprised me is…**