Lesson Can You Take the Pressure?

Quick Look

Grade Level: 8 (7-9)

Time Required: 45 minutes

Lesson Dependency: None

A photograph shows a hot air balloon in the blue sky.
Hot air balloons utilize pressure concepts in order to stay in flight.
copyright
Copyright © 2008 Jane82, Wikimedia Commons https://commons.wikimedia.org/wiki/File:Ballooning_trip_Morocco.JPG

Summary

Students are introduced to the concept of air pressure and use the associated activity to see how it is affected by different variables. They explore how air pressure creates force on an object. They study the relationship between air pressure and the velocity of moving air.

Engineering Connection

In designing airplanes, trains, cars, rockets and bicycles — nearly everything that moves through the air — engineers must understand Bernoulli's principle. The forces caused by moving air enable an airplane to fly or a train to slow. Understanding how Bernoulli's principle works enables engineers to take advantage of the nature of air pressure so their designs of these and many other applications function correctly, efficiently and safely.

Learning Objectives

After this lesson, students should be able to:

  • Describe basic facts about the history of flight.
  • Explain Bernoulli's principle.
  • Describe the concept of air pressure and how the movement of air influences its pressure.
  • Explain why engineers need to know about air pressure.

Introduction/Motivation

What is air? (Listen to student answers.) Air is a collection of molecules—very small particles that we cannot see—that cover the surface of the Earth. Air is comprised of 78% nitrogen, 21% oxygen, and the remaining 1% is mostly argon and carbon dioxide.

Blow onto your hand. What is happening? (Listen to student answers.) You are forcing air out of your lungs and it is hitting your hands.

Why doesn't air around the planet escape to space? (Listen to student answers.) Gravity pulls on air just like it pulls on a person and all objects. The weight of air is called air pressure and it pushes on us all the time. The existence of air pressure is what enables airplanes to fly.

Lesson Background and Concepts for Teachers

A Quick History of Human Flight

From the earliest times, people have marveled at the flight of birds and have wanted to fly as they do. According to Greek myth, Icarus flew too close to the sun on wings made of wax and he fell to his death. Many thousands of years later, it took a series of engineers to make this human dream of flight become a reality.

In his lifetime, Leonardo da Vinci (1452-1519), like other inventors of his time, had a fascination with the flapping flight of birds. He believed that "a bird is an instrument working according to a mathematical law. It lies within the power of humans to make this instrument with all of its motions." He designed several ornithopters, which are airplanes with flapping wings. He also designed a glider that resembles modern day hang-gliders.

In 1738, an Italian physicist named Daniel Bernoulli discovered that water and air move faster over curved surfaces than they do over flat ones. He also discovered that as the air or water moved faster, the pressure exerted by the fluid decreased. This critical discovery, known as Bernoulli's principle, paved the way for modern airplanes.

In 1903, a pair of bike shop owners from Ohio made the world's first powered flight at Kitty Hawk in North Carolina. Orville and Wilbur Wright had studied prior attempts to fly and realized that birds keep their balance by twisting their wings. In 1895, Wilbur designed a kite that reproduced this effect mechanically. Then, for several years after his kite invention, the two brothers experimented with flight using gliders. Wilbur also discovered the principle of combining rudder control with roll for smoother turns.

What Is Air Pressure?

Our atmosphere could be described as a sea of air that is about 80 kilometers thick surrounding the planet. All of that air piled on top of us is pulled down by gravity (see Figure 1). In fact, the air pushes against us in all directions. This is air pressure. Refer to the Air Pressure activity to have students learn more about the concept

A cartoon shows a walking boy with arrows pointing at him from all directions to represent air pressure pushing on him.
Figure 1. Air pressure acting on a person.
copyright
Copyright © 2003 Geoffrey Hill, College of Engineering, University of Colorado Boulder, using Microsoft clipart

How Does Air Pressure Work?

Air is pushing on all people and objects with a force of 14.7 pounds per square inch at sea level. (Disclaimer: Using the metric system, pressure is measured in newtons per square meter, which is called a pascal. With younger kids, it is easy to use pounds per square inch since the units are more familiar and the numbers are more reasonable. To convert pounds per square inch into pascals, multiply by 6,895.) As you go higher in elevation, less air is pushing down, so the air pressure is less. The change is not that great until you get very high. Heating the air, cooling it, or causing it to move can change the air's pressure. These differences in air pressure create wind, and make flight possible.

What Is Bernoulli's Principle?

Bernoulli discovered that the faster water moves over a surface, the lower the pressure the water exerts on that surface. In Figure 2, water flowing out of a bottle moves faster through the narrow neck than it does in the rest of the bottle. Students can practice this principle with the fun associated activity, Fun with Bernoulli. Bernoulli proved that the water pressure on the sides of the neck of the bottle were less than those on the sides of the rest of the bottle.

A diagram shows a fluid leaving a bottle. The fluid is moving faster at the narrow neck of the bottle than in the rest of the bottle.
Figure 2. Fluid moving out of a bottle.
copyright
Copyright © 2003 Tom Rutkowski, College of Engineering, University of Colorado Boulder

Bernoulli then found that these same principles apply to air as well. Bernoulli's principle simply states that the faster a fluid moves over a surface, the less it pushes on the surface (see Figure 3).

A three-part diagram shows three surfaces (represented by horizontal lines) with arrows of different sizes pointing down on and up at those surfaces (which might represent airplane wings). On the first surface, with no air movement, the air pressure is uniform over the entire surface. The second surface has air moving over the top of it, and thus the air pressure is less on the top of the surface than on the bottom. The air is moving even faster over the third surface, which means the air pressure on that surface is even lower.
Figure 3. The influence of air movement on air pressure.
copyright
Copyright © 2003 Geoffrey Hill, College of Engineering, University of Colorado Boulder

Engineers and the Bernoulli Princple

Engineers must fully understanding the concept represented by Bernoulli's principle when designing and testing airplane wings. They use this information to determine the amount of lift a wing will produce (how high an airplane can go). Engineers also use their knowledge of these principles for other projects relating to fluid flow. We will learn more about how Bernoulli's principle is related to flight in Lesson 2 of the Airplanes unit.

Lesson Closure

Ask students to explain air pressure. Then have them explain why one of the Bernoulli's principle activities they did worked in terms of air pressure. Make sure they understand Bernoulli's principle (as fluid velocity increases the pressure decreases).

Bonus question: What is a vacuum? (Answer: A vacuum is a volume with no air in it and therefore no air pressure.)

Vocabulary/Definitions

air pressure: The force exerted by air on an area. Units: metric is newtons per square meter; English is pounds per square inch.

Bernoulli's principle: A natural physical law that pressure exerted by a fluid decreases as the rate of flow increases.

fluid: Any substance in which the molecules can move around freely. A fluid is any gas or liquid, such as air and water.

force: A push or pull on an object.

lift: The force resulting from the difference in air pressure above and below the wings that causes aircraft to move upwards.

Assessment

Pre-Lesson Assessment

Discussion Questions: Solicit, integrate, and summarize student responses.

  • What is air? (Answer: Air is a collection of molecules—very small particles that we cannot see—that cover the surface of the Earth. Air is comprised of 78% nitrogen, 21% oxygen, and the remaining 1% is mostly argon and carbon dioxide.)
  • Why doesn't air escape into space? (Answer: Gravity)
  • Why don't you feel the air pressure? (Answer: Your body pushes out against the air to balance the air pressure.)

Post-Introduction Assessment

Voting: Ask a true/false question and have students vote by holding thumbs up for true and thumbs down for false. Tally the votes and write the totals on the board. Give the right answer.

  • True or False: Air pressure is caused by the rotation of the Earth. (Answer: False. Air pressure is caused by gravity pulling air towards the Earth.)
  • True or False: The faster air moves, the higher the pressure. (Answer: False. Pressure decreases as air velocity increases.)
  • True of False: Air pressure pushes in all directions. (Answer: True. Air pressure not only pushes down on us, but it also pushes from the sides and even from below.)

Lesson Summary Assessment

Bingo: Provide each student with a sheet of paper containing a list of the lesson vocabulary terms. Have each student walk around the room and find a student who can define one vocabulary term. Students must find a different student for each word. When a student has all terms completed s/he shouts "Bingo!" Continue until two or three students have bingo. Ask the students who shouted "Bingo!" to give definitions of the terms.

Lesson Extension Activities

Have students conduct library or Internet research to learn more about the Bernoulli principle and how it figures into everyday products and inventions.

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References

Hixson, B. K. Bernoulli's Book. Salt Lake City, UT: Wild Goose Company, 1991.

Physics Zone: Lessons on Forces. Science Joy Wagon. Accessed 2009. http://www.sciencejoywagon.com/physicszone/02forces/

Copyright

© 2004 by Regents of the University of Colorado

Contributors

Tom Rutkowski; Alex Conner; Geoffrey Hill; Malinda Schaefer Zarske; Janet Yowell

Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

Acknowledgements

The contents of this digital library curriculum were developed under grants from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and National Science Foundation (GK-12 grant no. 0338326). However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: May 28, 2019

Hands-on Activity Fun with Bernoulli

Quick Look

Grade Level: 6 (5-7)

Time Required: 30 minutes

Expendable Cost/Group: US $0.50

Group Size: 1

Activity Dependency: None

Photograph shows the take-off of a NASA rocket (space shuttle) beginning its launch to space.
Understanding air pressure is needed to successfully launch rockets.
copyright
Copyright © Pixabay http://pixabay.com/en/rocket-launch-rocket-take-off-nasa-67649/

Summary

While we know air exists around us all the time, we usually do not notice the air pressure. During this activity, students use Bernoulli's principle to manipulate air pressure so its influence can be seen on the objects around us.

Engineering Connection

Because they understand Bernoulli's principle, engineers manipulate air pressure in their designs to control and stabilize everything from rockets to helicopters to blimps. When designing airplane wings, engineers shape them so that they create lift. Even cars and trains are designed to take advantage of this principle, helping moving vehicles to stay on the ground at high speeds.

Learning Objectives

After this activity, students should be able to:

  • Explain that air pressure decreases as the speed of air increases.
  • Explain that air pressure acts in all directions (not just down).
  • Explain that engineers use their understanding of pressure differences to make airplanes fly.

Materials List

Each student needs:

  • 1 sheet of paper (new or recycled)
  • 2 round balloons
  • 2 pieces of string (18 inches long)
  • 2 small plastic cups
  • 2 straws
  • 1 ping pong ball
  • water
  • Fun with Bernoullil Worksheet

Worksheets and Attachments

Visit [www.teachengineering.org/curriculum/print/cub_airplanes_lesson01] to print or download.

Introduction/Motivation

When talking about baseball, why does a curveball curve? Why does an airplane fly? The reasons can be found in Bernoulli's principle, which states that the faster a fluid moves the less pressure it exerts. Different air velocities are present on different parts of a curveball as well as on the different parts of an airplane. Bernoulli's principle tells us that these differences in velocity mean differences in pressure exist as well. On a curveball, the difference in pressure causes the ball to move sideways. Engineers use their understanding of pressure differences to make airplanes fly.

For a system with little change in height, Bernoulli's equation can be written:

P + (v2/2g) = constant

Where P is the pressure, v is the velocity and g is gravity. Because this equation is always constant for a system, if the velocity increases, the pressure must decrease!

Procedure

Before the Activity

Gather materials and make copies of the Fun with Bernoulli Worksheet.

With the Students

Hand out the worksheets.

Part A: The Paper Tent

  1. Have students fold a piece of paper (lengthwise) in half and make a paper tent.
  2. Ask students to predict what will happen when they blow into the tent. Will it appear to get larger, will it remain unchanged, or will it bend down toward the table? (Alternately, have students turn their paper tents upside down and blow through the V-shaped paper.)
  3. Make sure students notice that the tent flattens.This is because the air moving through the inverted V has less pressure, so the higher pressure on the outside of the paper tent flattens the paper.
  4. Have students experiment with their paper tents, answer the relevant worksheet questions, and discuss their results.

Part B: Moving Balloons

  1. Blow up two balloons. Tie them off, and then attach a string to each one.
  2. Have students hold the two balloons together.
  3. Ask them to predict what will happen when they blow between the two balloons. Have students record their hypotheses in the space provided on the worksheet.
  4. Have students hold the balloons 4-6 inches apart and blow between them. If they hold the balloons too close together, the balloons simply move away from the student. The balloons must be sufficiently far apart so that students can blow between the balloons, not at the balloons.
  5. Expect students to see the balloons come together just like the paper V in Part A of the Procedures section.
  6. Have students complete the worksheet and discuss the results.

Part C: Magic Moving Ball

  1. Place two plastic cups about 6 inches apart.
  2. Place a ping pong ball in one of the cups.
  3. Ask the students to predict how to get the ball from one cup to the other without touching either the ball or cup.
  4. Have the students try a few of their ideas.
  5. Tell the students to gently blow across the top of the cup with the ball in it.
  6. The ball should jump from one cup to the next. This is because the air pressure moving across the top of the cup is less than the pressure inside the cup. The higher pressure inside the cup forces the ping pong ball to jump out of the cup.
  7. Have the students experiment with how far apart they can place the cups and still get the ping pong ball to jump from one to the other.

Part D: Bernoulli's Water Gun

  1. Give the students one cup filled with water and two straws.
  2. Have students place one of the straws in the water.
  3. Then, have students cut the second straw in half to use as a "blower."
  4. Ask the students to predict what will happen if they blow across the top of one straw in the water with the other straw.
  5. Have students blow across the top of the straw with the other straw.
  6. Expect the water to rise up in the first straw and blow across the table. This happens because the air blowing across the straw in the cup reduces the air pressure at that point. The normal pressure underneath pulls the water up the straw and the moving air blows the water out and across the room.
  7. Have students experiment with different straw lengths as the "blower."

Assessment

Pre-Activity Assessment

Discussion: Solicit, integrate and summarize student responses.

  • Review with students the Bernoulli principle. Make sure everyone understands the concept. (The faster a fluid moves the less pressure it exerts.)

Activity Embedded Assessment

Worksheet: Have students record measurements and follow along with the activity on their worksheets. After students have finished the worksheet, have them compare answers with their peers. Discuss as a class.

Post-Activity Assessment

Class Discussion: Have students engage in open discussion to suggest solutions to the following problem:

  • Given what we have learned, how does the Bernoulli principle relate to airplane flight? (Answer: If air moves faster on one side of an object, the air pressure decreases and the object will move in the direction of the faster moving air. This is how wings create lift and why the objects in this experiment move in the direction of the faster air.)

Troubleshooting Tips

  • In advance, cut the string pieces to speed up the activity.
  • Have a plan for the balloons after the activity is complete; otherwise, leaving the balloons with the students quickly becomes a distraction.

Activity Extensions

Have students search for "Bernoulli principle" on the Internet to find an online demonstration of how the Bernoulli principle works. One good site is: http://home.earthlink.net/~mmc1919/venturi.html

Have students blow with a straw between two empty soda cans laying on their sides. Expect the cans to roll together just like the balloons came together. Will this will work with any two objects? Have students investigate and write a paragraph summarizing their findings. (Answer: Most objects will do this unless the objects are too heavy to blow apart.)

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Other Related Information

Browse the NGSS Engineering-aligned Physics Curriculum hub for additional Physics and Physical Science curriculum featuring Engineering.

Copyright

© 2004 by Regents of the University of Colorado

Contributors

Tom Rutkowski; Alex Conner; Geoffrey Hill; Malinda Schaefer Zarske; Janet Yowell

Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

Acknowledgements

The contents of this digital library curriculum were developed under grants from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and the National Science Foundation (GK-12 grant no. 0338326). However, these contents do not necessarily represent the policies of the DOE or NSF, and you should not assume endorsement by the federal government.

Last modified: September 6, 2022

Hands-on Activity Air Pressure

Quick Look

Grade Level: 6 (5-7)

Time Required: 30 minutes

Expendable Cost/Group: US $0.00

Group Size: 1

Activity Dependency: None

An F-16J Fighting Falcon aircraft takes off at Bagram Air Field.
The air pressure exerted on this aircraft enables it to stay in the air.
copyright
Copyright © 2010 Afghanistan Matters, Wikimedia Commons http://commons.wikimedia.org/wiki/File:Sunset_Takeoff_(4292024430).jpg

Summary

Air pressure is pushing on us all the time although we do not usually notice it. In this activity, students learn about the units of pressure and get a sense of just how much air pressure is pushing on them.

Engineering Connection

Engineers take into consideration the existence of ambient air pressure in the design of everything from airplanes to chemicals. The weight of air is an important factor when developing the structure of aircraft and spacecraft. Environmental engineers pay careful attention to air pressure when designing wind turbines. Chemical engineers need to know how a chemical reacts in different air pressures. When designing anything that moves through the air, engineers analyze it to see how it reacts to air pressure.

Learning Objectives

After this activity, students should be able to:

  • Explain that the atmosphere exerts a pressure on objects.
  • Describe how the pressure of the atmosphere changes depending on where it is being measured (e.g., Denver or Boston), due to differences in altitude.
  • Use algebraic methods to explore values of air pressure.
  • Describe how engineers, when designing anything that moves through the air, must analyze it to see how it reacts to the air pressure.

Materials List

Each student needs:

Worksheets and Attachments

Visit [www.teachengineering.org/curriculum/print/cub_airplanes_lesson01] to print or download.

Introduction/Motivation

Have you ever felt the pressure on your ears when you dive to the bottom of a swimming pool? The pressure in a pool increases with depth just as air pressure increases as you go deeper into the sea of air (atmosphere).The bottom of the sea of air is represented by sea level, while higher elevations represent shallower parts of the atmosphere. As you move to shallower parts of the atmosphere, such as the top of Mount Everest, the pressure decreases.

Pressure is measured in different units. Scientists and engineers typically use the metric unit Pascal (Pa). A Pascal is defined as the pressure exerted by 1 Newton weight (1 kg under the Earth's force of gravity) resting on an area of 1 square meter. Let's go through some of the common units used to measure pressure and their equivalents. I'll write out four different units, but many additional units exist to describe the amount of pressure.

At sea level, the atmospheric air pressure can be represented as any of the following: (Write the units on the classroom board.)

  • 1.013 x 105 Pa (Pascal or N/m2)
  • 1 atm (atmosphere)
  • 760 mm Hg (millimeters of mercury)
  • 14.7 lb/in2 (psi – pounds force per square inch) (if 1-pound weight rests on 1-square inch of surface area, the pressure is 1 psi)

Procedure

Background

Pressure (P) is defined as the amount of force (F) applied per unit area (A) or as the ratio of force to area:

P= F/A (equation 1)

The pressure an object exerts can be calculated if its weight (the force of gravity on an object) and the contact surface area are known. For a given force (or weight), the pressure it applies increases as the contact area decreases. (To better understand this, hold a large book flat on your outstretched hand and notice how much pressure the book puts on it. Next, try to balance the book on the tip of your index finger. How much pressure does it seem to exert now?) It is also important to note that air pressure decreases with increasing altitude. It is helpful to think of the atmosphere as a swimming pool, with the water representing the air.

Remind students of Bernoulli's principle: the faster a fluid moves the less pressure it exerts. The Bernoulli equation is

P + 1/2 ρV2 + ρgh = constant

Where P is the pressure, v is the velocity, h is the height and g is gravity.

Before the Activity

Gather materials and make copies of the Air Pressure Worksheet (1 per student) and the Pressure vs. Altitude Graph (a few to share).

With the Students

  1. In Denver, CO, the Earth's atmosphere has a force of about 12 pounds per square inch (psi). For reference, a gallon of milk or water weighs about 8 pounds. Have students make a 1-inch by 1-inch square with their hands. Now ask the students what a 2 x 2 square looks like, and ask them how many pounds would be pressing down on that square. Solving for the force in equation 1, students can see that by multiplying the area (4 in2) by the pressure gives 48 pounds as an answer. (Note: multiply the length times the width to get the area of the square.)
  2. Ask students how many pounds would be pressing on a 3 x 3 square? (Answer: 108 lbs for an area of 9 in2.) A 4 x 4? (Answer: 192 lbs for an area of 16 in2.)
  3. Have students complete the Air Pressure Worksheet.
  4. Do the students see a pattern? What happens every time the area of the square increases by 1 in2? (Answer: The pounds of force increase by 12 for every 1 square inch increase in area. This is called a linear relationship. Linear means line; have students make a line graph plotting the area vs. the force so they see that it makes a straight line. See the worksheet for an example of the relationship between area and force.)

What happens every time the sides of the square are increased by 1 inch? (Answer: This is harder since the relationship is not linear. Every time you increase the length of the sides by one inch the force increases by more than 12 lbs. In fact, as the length of each side gets longer the increase in the force gets larger as well. When the length of the sides are 1 inch, the force is 12 lbs. If we increase the sides to 2 inches, the force becomes 48 lbs. This is an increase of 36 lbs. If we add another inch and make each side 3 inches, the force becomes 108 lbs, which is an increase of 60 lbs. If we plot the length of each side vs. the force, we see that the relationship is not linear. The line curves up, which is known as an exponential relationship.)

  1. The average pressure on a middle school student is 24,000 pounds! Ask the students why they do not feel the 24,000 pounds, and why they are not crushed. (Answer: The air inside the body [from breathing, through the skin, ears, etc.] balances out the pressure on the outside of the body.)
  2. The average force of the atmosphere at sea level is 15 lbs per square inch (almost two gallons of milk). Have students repeat their calculations for the sea level pressure. (Cities to use: New York City, 87ft; San Diego, 3 ft; and Boston, 10ft. All are very close to sea level.)
  3. Have students look at the Pressure vs. Altitude Graph and make pressure predictions for several places based on different altitudes. (For example: Chicago, IL [580 ft], Las Vegas, NV [2,030 ft], Leadville, CO [10,177 ft], Mt. Whitney, CA [14,495 ft], Mt. Everest [29,035 ft], airliner cruising at 30,000 ft.) Have students estimate the air pressure at 1 mile below sea level if no ocean water was present (this number is not on the graph, which means students must extend the line below the zero altitude line to estimate it).

Assessment

Pre-Activity Assessment

Discussion Question: Solicit, integrate and summarize student responses.

  • Review Bernoulli's principle. Make sure everyone understands how Bernoulli's principle relates to pressure. (The faster a fluid moves, the less pressure it exerts.)
  • Think about an airplane in motion. Is the air pressure only acting on the top of the plane? (Answer: No, it is acting on the entire surface of the plane.) Does this air pressure affect the speed of the plane? (Answer: Yes, at higher pressures, air is denser, and more air molecules exist to run into, and thus slow the plane down. This is a primary reason that planes fly at such high altitudes, even though ozone can be a problem for those inside the plane, compared to flying at 25,000 ft.)

Activity Embedded Assessment

Worksheet: Have students use the Air Pressure Worksheet to record measurements and follow along with the activity. After students have finished their worksheets, have them compare answers with their peers.

Post-Activity Assessment

Graphing: Have students use the information from their worksheets to create line graphs of the relationship between area and force. Plot area (in2) on the x-axis and force (ponds) on the y axis. Ask students or teams to explain what is happening in their graphs in their own words.

Activity Extensions

Have students complete an Air Pressure Worksheet for planets where the air pressure is different than on Earth. Examples are Jupiter (735,000,000 psi), Venus (1,325 psi), Mars (0.25 psi), Pluto (0.000147 psi). Have students discuss what kinds of challenges these pressures might impose on manned and unmanned missions to these planets.

Activity Scaling

For older students:

  • Rather than telling students that the amount of air pressure pushing on them is about 24,000 lbs, tell them the average surface area for an elementary school student is about 2000 in2 and have them calculate the pressure themselves.
  • Have students calculate the force for other areas such as 1 square foot (144 in2), a football field (approximately 8,000,000 in2).
  • Have students plot square inches vs. force on a graph.

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References

Atmospheric Pressure. It's a Breeze: How Air Pressure Affects You. NASA. Accessed 2004. (Simple air pressure experiments) http://kids.earth.nasa.gov/archive/air_pressure/

Copyright

© 2004 by Regents of the University of Colorado

Contributors

Tom Rutkowski; Alex Conner; Geoffrey Hill; Malinda Schaefer Zarske; Janet Yowell

Supporting Program

Integrated Teaching and Learning Program, College of Engineering, University of Colorado Boulder

Acknowledgements

The contents of this digital library curriculum were developed under grants from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and the National Science Foundation (GK-12 grant no. 0338326). However, these contents do not necessarily represent the policies of the DOE or NSF, and you should not assume endorsement by the federal government.

Last modified: February 9, 2024