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Table of Contents

1. Lesson Overview
2. Hypertext Outline of Lesson
3. Objectives
4. Time Allotment
5. NCTM Process Standards
6. NCTM Content Standards
7. Aeronautics Content
8. Prerequisite Skills
9. Vocabulary
10. Materials
11. Teacher Tips
12. Additional Activities
13. Accessibility

In the Airfoils Department, students learn how airfoils, or the shape of the cross section of a wing, are related to the amount of lift that wing is capable of generating. Mathematical concepts related to airfoils and lift are introduced and students are given experiences and opportunities to answer questions pertaining to these concepts (symmetry, chord, lines, function tables, patterns, square numbers, formulas, graphs of relationships). A culminating wind tunnel activity allows students to experiment with the variables of speed and airfoil coefficients of lift to see the effect on lift and drag.

This purpose of this outline is to help you navigate to specific parts of the lesson without having to go through every page. The section titles link to the first pages of that section, and the numbers in parentheses refer to the page number where that section starts.

• Airfoil definition (2)
• Symmetry (3)
• Asymmetrical shapes (7)
• Letters and symmetry (10)
• Symmetrical and asymmetrical airfoils (15)
• Chord lines (16)
• Airfoils and chord lines (19)
• String activity (20)
• Symmetrical airfoils (22)
• Asymmetrical airfoils (23)
• Coefficient of lift/Airfoil ratings (24)
• Direct relationships (27)
• Graphing direct relationships (30)
• Squared numbers and other relationships (33)
• Graph of a squared number (34)
• Equation for the lift of an airplane (36)
• Wind tunnel (38)

At the end of this lesson, students will:

• Be able to define and recognize symmetry, asymmetry and lines of symmetry.
• Be able to define and locate a chord line on a curved line and in an airfoil.
• Understand coefficient of lift.
• Understand the relationship between speed and lift.
• Understand and recognize squared numbers and whole number exponents.
• Be able to identify patterns in a function table and determine the rule for the pattern.
• Given a function table, be able to read a graph of that function and make predictions based on the graph.
• Understand the formula for lift and the variables that comprise lift: speed, coefficient of lift and wing area.
• Understand how squared numbers increase compared to how directly related numbers increase.
• Be able to read and interpret line graphs.
• Be able to recognize a direct relationship.
• Understand that a symmetrical airfoil has a low coefficient of lift compared to an asymmetrical airfoil.
• Understand that a plane with a strong engine and high cruising speed can use a symmetrical airfoil, while a plane with a low cruising speed must use an asymmetrical airfoil.
• Understand the effects of speed and the coefficient of lift on an imaginary plane through the use of a wind tunnel simulation.

30-40 minutes, depending on student's reading ability and familiarity with geometric terms and formulas.

Standard 1: Mathematics as Problem Solving

• Use problem solving approaches to investigate and understand mathematical content.
• Verify and interpret given results and generalize solutions and strategies to a new problem.

Standard 2: Mathematics as Communication

• Interpret and evaluate mathematical ideas presented in written and visual forms.
• Discuss mathematical ideas and make convincing arguments.

Standard 3: Mathematics as Reasoning

• Understand and apply reasoning to graphs.
• Make and evaluate mathematical arguments about how different values are related to each other.

Standard 4: Mathematical Connections

• Explore problems and describe results using graphical, physical and verbal math models.
• Apply mathematics to solve problems in science.
• Recognize the value of math in an applied technical situation

Standard 5: Number and Number Relationships

• Understand, represent and use numbers in exponential forms in real world situations.
• Represent numerical relationships in 2-dimensional graphs.

Standard 8: Patterns and Functions

• Describe and analyze a variety of patterns.
• Describe and represent relationships with graphs and rules.
• Analyze functional relationships to explain how a change in one quantity results in a change in another.
• Use patterns and functions to represent and solve number problems.

Standard 9: Algebra

• Understand the concept of equation and formula.
• Represent situations and number patterns with tables, graphs, verbal rules and explore the interrelationships of these representations.
• Analyze tables and graphs to identify relationships.
• Informally investigate nonlinear second-order equations.
• Apply algebraic methods tools to solve real world and mathematical problems.

Standard 10: Statistics

• Describe and represent data using tables and graphs.
• Read and interpret tables and graphs.

Standard 12: Geometry

• Identify, compare and classify geometric figures and concepts.
• Visualize 2-dimensional geometric figures.
• Develop an appreciation of geometry as a means of describing the physical world.

• Why Planes Fly
  • How airfoil shape and speed are related to amount of lift.
  • How a wind tunnel works and practice operating a wind tunnel.

• Students should have covered "Lift Off", one of the PlaneMath activities in Applying Flying.
• Students should have some basic knowledge of graphs and tables and how to interpret them.

Vocabulary words are linked to the activity pages on which they're defined.

• airfoil
• line of symmetry
• symmetrical
• mirror image
• asymmetrical
• chord line
• curved line
• coefficient of lift
• direct relationship
• x- and y-axis
• proportionally
• squared
• cruising speed

For the string activity:

• a 2-foot long string
• ruler

Other materials:

• graph paper and pencil (for graphing examples)
• paper shapes to test for symmetry through manipulation and folding

This lesson can be completed individually but will move faster and be more fun if two or more people work together. The lesson can be done in under an hour if the students are good readers. There are several good breaking places in this lesson--after symmetry lesson, after lift and airfoil (prior to function tables) and before the wind tunnel activity.

Pair up students if someone is unable to hold or manipulate objects. Students who are unable to write can provide verbal input on project or make choices during the activity. In the string activity, tape or anchor the string down at one end if a student is unable to use both hands.

1. Students create a data chart for the wind tunnel activity. Students fill in values for speed and lift as they explore the characteristics of the symmetrical and asymmetrical airfoils with the wind tunnel, and then determine what pattern exists. Students then describe in writing the relationship between speed and lift.

2. Students create models of airfoils using clay, paper mache or another malleable substance. Students make examples of symmetrical and asymmetrical airfoils, showing the chord line on each model.

3. Students draw symmetrical and asymmetrical shapes on pieces of paper and cut them out. Then, they test for symmetry by folding and manipulating the shapes to find the line of symmetry. Students may also use a mirror to view the mirror images of any half-shapes they might make.

4. Students visit an airplane museum, sketching the airfoil shape of different planes. They can also research the speed and function of these planes. Students then provide rationales for why different airplanes have different airfoil shapes.

Do you have ideas for other activities to use with this activity? Send your suggestions to us at planemath@infouse.com.

The interactive Shockwave portions of this activity, such as the wind tunnel, are accessible through both the keyboard and the mouse. Students can use the spacebar to cycle through all the entry options on the screen, which will be highlighted by a small yellow bar next to the option. Students then use the up or down arrows to change an option, or press Return or Enter to select a button.

All the pages maintain a consistent grid of 6 buttons along the bottom of the page, which should be accessible through a ClickIt! overlay for IntelliKeys. For more information on using assistive technology, please refer to the document "Making PlaneMath Accessible" on the main PlaneMath parent/teacher page.

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