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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 Wing Shape Department students continue their exploration and learning about wings. First, simple geometric shapes and area formulas are reviewed: parallel lines, angles, quadrilaterals, triangles, rectangles and trapezoids. As a follow up to airfoils students learn to calculate the wingspan, chord, area and aspect ratio of a wing. The students then learn how these values are related to the lift produced by the wing. Students learn about different wing shapes and how to calculate chord length and aspect ratio for those wings. A final activity allows students to change the wingspan and average chord length to obtain a specified wing area.

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.

• Three Dimensions (2)
• Wing Shape and Wing Area (3)
• Parallel and Non-Parallel Lines (4)
• Angles (7)
• Quadrilaterals and Triangles (18)
• Lift and Drag (26)
• Rectangular Wings (33)
• Aspect Ratio (39)
• Tapered Wings (47)
• Area of a Trapezoid (48)
• Aspect Ratio of Tapered Wings (54)
• Delta Wings (58)
• Area of a Triangle (59)
• Aspect Ratio of a Delta Wing (63)
• Wing Shape Design Activity (66)

At the end of this lesson, students will:

• Be able to define vertex and angle
• Learn that wingtips create a vortex which creates drag, the bigger the wing tip the more drag
• Learn that wing area tells us how much lift can be produced
• Be able to define and calculate the aspect ratio of a rectangular wing, given wingspan and chord
• Understand that wing area is directly proportional to lift as area goes up, so does lift
• Define root chord and tip chord
• Learn that wings produce drag and lift
• Measure an angle using a protractor
• Identify parallel and nonparallel
• Understand the term trade off in designing wings for strength and lift potential
• Convert simple fractions into decimals understand that higher aspect ratios give more lift
• Determine the area of a trapezoid (tapered wing)
• Given root and tip chord for a tapered wing, determine the area
• Identify the shape of swept back and delta wings
• Calculate the area of a triangle
• Determine the aspect ratio of a delta wing
• Be able to determine a specific wing area by changing wingspan and average chord length

30-40 minutes depending on student's reading ability.

Standard 1: Mathematics as Problem Solving

• Using problem solving approaches to investigate and understand mathematical content
• Students verify and interpret given results and generalize solutions and strategies to a new problem.

Standard 2: Mathematics as Communication

• Students use reading and viewing to interpret and evaluate mathematical ideas.
• Within a groups students will discuss mathematical ideas and make convincing arguments.

Standard 3: Mathematics as Reasoning

• Students understand and apply reasoning with graphs.
• Students make and evaluate mathematical arguments.

Standard 4: Mathematical Connections

• Students explore problems and describe results using graphical, physical and verbal math models.
• Students apply mathematics to solve problems in science.
• Students see the value of math in a applied vocational situation

Standard 5: Number and Number Relationships

• Understand, represent and use numbers in a variety of forms.
• Develop number sense for whole numbers, fractions and decimals in real-world situations and mathematical problem solving.
• Understand and apply ratios and proportions in a variety of situations
• Investigate relationships among fractions and decimals

Standard 6: Number Systems and Number Theory

• Develop and use order relations for whole numbers, fractions and decimals
• Extend understanding of whole numbers operations to fractions and decimals

  Standard 7: Computation and Estimation

  • Compute with whole numbers, fractions and decimals
  • Develop, analyze and explain methods for solving problems
  • Select and use an appropriate method for computing from among mental arithmetic, paper and pencil, calculator, computer.
  • Use computation, estimation and proportions to solve problems

  Standard 8: Patterns and Functions

  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.
  • Investigate nonlinear equations informally.
  • Apply algebraic methods tools to solve real world and mathematical problems.

  Standard 10: Statistics

  Standard 11: Probability

  Standard 12: Geometry

  • Identify compare and classify geometric figures. Visualize geometric figures.
  • Understand and apply geometric properties and relationships.
  • Develop an appreciation of geometry as a means of describing the physical world.

  Standard 13: Measurement

  • Extend understanding of the process of measurement.
  • Select appropriate tool and unit to measure to the degree of accuracy needed.
  • Extend understanding of the concepts of area and angle measure.
  • Develop formulas and procedures for determining measures to solve problems

• Why Planes Fly
  • Wing shape and speed related to amount of lift.
  • Use of a wind tunnel.
• History of Flight
  • Milestones in airplane design.

• Students should have covered the training activity "Airfoils" before trying "Wings".
• Students should have a basic knowledge or have been introduced to geometric figures and area formulas and the concept of proportion and ratio

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

• dimensional
• parallel
• vertex
• angles
• protractors
• obtuse
• acute
• quadrilaterals
• trapezoid
• drag
• vortex
• chord
• root chord
• tip chord
• area
• aspect ratio

• pencil and paper,
• calculator or assistive software package like MathPad

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--right before wing shape, lift and drag and before the wing design activity toward the end. Students who are unable to write can provide verbal input on project or make choices during activity.

1. Students visit an airplane museum, airport or an online site that shows airplanes. Have students draw all the different types of wing shapes and categorize them according to similarities.

2. Assign a different type plane to each group of 4-5 students in your room. Make sure the planes all have different wing shapes. Have students do research on the function of the planes and then report on why the planes have those particular wing shapes, aspect ratios, etc. related to the function of the plane (how high, how fast, what it does)

3. The wingshape of a plane is the shape of the wings visible from directly above or below it. What are the shapes of other objects from directly above? (i.e. cars, furniture, people) Draw a map of your classroom, home or neighborhood as it would appear from directly above. (See also "Birds Eye View" lesson from 4th grade Applying Flying.)

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 wing shape design activity, are accessible through both the keyboard and the mouse. Students can use the spacebar to cycle through all the options in the design activity, which will be highlighted by a small yellow bar. If the option toggles through different choices, students can use the up and down arrow keys to move through the choices. If the option is a button, students press Return or Enter to select that 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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