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- Lesson Overview
- Hypertext Outline of Lesson
- Objectives
- Time Allotment
- NCTM Process Standards
- NCTM Content Standards
- Aeronautics Content
- Prerequisite Skills
- Vocabulary
- Materials
- Teacher Tips
- Additional Activities
- Accessibility

In the Fuselage Department, students get a brief intro about the history, purpose and development of the fuselage, which is the main body of an airplane. Students are then guided into an exploration and discovery about the size and shape of a fuselage. Area of simple objects (square, rectangle, triangle) are taught (both through estimation and formulas) with their corresponding formulas as a lead in to volume. The area of a circle is introduced which leads to the formula for the volume of a cylinder. Shape, air pressure and drag are discussed as they pertain to a fuselage. The activity ends with a comparison table of 3 different types of planes, which students use to answer questions about "trade-offs" decisions during the design process.

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.

- Defintion/History (2)
- Shape (6)
- Air Pressure (10)
- Fuselage cross section/shape (11)
- Balloon Activity (12)
- Fuselage size (14)
- Area of a Rectangle(15)
- Volume of a Rectangular Prism (19)
- Area of a Circle (22)
- Average (24a)
- Radius and Diameter (28)
- Volume of a Cylinder (31)
- Trade-offs (35)

At the end of this lesson, students will:

- Be able to determine the area of a square by estimating with a grid and by using the formula for area.
- Understand the units for area and volume, along with their proper mathematical notation.
- Understand pi, radius, circumference and diameter.
- Understand and be able to use the formula for the area of a circle.
- Be able to estimate the area of a circle using grid estimation and an averaging technique.
- Understand and be able to use the formula for the volume of a cylinder.
- Be able to analyze information in a table to determine the trade-offs made in three different type planes.
- Understand and be able to define the terms wind resistance, drag and aerodynamics, and how they relate to a fuselage.

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

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.
- Apply a variety of strategies to solve problems.
- Acquire confidence in using mathematics meaningfully.
Standard 2: Mathematics as Communication

- Interpret and evaluate mathematical ideas presented in written and visual forms.
- Discuss mathematical ideas and make convincing arguments.
- Develop common understanding of definitions
- Appreciate the value of mathematical notation.
Standard 3: Mathematics as Reasoning

- Understand and apply reasoning with graphs.
- Make and evaluate mathematical arguments.
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 and decimal forms in real world situations.
- Develop number sense for whole numbers and decimals.
Standard 6: Number Systems and Number Theory

- Extend their understanding of whole number operations to decimals.
Standard 7: Computation and Estimation

- Compute with whole numbers and decimals.
- Analyze procedures for computation and techniques for estimation
- Select and use an appropriate method computing from mental arithmetic, calculator and paper and pencil.
- Use estimation to calculate the reasonableness of results.
Standard 8: Patterns and Functions

- Describe and represent relationships with tables and rules.
- Analyze functional relationships to explain how a change in one quantity results in a change in another.
Standard 9: Algebra

- Understand the concept of equation, variable and formula.
- Represent situations and number patterns with tables and equations and explore the interrelationships of these representations.
- Analyze tables to identify relationships.
- Apply algebraic methods tools to solve real world and mathematical problems.
Standard 10: Statistics

- Read and interpret tables and make inferences based on data analysis and evaluate arguments based on data analysis.
Standard 12: Geometry

- Identify, compare and classify geometric figures.
- Visualize geometric figures for spatial sense.
- Represent and solve problems using geometric models.
- Develop an appreciation of geometry as a means of describing the physical world.
Standard 13: Measurement

- Estimate and use measurements to describe and compare phenomena.
- Extend understanding of the concepts of area and volume.

- Why Planes Fly
- pressurized cabins
- aerodynamic shape of fuselage.

- History of Flight
- milestones in airplane design, specifically use of fuselage and the impact of shape and size on plane design as planes became faster and bigger.

- Student should be familiar with the terms area and volume and have some experience with equations and formulas.
- Students should be able to multiply 3 digit numbers with decimals.

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

- fuselage
- aerodynamics
- wind resistant
- cylindrical
- air pressure
- pressurized cabins
- area
- formula
- 2-dimensional
- 3-dimensional
- volume
- cylinder
- mean or average
- diameter
- radius
- trade-offs
- Mach
- sound barrier
- supersonic
- subsonic
- sonic booms

For helping with calculations:

- paper and pencil, calculator, or an assistive software package such as MathPad

This lesson can be completed individually but will move faster and be more fun if two or more people work together. A good breaking point is after the area section and before volume. Students who are unable to write can provide verbal input on project or make choices during activity. Print out a copy of the airplane comparison table for the students to refer to.

1. Allow students to experiment with finding the area and volume of other 2-D and 3-D shapes, using the methods taught in the area and volume sections. Use both regular and irregular shapes.

2. Groups of students can create questions for other groups from the airplane comparison table. Share and answer questions with the whole class.

3. Research the shapes, size and speed of planes that use the local airport. Arrange this information into a table format. What does this information in this table tell you about the airplanes?

4. Groups of students design a new student desk. Create a comparison chart for at least three features (volume of storage, height, area of base) of the different desks designed. Students draw comparisons about the different desks, noting any "trade-offs" made in any of the designs.

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

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.