# What Makes Airplanes Fly? History, Science and Applications ofAerodynamics

## CONTENTS OF CURRICULUM UNIT 90.07.10

- Narrative
- I. Introduction
- II. Rationale and General Objectives of the Unit
- III. Historical Overview of the Development of Aircraft4
- IV. The Mathematical Application
- II. An Introduction to Graph Theory
- Sample Lesson Plan 1
- Sample Lesson plan 2
- Sample Lesson Plan 3
- Sample Lesson Plan 4
- Sample Problems For Class Discussion
- Bibliography

### Unit Guide

## Historical Developments of the Aircraft Industry with Mathematical Applications

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## Sample Lesson Plan 3

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Topic: Vectors as a means of representing a Journey.
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Objectives: Students will be able
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- a) to read a network.
- b) to identify direct routes
- c) to identify direct routes in an arrow diagram.
- d) to use a table and a matrix to show direct routes.

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Prerequisites:
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- a) The ability to read and draw graph and digraphs.

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Skills and Concepts Presented:
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- a) Drawing and reading networks.
- b) Identifying direct routes.
- c) Drawing arrow diagrams
- d) Representing direct routes on a table.
- e) Writing simple matrices

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Developments:
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- 1. Review drawing graphs
- 2. Discussion of the network of street in their towns.
- 3. Use a map of flights between cities to show a network (ads in newspapers showing connections)
- 4. Use a simple network to show key points
- ____ a) towns represented by a dot (vertices)
- ____ b) routes represented by edges.
- 5. Define direct routes. Have students identify direct routes on maps shown.
- 6. Introduce the arrow diagrams as on way of representing direct routes.
- 7. Introduce the table as another way of representing routes
- 8. Introduce the matrix as a short cut to writing the routes.
- 9. Discuss and define the matrix.
- 10. Discussion and Questions from the matrix.
- ____ a) Deduce important features from the matrix.
- ____ ____ i) The leading diagonal
- ____ ____ ii) Is the matrix symmetrical with respect to the leading diagonal.
- ____ ____ iii) The diagonal in a directed matrix verses a undirected matrix
- ____ b) Provide drill and practice.

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Sample Problems:
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- 1. Copy the figure draw the arrow diagram.

*(figure available in print form)*

*(figure available in print form)*

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Questions:
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- 1. Why are there three routes from A to C?
- 2. Why are there 2 routes from A to B?
- 3. Show the leading diagonal.
- 4. Is the table symmetrical about the leading diagonal?

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Extension Activities
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- 1. Find the Matrix which describes these networks:

*(figure available in print form)*

*(figure available in print form)*

*(figure available in print form)*

*(figure available in print form)*

- 2. Find the matrices which describes the following network, then write a statement to describe the matrices.

*(figure available in print form)*

*(figure available in print form)*

*(figure available in print form)*

- 3. Given the matrix, can you draw the network? (work backwards)