# 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 1

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Topic: Introduction to Graph Theory.
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Objectives: Students will be able
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- a) to define a graph, and a null graph.
- b) to differentiate between graphs, null graphs and drawings that not graphs.
- c) to identify the parts of a graph.
- d) to identify a digraph.
- e) to use a graph to represent a problem.

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Prerequisites:
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- a) Plot points
- b) Draw straight lines.

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Skills and concepts presented:
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- a) Plotting and joining points
- b) organizing information.

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Developments:
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- 1. Develop a dialogue that would show the students the importance that drawing a diagram (in this case graphs) in solving problems.
- 2. Point out that graphs are considered as models and therefore can be used to represent a situation.
- 3. Introduce the airline problem from the content.
- 4. Introduce and explain each step:
- ____ i) how to organize the information in a table.
- ____ ii) introduce vertex or nodes and edges.
- ____ iii) give examples of null graphs, graphs, and drawings that are not graphs
- 5. Formally define graphs and digraphs.
- 6. Present drill and practice for students to use the information discussed.

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Possible Problems for discussion:
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- 1. In the following sketches identify the vertices and the edges.

*(figure available in print form)*

- 2. Identify each of the following as graphs or not a graph.

*(figure available in print form)*

- 3. a) Draw a complete graph with six vertices.
- ____ b) How many edges are there?
- 4. Draw a null graph with eight vertices.
- 5. Draw a graph to show the friendship relation among four individuals.

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Extension Activities:
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Draw a digraph to show the relationship between the statements.

- 1. Have students suggest possible problems from which they can produce a graph.
- 2. Have students draw a digraph of friendship relation in the classroom.
- 3. The following are dependent statements (implications exist between them):
- ____ a) The Wright brothers first flight.
- ____ b) Better aircrafts invented.
- ____ c) More jobs provided.
- ____ d) Standard of living improved.
- ____ e) More people take trips on airplanes.