DESCRIBING ANGLES IN AVIATION
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Jim Pelech and F Richard Singer III |
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Overview: Constructivist Learning Resources indicates five types of such resources, three of which are indicated below. This present resource is a simple Type 3 resource, with a Type 2 resource to be appended later. It compares describing angles in aviation to describing them in the standard coordinate system. One of the building blocks of navigation is the concept of describing the direction that a plane is flying. Navigation uses the concept of "heading" for this purpose. In the first set of activities students are given, a heading, and they are to convert it to a standard angle description. The next set reverses the procedure. In both cases, students are asked to explain in words how to do the procedure. By being asked to put the procedure in words, the student is given the opportunity to explicitly formulate a conceptual relationship. The third set of activities relates the concept of heading to pairs of cities.
Acknowledgements: This resource is a version of a small part of a much more extensive Type 1 resource on mathematics in aviation designed by Jim Pelech, Jim can be contacted at JPelech@ben.edu, and would be happy to communicate about his work.
Type 1: Plans designed to provide flexible group activities that involve some concepts and conceptual distinctions and that use situations which enhance concept construction
Type 2: Accounts of real or fictional interactions of groups engaged in constructivist learning activities that were designed to focus some cluster of conceptual distinctions
Type 3: Materials suggesting activities designed to focus on concepts and conceptual distinctions that a variety of users are likely to find manifest enough for some concept construction to occur
Background: This paper presupposes familiarity with the concept of an angle and how angles are described in a standard coordinate system, along with the ability to use a protractor to measure angle.
Objectives:
- Given standard description of an angle, find its heading
- Given a heading, find the standard description of this angle
- Find headings between two given cities
Materials: Activity Sheet, Map, Scissors, String
Note: The appendix merely contains imagined responses from 4 students to some of these activities. It will be expanded to a more fully developed Type 2 resource later. Anyone using these activities with a group of students is invited to send such an account.
ACTIVITIES:
MEASURING ANGLES IN AVIATION
Review Activity: Recall how angles are described in a standard system when their vertex is located at the origin. The initial side is the positive x-axis. For an angle with a positive measurement, rotate counterclockwise. For an angle with a negative measurement, rotate clockwise. Refer to the diagrams below and determine the standard angle measurement. Explain how you arrived at them.
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Headings: In aviation angles are measured differently. All angles are measured in a clockwise direction from due north. This gives angles which are nonnegative and less than 360 degrees. When the course of a plane follows a certain direction, that is called the heading of the plane.
Look at the example below.
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Activity 1a A plane’s heading is 210 degrees. What is the standard measure of the angle involved?
Activity 1b A plane has a heading of 320 degrees. What is the standard measure of the angle involved?
Activity 1c Convert from a heading to standard measure: A heading of 165 degrees, A heading of 325 degrees, A heading of 42 degrees.
Activity 1d State in words how to convert from a heading to standard measure. Discuss various ways this might be done.
Activity 2a Using standard measures, the angle is 220 degrees. Find its aviation heading.
Activity 2b Using standard measures, the angle is - 160 degrees. Find its aviation heading.
Activity 2c Find the heading that corresponds to 125 degrees and - 235 degrees. You are encouraged to draw the diagrams.
Activity 2c State in words how to convert from standard position measure to headings. Discuss various ways this might be done.
Activity 3a Go to a map, use a protractor and find the headings
between the two cities: from
Activity 3b Find the heading between Chicago and Houston. Using the pattern that you discussed above, predict the heading between Houston and Chicago.
Activity 3c Which system do you prefer, headings or standard? Explain why. Suppose that airplane pilots decided to vote for using standard system, for measuring angles. What would be the advantages? Disadvantages? Take into account the fact that the earth is not flat and the relationship between navigation and the stars.
Appendix: Student Discussion of the
Activities
Activity 1a A plane’s heading is 210 degrees. What is the standard measure of the angle involved?
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Bob: First we will draw the heading. |
To convert this to standard position, we must "start" at the positive x-axis. |
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Activity 1b A plane has a heading of 320 degrees. What is the standard measure of the angle involved?
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Activity 2a Using standard measures, the angle is 220 degrees. Find its aviation heading.
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Kay: Using standard measures, the angle is 220 degrees. Find its aviation heading. First draw this angle. Remember: the initial side is the positive x-axis, and the angle rotates counterclockwise. After this draw this angle and measure it starting from due north. |
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Activity 2b Using standard measures, the angle is - 160 degrees. Find its aviation heading.
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Jan: Using standard measures, the angle is - 160 degrees. Find its aviation heading. First draw this angle. Remember: the initial side is the positive x-axis, and this angle rotates 160 degrees clockwise. After this draw this angle and measure it starting from due north. So if a plane follows standard angle of - 160 degrees, has a heading of 250 degrees. |
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