High School Mathematics Problems from Alaska

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Azimuth and Arc Length
(Problem Two)

James H. Grey
jgrey@northstar.k12.ak.us

Standards: Measurement, Estimation and Computation, Functions and Relationships, Geometry, Statistics/Probability.

Performance Standards: A2.4.3, A3.4.1, A4.3.1, A5.4.1, A5.4.2, A5.4.7, A6.3.1, A6.4.1

Concepts: Angle Measure, Azimuth, and Arc Length.

Carnegie Chapter: No corresponding problem.

Overview: Focusing on making visual representations of geometric data.

Teacher Notes: Before doing this activity, I suggest having an anticipatory-set discussion with your students regarding where the sun rises and sets at different times of the year. Some leading questions and discovery-based dialogue will help with this set of problems.

Azimuth and Arc Length

Problem Two

The directions on a compass correspond to numbers from 0-360 as shown below, where North corresponds to 0 degrees, East is 90 degrees, South is 180 degrees, and West is 270 degrees. This is known as azimuth. If an object is at an azimuth of 45 degrees, it is in the Northeast direction relative to an observer.

The sun rises for the first time of the year in Barrow, Alaska on January 23. It rises (comes above the horizon) at an azimuth of 174.3 degrees and sets (goes below the horizon) at an azimuth of 185.9 degrees.

Write your answers to the following questions in the form of complete sentences.

1a. In what part of the sky, or in what direction does the sun first rise?

b. Where does it set?

2a. What is the mean of the sunrise and sunset azimuth?

b. What direction is it?

c. Explain why these answers might be surprising to some people.

Activity:

1. Find the wall of your classroom that most closely faces south. Put a sign on it labeling it 180 degrees. Label the other three walls accordingly.

2. Holding a protractor in a fixed position away from all four walls, run string lines from it every ten degrees toward the wall and label the wall accordingly. (Teacher note: It’s suggested that students make signs for every ten degrees that are large enough to be seen from any part of the room. Each sign should also have a mark to denote the position of the angle.) Viewed from above, your classroom will look something like this:

South Wall

1. Using a piece of colored yarn, tape the arc that the sun travels from its sunrise to sunset on January 23.

2. Using additional pieces of yarn and the table below, attach a piece to the wall for every-other day’s arc being sure to attach each end at the appropriate azimuth. (Teacher note: Labeling every day makes the wall look too "busy" and this may lead to confusion.) Label the day’s arc.

 Date Day Azimuth (Rise/Set) Date Day Azimuth (Rise/Set) Jan 23 23 174.3-185.9 Jan 28 28 158.6-201.5 Jan 24 24 169.5-190.8 Jan 29 29 156.6-203.6 Jan 25 25 165.9-194.0 Jan 30 30 154.7-205.5 Jan 26 26 163.2-196.8 Jan 31 31 152.7-207.4 Jan 27 27 160.8-199.4 Feb 1 32 151.0-209.3

1. Describe the impression you have of the arc that the sun covers each day.

4a. Using the attached polar coordinate graph paper and colored pencils, draw the arc for every-other day in the table below. Begin with the shortest arc closest to the center of the circle.

 Date Day Azimuth (Rise/Set) Date Day Azimuth (Rise/Set) March 16 75 93.2-267.5 March 22 81 85.7-274.9 March 17 76 91.9-268.7 March 23 82 84.5-276.3 March 18 77 90.7-270.1 March 24 83 83.2-277.4 March 19 78 89.4-271.2 March 25 84 82.0-278.6 March 20 79 88.2-272.4 March 26 85 80.8-279.9 March 21 80 87.0-273.7

1. Explain the difference between the information on the graph and the information on the wall, from part three.

1. Why do you think that it is different?

Click on image for a larger view

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Handbook for Culturally Responsive Science Curriculum by Sidney Stephens
Excerpt: "The information and insights contained in this document will be of interest to anyone involved in bringing local knowledge to bear in school curriculum. Drawing upon the efforts of many people over a period of several years, Sidney Stephens has managed to distill and synthesize the critical ingredients for making the teaching of science relevant and meaningful in culturally adaptable ways."

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