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High School Mathematics Problems from Alaska

A database of lessons and units searchable by content and cultural standards, cultural region and grade level. More units will be available soon. You can use Acrobat Reader to look at the PDF version of the Cover Sheet for the Units and Self-Assessment for Cultural Standards in Practice.

Azimuth and Arc Length
(Problem Two)


 James H. Grey


MS Word Download

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.


directions on a compass


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.





  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:
Holding a protractor

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.


Azimuth (Rise/Set)
Azimuth (Rise/Set)
Jan 23
Jan 28
Jan 24
Jan 29
Jan 25
Jan 30
Jan 26
Jan 31
Jan 27
Feb 1



  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.


Azimuth (Rise/Set)
Azimuth (Rise/Set)
March 16
March 22
March 17
March 23
March 18
March 24
March 19
March 25
March 20
March 26
March 21




  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

Table of Contents


AL-CAN Highway Problem

Cigarette Smoking

Commercial Salmon Fishing

Cutting Cordwood

Dog Lot Math

Firefighting Crew

Firewood Problem

Investigation of Herring Statistics

Jesse Thomas Moose Hide Problem

Predicting Plant Coverage

Problem 1 - Converting the Date to the Day of the Year

Problem 2 - Azimuth and Arc Length

Problem 3 - Reading a Graph: Azimuth Range vs. Day in Barrow, Alaska

Problem 4 - Writing an Equation: Altitude Range vs. Day in Barrow, Alaska

Problem 5 - Problem Setup and Pattern Recognition: Sunrise in Barrow, Alaska

Problem 6 - Extensions of Data: Barrow, AK Sunrise/Sunset Information

Problem 7 - Writing an Equation: Fundraiser

Problem 8 - Writing an Equation: Fundraiser Choosing a Distributor

Stanley Jonas Travel Problem

Subsistence Fishing on the Kenai

Tourist and Traveler Information



For the Teacher's Guides, please email Alaska Native Knowledge Network

Whouy Sze Kuinalth
"Teaching Our Many Grandchildren"
Tauhna Cauyalitahtug
(To Make a Drum)
Math Story Problems
St. Lawrence Island Rain Parka Winds and Weather Willow
Driftwood Snowshoes Moose
Plants of the Tundra Animal Classification for Yup'ik Region Rabbit Snaring
The Right Tool for the Job
Fishing Tools and Technology
Blackfish Family Tree
Medicinal Plants of the Kodiak Alutiiq Archipelago Beaver in Interior Alaska Digging and Preparing Spruce Roots
Moose in Interior Alaska Birds Around the Village  


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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Alaska Native Knowledge Network
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Last modified August 18, 2006