High School Mathematics Problems from Alaska

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Writing an Equation:
Altitude Range vs. Day in Barrow, Alaska
(Problem Four)

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

Standards: Numeration, Geometry, Functions and Relationships; Statistics/Probability.

Performance Standards: A1.3.2, A1.3.4, A4.3.1, A4.4.2, A4.3.3, A4.3.4, A4.3.5, A5.4.6, A6.3.2, A6.3.4

Concepts: Making predictions using given data, writing a linear equation from given data, and using a linear equation to predict later behavior.

Carnegie Chapter: Modeling Situations with Formulas, Tables of Values, and Graphs.

Overview: Focusing on writing and using the results of a linear equation.

Teacher Notes: It may help to do problem one first and carry over the information to this problem.

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

When sighting an object’s altitude in the sky relative to an observer, a compass-like system is also used. While facing an object on the horizon, it has an altitude of 0 degrees and an object straight overhead has an altitude of 90 degrees. The altitude of a visible object has an altitude between 0 and 90 degrees.

Altitude Range also shows the TOTAL arc the sun traverses in the sky each day. On February 14 the sun rises (comes above the horizon) at an altitude of 0 degrees and reaches its zenith (its highest altitude) at an altitude of 6.7 degrees in Barrow, Alaska. This gives an Altitude Range (the difference between the sunrise and sunset altitude) of 6.7 degrees for that day.

Avaiyak knows that the sun first rises above the horizon in Barrow, Alaska on January 23. Using a Global Positioning System (GPS), he notices that it rises at an azimuth of 174.3 degrees and sets at 185.9 degrees.

He also notices that it rises through an altitude of 0.6 degrees before setting.

In March, he takes the following data on the following days:

 Date Day Altitude Range Date Day Altitude Range March 16 75 17.9 March 22 81 20.3 March 17 76 18.3 March 23 82 20.7 March 18 77 18.7 March 24 83 21.1 March 19 78 19.1 March 25 84 21.5 March 20 79 19.5 March 26 85 21.9 March 21 80 19.9

Answer the following questions using complete sentences.

1. Convert the following dates to their respective day of the year:
1. February 2

2. March 1

3. March 15

4. March 27

5. March 28

6. April 21.

1. Given the data from above, predict the altitude range the following dates:
1. March 27.

2. March 28.

3. April 21.

4. What strategies did you use to find these answers?

1. What was the altitude range on:
1. March 15?

2. March 1?

3. February 2?

4. What strategy did you use to find these answers?

1. Using the data from numbers 2 and 3, which is the independent and dependent variable? Why?

1. Select from the data, numbers that form two ordered pairs in order to derive the equation for the altitude range with respect to the day.

1. On what day is the Altitude Range:
1. 17.1 degrees?

2. 23.5 degrees

3. 32.3 degrees

4. 95.1 degrees?

5. What methods did you use to find these answers?

1. Did your answer to question 6d make sense? Why or why not?

1. Using the information from question 6, on what date do each of these altitude ranges occur?

a.

b.

c.

d.

1. What are the logical limits to this equation? Why?

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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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