www.TeachEngineering.org: Free Curriculum for K-12

Contributed by: Integrated Teaching and Learning Program, College of Engineering, University of Colorado at Boulder

Colorado Math
- a. Add, subtract, multiply and divide rational numbers including integers, positive and negative fractions and decimals (Grade 8) [2009]
- b. Use representations of linear functions to analyze situations and solve problems (Grade 8) [2009]
Colorado Science
- b. Describe methods and equipment used to explore the solar system and beyond (Grade 8) [2009]
International Technology Education Association-ITEA STL Standards Technology
- C. Many inventions and innovations have evolved using slow and methodical processes of tests and refinements. (Grades 6 - 8) [2000]

Does this curriculum meet my state's standards?

A sextant's accuracy is expressed in "seconds of arc." A degree is divided into 60 minutes (noted as 60') and a minute is divided into 60 seconds (noted as 60''). A good thing to remember is that each minute of angular measurement represents a distance of one nautical mile. A sextant scale can generally read out to one-fifth or one-tenth of a minute — quite an accurate reading. But, that reading is not the final accuracy, as there are several corrections that must be made to that angle. At this point, the navigator must perform what amounts to a full page of corrections and calculations using astronomical tables and charts. The accuracy of the correction values affects the final result and the calculations offer ample opportunity for human error.

Depending on the accuracy of the corrections, the final angular measurement could easily be off by several minutes or more, so most navigators (assuming they were skilled and had good weather) could expect at best an accuracy of within a few miles.

Today, refined manufacturing techniques and robust materials make sextants last longer but do not actually improve the accuracy of the tool, which is limited by the fuzzy edges of celestial objects. Skill in use and better understanding of the math and geometry involved can improve accuracy slightly (compared to the past). But, there is still the large chance of error in the many pages of calculations needed! This is where a modern advantage finally comes in — the computer.

When used properly by an experienced navigator and under ideal weather conditions, a well-made sextant can measure an angle with precision to the nearest ten seconds of arc (10 seconds of a degree is about 0.003 degrees of a 360-degree circle!). A computer can do the corrections and calculations quickly, and an accuracy of 0.2 miles in final position is possible. More likely, it will be about twice that under normal weather conditions (0.4 miles), and in poor conditions, it may still be 1-2 miles off. This is no better than measurements taken in good weather conditions hundreds of years ago, but thanks to the computer, navigators no longer have to do all that math by hand.

A picture describing a source of error when using a sextant: measuring an angle between the horizon and the sun incorrectly by looking down at the horizon.

Figure 1. How a sextant works.
click for copyright

Sextant Use and Error

The sextant is a high precision instrument. Caution must be used when handling a sextant, as even shaking it mildy might cause damage. The movable arm has an arc range of 60 degrees, and this is why it is called a sextant. You double this measurement to 120 degrees to find your altitude angle. Every sextant has an inherent error, which is called its offset. Sextants can be calibrated to determine their offset. Once the offset is known, you can correct for the error of the sextant calculations are performed.

Besides the sextant offset, there are many other sources of error. Int this activity, we will look at two sources of error when using a sextant.

If you look at the picture above, you can see that the person who looks at the horizon is not actually looking straight, but down a bit. This is because of the earth is round, not flat. The angle that you look down depends on how tall you are. If you were on top of a building that was in the middle of a big field, you would have to look down quite a bit to see the horizon. If you were lying on your stomach in the field, you would not have to look down at all (this is illustrated in the Sextant Corrections Worksheet). It is easier to use a sextant when you are standing, so the angle that you are measuring is actually larger than the true altitude. This error is called the "dip of the sea horizon." Luckily, it is easy to figure out using the following formula:

DIP equals one point seven five three times the squareroot of H.

H is the height of your eye in meters and DIP is the correction in minutes of arc. Subtract this from the angle you measure off the sextant.

Another source of error is the refraction effect of the atmosphere. The atmosphere of the earth bends the light coming from the sun. The sun might be below the horizon, but the atmosphere will bend the sunlight so that you still see it. Just like the DIP, this makes the altitude seem larger than it is. The amount of bending depends on the atmospheric pressure, the temperature and your altitude. A good approximation for this error is:

Delta equals

The triangle is called delta and is in minutes of arc

P is the atmospheric pressure in millibars (1 atm = 1013 millibars)

T is the temperature in degrees Kelvin

Alt is the altitude in degrees (reading from the sextant corrected for dip)

Before the Activity

Print out enough Sextant Corrections Worksheets for each group or individual.
The MS Excel® files are write-protected against changes (with the exception of the data entry boxes), but the protection can be removed, if necessary. If students are fairly computer savvy, a password may be added to further guard against file corruption (but, if you are going to do this, make the change and save the file before loading onto each student computer). To add a password to the file, the following instructions are provided:

On the Tools menu, point to Protection, and then click Unprotect Sheet.
Then again, from the Tools menu, point to Protection, and then click Protect Sheet.
When prompted, leave all boxes checked and enter a desired protection password for the worksheet. Passwords are case sensitive. To unprotect the sheet again, you must type the password exactly as it was created, including uppercase and lowercase letters.

Load the Sextant Corrections Excel File onto all computers and put it in an easy to access place (Desktop, for instance), or better yet, have it opened up when students arrive.

Note: In the "Refraction of the Atmosphere" section, the Temperature and Pressure data boxes are not protected. This is to allow the option of investigating these variables, but they are not highlighted to keep the basic lesson more focused. See Activity Scaling section.

With the Students

Before students go to the computers:

Divide class into groups (depending on the number of computers available) and give each student or group the Sextant Corrections Worksheet.
Discuss the concepts of the "Dip of the Sea" correction. If students have calculators have them check the 2-meter height example answer.
Discuss the concepts of the "Refraction of the Atmosphere" correction. Reassure them they will NOT have to do this calculation by hand. Emphasize that the computer will be doing that calculation every time they put in a new number. This allows students to try many angles and look for trends in the results.

At the Computers:

Have students do the "Dip of the Sea" correction and answer the questions on the worksheet.
Have students do the "Refraction of the Atmosphere" correction and answer the questions at the bottom of the Worksheet.
Each group or individual should turn in the worksheet when complete. No print out is needed.
Try doing the Refraction example equation. The 10 degrees Celsius temperature must be converted to Kelvin (283.15 degrees) when used in the equation, and all other values are as shown. Note the results are given in minutes of arc, and there are 60 minutes of arc in 1 degree.

Pre-Activity Assessment

Discussion Questions: Solicit, integrate and summarize student responses.

Who would believe me if I told you that when you are looking at a sunset, the sun has actually already set? Encourage discussion: How much can we trust our eyes? Is the sun setting or are we? How could the sun have already set if we can still see it? (Answer: It is a true statement because the Earth's atmosphere refracts (bends) the rays of sunlight over the horizon, allowing us to still see the rays for a while after the sun has geometrically set!

Activity Embedded Assessment

Worksheet/Computer Calculations: Have the student complete the activity worksheet; review their answers to gauge their mastery of the subject.

Students follow and complete the Excel file and worksheet.

Post-Activity Assessment

Questions/Answers: Ask the students and discuss as a class:

Would someone using a sextant on the moon need to make these same corrections? Why or why not? How might they be different? (Answer: The horizon dip effect would still need to be corrected on the moon and it would be larger because the moon is smaller than Earth; therefore, its horizon "dips" away even faster than Earth's. Imagine standing on a basketball! Looking down, you can see almost 90 degrees around the horizon of the ball. This is a HUGE dip error. The refraction of the atmosphere correction would not be needed since the moon has almost no atmosphere.)