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Show Me the Way to Go Home

One of the fundamental problems faced by the early explorers was figuring out where they were and which direction they were heading. Magnetic compasses were rare and extremely inaccurate – much too inaccurate to use for long-distance travel (and the Earth’s magnetic north pole is not at the geographical north pole anyway!). Travelers quickly realized that during the day the motion of the Sun could tell them which direction was north. Knowing the direction of north allowed them to travel in any direction they needed and still keep up with where they were.

In this activity, students will learn how to determine the direction of true north using only a shadow cast by the Sun. If the students have performed Marsdial Activity #1, they will already be well acquainted with the materials they will be using in this activity.

Grade Levels: 6-8

Time Frame: 30 minutes plus at least one measurement approximately 1-1.5 hours before and after local solar noon

Objectives

Educational:
Students will develop skills in working with geometric measuring tools in a real-world application by measuring shadow lengths and angles and by using this information to achieve the goal of finding direction.

Real World Application:
Students will learn how to find true north so that they can begin to navigate without the use of modern technology.

National Council of Mathematics Teachers Principles and Standards:

Geometry: Use Visualization to Solve Problems

  • draw geometric objects with specified properties, such as side lengths or angle measures;
  • recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Measurement: Understand Measurable Attributes

  • understand both metric and customary systems of measurement;
  • understand relationships among units and convert from one unit to another within the same system.

Measure: Apply Appropriate Tools, Techniques, and Formulae

  • select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

What You Do

The general procedure is very simple. Beginning about one to one and half hours before local solar noon (which will occur between 11:30 AM and 12:30 PM, the exact time isn’t important), place the pencil perpendicular to the ground in a marked spot (either on a piece of paper or using sidewalk chalk) and trace its shadow. Accuracy is very important. Measure the length of the shadow in either English or S.I. (metric) units. As time passes, the shadow length will change. Measure the shadow periodically over the next three hours until the shadow length is again exactly equal to the initial measurement. Trace the shadow as before.

The line which bisects the angle formed by the two traced shadows is exactly true north! The simplest way to bisect the angle is to simply measure the angle with a protractor and draw a line at exactly half that angle. The method works because the shadow length is always symmetrical about local solar noon (Geometry teachers may want to discuss periodic functions in conjunction with this activity). You should start measuring at least an hour before solar noon because as solar noon approaches, the changes in the shadow length become much smaller and therefore require much more accuracy in measurement. There are a number of extensions to this activity that could be made as well; see the Extensions section below.

Assessment

Using a known reference (a magnetic compass will not show true north), figure out in advance where true north is. A GPS system will show you, or you can simply do this activity yourself ahead of time – it’s extremely accurate! Compare how close your students came to finding true north.

Extensions
  • Geometry students can use the compass procedure to bisect the angle by construction. Using this method and a stick which has been cut to be exactly the initial shadow length, students can find true north in exactly the same way the ancients did!
  • The time at which the shadow points to true north is at local solar noon. By recording the clock time at which this occurs, the students can figure out how many degrees of longitude they are from the “reference longitude” that establishes their time zone. Four minutes of difference represents one degree of longitude away from the reference longitude.
  • The approximate direction of true north can be found any time by rotating a sundial until the shadow on the dial points to the current time. The 12:00 marking will then point to true north. The problem with this is that local solar time and clock time are not the same, so you are introducing a good bit of error. Have your students do both methods and compare their results. This method is much faster, but is less accurate. Is the trade-off worth it?

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  What You Need
  • Unsharpened #2 pencil per student team
  • Measuring device (ruler, tape measure, meter stick) in either English or metric (S.I.) units
  • Protractor
  • Student worksheet
  • A sunny day!

Optional Materials:

  • Compass (the geometric kind!) to bisect the angle without the use of a protractor
  • Watch or clock
 
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Resources
  • Maestro, Betsy. (1999). The story of clocks and calendars: Marking a millennium. New York, NY: Lothrop, Lee, & Shepard Books.
  • Jespersen, James & Fitz-Randolph, Jane. (1999). From sundials to atomic clocks: Understanding time and frequency. Mineola, NY: Dover Publications, Inc.
Vocabulary
  • True north
  • Local solar moon
  • S.I. units
Handouts

Data Log

Credits

Keith Watt, M.A., M.S.
ASU Mars Education Program
Mars Space Flight Facility
Arizona State University
marsed@asu.edu
(480) 965-1788