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

**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.

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.

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.

- 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?