The earliest and simplest form of a sundial was called a gnomon. It was
simply a stick that was stuck in the ground. Early people would thrust
a stick, cut to a standard size (usually about the length of the
forearm) in the ground and then measure the shadow length, usually
hand-widths. They knew how many hand-widths corresponded to local noon
for their area, so they could predict how much of the day was left
before sunset. This was critical information, as life got very
dangerous after dark, and it was important that travelers find a safe
place to rest well before nightfall.
The method worked, and it worked surprisingly well. The only numbers
the traveler really had to remember were the number of hand-widths the
stick’s shadow measured at noon and the number of hand-widths the
stick’s shadow measured in the late afternoon. The method was not
without its drawbacks, however. The length of the shadow changed with
time of year, so it could never be used as a clock in the truest sense
of the word. While rarely an issue for early travelers who did not go
far from their homes, the length of the shadow varied with latitude as
well. Nevertheless, the method was simple, used readily-available
materials, and did the job for which it was intended: help get people
into shelter in plenty of time to avoid searching in the dark. In this
activity, your students will recreate this ancient method, but will
refine it to be (within its limitations) a much more accurate means of
keeping time that was available to early travelers. Your students will
be able to amaze and impress their friends with their “wilderness
survival” skills!
The intent of this activity is to give your students a chance to apply
their measurement, graphing, and analysis skills in a real-world
context, but sundials have the added attraction of being fascinating as
well as a lot of fun!
Grade Levels: 6-8
Time Frame: 40 minutes plus at least two measurements outside of class
Educational:
The students will learn to plot collected shadow-length data on a graph
and will extrapolate predictions and measurements from that graph.
Real World Application:
The students will gain an understanding of how local solar time differs
from clock.
National Council of Mathematics Teachers Principles and Standards:
Algebra: Understanding Patterns
- represent, analyze, and generalize a variety of patterns with tables,
graphs, words, and, when possible, symbolic rules;
- identify functions as linear or nonlinear and contrast their
properties from tables, graphs, or equations.
Algebra: Use Mathematical Models
- model and solve contextualized problems using various representations,
such as graphs, tables, and equations.
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.
Data Analysis: Collect, Organize, and Display Relevant Data
- select, create, and use appropriate graphical representations of data.
Measurements will have to be taken throughout the course of the day (or
several days, as long as they aren’t spaced too far apart). The more
measurements your students make, the more accurate their final graph
will be. At a minimum they should take one reading early in the morning
before school, one at noon, and one in the late afternoon. While three
data points will show that the curve is not linear, they will not be
sufficient to really get an accurate curve. Five data points is really
the functional minimum, but obviously the more data points your students
have, the more accurate their graph will be – an important general
principle that is worth pointing out to them!
Once all the measurements have been made and recorded on the student
Data Log, have your students convert from English units to metric (S.I.)
units (or vice versa, if they measured in S.I. units to begin with).
This serves the dual purpose of giving your students practice in
computational skills as well as giving them a better intuitive feel for
the S.I. system of measurement by directly comparing it to the English
system.
When all the conversions are complete, have your students plot both the
English and the S.I. data points on the same graph. Have them
extrapolate a “best fit” curve which passes through all of the data
points. They should notice that the English and S.I. curves have
exactly the same shape, but one curve has been translated (shifted) a
specific distance on the graph. Point out to your students that this is
always the result when a linear translation is applied to a curve.
Anytime we see two curves that are the same shape but offset from one
another, we know that they are related by a linear function.
Once their graph is complete, have your students measure the length of
their pencil’s shadow at some different time of day. By finding the
shadow length on the curve that was plotted, they can then read off the
local solar time. Note that this time may differ slightly from “clock
time” because we assume everyone in the same time zone has the same
local time – which is never true! The “true” local solar time can be as
much as 30 minutes from the local solar time on the other side of a time
zone. The local solar time is really the “actual” time, but we have
established time zones mostly for convenience’s sake. If your students
are interested in this phenomenon, please consult the references listed
below for more fascinating information about the history of timekeeping!
The students will use the attached Data Log, questions, and graph during
the activity. These instruments are also intended for the teacher to
use for assessment.
- Will the curve be the same throughout the year? Why or why not?
- Will the curve be the same everywhere on Earth? Why or why not?
- How are the English and S.I. curves the same? How are they different?
- How do you think the curves would change if you used a different-sized
pencil to cast the shadow?
- The shape of the curve will be the same, but the exact positioning of
the curve on the graph will not. This is because the tilt of the Earth
will cause shadows to be longer in the winter than in the summer – a
fact which could be directly read from the graph.
- No, different latitudes receive sunlight more or less directly than
other latitudes. Because the angle of the sun is different, the shadow
length will be different.
- They are identical, except that one has been translated on the graph
relative to the other.
- A longer pencil would cast a longer shadow, so the curve would have the
same general shape, but the “peaks” (longest shadow lengths) would be
greater.
