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Just Me and My Shadow

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

Objectives

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.

What You Do

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!

Assessment

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.

Questions
  1. Will the curve be the same throughout the year? Why or why not?
  2. Will the curve be the same everywhere on Earth? Why or why not?
  3. How are the English and S.I. curves the same? How are they different?
  4. How do you think the curves would change if you used a different-sized pencil to cast the shadow?
Answers
  1. 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.
  2. 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.
  3. They are identical, except that one has been translated on the graph relative to the other.
  4. 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.

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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
  • Student worksheet and blank graph
  • A sunny day!

 
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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
  • Gnomon
  • Solar time
  • Clock time
  • S.I. units
Extensions

Students can use this activity as a preparation for constructing their own sundials, using the references listed in the Resources section.

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