Tell Time with your Feet

(a feet-on math lesson for K-8 students)

The address of this document on the World Wide Web is
http://www.math.csusb.edu/faculty/susan/timefeet.html

How to tell time with your feet

Preparation

• Get one or more copies of the shadow table for your location. You can get a custom table from the World Wide Web page listed below the title by clicking the underlined words. You will need to know the latitude and longitude of your location. (You can also this find on the Web. From this linked page, you also get times of sunrise and sunset for your location.) Having trouble getting the right table? Here's more detailed help.
• Of course, this activity only works on a sunny day.

The activity

1. Measure your shadow by pacing it off with your feet, toe to heel. (How can you do this? When you walk, your shadow moves with you.) Remember to stand up straight.
2. Find the closest day of tight.
3. Find the closest day of the year on the shadow table.
4. On the top line (bold face) for the closest day, find the number closest number to your shadow length.
5. One of the two numbers underneath will be (approximately) the time. The first line is times in the morning; the second line for times in the afternoon. The first time, "noon", is not 12:00 clock time, but the time the sun is highest. Similarly, "dusk" is when the sun is just below the horizon, in the morning and the evening.

Questions

1. Check your answer with a clock. Was it close? If it wasn't close, how can you explain the difference?
2. How does this work? In particular:
3. How many feet long was your shadow? Compare with everyone else's answer.
4. Have a very tall person measure their shadow and compare their measurement with the measurement for a short person. Were they almost the same? How could this be?

Extensions

1. Make a bar graph of all the shadow lengths for the class. Find the average shadow length for the class. (Good opportunity for discussing the meaning of average.)
2. Patterns over a day.
• Have the class repeat the shadow activity at several different times during the day.
• Before measuring, predict what the results will be.
• When will shadows be longest? Shortest?
• ll shadows be longest? Shortest?
• Make a graph on graph paper showing shadow length at the times you measured. (Make sure to space times evenly on the horizontal axis, even if you don't do the measuring at even intervals.) See attached form. Before making the graph, predict its shape. In particular, predict the values in between the values you measured.
• Use the values on the table to make a graph, and compare to the graph made from your actual measurements.
3. Patterns over a year.
• Pick a specific time of day, and do the shadow measurement at days throughtout the year.
• Before measuring, and throughout the year, predict what the results will be.
• When will shadows be longest? Shortest?
• Make a graph on graph paper showing shadow length on the days you measured. (Make sure to space the days of the year evenly on the horizontal axis. If you don't do the measuring at even intervals, the points you plot on the graph won't be spaced evenly. This is fine.) Throughout the year, predict the shape of the graph.
• Use the values on the table to make a graph, and compare to the graph made from your actual measurements.

Everyone is 6 feet tall: a related activity

In this activity, each person will measure his/her height with his/her own feet (NOT the feet on a ruler). It has been found that everyone is about 6 of their own feet tall.

For younger children

Preparation

• Tape a big sheet of paper on the wall so that children can mark their head heights.
• Each child needs a sheet of paper, pencil or crayon, and scissors for the foot cutout.
• Get a piece of class-size graph paper for the bar graph.

The activity

1. Trace around your foot on a piece a paper and cut out the foot. Write your name on it.
2. Have someone help you mark your height and name on a paper on the wall, and use your foot cutout to measure how tall you are, measuring from the floor to your head mark.
3. Compare your result with everyone else's, by making a list, an ordered list, and/or a bar graph. Was everyone's measurement close to 6 feet?
4. Originally, long ago, the foot on a ruler was the length of the king's foot. Was this a good way to measure things? Why or why not?

For older students

Preparation

• Tape a big sheet of paper on the wall so that children can mark their head heights.
• Each child needs a sheet of pa heights.
• Each child needs a sheet of paper, pencil, and ruler for measuring his/her foot and height.
• Calculators are optional if students can divide decimals by hand.
• Need class-size graph paper or transparencies for the bar graph.

The activity

1. Trace around your foot on a piece a paper. Measure its length and record the result. Centimeters are more convenient, or inches and tenths of an inch. (Why? You're going to divide by this number later.)
2. Have someone help you mark your height and name on a paper on the wall, and measure your height using the same units.
3. Find the ratio of your height to foot length by dividing.
4. Compare your result with everyone else's, by making a list, an ordered list, and/or a bar graph. Was everyone's measurement close to 6 feet? Find the average for the class. How far did results vary from the average, above and below? What would it mean if someone's result was 5.7? What if it was 6.4?
5. Would the result be different for kids of a different age? Cooperate with another class and find out, or make a whole-school project.
6. Do the same measurements for some "realistic" dolls, including a Barbie. How do they compare to measurements for real people?

Another related activity

1. Have everyone take off their right shoe.
2. Arrange the shoes by length; the shortest shoe ie shoes by length; the shortest shoe is at one end, the longest at the other.
3. Arrange everyone by height.
4. Does the order of the shoes match the order of the people? Are there any big discrepancies, or is it close?

Overview of the math and science

Shadow math/science facts, elementary level

• Short people have short feet, tall people have long feet.
• In the morning and evening, the sun is low in the sky and shadows are long. Around noon, the sun is high and shadows are short.
• In the winter, the sun is lower all day than in the summer. As the year progresses (starting on the shortest day, Dec. 21) the sun gets higher and higher, until it's at its highest point on the first day of summer (June 21). Then it gets lower again.
• In the winter, days are shorter than in the summer. As the year progresses (starting on the shortest day, Dec. 21) days get longer, until the longest day, which is the first day of summer (June 21). Then days get shorter again.

Shadow math facts, intermediate level

• To see how the angle of the sun changes during a day and a year, do a demonstrng a day and a year, do a demonstration with a globe and a flashlight. Most globes are on a stand that is inclined at the proper angle.
• Here are two diagrams that show how the angle of the sun and the length of your shadow are related. For precise relations between distances and angles, you have to use trigonometry.
  

In the first diagram, the sun is fairly high, and the shadow is short.

The facts behind the table, advanced level

The table was made using some rather complicated geometry and trigonometry, based on knowledge on the positions of the earth and sun and how they change in time. It was done by Stuart Levy of the Geometry Center in Minneapolis (thanks, Stuart!) Here is Stuart's explanation of how the calculations work.