Sticks, Stones, and Rope Activity
Ancient peoples accomplished many practical tasks, such as building houses and measuring the sizes of fields, using fewer tools than we have today. Explain how you could complete each of the following tasks using just sticks, stones, rope, and any number of friends.
1. The first step in building a house is to mark out the foundation.
a. Mark out a circular foundation for the house you plan to build.
b. Mark out a rectangular foundation for your house.
2. Use the formula for the area of a triangle to find the area of an irregularly shaped field, such as the one shown below. Describe how you could use the Egyptian cubit as your unit of measurement. (A cubit is the distance from a person’s elbow to the end of her middle finger. The area of a field would be measured in square cubits.)
Objective: This activity is designed to allow students to relive ancient human history, and to encourage them both to review and to think beyond their own conceptions of geometric figures and methods for constructing and measuring them. The solutions described below include historical methods for constructing and measuring geometric figures.
Materials: Equip students with sticks, pebbles, and string, or let them use their imaginations to invent units of measurement!
How to Use: Students should work in small groups to complete this activity. Read the Solutions section below beforehand so that you are prepared to offer students hints and suggestions, and so that you can discuss with them the histories of their own solutions afterwards. Be prepared to see novel solutions to age-old problems.
Solutions: Please note that these are open-ended problems, each with several viable solutions.
1a. Try these ideas:
i. Stretch a length of rope between you and a friend, who will serve as the pivot. Keepthe rope stretched tight as you walk around your friend, marking the circumference of a circle on the ground with your feet or a stick or pebbles.
ii. Have your friends form a circle by holding hands and stretching out as far as they can go. (Try it: it works!)
Note that these two approaches represent two different conceptualizations of circles, one in terms of radius and the other in terms of circumference.
1b. Try these four ideas:
i. Forma parallelogram with two pairs of sticks of equal lengths. Instead of sticks, you could use lengths of rope stretched straight, which would be easier to move around. To form a rectangle, adjust the sticks until the diagonals, as measured with stretched rope, are equal. See Figure 1. This method of constructing rectangular foundations is used today in various countries, including the United States. According to the ethnomathematician and educator Paulus Gerdes, it is used in some areas of Mozambique (Gerdes, 94-95).
ii. Gerdes gives another traditional Mozambican method for constructing a rectangular foundation using only ropes and a stick (Gerdes, 94-95). Tie together two ropes of equal length at their midpoints, then attach one end of each rope to the ends of a stick. The stick forms one side of the rectangle. Then stretch the ropes out so that both ropes are straight; the loose ends of the ropes now form the other two corners (vertices) of the rectangle. See Figure 2.
iii. Tie equally spaced knots in your rope (or stick twigs through the rope to mark off equal sections), then form a parallelogram with sides of lengths, say, 6 and 10units. Right the parallelogram using the equal-diagonals method described in(a) or by forming a 3-4-5 triangle (a right triangle!) with another piece of rope. Or, you could use two 5-12-13 right triangles to form a 5 x 12 rectangle(see Figure 3), or twelve 3-4-5 triangles to form a 6 x 12 rectangle, or . . ..The method of using 3-4-5 right triangles to form right angles has been attributed to the ancient Egyptians.
iv. Here’s another method of constructing right angles, which students may have learned in geometry class. Using a fairly long stick or a fixed length of rope as the radius, construct two circles of the same radius, as shown in Figure 4. The straight line connecting the intersection points is perpendicular to the line on which the circle centers lie.
To construct the perpendicular at a particular point, make sure that the circle centers are equidistant from that point. Note that if the centers of the circles are r units apart, where r is the radius of each circle, then connecting the centers and the intersection points with straight lines forms two equilateral triangles. These are standard compass and straightedge constructions, known to the classical Greeks of 600-300 BCE and to most high school geometry students!
2. Here is an idea for finding area believed to have been used by the ancient Egyptians:
Divide the region into triangles, as in Figure 5. (Other triangulations are possible.) Mark off cubits on your rope with your forearm, tying knots or sticking a twig through the rope to mark each cubit. Measure the base and altitude of each triangle with your rope. (Use one of the ideas from part 1babove to form right angles.) Compute the area of each triangle using the triangle area formula. Note that your area units will be square cubits. (To check your work, use a different side of the triangle for the base.) If you are working with toothpicks and string to measure the area of a region outlined on a piece of paper, then mark off "cubits" on your "rope” with a stick or toothpick.
References: Activity from Lengths, Areas, and Volumes, by J. Beery, C. Dolezal, A. Sauk, and L. Shuey, in Historical Modules for the Teaching and Learning of Secondary Mathematics, Mathematical Association of America, Washington, D.C., 2003.
Gerdes, Paulus, Geometry from Africa: Mathematical and Educational Explorations, Mathematical Association of America, Washington, D.C., 1999.