Using Problems from the History of Mathematics - Assembling Thematic Sequences of Problems

Author(s):
Frank Swetz (The Pennsylvania State University)

Historical sequences of problems from different time periods and cultures can be assembled and assigned as exercises for students to solve and compare. For example, facility in the use of the Pythagorean theorem was valued in all societies:

• A beam of length 30 feet stands against a wall. The upper end has slipped down a distance of 6 feet. How far did the lower end move? (Babylonia, 1800-1600 B.C.E.). [Answer: 18 ft.]
• The height of a wall is 10 feet. A pole of unknown length leans against the wall so that its top is even with the top of the wall. If the bottom of the pole is moved 1 foot farther from the wall, the pole will fall to the ground. What is the length of the pole? (China, 300 B.C.E.) [Answer: 50.5 ft.]

More mathematically mature societies puzzled over indeterminate equations:

• 576 coins have been paid for the purchase of 78 bamboo poles. It is desired to calculate prices for large and small poles. How much is the price of each? (China, C.E. 300) [An answer: 48 small poles at 7 coins each; 30 large poles at 8 coins each]
• A hundred bushels of grain are distributed among 100 persons in such a way that each man received 3 bushels, each woman 2 bushels, and each child half a bushel. How many men, women, and children are there? (Europe, 775) [An answer: 20 men, 0 women, and 80 children]

At times, in instructional situations, the use of historical problems reinforces and clarifies the concept being taught. For example, a discussion on the technique of "completing the square" to find the roots of a quadratic equation is enhanced by reference to actual Babylonian problems from which the concept originated (McMillan, 1984). Consider a problem from 2000 B.C.E.:

If the side of the square in question is taken to be x, then the problem becomes one of solving the equation

x2 + (2/3)x = 35/60

which in Babylonian methodology would be depicted as shown here:

Thus by geometrically completing the square, the new area (see below)is seen to be 35/60 + 1/9 and the algebraic solution situation becomes one of solving

(x + 1/3)2 = 35/60 + 1/9.

Assisted by the use of a hand calculator, modern students would find that x = 1/2 or x = -7/6, and the positive root agrees with the solution found by the Babylonians.

Frank Swetz (The Pennsylvania State University), "Using Problems from the History of Mathematics - Assembling Thematic Sequences of Problems," Convergence (June 2010), DOI:10.4169/loci002055