You might be surprised to find an activity on algebra in a module featuring the history of length, area, and volume. However, as we shall see, early practitioners of mathematics thought of numbers, arithmetic, and, yes, even algebra, almost entirely geometrically. The Greek mathematician Euclid, in Book VII of his famous mathematical compendium, the Elements, defines products of two and three numbers as follows (Heath, vol. 2, 278).
Definition VII.16: And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.
Definition VII.17: And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.
Thus, the product of two positive numbers is the area of a rectangle with the lengths of its sides equal to the numbers, while the product of three positive numbers is the volume of a rectangular solid with the lengths of its sides equal to the numbers, as shown in Figure 1.
Euclid then gives the special cases of a square number and a cube.
Definition VII.18: A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
Definition VII.19: And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
For Euclid, a square number is the area of a square figure, and a number thatŐs a cube is the volume of a geometric cube, as in Figure 2.
Although Euclid wrote the Elements in approximately 300 BCE, these definitions go back at least as far as the Greek mathematician and mystic Pythagoras (approximately 580-500 BCE) and his followers, the Pythagoreans. The tendency to visualize arithmetic and algebraic operations geometrically persisted at least to the middle of the sixteenth century. While it resulted in the discovery and proof of many important mathematical formulas, it had obvious limitations. How does one visualize the product of four or more positive numbers? For that matter, how does one visualize products of negative numbers or of imaginary (complex) numbers?
By Ňgeometric algebra" we mean establishing algebraic formulas and solving algebraic equations by computing areas of plane figures or volumes of solid figures. For example, Figure 3 illustrates the algebraic identity for positive numbers a and b. Notice that, as a square with side length the figure has area On the other hand, when dissected into four rectangles as shown in Figure 3, the figure has area = Hence, since and give the area of the same figure, we have Euclid states this result as Proposition 4 of Book II of the Elements (Heath, vol. 1, 379).
Proposition II.4: If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.
Problems from EuclidŐs Elements:
1. Illustrate each algebraic identity using a diagram like that used to illustrate (Figure 3).Explain how to compute the area of your diagram in two different ways in order to obtain the identity. Assume that a, b, c, and d are positive numbers(Avelsgaard, 19).
Parts (a) and (b) are special cases of Proposition 1 of Book II of Euclid's Elements.
2. Illustrate the algebraic identity using the two diagrams shown in Figure 4 (Avelsgaard, 19). Assume that a and b are positive numbers with a > b. Begin by showing that the area of the rectangle in Figure 4a is Then shade part of Figure 4b so that a region with area remains. The next step is to show that the two figures have the same area without doing any algebra. To accomplish this, move one piece of one of the figures (either figure works!) to obtain the other figure.
Sir Thomas Heath, a renowned translator of Euclid's Elements, points out that the identity is implied by each of Propositions 5 and 6 of Book II of Euclid's Elements (Heath, vol. 1, 383).
3. Illustrate the identity = using a cube(Serra, 541). Explain how to compute the volume of your cube in two different ways in order to obtain the identity. Assume that a and b are positive numbers.
Objective: Students will see how Euclid derived algebraic identities using geometric figures.
Materials: For Problems 1 and 2, you may wish to have students use cardboard, wooden, or plastic tiles. For Problem 3, you might construct a model from wood or Styrofoam.
How to Use: Have students work on this activity individually or in small groups. They may need extra hints for Problems 2 and3.
Extension Activity: Have students use Figure 4b to explain why
Solutions: 1a. 1b.
2. Since Figure 4a has height and width then its area is Now shade the lower right-hand corner of Figure 4b, as shown in Figure 4bŐ, to obtain a region with area
To see that Figures 4a and 4bŐ have the same area, students may move the lower left rectangle of Figure 4bŐto the right side of the figure to obtain a copy of Figure 4a, as shown in Figure 4c. Or, they may move the rectangle from the right side of Figure 4a to the bottom of the figure to obtain a copy of Figure 4bŐ, as shown in Figure 4d.
3. As a cube with side length the solid has volume When dissected into eight rectangular solids, as shown in the figure below, the volume is
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.
Avelsgaard, Carol, History of Algebra Laboratory Exercises (unpublished classroom activities), Middlesex Community College, Middlesex, New Jersey,1996.
Heath, Sir Thomas, Euclid: The Elements(3 vols.), St. John's College Press, Annapolis, Maryland, 1947.
Serra, Michael, Discovering Geometry: An Inductive Approach, Key Curriculum Press, Berkeley, California, 1997.