In the toothpick problem, we used a variable to describe the relationship between the number of toothpicks and the number of triangles in the pattern. The use of variables is the most familiar part of working algebraically. We have also seen in the Eric the Sheep problem that algebraic thinking does not always require using variables. The concept of variable, however, is an important one in algebra. In fact, there are many meanings for variables and how they are used in mathematics.
Mathematician Zal Usiskin has outlined four conceptions of algebra based on different uses of variable:
Conception 1: Algebra as generalized arithmetic. Here, variables are indeterminates-- they do not have specific values, but allow you to analyze operations like multiplication and addition.
Example: The sum of 2 even numbers is even: 2a + 2b = 2(a + b).
Example: Any number times zero is zero: 0 x n = 0
Conception 2: Algebra as a study of procedures for solving certain kinds of problems. Here, the variables are unknowns, and you want to solve for them.
Example: When 4 is added to 9 times a certain number, the sum is 40. Find the number. We represent this as 4 + 9n = 40, and n is the unknown solution.
Example: You get paid $10 per hour and earned $30 in tips. In total, you made $380 last week. How many hours did you work? Here, the unknown is the number of hours worked.
Conception 3: Algebra as the study of relationships among quantities. Here the variables really vary, and you look at how changes in one variable affect the others.
Example: In a rectangle, area is length times width: A = L x W.
Example: What happens to the value of 1/x as x gets larger and larger?
Example: In Part B, what happens to the number of toothpicks as the number of triangles increases?
Conception 4: Algebra as the study of structures. This conception of algebra explores the nature of numbers and operations, and we will explore this conception in greater detail in Session 9.
It is clear from these descriptions that variables can be used in different ways and for different purposes. The relative importance given to these multiple uses of variables affects the purposes for which algebra is used as well.