Part 1: Translating Words Into Symbols
Translating words into symbols is equivalent to modeling a situation using an equation and variables. Similarly, algebraic equations and inequalities can represent the quantitative relationship between two or more objects.
Variables have important roles in algebra. Often, they serve as placeholders in equations for which there are unknown quantities. In such cases, finding the specific value of the variable for which an equation is true yields the solution to the problem. Students are likely to be familiar with this from early elementary school, when they filled in the square to make a statement such as 5 + = 13 true.
One example of a problem that uses a variable in this way involves finding the width of a rectangle when the area and length are known. If the area is 36 square units and the length is 9 units, an equation for finding the width (w) of the rectangle is 9w = 36. In this situation, the variable w does not vary; it is a placeholder representing 4 units.
Another use of variables is to represent quantities that truly vary. In the area formula 36 = lw, the value of each variable depends on the value of the other. Consequently, as the value of one variable changes, the values of the other variables change, too. Used in this manner, variables serve several purposes that we will explore in the Role in the Curriculum section of this workshop.
- If a runner jogs one mile in eight minutes, the number of miles the
runner covers in t minutes can be represented by the expression
Therefore, the equation
could be used to determine how long it took to run
More generally, if the distance run is d miles, the linear function
can be used.
- To determine the number of 1-foot tiles needed to construct a border around a pool that measures l feet by w feet, consider that the two lengths need l tiles each, the two widths need w tiles each, and four tiles are needed at the corners. The equation could be 2l + 2w + 4. Equivalently, however, the following expressions also represent the number of tiles needed: 2(l + w + 2), (l + 2)(w + 2) - lw and 2(l + 1) + 2(w + 1).
Variable: A symbol used to represent an unspecified member of some set. A variable is a "place holder" or a "blank" for the name of some member of the set. Any member of the set is a value of the variable and the set itself is the range of the variable. If the set has only one member, the variable is a constant. The symbols x and y in the expression x2 - y2 = (x + y)(x - y) are variables that represent unspecified numbers in the sense that the equality is true whatever numbers may be put in the places held by x and y.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary [4th edition]. New York: Chapman and Hall, 1976.)
Role in the Curriculum
Variables, expressions, and equations are important parts of the algebra curriculum. The National Council of Teachers of Mathematics (NCTM) states:
Students' understanding of variable should go far beyond simply recognizing that letters can be used to stand for unknown numbers in equations ... The following equations illustrate several uses of variable encountered in [algebra]:
Students should be able to relate variable expressions and equations to other forms of representations, such as tables, graphs, and verbal descriptions. One way of developing competency in this area is to use a functional approach. Present students with a table of values and have them generate a function to describe the relationship. For instance, consider the following table:
The first equation illustrates the role of variable as "place holder:" x is simply taking the place of a specific number that can be found by solving the equation. The use of variable in denoting a generalized arithmetic pattern is shown in the second equation; it represents an identity when t takes on any real value except 0. The third equation is a formula, with A, L, and W representing the area, length, and width, respectively, of a rectangle. The third and fourth equations offer examples of covariation: in the fourth equation, as x takes on different values, y also varies.
(Principles and Standards for School Mathematics, NCTM, 2000, p. 224)
From this table, students should notice that the y value is always equal to one more than three times the x value. Stating this relationship using variables, y = 3x + 1.
In a similar manner, Janel Green used a real-world problem involving the number of tiles needed to form a border around a rectangular pool. This problem provides a context in which students can use variables to represent a situation. NCTM recommends the pool-border problem as a means of developing fluency in using various representations:
Students should become flexible in recognizing equivalent forms of linear equations and expressions. This flexibility can emerge as students gain experience with multiple ways of representing a contextualized problem. For example, consider the following problem, which is adapted from Ferrini-Mundy, Lappan, and Phillips (1997):
Both examples above give students the opportunity to translate numerical situations into symbolic expressions. However, it is also important for students to translate symbolic expressions into words. For instance, students might be given the expression 6x + 3 and asked to state a situation for which that expression could serve as a representation. A possible response might be, "Harriet has three beads. She can buy more in packets of six beads each. If she buys x packets, she will have 6x + 3 beads." Exercises of this type could be extended to include more complex expressions and equations. In most classrooms, students typically gain extensive experience translating situations into symbolic expressions, but they generally don't encounter opportunities in the other direction. Experience with both types of situations - generating expressions for particular situations, and generating situations for particular expressions - is known as "bi-directional practice," and such practice is crucial for developing in-depth understanding.
A rectangular pool is to be surrounded by a ceramic-tile border. The border will be one tile wide all around. Explain in words, with numbers or tables, visually, and with symbols the number of tiles that will be needed for pools of various lengths and widths.
(PSSM, p. 282)
Effectively translating words into symbols involves recognizing equivalent forms of the same relationship. Representing the same situation in more than one way provides opportunities for students to understand equivalent algebraic expressions. The pool problem Janel Green used is one such example. NCTM states:
Complex symbolic expressions also can be examined, such as the equivalence of 4 + 2L + 2W and (L + 2)(W + 2) - LW when representing the number of unit tiles to be placed along the border of a rectangular pool with length L units and width W units. (PSSM, p. 225)
Once students have found more than one expression that describes the number of tiles needed, they should be asked to find as many equivalent expressions as possible, and to discuss reasons why the expressions are equivalent.
In addition to meeting the NCTM Algebra Standard, the ability to interpret and describe situations in various ways helps students attain the goals of the Representation Standard. By the end of high school, students are expected to understand various representations of the same relationship and effectively represent situations using tables, graphs, and symbolic expressions. NCTM explains this further:
By working on problems like the "tiled pool" problem, students gain experience in relating symbolic representations of situations and relationships to other representations, such as tables and graphs. They also see that several apparently different symbolic expressions often can be used to represent the same relationship between quantities or variables in a situation. The latter observation sets the stage for students to understand equivalent symbolic expressions as different symbolic forms that represent the same relationship. In the 'tiled pool' problem, for example, a class could discuss why the four expressions obtained for the total number of tiles should be equivalent. They could then examine ways to demonstrate the equivalence symbolically. For example, they might observe from their sketches that adding two lengths to two widths (2L + 2W) is actually the same as adding the length and width and then doubling: 2(L + W). They should recognize this pictorial representation for the distributive property of multiplication over addition - a useful tool in rewriting variable expressions and solving equations. In this way, teachers may be able to develop approaches to algebraic symbol manipulation that are meaningful to students.
(PSSM, p. 283)
Students can solve some equations by examination or guess-and-check procedures; other equations may warrant a paper-and-pencil solution or possibly the use of technology and algebra software. A major goal of algebra is for students to acquire fluency with symbols, expressions, and equations, and to be able to represent various situations using algebraic expressions and equations. In this way, they will be able to determine which method of solution is most appropriate for a given problem.