Foundations of Algebra

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What Is It?

Algebraic thinking involves finding and describing patterns, making generalizations about numbers, using symbols and models to represent patterns, quantitative relationships, and changes over time. Here are some typical algebraic expressions:

Distance = Rate x Time
total Apples = num Trees x apples Per Tree
a2 + b2 = c2

Why Is It Important?

The National Council for Teachers of Mathematics (NCTM) has extended the algebra standards to pre-kindergarten, citing research that algebraic concepts need to be grounded in extensive experience and developed over a long time (Sfard, 1991).

Teaching these concepts early using the language in which most of mathematics is communicated provides students with a solid foundation for understanding more ambitious mathematical thinking in the higher grades. Younger student should start learning algebraic concepts such as patterns, multiple representations, and modeling mathematical relationships, such as change over time. Expanding the amount of time students have to explore algebraic concepts and abstract ways of thinking increases their chances of success. NCTM considers algebraic representation a prerequisite to formal work in "virtually all mathematical subjects including statistics, linear algebra, discrete mathematics, and calculus" (National Council of Teachers of Mathematics, 1989).

How Can You Make It Happen?

The important algebraic concepts for elementary students to understand are variables, patterns and relationships, equality, constants, and change.


Variables can represent a range of values-numbers that vary-or for an unknown value. Variables typically are represented by italic symbols or letters, such as x, y, or more usefully in words, such as numApples. For example, consider the following problem:

If each tree in an apple orchard produces an average of 325 apples each season, then what is the total number of apples an orchard produces each season?

Start by representing this problem in words.

The total number of apples depends on the number of trees, which can be different for different orchards. If we want to find out how many apples there are, we have to know the number of trees in the orchard. The number of apples is equal to the number of trees times 326.

Once students have an understanding of how to state the mathematical relationships in words, then have them represent their thinking by using variables. The following are simple mathematical models of the relationship between apples and trees in an orchard.

number of apples = number of trees x 325
numApples = numTrees x 325

Variables can be used to represent specific values. For example, if we know the number of trees is 100, we can substitute that information in the expression.

numApples = 100 x 325
numApples = 32,500

When variables represent specific values, they are subject to the same principles and rules of mathematics as numbers. For this reason, variables can be used in mathematical expressions to describe all manner of patterns, relations, or functions. For example, variables can be divided.

numApples⁄numTrees = 325

Encourage students to be clear about what each variable represents (the weight, length, cost, etc.) and to use variable names that convey the meaning of the values they represent. For example, in making mathematical models understandable, this expression:

Total cost = price per person X number of tickets

might be easier to understand than this expression:

c = p x t

Some single-letter variables names are conventional such as x and y for horizontal and vertical components of a point on a graph, r for radius, C for circumference, and a, b, and c for base, height, and hypotenuse of a right triangle.

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