One good way to describe mathematics is that it provides a way to understand and describe the patterns and structures that we can see, and that we can imagine. It does this by revealing the underlying relationships, and how they arise and are changed by various transformations.

For example, we can consider 2 × 3 as a transformation (doubling), and the "answer" 2 × 3 = 6 as a relationship. This example is a very specific, and so is not very useful (unless you just happen to need to give three things to two people). Algebra gives us a way to go beyond this kind of very specific mathematical statement, and instead describe underlying, general rules. In this case, we can write the related algebraic expression 2x to describe doubling—but not just doubling 3 as our arithmetic expression described, but doubling any number.

This is the power of algebra. The algebraic expression above describes doubling for all numbers at once. It doesn't matter if they are big, small, negative, integers, or fractions. By using an x or y (i.e. a pronumeral) in place of a specific number, we have turned an arithmetic statement into algebraic one. If y = 2x, then y will be the double of x whatever number we choose for x.

Algebra can be considered the language of mathematics. It is how we write down relationships, and gives us the rules so we can transform them and employ them in problem solving. The activity below employs various visual patterns, constructed with matchsticks. Drag the matches around to study each pattern and work out its underlying mathematical nature. See if you can describe each pattern using algebra.

Matchstick Patterns
Using algebra to describe patterns mathematically
Choose a basic shape from the drop-down menu, and hit the Draw button. Drag additional matchsticks from the left to extend and study the pattern.

Basic shape:

Fill out as many entries in the table below as you can. Can you discover an algebraic expression for the number of matchsticks needed to make n copies of the basic shape?
Copies of basic shape Number of matches used Copies of basic shape Number of matches used
15
210
3100
4n

Try to make some patterns of your own using other basic shapes and combinations.