So far in math, you've worked with concrete numbers like 5, 18, and 3.14. Arithmetic gives us the rules for adding, subtracting, multiplying, and dividing these numbers. But what happens when we want to talk about a number without knowing exactly what it is?

This is where algebra begins. Instead of using only specific numbers, we start using symbols to represent numbers. These symbols are called variables.

Think of a variable as a placeholder or a box. You might not know what's in the box, but you know it holds a number. We can still talk about it and describe what we might do with it. The most common letters used for variables are $x$, $y$, and $z$, but any letter can be used.

Using variables allows us to create general rules. In arithmetic, you know that $3 + 4 = 4 + 3$. You also know that $8 + 2 = 2 + 8$. We could list examples all day, but with variables, we can state the rule for all numbers at once.

This rule, known as the commutative property of addition, is a core idea in math. Variables make it simple and clear to express.

Once we have variables, we can combine them with numbers and operations to form algebraic expressions. An expression is like a mathematical phrase that doesn't have an equals sign. It's a recipe for a calculation.

For example, if you have some apples, but we don't know how many, we can use the variable $a$ to represent the number of apples. If someone gives you 5 more, you now have $a + 5$ apples. This is an algebraic expression.

Here are a few more examples of turning words into expressions:

Notice that in algebra, we often write multiplication by putting a number right next to a variable, like $2z$. This means "2 times $z$." It's a shorthand that makes expressions cleaner and easier to read. The number in front of the variable is called a coefficient.

Understanding variables and expressions is the first big step into algebra. It shifts thinking from doing calculations with known numbers to describing relationships and patterns that work for any number.

Ready to check your understanding? Let's try a few questions.

With this foundation, you're ready to explore how these expressions can be used to build and solve equations.