Writing Expressions with Numbers and Letters

Where Did Expressions Come From?

People have been doing math for thousands of years. But for most of that time, they wrote everything out in words! Imagine writing "take a number, double it, then add five" every single time instead of just writing 2n + 5. Using letters and symbols to shorten math took a long journey. Let's look at a few key moments.

So the big question people kept trying to answer was: How can we write math ideas quickly and clearly so anyone can understand them? That is exactly what you will learn to do in this lesson — turn everyday situations into short, powerful mathematical expressions.

Core Ideas You Need to Know

Before we start writing expressions, let's nail down four key ideas. Think of these as your toolbox — every expression you build will use at least one of these tools.

See It: Anatomy of an Expression

Let's break apart the expression 3x + 7 and see what every piece does. The diagram below labels each part so you can spot them in any expression.

Notice how the expression has two terms: the first term is 3x (a coefficient times a variable), and the second term is 7 (just a constant). The + sign is the operation that connects them. A coefficient is the number in front of a variable that tells you how many times to multiply. Here, the coefficient 3 means "multiply x by 3."

The cool part is that x can be any number. If x = 2, then 3x + 7 equals 3 × 2 + 7 = 13. If x = 10, then 3x + 7 = 3 × 10 + 7 = 37. The expression stays the same — only the answer changes when we swap in different values for the variable.

Building Expressions Step by Step

The real power of expressions is turning everyday English into math. Every time you see a word problem, you're basically being a translator — going from words to math. Here are the four operations and how they look as expressions.

Here is an important shortcut to remember: when you multiply a number by a variable, you drop the × sign and just write them together. So "4 times n" becomes 4n, not 4 × n. This is called implied multiplication, and it makes expressions easier to read.

You can also combine operations. "Three times a number, plus eight" becomes 3n + 8. "Ten minus twice a number" becomes 10 − 2n. You read the English phrase and translate each piece, one chunk at a time.

The Translation Cheat Sheet

Below is a handy reference table. The left column has common English phrases. The right column shows the matching math expression. Study this table — once you know these patterns, writing expressions becomes much faster.

How to Translate: A Step-by-Step Flowchart

Use this flowchart every time you need to turn a word problem into an expression. Start at the top and follow the arrows!

Watch out for tricky phrases like "5 less than a number." It sounds like 5 − n, but it actually means n − 5. The "less than" tells you to take 5 away from the number. Always ask yourself: "What am I starting with, and what am I doing to it?"

Worked Example

Let's walk through a complete example together, using the flowchart from Section 5.

Common Mistakes (and How to Avoid Them)

Everyone makes mistakes when learning to write expressions. Here are the most common ones so you can dodge them.

Looking Ahead: From Expressions to Equations

Right now, you're learning to write expressions — math phrases without an equals sign. But very soon, you'll use these same skills to write equations, which include an equals sign and let you solve for the unknown variable. Expressions are the building blocks of equations!

Think of it this way: an expression is a question, and an equation is a statement. You need to be great at writing expressions before you can master equations. Everything you learn in this lesson will be used again and again in 7th grade, 8th grade, and beyond!

Practice Problems

Try these five problems on your own. Click "Show Answer" when you're ready to check your work. The problems get a little harder as you go — that's on purpose!

Lesson Summary