Writing Expressions with Numbers and Letters - 6th Grade Math

SECTION 1

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.

~1800 BCE

Ancient Babylonians solved math problems using words carved into clay tablets. They described unknowns as "the length" or "the width" instead of using letters.

~250 CE

A Greek mathematician named Diophantus started using abbreviations for unknowns. He used a Greek letter (similar to "s") for the unknown number. This was a big step!

~820 CE

Al-Khwarizmi, a scholar in Baghdad, wrote one of the first algebra textbooks. The word "algebra" actually comes from the Arabic title of his book. He still used words, but he organized the rules clearly.

1637 CE

René Descartes, a French mathematician, popularized using the letters x, y, and z for unknowns. This is the style we still use today!

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.

SECTION 2

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.

What Is an Expression?

An expression is a math phrase that combines numbers, letters (called variables), and operations like +, −, ×, and ÷. Unlike an equation, it does not have an equals sign.

What Is a Variable?

A variable is a letter that stands for a number we don't know yet. Common choices are x, n, and y, but any letter works. It's like a blank space waiting for a value.

What Is a Constant?

A constant is a number that never changes. In the expression 3x + 7, the number 7 is a constant. It always stays 7, no matter what x is.

What Are Operations?

Operations tell you what to do with the numbers and variables. The four main operations are addition (+), subtraction (−), multiplication (×), and division (÷).

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Key Takeaway

Think of a mathematical expression like a recipe. The numbers are your ingredients (2 cups of flour, 5 eggs). The variable is a mystery ingredient you'll figure out later. And the operations (+, −, ×, ÷) are your instructions — mix, fold, stir. Put them together and you get a recipe that describes exactly what to do!

SECTION 3

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.

SECTION 4

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.

ADDITION

n + 5

"a number plus five" — the variable n represents the unknown number

SUBTRACTION

n − 3

"a number minus three" — we subtract 3 from the unknown

MULTIPLICATION

4n

"four times a number" — in algebra, we write 4n instead of 4 × n

DIVISION

n ÷ 2 or n/2

"a number divided by two" — both ways of writing mean the same thing

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.

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Key Takeaway

Translating words into expressions is like translating English to Spanish — there are key vocabulary words you need to know. "Sum" means add, "difference" means subtract, "product" means multiply, and "quotient" means divide. Learn those four words and you can translate almost anything!

SECTION 5

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.

English Phrase	Operation	Expression
a number plus 9	Addition	`n + 9`
the sum of a number and 12	Addition	`n + 12`
7 more than a number	Addition	`n + 7`
a number increased by 4	Addition	`n + 4`
a number minus 6	Subtraction	`n − 6`
5 less than a number	Subtraction	`n − 5`
the difference of a number and 8	Subtraction	`n − 8`
a number decreased by 3	Subtraction	`n − 3`
twice a number	Multiplication	`2n`
the product of 6 and a number	Multiplication	`6n`
triple a number	Multiplication	`3n`
a number divided by 4	Division	`n ÷ 4`
the quotient of a number and 10	Division	`n ÷ 10`
half of a number	Division	`n ÷ 2`

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?"

SECTION 6

Worked Example

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

Maya's Babysitting Earnings

1
Problem

Maya earns $8 per hour babysitting. She also got a $15 tip. Write an expression for the total amount of money Maya earns if she babysits for h hours.
2
Step 1 — Find the Unknown

We don't know how many hours Maya works. That is our unknown quantity.
3
Step 2 — Choose a Variable

The problem already uses h for hours. Let's go with that.
4
Step 3 — Spot the Operations

Maya earns $8 per hour. The word "per" tells us to multiply: 8 × h, which we write as 8h. She also gets a $15 tip, so we add 15.
5
Step 4 — Write the Expression

This expression says: "Multiply the number of hours by 8, then add 15."

8h + 15
6
Step 5 — Quick Check

Let's test it. If Maya works 3 hours: 8(3) + 15 = 24 + 15 = 39

That means she earns $39. That makes sense — $24 from babysitting plus a $15 tip. ✓

SECTION 7

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.

Mistake	Why It's Wrong	Correct Version
Writing "5 less than n" as 5 − n	"Less than" means subtract from the number, not subtract the number.	n − 5
Writing "the product of 3 and n" as 3 + n	"Product" means multiply, not add.	3n
Writing 4 × n instead of 4n	Not technically wrong, but in algebra we drop the × sign and write them together.	4n
Forgetting that order matters for subtraction and division	n − 3 is not the same as 3 − n. Be careful about which number goes first.	Read the phrase carefully to pick the right order.
Using the word "and" to mean addition	"The sum of 4 and n" uses "and" for addition. But "the product of 4 and n" uses "and" for multiplication. Look at the key word before "and."	Focus on the operation word, not "and."

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Key Takeaway

When the phrase says "less than" or "subtracted from," the order flips! Think about it with real life: "10 dollars less than your paycheck" means you start with your paycheck and take away 10, not the other way around. So it's paycheck − 10, which is p − 10.

SECTION 8

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!

Feature	Expression	Equation (Coming Soon!)
Has an equals sign?	No	Yes
Example	`3x + 7`	`3x + 7 = 22`
Can you solve for x?	Not by itself	Yes — x = 5
What it does	Describes a calculation	States that two things are equal
Everyday example	"the cost of n apples at $2 each"	"the cost of n apples at $2 each is $10"

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!

SECTION 9

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!

Question 1: PROBLEM 1 — CONCEPTUAL

What is the difference between a variable and a constant? Give an example of each from the expression 5m + 12.

Show answer

A variable is a letter that stands for an unknown number. In 5m + 12, the variable is m. A constant is a fixed number that doesn't change. In 5m + 12, the constant is 12. (The number 5 is called the coefficient — it tells you how many times to multiply m.) Nice work if you got all three parts!

Question 2: PROBLEM 2 — BASIC

Write a math expression for each phrase: (a) The sum of a number and 14 (b) A number divided by 6 (c) Nine times a number

Show answer

(a) n + 14 — "Sum" means addition, so you add 14 to the variable. (b) n ÷ 6 (or n/6) — "Divided by" means division. (c) 9n — "Nine times a number" means multiply. Remember, we drop the × sign and write 9n.

Question 3: PROBLEM 3 — INTERMEDIATE

Write a math expression for: "Seven less than three times a number." Be careful with the order!

Show answer

The answer is 3n − 7. Here's how to break it down: "Three times a number" translates to 3n. "Seven less than" means you subtract 7 from that result. So the expression is 3n − 7. A common mistake is writing 7 − 3n, but that would mean "three times a number, subtracted from seven," which is a different phrase.

Question 4: PROBLEM 4 — APPLIED / WORD PROBLEM

You and your friends are buying pizza. Each pizza costs $11, and there's also a $4 delivery fee. Write an expression for the total cost if you order p pizzas. Then find the total cost if you order 3 pizzas.

Show answer

The expression is 11p + 4. Each pizza costs $11, so p pizzas cost 11 × p = 11p. The $4 delivery fee is added on, giving us 11p + 4. To find the cost for 3 pizzas, substitute p = 3: 11(3) + 4 = 33 + 4 = $37. Great job if you remembered to add the delivery fee on top of the pizza cost!

Question 5: PROBLEM 5 — CHALLENGE

A rectangle's length is 4 centimeters more than its width. Let w represent the width. Write an expression for the perimeter (total distance around) of the rectangle. (Hint: Perimeter = 2 × length + 2 × width.)

Show answer

Since the width is w, and the length is "4 more than the width," the length is w + 4. Perimeter = 2 × length + 2 × width = 2(w + 4) + 2w = 2w + 8 + 2w = 4w + 8. This is a great stretch problem! You had to write an expression for the length first (w + 4), then plug it into the perimeter formula. If you got 4w + 8, awesome work — you're thinking like an algebraist!

SUMMARY

Lesson Summary

Summary

In this lesson, you learned how to write mathematical expressions that use numbers, operations, and variables (letters that stand for unknown numbers). An expression like 3n + 7 is made up of terms connected by operations. Each term can be a constant (a fixed number like 7), a variable (like n), or a variable multiplied by a coefficient (like 3 in 3n).

You practiced translating English phrases into math by matching key vocabulary words to operations: "sum" means addition, "difference" means subtraction, "product" means multiplication, and "quotient" means division. You also learned to watch out for tricky order-switching phrases like "less than." These expression-writing skills are the foundation for everything that comes next in algebra — including equations, which add an equals sign so you can solve for the unknown. Keep practicing, and writing expressions will start to feel as natural as writing sentences!