Writing Algebraic Expressions

Learn how to translate words and real-world situations into algebraic expressions using variables and operations.

Intermediate25 minLesson
Printable worksheet

Definition

Writing algebraic expressions means translating words into math using numbers, variables, and operations.
A variable (like xx, nn, or yy) represents an unknown number.

Key Phrases to Know

OperationKey WordsExample PhraseExpression
Addition (++)sum, plus, more than, increased by, totalfive more than a numberx+5x + 5
Subtraction (−-)difference, minus, less than, decreased by, fewera number decreased by 3x−3x - 3
Multiplication (×\times)product, times, of, twice, tripletwice a number2x2x
Division (÷\div)quotient, divided by, ratio, pera number divided by 4x4\frac{x}{4}
Important: In "less than" and "more than" phrases, the order matters!
  • "5 more than xx" means x+5x + 5
  • "5 less than xx" means x−5x - 5 (not 5−x5 - x)

Try it now

Which expression represents "the sum of a number and 8"?

Worked Examples

Write an expression for: "The sum of a number and 12"
1

Identify the operation

"Sum" means addition (++) → Operation: ++
2

Identify the quantities

"A number" (unknown) and 12 → Variable and 12
3

Choose a variable

Let nn represent "a number" → nn
4

Write the expression

Sum means add: n+12n + 12

Common Mistakes

Writing "5 less than xx" as 5−x5 - x instead of x−5x - 5

Why it's wrong: "Less than" reverses the order! "5 less than xx" means start with xx and take away 5.

Correct: "5 less than xx" = x−5x - 5. Think: "I have xx dollars and spend 5" leaves x−5x - 5.

Forgetting to use parentheses when needed

Why it's wrong: "Twice the sum of xx and 3" needs parentheses to show you add first.

Correct: "Twice the sum of xx and 3" = 2(x+3)2(x + 3), not 2x+32x + 3

Confusing "a number" with a specific number

Why it's wrong: "A number" means an unknown value that should be a variable.

Correct: Always use a variable (xx, nn, yy) when you see "a number," "some number," or "an unknown."

Interactive Visual

Function graph

0123456020406080ty

Tap or hover over the graph to read off values.

12t + 5

Table of values

t01234
y517294153

Algebra tiles: combining like terms

Each tile is a term: big squares are x², long strips are x, small squares are 1. Red tiles are negative.

3x3x
44

3x+43x + 4

Every term becomes tiles: the coefficient tells how many.

=x2= x^2=x= x=1= 1negative

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
Which expression represents "the sum of a number and 8"?

Why It Matters

Writing algebraic expressions is the foundation of algebra! Here's why it matters:
  • Problem Solving: Real-world problems are written in words. To solve them, you must first translate them into mathematical expressions.
  • Communication: Expressions let you describe patterns and relationships concisely. Instead of saying "add 5 to whatever number you have," you write x+5x + 5.
  • Programming: Every app, game, and website uses expressions. When a game calculates your score as "base points times bonus multiplier," that's p×mp \times m!
  • Science: Formulas like distance = speed ×\times time (d=std = st) are algebraic expressions describing how the world works.

Real World Applications

Shopping and Budgeting

Stores use expressions to calculate totals with discounts, tax, and multiple items.

Example:

If shirts cost 15 dollars each and you have a 10 dollar coupon, the cost for ss shirts is 15s−1015s - 10.

1Try It Yourself

A streaming service charges 8 dollars per month plus a one-time 20 dollar setup fee.

Write an expression for the total cost after mm months.

Step 1: Write the mathematical expression

Monthly cost times months plus setup fee:

Sports Statistics

Athletes and coaches use expressions to track performance and predict outcomes.

Example:

If a basketball player scores pp points per game, their total after gg games is pgpg.

2Try It Yourself

A runner improves by 30 seconds each week. Their initial time was 8 minutes (480 seconds).

Write an expression for their time after ww weeks of training.

Step 1: Write the mathematical expression

Initial time minus improvement per week times weeks:

Cooking and Recipes

Recipes are scaled using expressions when cooking for different numbers of people.

Example:

A recipe uses 2 cups of flour per person. For pp people, you need 2p2p cups.

3Try It Yourself

A pizza recipe needs 3 cups of cheese for every 2 pizzas, plus 1 extra cup for the crust.

Write an expression for the cheese needed for nn pizzas (where nn is even).

Step 1: Write the mathematical expression

Cheese per 2 pizzas times number of pairs plus extra:

Key Takeaways

  • 1Algebraic expressions translate words into math using variables and operations
  • 2Key words signal operations: sum (+), difference (-), product (x), quotient (/)
  • 3Watch word order carefully: "5 less than xx" means x−5x - 5, not 5−x5 - x
  • 4Use parentheses to group operations: "twice the sum" = 2(x+y)2(x + y)
  • 5Variables represent unknown or changing quantities in real situations

Frequently Asked Questions

No! You can use any letter. However, some letters are traditional: tt for time, dd for distance, nn for "a number." Use whatever makes sense for the problem.
No! You can use any letter. However, some letters are traditional: tt for time, dd for distance, nn for "a number." Use whatever makes sense for the problem.
In algebra, we often skip the multiplication sign between a number and a variable. Writing 2x2x is the same as 2×x2 \times x. This makes expressions cleaner and easier to read.
Use parentheses when you need to do an operation on a group. "Twice the sum of xx and 5" needs parentheses: 2(x+5)2(x + 5). Without them, 2x+52x + 5 means "twice xx, then add 5" - a different thing!

Glossary

Algebraic expression
A combination of numbers, variables, and operations (like 3x+53x + 5)
Variable
A letter that represents an unknown or changing number (like xx, nn, or tt)
Coefficient
The number multiplied by a variable (in 3x3x, the coefficient is 3)
Constant
A number without a variable (in 2x+52x + 5, the constant is 5)
Term
A single part of an expression separated by + or - (in 3x+53x + 5, the terms are 3x3x and 55)
Today's challengeOne problem, three tries

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