Writing Algebraic Expressions

For Elementary Students

What is an Expression?

An expression is a math phrase that uses numbers, variables, and operation symbols (+, −, ×, ÷).

Examples:

• 5 + 3
• x + 7
• 2 × n

No equals sign! (That would be an equation, not an expression.)

What is a Variable?

A variable is a letter that stands for an unknown number.

Think about it like this: It's a placeholder — like a blank space you'll fill in later!

Common variables: x, y, n, a, b

Example: "x + 5" means "some number plus 5"

Translating Words to Math

Certain words tell you which operation to use:

Example 1: Addition

Words: "a number plus 8"

Variable: Let's use n for "a number"

Expression: n + 8

Example 2: Subtraction

Words: "10 minus a number"

Variable: Use x for "a number"

Expression: 10 − x

Important: Order matters! "10 minus x" is 10 − x, not x − 10!

Example 3: Multiplication

Words: "5 times a number"

Variable: Use n

Expression: 5 × n or 5n

Note: In algebra, we usually write 5n instead of 5 × n.

Example 4: Division

Words: "a number divided by 3"

Variable: Use x

Expression: x ÷ 3 or x/3

For Junior High Students

Writing Multiplication Without the × Symbol

In algebra, we avoid using × because it looks like the variable x!

Instead of: 5 × n

Write: 5n (number next to variable)

Instead of: x × y

Write: xy (variables next to each other)

Order Matters for Subtraction and Division

Subtraction:

• "5 less than x" → x − 5 (start with x)
• "x less than 5" → 5 − x (start with 5)

Division:

• "x divided by 4" → x/4 or x ÷ 4
• "4 divided by x" → 4/x or 4 ÷ x

Watch the order carefully!

Two-Operation Expressions

Example: "3 more than twice a number"

Step 1: Identify the operations

• "twice a number" → 2n (multiplication)
• "3 more than" → add 3

Step 2: Combine

• 2n + 3

Example: "5 less than the product of 4 and x"

Step 1: Identify operations

• "product of 4 and x" → 4x
• "5 less than" → subtract 5

Step 2: Combine

• 4x − 5

Common Phrases

"More Than" and "Less Than" (Tricky!)

These phrases reverse the order!

"5 more than x" → x + 5 (not 5 + x, but both are equal)

"5 less than x" → x − 5 (NOT 5 − x — these are different!)

Example:

• "3 less than 10" → 10 − 3 = 7
• "10 less than 3" → 3 − 10 = −7 (different!)

Parentheses in Expressions

Use parentheses to group operations.

Example: "3 times the sum of x and 5"

Step 1: "sum of x and 5" → x + 5

Step 2: "3 times" that whole sum → 3(x + 5)

Important: 3(x + 5) is NOT the same as 3x + 5!

Real-World Expressions

Example: "The cost of n notebooks at $4 each"

Expression: 4n (price × quantity)

Example: "The total cost of a $20 shirt plus a $15 pair of jeans"

Expression: 20 + 15 = 35 (or just 35 if you simplify)

Example: "Your age in 5 years, if you're x years old now"

Expression: x + 5

Example: "The perimeter of a square with side length s"

Expression: 4s (4 sides, each length s)

Variables Can Represent Anything

Variables don't have to be numbers! They can represent:

• Age: a
• Time: t
• Distance: d
• Cost: c
• Number of items: n

Evaluating vs Writing

Writing an expression: Translating words → math

Evaluating an expression: Finding the value when you know what the variable equals

Example: Write "5 more than x" → x + 5

Then, if x = 7, evaluate: 7 + 5 = 12

We will learn more about evaluating expressions later.

Practice