Writing Expressions

Learn to translate words into mathematical expressions! This crucial skill bridges everyday language and algebra, setting you up for success in higher mathematics.

What Is an Expression?

An algebraic expression is a mathematical phrase that contains numbers, variables, and operation symbols. Unlike an equation, it has no equal sign.

Examples of Expressions:

• 5x + 3
• 2n - 7
• 4(a + 2)
• x/3 + 10

Not Expressions (these are equations):

• 5x + 3 = 18
• 2n - 7 = 15

Understanding Variables

A variable is a letter that represents an unknown number. Variables let us write general rules and formulas.

Common variables: x, y, n, a, b, t What they represent: Any number (the value can vary, hence "variable")

Example: In "5x," the variable is x, and 5 is the coefficient (the number multiplied by x).

Translating Words to Expressions

Addition Phrases

When you see these words, think addition (+):

Key Words:

• sum
• plus
• more than
• increased by
• added to
• total
• altogether

Examples:

• "The sum of x and 8" → x + 8
• "5 more than a number n" → n + 5
• "y increased by 12" → y + 12
• "3 plus twice a number" → 3 + 2x (or 2x + 3)

Order Matters? For addition, order doesn't affect the answer:

• "x plus 7" = "7 plus x" = x + 7 = 7 + x

Subtraction Phrases

When you see these words, think subtraction (-):

Key Words:

• difference
• minus
• less than
• decreased by
• subtracted from
• take away
• fewer than

Examples:

• "The difference between x and 9" → x - 9
• "5 less than a number n" → n - 5
• "y decreased by 3" → y - 3
• "8 minus a number" → 8 - x

Order Matters! For subtraction, order is crucial:

• "5 less than x" → x - 5 (NOT 5 - x)
• "x minus 5" → x - 5
• "5 subtracted from x" → x - 5 (start with x, subtract 5)

Trick: With "less than" and "subtracted from," reverse the order!

• "7 less than n" means start with n, then subtract 7 → n - 7
• "10 subtracted from y" means start with y, then subtract 10 → y - 10

Multiplication Phrases

When you see these words, think multiplication (× or ·):

Key Words:

• product
• times
• multiplied by
• of (when working with fractions/percents)
• twice (means ×2)
• triple (means ×3)

Examples:

• "The product of 6 and x" → 6x
• "4 times a number n" → 4n
• "y multiplied by 5" → 5y
• "Twice a number" → 2x
• "One-half of x" → (1/2)x or x/2

Note: In algebra, we usually write multiplication without the × symbol:

• 5 × x = 5x
• 3 × (n + 2) = 3(n + 2)

Division Phrases

When you see these words, think division (÷ or /):

Key Words:

• quotient
• divided by
• ratio
• per
• split

Examples:

• "x divided by 4" → x/4 or x ÷ 4
• "The quotient of 20 and n" → 20/n
• "y split into 5 equal parts" → y/5
• "24 per x" → 24/x

Order Matters! For division, order is critical:

• "x divided by 5" → x/5 (NOT 5/x)
• "The quotient of 12 and n" → 12/n (first number on top)

Multi-Step Expressions

Often, you'll translate phrases that require multiple operations.

Example 1: Two Operations

Phrase: "Five more than twice a number"

Step 1: Identify the parts

• "twice a number" → 2x
• "five more than" → add 5

Step 2: Write the expression 2x + 5

Example 2: With Grouping

Phrase: "Three times the sum of x and 4"

Step 1: Identify the parts

• "the sum of x and 4" → x + 4
• "three times" → multiply by 3

Step 2: Write with parentheses (grouping is important!) 3(x + 4)

Note: This is different from 3x + 4!

• 3(x + 4) = 3x + 12
• 3x + 4 stays as is

Example 3: Complex Expression

Phrase: "Eight less than the product of 6 and a number"

Step 1: Identify the parts

• "the product of 6 and a number" → 6n
• "eight less than" → subtract 8 (remember to reverse!)

Step 2: Write the expression 6n - 8

Not 8 - 6n! ("Less than" means reverse the order)

Common Patterns to Recognize

Consecutive Numbers

Consecutive integers: n, n+1, n+2 Example: If n = 5, then the next two integers are 6 and 7

Consecutive even integers: n, n+2, n+4 Example: If n = 10, then 12 and 14 follow

Consecutive odd integers: n, n+2, n+4 Example: If n = 7, then 9 and 11 follow

Age Problems

Current age: a Age 5 years from now: a + 5 Age 3 years ago: a - 3 Twice someone's age: 2a

Geometry Formulas

Perimeter of rectangle: 2l + 2w (where l = length, w = width) Area of rectangle: lw Area of triangle: (1/2)bh or bh/2 (where b = base, h = height)

Real-World Applications

Shopping Problem

Situation: Notebooks cost $3 each, and you buy x notebooks.

Expression for total cost: 3x

If you have a $2 coupon: 3x - 2

Distance Problem

Situation: A car travels at 60 mph for t hours.

Expression for distance: 60t

If the car already traveled 20 miles: 60t + 20

Sharing Problem

Situation: You have d dollars to split equally among 4 friends.

Expression for each person's share: d/4

Writing Expressions from Tables

Sometimes you identify patterns in tables and write expressions.

Example Table

| x | y | |---|---| | 1 | 5 | | 2 | 7 | | 3 | 9 | | 4 | 11|

Pattern: y is always 3 more than twice x

Expression: y = 2x + 3

Check: When x = 1, y = 2(1) + 3 = 5 ✓

Common Mistakes to Avoid

Mistake 1: Order confusion with "less than" Wrong: "5 less than x" → 5 - x Right: "5 less than x" → x - 5

Mistake 2: Forgetting parentheses "Twice the sum of x and 3" Wrong: 2x + 3 Right: 2(x + 3)

Mistake 3: Confusing "of" and "more than"

• "5 more than x" → x + 5 (addition)
• "Half of x" → x/2 or (1/2)x (multiplication)

Mistake 4: Writing equations instead of expressions If the problem says "find an expression," don't include = Expression: 3x + 7 ✓ Equation: 3x + 7 = 22 (only if problem mentions "equals")

Mistake 5: Variable confusion Be consistent! If you start with n for "a number," don't switch to x halfway through.

Practice Strategy

Step 1: Identify the operation words Circle or underline key words (sum, product, less than, etc.)

Step 2: Determine the order Watch for "less than" and "subtracted from" - these reverse order!

Step 3: Check for grouping Look for phrases like "the sum of" or "the product of" that need parentheses

Step 4: Write the expression Use clear variable names and proper mathematical notation

Step 5: Test your expression Plug in a number and see if it makes sense

Connection to Equations

Writing expressions is the first step toward solving equations!

Expression: 3x + 7 (a phrase) Equation: 3x + 7 = 22 (a complete sentence with equals)

In the next topic, you'll learn to evaluate these expressions by substituting values for variables!

Master writing expressions and you've unlocked a powerful tool for solving real-world problems with mathematics!