Variables and Expressions

For Elementary Students

What Is a Variable?

A variable is a letter that stands for a number we don't know yet!

Think about it like this: It's like a mystery box! The letter (like x or n) is the box, and inside is a number we're trying to find!

Common variables:

• x (most common!)
• y
• n (for "number")
• a, b, c

Why Use Variables?

Example: "I'm thinking of a number. If I add 5 to it, I get 12. What's my number?"

Instead of saying all that, we can write: x + 5 = 12

• x = the mystery number
• Much shorter and easier!

What Is an Expression?

An expression is like a math phrase using numbers, variables, and operation signs (+, −, ×, ÷).

Examples:

• x + 3 (x plus 3)
• 2n (2 times n)
• y − 7 (y minus 7)
• a ÷ 4 or a/4 (a divided by 4)

No equals sign! If there's an = sign, it's an equation (we'll learn that later).

Translating Words to Expressions

Let's turn word problems into math!

Example 1: "A number plus 5"

• Variable: let's use n for "number"
• Operation: plus
• Expression: n + 5 ✓

Example 2: "Twice a number"

• "Twice" means 2 times
• Expression: 2n ✓

Example 3: "3 more than a number"

• "More than" means add
• Expression: n + 3 ✓

Common Phrases Chart

Tricky One: "Less Than"

WATCH OUT! "Less than" flips the order!

Example: "5 less than a number"

• Think: Start with the number, then subtract 5
• Expression: n − 5 ✓
• NOT 5 − n ✗

Parts of an Expression

In the expression 3x + 7:

3x + 7
↑  ↑ ↑
│  │ └─ constant (plain number)
│  └─── plus sign
└────── term with variable

• 3x = 3 times x
• 7 = just a number (called a constant)

Evaluating Expressions

Evaluating means: plug in a number for the variable and calculate!

Example: Evaluate x + 4 when x = 5

Step 1: Replace x with 5
        5 + 4
Step 2: Calculate
        9

Answer: 9 ✓

Example 2: Evaluate 2n when n = 8

Replace n with 8:  2 × 8 = 16

Answer: 16 ✓

For Junior High Students

Understanding Variables

A variable is a symbol (usually a letter) that represents an unknown or changeable value.

Purpose of variables:

• Represent unknown quantities
• Express general relationships
• Formulate equations and functions

Vocabulary:

• Variable: A letter representing a number (e.g., x, y, n)
• Constant: A fixed numerical value (e.g., 5, −3, 2.7)
• Coefficient: The number multiplied by a variable (in 3x, the coefficient is 3)
• Term: A single number, variable, or product (e.g., 5, x, 3y)
• Expression: A mathematical phrase combining terms (e.g., 2x + 7)

Algebraic Expressions

An algebraic expression is a combination of:

• Variables
• Constants
• Operations (+, −, ×, ÷)
• Grouping symbols (parentheses)

Key characteristic: No equals sign (that would make it an equation)

Examples:

• x + 5 (sum)
• 3y − 2 (difference)
• 4n (product)
• a/2 (quotient)
• 2(x + 3) (with grouping)

Anatomy of an Expression

Example: 5x + 3

Example: 3a − 7 + 2a

• Terms: 3a, −7, 2a (separated by + or −)
• Variable terms: 3a, 2a
• Constant term: −7
• Coefficients: 3 (of a), 2 (of a)

Writing Expressions from Words

Translate verbal phrases into algebraic expressions.

Addition:

• "Sum of x and 5" → x + 5
• "3 more than n" → n + 3
• "Increased by 8" → x + 8

Subtraction:

• "Difference of y and 4" → y − 4
• "7 less than a number" → n − 7
• "Decreased by 2" → x − 2

Multiplication:

• "Product of 6 and x" → 6x
• "Twice a number" → 2n
• "Triple y" → 3y

Division:

• "Quotient of a and 4" → a/4 or a ÷ 4
• "Half of n" → n/2
• "x divided by 5" → x/5

Combined operations:

• "5 more than twice a number" → 2n + 5
• "3 less than the product of 4 and y" → 4y − 3
• "Half of x, plus 7" → x/2 + 7

Order Matters: "Less Than" and "Divided By"

"Less than" reverses order:

"5 less than n" means start with n, then subtract 5

n − 5  ✓
NOT 5 − n  ✗

"Divided by" also matters:

"x divided by 3" means x is being divided

x/3  ✓
NOT 3/x  ✗

Example: "The difference of 10 and x"

• "Difference of A and B" means A − B
• Answer: 10 − x ✓

Evaluating Expressions

Evaluating an expression means substituting a specific value for the variable and computing the result.

Process:

1. Substitute the given value for each variable
2. Follow order of operations (PEMDAS)
3. Simplify to a single number

Example 1: Evaluate 3x + 4 when x = 5

= 3(5) + 4
= 15 + 4
= 19

Example 2: Evaluate 2y − 7 when y = 10

= 2(10) − 7
= 20 − 7
= 13

Example 3: Evaluate (a + 3)/2 when a = 9

= (9 + 3)/2
= 12/2
= 6

Example 4: Evaluate 5n − 2n + 8 when n = 4

= 5(4) − 2(4) + 8
= 20 − 8 + 8
= 20

Coefficients and Constants

Coefficient: The numerical factor of a term containing a variable

In 7x, the coefficient is 7 In x, the coefficient is 1 (understood, not written) In −3y, the coefficient is −3

Constant: A term with no variable

In x + 5, the constant is 5 In 3n − 8, the constant is −8

Example: Identify coefficients and constants in 4x + 9 − 2y

• Coefficient of x: 4
• Coefficient of y: −2
• Constant: 9

Like Terms

Like terms have the same variable(s) raised to the same power(s).

Examples of like terms:

• 3x and 5x (both have x)
• 2y and −7y (both have y)
• 8 and −3 (both are constants)

Examples of unlike terms:

• 3x and 3y (different variables)
• x² and x (different powers)

Combining Like Terms

Add or subtract the coefficients of like terms.

Example 1: 3x + 5x

= (3 + 5)x
= 8x

Example 2: 7y − 2y + 4

= (7 − 2)y + 4
= 5y + 4

Example 3: 4a + 6 + 2a − 3

Combine a terms: 4a + 2a = 6a
Combine constants: 6 − 3 = 3
Result: 6a + 3

Example 4: 5x + 3y − 2x + 7y

Combine x terms: 5x − 2x = 3x
Combine y terms: 3y + 7y = 10y
Result: 3x + 10y

The Distributive Property

Property: a(b + c) = ab + ac

Example: 3(x + 4)

= 3 · x + 3 · 4
= 3x + 12

Example: 5(2n − 3)

= 5 · 2n − 5 · 3
= 10n − 15

Real-Life Applications

Geometry: Perimeter of a rectangle with length l and width w

P = 2l + 2w

Shopping: Total cost of n items at $3 each plus $5 shipping

C = 3n + 5

Temperature: Converting Celsius C to Fahrenheit

F = 9C/5 + 32

Travel: Distance traveling at 60 mph for t hours

d = 60t

Common Mistakes

Mistake 1: Writing "5 less than x" as 5 − x

❌ 5 − x ✓ x − 5

Mistake 2: Forgetting the multiplication sign

❌ Writing 2x as 2 + x ✓ 2x means 2 × x

Mistake 3: Combining unlike terms

❌ 3x + 2y = 5xy ✓ 3x + 2y (cannot combine)

Mistake 4: Incorrect coefficient

❌ Coefficient of x is 0 ✓ Coefficient of x is 1 (written as just x)

Tips for Success

Tip 1: Read word problems carefully — order matters!

Tip 2: When evaluating, use parentheses when substituting: 2(5) not 25

Tip 3: Only combine like terms (same variable and power)

Tip 4: Remember: 2x means 2 × x, not 2 + x

Tip 5: Check your translation by reading it back in words

Tip 6: Use PEMDAS when evaluating expressions

Practice