Order of Operations

For Elementary Students

What Is the Order of Operations?

When you have a math problem with more than one operation (like + and ×), you need to follow special rules about which one to do first!

Think about it like this: It's like following a recipe — if you mix the ingredients in the wrong order, you get a different result!

Why Does Order Matter?

Example: 3 + 4 × 2 = ?

If you add first:

• 3 + 4 = 7
• 7 × 2 = 14 ❌ WRONG!

If you multiply first:

• 4 × 2 = 8
• 3 + 8 = 11 ✓ CORRECT!

The right answer is 11, not 14!

The Magic Word: PEMDAS

PEMDAS tells you the order:

P  →  Parentheses ( )
E  →  Exponents (powers like 2²)
M  →  Multiplication ×
D  →  Division ÷
A  →  Addition +
S  →  Subtraction −

Memory trick: "Please Excuse My Dear Aunt Sally"

Step 1: Parentheses First!

Always do what's inside parentheses ( ) first.

Example: (5 + 3) × 2 = ?

Step 1: Parentheses: 5 + 3 = 8

Step 2: Multiply: 8 × 2 = 16

Answer: 16

Step 2: Multiplication and Division (Left to Right)

After parentheses, do multiplication (×) and division (÷) from left to right.

Example: 10 − 2 × 3 = ?

Step 1: Multiply first: 2 × 3 = 6

Step 2: Subtract: 10 − 6 = 4

Answer: 4

Step 3: Addition and Subtraction (Left to Right)

Last, do addition (+) and subtraction (−) from left to right.

Example: 8 + 2 − 5 = ?

Step 1: Add first (left to right): 8 + 2 = 10

Step 2: Subtract: 10 − 5 = 5

Answer: 5

A Full Example

Problem: 15 − (2 + 3) × 2 = ?

Step 1: Parentheses first!

• 2 + 3 = 5
• Now we have: 15 − 5 × 2

Step 2: Multiply

• 5 × 2 = 10
• Now we have: 15 − 10

Step 3: Subtract

• 15 − 10 = 5

Answer: 5

Remember the Rule!

Multiplication and division come BEFORE addition and subtraction!

Example: 6 + 3 × 2

Don't add first! Multiply first!

• 3 × 2 = 6
• 6 + 6 = 12 ✓

For Junior High Students

Understanding the Order of Operations

When you have a mathematical expression with multiple operations, you need to follow a specific order to get the correct answer. This standard ensures everyone gets the same result.

Vocabulary:

• Expression — a math statement with numbers and operations (like 3 + 4 × 2)
• Evaluate — to calculate the value of an expression

PEMDAS (or BODMAS)

PEMDAS is a memory trick for the order of operations:

1. Parentheses (or Brackets)
2. Exponents (or Orders)
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

Memory tricks:

• "Please Excuse My Dear Aunt Sally"
• "Big Elephants Destroy Mice And Snails" (for BEDMAS)

Important: M and D have the same priority — work left to right. Same with A and S!

The Rules in Detail

1. Parentheses / Brackets — always evaluate what's inside first
2. Exponents / Orders — powers and roots come next
3. Multiplication & Division — work left to right (they have equal priority)
4. Addition & Subtraction — work left to right (they have equal priority)

Multiplication and division are at the same level. Always resolve them from left to right — don't assume multiplication always comes before division.

Step-by-Step Process

1. Parentheses First

Do anything inside parentheses (or brackets) before anything else.

Example: (3 + 5) × 2

Step 1: Solve inside parentheses: 3 + 5 = 8

Step 2: Multiply: 8 × 2 = 16

Nested parentheses: Work from innermost to outermost

Example: 2 × (3 + (4 − 1))

Step 1: Innermost: 4 − 1 = 3

Step 2: Next level: 3 + 3 = 6

Step 3: Multiply: 2 × 6 = 12

2. Exponents Next

Handle any exponents (powers) after parentheses.

Example: 2 + 3²

Step 1: Exponent: 3² = 9

Step 2: Add: 2 + 9 = 11

Example with both: (1 + 2)² × 3

Step 1: Parentheses: 1 + 2 = 3

Step 2: Exponent: 3² = 9

Step 3: Multiply: 9 × 3 = 27

3. Multiplication and Division (Left to Right)

These have the same priority, so work from left to right.

Example: 8 ÷ 4 × 2

Step 1: Work left to right: 8 ÷ 4 = 2

Step 2: Continue: 2 × 2 = 4

Important: Don't do all multiplication before division! Work left to right.

Correct (left to right): (8 ÷ 4) × 2 = 2 × 2 = 4 ✓

Wrong (multiplication first): 8 ÷ (4 × 2) = 8 ÷ 8 = 1 ❌

Example: 20 ÷ 4 × 5 ÷ 2

Go left to right:

• 20 ÷ 4 = 5
• 5 × 5 = 25
• 25 ÷ 2 = 12.5

4. Addition and Subtraction (Left to Right)

These also have the same priority, so work from left to right.

Example: 10 − 3 + 5

Step 1: Work left to right: 10 − 3 = 7

Step 2: Continue: 7 + 5 = 12

Example: 15 + 8 − 6 + 3

Go left to right:

• 15 + 8 = 23
• 23 − 6 = 17
• 17 + 3 = 20

Full Worked Example

Evaluate 3 + 6 × (5 + 4) ÷ 3 − 7:

Step 1: Parentheses: 5 + 4 = 9 → 3 + 6 × 9 ÷ 3 − 7

Step 2: Multiplication (left to right): 6 × 9 = 54 → 3 + 54 ÷ 3 − 7

Step 3: Division: 54 ÷ 3 = 18 → 3 + 18 − 7

Step 4: Addition (left to right): 3 + 18 = 21 → 21 − 7

Step 5: Subtraction: 21 − 7 = 14

Answer: 14

Common Mistakes

Mistake 1: Doing all multiplication before division

Many students read PEMDAS and think multiplication always comes before division. It doesn't. They share the same priority and are evaluated left to right.

❌ 8 ÷ 4 × 2 = 8 ÷ (4 × 2) = 8 ÷ 8 = 1 WRONG

✓ 8 ÷ 4 × 2 = (8 ÷ 4) × 2 = 2 × 2 = 4 CORRECT

Mistake 2: Doing all addition before subtraction

❌ 10 − 3 + 5 = 10 − (3 + 5) = 10 − 8 = 2 WRONG

✓ 10 − 3 + 5 = (10 − 3) + 5 = 7 + 5 = 12 CORRECT

Mistake 3: Adding before multiplying

❌ 6 + 2 × 3 = (6 + 2) × 3 = 24 WRONG

✓ 6 + 2 × 3 = 6 + (2 × 3) = 6 + 6 = 12 CORRECT

Using Parentheses to Change the Order

You can use parentheses to override the normal order!

Without parentheses: 5 + 3 × 2 = 5 + 6 = 11

With parentheses: (5 + 3) × 2 = 8 × 2 = 16

Parentheses force that operation to happen first.

Why It Matters

Without a standard order, the same expression could have different answers!

Expression: 6 + 2 × 3

• Interpretation 1: (6 + 2) × 3 = 8 × 3 = 24
• Interpretation 2: 6 + (2 × 3) = 6 + 6 = 12

The correct answer following PEMDAS: 12

Math would be chaos without agreed-upon rules!

Real-Life Uses

Shopping: "Buy 3 items at $5 each, plus a $2 bag"

• Expression: 3 × 5 + 2
• Calculate: 15 + 2 = $17 (multiply first!)

Recipes: "Divide 12 cups among 3 bowls, then add 2 more cups to each"

• Expression: 12 ÷ 3 + 2
• Calculate: 4 + 2 = 6 cups per bowl

Construction: "A room is (4 + 3) feet wide and needs 2 boards per foot"

• Expression: (4 + 3) × 2
• Calculate: 7 × 2 = 14 boards

Practice