How to Teach Order of Operations (Beyond PEMDAS)

"Please Excuse My Dear Aunt Sally." Your child can probably recite the PEMDAS mnemonic. But many children who know PEMDAS still get order of operations problems wrong — because the mnemonic creates two common misconceptions:

1. Multiplication always comes before division
2. Addition always comes before subtraction

Both are wrong. PEMDAS is a useful memory aid, but it needs careful teaching to avoid these traps.

What order of operations actually means

When a mathematical expression has multiple operations, we need a shared agreement about which to do first. Without this, 3 + 4 × 2 could mean 14 (multiply first) or 14 (multiply first) or 10 (add first, then multiply) — but only one answer is correct.

The actual order:

1. Parentheses (and other grouping symbols): Do whatever is inside first
2. Exponents: Powers and roots
3. Multiplication and Division: Left to right, whichever comes first
4. Addition and Subtraction: Left to right, whichever comes first

The critical detail: multiplication and division are the same priority. Addition and subtraction are the same priority. Within each level, you go left to right.

Key Insight: PEMDAS makes it look like there are six levels. There are actually four. M and D are on the same level. A and S are on the same level. Within each level, you work left to right.

The misconception that breaks everything

Consider: 8 - 3 + 2

A child who thinks addition comes before subtraction (because A comes before S in PEMDAS) will do:

• 3 + 2 = 5, then 8 - 5 = 3 ← Wrong

The correct approach: left to right

• 8 - 3 = 5, then 5 + 2 = 7 ← Correct

Similarly: 12 ÷ 3 × 2

A child who thinks multiplication comes before division will do:

• 3 × 2 = 6, then 12 ÷ 6 = 2 ← Wrong

Correct: left to right

• 12 ÷ 3 = 4, then 4 × 2 = 8 ← Correct

Teaching the correct model

Instead of PEMDAS as a straight line, teach it as levels:

Level 1: Parentheses Level 2: Exponents Level 3: Multiplication AND Division (left to right) Level 4: Addition AND Subtraction (left to right)

Some teachers use GEMS (Grouping, Exponents, Multiply/Divide, Subtract/Add) or just teach the four levels explicitly.

Start with parentheses

Parentheses are the most powerful rule and the most intuitive:

• "Parentheses mean: do this first."
• 3 × (4 + 2) = 3 × 6 = 18
• (3 × 4) + 2 = 12 + 2 = 14

Show your child that parentheses change the answer by changing which operation happens first. This makes the purpose of order of operations visible — it matters which thing you do first.

Then add multiplication vs. addition

After parentheses, introduce the main rule: multiplication and division happen before addition and subtraction.

• 3 + 4 × 2 = 3 + 8 = 11 (not 14)
• 10 - 2 × 3 = 10 - 6 = 4 (not 24)

Why does multiplication go first? It is a convention — a shared agreement — but there is a logical reason: multiplication is repeated addition. 3 + 4 × 2 means "3 plus two groups of 4," which is 3 + 8 = 11.

Practice with purpose

Wrong vs. right comparison:

• Show: "Someone said 5 + 3 × 2 = 16. Are they correct?" (No — 5 + 6 = 11)
• "What mistake did they make?" (Added before multiplying)

Insert parentheses:

• "Use parentheses to make 3 + 4 × 2 = 14." → (3 + 4) × 2 = 14
• "Use parentheses to make 3 + 4 × 2 = 11." → 3 + (4 × 2) = 11 (or no parentheses needed)

This exercise builds understanding of how parentheses control the order.

Multi-step expressions:

• 2 + 3 × (8 - 5) = 2 + 3 × 3 = 2 + 9 = 11
• Walk through each step: "Parentheses first: 8 - 5 = 3. Then multiply: 3 × 3 = 9. Then add: 2 + 9 = 11."

Common mistakes

Multiplication before division (always): They see 12 ÷ 3 × 2 and multiply first. Remind: same level, left to right.

Skipping parentheses: They see 4 × (3 + 2) and start with 4 × 3. Parentheses always come first.

Thinking PEMDAS is a math law: PEMDAS is a convention, not a law of nature. Different countries use different mnemonics (BODMAS, BEDMAS) but the rules are the same.

Key Insight: Order of operations is not about memorizing a sequence — it is about understanding that without an agreed-upon order, expressions would be ambiguous. The rules exist so that everyone gets the same answer from the same expression.

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Order of operations is a convention that makes mathematical communication unambiguous. Teach it as four levels, not six steps. Emphasize that M/D and A/S are same-level pairs resolved left to right. And let parentheses be the powerful tool they are — the way your child controls which operation happens first.

If you want a system that teaches order of operations with the correct model — and catches the PEMDAS misconceptions before they solidify — that is what Lumastery does.