Order of Operations
Your 5th-grade student calculates 3 + 4 × 2 as 14 instead of 11, revealing a critical gap in understanding order of operations. This foundational concept, aligned with CCSS 5.OA and CCSS 6.EE standards, determines whether students solve multi-step problems correctly or fall into systematic calculation errors.
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Why it matters
Order of operations prevents mathematical ambiguity in real-world calculations. When a carpenter calculates material costs using 15 + 8 × 12 boards at different prices, the correct answer of 111 dollars differs significantly from the incorrect 276 dollars. Engineers designing bridge supports rely on PEMDAS to ensure structural calculations like 50 + 20 × 3 yield 110 pounds of force, not 210. Financial professionals calculating compound interest use expressions like 1000 × (1.05)² + 200, where incorrect order produces vastly different investment outcomes. Students who master PEMDAS in elementary grades build confidence for algebra, where expressions like 3x + 2(x - 5) require systematic evaluation. This skill transfers directly to programming, scientific formulas, and any field requiring precise mathematical communication.
How to solve order of operations
Order of Operations (PEMDAS)
- Parentheses first.
- Then exponents.
- Then multiplication and division (left to right).
- Then addition and subtraction (left to right).
Example: 3 + 4 × 2 = 3 + 8 = 11 (not 14).
Worked examples
True or false: 2 + 1 × 2 = 6
Answer: False (4)
- Multiply first → 1 × 2 = 2 — Multiplication before addition (PEMDAS).
- Then add → 2 + 2 = 4 — Now add the remaining term.
- Verify → 2 + 1 × 2 = 4 ✓ — Check the answer.
True or false: 5 + 5 × 10 = 100
Answer: False (55)
- Multiply first → 5 × 10 = 50 — Multiplication before addition.
- Then add → 5 + 50 = 55 — Add the remaining.
- Verify → 5 + 5 × 10 = 55 ✓ — Check.
Notebooks cost $6.00 each. You buy 2 for yourself and 3 for a friend, then use a $2.00 coupon. What do you pay?
Answer: $28.00
- Total notebooks → 2 + 3 = 5 — Add quantities first (parentheses).
- Total before discount → 6 × 5 = 30 — Multiply price by total count.
- Apply coupon → 30 − 2 = 28 — Subtract the coupon.
- Verify → 6 × (2 + 3) − 2 = 28 ✓ — Check.
Common mistakes
- ✗Students calculate left to right without considering operation priority, writing 6 + 4 × 2 = 20 instead of 14.
- ✗When solving 8 - 3 × 2, students often compute 8 - 3 = 5, then 5 × 2 = 10, rather than 3 × 2 = 6, then 8 - 6 = 2.
- ✗Students mishandle parentheses in expressions like 4 × (3 + 2), calculating 4 × 3 + 2 = 14 instead of 4 × 5 = 20.
- ✗With exponents present, students solve 2 + 3² as (2 + 3)² = 25 instead of 2 + 9 = 11.
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