Order of Operations (PEMDAS)
Parentheses
Do whatever is inside grouping symbols first — round brackets ( ), square brackets [ ], and the bar of a fraction all count.
Exponents
Powers and roots come next: 3² means 3 × 3 = 9.
Multiplication & Division
Equal rank, worked left to right — whichever appears first. Multiplication does not outrank division.
Addition & Subtraction
Also equal rank, also left to right. These always come last.
A worked example, step by step
Take 5 + (8 − 3)² × 2 ÷ 5 and walk the ladder top to bottom:
| Step | Expression | What happened |
|---|---|---|
| Parentheses | 5 + (8 − 3)² × 2 ÷ 5 → 5 + 5² × 2 ÷ 5 | 8 − 3 = 5 |
| Exponents | 5 + 5² × 2 ÷ 5 → 5 + 25 × 2 ÷ 5 | 5² = 25 |
| Multiply (leftmost first) | 5 + 25 × 2 ÷ 5 → 5 + 50 ÷ 5 | 25 × 2 = 50 |
| Divide | 5 + 50 ÷ 5 → 5 + 10 | 50 ÷ 5 = 10 |
| Add | 5 + 10 = 15 | done |
Notice the multiplication and division were handled as one left-to-right sweep — we multiplied first only because it appeared first, not because M comes before D in the acronym.
The three traps that cause almost every wrong answer
1. Treating M as “before” D
8 ÷ 2 × 4 = 16, not 1. Multiplication and division are the same rank; you sweep left to right. Reading PEMDAS as “multiply, then divide” gives 8 ÷ 8 = 1 — the single most common order-of-operations mistake (it’s the engine behind every viral “only 1% get this right” math post).
2. Treating A as “before” S
10 − 4 + 3 = 9, not 3. Same rule, bottom rung: addition and subtraction sweep left to right. Adding first (4 + 3 = 7, then 10 − 7) silently regroups the expression.
3. The exponent-and-minus-sign trap
−3² = −9, but (−3)² = 9. An exponent binds before a leading minus sign, so −3² means −(3 × 3). If the negative number itself should be squared, it needs its own parentheses. This one matters constantly in pre-algebra.
PEMDAS, BODMAS, GEMS — same rules, different mnemonics
Depending on where and when you went to school you may have learned PEMDAS (“Please Excuse My Dear Aunt Sally”), BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction — common in the UK and India), or GEMS (Groupings, Exponents, Multiply/divide, Subtract/add). They describe the same mathematics. BODMAS putting D before M changes nothing — because that middle rung is always one left-to-right sweep, whatever order the letters imply.
Try five — answers hidden until you click
3 + 6 × 5 − 6 = ?
Multiply first: 6 × 5 = 30. Then left to right: 3 + 30 = 33, 33 − 6 = 27.
36 ÷ 6 × 2 = ?
Trap #1! Left to right: 36 ÷ 6 = 6, then 6 × 2 = 12 (not 3).
(5 + 4) × (7 − 1) = ?
Both parentheses first: 9 × 6 = 54.
2³ + 4 × 5 = ?
Exponent first: 2³ = 8. Multiply: 4 × 5 = 20. Add: 28.
20 − 12 ÷ (2 + 2) = ?
Parentheses: 2 + 2 = 4. Divide: 12 ÷ 4 = 3. Subtract: 20 − 3 = 17.
Rules are easy — speed takes practice.
Num Drill’s pre-algebra levels mix order-of-operations expressions with solve-for-x problems and adapt to whatever your child misses. Free, no login, about a minute per quiz.
Start a free quiz See the pre-algebra levelsHow to teach it (without the tears)
- Teach the ladder as four rungs, not six letters. The two “shared rungs” (MD and AS) are the whole game — most kids who miss problems know the acronym fine and still treat it as six ordered steps.
- Make them write one step per line. Rewriting the whole expression after each single operation catches nearly all slips, and it’s exactly how the practice worksheet is spaced.
- Weaponize the viral posts. Those “8 ÷ 2(2+2) = ?” arguments online are genuinely ambiguous to adults — ask your child to referee one using the left-to-right rule. Being the person at the table who can settle it is great motivation.
- Then drill the arithmetic underneath. A child who stalls on 25 × 2 loses the thread of the bigger problem. Fast fact recall is what makes multi-step work feel easy.
Related: pre-algebra practice · solve for x · 6th grade math · 5th grade math · printable worksheets · multiplication chart