Master the Order of Operations: 24 Flashcards on PEMDAS, Implied Multiplication and Notation Traps
Learn the order of operations with deck TE9011: the four-level hierarchy, PEMDAS traps, implied multiplication and the ambiguities nobody warns you about.
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Deck TE9011 — Order of Operations (PEMDAS) is 24 cards, and it is really two decks stacked on each other. Cards 1-13 are the schoolroom rules. Cards 14-24 are the fine print mathematicians argue about: implied multiplication, radical vinculums, right-associative exponent towers, and the reason 8 ÷ 2(2 + 2) starts fights online. Knowing which half you are in is the trick to studying it.
Study this deck: Order of Operations (PEMDAS)
1 / 10Order of operations
Tap the card for the answer · swipe right if you knew it, left if not
Nine cards from the deck
| Front | Back |
|---|---|
| Four levels of the standard order of operations | 1) Parentheses, 2) Exponentiation, 3) Multiplication and division, 4) Addition and subtraction |
| Value of 3 × 5² | 75 (exponentiation before multiplication: 3 × 25 = 75) |
| How is -3² typically interpreted in written mathematics? | As -(3²) = -9 (exponentiation before negation) |
| Standard direction for evaluating operations of the same precedence | From left to right |
| Implied multiplication | Multiplication denoted by juxtaposition (e.g., 2x instead of 2·x) |
| How is 1/2n often interpreted in academic literature? | As 1/(2·n) rather than (1/2)·n, giving implied multiplication higher precedence than division |
| Ambiguity in expressions like 8 ÷ 2(2 + 2) | Either 8 ÷ [2·(2 + 2)] = 1 or (8 ÷ 2)·(2 + 2) = 16, depending on whether implied multiplication has higher precedence |
| How is a^b^c typically evaluated? | As a^(b^c), working from top down (right to left) |
| German mnemonic for order of operations | Punktrechnung vor Strichrechnung — dot operations before line operations |
What you actually cover
Precedence and operator associativity, left-associative vs right-associative levels, PEMDAS and the Please Excuse My Dear Aunt Sally phrase, brackets [ ] and braces { } for nesting, infix notation, the radical symbol and its vinculum, function-argument conventions (sin 3x read as sin(3x)), multiplication by juxtaposition, unary negation vs exponentiation, the four worked evaluations (1 + 2 × 3, 3 + 5², 3 × 5², (2 + 3) × 4), why the conventions exist at all, and the standard criticism of mnemonic acronyms.
Two things to fix before your first session
Split the deck at card 14. Everything from radical symbol onward is typographic convention, not arithmetic. Suspend cards 14-24, get the four-level hierarchy automatic, then unsuspend them as a separate topic.
Add the mnemonic you were actually taught. The deck gives PEMDAS (US) and Punktrechnung vor Strichrechnung (German) but never BODMAS or BIDMAS, used across the UK, India and Australia, where B is Brackets and O is Orders. Write one card mapping the three: BODMAS sets its own trap, since Division before Multiplication misleads exactly like PEMDAS's MD. Card 22 names the failure mode — students read the mnemonic as addition first, subtraction afterward instead of equal precedence, left to right. That card is worth more than the other 23.
Housekeeping: cards 13 and 23 teach the same fact twice. Keep Operator associativity — the term you will meet again in programming — and delete the other.
The deck's real weakness
Only four of 24 cards ask you to compute anything. Flashcards store a rule well and build calculation speed badly, so treat TE9011 as a rule index and get your reps elsewhere: write ten mixed expressions of your own, each combining a bracket, an exponent and a division, and check them against the hierarchy card. If you can do -2² + 6 ÷ 3 × 2 without hesitating, the deck has done its job.
Find the deck as TE9011 — Order of Operations (PEMDAS) in your library on Flip It.
More arithmetic study guides: Fractions, Decimals and Percentages and Ratios and Proportions.
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