Order of Operations with Whole Numbers
Warm-up
Write on the board and collect answers silently on whiteboards. The room splits: 50 and 26. Don't referee yet — instead ask both camps to defend their reading of the expression aloud.
The stakes become clear fast: if an expression can mean two things, mathematics can't be written down reliably. Today's lesson is the treaty that ends the ambiguity — and it's a CONVENTION, agreed rules of reading, not a law of nature.
Explore
Give pairs a calculator-free sorting task: eight expressions, each evaluated two ways (following left-to-right vs. following the operation-priority rules), with the class predicting which answer a scientific calculator will give. Then verify three of them on a real calculator.
Second structure hunt: "make the target" — insert parentheses into to hit as many different values as possible (; ; ; ). Parentheses aren't decoration; they're steering.
Formalize
Formalize the convention: parentheses first, then exponents, then multiplication and division together left to right, then addition and subtraction together left to right. The two "together" clauses are where errors live:
Neither PEMDAS nor BEDMAS means "multiplication before division" — M and D are one tier, read left to right, as are A and S. The mnemonic's ladder shape misleads; draw the four tiers as four floors of a building with M·D sharing a floor and A·S sharing the ground floor.
Practice
Practice: evaluate eight expressions of climbing complexity (including one with an exponent and nested parentheses), then two "error autopsy" items — a fictional student's worked line with one order slip to find and fix.
Exit ticket: evaluate showing one tier per line. (.)
Exit ticket
Practice: evaluate eight expressions of climbing complexity (including one with an exponent and nested parentheses), then two "error autopsy" items — a fictional student's worked line with one order slip to find and fix.
Exit ticket: evaluate showing one tier per line. (.)
Step 1 — exponents (no parentheses to do first): , giving .
Step 2 — multiplication and division, left to right: first, then , giving .
Step 3 — addition: .
Step 4: Audit the danger point: doing before (right to left) gives , then — wrong. Left-to-right within the tier isn't optional politeness; it changes answers.
The story: "A team buys 4 pizzas at \$9 each and pays a \$6 delivery fee. Split the total among 6 players — what does each pay?"
Step 1: Build the expression inside-out: cost of pizzas ; plus fee → ; split six ways — the WHOLE total splits, so parentheses must gather it: .
Step 2: Evaluate by tiers: inside first: ; then . Each pays $7.
Step 3: Show what the missing parentheses would claim: — a story where only the fee gets split. Parentheses carry MEANING from the story into the symbols; leaving them out changes the story.
Step 1: Check line one: parentheses ✓ and exponent ✓. So far honest: .
Step 2: Check line two: the student added BEFORE multiplying — addition jumped the multiplication tier. There's the slip.
Step 3: Repair from the honest line: .
Step 4: Name the pattern for the error journal: left-to-right reading FEELS natural (we read words that way), but expressions are read by TIER first, position second. The autopsy habit — verify each line against the tier rules, find the first breach, repair from there — is exactly how students should check their own multi-line work.