Order of Operations (PEMDAS) — A Parent's Guide

"Please Excuse My Dear Aunt Sally" Sounds Simple — It Isn't

PEMDAS is one of those math topics that looks tidy in a textbook and breaks open the moment a real expression appears. The acronym is fine. The problem is what the acronym hides: that multiplication and division aren't actually two ranked steps but one tier, and that the same is true for addition and subtraction. Most kids learn the mnemonic and never hear the footnote — so they make the same predictable mistakes for years.

The good news: once the two missing pieces click, the order of operations stops being a trap. It becomes a checklist a child can run reliably on any expression. The trick is making the missing pieces explicit and walking through enough examples that the pattern locks in.

This article does not introduce the rule — your child has already met it. It explains where it usually goes wrong and what to do about it.

What's Actually Going On

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. The expression $6 + 2 \times 3$ equals 12, not 24, because multiplication comes before addition. So far, so clean.

Here's what the mnemonic hides.

The two-tier rule that gets lost. Multiplication and division share the same tier. So do addition and subtraction. When an expression has both multiplication and division (with no parentheses or exponents in the way), you do not do all the multiplication first — you go left to right. So $12 \div 4 \times 3$ equals $(12 \div 4) \times 3 = 9$, not $12 \div (4 \times 3) = 1$.

The same is true for $+$ and $-$. $10 - 4 + 2 = (10 - 4) + 2 = 8$, not $10 - (4 + 2) = 4$.

If your child treats PEMDAS as six ranked steps (do all M before any D, all A before any S), they will get specific problems wrong in a predictable pattern.

The parenthesis trap. When parentheses contain an expression — not just one number — you still apply PEMDAS inside the parentheses, then handle what's outside. $2(3 + 5)$ means $2 \times 8 = 16$. Kids who learned "do the parentheses first" sometimes write $2 \times 3 + 5 = 11$ by distributing badly. The rule is: finish what's inside the parentheses, then multiply outside.

Implicit multiplication. $2(3) = 6$. The missing $\times$ sign is the source of more confusion than parents realise. A child who reads $2x$ as "two x" needs to know that the implied multiplication has the same priority as a written $\times$ — usually. (Some textbooks treat $1 \div 2x$ as $1 \div (2x)$, giving implicit multiplication a tighter grip. This is genuinely ambiguous; we'll flag where it bites your child.)

PEMDAS, BODMAS, BIDMAS, BEDMAS, GEMDAS — Same Rule, Different Letters

Regional curricula use different mnemonics. They all describe the same four-tier rule.

The D-before-M ordering in BODMAS confuses many students into thinking division strictly precedes multiplication. It does not. They are the same tier and are evaluated left to right — whichever appears first. Mathematicians treat division as multiplication by the reciprocal, which is why they share the tier. The same logic puts subtraction on the same tier as addition (subtraction is addition of a negative).

If your child switched curricula — moved from a US school to a UK one, or from India to Australia — the letters may have changed but the rule did not.

Patterns to Watch For

These are the exact signals that a child has memorised the mnemonic but not internalised the structure.

• They get $12 \div 4 \times 3 = 1$ (treated all M before D), instead of 9.

• They get $10 - 4 + 2 = 4$ (treated all A before S), instead of 8.

• They get $2(3 + 5) = 11$ (distributed the 2 before adding inside), instead of 16.

• They can do problems with one operation type fine, but lose accuracy the moment an expression mixes multiplication and division.

• They argue passionately about which way is right — confidence is high, error rate is also high. (This is the memorizer archetype. The mnemonic is doing the thinking instead of the child.)

• They get the right answer on simple expressions but freeze when negative numbers or exponents enter.

A child who has only the mnemonic gets the easy expressions right and the medium ones wrong. A child who actually understands the order gets a wider range right and asks better questions when stuck.

Walk Through — Ten Worked Examples (Wrong Path Shown First)

The fastest way to make the four-tier rule stick is to walk through expressions where the intuitive answer is wrong. Show the slip, then show the fix.

Example 1 — Quick. $6 + 2 \times 3$

Wrong path. Read left to right: $6 + 2 = 8$, then $8 \times 3 = 24$. Correct. Multiplication is on a higher tier than addition. $2 \times 3 = 6$ first. Then $6 + 6 = 12$. Final answer: 12.

Example 2 — Quick. $20 - 5 \times 2$

Wrong path. Left to right: $20 - 5 = 15$, then $15 \times 2 = 30$. Correct. Multiplication first: $5 \times 2 = 10$. Then $20 - 10 = 10$. Final answer: 10.

Example 3 — Standard (the left-to-right trap). $12 \div 4 \times 3$

Wrong path. "M comes before D in PEMDAS, so do multiplication first": $4 \times 3 = 12$, then $12 \div 12 = 1$. Correct. Multiplication and division share a tier — go left to right. $12 \div 4 = 3$ first. Then $3 \times 3 = 9$. Final answer: 9.

Example 4 — Standard (the subtraction trap). $10 - 4 + 2$

Wrong path. "A comes before S, so do addition first": $4 + 2 = 6$, then $10 - 6 = 4$. Correct. Addition and subtraction share a tier — left to right. $10 - 4 = 6$ first. Then $6 + 2 = 8$. Final answer: 8.

Example 5 — Standard (the parenthesis trap). $2(3 + 5)$

Wrong path. Distribute too early: $2 \times 3 + 5 = 6 + 5 = 11$. Correct. Finish inside the parentheses first: $3 + 5 = 8$. Then multiply: $2 \times 8 = 16$. Final answer: 16.

Example 6 — Standard (mixing tiers cleanly). $5 + 4 \times (10 - 6)$

Wrong path. Multiply before the parenthesis is finished: $4 \times 10 = 40$, then $5 + 40 - 6 = 39$. Correct. Parentheses first: $10 - 6 = 4$. Then multiplication: $4 \times 4 = 16$. Then addition: $5 + 16 = 21$. Final answer: 21.

Example 7 — Standard (exponent before multiplication). $3^2 + 4 \times 2$

Wrong path. Multiply first: $4 \times 2 = 8$, then $3^2 = 9$, then $9 + 8 = 17$. (Correct value — but the wrong reasoning path. Many students confidently get this answer for the wrong reason and stumble when exponents appear inside a different position.) Correct. Exponents first: $3^2 = 9$. Then multiplication: $4 \times 2 = 8$. Then addition: $9 + 8 = 17$. Final answer: 17.

Example 8 — Standard (nested parentheses). $2 \times [3 + (4 - 1)^2]$

Wrong path. Skip the inner parenthesis: $2 \times [3 + 4 - 1^2] = 2 \times [3 + 4 - 1] = 2 \times 6 = 12$. Correct. Innermost parenthesis first: $4 - 1 = 3$. Then the exponent: $3^2 = 9$. Then the outer bracket: $3 + 9 = 12$. Then the multiplication: $2 \times 12 = 24$. Final answer: 24.

Example 9 — Stretch (with negatives and division). $-6 + 18 \div (-3) \times 2$

Wrong path. Process the division and multiplication separately, then add: $18 \div (-3) = -6$, $(-3) \times 2 = -6$, then $-6 + (-6) = -12$. (Multiplication done after the division on its own, not left to right with it.) Correct. Division and multiplication left to right: $18 \div (-3) = -6$ first. Then $(-6) \times 2 = -12$. Then $-6 + (-12) = -18$. Final answer: $-18$.

Example 10 — Stretch (the viral one). $8 \div 2(2 + 2)$

Wrong path A. Treat $2(2+2)$ as a single term (implicit multiplication binds tighter): $2(2+2) = 2 \times 4 = 8$. Then $8 \div 8 = 1$. Wrong path B. Strict left-to-right with explicit multiplication priority: $8 \div 2 = 4$, then $(2+2) = 4$, then $4 \times 4 = 16$.

This expression is genuinely ambiguous in conventional notation. Most textbooks resolve it to 16 by applying standard PEMDAS (do the parenthesis first, then divide/multiply left to right). Many handheld calculators give 1 because they treat implicit multiplication (the missing $\times$ sign) as tighter than explicit multiplication.

The honest answer for parents. Write the expression with extra parentheses to remove the ambiguity. Either $(8 \div 2)(2+2) = 16$ or $8 \div (2(2+2)) = 1$. Mathematicians don't write expressions like this in real work — the ambiguity is exactly why.

The pattern across all ten. Every wrong path comes from the same mistake: treating PEMDAS as six ranked steps instead of four tiers with left-to-right within each tier.

What to Do (Concrete Actions)

Specific things a parent can do this week.

• Write $12 \div 4 \times 3$ on paper and ask them to solve it twice — once as "all M first," once as "left to right." Have them check on a calculator. The reveal does the work. Most kids didn't realise the order matters and walk away with the actual rule for life.

• Replace the mnemonic with a four-tier summary. "Parentheses, then exponents, then $\times$ and $\div$ left to right, then $+$ and $-$ left to right." It fits on an index card. Stick it on the fridge if needed.

• Hunt for the trap problem. Give them three expressions a week that look like they obey PEMDAS but really test the left-to-right rule: $8 \div 2 \times 4$, $20 - 6 + 1$, $3 \times 4 \div 6$. The ones the internet argues about (e.g., $8 \div 2(2+2)$) are good conversation starters about implicit multiplication.

• Make them write each step. A child who rushes through PEMDAS in their head makes the predictable mistake. A child who writes each intermediate step doesn't. The wrong instinct — "just do it in your head" — is the source of half the errors.

• Use a calculator strategically. Not for getting the answer. For checking their reasoning. A child who predicts the calculator's output before pressing enter is doing the math. A child who presses first is hoping.

When to Bring in Outside Help

The honest signals.

• The mistakes persist past Grade 6. PEMDAS lives at the heart of algebra. A child still getting order-of-operations wrong in Grade 7 will fail simple algebra problems for reasons that have nothing to do with algebra. That's the threshold.

• They get the right answer but can't explain why. This is the memorizer signal. They'll hit a ceiling in middle school when problems require reasoning, not pattern-matching.

• They've developed a fear of "tricky" problems. Order-of-operations expressions designed to expose the mnemonic gap (the famous viral ones) often hit a child as gotcha problems. If your child now avoids them or insists they're "unfair," the underlying gap is real.

A structured math program — Bhanzu, a Singapore-method tutor, a school-recommended specialist — becomes worth the investment once one of those thresholds is hit. Most kids reach genuine understanding through the kitchen-table work above. Some need a patient 1:1 setting to undo the mnemonic-trained reflex.

Conclusion

• The order of operations in math is four tiers, not six steps: P, E, then ($\times$/$\div$ left to right), then ($+$/$-$ left to right).

• The two-tier rule is what the mnemonic hides — teach it explicitly with paired examples.

• Most errors come from rushing through M/D and A/S as if they were ranked.

• Writing each step out catches almost every mistake.

• Implicit multiplication is genuinely ambiguous — write parentheses to avoid the trap.

• Outside help is worth it once order-of-operations errors persist into algebra territory.