What are common mistakes when combining addition and multiplication in one expression?
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Executive summary
Students and adult solvers repeatedly trip over expressions that mix addition and multiplication because they either ignore the formal order of operations or misread the mnemonic tools meant to help—most notably doing addition before multiplication in examples like 2 + 3 × 4, or treating multiplication as always preceding division and addition as always preceding subtraction instead of applying the left‑to‑right rule [1] [2] [3]. Instructional materials and classroom practice both highlight predictable pitfalls—misapplied PEMDAS, neglected parentheses, hidden negatives, and sloppy translation of subtraction to adding a negative—that produce incorrect results and confusion [4] [5] [6].
1. Misreading PEMDAS: thinking M always beats D and A always beats S
A persistent mistake is treating the letters in PEMDAS as a strict stack rather than as paired steps: multiplication and division have equal precedence and must be handled left to right, as do addition and subtraction, yet learners often do multiplication before division or addition before subtraction simply because of the order of letters in the mnemonic [2] [4] [7].
2. Doing addition before multiplication in mixed expressions
The classic error—adding in 2 + 3 × 4 to get 20 instead of first multiplying to get 14—appears across curricula and remediation resources; guides repeatedly use this exact example to show why following the order of operations matters and why multiplication/division come before addition/subtraction unless parentheses indicate otherwise [1] [8] [9].
3. Ignoring left‑to‑right evaluation within equal‑priority operations
Even when students know multiplication and division are the same “level,” they sometimes fail to process them left to right, producing inconsistent answers on strings like 12 ÷ 3 × 2; authoritative explanations recommend treating those operations in the order they appear, not by elevating one operation over the other [3] [10] [9].
4. Negatives, subtraction, and the invisibility of “adding a negative”
Mixing subtraction and multiplication frequently exposes deeper notation misunderstandings: the negative sign is an operation, not a mere marker, and replacing subtraction with addition of a negative can both simplify manipulation and reveal order issues (e.g., 5 + 3 − 6 becomes 5 + 3 + (−6)), but students may not apply this consistently, especially when negative signs combine with exponents or with multiplication [5] [6].
5. Parentheses, implied grouping, and careless notation
Ambiguity of grouping—failing to use parentheses to force the intended order—causes many errors; calculators and computer systems ignore whitespace, so sloppy spacing or omitted brackets can change an expression’s value, and teachers stress adding explicit parentheses to avoid misinterpretation [9] [10] [6].
6. Other recurring traps: exponents, distribution, and rushed work
Exponents must be handled before multiplication and addition, and neglecting them or misapplying distribution (e.g., misunderstanding whether to square a sum or a single term) leads to mistakes; many educators also point to simple haste—skipping steps, not checking with parentheses or rewriting the problem—as a common cause of wrong answers [11] [12] [8].
7. Why the confusion persists and what the pedagogical debates reveal
Part of the ongoing confusion is pedagogical: memorable mnemonics like PEMDAS help recall but can embed misleading impressions (that M always precedes D), and different regional acronyms (BODMAS, BEDMAS, GEMDAS) plus informal teaching practices create mixed messages; some commentators argue the remedy is emphasizing left‑to‑right processing and teaching algebraic rewrites (turning subtraction into adding negatives) rather than drilling acronyms alone [2] [4] [5].
8. Practical checks to avoid the mistakes
To prevent errors, authoritative sources recommend: insert parentheses to show intent, perform multiplication/division then addition/subtraction left to right, rewrite subtractions as adding negatives when helpful, evaluate exponents first, and slow down to check each transformation—strategies repeatedly endorsed in guides and error‑finding worksheets [10] [5] [13].