Are there common mistakes when evaluating expressions like 143 + 305 * 70?
Executive summary
When evaluating 143 + 305 * 70 the usual error is ignoring the order of operations and adding before multiplying; the correct step is to compute 305 × 70 = 21,350 and then add 143 to get 21,493 (available sources describe the order-of-operations rule and common mistakes but do not compute this exact numeric result) [1] [2]. Education research shows a large fraction of people — including teachers and adults — hold misconceptions about operation precedence that produce predictable errors on multi‑term problems (roughly one third to one half in some studies) [3].
1. The single common trap: PEMDAS misapplied
Students and adults routinely apply PEMDAS incorrectly by performing addition before multiplication when faced with mixed operations; authoritative teaching pages advise doing multiplication before addition (PEMDAS/BODMAS) or, more precisely, doing multiplication and division left to right before addition and subtraction [1] [2].
2. What typically goes wrong with 143 + 305 * 70
A common error is to compute 143 + 305 first and then multiply by 70, yielding a much larger wrong answer. Teaching resources repeatedly flag this pattern — adding terms out of order — as an elementary but frequent mistake that stems from rushing or weakly internalized order‑of‑operations rules [2] [4].
3. Why the mistake persists: cognitive habits and format
Research shows many adults retain misconceptions about associativity and precedence; studies found roughly one third to one half of participants display specific misconceptions that lead to errors on multi‑term arithmetic [3]. Classroom and tutoring blogs echo that speed, sloppy notation and mental‑math shortcuts encourage these slips [4] [5].
4. Calculator and notation pitfalls
Different calculators and programming languages can interpret ambiguous inline notation inconsistently unless parentheses clarify intent; pedagogy sites recommend always using grouping symbols when there is any doubt [6] [7]. Teaching materials advise writing intermediate steps to prevent misreading your own work [5] [8].
5. How teachers and materials suggest avoiding the error
Common classroom strategies include: explicitly teaching multiplication before addition, insisting on left‑to‑right rules for operations of equal precedence, using parentheses to disambiguate, and giving error‑analysis practice where students find and correct mistakes [1] [9] [10]. Many tutoring blogs also recommend slowing down, writing each step, and estimating to check reasonableness [4] [11].
6. Alternate viewpoints and ambiguities to note
While PEMDAS is widely taught, some sources emphasize nuances — for example, multiplication and division share precedence and are evaluated left to right; similarly addition and subtraction share precedence — and that misstatements of the acronym can cause errors [6] [9]. There is also debate in pedagogical writing over mnemonic emphasis versus teaching the underlying reasons for the conventions [9].
7. Concrete checking techniques teachers recommend
Practices that reduce mistakes include: estimating the answer first (to detect wildly wrong results), substituting simpler analogous numbers to test method, forcing students to write out multiplication first in mixed expressions, and using error‑finding activities so learners internalize common slip patterns [5] [10] [12].
8. What the provided sources do not say
Available sources do not show this exact calculation step‑by‑step for 143 + 305 * 70 nor do they give the explicit numeric answer 21,493. They instead document the general rule, common error patterns, pedagogical fixes and empirical findings about misconceptions [1] [3] [8].
Limitations: this summary draws only from the supplied documents about order‑of‑operations errors, classroom advice and research on misconceptions; there are many additional textbooks and standards documents that also discuss these topics but are not included here [6] [13].