The standard way people teach this stuff usually trips students up
I kept seeing the same mistakes over and over in worksheets and online assignments, so I stopped trying to make it sound impressive and just started explaining it the way it actually works in practice. Order Of Operation Math Problems follow a hierarchy that most people learn as PEMDAS, but the letters alone don't mean much if you don't understand what each one is actually controlling. The order goes Parentheses first, then Exponents, then Multiplication and Division (which share the same tier and are handled left to right), then Addition and Subtraction (same thing). That's it. The whole system is built around preventing ambiguity when you write an expression on a single line.
Working Through Order Of Operation Math Problems Step By Step
Take a problem like 3 + 4 × 2² (6 ÷ 3). You don't just go left to right because that would give you a completely wrong answer. You start with the parentheses: 6 ÷ 3 equals 2, so the expression becomes 3 + 4 × 2² 2. Then exponents: 2² is 4, giving you 3 + 4 × 4 2. Then multiplication before addition and subtraction: 4 × 4 is 16, so you're left with 3 + 16 2, which is 17. Left to right from the start would've given you 49, which is nowhere close. Here's where most people mess up. The "left to right" rule for multiplication and division trips almost everyone who's teaching this at an intro level. If you have something like 12 ÷ 3 × 2, you do the division first because it appears first when reading left to right, and that gives you 8. A lot of students see the multiplication symbol and immediately multiply 3 by 2, getting 12 ÷ 6 which equals 2. Wrong. The symbols are equal in priority. The leftmost one wins. I ran into a problem last year in a tutoring session that really highlighted how messy this gets in practice. A student had 8 ÷ 2(2 + 2) and got two different answers depending on who they asked. The expression is technically ambiguous without additional convention. Some calculators and textbooks treat the implicit multiplication after the parentheses as having higher priority than the division, which gives you 8 ÷ 8 = 1. Others treat it purely left to right: 8 ÷ 2 = 4, then 4 × 4 = 16. I worked around it by converting everything to fraction notation on the board so the intended grouping was visually clear. The student's textbook used the first convention, which is why they kept getting the answer marked wrong even though their arithmetic was fine. That's the real problem with these exercises: the notation itself sometimes encodes assumptions about priority that aren't stated anywhere in the PEMDAS acronym.
Another thing that catches people off guard involves negative numbers and exponents. Write 3² on a test and ask what it equals. Half the class will say 9 because they compute (3)². The correct answer is 9. The exponent applies only to the 3, not to the negative sign in front of it, unless parentheses are explicitly there. This comes up constantly and it's worth drilling early because it doesn't get easier with practice. You just have to remember the convention. Fractored expressions add another layer. When you see something like (5 2)² + 6 ÷ 3, you have to resolve the parentheses before touching the exponent, then the division, then the addition. The nesting depth doesn't change the rules. Inner parentheses always get solved first, then you work outward. I once saw a student skip the inner grouping entirely because they were rushing and ended up subtracting inside the exponent, which is impossible to justify mathematically. It happens more often than you'd think on timed tests. The biggest practical limitation of relying on PEMDAS as the sole teaching tool is that it breaks down with more complex algebraic notation. It works fine for arithmetic, but once you introduce variables, hidden operations, or nested fractions, the acronym becomes insufficient. At that point you need the more general concept of operator precedence, which is exactly how programming languages handle it too. The rules are the same, but the edge cases multiply quickly.
Get the Full Details
If you're looking for printable worksheets or practice sets, searching for "order of operations worksheets PDF" will turn up free resources from sites like K5 Learning, Math-Drills, and CommonCoreSheets. Those tend to be reliable and cover the full range from basic four-operation problems to ones with exponents and nested grouping symbols. There's no single authoritative source, but those three will get you through a semester of practice material without repeating content.
