[Rescued from my old blog.]
Marjorie in AZ asked a terrific question on the (now defunct) AHFH Math forum:
“…I have always been taught that the order of operations (Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction) means that you work a problem in that order. All parenthesis first, then all exponents, then all multiplication from left to right, then all division from left to right, etc. …”
Many people are confused with order of operations, and it is often poorly taught. I’m afraid that Marjorie has fallen victim to a poor teacher — or at least, to a teacher who didn’t fully understand math. Rather than thinking of a strict “PEMDAS” progression, think of a series of stair steps, with the inverse operations being on the same level.
First do whatever is inside parentheses or brackets, from the innermost set outward. Parentheses are used when necessary to override the ordinary order of operations, to communicate that certain things within your equation belong together. Within each set of parentheses or brackets, follow the normal order of operations. Be aware that some other mathematical symbols, such as the radical sign and the fraction bar, double as an understood set of parentheses. That is, if you have a complicated algebra fraction, you need to treat it as if there were brackets around the entire numerator and around the entire denominator.
Exponents and Roots
These are inverse operations, which means they are just two different ways of looking at the same mathematical concept. They are the highest level in the normal order of operations, indicating repeated multiplication. Exponents and roots all need to be calculated before any multiplication, addition, etc., unless there are parentheses that dictate otherwise. On any single level of operation (such as this exponents-and-roots level), work from left to right through your equation.
Multiplication and Division
These are inverse operations, which means they are virtually interchangeable. Division could be written as multiplication: think of the rule for dividing fractions. This is the middle level in the normal order of operations, indicating repeated addition. Multiplications and divisions all need to be done before any addition or subtraction, unless there are parentheses that dictate otherwise. On any single level of operation (such as this multiplication-and-division level), work from left to right through your equation.
[Hint: In a complex equation, it is a good idea to change all the divisions into multiplying-by-the-reciprocal before doing any calculation.]
Addition and Subtraction
These are inverse operations, too. They are the lowest level in the normal order of operations, and they all need to be done last, unless there are parentheses that dictate otherwise. On any single level of operation, work from left to right through your equation.
[Hint: In a complex equation, it is a good idea to change all subtractions into adding-the-opposite before doing any calculation.]
Notice that we do the repeated operations first. Powers are repeated multiplication, and multiplication is repeated addition. We could, if we felt masochistic, try to convert the powers into multiplications, and then convert the multiplications into additions — and then we wouldn’t have to worry about what order to do everything in. Addition is both associative and commutative, which means it can be done in any order. But I don’t even want to think about what a mess an equation like that would be, especially with algebra variables. Yuck! Much better to learn and follow the standard order.
Students often feel that order of operations is an arbitrary rule, like driving on the right (or left, depending on where you live) side of the road. Like driving on the proper side of the road, however, such arbitrariness does NOT mean that the rule is unimportant. If you think of it as I’ve described here, you may save yourself and your students a few mathematical wrecks!