Estimated reading time: 3 minutes, 39 seconds

If you’re on Facebook, you’ve probably seen one of a variety of graphics like the one below:

The idea is solve the problem and then post your answer. From what I’ve observed, about half of the respondents get the answer correct, while the other half come to the wrong answer. The root of this problem? The order of operations.

Unlike reading English, arithmetic is not performed from left to right. There is a particular order in which the addition, subtraction, multiplication and division (not to mention parentheses and exponents) must be done. And for most of us old timers, that order is represented by the acronym PEMDAS (or its variations).

P – parentheses

E – exponents

M – multiplication

D – division

A – addition

S – subtraction

I learned the mnemonic “Please Excuse My Dear Aunt Sally” to help me remember the order of operations.

The idea is simple: to solve an arithmetic problem (or simplify an algebraic expression), you address any operations inside parentheses (or brackets) first. Then exponents, then multiplication and/or division and finally addition and/or subtraction.

But there really are a lot of problems with this process. First off, because multiplication and division are inverses (they undo one another), it’s perfectly legal to divide before you multiply. Same thing goes for addition and subtraction. That means that PEDMAS, PEDMSA and PEMDSA are also acceptable acronyms. (Not so black and white any more, eh?)

Second, there are times when parentheses are implied. Take a look:

If you’re taking PEMDAS literally, you might be tempted to divide 6 by 3 and then 2 by 1 before adding.

Problem is, there are parentheses *implied,* simply because the problem includes addition in the numerator (top) and denominator (bottom) of the fraction. The correct way to solve this problem is this:

So in the end, PEMDAS may cause more confusion. Of course, as long-time Math for Grownups readers should know, there is more than one way to skin a math problem. Okay, okay. That doesn’t mean there is more than one order of operations. BUT really smart math educators have come up with a new way of *teaching* the order of operations. It’s called the Boss Triangle or the hierarchy-of-operations triangle. (*Boss triangle* is so much more catchy!)

The idea is simple: exponents (powers) are the boss of multiplication, division, addition and subtraction. Multiplication and division are the boss of addition and subtraction. The boss always goes first. But since multiplication and division are grouped (as are addition and subtraction), those operations have equal power. So either of the pair can go first.

So what about parentheses (or brackets)? Take a close look at what is represented in the triangle. If you noticed that it’s only *operations*, give yourself a gold star. Parentheses are not operations, but they are containers for operations. Take a look at the following:

Do you really have to do what’s in the parentheses first? Or will you get the same answer if you find 3 x 2 first? The parentheses aren’t really about order. They’re about grouping. You don’t want to find 4 + 3, in this case, because 4 is part of the grouping (7 – 1 x 4). (Don’t believe me? Try doing the operations in this problem in different order. Because of where the parentheses are placed, you’re bound to get the correct answer more than once.)

And there you have it — the Boss Triangle and a new way to think of the order of operations. There are many different reasons this new process may be easier for some children. Here are just a few:

1. Visually inclined students have a tool that suits their learning style.

2. Students begin to associate what I call the “couple operations” and what real math teachers call “inverse operations”: multiplication and division *and *addition and subtraction. This helps considerably when students begin adding and subtracting integers (positive and negative numbers) later on.

3. Pointing out that couple operations (x and ÷, + and -) have equal power allows students much more flexibility in computing complex calculations and simplifying algebraic expressions.

Even better, knowing about the Boss Triangle can help parents better understand their own child’s math assignments — especially if they’re not depending on PEMDAS.

*So what do you think? Does the Boss Triangle make sense to you? Or do you prefer PEMDAS? Share your thoughts in the comments section.*

Denise says

I like that the boss triangle gives us another way to look at the order of operations but I can’t seem to get past the ingrained PEDMAS. 🙂

Laura says

And herein lies the beauty of math: You can use PEMDAS to your heart’s content! That’s how my brain understands the order of operations, too. Now that I’ve read a little more about the process behind learning PEMDAS, I wonder if I was really, really confused in algebra class. As I mentioned to someone else this morning, in an email, the boss triangle feels like a long-term investment — it takes into account how the students will be manipulating more complex algebraic expressions, including factoring quadratic equations, using FOIL (multiplying binomials) and even use the distributive property.

But if PEMDAS works for you — go for it!

Laura

Travis says

Upon seeing you mention FOIL, I hope (not sure if you have already) that you have a post about the problem with FOIL. Cause its great for (2x+3y)(3x+6y) but FOIL falls short when it comes to (2x+3y)(3x+6y+3z). And unfortunately teachers, at least when I was in school, largely dropped the ball on multiplying polynomials greater than 2.

Laura says

I have not written about FOIL yet. It is clearly not a foil-proof method, as you demonstrate. Ha! When I learned (and taught) FOIL I was told it was strictly for multiplying two binomials.

For those who aren’t aware of FOIL, this is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. Unlike PEMDAS, FOIL does not require a certain order, but simply offers an organized way to make sure that each term in the first binomial is multiplied by each of the terms in the second binomial: the first two terms are multiplied, then the outer terms are multiplied, then the inner terms and finally the last two terms.

Laura

Shawn says

Ikr! And why should I or do I have to if it works!!!!!!

Megan says

Also, the math world has been discussing GEMS instead of PEMDAS as a way to use mathematical language to explain the steps of PEMDAS (and to simplify it!)

http://randomcontact.blogspot.com/2008/08/forget-aunt-sally-ive-got-gems-for-you.html

Laura says

That’s another terrific idea, Megan. GEMS = Grouping, Exponents, Multiplication/Division, Subtraction/Addition. It’s still PEMDAS, but it underscores the idea that the inverse operatons (x and ÷, and + and -) can be done in either order.

Laura

Siouxgeonz says

Some students are going to have trouble with having two separate abstract ideas for figuring out “what to do next.” I think the grouping concept could be included in the visual (without saying something like “of course, the fraction bar acts as a vinculum,” and watching that entire class of pre-algebra students make their “okay, I’ll pretend I’m following you!” look and making that tutor observing, me, go downstairs and look up what vinculum means ;))

Laura says

Yep, some students are going to have trouble — no matter what model is used. And that’s why teaching math (and learning math) is so hard. I wonder though, if using the Boss Triangle in the early grades — elementary school — would be more useful than PEMDAS, thus preventing algebra students from having to pretend they know what’s going on. I’d love to hear from an elementary teacher who is using the Boss Triangle.

Thanks for chiming in!

Laura

guatdidusay says

I agree very much.

Jori says

Oh, I like the triangle I use arrows to indicate A/S M/D have equal powers. 😀 Triangle shows what I am trying to explain better.

Carol says

I teach remedial math at a community college and spend a good amount of time undoing the damage that PEMDAS has done: students are inclined to put multiplication before division and addition before subtraction.

Robi B. says

I always learned that it is “power before dot before line operations.” where line operations are the weakest.

Multiplication and division are dot operations (5·5÷5),

addition and subtraction are line operations (5+5-5).

shouldn’t be too confusing 😉

Connie says

Still not seeing the WHY here. It’s all different answer-getting techniques. We need that too, but they matter less if students have conceptual understanding.

There is a natural order to operations, not a list of rules invented by a math professor somewhere. If you know that 3 + 7 x 5 represents 3 pennies and 7 nickels, the answer is obviously 38 instead of 50. We can only add like items, so nickels need to be converted before then can be added to the pennies.

If instead, the context is two people, one with 3 nickels, and the other with 7 nickels, the expression would read: (3 + 7) x 5. Grouping symbols need to be added to force a change to the natural order.

Exponents are simply another form of multiplication – converting units.

Just read Ch. 5 of Como Molina’s The Problem with Math is English or I wouldn’t have had anything to add here!

Laura says

Your example of nickels and pennies is spot on, Connie. Perhaps from Molina’s book? Nice example! Thanks to both of you for commenting.

Laura

Heather says

Thank you.

To say that we can ignore grouping is ridiculous because clearly 5÷2+3 is not the same as 5÷(2+3).

It was really bothering me that so many comments above were praising this article and did not notice this crucial flaw.

Shawn says

Ikr! I’ve noticed no one had responded to your statement.. It just doesn’t make sense to use anything but PEMDAS.. America is making our kids dumb.. On purpose at that! That saddens me add as an educator and a parent 🙁

Como Molina says

Howdy Connie!

I am the author of The Problem with Math is English. I am so glad that you see the conceptual side of the order of operations and I hope that you learned a lot from reading the book.

S says

Clarification Needed:

So if M/D and A/S have equal powers then which one would you solve first in order to get the correct answer if you find them together within the same equation? Wouldn’t the simplest answer be using the “rule” of reading/solving going from left to right when one encounters M and D, or A and S within the same equation (although M/D would be solved before A/S get my drift)? For example:

1+2-3 = 3-3 = 0

1-2+3 = -1+3 = 2

Since Addition and Subtraction are equals you would then solve the problem going from left to right regardless of which one was in front/back of the other. However, if you used the PEMDAS very strictly you could come up with a different answer such as this:

1-2+3 = 5-1 = 4 Because you could interpret PEMDAS as meaning that Addition comes BEFORE Subtraction in all circumstances, but this is not the case. If you are using the Pyramid method then you still need a further explanation of what to do when you encounter M/D or A/S in the same equation.

So, really I think you need to add an additional explanation stating the if M/D and A/S are created equal when encountered within an equation the next step is solving the equation going from left to right. So in other words, solving the equation going from left to right trumps order of operation if all you have left to calculate comes down to M/D or A/S. Does that make sense? If there is a clearer way to explain this please respond.

S says

Oops I made a mistake, the answer to the 2nd example of 1-2+3 = 1-5 = -4 if doing Addition BEFORE Subtraction.

N says

Your mistake is thinking the 2 is positive. You do not need to do Addition and Subtraction from right to left, what you need to do is keep your signs straight.

The numbers are 1, -2 and 3. You can add them together in any order you wish. This is referred to as the cumulative property.

Marc says

I’ll come to S’s defense. If we were to adopt the convention that addition always precedes subtraction, or (even worse) that it be left to the one performing the computation to decide, then one can argue that the 2 is not negative. It is positive. The – sign before the 2 is a subtraction operation, not a negation operation. It is tempting to those versed in the universally agreed upon order of operations to mentally see 1-2+3 as 1+(-2)+3, but in doing so, those persons are instinctively traversing the operations from left to right. Supposing instead that the addition operation already present in 1-2+3 precedes the subtraction, and therefore also precedes the reformulation of subtraction as addition of the opposite, then 1-2+3 can only be interpretted as either 1-(2+3) or 1+-(2+3). The 2 is positive in either case. That said, I am in agreement with S that the boss triangle leaves out the essential convention of left-to-right traversal through operations of otherwise equal precedence. It is only if this rule is included that N’s so called “cumulative property” necessarily holds.

Daniel Aplet says

There are many stupid things in this world and this system of math is high on the list,These examples take simple math and make problems out on them,The order of operations is from left to right,it one wants a different order all they have to do is write it that way.I have seen many problems on Facebook where at the end (what ever number that may be) you multiply it by 0,(and any number 0 times is 0) then they just as 10,what you you think the answer is going to be???

Daniel Aplet says

I agree,this Pemdas system is crazy.Order of operations is from left to right.These problems i have seen on Facebook always put brackets when there are none,if you going to do that what determines where the bracket should be?And would one have a long equation and jump anywhere within it and go wherever? One can come up with a lot of different answers,and as i see on Facebook people do.In an example here they give a long equation and start at the end and go back,why would anyone with a brain do that.

Jane Ambali Jamon says

Tnx for this….I’d learn so much also at PEMDAS

bill says

Pretty cool but I’ll stick with PEMDAS. The BOSS thing seems to require that you remember a picture. Not good (for me at least). Thanks for the article. It was a really good explanation.

Jenny says

Since there are no quotations here, should one simply work left to right and ignore PEMDAS?

3-3×6+2=

AC says

Last night i finally figured out why i sat in remedial math in college, and kept making super low scores on the tests. I’m talking 48, and 52 scores low. I couldn’t understand why my problems never came out correct. Apparently i wasn’t paying attention in middle school, and kept trying to go in order. Meaning multiplication always before division, and addition always before subtraction.

I even joined a tutoring group and couldn’t grasp this concept.