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

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.

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.