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PEMDAS

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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.

Last week, we had some fun with the order of operations at the Math for Grownups facebook page.* Turns out remembering the order that you should multiply, add, etc. in a math problem is a tough thing for adults to remember. Imagine how kids feel! But this is a really simply thing that you can apply to your everyday life — all the while, reminding your kid how it goes.

First off, here’s the problem that we considered on facebook last week:2 • 3 + 2 • 5 – 2 = ?The answer choices were 38 and 14.I would say that the responses split pretty evenly. Lots of folks chose the incorrect answer first and then realized their mistakes.

So what’s the correct answer? 14. Why? Because of the order of operations. A lot of us learned the order of operations — or the set of rules that establishes the order we add, subtract, multiply, divide, etc. — with a simple mnemonic:Please Excuse My Dear Aunt SallyORParentheses, Exponents, Multiplication, Division, Addition, SubtractionORPEMDAS

(Before going further, I must acknowledge that there are some problems with this approach. First off, it doesn’t really matter if you add before your subtract or multiply before you divide. Those operations can be done in either order with no problem. Second, many teachers are approaching this differently, a topic that I’ll explore in September.)

If you do the operations in the wrong order — add before you multiply, for example — you’ll get the wrong answer. And that’s how people got 38, instead of 14. They simply did the math from left to right, without regard to the operations.CORRECT2 • 3 + 2 • 5 – 2 = ?6 + 2 • 5 – 2 = ?6 + 10 – 2 = ?16 – 2 = 14INCORRECT2 • 3 + 2 • 5 – 2 = ?6 + 2 • 5 – 2 = ?8 • 5 – 2 = ?40 – 2 = 38

All of this is well and good, but what does it have to do with the real world? How often are you faced with finding an answer to a problem like the one above? And that’s exactly what one reader asked me. So I promised to explain things using a real-world problem.

Thing is, you do these kinds of problems all day long, without even thinking of the order of operations. And that’s because you’re not writing out equations to solve problems. You’re simply using good old common sense.

Let’s say you’re going back-to-school shopping with your child. He’s chosen a pair of pants that are $15 and five uniform shirts that cost $12 each. But the pants are $5 off. What’s the total (without tax)?

You probably won’t write an equation out for this, right? (I wouldn’t.) Instead, you’d probably just do the math in your head or scribble some of the calculations on a scrap piece of paper or use your calculator. So here goes:

First the shirts: there are five of them at $12 each. That’s a total of $60, because 5 • 12 is 60.

Now for the pants: all you need to do here is subtract: 15 – 5 = 10. The pants total $10.

Finally, add the cost of the pants and the cost of the shirts: $10 + $60 = $70.

The above should have been super easy for most of us. And — surprise! surprise! — it used the order of operations. Here’s how:15 – 5 + 5 • 12 = ?The order of operations says you must multiply before you can add:

15 – 5 + 60 = ?

Then you can add and subtract:

10 + 60 = 70

There are other ways to set up this equation. In fact, I would use parentheses, simply because I want to keep the pants’ and shirts’ calculations separate in my mind:

(15 – 5) + (5 • 10) = ?

The result is the same, because the process follows the order of operations — do what’s inside the parentheses first and then add.

UPDATE: A reader asked if I’d also show how this problem can be done wrong. So here goes! When you do the operations in the wrong order, you won’t get $70.15 – 5 + 5 • 12 = ?10 + 5 • 12 = ?

15 • 12 = 180

That’s more than twice as much as the actual total!

Try this with your kid. You can make it more complex by figuring out the tax. And there are lots of different settings in which this works — from shopping to figuring the tip in a restaurant and then splitting the tab to dividing up plants in the garden.  Just about any complex math problem that involves different operations requires PEMDAS. And that’s something all kids need to know about.

When have you used PEMDAS in your everyday life? Did this example spark some ideas? Think about the math that you did yesterday — or today — and share your examples in the comments section.

*Have you liked the Math for Grownups facebook page yet? What’s stopping you? We’re having great conversations about the math in our everyday lives. And I ask questions of my dear readers. Come answer them!