"When you're curious, you find lots of interesting things to do." ~ Walt Disney

Aug

14

Filed Under (Mathematics) by Philip McIntosh on August 14, 2010 and tagged GEMDAS, order of operations, PEMDAS

It was at the first meeting of the Challenger Middle School Mathematics Department that I first heard of GEMDAS. What an obviously good idea!

**Remember PEMDAS**

PEMDAS is a mnemonic device used to remember the order of operations. The order of operations is a set of rules that specify in which order to perform the mathematical operations in an expression. Such a convention is necessary so that everyone in the world gets the same results when working a math problem (assuming they don’t make a mistake!).

PEMDAS is also remembered as “**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally” (what for, no one is quite sure). It means that when evaluating an expression that includes more than one operator, anything in **p**arentheses is evaluated first, followed by **e**xponents, then **m**ultiplication and/or **d**ivision (whichever comes first, left to right), and finally **a**ddition and/or **s**ubtraction (whichever comes first, left to right). That is the order of operations.

**GEMDAS is Better**

At the aforementioned department meeting, Ms. Hertzog, a math teacher at Challenger, said something about GEMDAS being superior to PEMDAS because with PEMDAS some learners get it stuck in their heads that **p**arentheses are the only grouping symbols that need to be taken into account, or else they get confused when some other grouping symbol is used instead of parentheses. She made no claim to inventing GEMDAS, but apparently heard about it at a workshop somewhere.

In GEMDAS, the **G** stands for **G**rouping symbol, and all the other letters keep the same meanings that they have in PEMDAS. So with GEMDAS, learners are better able to keep in mind that ALL expressions in, on, or under grouping symbols need to be evaluated first. Grouping symbols include *parentheses *(), *brackets *[], *braces *{}, the *vinculum *^{____} (which is the technical name for what most people call a *fraction bar*, which is also a repetition symbol used in decimal notation) , and the *radical *√ (also known as the *square root symbol*).

Google hits for PEMDAS outnumber hits for GEMDAS by a bit more than 10 to 1 (40,400 to 3,030, as of the date of this posting) so clearly PEMDAS is still a more popular mnemonic that GEMDAS.

I for one am switching over to using and teaching GEMDAS, and I have the impression that all the math teachers at my school are going to do the same so we are using a common vocabulary. It is traditional to teach and use PEMDAS, but tradition is no reason to keep an old idea around when a better alternative exists. It’s time to get rid of PEMDAS. Long live GEMDAS!

August 14th, 2010 at 9:32 PM

Sure does generalize the word “parentheses”. Now we just need a fun little phrase for this acronym “GEMDAS” (Unless there already is one) …Giving everyone my delicious apple sauce…I don’t know. I bet there are some other ideas for said word.

August 16th, 2010 at 1:39 PM

Thanks for checking it out April. I like “Give Everyone My Delicious Apple Sauce!” or “Give Everyone More Delicious Apple Sauce”. Any other ideas out there?

August 18th, 2010 at 4:02 PM

Try this one:

Good Education Makes Doing Algebra Simple

August 18th, 2010 at 4:26 PM

I think that is an excellent mnemonic for remembering GEMDAS!

August 26th, 2010 at 8:14 PM

Hey mr.mcintosh. So how is gemdas supposed to help you?

isnt please excuse my dear aunt sally still ok?

September 7th, 2010 at 7:14 AM

Just read your post today. It’s funny, I just taught order of operations to my students last week and suggested we use GEMDAS rather than PEMDAS so we were sure to address all grouping symbols and not just parentheses. I find no matter how much I emphasize it I still find students performing M before D and A before S rather than working out M and D from left to right before A and S from left to right. Perhaps we need GE(MD)(AS) or GE (DM)(SA).

September 7th, 2010 at 12:02 PM

Yeah it’s still “okay” but just not as good as GEMDAS for the reasons I outlined in the post. If you are able to wrap your head around the important idea that the P does not only apply to parentheses then stick with it.

September 7th, 2010 at 12:04 PM

Thanks for reading Zach. You point out another problem with the mnemonic method. It goes to show that there is no substitute for true understanding in place of blind use of a mnemonic device.

September 14th, 2011 at 1:24 PM

My pre algebra teacher is teaching this right now and this really helped thanks!

February 27th, 2012 at 11:13 AM

I believe it needs to change completely. By using either one we are teaching 6 steps instead of 4 steps.

We use Gary Enjoys MaD AntS. This is only 4 words.

Grouping symbols, Exponents, and so forth. We put this on a poster with a boy burning ants with a magnifying glass. If we keep using PEMDAS or GEMDAS we are still teaching six steps instead of 4 steps.

February 27th, 2012 at 2:52 PM

Thank you for sharing your ideas Leslie.

Well, that poster about the ants is inappropriate, but I can see how it would definitely work!

I am not convinced there is any advantage to your method. There is no way to shorten the number of steps in evaluating an expression using the order of operations. The number of steps is the number of steps, no matter what method we use to teach, learn and remember it.

September 18th, 2012 at 5:01 AM

I was looking for a mneomic for GEMDAS when my 5th grader came home with it. Another trick he was using was drawing vertical columns down on either side off addition and subtraction symbols so that everything else is pretty much in their own group. Then, after doing the proper grouping symbols it is all just left-to-right and the vertical bars assist in doing addition and subtraction last. Nice.

Now if only he’d get the implied multipication when a number is outside a group. Example: 5(66-64)

November 20th, 2012 at 12:14 PM

GEMDAS is nice, but like Leslie I think it still adds to confusion. I think Leslie could possibly do better by saying MuDdy AntS instead of MaD AntS, since there may be confusion about the A in MaD standing for Addition. Plus, this loses the ant cruelty aspect of her mnemonic device…unless by muddy you mean

November 20th, 2012 at 12:15 PM

…chocolate covered.

April 10th, 2013 at 12:47 PM

Is / a grouping symbol? What are the rules behind that? If it is, then is 4/2^2 supposed to evaluate the division first or the items on either side first? If its the items on either side, then this is 4/4 = 1. Otherwise, if we look at this as 4/2*2 is the 2*2 part of the group or just 2? In other words can we rewrite as 2*4/2 or as 2/2*4? Please provide a link to something that explains how to handle the various division symbols as grouping symbols and whether 4/2^2 is not the same as 4/2*2. I guess a/bc could = c/ba or it could = a/(bc)depending on perspective, but what about a/b where there is a leading 1? IN this case we actually have a/b = a/1b, so based on a/bc = ac/b, we can say a/1b=ab/1? I don’t think so. You tell me.

June 23rd, 2013 at 9:35 AM

I have heard it called that and I agree. It means “simplify both the top and bottom before proceeding.” 4/2^2 means you cannot do anything with the 4 or the 2^2 until the expression is simplified because they are in a “group.” You have to square 2 then divide that result into 2 before anything else can be done with that expression.

November 21st, 2013 at 10:11 PM

there is green elephants make dandy apple sauce

July 22nd, 2014 at 7:12 AM

GEMDAS is still a faulty way of teaching order of op’s. Multiplication and division are on the same level as well as addition and subtraction. Each should be calculated from left to right. Of course there is associativity and commutative properties and distributive. All of which are inherent in operations if one understand’s their group theory. PEMDAS and GEMDAS still fall short of what you/we should be teaching. Teach the math, not faulty and incomplete rules that frustrate everyone.

July 22nd, 2014 at 9:03 AM

Sure it’s faulty. Just like pretty much everything else we do. But, you have to do “something” and I think it is better to at least have some sort of mnemonic like this to work from rather than nothing at all. If you have an idea of how the order of operations can be taught better without ever mentioning GEMDAS or the like, I would love to hear about it. Of course we “teach the math.”

How about GE[MD][AS], with the brackets indicating the L-R orders. Or perhaps overlining the MD and the AS.

August 19th, 2014 at 11:47 AM

Quick question, does absolute value also fall in that category of grouping along with parenthesis and brackets?

August 19th, 2014 at 12:06 PM

Great question. Yes, because nothing can go through the absolute value symbol unless you can somehow simplify and resolve what is enclosed in the absolute value bars. Sometimes you have to accept that there is more than one result.