Help me understand order of operations

[Farrar]

So, I understand what it is and how it works. What I have never understood is why. And as a result I'm forever forgetting it. I'm pretty good at checking myself when an answer makes no sense and I often forget algorithms for things I don't use that often but I find it pretty easy to reverse engineer it, if that makes sense, because I usually understand what's being asked. (A good example is that problem in the Liping Ma book about dividing by a fraction - I wasn't positive what the procedure was, but it was easy for me to visualize the simple math being done, so I got the right answer, and then checked it against the algorithm as I remembered it then read the answer and it turned out to be correct). But I don't have that innate sense with order of operations so it always passes my reality check even when it's totally wrong. Ds had a problem that required it today (I remembered it in time, but I could have easily forgotten it) and I realized that they'll only have more. Sigh. So, anyone?

[Bang!Zoom!]

brightstorm.com has very helpful simple videos

 

khan & cool math are good for this for very beginner understanding as well

 

http://www.brightstorm.com/math/algebra/pre-algebra/order-of-operations/

[Farrar]

But that's what I mean. That video is well done. It explains HOW it works, which I get, even if I forget it. But why is that the order? The only thing I was ever taught was "it just is." But that can't be all there is to it.

[TracyP]

But that's what I mean. That video is well done. It explains HOW it works, which I get, even if I forget it. But why is that the order? The only thing I was ever taught was "it just is." But that can't be all there is to it.

 

I just read that the answer actually is that "it just is". They said that mathematicians decided on this order so there would be ease of understanding (not the term/word I want:tongue_smilie:) between themselves. I *think* I saw this in MM but I'm not sure.

[Bang!Zoom!]

Ah, you are looking for the history/development/acceptance of rules of the Order of Operations?

[Farrar]

Oh great. So what you're saying is that there's no way to intuit it if you forget it. I'm totally screwed.:lol:

[Farrar]

Not the acceptance or the history or anything... I just thought there had to be an underlying mathematical reason for doing it in that order and not in a different order. But maybe there isn't?!?

[Dana]

But that's what I mean. That video is well done. It explains HOW it works, which I get, even if I forget it. But why is that the order? The only thing I was ever taught was "it just is." But that can't be all there is to it.

 

I explain it as shortcuts.

 

This works at a basic level... there's a bit more to it higher up - but this explanation works for getting an understanding of why.

 

Say you have 5+3x4

 

We do the multiplication first since we see multiplication as a shortcut for repeated addition... so if we were to expand the problem, it'd be

5+4+4+4

 

Exponents are a shortcut for repeated multiplication, so you'd expand the exponent as multiplication, then expand it as addition.

 

Parentheses (or symbols of grouping) simply say DO THIS FIRST!

 

So, order of operations is

(1) Symbols of grouping (DO THIS FIRST!)

(2) Exponents (shortcut for multiplication)

(3) Multiplication and division as they appear left to right (multiplication is a shortcut for repeated addition... and division doesn't exist... only multiplication... which is why you'd do division before multiplication when reading left to right)

(4) Addition/subtraction left to right (similarly, subtraction doesn't exist...only addition (define subtraction as addition of the opposite)).

 

Thinking of the operations as shortcuts does help with WHY the order is what it is - and I really hate the mnemonics because they don't explain why at all - and they have students arguing multiplication before division rather than seeing them at an equal level.

[Miss Tick]

Wait, are you specifically talking about dividing fractions?

 

Is it because you are writing it 1/2 (division symbol) 1/2 and then can't remember to invert and multiply because there isn't a visual clue about whether the inversion or multiplication happens first? If so, what if you write it with 1/2 perched way up over the other 1/2 as a huge compound fraction, then is it more clear (because there isn't much else you can do with that super tall fraction...)?

 

I read Liping Ma's book a while ago, I remember the fractions were more unusual than 1/2 and 1/2 but I don't recall what they were.

 

Or perhaps I'm taking your question too literally and you are looking for some grander "order of operations"...

[Beth in SW WA]

Thinking of the operations as shortcuts does help with WHY the order is what it is - and I really hate the mnemonics because they don't explain why at all - and they have students arguing multiplication before division rather than seeing them at an equal level.

 

:iagree:

Singapore HIG says NOT to use this mnemonic: Please Excuse My Dear Aunt Sally.

Although I'm tempted. :)

[Farrar]

The dividing fractions thing was just my example of how I'm always forgetting how to do math, but I can usually remember it because I feel like I have a decent (if not always great) grasp on what I'm doing. But I totally lack that on order of operations.

 

That helps, Dana. That makes a lot more sense to me that "some old, dead math guys got together and agreed on it."

[Lang Syne Boardie]

But if a child can be taught the mnemonic he can also be taught the rules of it. He can learn that multiplication and division are solved as you come to them, neither before the other, and the same for addition and subtraction. I don't remember having any problem grasping that concept. My children have easily understood it, as well.

 

Please

Excuse

My Dear

Aunt Sally

 

I don't think the problem is with the mnemonic. The problem occurs if the teacher only teaches half the lesson.

 

In any subject, mnemonics should help the student recall what he was taught to understand. PEMDAS should be the last thing learned, after the student understands what he's doing.

 

Sometimes (remember the giant McDonalds Cheeseburgers thread?) it seems to me that teachers start with the mnemonic and hope the child will understand what he's doing after he goes through the motions many times. That's backwards.

[Beth in SW WA]

But if a child can be taught the mnemonic he can also be taught the rules of it. He can learn that multiplication and division are solved as you come to them, neither before the other, and the same for addition and subtraction. I don't remember having any problem grasping that concept. My children have easily understood it, as well.

 

Please

Excuse

My Dear

Aunt Sally

 

I don't think the problem is with the mnemonic. The problem occurs if the teacher only teaches half the lesson.

 

I wonder why the author of the HIG warned against it.

 

In any subject, mnemonics should help the student recall what he was taught to understand. PEMDAS should be the last thing learned, after the student understands what he's doing.

 

:iagree:

 

Sometimes (remember the giant McDonalds Cheeseburgers thread?) it seems to me that teachers start with the mnemonic and hope the child will understand what he's doing after he goes through the motions many times. That's backwards.

 

I loved that cheeseburgers thread. Link?

[SonshineLearner]

:iagree:

Singapore HIG says NOT to use this mnemonic: Please Excuse My Dear Aunt Sally.

Although I'm tempted. :)

 

Oops, why wasn't I suppose to use this? My son is 8, but we've had to explain order, as he makes up huge math problems for us. How does Singapore say to explain it? Ooops....

[Dana]

Singapore HIG says NOT to use this mnemonic: Please Excuse My Dear Aunt Sally.

Although I'm tempted. :)

 

I've seen it really cause problems.

I've had students argue with me about it because MD means multiplication first.

It's slower to walk through the steps... but the mnemonic can get them in trouble later. Don't put up roadblocks... even though you are tempted now!

 

That helps, Dana. That makes a lot more sense to me that "some old, dead math guys got together and agreed on it."

 

Purplemath is also a good site for explanations.

 

Some of math IS simply that "common agreement is this, so this is what it is".

 

I recently read an explanation/discussion of order of operations that strongly disagrees with the way I explain it given that it breaks down with numbers like fractions, decimals, and irrational numbers (you can't talk about multiplication as repeated addition for instance). But I think that looking at order of ops as shortcuts helps give a reason for WHY the convention is what it is and not "just because" :)

 

(And I just managed to multiquote! Yea! :D)

[Farrar]

I never learned the mnemonic for order of operations (well, the Aunt Sally thing does sound vaguely familiar, but I know I didn't get it in school) - but I did just learn the procedural method. I really only ever learned the procedural method for anything. Mostly I could get the underlying concepts, but this is a case where I never did. I've really learned that for me, if I can't remember WHY something works in math, I never remember how to make it work anyway.

 

Of course, SM tells you not to use the mnemonic. SM is like, always right, right? :D

[Dana]

In any subject, mnemonics should help the student recall what he was taught to understand. PEMDAS should be the last thing learned, after the student understands what he's doing.

 

Sometimes (remember the giant McDonalds Cheeseburgers thread?) it seems to me that teachers start with the mnemonic and hope the child will understand what he's doing after he goes through the motions many times. That's backwards.

 

I agree with this.

Unfortunately, I hear so so so so many times from my students (I teach developmental math at a cc) that they are just told the mnemonic and not told why. In some cases they've been told things completely wrong (I heard from a district math specialist that she SAW a teacher tell students they didn't need the distributive property because they could use order of operations.)

 

I see PEMDAS as a mnemonic introduce problems - even with slashes: P/E/MD/AS. It may be just given the type of student I see, but I shudder any time I hear someone say PEMDAS. Some students manage to use it correctly - but we're only talking about 4 steps to remember. I don't see a need for a mnemonic for that. (And I see the problems with it every semester!)

[Miss Tick]

What if we made the mnemonic "Please excuse mad, dangerous aunt sally." Then it would still make sense if it were "Please excuse dangerous, mad aunt sally."

[Farrar]

But shouldn't it also be Sally Aunt in that case, which doesn't make sense?

[Lang Syne Boardie]

I loved that cheeseburgers thread. Link?

 

I'll see if I can find it. I loved that thread, too!

 

I've never used Singapore, but I'm with farrarwilliams. Aren't they always right? LOL

 

I wonder if it was just about the fear that the mnemonic would be assimilated but the details and concepts would not.

 

I have to constantly remind myself that I have no clue how to teach remedial math or how to teach a classroom full of kids with varying abilities. My children are all able to learn anything I throw at them, so if I tell them that multiplication doesn't come before division (and show them why) they just say, "Oh. OK, I see that." And the mnemonic is not a stumbling block.

 

Maybe Singapore doesn't want anybody hanging onto the mnemonic because there's no way a whole class will all understand what they're doing. (And no guarantee that the teacher really gets it, either, TBH.)

[Lang Syne Boardie]

Found it! Does McDonalds Sell Cheeseburgers?

[Lang Syne Boardie]

I agree with this.

Unfortunately, I hear so so so so many times from my students (I teach developmental math at a cc) that they are just told the mnemonic and not told why. In some cases they've been told things completely wrong (I heard from a district math specialist that she SAW a teacher tell students they didn't need the distributive property because they could use order of operations.)

 

I see PEMDAS as a mnemonic introduce problems - even with slashes: P/E/MD/AS. It may be just given the type of student I see, but I shudder any time I hear someone say PEMDAS. Some students manage to use it correctly - but we're only talking about 4 steps to remember. I don't see a need for a mnemonic for that. (And I see the problems with it every semester!)

 

Wow. (the bolded) Algebra should be lots and lots of fun for those kids.

 

Since Singapore is designed for classroom use their experiences have probably mirrored yours.

[Farrar]

Hey, I remember that thread! And I learned from looking at it that Bill is apparently banned again, so I guess he won't be weighing in here. Somehow I always miss the banning threads.

 

Long division is another example of something I've completely forgotten the procedure for, but I can still do, only slowly, because I know I understand what the heck I'm actually doing.

[Dana]

Wow. (the bolded) Algebra should be lots and lots of fun for those kids.

 

Yeah. I was astounded.

 

She said she asked the teacher (privately) what she'd do if there were a variable... 3(x+5) rather than 3(4+5).

The teacher said, "Oh! I hadn't thought of that!" :001_huh:

 

Hopefully she made the change.

 

But my poor son is going to LEARN the math REALLY REALLY WELL!!!

I tell him he gets punished for actions of my students. Poor kid. :D

[Dana]

Long division is another example of something I've completely forgotten the procedure for, but I can still do, only slowly, because I know I understand what the heck I'm actually doing.

 

Base 10 blocks are great for showing what's happening.. then speed picks up with the algorithm.

 

(Banning is generally on the general board.)

[letsplaymath]

So, I understand what it is and how it works. What I have never understood is why. And as a result I'm forever forgetting it. But I don't have that innate sense with order of operations so it always passes my reality check even when it's totally wrong. Ds had a problem that required it today (I remembered it in time, but I could have easily forgotten it) and I realized that they'll only have more. Sigh. So, anyone?

 

The order of operations is what it is because multiplication (or division) distributes over addition (or subtraction), and not the other way around.

 

That is, if you write: 3x(2+6),

It's the same as writing (3x2)+(3x6),

And it's also the same as: 3x8.

3x(2+6)=(3x2)+(3x6)

BUT

If you write: 3+(2x6),

It is NOT the same as (3+2)x(3+6).

[some beginning algebra students do try to distribute addition that way, but if they thought about familiar numbers, they would see that it can't work.]

 

Because addition does not distribute over multiplication, we can write that last sample 3+(2x6) without the parentheses. They make no difference.

3+(2x6)=3+2x6

Therefore, when you don't have parentheses grouping the addition together, the multiplication (or division) only applies to whatever it is next to. But when there are parentheses, then it will distribute.

 

As for why the distributive property works that way, the easiest way to see it is by drawing a rectangle:

Euclid’s Geometric Algebra

[ChandlerMom]

You can reason why it is a good way to do things, but it still comes down to the idea that symbolic math is a language and like any language it requires rules of syntax. The order of operations is the fundamental syntax of symbolic math.

 

We teach our kids all sorts of rules about word order in english and pluralization, tenses, etc. And there are usually GOOD reasons we do it that way. But other languages, like latin or chinese use very different syntaxes that also make sense.

[regentrude]

As far as mnemonics go, I like the one kids are taught in German schools. in Germany, the symbol used for division is a colon : and the multiplicaton symbol a dot. The kids learn

"Punktrechnung vor Strichrechnung" which translates into

"dot operations (i.e. multiplication and division) before line operations (i.e. plus and minus)"

 

Just wanted to share this in case somebody finds it helpful.

It eliminates the false impression PEMDAS gives that multiplication should be done before division by grouping them together as "dot operations".

[Dana]

As far as mnemonics go, I like the one kids are taught in German schools. in Germany, the symbol used for division is a colon : and the multiplicaton symbol a dot. The kids learn

"Punktrechnung vor Strichrechnung" which translates into

"dot operations (i.e. multiplication and division) before line operations (i.e. plus and minus)"

 

Just wanted to share this in case somebody finds it helpful.

It eliminates the false impression PEMDAS gives that multiplication should be done before division by grouping them together as "dot operations".

 

COOL!!!

 

That also explains why ratios are done with a colon. I've never gotten why that was :)

[boscopup]

As far as mnemonics go, I like the one kids are taught in German schools. in Germany, the symbol used for division is a colon : and the multiplicaton symbol a dot. The kids learn

"Punktrechnung vor Strichrechnung" which translates into

"dot operations (i.e. multiplication and division) before line operations (i.e. plus and minus)"

 

That is so much easier than the silly acronym used here in the US! :tongue_smilie:

 

I never actually learned the acronym. I couldn't remember it. I just remembered the order itself. That was easier for me than the acronym. Parenthesis came first because... well, that's the whole point of them being there at all. Exponents came next. Multiplication and division came next (and they're just the inverse of each other, so it's easy to remember that they go together), and then addition and subtraction (again, inverse of each other, so easy to remember). And you go left to right, the same way you do when you read English words. I guess I always thought of the operation order as being from most difficult to least difficult (meaning, we learned exponents after we learned multiplication and division, and we learned multiplication and division after we learned addition and subtraction).

 

In MM, my son had to do several order of operation problems where the parenthesis were changed between similar problems to illustrate how the results can be vastly different if you do the operations in different order. I think that helped drive it home.

[morosophe]

I learned Aunt Sally, but I also understood WHY Aunt Sally, although not to as great a degree as letsplaymath, above. (Actually, I think they might have mentioned something like that in class, now that I think about it, but it was back in junior high, so I'm not too clear, all right?)

 

But, yes, parentheses first because... that's what they're for.

 

Exponents next, because an exponent clearly only relates to the number it's next to. (Unless it's there for a quantity in parentheses, of course.)

 

Multiplication and division next because they're higher-order operations, (which is another way of saying that whole "shortcut" thing talked about above) and that's just how mathematicians decided to do it. (Plus, that distribution thing letsplaymath was talking about, I guess.)

 

And good old addition and subtraction last, because they're what's left.

 

So, Aunt Sally didn't really cripple ME. (Now, the fact that my very educated mother kept insisting on just serving us nine pizzas, and yet Pluto's not a planet anymore... That has caused me some problems, or at least sorrow over Pluto. Please excuse my very educated yet misguided mother. Suddenly vaguely Oriental caricature of a poster says, Going to bed sooner is wise.)

[Barb_]

Say "groups of" when you see the X sign and use the word "and" for addition as well as "is" for equals and it becomes clear why operations should be done in order.

 

We read from left to right, so

 

3x5+2x7+8=

 

Is the same as saying 3 "groups of" 5 "and" 2 "groups of" 7 and 8 individual items are 37.

 

Read it as 3 groups of five and 2 individual items...then you have to multiply the result by 7 and the whole thing is muddled.

 

I always tell my kids to do the groups first. We are either taking apart the groups (dividing) or putting them together (multiplying), but either way, we get rid of all the groups and translate them into individuals before we can begin adding and subtracting.

 

Barb

[justasque]

Oh great. So what you're saying is that there's no way to intuit it if you forget it. I'm totally screwed.:lol:

 

I think a lot of it is just DOING it, over and over again, until your eye just SEES it. Sort of like driving - when you first learn to turn you have to really think through putting on the signal, and slowing down, and turning the wheel, and going round the curve, then straightening the wheel, etc. But once you've got it in "muscle memory", you just think "turn right" and you do all the steps without needing to break it down.

 

When I see a math problem, my eyes just "see" it the right way. PEMDAS, or whatever method you use, is just for the "training the eye" phase. After using it a bazillion times, PEMDAS isn't needed - the training kicks in.

 

FWIW, I teach my students to think of it as PE(MD)(AS), and make sure they get lots of examples where the answer will be wrong if they use PEMDAS instead, if you KWIM.