Math ? - Order of operations

[Pip]

I learned the acronym Please Excuse My Dear Aunt Sally to do the order of operations. However, I have noticed that sometimes the order changes for no apparent reason. I am hoping someone can help me, because my dd is also confused.

 

We are using Saxon Algebra 1/2 with the DIVE cd, but I also noticed this when doing CLE math.

 

6+2-4x3.

 

I would do the 4x3 first and then do the addition and subtraction. However, the solutions manual and answer key would show going straight down the line. The order of operations WAS followed up to this point if there were parentheses. Once the parentheses were eliminated, the equations was done in the order written.

 

However, now we are seeing problems done in the normal order of operations. We are doing solving for x with fractions, no parentheses. Now dd is getting the answers wrong, because in order to get them right previously, she did them left to right. Now she is expected to change gears and do the order of operations correctly to get the right answer.

 

I am NOT math illiterate. I have pored through the books trying to find the reason behind this. I have run this by friends who agree the standard order of operation should be followed. This is not a typo in the answers becauses it has come up numerous occasions.

 

What am I missing. It is probably painfully obvious.

[Sunshine State Sue]

I have never heard of going left to right and ignoring PEMDAS. Never. I have a degree in math.

[HSHS]

Is there something that specifically says ignore the order of operations for this part, or has only part of the order of operations been covered & they're not expecting the student to know the rest of it & so working without it...? Other than that I agree that it's a little odd. :001_huh:

[Beth in Central TX]

We recently watched The Teaching Company Algebra I DVD lesson on order of operations. Prof. Sellers taught PEDMAS just like I was taught in school and just like it is taught in our Modern Algebra Book 1 by Dolciani.

[Robin in DFW]

which is why we decided to use a more traditional textbook.

 

Math is math is math...

 

We just covered this in BCM...PEMDAS...it is what it is...there is no variation unless the directions specifically asked for something else.

 

hth,

Robin

[Pip]

Well, I think I will just tell dd to follow PEDMAS and if the solutions show another answer using another method, I'll ignore it.

 

Thanks.

[Susan C.]

Multiplication and division are grouped together, and you go from left to right, so if division comes first, you do the division. The same goes for addition and subtraction. The wording is multiplication and division, from left to right; and addition and subtraction, from left to right.

 

HTH

[Georgia On My Mind]

:iagree:PEMDAS is correct not PEDMAS. I am a degreed teacher and home schooler; I have taught Saxon 5/4 through Alg II and it is not taught differently in any of these textbooks.

[kiana]

Since multiplication and division are done at the same time, left to right, it doesn't really matter whether you say DM or MD, as long as the student is aware that they have the same precedence.

 

If they were doing the addition first, then the multiplication, that's absolutely wrong -- this should not even be shown to someone who has not learned the order of operations, as eliminating bad habits is far more difficult than not forming them in the first place. I have absolutely no idea why a curriculum would show it this way. I agree with your solution as to telling your dd to follow order of operations rules from the start and ignoring the key if it is wrong.

[kiana]

Also: quoting from wikipedia, with respect to multiplication/division and addition/subtraction having the same order of precedence.

 

There is a new mnemonic featured in Danica McKellar's books Math Doesn't Suck and Kiss My Math that does address this very issue: "Pandas Eat: Mustard on Dumplings, and Apples with Spice." The intention being that Mustard and Dumplings is a "dinner course" and that Apples and Spice is a "dessert course." Then it becomes not a linear string of operations to do one after the other, but rather the "dinner course" operations are considered together and performed left to right, and then addition and subtraction are considered together, again performed again left to right.

 

I had never heard of this before, but consider it interesting.

[tex-mex]

Also: quoting from wikipedia, with respect to multiplication/division and addition/subtraction having the same order of precedence.

 

There is a new mnemonic featured in Danica McKellar's books Math Doesn't Suck and Kiss My Math that does address this very issue: "Pandas Eat: Mustard on Dumplings, and Apples with Spice." The intention being that Mustard and Dumplings is a "dinner course" and that Apples and Spice is a "dessert course." Then it becomes not a linear string of operations to do one after the other, but rather the "dinner course" operations are considered together and performed left to right, and then addition and subtraction are considered together, again performed again left to right.

 

I had never heard of this before, but consider it interesting.

 

We've always stressed PEMDAS... love the food idea!! TT uses this thought but not like a fun learning tool as Ms. Danica. Just learned something. :D

[LinRTX]

I have the 2nd edition of Saxon Algebra 1/2. Looking through the book it clearly states that multiplication should be done before addition. What edition do you have? What lesson number are you on?

 

Linda

[Jann in TX]

The "Order of Operations" or what some people call PEMDAS or PEDMAS is standard across the board.

 

Parenthesis followed by exponents are worked first,

Multiplication and Division from left to right

Addition and subtraction from left to right.

 

It does not matter if you do the multiplication or the division first...

 

5 + 3 X 4 / 2 + 7

 

I would personally work it : 5 +[ (3 x 4) / 2 ] + 7

 

With regards to the 3x4/2 it does not matter if you multiply the 3 and the 4 first then divide by 2 or if you choose to divide the 4 by 2 then multiply by 3 the answer is '6' either way...

 

You have to read the directions very carefully in Saxon 87 and Algebra 1/2. They are trying to demonstrate the reason for an 'order of operations'... I vaguely remember a section where they specifically asked the students to ignore the 'order of operations' and just work the problem straight from left to right.... perhaps this is what the original poster was dealing with...

 

In reality even WITHOUT parenthesis the 'Order of Operations' is ALWAYS used. I tell my classes this is so that scientists from around the world will all agree on the SAME answer. I also encourage my students to WRITE IN THE PARENTHESIS/BRACKETS so they do not drop negatives...

[Beth in Central TX]

:iagree:PEMDAS is correct not PEDMAS. I am a degreed teacher and home schooler; I have taught Saxon 5/4 through Alg II and it is not taught differently in any of these textbooks.

 

It was typo on my part because I do multiplication first. I'm glad to know that both acronyms are correct.

[Georgia On My Mind]

:001_smile:I like the new chunking acronym mentioned above. It adds another dimension of learning to the process. That will help students who have short term memory issues to make it more meaningful and then it will get into their long term memory for easier recall. That's how we remember anything new; we must attach meaning to it. Write the last statement down somewhere you can find it easily it is a very important strategy for learning anything. Think about it yourself. That is how you can remember names, dates, events, obscure facts, etc. We remember nothing that doesn't have meaning to us. Ask your students what something means to them and help them remember it in that way. It's called making a connection in some programs.

[Dana]

The "Order of Operations" or what some people call PEMDAS or PEDMAS is standard across the board.

 

Parenthesis followed by exponents are worked first,

Multiplication and Division from left to right

Addition and subtraction from left to right.

 

It does not matter if you do the multiplication or the division first...

 

5 + 3 X 4 / 2 + 7

 

I would personally work it : 5 +[ (3 x 4) / 2 ] + 7

 

With regards to the 3x4/2 it does not matter if you multiply the 3 and the 4 first then divide by 2 or if you choose to divide the 4 by 2 then multiply by 3 the answer is '6' either way...

 

 

But in a different situation, it does matter what order you do the multiplication and division in.

 

I use the example of 12 / 3 X 2

The correct answer is 8 (do division, then multiplication)

I see a lot of students say 2 is the answer because they think "multiplication, then division" rather than "multiplication and division from left to right".

 

(Just one of my pet peeves, so I have to add the clarification :) )

[Georgia On My Mind]

:001_smile:I like the new chunking acronym mentioned above. It adds another dimension of learning to the process. That will help students who have short term memory issues to make it more meaningful and then it will get into their long term memory for easier recall. That's how we remember anything new; we must attach meaning to it. Write the last statement down somewhere you can find it easily it is a very important strategy for learning anything. Think about it yourself. That is how you can remember names, dates, events, obscure facts, etc.

[Jann in TX]

I hate rearranging numbers--too dangerous (and too many dropped negatives) so I teach Left to Right... and that is the way it is. Period.

 

I still have nightmares of my first year teaching--150 inner-city 9th and 10th grade math students... I learned the HARD way to teach 'proper form' in answers and to stick with PEMDAS... imagine we are grading homework in class the answer I read is "x + y + z" immediately hands fly up--Mrs P. Sally wrote "y + x + z" is that correct too? ... all the variations possible on EVERY QUESTION we graded... again I learned by fire!

[emzhengjiu]

I agree with you but in this case wouldn't the answer be the same either way? 12/3 x 2 is 12/3 x 2/1 = 24/3 = 8.

 

 

Ann

[Jann in TX]

But many would argue that changing the 2 to 2/1 is 'adding to the problem'... instead of using it the way it is written... Unfortunately most students are not aware that the 'real' division sign is a fraction bar!--so the 12/3 is really a fraction and you always multiply fractions by other fractions!

[kiana]

I agree with you but in this case wouldn't the answer be the same either way? 12/3 x 2 is 12/3 x 2/1 = 24/3 = 8.

 

 

Ann

 

The problem is that many students (at all levels) will say 12/3x2 = 12/6 = 2, because, in their heads, multiplication comes before division -> you do 3x2 before you do the division.

 

Clearly this is wrong, but it's something to watch out for because somehow, many students arrive at university still believing this. They also don't understand that if you're entering it into a calculator, if you really DO mean 12/(3*2), you need parentheses because the calculator strictly follows the order of operations.

[Julie in MN]

The problem is that many students (at all levels) will say 12/3x2 = 12/6 = 2, because, in their heads, multiplication comes before division -> you do 3x2 before you do the division.

 

Clearly this is wrong, but it's something to watch out for because somehow, many students arrive at university still believing this. They also don't understand that if you're entering it into a calculator, if you really DO mean 12/(3*2), you need parentheses because the calculator strictly follows the order of operations.

 

Wouldn't that only happen if the problem were typed like you have it? In a math book, wouldn't it look closer to 12 x 2, with the 3 typed underneath?

 

I think the ability to move terms around an equation is essential to really doing math, rather than just doing the workbook. I love how Math Relief teaches that first. A "term" is clearly identified and all parts labeled (including "assumed" parts such as an exponent of 1 or a + sign). Then they are moved around.

 

MR also teaches the proper order of an answer (alphabetical terms, highest exponent first, etc.). So the correct order of an answer is always clear. The only question a student should have is how many points are lost for giving a correct answer in the wrong order...

 

Julie

[kiana]

Wouldn't that only happen if the problem were typed like you have it? In a math book, wouldn't it look closer to 12 x 2, with the 3 typed underneath?

 

Julie

 

In a book, it may or may not be written that way, depending on what they're trying to teach. If they're trying to teach order of operations, for example, it may well be written like that.

 

Another reason, though, that a student should be able to use order of operations in that format is because this is how a calculator enters them. Expanding on my previous statement -- it is quite frequent for students to get the wrong numerical answer because they have entered 12/2*3 or 12/2+4 into their calculator instead of 12/(2*3) or 12/(2+4). This applies not only with simple fractions but with far more complex operations.

[Julie in MN]

Another reason, though, that a student should be able to use order of operations in that format is because this is how a calculator enters them.

 

Yes <sigh> I suppose I need to teach my son to use a calculator some day. His math team was using them & he never used his, but then a lot of competitions stopped allowing them anyways, because of new cheating techniques I hear. I suppose I push it off since I'm assuming it will take little time, as he manages to type numbers into computers, ipods, cell phones, and the like :tongue_smilie:

[Dana]

 

Another reason, though, that a student should be able to use order of operations in that format is because this is how a calculator enters them.

 

Again, just as a heads-up, it depends on the calculator!

The current generation of TI calculators do order of operations so a student can enter the problem just like it looks and be correct, but not all brands do this - and older models handle the arithmetic based on entry.

 

I demonstrate this one with -3^2. Without parentheses, we have 3^2 and then take the opposite of it, giving -9. Some calculator models say the answer is +9 - even without parentheses.

 

It makes for interesting discussions - and does really show the importance of knowing what you're doing and why - and not just blindly following technology.

[emzhengjiu]

Is the problem then the order of operation or are the students not being taught mathematics? Although I can see how easy it would be to make a mistake when inputing numbers into the calculator. I bought my daughters calculators that show the actual equation entered.

 

Ann

[Storm Bay]

I don't know what that E is for in the acronyms, but it should be first everything in parenthesis (and if they are like this 2 + 4[9/3 -7(3*2)], it would be done in this order (yes, I know there are ways to save time, but I'm going in strict order):

 

2 + 4[9/3 + 7(3*2)]=

2 + 4[9/3 +7(6)]=

2 + 4[3 + 24]=

2 + 4[27]=

2 + 108=

110

 

Of course, if it were 7(3+2) you could either make it 7*5, or you could multiply the 7 through first and make it 21 + 14. The latter is important when you get to Algebra and have things like 7(x+2).

 

Also, you can have more complicated things within the parenthesis, but always within each parenthesis you would multiply & divide before you would add and subtract, and left to right!

 

fwiw, my dc don't use calculators as a rule yet, but can't you add parentheses with calculators?

Edited January 25, 2010 by Karin

[LinRTX]

The E is for exponents.

 

Linda

[Dana]

Of course, if it were 7(3+2) you could either make it 7*5, or you could multiply the 7 through first and make it 21 + 14. The latter is important when you get to Algebra and have things like 7(x+2).

 

 

Actually, if you're doing order of operations, you could not distribute the 7 through. Then you're using the distributive property and not using order of operations.

 

The distributive property is essential with algebra - but order of operations is how you check solutions to equations. I've had students solve an equation incorrectly (making an error with distribution: for example saying -2(x-3) = -2x - 6 by not changing the sign), then checking the solution using order of operations would catch that there was a mistake. The student distributed incorrectly again during the check and claimed that their answer was correct.

 

So order of operations is important for checking work.

 

As for students not understanding - the mnemonic of PEMDAS is good, but students who are taught the mnemonic but without the understanding - or without the stressing that it's P/E/MD/AS - will get the problem I mentioned wrong - even with a calculator in front of them - because they multiply before dividing.

 

And yes, there are a lot (sigh) of students who are not taught math correctly. I teach at the community college (generally basic algebra) and have seen it all. We're starting rational expressions and equations and I guarantee you that there are many students who can't add two fractions correctly without a calculator.

[SonshineLearner]

And yes, there are a lot (sigh) of students who are not taught math correctly. I teach at the community college (generally basic algebra) and have seen it all. We're starting rational expressions and equations and I guarantee you that there are many students who can't add two fractions correctly without a calculator.

 

I'll show that I don't know how to use a calculator here. I don't know how to add fractions using a calculator. I can't see why you'd need to use a calculator. Fractions are fun to add on paper.....

 

Now.... all of the algebra makes my brain kinda burst.... but I'm gonna keep doing math with my daughter... and hopefully some day, I'll be Well Educated:-)

 

All the talk about students showing up at College without basic math down makes me scared!! I don't know how you make it through 3 or 4 yrs of High School Math without getting basic math down.

 

Hmmm

[KAR120C]

As for students not understanding - the mnemonic of PEMDAS is good, but students who are taught the mnemonic but without the understanding - or without the stressing that it's P/E/MD/AS - will get the problem I mentioned wrong - even with a calculator in front of them - because they multiply before dividing.

Getting slightly off topic but I thought I'd throw in that I've always taught DS that it was "PEMA" -- parentheses, exponents, multiplication, addition. And that exponents include square roots (fractional exponents), multiplication includes division (multiplying an inverse), and addition includes subtraction (adding a negative). That seems to have kept it simple and clear.

[Dana]

I'll show that I don't know how to use a calculator here. I don't know how to add fractions using a calculator. I can't see why you'd need to use a calculator. Fractions are fun to add on paper.....

 

Now.... all of the algebra makes my brain kinda burst.... but I'm gonna keep doing math with my daughter... and hopefully some day, I'll be Well Educated:-)

 

All the talk about students showing up at College without basic math down makes me scared!! I don't know how you make it through 3 or 4 yrs of High School Math without getting basic math down.

 

Hmmm

 

They teach students to use the calculator only (see all the threads on EveryDay Math). The appalling thing for me is that I teach material that's at an Algebra I level - and I have students testing into it straight from high school who have graduated high school with the 3 math credits required - of which Algebra I is the lowest course they can get high school credit for.

 

It's quite a challenge because you not only have to teach them the math, you have to convince them that they need study skills and the approaches they used in high school will not let them pass my course. I'm not limited to the number of Fs I can assign and I will assign you the grade you earn. (Local public schools in a couple of districts are not allowed to assign a grade of below a 50 - so as not to discourage the students.)

 

Just back from 3 hours of teaching, so a bit burned out. First test on Wednesday and grading them is always discouraging. Oh well. It was cool when writing the test to have ds (7) look over my shoulder and give the answer to one problem :lol:

[Carol in Cal.]

...and am now teaching Algebra 1. They do teach orders of operations, and apply it consistently, except when specifically told not to for a particular reason.

 

The only major exception is the mental math, which is chained problems that you do orally. IIRC those are all just worked left to right.

[Julie in MN]

Actually, if you're doing order of operations, you could not distribute the 7 through. Then you're using the distributive property and not using order of operations.

 

The distributive property is essential with algebra - but order of operations is how you check solutions to equations. I've had students solve an equation incorrectly (making an error with distribution: for example saying -2(x-3) = -2x - 6 by not changing the sign), then checking the solution using order of operations would catch that there was a mistake. The student distributed incorrectly again during the check and claimed that their answer was correct.

 

So order of operations is important for checking work.

 

As for students not understanding - the mnemonic of PEMDAS is good, but students who are taught the mnemonic but without the understanding - or without the stressing that it's P/E/MD/AS - will get the problem I mentioned wrong - even with a calculator in front of them - because they multiply before dividing.

 

And yes, there are a lot (sigh) of students who are not taught math correctly. I teach at the community college (generally basic algebra) and have seen it all. We're starting rational expressions and equations and I guarantee you that there are many students who can't add two fractions correctly without a calculator.

 

I'm getting more and more confused by the dichotomy on this thread -- bemoaning that students don't understand math on the one hand, and on the other hand not allowing them to use the distributive property because they might get confused? Or restricting students from using higher math skills just because it's hard to read on a calculator? I mean, I don't care if my students know what it's called, but if they know how to distribute, then why would I say they can't just because they made (or might make) an error? Trying it in different situations to see what happens and making some mistakes along the way is how you learn! I personally would commend the student for trying to distribute, and tell them to notice where they went wrong -- and look out for making that error in the future.

 

In my particular homeschool, we wouldn't artificially hold our kids back from trying to solve a problem using a good skill, and on the other hand we wouldn't move forward into complex topics if students were clueless about the basic math they were using. What's the point of memorizing steps they don't understand?

 

I suppose in group schooling, the class must go on according to plan. But does that really apply to homeschooling?

 

:confused: Julie

[Dana]

I'm getting more and more confused by the dichotomy on this thread -- bemoaning that students don't understand math on the one hand, and on the other hand not allowing them to use the distributive property because they might get confused? Or restricting students from using higher math skills just because it's hard to read on a calculator? I mean, I don't care if my students know what it's called, but if they know how to distribute, then why would I say they can't just because they made (or might make) an error? Trying it in different situations to see what happens and making some mistakes along the way is how you learn! I personally would commend the student for trying to distribute, and tell them to notice where they went wrong -- and look out for making that error in the future.

 

In my particular homeschool, we wouldn't artificially hold our kids back from trying to solve a problem using a good skill, and on the other hand we wouldn't move forward into complex topics if students were clueless about the basic math they were using. What's the point of memorizing steps they don't understand?

 

I suppose in group schooling, the class must go on according to plan. But does that really apply to homeschooling?

 

:confused: Julie

 

I'm homeschooling my 7 yo son. I teach math at a community college as well. We're really on a fixed schedule there and in college most of the student learning needs to happen outside the classroom. That's where they can play around and see what happens if they attempt with a different method.

 

As far as order of operations goes, my point is that order of operations is there for a reason (I use the analogy that you're doing shortcuts: multiplication is a shortcut for addition, exponents are a shorthand for writing multiplication). Using the distributive property, although it gives an equivalent answer if done correctly, is not following the order of operations and can cause problems (and is wrong in terms of procedure if what I'm trying to see if you understand is order of operations). It isn't a fear that a student will get confused if they use the distributive property (which is necessary in many places), it's that in demonstrating order of operations, you must use the order.

 

My student illustration was with someone who used the distributive property incorrectly both when solving an equation and checking the equation. That defeats the purpose of using order of operations to check your solution. If she had used order of ops, she'd have caught her distributive error.

 

I've taught at the cc for over 13 years now and have seen so many students who have seen a lot of math in their schooling but who have only been taught memorization and not understanding. With Order of Ops, they've been taught PEMDAS and not the idea of shortcuts, and most of them will multiply before dividing because the mnemonic is MD. I've had students argue with me that -3^2 is 9 because "that's what my calculator says".

[angela in ohio]

I have the 2nd edition of Saxon Algebra 1/2. Looking through the book it clearly states that multiplication should be done before addition. What edition do you have? What lesson number are you on?

 

Linda

 

Yes, if you give us a lesson and problem number, maybe we can help (I teach Alg 1/2.)

[Storm Bay]

Actually, if you're doing order of operations, you could not distribute the 7 through. Then you're using the distributive property and not using order of operations.

 

The distributive property is essential with algebra - but order of operations is how you check solutions to equations. .

 

 

Right. I'm up to my ears in pre-algebra, geometry and so I thought of that because I was explaining something to my dd who is in pre-algebra who was trying to do everything in her head. When I showed her the distributive property with only digits instead of with a digit and a letter, it clicked for her. However, done properly, it still works even if does technically mess up the order of operations.

 

The calculator problem is a big one. A girl down the street had always done well in math until Algebra. Her complaint? That the teacher had them do everything on the calculator so she wasn't able to understand it.

[mcconnellboys]

If I'm reading you correctly, I would say that is a mistake on their part. Standard order of operations is ALWAYS followed; there are no exceptions to it.

[Storm Bay]

If I'm reading you correctly, I would say that is a mistake on their part. Standard order of operations is ALWAYS followed; there are no exceptions to it.

 

Except in Algebra when you have to use the distributive property at times with things such as 3(4x +10)=90 because you won't be able to solve it otherwise. Yes, this is easy enough that many of you can solve it in your heads, but you get the point.

[nutmeg]

Except in Algebra when you have to use the distributive property at times with things such as 3(4x +10)=90 because you won't be able to solve it otherwise.

 

The way I explained it to my dd (and please correct me if I am wrong!) is to start with PEMDAS

 

P - Are there parantheses? Yes. Is there anything we can do inside the parantheses? No, we cannot add 4x+10, so we move on...

 

E - Are there exponents? Nope, we move on...

 

M/D - Yes, we can multiply using the distributive property...

 

A/S - No, move on...

 

Then solve for X

[Storm Bay]

The way I explained it to my dd (and please correct me if I am wrong!) is to start with PEMDAS

 

P - Are there parantheses? Yes. Is there anything we can do inside the parantheses? No, we cannot add 4x+10, so we move on...

 

E - Are there exponents? Nope, we move on...

 

M/D - Yes, we can multiply using the distributive property...

 

A/S - No, move on...

 

Then solve for X

 

 

Thanks. I didn't need this for my eldest, who got this on her own, but this may be necessary for my second who has been trying to solve her pre-Algebra in her head and has messed up on this with the parentheses a few times. This is the first time I recall reading about PEMDAS in my entire life. ETA I mean the abbreviation, of course!

[Murphy101]

I LOVE that food example! My boys will remember that much better than anything about any aunt sally!:lol: