posted
This is the lesson I review the most often with my kids. I have taught proper order of operations hundreds of time. Somethings have to be defined and this is how we have defined them and having a standard amongst everyone is necessary for sanity. Honestly, I find it ridiculous that this is even a discussion.
Posts: 2223 | Registered: Mar 2008
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posted
I kept getting 72. I couldn't tell on this little screen that it was a division symbol. I thought it was +. Posts: 891 | Registered: Feb 2010
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posted
Wow, what a pointless "problem" for people to be spending their time and energy arguing about. It's not even a question about math, really. It's a question about the notation we use to describe math, which is a matter of arbitrary convention.
Posts: 4600 | Registered: Mar 2000
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posted
well, it is interesting for me; a year or two ago our state math competition league had this argument because a problem writer (not me!) had written an expression in a format similar to this (thought I can't remember the specific expression) and different graphing calculators evaluated it differently!
I always make sure my competition math problems use a horizontal bar for division, as that avoids the confusion.
On a related note, I get emails from elementary teachers all over the world complaining that my "Order of Operations" game ( http://www.theproblemsite.com/games/onetoten2.asp ) doesn't work properly, because they've never been taught that M & D are on the same level of priority, as are A & S.
And these are the people teaching Order of Operations to our kids. Really scary. Posts: 324 | Registered: Mar 2008
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posted
Eh. I'm all for having competent math teachers, but there's such a thing as a priority. If the answer depends on getting the order of operations right, UR DOIN IT RONG. Put in some dang parens.
Posts: 10645 | Registered: Jul 2004
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posted
I teach BIDMAS (or BEDMAS or PEDMAS or PIDMAS depending on which words you use, sigh) to my maths classes.
I see how two is reached, and agree that it could possibly be seen as ambiguous, but in order to have a general consensus on how to solve mathematical notation in general I agree the answer should be 288.
Defining it different is fine, but it means you're talking a different mathematical language and, chances are, your spaceship is going to miss Neptune.
Posts: 8473 | Registered: Apr 2003
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quote:Edited also to add in responce (sic) to Glenn:
Except this is completely different than the topic at hand. We can prove that pi =/= 3. Proving that something previously thought to be correct is not correct will cause mathematical conventions to change, but that's not ignoring something "when it suits you", it's ignoring something because it's wrong.
You ignore i, pi vs. Tau, and the parallel postulate, and focus on my hyperbole? (that's a joke, son)
Mathematical conventions like order of operations are just that: conventions. One might just as well say that Hebrew must be read left to right because it's the correct way to read.
Mathematical facts are determined not by convention, but by proof. And one of the standard ways to begin a proof is to assume that a convention is untrue. i assumes that there is a square root of -1. To date, no one has found a value for i, but we've found i very valuable. Likewise for the parallel postulate. Assume it's not true, and we get entirely valid (and useful) geometries that Euclid never dreamed of.
In this case, order of operations in ENTIRELY convention. It's not even axiomatic, it's just accepted by agreement. We could change the rules at any time, as long as we agree on new rules. But the current rules don't eliminate ambiguity. They're better than nothing, but they aren't perfect. We can prove that order of operations is important, but we can't prove that they are true.
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I think you are misunderstanding the debate on Pi vs Tau. Just because C = τr doesn't mean that C = 2πr is somehow incorrect. Those in favor of Tau are simply suggesting that we should default to using Tau in those equations because Pi is a bit clunkier. I don't see how this has anything to do with ignoring the standard order of operations.
quote:But the current rules don't eliminate ambiguity.
Can you give an example where they don't?
Edit: I missed where you expand on why you bring up i, but I still don't understand how that's relevant.
Added:
quote:One might just as well say that Hebrew must be read left to right because it's the correct way to read.
That's actually a great analogy for this, but it doesn't help your point. People who write in Hebrew have decided that it is to be read right to left, in a pretty analogous way to how mathematicians decided on the standard order of operations. What you are suggesting is that we should ignore standard conventions and write Hebrew left to right because we feel like it. You are the one suggesting we break from current conventions, not us.
Posts: 5656 | Registered: Oct 1999
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Microsoft Excel says the answer is... let's see... Ah! "Formula error"!
Seeing as Microsoft is the foremost expert on everything, I conclude that the question is a trick question and there is no acceptable answer. That is unless "Formula error" is one of the available choices in a multiple choice question.
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posted
One of the comments here argues that since we wouldn't evaluate 48÷2x as 48÷2×x, we should also treat 2(12) as a distinct entity. I just don't know what the rule is here. Are variables treated differently than numbers? Maybe it is 2.
I went to the Order of operations article at Wikipedia, and there's nothing there that would help. I posted a note on the discussion page, though, and maybe someone there will have an answer.
Posts: 12266 | Registered: Jul 2005
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quote:But the current rules don't eliminate ambiguity.
They do in this case.
Assuming standard mathematical notation, there is one and only one solution for 48÷2(9+3). It is completely unambiguous. If you get any other answer than the correct one, then you made an error in reading the mathematical notation.
What it lacks is clarity -- it's far too easy for a human to misread it, as evidence by the multiple people who are comfortable and experienced with mathematical notation getting it wrong.
I'm glad that Samp posted this. It's interesting. I got it wrong at first as well, but as soon as I realized what I had done wrong, the correct answer was obvious.
Posts: 16551 | Registered: Feb 2003
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The ÷ sign seems to be confusing you. Replace it with "/" and it becomes clear.
48/2x = 24x
We don't treat variables any differently than numbers. If we did, we'd have to chuck the whole system.
Added:
If the above isn't clear, consider 3/4x. That is (3/4)x, never 3/(4x). So if x is 5, then 3/4x is 15/4, not 3/20.
Posts: 5656 | Registered: Oct 1999
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posted
Given that you always solve math problems left to right, you should solve it so you end up with a multiplication by x or you should write it as 48/(2x).
However, I can see this ambiguity, because I think this is how Lisa is seeing it, although I don't think you can solve this as written without knowing x (although this is something I'm not sure about).
48 ____ = 2x
This is the equivalent of 48/(2x) = NOT 48/2x, although I can definitely see that there is some ambiguity (it's a poorly written question there for those who don't view / as distinct from fraction notation.
The real problem written with a fraction looks like this:
48 (____) x = 2
I agree the distinction is not intuitive, but I think that, like the totally infamous airplane on a treadmill question, the ambiguity lies on the issues in the problem, rather than the issues in the actual conventions of our notation.
Posts: 8473 | Registered: Apr 2003
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quote:Originally posted by King of Men: Eh. I'm all for having competent math teachers, but there's such a thing as a priority. If the answer depends on getting the order of operations right, UR DOIN IT RONG. Put in some dang parens.
I disagree. I teach algebra, and when I have students who have been taught Order of Operations incorrectly, I have to re-teach it, otherwise I get students who will simplify this incorrectly:
2x - x + 5x
And they say it's -4x. Why? Because you do addition before subtraction, so it's 2x - (x + 5x).
And I'm sorry, but I'm not going to ask my students to insert dang parens in every problem to force them to read it as:
(2x - x) + 5x
That's a waste of time and ink and effort.
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quote:Originally posted by Jeorge: Because you do addition before subtraction,
Since when? Addition and subtraction are done from left to right, without any regard for which is first.
Posts: 12266 | Registered: Jul 2005
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quote:Originally posted by Jeorge: Because you do addition before subtraction,
Since when? Addition and subtraction are done from left to right, without any regard for which is first.
Did you read my posts in their entirety? Because that was exactly my point! There are teachers all over the place who teach it incorrectly, so by the time I get students in high school, they have it ingrained that you do addition before subtraction, even though that's not correct.
At first I thought they just were missing the point of what their teachers were saying, until I built that math game I linked earlier, and all of a sudden started getting emails from elementary school teachers all over the place complaining that my game didn't work right, because it didn't do addition before subtraction. Posts: 324 | Registered: Mar 2008
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posted
If you have a school teacher who isn't strong in math, but has to teach math anyway (which typically happens more in the elementary grades) they do a cursory reading on things, and then say, "Oh, I've got it...PEMDAS means you do operations in that order, so multiplication is before division..."
And they teach it to their students that way, and no one is ever the wiser until the student gets to an Algebra class and their understanding of it gets turned upside down.
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posted
As someone who teached BEDMAS for the first time, I can say it may not necessarily be what they are taught, only what they have learned.
The idea of BEDMAS as an order mnemonic is misleading. Lots of kids, despite my off-the-top explanation that DM and AS go simply from right to left in any order still stick to the idea that division comes before multiplication and addition before subtraction because that's what BEDMAS (or PIDMAS or PEDMAS or BIDMAS) makes it look like.
And some kids have difficulty getting over the mnemonic. Sigh.
Posts: 8473 | Registered: Apr 2003
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quote:Originally posted by Teshi: I can say it may not necessarily be what they are taught, only what they have learned.
Yeah, and sometimes it is simply mis-learning, but I've had enough inquiries from teachers to know that it's also mis-learning on the part of the teacher.
I don't know how to display this properly in the forum, but I always put MD vertically and AS vertically when spelling out PEMDAS, to visually show that M and D, from left to right, are at the same level, as are Addition and Subtraction.
Posts: 324 | Registered: Mar 2008
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quote:I think you are misunderstanding the debate on Pi vs Tau.
I think you are misunderstanding my argument. Pi is a convention. It is currently argued (we'll see if the convention changes) that Pi is inelegant, and the convention should be changed so that it is easier to understand.
quote:What you are suggesting is that we should ignore standard conventions
Ah. this proves you don't understand my argument. I'm not suggesting that we ignore conventions. I'm suggest we improve them.
quote: quote:But the current rules don't eliminate ambiguity.
Can you give an example where they don't?
Yes. This one. It's pretty obvious that the question was asked to begin with because the OP recognized that many people would find the problem ambiguous. And then, a Ph.D physicist (among others) gave the "wrong" answer. As I said in my first post in this thread, the problem here is that these numbers are laid out purely for arithmetic's sake. Assuming that there is a real world problem that these numbers represent, investigating that problem may well reveal that the arithmetic is not laid out in accordance with the rules of order of operations. Order of operations is irrelevant. What's important (in all math problems) is reality.
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quote:What it lacks is clarity -- it's far too easy for a human to misread it, as evidence by the multiple people who are comfortable and experienced with mathematical notation getting it wrong.
posted
I get what you are saying Glen. Since this problem is a strictly mathematical one (we have no context for what real world problem it is trying to solve), we do not know if the standard order of operations is justified or not. All we can do is try to follow the conventions to get the answer our teacher expects on the test.
If we did have a context for the problem, we might find that conventional order of operations gives the wrong answer to the real world situation. However, I would like to point out that if this is the case, I feel it is more an error on the person that wrote the problem in the first place, not an inherent error in the convention.
Posts: 891 | Registered: Feb 2010
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posted
After pondering this question, I've come to the conclusion that the correct answer is 2. And that positioning two expressions adjacent to one another in order to signify multiplication carries with it an implicit grouping. The reason the contents of parentheses are evaluated first is that they signify grouping. This notation carries with it an implicit grouping.
My blog post about this actually started out by saying that either 2 or 288 is theoretically correct, and that it depends on how you handle the vagueness of the standard order of operations. But by the time I reached the end of the post, I'd concluded that this is wrong, so I went back and edited the beginning.
The truth is, anyone using an equation like that probably should use another set of parentheses in order to avoid confusion, but lacking that, the answer has to be 2.
Posts: 12266 | Registered: Jul 2005
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quote:Originally posted by Lisa: After pondering this question, I've come to the conclusion that the correct answer is 2.
What if this equation was for determining the minimum safe number of bolts needed for a bridge section and the correct answer in reality is 288? You just built a bridge that will collapse and kill many people when those 2 bolts fail.
I think this is what some people were talking about with the difference between mathematics as pure abstraction and real world situations.
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quote:Originally posted by scholarette: There is no ambiguity. 2 is wrong. 288 is right.
Scholarette is right. The rules are clear. First you do the operation that is in parentheses, then you do exponents, then multiplication (in the order in which its written), then addition (in the order in which its written.
Division and multiplication are the same thing as are addition and subtraction, so they have the same priority and are always done in the order in which they are written (regardless of how they are written). I learned those rules in 7th or 8th grade, they've been invaluable to me writing computer programs.
Posts: 12591 | Registered: Jan 2000
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quote:Originally posted by Lisa: After pondering this question, I've come to the conclusion that the correct answer is 2.
What if this equation was for determining the minimum safe number of bolts needed for a bridge section and the correct answer in reality is 288? You just built a bridge that will collapse and kill many people when those 2 bolts fail.
If you're doing a mechanics problem, you just don't put it that way. This question can really only arise theoretically or in a poorly written math text.
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quote:positioning two expressions adjacent to one another in order to signify multiplication carries with it an implicit grouping
If you're doing any implying or inferring, you're not using standard mathematical notation. If you follow the rules of standard notation, there is one and only one answer.
Posts: 16551 | Registered: Feb 2003
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quote:What it lacks is clarity -- it's far too easy for a human to misread it, as evidence by the multiple people who are comfortable and experienced with mathematical notation getting it wrong.
Semantics.
The two concepts usually go hand in hand, but the differences between them are pretty important in this situation.
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quote:positioning two expressions adjacent to one another in order to signify multiplication carries with it an implicit grouping
If you're doing any implying or inferring, you're not using standard mathematical notation. If you follow the rules of standard notation, there is one and only one answer.
So then you'd say that 48/2x (or 48÷2x) is equal to 24x? And not 24/x?
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quote:If you're doing a mechanics problem, you just don't put it that way. This question can really only arise theoretically or in a poorly written math text.
This problem arises all the time and has nothing to do with poorly written math texts. It comes up whenever you need to enter a long mathmatical formula into a computer. The solution to this problem is NOT to use more brackets. It is far too easy to make errors when you use lots of nested parentheses and far too difficult to find those errors. The standard student approach seems to be "When in doubt, add some more brackets. I've helped hundreds of engineering students debug/troubleshoot computer problems and virtually without exception, the more parentheses they've used, the more likely they are too have an error in their formula and the harder it will be to find that error.
The fact that some calculators (and possible other computers) aren't following the standard order of operations is really scary and dangerous.
Posts: 12591 | Registered: Jan 2000
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posted
First, I'd say that 48/2x is a terrible way of writing it. In fact, it is the first thing I said in this thread.
Despite how used we are to expressions like 2x being considered a single thing, it's really just a shorthand for 2 * x, which bring us to 48 / 2 * x.
And since multiplication and division have the same priority, standard mathematical notation says that you have to resolve it from left to right, giving us 24 * x.
If you assume that it's 48/(2x), then you're assuming that the person who wrote it wrote it wrong (that is, not following standard mathematical notation for what he intended). Which, honestly, is a pretty good assumption. It's a terrible expression.
Posts: 16551 | Registered: Feb 2003
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posted
That's the rule (24x). I think you have a reasonable case that many people would interpret it otherwise, but that's not how the rules work. The currently accepted way of interpreting mathematical expressions would render it as 24x, full stop. Of course, the real rule in such a situation is to not write things in such an annoying way. You'd never see somebody who writes down mathematical expressions in the normal course of things ever write an expression so annoyingly.
Posts: 15770 | Registered: Dec 2001
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quote:If you're doing a mechanics problem, you just don't put it that way. This question can really only arise theoretically or in a poorly written math text.
This problem arises all the time and has nothing to do with poorly written math texts. It comes up whenever you need to enter a long mathmatical formula into a computer. The solution to this problem is NOT to use more brackets. It is far too easy to make errors when you use lots of nested parentheses and far too difficult to find those errors. The standard student approach seems to be "When in doubt, add some more brackets. I've helped hundreds of engineering students debug/troubleshoot computer problems and virtually without exception, the more parentheses they've used, the more likely they are too have an error in their formula and the harder it will be to find that error.
Rabbit makes a point -- I run into this sort of thing all the time in programming.
In my experience, it's far better to use parentheses than to rely on the order of operations being left-to right. Not because the left-to-right order of operations cannot be depended on, but because it makes the expression much more difficult for humans to read and decipher.
But in cases where there are enough parentheses that it becomes difficult to see what's happening, it's usually easy enough to create new variables to simplify things.
I have zero qualms depending on the other order of oprations. If you don't instinctively know that multiplication happens before addition, you've got huge problems as a programmer.
Posts: 16551 | Registered: Feb 2003
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quote:That's the rule (24x). I think you have a reasonable case that many people would interpret it otherwise, but that's not how the rules work. The currently accepted way of interpreting mathematical expressions would render it as 24x, full stop. Of course, the real rule in such a situation is to not write things in such an annoying way. You'd never see somebody who writes down mathematical expressions in the normal course of things ever write an expression so annoyingly.
While this is true of the example in the subject bar, I did see this sort of thing pretty commonly in my way to my Math minor:
1/4x^3 + 3/4x^2 + 1/2x + 2/3
Here the professor could have written:
(1/4)x^3 + (3/4)x^2 + (1/2)x + 2/3
But its not necessary to do so when you assume your students are familiar with the order of operations.
The example in the subject title is annoyingly unclear, but 3/4x being (3/4)x and not 3/(4x) is something you needed to know off the top of your head for the curriculum I encountered.
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posted
But how many curriculums still write fractions like that, as a single line, and not as a proper fraction with a horizontal dividing line, where numerator and denominator are clear?
Posts: 3486 | Registered: Sep 2002
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quote:n my experience, it's far better to use parentheses than to rely on the order of operations being left-to right. Not because the left-to-right order of operations cannot be depended on, but because it makes the expression much more difficult for humans to read and decipher.
My experience leads me to the opposite conclusion. The more parentheses people use, the harder it is to decipher a long formula in code. It's too easy to mistake which close parenthesis goes with which open parenthesis. Yes, you can break it up by introducing new variables that play the same role, but when the formula has some physical meaning (as they nearly always do in engineering) introducing dummy variables makes it harder rather than easier to decipher what's going on.
Posts: 12591 | Registered: Jan 2000
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quote:Originally posted by Nighthawk: a proper fraction
I'm assuming you are not using the proper mathematical definition of a proper fraction? As it would be quite improper to imply that this does not properly apply to improper fractions as well...
Posts: 324 | Registered: Mar 2008
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quote:Originally posted by Nighthawk: But how many curriculums still write fractions like that, as a single line, and not as a proper fraction with a horizontal dividing line, where numerator and denominator are clear?
That's irrelevant. If you need to enter the formula into a computer program, it's still most commonly done in a single line. Since being able to enter mathematical expression into computer programs is an important skill, people need to know how to do it.
Posts: 12591 | Registered: Jan 2000
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quote:Originally posted by Nighthawk: a proper fraction
I'm assuming you are not using the proper mathematical definition of a proper fraction? As it would be quite improper to imply that this does not properly apply to improper fractions as well...
I'm not using the mathematical term. I mean "proper" in terms of display, as far as it is written: numerator and denominator.
quote:Originally posted by The Rabbit:
quote:Originally posted by Nighthawk: But how many curriculums still write fractions like that, as a single line, and not as a proper fraction with a horizontal dividing line, where numerator and denominator are clear?
That's irrelevant. If you need to enter the formula into a computer program, it's still most commonly done in a single line. Since being able to enter mathematical expression into computer programs is an important skill, people need to know how to do it.
But entering it as written above, at least in most of the programs I know of, will yield an error. Can't enter it exactly as written in Excel or most of the programming languages I know. And as far as I know (it's been a while since I use it) programs like Mathematica explicitly require entering it in fraction form.
Posts: 3486 | Registered: Sep 2002
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quote:But entering it as written above, at least in most of the programs I know of, will yield an error.
That's really irrelevant. The terms 3x2, 3*2, 3(2), and 3*(2) mean exactly the same thing. 3/2,3÷2 and 3 over 2 mean exactly the same thing. The fact that some computer programs will accept only a subset of those terms is why its important to understand that they mean exactly the same thing. The fact that we calculate so many things using computer programs is a large part of why its important to understand the standard rules for the order of calculations.
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Because that was exactly my point! There are teachers all over the place who teach it incorrectly, so by the time I get students in high school, they have it ingrained that you do addition before subtraction, even though that's not correct.![[Frown]](frown.gif)
