Implicit multiplication sign

Harry_the_cat

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I just saw on Facebook a post asking to solve \(\displaystyle 6 \div2(1+2)\) with many different answers proffered (many obviously incorrect).

I usually do not respond to these posts but enjoy seeing incorrect answers being vigorously defended by some people.

This was my response this time:

The issue with this question is your interpretation of the "missing" multiplication sign. Does it mean: \(\displaystyle 6\div 2(1+2)=3*3=9\) or does it mean \(\displaystyle 6\div(2(1+2))=6\div6=1\) ? A "missing" or implicit multiplication sign implies a second set of brackets, so the correct answer is 1. If the multiplication sign was explicit, then the correct answer would be 9. So in this case, 1 is the correct answer.

Am I correct?
 
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MarkFL

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I would add within brackets first to get:

\(\displaystyle 6\div2\cdot3\)

And then do the multiplication/division from left to right:

\(\displaystyle 3\cdot3\)

\(\displaystyle 9\)
 

Dr.Peterson

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I just saw on Facebook a post asking to solve \(\displaystyle 6 \div2(1+2)\) with many different answers proffered (many obviously incorrect).

I usually do not respond to these posts but enjoy seeing incorrect answers being vigorously defended by some people.

This was my response this time:

The issue with this question is your interpretation of the "missing" multiplication sign. Does it mean: \(\displaystyle 6\div 2(1+2)=33=9\) or does it mean \(\displaystyle 6\div(2(1+2))=6\div6=1\) ? A "missing" or implicit multiplication sign implies a second set of brackets, so the correct answer is 1. If the multiplication sign was explicit, then the correct answer would be 9. So in this case, 1 is the correct answer.

Am I correct?
On what grounds do you say, 'a "missing" or implicit multiplication sign implies a second set of brackets'? There are valid and invalid reasons for that claim.

My answer to the question has been that both answers are justified, depending on who taught you; so this sort of expression is ultimately ambiguous at some level, and should never be written (unless the intent is to confuse or to cause conflict -- and then it shouldn't be written anyway).
 

Harry_the_cat

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On what grounds do you say, 'a "missing" or implicit multiplication sign implies a second set of brackets'? There are valid and invalid reasons for that claim.
I was thinking of something like \(\displaystyle 6\div 2sin(x)\). Is that \(\displaystyle 3sin(x)\) or \(\displaystyle \frac{3}{sin(x)}\) ? I would say the second option.

I agree that expressions should always be written unambiguously. The expressions we often see on FB are deliberately ambiguous.
 

Dr.Peterson

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Here's the issue: The PEMDAS rules (or equivalents, such as BODMAS) taught in many schools state that multiplications and divisions are done from left to right as they come. (Many students misunderstand that, of course, thinking it means either M then D, or D then M.) So 6÷2(1+2), following the rule, means (6÷2)(1+2) = 3(1+2) = 9, and a÷bc means (a÷b)c.

But students reading such an expression tend to see the juxtaposition as clinging tightly together, belonging in the denominator as a unit, so they assume (without having been taught it) that 6÷2(1+2) = 6÷(2(1+2)) and a÷bc = a÷(bc). Similarly, some distribute in an example like yours, without noticing that they are assuming the 2 is what is to be distributed, and not the entire 6÷2.

In fact, this is so natural that many mathematicians do it (if they ever run across such an expression), and some textbooks teach it explicitly: Multiplication implied by juxtaposition is done before division. I think such a rule is actually very reasonable; but when it isn't taught, it is only a feeling, not a rule, and can't be the basis of a decision. This extra rule is taught here by our own stapel, calling it a general consensus. If only textbook authors would agree. But even then, students would still be confused, and it would still be best to avoid writing it.

That's why I like to ask for the grounds of one's opinion. If you've been taught it (and in fact I hope you have), then what you said is fine. But if not ...
 

Otis

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… PEMDAS rules … taught in many schools state that multiplications and divisions are done from left to right as they come. (Many students misunderstand that, of course
Why "of course"? Do you think PEMDAS is poorly taught, in general?


… students reading such an expression tend to see the juxtaposition as clinging tightly together …
I think the reasons for student misunderstandings are complex, but I suppose in a tutoring environment (where the expectation is that students are struggling) we ought to assume the worse. I'll need to be more specific with notation (in my posts) going forward. 🤔
\(\displaystyle \;\)
 

Dr.Peterson

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Why "of course"? Do you think PEMDAS is poorly taught, in general?
Actually, I probably meant "of course, because we've all observed it" or "of course, because that's what students do". Or maybe just, "because PEMDAS looks like it's saying M before D, and teachers themselves don't always know it doesn't mean that." So, yes.

I think the reasons for student misunderstandings are complex, but I suppose in a tutoring environment (where the expectation is that students are struggling) we ought to assume the worse. I'll need to be more specific with notation (in my posts) going forward. 🤔
\(\displaystyle \;\)
And, of course, in a teaching environment, where either the whole thing is new to them, or they might have been taught (or learned, which is different) incorrectly in a previous exposure to it, we also have to be careful.
 

Jomo

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I NEVER teach PEMDAS as I think it does not work well with students. Here is what I do.
Please let me know what you think?
 
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Dr.Peterson

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I NEVER teach PEMDAS as I think it does not work. Here is what I do.
Please let me know what you think?
Yes, that looks a lot like what I prefer.

I never mention PEMDAS until after I've talked about the process in general (saying that PEMDAS, which they have usually heard of so I can't ignore it, is just a reminder of the rules, not a complete statement of them). I try to focus on understanding over memorized rules, which often means telling students to forget mnemonics they've learned (or at least set them aside).

At least for some examples, I take the top-down or outside-in approach of the video, starting with the big picture (finding what things are being added, circling them to call attention to them), and then moving in to the details of multiplication and powers, eventually finding what will have to be done first. Once the "grammar" of an expression is clear, we can carry out the details, working back from the inside to the outside. This can avoid several common errors that come from looking for E, then M and D, and so on.

One thing I do differently is to start with parenthesized examples before doing complicated unparenthesized expressions; parentheses are, in effect, parts of the expression that are already circled for us. Order of operations rules are ways to decide the appropriate order when no one is there to tell us.
 

Otis

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… Order of operations rules are ways to decide the appropriate order when no one is there to tell us.
As long as everyone's on the same page, heh.

I forgot to mention in my last post that I've seen classrooms designate symbol ÷ as a grouping symbol. That's not really mainstream; and they used GEMDAS.
 
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Denis

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Wolfish solution is 9:

So is George Google's:
 

Otis

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Wolfish solution is 9 …

So is George Google's…
Yup. Software must follow the Order of Operations as written. Software has been properly "taught". 😉
 

Dr.Peterson

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Yes, most software today agrees; but there have been at least some calculators that have taken the other interpretation. (My impression was that teachers forced TI to change their behavior to what's usually taught, even though at least some mathematicians considered what was taught to be wrong.)

That's another reason to consider this, at best, ambiguous.
 

Otis

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I can't say what's usually taught; I don't know what most classrooms are doing anymore. I need to be more mindful of using what I've long considered to be "unnecessary grouping symbols". 😎
 
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