These 2 are the same
56 - 24 / 3 + 12 = 60
56 - 24 ÷ 3 + 12

this is not allowed, you can't put a parentheses into an expression in any ole place that you want, in this case it does not change the answer though but it is only serandipity that the answer doesn't change
56 - (24 / 3 + 12) = 36

you can put parenthesis where it does not change the problem
56-(24/3) +12 is allowed, because you would need to do the division first anyway
56+(-24/3) +12 is also allowed, here you added a parenthesis that didn't change any thing and used the unary minus in place of the binary minus, also allowed, in the original, this does not change the problem, adding a negative is the same as subtracting the same positive . It does change what is possible as a next step. because now the minus is inside the parentheses leaving only addition outside

56+12+(-24/3)here the above expression which is the same as the original used the commutative property of addition to change the order of processing, the answer does not change. Only addition and multiplication(of the common binary operations) are commutative. Neither subtraction nor division is commutative.
(-24/3) +12+56 or 12+56+ (-24/3) are also possible rewrites because of CPA

This changes the meaning of the original and is not allowed as a transformation of the original, it may or may not have been what was intended, but this was a math test not a ESP test.
(56-24) / (3+12)
this also changes the meaning of the original, it is the same as the one above and is not allowed as a transformation of the original
56 - 24
3 + 12

The mnemonics PEMDAS, BODMAS, BEDMAS, and BIDMAS, and associated "rules" are just conventions to make writing and reading math equations possible without having to use an excessive number of parenthesis or brackets or having to explain the rules each and every time you write something down.

The problem with the simple mnemonics is that they lead to misunderstandings. Equal precedence is not inherently obvious to the layman. Even those that know about precedence can have a problem with writing clear equations. For example, what does this mean?

1/3x

Is it 1/3 of x or is it 1 divided by 3x? Many people that know the rules would write that expression when they meant to write:

(1/3)x

Even knowing the rules, people should strive for clarity in writing equations. So this equation:

56 - 24 / 3 + 12 = ?

Should be written:

56 - (24 / 3) + 12 = ?

Unless, of course, you want to post it up on Facebook and start an argument.

FYI, Noah did say that the equation was actually written as: 56 - 24 ÷ 3 + 12 =, but that he didn't know how to get the ÷ symbol, so he used the /.

Noah, just FYI, I know how to get the ÷, but I don't off of the top of my head know the number for the symbol, so to get it in my previous post, I just did a google search for division symbol and copy/pasted into my post.

The "÷" or "/" makes no difference.

So was the math teacher hot or just blonde?

Yep

56 - 24 / 3 + 12

56-8+12

=60

60

There is absolutely no room for interpretation in math.

I have seen some ****ty math teachers but that knocks it out of the park!

Of course there is. Why? Because there are no inviolable rules regarding math notation.

BS. There are international rules of math. I have studied abroad and literally worked with people from every continent. They all speak the same math. It is wonderful (compared to other non-science disciplines, i.e. law, which are very country specific).

Seriously, you could give that math "problem" to scientists across the globe and the answer will be 60. And I am afraid in places such as Europe or Asia, there would be zero discussion on "interpretation". This is 5th grade math man.

G

4th grade. My daughter's in fourth grade.

Not BS.

Yes, that math problem is easy because those basic conventions have been agreed upon for a long time. But not all "conventions" are so set in stone. Read these:

https://artofproblemsolving.com/wiki/index.php?title=Mathematical_convention
Conventions
https://math.berkeley.edu/~wu/order5.pdf
https://en.wikipedia.org/wiki/Ambiguity#Mathematical_notation
Ambiguous PEMDAS
Common Errors in College Math

Do you want more??

I didn't read past the second link but nowhere do I see ambiguity what the answer to the math problem posted is. It is 60 and it will be if you ask anyone past 4th grade (apparently!) in the world. Maybe I am misunderstanding what the discussion is about here?

The second article points out some silly rules on left to right and distinguishing between addition/subtraction and multiplication/division. This should not be taught and I don't recall being taught this myself. That said, sometimes getting kids going without understanding the deeper reasons is not a big issue, it will all come together by algebra 2.

What is a big issue, and I have seen this in my kid's school, is that elementary and middle school teachers often have no clue about math and memorize the rules, passing them straight to the kids. A simple brain fart will have it all come undone as seen in the OP's example.

G

Wow. Did I say there was any any answer other than 60 for the equation the OP posted? No I did not. I did say that it could be made more clear for the layman.

My "no inviolable rules regarding math notation" text was in response to your blanket statement quoted here:

This statement of yours is false.

We didn't learn the PREMDAS thing in school in England back in the 50's and 60's and I dont think my daughter did in the 90's; however the answer, 60, only needs to be obtained via careful reading of the question.

agreed with averybody that the answer is 60, however I think it is good form to write mathematical expressions to eliminate any chance of ambiguity

Scott, in all the links the authors are merely expressing the idea that not all readers have learned or use the mathematically correct interpretation of common expressions, not that there isn't a common inviolate set of rules.

If you study math at a deeper level than is common is HS or even undergraduate math the reasons become more obvious. There are symmetry, associative, commutative, distributive, identity and inverse axioms that make up the mathematical concept of a Field which includes a group of #s in this case Real #s, The Field Axioms define what can be done on a given set of data and in what order. You need to understand the rules for the data set and the field axioms that apply to the data set. It is boiled down to overly simplified mnemonics in HS and undergrad work. This is generally adequate for most purposes but can fail at surprising places to those that haven't fully integrated all of the rules, both the obvious overt ones and more subtle underlying ones..

Wong, in the second link notes that simplicity of presentation often precludes including all of the parentheses that would eliminate all of the ambiguity. He goes on to urge clear unambiguous notation. This is what all good math teachers do, and what many less gifted math teachers don't do and what causes no end of difficulty for students(and teachers) who know most but not all the rules. In the absence of parentheses there are rules that tell the solver what to do and in what order, some obviously don't know or don't follow them.

The fault in the original question was w/ the teacher he/she intended something other than what she wrote, and then compounded the fault by not acknowledging the mistake.

Bill,

Yes, I understand all of that. The issue is that the difference between the layman and the high level academic is huge. Also, conventions change over time as mathematics evolves. Regardless, what I said is true. The conventions are not set in stone. The math itself is inviolable, but the way we write it down and interpret that writing is not. aigel is putting to much faith in the conventions being absolute and not ambiguous.

Yes but it all comes down to either knowing and following all the rules or not, in this thread there is no higher level esoterica involved

there is and has been evolution but as far as the HS math discussed here

again the HS math discussed here is well defined and not subject to debate except by those that haven't fully integrated it.

It is true that other countries have different notations, these are entirely process/organization oriented and not presentation oriented(by that I mean the way they show things like the process of long division or manually calculating a root). They would evaluate the expression here using the same rules as us.

Bottom line is the teacher erred and wouldn't own up to it. It wasn't a matter of ambiguity except in his/her own mind. In every one of the controversial math questions i've seen here and at my job the error has always been that the presenter wrote something different from what was intended often because of a lack of understanding of concepts, and then refused to acknowledge the error. The defense often includes semantics like 'ambiguous' designed to put the burden on someone else.

perhaps you can present some notation that you feel is ambiguous, that can then be discussed.

Please look at this one again:

Ambiguous PEMDAS

again there is absolutely no ambiguity, there are only people that don't understand or follow the rules
from the link,
What is 2x/3y-1 if x=9 and y=2 ? the author argues 'It is not clear what the textbook had intended with the 3y' that is a specious argument as the solver must only consider what is written, again Math not ESP, 'One can however also argue that "3y" is a unit which belongs together.' again a specious argument designed to deflect responsibility from knowing the rules, 3y is a unit is nonsense, what is written is 3 multiplied by y

Further using the arguement that various computer programs give different results merely shows that the coder wrote faulty code

first note that there are no parenthesis, the only way to add a parenthesis where one did not exist is to use the substitution principle as outlined below or where by adding the parenthesis the sense of the problem is not changed
an exaple of the latter is to rewrite 3x as 3(x)

so step 1 is to substitute equivalent values for the variables, the rule to do that is to first put parenthesis where the variable was
2( )/3( ) - 1 then fill the parenthesis w/ the equivalent value or equivalent expression.
2(9)/3(2) - 1 now eliminate the parenthesis by multiplication, this is part of the P in PEMDAS
18/6 - 1 now divide, this is the D in PEMDAS
3 - 1 now subtract
2

there alternate pedagogies to PEMDAS that do the same thing
one is to always isolate terms from each other and to simplify the terms in isolation, terms quite simply are separated from each other by + or - which are not inside a parenthesis or by comparatives like =, > or < etc.

using this method 2x/3y is considered in isolation
from there the same substitution principle is used and then since there is nothing to be done inside the parenthesis and the exponent on the parenthesis is 1 the next step is to multiple to remove each parenthesis. There is certainly a lack of full uderstanding by many about the full extent of the p in PEMDAS, limiting the discussion to only Algebra and excluding tri or other higher functions

P alone has several steps that are often not considered
step 1 of P is to do what you can inside the parenthesis, again this is done in isolation
step 2 is to do any exponents on the parenthesis, there is always an exponent even if it is only 1 which doesn't change anything most of the time so like extra parenthesis is rarley written for simplicity.
step 3 is to multiply the parenthesis bu it's numerical coefficient, as w/ the exponent 1 there is always a numerical coefficient even if it is 1 which again is often not written but which serves the very important function of eliminating the parenthesis from the problem

This expression shows the usually unwrtitten but understood 1's, here the exponent 1 does notheing but the coefficient 1 when miultiplied by using the distributive property eliminates the parenthesis
http://forums.pelicanparts.com/uploa...1478058321.gif
it is equivalent to
1(x - 1) and (x - 1) and x - 1

there is a third one that would be considered if this was a ratioanl expression, but since it's not I've ommited discussion of that

then the 2 terms are integrated to get the final answer

What does the set of natural #s have to do w/ this discussion?

if one limits ones solution set to the set of naturals, that doesn't change any of the actual math involved, it merely adds the possibility of a null answer should the result be a rational # such as 1/4 or 0

Bill,

The set of natural numbers has some ambiguity depending on what discipline is defining it. Because of that, extra clarity is often called for to avoid confusion.

I have never said the "actual math" ever changes. Just how we write it and how we interpret what is written. Ambiguity exists otherwise there would not be pages and pages written about it all over the Internet and we wouldn't have these arguments all over the Internet about equations and their results. Just because there shouldn't be any ambiguity doesn't mean there isn't any.

the set of natural #s has no ambiguity it is a well defined set
it is commonly represented in set notation as {1,2,3,...}, the ellipsis(...) means to continue in the same, manner is also well defined and standard

whole #s is also a well defined set, {0,1,2,3,...}

integers is the set, {..., -2, -1, 0, 1, 2, ... }

Rational #s is another well defined set, this one has to be defined by rule rather than iteration, {any # that can be represented as an integer divided by a natural #} alternately {any # that can be written as the ratio of an integer and a natural #}

irrational #s is the next set that needs to be defined, again by rule, {any # that cannot be written as an integer divided by a natural #} or {any # that cannot be expressed as the ratio of an integer and a natural #}

next is the real #s, { set of rational #s + the set of irrational#} this last is what makes up a Real # line, such as an axis used in graphing

there are further # sets such as {Complex #s}, {Diophantine #s}, {Transcendental #s} etc. that can also be defined

Anyone that claims other wise is making their own idiosyncratic rule or definition.

Bill, you a super smart guy and explain things like few can. Does math come naturally for you? What I mean is...do you invariably arrive at the correct answer without conciously thinking about those silly acronyms? I don't recall ever needing the crutches, was always exceptional in math, but could never "show my work" if teachers required it :(...don't have a clue as to how my brain functions sometimes :). But I could arrive at the correct answer better than 99.99% of those that don't have "the gift".

Can't explain why 1+1=2. (Base 10).
Or very complex equations either....

It just is :)

Well, that equation works Base 3 and up! SmileWavy

<iframe width="560" height="315" src="https://www.youtube.com/embed/UIKGV2cTgqA?list=RDUIKGV2cTgqA" frameborder="0" allowfullscreen></iframe>

Local news, true story....

Kid gets a math problem wrong...

5 x 3 = x (must show your work)

Answer:

15 (kid shows 5 + 5 + 5)

WRONG answer...

Teacher wanted to see 3 + 3 + 3 + 3 + 3

Idiots :(

Huh???

5 x 3 = x

5 x 3 - x = 0

15 x - x = 0

14 x = 0

x = 0

Oh brother.....

5 • 3 = x

5 • 3 - x = 0

15 - x = 0

I'll let you simplify from there. You should be able to get the right answer now.

Relax Francis :p

Not only do I suck at explaining equations, I can't even write them clearly either :)

The story on the news was (as I recall)

5 x 3 =

Still is silly that the teacher said that 15 was the wrong answer....and yep, I remember hearing Common Core Math in the story :(

You explained fine. I was just being a wise-azz.:cool:

How about

5 x 3 = ?

So "question mark" equals 15 now? :D

The teacher needs to consult with Ma and Pa Kettle:
https://www.youtube.com/watch?v=t8XMeocLflc

Bud Abbott Weighs in:
https://www.youtube.com/watch?v=MS2aEfbEi7s

Yep, "?" is our new variable. ;)

So...in my world

5 ? 3 = ?

Shaddup already Esel :)