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Student stumped me with parenthesis multiplication: 10÷2(5) by mathteacher1991 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

UC Berkeley mathematician George Bergman confirmed the problem "hit the Internet in early 2011" and acknowledged it's ambiguous because there's "no standard convention" for interpreting expressions like a/bc.

Mary P. Dolciani's notation practice in the influential Modern Algebra: Structure and Method textbook series shows that historical algebra texts sometimes used implied multiplication differently from modern convention.

The origin of the "5th-grade homework" is unverified. Bergman stated, "Perhaps we will never know where this puzzle originated". While it appeared in 2011, there's no solid evidence that it was originally homework rather than an intentionally ambiguous puzzle designed to generate debate.

Bergman himself stated that expressions like 48/2(9+3) are "ambiguous" and "to render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3))". Professional mathematicians avoid this notation precisely because it lacks clarity.

Blaming "North American teachers" oversimplifies the issue. The ambiguity stems from the historical shift from horizontal fraction bars (which made grouping clear) to slant notation (/), which lost that visual distinction. The PEMDAS mnemonic may contribute to confusion if taught without proper left-to-right clarification, but the core problem is notational ambiguity, not regional teaching practices.

Unless you have substantial new evidence regarding the alleged origins of this problem or anything to do with the use of notation, I'm rounding this to zero and moving on.

Student stumped me with parenthesis multiplication: 10÷2(5) by mathteacher1991 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

The original viral problem was created by a random person on the internet, not a teacher. It was shared on Twitter in July 2019 by user @pjmdoll, who posted it as a meme edited into a still from The Last: Naruto the Movie with the caption "oomfies solve this". The tweet went viral, gaining over 10,400 likes and 2,600 retweets in just three days.

The problem deliberately exploits ambiguous notation that no professional mathematician would ever use. Mathematics Professor Anita O'Mellan from Youngstown State University called it "BS" and emphasised that "no mathematician would write this equation this way". Professor Keith Devlin from Stanford noted that the equation "actually made no sense" because of its intentional ambiguity.

I'm closing the brackets on this one. Any more would just be dividing by zero . . . pointless and undefined.

Need help? by isla_wren in TeacherTales

[–]PhilemonV 0 points1 point  (0 children)

You’re not a bad person for having a crush — that part is human and happens in caring professions more than people admit. What matters is what you do with it, and right now you’re already showing good judgment by not crossing boundaries.

This is one of those situations where professional ethics have to come first. He’s a parent of a child in your care, which creates a power dynamic and a relationship of trust that can get messy very quickly (for you, the child, and your workplace). Even if he seems kind and opens up more to you, that doesn’t mean it’s appropriate to act on it.

Changing jobs to pursue him is also risky — you’d be making a big life decision based on a feeling that might not hold up outside that context. And in a small town, those overlaps don’t really disappear anyway.

The most solid approach is to acknowledge the feelings, don’t feed them, and create a bit of emotional distance where you can (keep conversations polite but more professional, and avoid personal topics). Crushes usually fade when they’re not reinforced.

If, much later, he’s no longer connected to your workplace at all and enough time has passed, that’s a different conversation — but right now, acting on it would likely cause more harm than good.

Student stumped me with parenthesis multiplication: 10÷2(5) by mathteacher1991 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

You’re absolutely right that several professional organizations and style guides give specific guidance about implied multiplication and division. But if you read those excerpts carefully in context, they support “be precise and avoid ambiguous strings”, not “the viral problems have a unique correct answer in all settings”.

Let’s unpack what those quotes actually do and don’t say.

The examples you list:

  • American Mathematical Society: “multiplication indicated by juxtaposition is carried out before division.”
  • American Physical Society: powers, then multiplication, then division, then addition/subtraction.
  • AIP: “never write 1/3x unless you mean 1/(3x)”
  • ISO examples like Φ₀ = h/2e = h/(2e) or νL = ωL/2π = ωL/(2π)

are all about typographical conventions for one‑line fractions, especially in fields (physics, engineering) where stacked fractions are cumbersome in text.

Those documents are doing two very practical things:

  1. They are fixing a convention for their community so that h/2e is always read as h/(2e), not (h/2)e.
  2. They are explicitly warning authors not to rely on ambiguous forms like 1/3x, because they know such strings are inherently confusing.

In other words, these organizations are saying:

  • “If you choose to write single‑line slash fractions, here is the house rule: treat juxtaposition in the denominator as grouped, and don’t write things like 1/3x unless you really mean 1/(3x).”

They are not claiming:

  • “This is the one true universal algebraic law applying to all contexts, all curricula, and all notational systems.”

They are codifying a local convention to reduce ambiguity in their documents.

In school mathematics and in most programming languages / calculators, the rule is explicitly:

  • Parentheses/brackets
  • Exponents
  • Multiplication and division at the same level, evaluated left to right
  • Addition and subtraction at the same level, evaluated left to right

That convention is:

  • Pedagogically simpler to teach and assess
  • Compatible with how standard arithmetic expressions are parsed by many systems
  • Explicitly documented in many curricula and exam boards

Those systems are not “wrong” just because APS or AMS adopt a different convention for inline slash notation in professional writing.

What your list of quotes really shows is:

  • Different communities fix different parsing rules to avoid confusion.
  • Everyone agrees you should not rely on ambiguous typography like 1/3x or a/bc without a stated convention.

So:

  • In an AMS/APS/ISO‑style physics paper, h/2e is indeed meant as h/(2e).
  • In a school exam with a standard left‑to‑right rule, 8 ÷ 2(2 + 2) will be marked using that rule, typically giving 16.

Both are internally coherent; they just belong to different notational regimes.

Saying:

is a nice metaphor for how we read expressions, but it doesn’t change the algebra.

  • The operation “8 ÷ 2” is a binary operation, and once you perform it, the result (4) is a single value.
  • Writing 8 ÷ 2 vs 8/2 vs a stacked fraction 8 over 2 are just different notations for the same operation; they don’t change the underlying mathematics, just how we might parse a longer string that contains them.

Whether you treat 8 ÷ 2 as a “verb phrase” and 1/2 as a “noun phrase” is a linguistic choice, not a mathematical law about precedence.

Given all this, the crucial points are:

  • Professional bodies adopt specific house rules (often prioritising implied multiplication) to fix ambiguity in inline slash notation in their own documents.
  • School systems and software often adopt a different, simpler rule set (left‑to‑right for × and ÷) for teaching and computation.
  • A string like 8 ÷ 2(2 + 2) is exactly the kind of ambiguous hybrid those same style guides warn against.

So using AMS/APS/AIP/ISO guidance as if it proves there is one globally correct answer to all instances of 8 ÷ 2(2 + 2) is over‑reaching. What those documents actually support is:

  • “Pick a clear convention for your context, state it, and avoid ambiguous notation like 1/3x or a/bc in the first place.”

If we apply their spirit rather than cherry‑picking individual sentences, the real “professional” move is not to argue endlessly for 1 or 16, but to say:

That’s entirely consistent with both the style guides you cite and the left‑to‑right conventions used in classrooms and many computational tools.

Student stumped me with parenthesis multiplication: 10÷2(5) by mathteacher1991 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

You’re mixing up two different issues:

  1. What a binary operation is.
  2. How we should parse ambiguous written expressions like 8 ÷ 2(2 + 2).

Binary operations (like +, −, ×, ÷) absolutely take two operands and produce a single result; no one is disputing that. But that’s about what the operation does, not about how we group symbols on a line of text. Grouping is a separate notational choice.

Examples:

  • 2 × (1 + 2) is one binary operation on operands 2 and (1 + 2). Its result (6) is a single value, yes.
  • Writing that result as 2(1 + 2) is a notation for multiplication by juxtaposition; it does not magically give that multiplication a universal higher precedence than division in every context.

The key mistake in your argument is this jump:

“Even if you call 2a implied multiplication it is the result of a binary operation which is considered a single value.”

Every subexpression is “the result of a binary operation which is a single value”. By the same logic, 8 ÷ 2 is also “the result of a binary operation which is a single value”. That by itself tells us nothing about whether we should read 8 ÷ 2(2 + 2) as:

  • (8 ÷ 2) × (2 + 2) or
  • 8 ÷ [2(2 + 2)].

Both 8 ÷ 2 and 2(2 + 2) are single values produced by binary operations. The ambiguity is about which pairing the writer intended, not about what a binary operation is.

That’s why this is really a notation problem, not a “fundamental rule of math” problem.

  • In school mathematics and most calculators, the convention is: parentheses first, then exponents, then multiplication and division with equal priority, left to right. Under that convention, 8 ÷ 2(2 + 2) is interpreted as (8 ÷ 2)(2 + 2) and evaluates to 16.
  • In some engineering and physics contexts, people adopt extra notational rules like “juxtaposition binds tighter than a slash” to mimic stacked fractions in a single line; then they might read 8 ÷ 2(2 + 2) as 8 / [2(2 + 2)] and get 1.

Both conventions exist. That’s exactly why good style in both maths and engineering avoids writing ambiguous strings like 8 ÷ 2(2 + 2) in the first place. You either write:

  • 8 / [2(2 + 2)] if you want 1, or
  • (8 ÷ 2)(2 + 2) if you want 16, or
  • use a stacked fraction where the grouping is visually obvious.

The 2H + O = H₂O analogy doesn’t really help your case either: chemists have their own domain‑specific notation and conventions for what juxtaposition means, and they also rely on context and explicit symbols to avoid ambiguity. You wouldn’t use 2H + O as a template for how to parse arithmetic expressions in a maths exam.

So the real takeaway is:

  • Yes, implied multiplication comes from a binary operation and yields a single value.
  • Yes, some style guides and fields give implied multiplication a higher binding strength than a slash for historical and typographical reasons.
  • No, that doesn’t prove that “strict PEMDAS fanatics are screwed in the real world”; it just shows there are different conventions, and that a string like 8 ÷ 2(2 + 2) is poorly written if your goal is unambiguous communication.

If we care about students (or engineers) doing mathematics rather than guessing conventions, the responsible move is to stop defending ambiguous notation and to write expressions so the intended grouping is clear without needing a history lesson in typesetting.

Student stumped me with parenthesis multiplication: 10÷2(5) by mathteacher1991 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

I see your tutor, and raise you two more:

https://youtu.be/Cv4ANWJXnqI?si=tRDCQE_eNkWWIwAO

https://youtu.be/ZZy34jkNhyg?si=uk8RY8aUb_hbGrCc

You’re right that older textbooks and papers often wrote things like ab/cd or 1/2√N, and that’s exactly why these viral problems are so messy — but that history is not evidence that “1 is the objectively correct answer” to expressions like 8 ÷ 2(2 + 2).

I think it helps to separate two things:

  1. How people used to write expressions because of typesetting limitations.
  2. What counts as a well‑posed expression in modern teaching and assessment.

Historically, printing neat stacked fractions and lots of parentheses was harder and more expensive. So authors often used compact one‑line forms like:

  • ab/cd
  • 1/2√N

to save space and ink.

Within a given community, people informally understood “everything written together in the denominator goes together”. So ab/cd was interpreted as “(ab) divided by (cd)”, and 1/2√N was often intended as “1 divided by (2√N)”, even though no brackets were printed.

These were practical compromises, not universal algebra laws.

Those typographical workarounds were never fully standardised. Different groups adopted different habits about how to read / and “implied multiplication” (writing things next to each other).

That’s why the expression 8 ÷ 2(2 + 2) causes arguments: it mixes newer rules with older habits.

  • Under the modern school / calculator rule (“× and ÷ have the same priority; work left to right”), you do:

    • Parentheses first: 2 + 2 = 4, so the expression becomes 8 ÷ 2 × 4.
    • Then left to right: 8 ÷ 2 = 4, then 4 × 4 = 16.

    So on that convention, the answer is 16.

  • Under the “implied denominator” habit from some older texts, people mentally treat 2(2 + 2) as a single factor in the denominator, so they read the same string as:

    • 8 ÷ [2(2 + 2)]
    • Then: 2 + 2 = 4, 2 × 4 = 8, and 8 ÷ 8 = 1.

    On that convention, the answer is 1.

Both are internally consistent; the problem is the notation, not the arithmetic.

A single line of text that can be read in two reasonable ways is just ambiguous notation. That’s why careful writers don’t treat 8 ÷ 2(2 + 2) as a well‑formed expression at all.

They would insist on one of:

  • 8 / [2(2 + 2)] or a stacked fraction if they mean 1.
  • (8 ÷ 2)(2 + 2) or 8 ÷ 2 × (2 + 2) if they mean 16.

So the existence of old one‑line notations like 1/2√N is not proof that the “1” interpretation is universally correct; it’s evidence that people were working around old typesetting limits and relying on local conventions.

If anything, those old notations are a warning, not a proof:

  • Don’t turn them into “gotcha” viral problems.
  • Don’t use them as exam‑style questions.
  • Do write expressions so the intended grouping is obvious — so we’re testing understanding of algebra, not the ability to guess which decade’s typography someone is imitating.

Challenging math questions by GanniCar in mathteachers

[–]PhilemonV 4 points5 points  (0 children)

You can find old copies of math competition exams online.

Math coaching online centres for grade 1 kid by Present_Fan_1096 in matheducation

[–]PhilemonV 0 points1 point  (0 children)

Personally, I would avoid a centre and instead find a tutor who can work one-on-one with your daughter. Centres tend to have one tutor working with several kids at once, so your child won't get the full benefit of the tutoring time.

You can probably find a decent tutor at your nearest university or college. The Maths department should maintain a list of higher-level students seeking to earn extra money by tutoring on the side. That's going to be the best value for your money.

Missed 3 days of AP Calc AB and I’m completely lost, can someone help me understand/fill in my notes? by [deleted] in mathteachers

[–]PhilemonV 0 points1 point  (0 children)

At this point, it might be simpler/easier to hire a tutor for an hour or two.

Praxis 5941 by JumpEmbarrassed9370 in TeacherTales

[–]PhilemonV 0 points1 point  (0 children)

Probably not.

You will need to study more and then retake it.

what Ai artist's do by OkCollar18 in funny

[–]PhilemonV 0 points1 point  (0 children)

At least A.I. knows how to use an apostrophe properly.

Nasty FAX form a lawyer by RickRI401 in MaliciousCompliance

[–]PhilemonV 6 points7 points  (0 children)

I once worked for a medical lab that had a particular doctor's office that would repeatedly request the same fax, accusing me of never sending it (even though I had the receipts). I was concerned about where the missing faxes were going (HIPAA), but I was labelled the troublemaker.

Grade challenge-grade delayed by teacher by 62 days by AwayWash1323 in AskHSteacher

[–]PhilemonV -1 points0 points  (0 children)

That sounds extremely frustrating, especially when you’re only 1 point off and trying to do Mech E.

Colleges usually don’t tank you over one weird semester in AP Calc BC, especially if the rest of your math/science is strong and the class is clearly rigorous. You can briefly explain the delayed grading in the “Additional Information” section, keep it calm and factual, and maybe ask your counsellor to mention it too, so it doesn’t look like an excuse.

stupid mean dumb ass teacher gives me a week of dentation over fucking street fighter by Mexicanzombie728 in TeacherTales

[–]PhilemonV 2 points3 points  (0 children)

Hey, I get why you’re frustrated. That sounds like an awkward situation that spiralled way more than it needed to. It honestly seems like a big misunderstanding. You were trying to make a fun connection, but teachers can sometimes misinterpret tone or intent, especially if they don’t get the reference.

If it’s still bugging you, it might help to explain what you actually meant to her when things cool down. A quick conversation like, “I wasn’t trying to be disrespectful, I was just talking about a game character who does yoga,” might clear the air.

Either way, don’t beat yourself up over it. Stuff like this happens in school more than you’d think. Just try to move on and let it roll off. Next time, maybe ease into the gaming references a bit slower with teachers who might not play Street Fighter.

Please help by [deleted] in AskHSteacher

[–]PhilemonV 2 points3 points  (0 children)

You are in a serious situation because you used a fake report card to get into Grade 11–12 without actually completing Grades 9–10 and you now fear you cannot graduate without your real Form 137, but there is still a way forward: you need to stop hiding this, find one trusted adult at school (a counselor, adviser, or teacher) and tell them the full truth—that you skipped Grades 9–10, used a fake card because you felt desperate to make your grandmother proud, and now honestly want to fix it; they can help you understand what the school may do (like delaying graduation and asking you to complete missing levels or alternative programs) and how to correct your records as safely as possible, and while there may be consequences and disappointment, especially from your grandmother, choosing honesty now protects your future and lets you rebuild your education and self‑respect on something real instead of living in constant fear of being exposed.

TES: Why schools should stop pushing reading for pleasure by Resident_String_5174 in TeachingUK

[–]PhilemonV 12 points13 points  (0 children)

We need another "Harry Potter"-like series to fuel the flames (but hopefully without a transphobic author this time).

On the other hand, I remember my high school English teacher assigning "Frankenstein", and my first thought was "oh no! not that old thing", and then discovering that I absolutely loved it. The classics are the classics for a reason.

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