the one on the right is correct

No, it isn't.

8/2×(2+2)

...isn't the same thing as 8/2(2+2). You separated the term in the denominator, leading the (2+2) to get flipped into the numerator, hence wrong answer.

Turns out I’m wrong, but I haven’t been told how or why. I’m willing to learn if people actually tell me

Well, I don't know what you said originally, so I don't know what it is you were told was wrong - 1 or 16? 😂 The correct answer is 1.

Anyhow, I have an order of operations thread which covers literally everything there is to know about it (including covering all the common mistakes and false claims made by some). It includes textbook references, historical Maths documents, worked examples, proofs, memes, the works! I'm a high school Maths teacher/tutor - I've taught this topic many times.

the problem with your logic is that it’s using rules designed for primary school

Actually The Distributive Law is taught in Year 7. The Primary School rule, which doesn't include brackets with coefficients, is only the intermediate step.

many decades ago. Not a rigorous mathematical convention

It's an actual rule which is centuries old.

mathematicians frequently use juxtaposition to indicate multiplication

It's not multiplication - it's either The Distributive Law or Terms, which are 2 separate rules.

an operation that doesn’t get used in primary school

Yes, as I said it's taught in Year 7.

It's 1

M does not get priority over D

And M refers literally to multiplication signs, of which there are none, and Brackets has priority over everything.

8/2(2+2) =8/(2x2+2x2) =8/8 =1

Yes correct! I just had someone else here claim you couldn't do it with RPN - took me no time at all to show he was wrong about that! 😂

you could solve for the parentheses by multiplying 2+2 by two, giving you 8/8 quicker and still yielding 1. I’m now having more doubts about my math capabilities

No, that's the correct way to do it, as per The Distributive Law.

both are right

No, only 1 is right. If you get 16 then you did division before finishing solving brackets.

8/2(4) - Parentheses

8/8 - Multiplication

Correct steps, but wrong names. Where you said "multiplication" is actually still parentheses - that first step isn't finished until you have removed them (which isn't until after you have distributed and simplified, which you did do correctly).

Modern phone apps seem to be notorious for getting order of operations wrong

Yes, I know, and as a Maths teacher I am well and truly sick of hearing "but Google says...", and so I wrote this thread to try and get developers to fix their damn calculators.

What order you do your exponents in is another ambiguity though

No it isn't - top down.

Sorry but both my phone calculator and TI-84 calculate 1/2X

...and they're both wrong, because they are disobeying the order of operations rules. Almost all e-calculators are wrong, whereas almost all physical calculators do it correctly (the notable exception being Texas Instruments).

You are saying that an implied operation has higher priority than one which I am defining as part of the equation with an operator? Bogus. I don’t buy it. Seriously when was this decided?

The rules of Terms and The Distributive Law, somewhere between 100-400 years ago, as per Maths textbooks of any age. Operators separate terms.

I am no mathematics expert... never heard this “rule” before.

I'm a High School Maths teacher/tutor, and have taught it many times.

The blog post claims it is popular in academy

The blog post also completely ignores what is actually taught in high school - as found in Year 7-8 Maths textbooks - which indicates how much credibility you should attach to the blog post - none.

It comes up like that the first time, but then changes once you hit the switch a few times