Fair 'nuff! And my condolences to your niece… my wife keeps saying, “at least I’m not an elementary school teacher! My middle schoolers are enough of a handful, and I see each class for 90 minutes every other day. Can you even imagine spending 6 hours on Zoom with the same 5 year olds every day?” And no… no, I cannot.
You’re right, some bad work is done by good workers in a bad situation. Unfortunately there are also plenty of terrible teachers in this bad situation, too.
And terrible parents – the other day my wife was interrupted by a kid (who had dropped off the call for 5 minutes) stopping her in the middle of a sentence and asking her to start the topic over, because his mom called him away from class for 10 minutes to help her unload groceries (!!!).
If we assume there’s a transcription error in how it was delivered to the kids and the problem was originally typeset as:
8 + 4^2
----------
7/2 - 2
Then that reduces to (8+16) / (3.5 - 2) or 24 / 1.5 which is 16. For teaching kids simple order of operations and basic fractions, that’s about the right degree of difficulty.
So that’s probably what was intended. But “probably” is all the close we’re going to get.
I believe 16 was one of the answers, and IIRC the other whole number answers were 8 and 20. No idea what the “right” answer was - my granddaughter just picked the answer with the mixed number (which was wrong) just to get past the question and move on.
By the way I asked her today if she could get back into the quiz and she said no. Apparently once she completes it then it disappears from her assignment list. So no chance of getting a screenshot, sorry.
I would follow up with the teacher and let them know that this question was confusing because it was unclear what the ÷ was meant to represent and that the entire phrase before it is the numerator, not just the bit in parentheses.
If it’s a category 1 teacher who is really trying but just made a mistake converting to a digital format, then they’ll be happy you brought it to their attention. If it’s a category 2 teacher they’ll be annoyed, but they deserve some annoyance in their lives.
There is no unambiguously “right” answer to a badly posed problem. We can certainly mechanically follow the rules for evaluation. But once you see something truly defective in a problem statement you can’t generate an answer. Once you see something suspicious in a problem statement you may still be able to execute the evaluation algorithm without error. But at that point your confidence should collapse.
I’d compare it to the concept of errors and warnings in compiled computer programming languages. An error signals a syntax fault that prevents meaningful evaluation. A warning indicates a lesser syntax fault that suggests what although what you wrote is do-able, it may be ambiguous or it probably is not what you meant.
The OP said
The equation was: 8 + (4)^2 ÷ (7/2 - 2)
Then later admitted that his recollection of the parens around the 4 was faulty.
So his best recollection of the problem as written was/is:
8 + 4^2 ÷ (7/2 - 2)
Following the order of evaluation rules that evaluates as
8 + 16 / 1.5 = 8 + 10.666 or 18.666 or equivalently 18 & 2/3rds.
But the fact the problem contains both a ÷ and a / with both representing division is a warning flag that something is maybe not as simple as it appears. There’s a backstory here, a backstory we can only guess at.
PEMDAS:
IANA teacher. But fundamentally that’s a flawed acronym. Because it implies that multiplication prioritizes before division and that addition prioritizes before subtraction. Which is false.
It’s parenthesis, exponentiation, [multiplication or division], [addition or subtraction].
As well, it’s required that items of equal precedence are evaluated from left to right, but that’s not documented in the acronym.
Finally, overlaid on all of that for typeset problems is that the visual arrangement of the equation is treated as another layer of parenthesis-like ordering rules which override any actual visual parentheses. Here’s a more or less random wiki example of a fairly simple formula where there are two major grouping instructions that are implicit in the “shape” of the equation.
How do we know to do the second subtraction in the denominator before taking the square root? Because of the length of the square root symbol.
How do we know to separately complete all the operations above and below the big line before we perform the division the big line represents?
One could argue that the bit inside the square root is equivalent to (D^2 - d^2)^0.5 and when rewritten in this way it is clear that you should be doing the inside first; by that same logic the numerator and denominator have an implied parentheses around each of them as well (and when the teacher converted the formula from NUMERATOR over DENOMINATOR to using the ÷ sign, they screwed up by leaving out the implied parentheses)
Hence I propose a new acronym;
POPE (Parentheses Or Parentheses Equivalent)
E (exponents)
M.D. (multiple and divide)
AS (addition and subtraction)
AEBETLTBR (all else being equal, top left to bottom right)
I’m sure this will prove much less confusing for our school kids!
It is just a mnemonic. I don’t think it was meant to perfectly represent operations and expected the person to understand what the real order was.
At least that is how I understood it. I never had a problem knowing that multiplication and division were grouped together and done left to right. I just used it as the easy to remember thing it is to guide me.
All good. For me in general mnemonics are harmful not helpful.
They’re simply another arbitrary factoid to remember. You also need to remember how to decode it into words. And then how to convert that shorthand representation into the real thing. And remember what situation(s) it applies to.
I’m much better at understanding why multiplication sensibly comes before addition. Which also explains why multiplication and division are at the same level. And once I know the why I tend not to forget it. Or in a pinch can regenerate it from other more basic knowledge.
Mentally a piece of knowledge with lots of why’s is like a burr with lots of little hooks. It connects tenaciously to lots of stuff around it and doesn’t fall off easily. To me an arbitrary “mnemonic” is like smooth pebble; connected to nothing for any logical reason except “because I said so” it promptly sinks unnoticed into the ooze of forgotten trivia never to be seen again.
To put that statement into perspective, when I saw PEMDAS in your post I had to figure out what it probably meant from first priniciples. I recalled reading about some word that I thought maybe sorta looked like that in previous SDMB threads on order of evaluation follies. Of which we have had many over the years. That plus the topic of this thread triggered that PEMDAS probably was something about order of evaluation. But truly I have no recollection of ever having known it or being taught it as a kid. It’s a new experience every time I see it. To me it’s eminently forgettable precisely because it has zero why’s to connect it to anything. Not even to math as a category of human thought.
Note I’m not suggesting “my” way is the “right” way. It’s just interesting to me that some folks learn best one way and some learn best another way. As I said in another thread today, human cognition is strange. And wonderful.
I second those who have said the student should contact the teacher.
I teach math at a community college. I’ve been using an online homework systems for years. I author some of my own problems, and use some that others have written. Sometimes students discover an issue where they think they got the answer correct, but the system marked them wrong. And when I look into it, I discover that there is an error in the code or something. I then am able to fix the problem (if its mine) or notify the problem’s author. And in either case, I can override the student’s score to give them the points for the problem.
If no one tells this teacher there’s an issue, then the problem won’t get fixed, and students could be missing out on points that are rightfully theirs!
After you completed the second step, I don’t think that the parentheses should’ve been moved to include the 16.
Once you subtract 2 from 3.5, those parentheses are “eliminated” and we’re left with, 8 + 16 ÷ 1.5. Then we go left to right and the answer is 16.
If they wanted us to divide 16 by 1.5, there would’ve been brackets:
8 + [4^2 ÷ (7/2 -2)]
That’s what I remember from math class anyway.
Maybe there were brackets and the OP accidentally omitted them, but since 18 2/3 wasn’t an available answer on the quiz and it’s possible that 16 was, I think it’s likely that question was written correctly and the correct answer was available on the quiz.
