I heard the punchline with the addition, “Cornbread are square!”
In fact many of the tougher engineering courses I took were designed so that there were very few formula to learn. You really did have to understand everything from first principles. A blessing. Good thing there are many ways to chop an onion.
Not sure it would have even occurred to me to integrate r dθ. Clever.
In addition to all of the required math courses for my engineering education, taking at least one elective math course was sort of expected. So I took a vector calculus course which focused on path and surface functions. I probably picked it up there.
I met with a financial advisor before I retired. He was an arrogant jerk, but I still met with him because even a jerk might be able to teach me something new. He was absolutely certain that his calculations were better than mine. They were not. He thought that his ability to use a formula for a financial calculation would impress me. What I didn’t say to him, but now wish I did, was “You might be able to use that formula, but give me a blank sheet of paper and I can derive it”. What I really wanted to do was punch him in the face.
Because it’s r dr dθ. Integrating once, you are left with \frac{1}{2}r^2\,d\theta or 2\pi r\,dr, depending on which variable you pick.
Straight lines seem pretty useful, though.
Straight lines are useful. Slopes? Like I said, I’ve not had the opportunity to use them since I learned them in seventh grade.
Which isn’t to say they shouldn’t be taught, just that not everyone will use them.
Of course Canadians will do better on a test like that. The American test gives you room dimensions in feet and carpet prices in dollars per square yard. The (hypothetical) Canadian test gives you room dimensions in metres and carpet prices in dollars per square metre. Yes, I’m jealous.
When I open my own carpet business here in the good ole US of A, I intend to measure rooms in firkins per acre (a unit of length) and sell carpet by the gallon per cubit (a unit of area). That will confuse even the numerate 4%, so I can charge as much as I want.
True, but blaming autocorrect.
Very similar to the “How many are going to St. Ives?” riddle.
I think the literacy part of the question is that if you read the problem and understand it at the start you realize that you don’t need to multiply 12 by 9 to get the area, then divide by 9 to convert to square yards. The multiply by 9 and divide by 9 cancel each other out, so you can skip both operations.
Ever curious, I put this riddle to GPT 3.5. Its behaviour was rather amusingly human-like. The first response was that there’s not enough information to answer the question. Then I said, “look at the first phrase in the first sentence and try again”. You could almost see a light bulb go on in its little artificial brain and suddenly it understood!
I hope you’re right because that’s what I got too.
A square with side length r inscribes a square covering 1/4 of a circle. 4 of those squares would form a square strictly covering the circle with an area of 4r^2. Cut the 4 corners off the square midpoint to midpoint and you have a diamond contained exactly within the circle with an area of 2r^2. Thus, a circle must have an area 2r^2 < area < 4r^2. Of the 3 possible options listed above, only pi fits in that range.
The ability to mentally calculate and develop intuitons for what order of magnitude a correct answer should be is, IMHO, the most foundational and most poorly taught part of numeracy. Part of the problem with the carpet question is that it’s very hard to gut check whether your answer is in the ballpark of reasonable because pricing carpet is far outside of most people’s everyday experience and the question is old enough that inflation wrecks your intuition.
I once read a library book called (maybe) ‘The Art of Approximation’. One of the most important books of my engineering study years, judged by how much I used what I remembered, and even today I know it taught me where to focus in problems: what did and didn’t matter in getting a trustworthy ballpark figure.
Edit: I hasten to add that it’s strength was in not being a theory book, the author addressed many (most) common electronic engineering and maths problems, and showed how to approximate them with confidence.
This is how not-so-smart people get their maths right: they check that their result makes sense.
Can’t find this book anywhere now, on-line or off, all I know it that it was a big silver hardback (US-sized textbook). Any ideas?
I’d like to reconnect with this old friend!
A “math” problem?
This is an imperial units problem.
And that’s why it’s important to keep the units in the math(s). But smart people do that too.
I must admit that I would probably be counted among those who fail that test question. Not because I am innumerate (I am an electrical engineer with three decades of solving problems that are not merely basic arithmetic) but because long experience of solving real world problems has degraded my skills at test problems.. The last time I had to solve arithmetical test problems at all was in boating license class twenty years ago. Test problems give exact as much information as you need to solve them; real world problems usually miss data plus often you first have to quiz your interlocutor first about what exactly they want to know, because most people could not express themselves unambiguously to save their life.
I’ll leave aside the foot/yard angle. I might have totally missed it because I work in a metric country and using different units of length for length and area looks crazy to me.
The first thing I notice about the ad is that it’s false advertising. 41% off 15.99 is not 9.49 but 9.43 (9.4341, actually).
Next thing: obviously you can only carpet a rectangular room without waste if one of the two dimensions is a multiple of roll width. That width is not given; this alone precludes answering the test question with a single number. So a much better way to state the problem would be to add “Assume … % waste by cutting.”
Also I need to do what the doorway dimensions are (I assume you’d want the carpet to end under a rail that aligns with the door, wouldn’t you?)
Lastly, I’d be thrown off by the term ‘with padding’. Googling for it it seems to be a material for the carpet to rest on (for grossly uneven floors?); reading the ad I might imagine that waste would somehow be included in the price.
(Something not connected to the test question is something that I wonder about: if square yards and square feet are available as units, why does the store quote a price per square yard if the price per square foot would be lower by a factor of 9?)
Traditional in the carpet business. I suppose as a textile carpet was commonly measured in yards like cloth. No idea why either was measured in yards to start with except perhaps 3 foot wide cloth was common and easy to describe in square yards.
TIL that a shag carpet is not a carpet you use for shagging. Good to know. Thank you, DuckDuckGo.
I did not know that there were “word problems” as opposite to normal problems either. Is that a US-American thing, like not liking writing in cursive?
Concerning the OP, after reading today that a poison expert (what I would call a toxicologist, which is, it seems, too complicated for Ars Technica articles) allegedly poisoned his wife with colchicine, and, according to the article
The medical literature indicates that a lethal oral dose of colchicine is anything that exceeds 0.8 mg per kilogram. The police investigation that looked at Connor Bowman’s Internet history found “on August 10, Bowman was converting [Betty’s] weight to kilograms and is multiplying it by 0.8,”
I believe anything is possible. There will come a day when the officially recognized standard procedure to solve this kind of questions (the carpet problem, not the murder thing, at least as long as you don’t want to get caught) will be “ask ChatGPT”.
I also believe that the living room is much too small, that you cannot solve this problem not knowing the width of the rolls the carpet is sold, and that the whole problem would not arise if the metric system had been applied consistently from the start. And the investigator was supposed to be French? 
Très bizarre, ça!
I don’t know if it’s a US thing exactly - basically “word problem” is used to distinguish a problem where the information needed to solve the problem is given in ordinary language and part of what is being “tested” is the ability to determine which mathematical operations are needed to answer the question.
This is a “word problem”
Jack has 8 cats and 2 dogs. Jill has 7 cats and 4 dogs. How many dogs are there in all?
and this is not
4 + 2 = ____