So my second grader brought home these 2 riddle-type questions on a math worksheet and I’m soliciting opinions on how others would answer, and why.
This was a week ago, they’ve already been answered, turned in, and graded so no one will be helping a seven year old cheat here. However, the answers I insisted he use–which he vehemently protested–ended up being marked wrong. Aghast So now I’m in hot water with the kidlet. I’m not saying what we answered, to prevent muddying the waters, but I’m dying for a little validation here.
Mrs. Carter saw some animals in her garden.
All of them were chickens except one.
All of them were roosters except one.
Can you figure out how many animals are in her garden?
Mr. Carter had some animals in his pasture.
All are cows except two.
All are horses except two.
All are sheep except two.
Can you figure out how many animals are in the pasture?
[spoiler]Either one of two choices for each question
1a. One animal; not a chicken or a rooster
1b. Two animals, one chicken and one rooster
2a. Two animals, neither of which are cows, horses or sheep
2b. Three animals, one cow, one horse, one sheep
My mom taught grade 2, and that’s the kind of “logic” which would have been expected from her class. They aren’t good riddles, but these are 7 year olds… it’s a step towards logical thinking, but not the destination! If you answered either of these choices, and the teacher marked them incorrect, then she’s just grading from a book and not thinking about the work she’s assigning to the kids.
[/spoiler]
Mrs. Carter saw some animals in her garden.
All of them were chickens except one.
All of them were roosters except one.
Can you figure out how many animals are in her garden?
Since all roosters are chickens, there are only two animals – one a rooster (which is a chicken), and one that is not a chicken.
Mr. Carter had some animals in his pasture.
All are cows except two.
All are horses except two.
All are sheep except two.
Can you figure out how many animals are in the pasture?
[/QUOTE]
Three animals – one cow (and two not-cows), one horse (and two not-horses), and one sheep (and two not-sheep).
Mrs. Carter saw some animals in her garden.
Can 1 be a valid answer when it’s worded this way? Am I overthinking it?
On preview-- that first answer is great robardin–and here I was afraid *I *was reading too much into it.
I saw your pre-edit post, mnemosyne. But don’t worry; your secret’s safe with me.
To belladonna: 1 can be a valid answer when it’s worded that way. It can also not be. It depends on the conventions one is working with… which have perhaps not been explicitly set. Alas.
I don’t think some of these fit the puzzle statements:
[spoiler]
1a. - There are “some animals”, which is to say, more than one.
2a. - “All are cows, except two” would not admit “none are cows”, at least not by my interpretation of “all” at a second grade level. Even by standard set theory, the null set is an empty set.[/spoiler]
[spoiler]We had a debate on this before: in standard mathematical jargon, if there are precisely two things which aren’t Xs, it is legitimate to say “all are Xs, except for two”, even if there are no Xs. You mention set theory, which is perhaps needlessly obfuscating, but in standard set theoretic jargon, one can say “All Xs have property P” even when the set of Xs is empty. In other contexts, of course, the conventions might be different. Since ordinary language is not so heavily formalized, people have different interpretations of how to interpret it.
[spoiler]Just take each sentence, take what you know you to be true, then add it up.
All of them were chickens except one.
=definitely one animal not a chicken; indeterminate number of chickens
All of them were roosters except one.
= definitely one rooster, which is presumably the aforementioned chicken; we now know that the number of chickens is one.
So one rooster plus one something else = two animals.
All are cows except two.
= one cow (minimum) plus two non-cows
All are horses except two, one of which we know is a cow.
= One horse (minimum) plus two non-horses
All are sheep except two.
= One sheep (minimum) plus two non-sheep, one of which we know is a cow, one of which we know is a horse. We also know that it really is just one sheep, since the other lines also say ‘except two.’
= 3 animals.
That’d be quite hard for my daughter now, partly because she’s not good at logic, partly because she probably wouldn’t know that a rooster is a chicken. [/spoiler]
Mrs. Carter saw some animals in her garden.
All of them were chickens except one.
All of them were roosters except one.
Can you figure out how many animals are in her garden?
I first answered:
Since all roosters are chickens, there are only two animals – one a rooster (which is a chicken), and one that is not a chicken.
On second thought, I think the question is meant to have this answer, but by my logic and given the statement,
Since all roosters are chickens, there can be any number of animals of a non-chicken kind (say, geese or ducks) in addition to a single rooster. The statement allows for any number of not-chickens, “except one”, which is a rooster.
I think the intended statement was “they’re all roosters, except for one; they’re all hens, except for one”.
Heh. If there were any forum where this would have spawned its own discussion topic it’d be the SDMB
But yes, that’s why I caveatted my statement as “at a second grade level”, even without having been a part of that other thread. I can see there’s wiggle room for argument at a somewhat high level, but NOT at a second grade level, IMHO.
That sounds like when people make jokes like ‘everyone in the US is happy with paying more taxes, except for everyone in the US.’ It sounds more like rhetoric than logic.
One could argue it’s the same convention which would allow the use of “people” in a plural way to include cases of a single person. The difference here though is one is not saying “People are people”, but describing a specific scenario with a plural term (“there are animals…”), which IMHO does NOT admit a single value. Maybe it’s 'cause I’ve done too many crosswords but this is deeply set in my bones. If you state “I saw some animals”, and then end up admitting there had been only one, you are guilty of lying, not of clever trickery.
My wife looked up chicken to make sure they weren’t just female (they aren’t), but she found that roosters are not necessarily chickens, the term can refer to other adult male domestic fowls or various other birds.
Frankly, I don’t think such things matter to a second grade teacher, or to second graders for that matter.
I think, at least when looking for the intended replies, that the simpler the interpretation of the wording is, the better. These things weren’t written with trick-minded adults in mind, after all.
Well, I have tried to, at all times, point out just how sensitive one needs to be to the linguistic conventions set by the particular context, which, unfortunately, may not actually be unambiguous. Still, I had to explicitly mention the conventions of mathematics and set theory because it was going to happen eventually (particularly because, hey, robardin did it first. :)).
I think it’s entirely reasonable to worry about the ambiguity here, even for second-graders, incidentally; I don’t think they will slavishly fall one way or another as to whether “there are two pigs” and “there’s a cow, a horse, and a sheep” count for the second problem.
But if all roosters are chickens, then you’re basically not refining or restricting the first statement with the second with respect to the unknown set of animals.
“All are chickens, except one”
“All are roosters, except one” (OK, so the chicken’s a rooster)
And yes I said it backwards – I meant there could be any number of ROOSTERS and one not-rooster. 1000 roosters and a goose, or 10,000 roosters and a duck, whatever.
