What would you do on this math question?

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MahoningQuizBowler
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What would you do on this math question?

Post by MahoningQuizBowler »

I'm sitting here in Massillon, Ohio and this issue just came up in the consolation bracket match that eliminated the league champion. They aren't happy...

The question is: Evaluate i raised to the negative 15th power.
The answer written is: i
The answer given was: 1/-i

Issues raised include whether or not this is an acceptable alternate and if the word evaluate automatically indicates simplification of answers down to the most acceptable form.

BTW, the protest jury consisted of four math teachers, two of whom were involved in the match. The other two split on the issues raised.

Nothing can be done now--the losing team left--but if it happens again, I want to know.

Thanks in advance.

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Post by jewtemplar »

I would prompt, but rules usually specify whether simplification is necessary. I don't think the word evaluate has much to do with it. The safest thing is obviously to stick with the rules. If they're unclear, prompting makes sense, since the answer was technically correct.

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Post by zwtipp »

EDIT: My stupidity.
Last edited by zwtipp on Thu Jan 27, 2005 12:22 am, edited 1 time in total.
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Post by MLafer »

1/(-i) = (1(i))/((-i)(i)) = i/(-(i)^2) = i/1 = i

Edit: i would have accepted the answer. prompting on math questions is pretty brutal if you only give a few extra seconds to extract the simplified form.
Last edited by MLafer on Thu Jan 27, 2005 12:23 am, edited 1 time in total.

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Post by Captain Sinico »

Rationalize the fraction, dude:
1/(-i) = -i*1/((-i)*(-i)) = -i/(i*i) = -i/(-1) = i.
If the question is as given and there are no ancillary, applicable rules that state that only the "simplest" form is acceptable, I would accept 1/(-i) without hesitation.

MaS

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Post by MahoningQuizBowler »

Let it be known that I"m not a math teacher, so that's why I'm asking. I know better now.

The protest didn't come until like six minutes after the match, that's why it didn't go through. I'll pass this thread on to the TD and she can send it on to the teams if she chooses.

Thanks for the help.

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Post by dtaylor4 »

ImmaculateDeception wrote:Rationalize the fraction, dude:
1/(-i) = -i*1/((-i)*(-i)) = -i/(i*i) = -i/(-1) = i.
If the question is as given and there are no ancillary, applicable rules that state that only the "simplest" form is acceptable, I would accept 1/(-i) without hesitation.

MaS
if all requirements for a challenge are met, i would give it to him because the only way i see him giving that answer is that the first thing out of his mouth was "1" and he added "over i" to try to correct himself

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Protests

Post by jeffburns1 »

Question: I'm not familiar with Ohio rules, but is it actually possible to make a protest after a match in other high school formats? In Georgia, we're pretty consistent that no protest may be considered after the reader has started the next question.

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Post by STPickrell »

This is from the "Commissioner's Interpretations" of the VHSL Scholastic Bowl rules (it takes 2+ years to make a formal change to the VHSL handbook and this is something we can send out fairly quickly.)

If questions are raised about the correctness of an answer and there is easy agreement of players, coaches and/or judges on the appropriate response, the issue may be resolved immediately and the score adjusted as appropriate. If agreement is not easy or immediate (10-15 seconds), once a decision has been made, a coach may appeal. A player active in the game may prompt her/his coach to appeal. No more than one additional question may be asked prior to an appeal. If the appeal comes at the end of a round, it must be made prior to the beginning of the next round. An appeal will be resolved only at the end of the match, if the outcome of the match could be changed by the decision. (126-9-10)


Under the old rules, an appeal could only be made at the end of the period. We got that changed two years ago.

That passed by a fairly narrow margin in a coaches' survey and a fairly wide margin among the active coaches.

We put in the "easy agreement" thing to resolve a situation where it's fairly obvious and we just forgot to put in an alternate acceptable answer (like "7 is less than x" when the answer says "x is greater than 7.") I don't see why that sort of thing has to be shoved off until the end of a match.

Also, I put in the "no more than one additional question may be asked" bit to prevent a reader from doing what Pittsburgh tried to do during the AFC title game and rush off the next question. I figured the usual score-check ritual that occurs between periods would be enough time for a team to get its act together and make an appeal though.
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Post by Irreligion in Bangladesh »

The Illinois perspective: Wrong. Not fully simplified, as everything on every math problem must be 100% simplified, unless explicitly asked for otherwise. You can perform operations on 1/-i to reach a simpler answer (as the simplest answers for i-type simplifications like this are 1, -1, i, -i, and the answer is one of those), therefore it is not simplified and not correct. No prompts, no fuss, as many moderators wouldn't know the similiarity and count it wrong immediately.

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Post by quizbowlmike »

Speaking as a former college precalculus TA, a class where i raised to powers is covered, the answer is wrong. In precalc, you are always suppose to rationalize the demoninators, which clearly isnt the case with the sqrt(-1) down there.

But, as a calculus TA, that would be acceptable. Generally, in calc no one cares what's in the denominator.

If i was a reader, i probably would have prompted. Generally when there is a problem of i raised to something, the answer is always given in the form of 1, -1, i, or -i. If nothing more was given on the prompt, I would have said it was wrong. It would be like giving and answer in spanish when the question is clearly in english.
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Post by NotBhan »

DaGeneral wrote:
if all requirements for a challenge are met, i would give it to him because the only way i see him giving that answer is that the first thing out of his mouth was "1" and he added "over i" to try to correct himself

I think that a more likely reason for the structure of the answer given is that the question asked for i to the negative 15th. Since i to the 15th is -i, then i to the negative 15th is 1/(-i).

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Post by rocket »

i think in this case it should have been prompted since the question said negative power, which most people would just take the answer and put it under 1. but in this case, it could be simplified further, however, a prompt seems to me to be more justified.

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Post by jewtemplar »

quizbowlmike wrote:Generally when there is a problem of i raised to something, the answer is always given in the form of 1, -1, i, or -i. If nothing more was given on the prompt, I would have said it was wrong. It would be like giving and answer in spanish when the question is clearly in english.
I see no justification for this. Rationalizing denominators is a convention and nothing more. It has no bearing on the mathematical correctness of the answer, and in a perfect world of math-literate moderators, such simplification would not be required. Obviously, the easy answer is to state clearly in the rules, and repeat in the questions that answers must be fully simplified. What did the tournament rules say, anyway?

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Post by MahoningQuizBowler »

That's the major problem. There was no mention of this topic in the tournament rules, nor are there in the state rules. It is up to the question writer, really, to state whether or not the answer should be simplified or not. This writer sometimes does and sometimes does not state that simplification is necessary.

In this particular tournament, there were no specific rules in terms of when a protest could be filed. I have to make it clear I was not the tournament director here, just a reader, but my experience with sports officiating influenced me here. The protest never came up DURING the match; it was only after the match, when the coach was looking over the questions trying to find a place where her kids could have scored but didn't, that this became an issue.

It was this point that eventually ended the argument. The only good that came from this is that the team that got the short end of this stick was indeed the league champion, and will still get to play in the Regional tournament.

Lost in all this is that the team that won the round on its face, Canton McKinley, is a large public city school that did fantastically well and were unlucky not to advance further--they were defeated by Canton Central Catholic in the match immediately following this incident.

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Post by jewtemplar »

If there's no mention whatsoever, then the decision's even easier. You accept an answer that is correct. Pardon my repeated postings.

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Post by pakman044 »

jewtemplar wrote:
quizbowlmike wrote:Generally when there is a problem of i raised to something, the answer is always given in the form of 1, -1, i, or -i. If nothing more was given on the prompt, I would have said it was wrong. It would be like giving and answer in spanish when the question is clearly in english.
I see no justification for this. Rationalizing denominators is a convention and nothing more. It has no bearing on the mathematical correctness of the answer, and in a perfect world of math-literate moderators, such simplification would not be required. Obviously, the easy answer is to state clearly in the rules, and repeat in the questions that answers must be fully simplified. What did the tournament rules say, anyway?
I must agree. Often times on mathematics contests, you will often see answers expressed non simplest radical form, such as 1/sqrt(2) and similar.

Even saying that everything must be in simplifiest form is not necessarily good enough. Just as a lower level application, it's sort of like 3/2 and 1 1/2. Strictly speaking, 3/2 isn't completely simplified because it's an improper fraction. However, most everyone uses the improper fraction as the simplest form. In a similar fashion, you can argue whether a factored answer is simplified, and many teachers/professors will give different answers based on the application requested. There's not an easy answer to this question, and devolves upon the question writer, TD, and rules-writers to clarify these situations, or the moderator will be required to make a spot decision.

I must admit that as a moderator I probably would have ruled the answer incorrect on hearing it. Even though I have taken a goodly number of math courses in college (I'm on number five at the moment), I would not have recognized the equivalence without doing it on paper first. If the answer had been challenged, and I reached the point where I saw that -1/i was mathematically equivalent to i, *even* with rules saying that all answers must be simplified, I would probably have to accept the answer because the rules are not clear enough on the issue (and in many formats, rules interpretations are the final decision of the moderator and cannot be appealed out of the room).

If there is any "blame" to apply in this situation, I would think that it would lie with the question writer. The question writer has to have explored the question fully enough, determined possible alternative answers, and give the moderator instructions about how to handle them. Otherwise, conundrums like this happen!

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Post by Stained Diviner »

Be careful about blaming the question writer. Most problems that arise during the match are the fault of the writer, but it is impossible for someone writing over one hundred questions to anticipate every reasonable response to every question. You can find i^-15 or something very similar to it in plenty of Algebra II and Precalculus textbooks, and every single one will tell you that the answer is i without alternatives. Ideally, the writer would have given instructions on what to do with -1/i, but most writers would not anticipate such an answer.

It was certainly reasonable (though perhaps not ideal) for the moderator to not accept the answer at first. Because the protest was filed too late, that's the end of it. It is also reasonable to expect simplified answers because the whole point of the problem is to simplify i^-15.

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Post by pakman044 »

Oh, I agree it's not obvious to the question writer in that situation. But in the end since they are likely to have more expertise than anyone else with the question at hand, so they have to be the ones to examine all the permutations (them and whoever edited the questions). In this situation, if you *had* to assign blame, they would be more likely to get some, but it's still hard to give it to them. Is there anyone who actually could anticipate this type of answer?

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Post by Admiral »

styxman wrote:The Illinois perspective: Wrong. Not fully simplified, as everything on every math problem must be 100% simplified, unless explicitly asked for otherwise. You can perform operations on 1/-i to reach a simpler answer (as the simplest answers for i-type simplifications like this are 1, -1, i, -i, and the answer is one of those), therefore it is not simplified and not correct. No prompts, no fuss, as many moderators wouldn't know the similiarity and count it wrong immediately.
I must disagree with my fellow Illinoisian. I picked up the Illinois rule book, and in fact coaches may challenge such a question, and can even request the player perform the computation to confirm the validity of their claim. I have seen coaches use this before, and are well within the rules. While Illinois has its share of bad moderators, I think most that don't live in the 1970's are pretty good.

I know that it used to be the case that challenges were not permitted, but that changed some years ago. I would ask the moderator for permission to work out the problem and verify my claim that it was correct.
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Post by Captain Sinico »

Please note that the question requests an evaluation without further stipulating what type of evaluation. Further, there is no applicable rule which would allow a request for "evaluation" to be justly interpreted as one for "reduction to simplest form" (whatever that might be, in this case, conventionally i.) Therefore, 1/(-i) is a correct response to the question and ought to have been accepted or, at very least, issued a directed prompt for simplification.
This issue is, however, entirely the fault of the question writer, as there is an infinitude of acceptable responses to this question as posed, but the writer has implicitly demanded only one of the answers, by making only that one acceptable, without explicitly requesting that one from the players. While it is broadly conventional to say that the simplest answer to this question is i, there is no reason to assume that this convention is universally applicable (in fact, it is not even close to being so) and, moreover, there is no reason to accept only the simplest answer, even if we could establish to satisfaction what that is exactly.
The only satisfactory resolution to this issue is to write questions that request a single, clearly-defined answer or small set of explicitly acceptable answers. For example, this question might be re-written as follows: "In a term of the first four positive integer powers of i, evaluate i to the negative fifteenth power, where i is the imaginary unit; the square root of negative one." The only acceptable answers would then be i or i to the first power. This issue also might be imperfectly resolved by promulgating a rule requiring "all computed answers in simplest form" or something along those lines, though, as I said, the simplest form is defined entirely (and often ambiguously) by generally non-universal conventions.

MaS

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Post by G_Unit »

ImmaculateDeception wrote:This issue is, however, entirely the fault of the question writer, as there is an infinitude of acceptable responses to this question as posed, but the writer has implicitly demanded only one of the answers, by making only that one acceptable, without explicitly requesting that one from the players.
I agree, that the writer is primarily to blame for not explicitly asking for a format to the answer. If i is acceptable, how about answering i ^ 5? Or, how about being a jackass and using the Euler Formula to declare the answer to be e to the power of (i*pi)/2?

The ambiguity is first and foremost the writer's problem, and a team should never be blamed for that. Personally, I might have used the same reasoning Raj did earlier to get 1/-i.

My answer: accept the answer immediately. At the very least, prompt, unless the moderator is truly skeptical of the equality of answers. In that case, I'd just put forth the usual protest and resolve later.

P.S. - Credit goes to ImmaculateDeception for the new, improved form of the question, no matter how ugly it would actually be to use in a match.

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Post by Stained Diviner »

Some (though not all) of the blame should go to the team that answered -1/i. Anybody who has ever been to a scholastic bowl tournament knows that questions and moderators are fallible. Teams need to learn when to challenge a ruling. I know that it is difficult to do when there are only a few seconds before the next question, and I know that nobody wants to be a pain by challenging every answer, but that is how it works. If you really think you're right, then you have to speak up quickly and respectfully. If you fail to do so, it can cost you.

I write a lot of questions, and I leave a lot of notes as to when to accept a half-answer, when to prompt, and when to just say no. If I had written this question before I saw this discussion, then I would not have considered the possibility that anybody would have answered -1/i. I am a math teacher, and I understand perfectly the arguments above and probably would have prompted had I been moderating, but I would not have anticipated it on my own.

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Post by Irreligion in Bangladesh »

Admiral, or anyone with an Illinois rule book handy, what's the status on blitzing? I've asked both varsity and freshsoph coaches, along with some kids at a different school, and no one can say definitively how much, if at all, you can give extra information. If you could help, I'd appreciate it.

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Post by dtaylor4 »

styxman wrote:Admiral, or anyone with an Illinois rule book handy, what's the status on blitzing? I've asked both varsity and freshsoph coaches, along with some kids at a different school, and no one can say definitively how much, if at all, you can give extra information. If you could help, I'd appreciate it.
my interpretation is that when you blitz (or as i call it, fish), all information stated in the blitz must be accurate or it is counted incorrect

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Post by Tegan »

styxman wrote:Admiral, or anyone with an Illinois rule book handy, what's the status on blitzing? I've asked both varsity and freshsoph coaches, along with some kids at a different school, and no one can say definitively how much, if at all, you can give extra information. If you could help, I'd appreciate it.
If I may quote the sacred parchment written by the Grande Inquisitors from Bloomington (the IHSA):

Rule 4-C-2: If a player gives additional correct information that pertains to the correct answer, without pause, up to a total of two pieces of information, the answer is ruled correct. If a player gives addtional correct information that has no direct bearing on the question or the answer, or a player includes additional incorrect information, the answer will be ruled incorrect............this rule may apply to, but is not limited to, creation/creator situations.

This rule was added for the 2001-02 season.....the problem is that very few coaches/officials bothe to read the rule book that comes out every year for changes.

And whoever "Admiral" is......chop-chop with the I.D. check!

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Post by Irreligion in Bangladesh »

Excellent. Thanks Tegan! I now have no fear of buzzing in on a question beginning with a main charac.....oh wait, that's right. Illinois questions don't start with those, they start with "Name the author of.". Helpful just the same though. :)

If you think :chip: questions are bad, just play a round of IHSA questions. You ain't seen nothin' yet.

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Post by Tegan »

styxman wrote:Excellent. Thanks Tegan! I now have no fear of buzzing in on a question beginning with a main charac.....oh wait, that's right. Illinois questions don't start with those, they start with "Name the author of.". Helpful just the same though. :)
If you think :chip: questions are bad, just play a round of IHSA questions. You ain't seen nothin' yet.
I haven't seen any of Chip's questions (though I have heard plenty about them).

Before your time.....back when IMSA ruled the state, the questions were pretty bad. A few years ago, a couple of coaches got together and tried a new system (hardly perfect, but light years better). One of the big problems were these ambiguous lead ins. This Spring, check out the lead ins on the IHSA Regional (and beyond) questions. The ambiguity should (keyword: should) not be there. My opinion is that the ambiguous lead ins are more found in tournaments where you don't want to see the best team win (or at least too easily).

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Post by dtaylor4 »

At a tournament this year that DMac attended, there were a few, not a lot, but a few lit questions that started off "George and Lennie Small..." *buzz*
that are blitzable, but for when it leads off with a title, the way to beat others is through recognition within a few syllables i.e. "When Lil..." *buzz*
or "I Know..." *buzz*. In Illinois, anticipation wins games

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Post by zwtipp »

Most of the steinbeck questions out there seem to start out with something about Salinas, CA. It's getting old, but it's a nice power opportunity.
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Post by dtaylor4 »

zwtipp wrote:Most of the steinbeck questions out there seem to start out with something about Salinas, CA. It's getting old, but it's a nice power opportunity.
almost all tournaments here are straight tossup/bonus with no powers or negs

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Post by zwtipp »

so are most of the ones here, but when the are powers salinas is always before the mark.
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