Computational math in high school quizbowl

[STPickrell]

For what it's worth the VHSL executive committee is voting on making math questions either be 10, 20, or 30 seconds (and announce it at the start of the question.) If someone from Illinois could give me some *good* examples of "30-second math" I would be most appreciative.

[Irreligion in Bangladesh]

Remember that Illinois is way, WAY above the rest of the nation in terms of converting math questions because 20% of our format is math and we're given 30 seconds to do it. Combine the heavy amount of questions an Illinois team will face, the nearly excessive time alloted to a player, and the fact that the computational canon is so restricted, and you get teams that have cold-blooded math geniuses in terms of quizbowl. In the rest of the nation, where math is not 20% of the distribution, there isn't as much of a need to get math correctly. Matches hinge on a math player in Illinois, especially in non-pyramidal, single-clue questions that are often simply inaccessible. You have to get the math - you can't count on anything else to be there for you. If it is there, it's a buzzer beater anyway, and it doesn't matter how much you study.

Illinois is sufficiently math heavy not just to have seen those tossups on solid of revolution and area between two curves, but to seen them CONVERTED, not just by the top two or three players in the state, but by the math player on teams as low as third or fourth place in the small school sectionals.

Basically, all of this boils down to the oft-regurgitated fact that Illinois format is so screwed up compared to the rest of the nation, and you really should stop listening to us about math because (I hope) we're really close to ditching a good amount of what makes Illinois what it is, and that we'll throw computational math out with the bathwater pretty soon. So, the whole point of this post is to provide some good examples of 30 second math.

From New Trier's website, their Varsity tournament from two years ago.

Give all the values of theta in degrees, from 0 to 360, for which sine of two theta is one.
Answer: 45 degrees and 225 degrees (must have both answers and no others)
Note - this one could even be a little harder by Illinois standards.

You play a game with four fair six-sided dice. You pay a dollar, and then roll all four. If all four dice show the same number, you win three hundred dollars. Otherwise, you win nothing. What is the average amount of money, in dollars, that you should gain each roll? It may help you to know that 216 is 18 times 12.
Answer: 7/18 dollars (prompt for dollars, accept .38 repeating)

For the following problem, use 39 grams as the molar mass of potassium, and 127 grams for the molar mass of iodine. Iodine gas reacts with potassium metal to make potassium iodide. If 166 grams of potassium iodide are formed, how many liters of iodine gas reacted, at standard temperature and pressure? Give three significant figures.
Answer: 11.2 liters (prompt for liters)
Note - science is even pretty doable with 30 seconds.

Find the area bounded by the pentagon with vertices, in order around the pentagon, (0,2), (3,4), (1,-1), (-2,-2), and (-1,1).
Answer: 11.5 square units (accept 23/2; units not necessary)
Note - a good example of Illinois' crazy math. I don't even think a non-computational tossup with answer "Shoelace theorem" would have been converted very well elsewhere.

[Tegan]

Remember that Illinois is way, WAY above the rest of the nation in terms of converting math questions

I would disagree ..... at NAQT .... while a few of the big math studs were well enough ..... it was difficult to adjust from hard core crank-it-out math to math with some quick trick ..... Illinois DOES tend to do good at PAC format because that math is hard core crank-it-out math

Basically, all of this boils down to the oft-regurgitated fact that Illinois format is so screwed up compared to the rest of the nation, [sic] (I hope) we're really close to ditching a good amount of what makes Illinois what it is

Think the opposite ..... there is more support on the IHSA Advisory Board than ever before to steer Illinois to less academically motivated questions. After the meeting in '07, I was half surprised that pyramidal questions survived. Attempts to dump such nontrivial subjects as drivers education was flatly rejected.

[Saiem]

A shoelace theorem question would be absolutely delightful in an NAQT format. Thats a real math type topic.

Or even a question that involved vectors with the same motivation.

How do people feel about multiple questions proposed testing the same concept, and with the same numerical answer?

"Given three points in the coordinate plane (1,1), (I,4), (5,1), find the area of the triangle formed by these points. This can be done by applying the shoelace theorem. This can be done by forming the three points into two dimensional vectors and centering them about the origin and proceeding to take the magnitude of the cross product over two. This can also be done using the distance formula, and applying heron's formula. However, most simply this can be done by plotting the points and realizing that it is a 3-4-5 right triangle. For ten points, find the area of the triangle bounded by the three points?"

Thats like... 4 clues. Best I can do on a shoelace theorem question.

IMPORTANT NOTE: ALWAYS PLACE COORDINATES IN CLOCKWISE FASHION.

It is utterly cruel to do otherwise. The shoelace theorem will screw up, and you'll want to punch a wall.. or the closest teammate.

Feel free to critique my above tossup. It was written rather spur of the moment, and I wrote down all the ways I could think of. You'd need to be very careful about having 4 coordinates, cause it severely limits the clues if they're not collinear. Though that'd definitely limit the amount of bruteforce until the end of the question.

Imagine a rectangle rotated 30 degrees about the x-axis. It'd make the shoelace theorem, vectors and the like very valuable clues if they can apply them. It wouldn't slow down people who know math, but it would severely hinder people until the end of the question. It'd be interesting, at least. I want to know what you all think.

[Sen. Estes Kefauver (D-TN)]

WTF is the shoelace theorem, and instead of saying "Use the shoelace theorem", why dont you say "you can solve it by using the shoelace theorem by [instructions for direct application].

[Saiem]

Basically allows you to compute the area of any figure in coordinates real quick.

i.e. Find the area of the figure bounded by the points (1,1), (1,4), and (5,1)

I'd quickly just line them up like this:
1 1
1 4
5 1
1 1

absolute value [ (1*4 +1*1 + 5*1) - ( 1*1 + 4*5 + 1*1)]
divided by two. = 6.

Multiply each number in the left column by the number corresponding to one below its line. I wrote the first coordinate at the bottom since the last coordinate connects it, and therefore also has to be multiplied as well.

its a pretty simple and fast process that can be extended to any number of coordinates. look it up. There isn't really a nice succint way to tell someone how to apply it. its more like heres how you do it.. remember it forever.

[Sen. Estes Kefauver (D-TN)]

But I think the point we've made on this discussion is that you shouldn't just namedrop a theorem in a question. If you're gonna put there, you need to then explain its application in context of the question to allow people who do know what it is to start working on it, and to allow people who don't know what it is to learn it and then start working on it a little later, once they've learned. That is what can help create the pyramidality.

[Irreligion in Bangladesh]

But I think the point we've made on this discussion is that you shouldn't just namedrop a theorem in a question. If you're gonna put there, you need to then explain its application in context of the question to allow people who do know what it is to start working on it, and to allow people who don't know what it is to learn it and then start working on it a little later, once they've learned. That is what can help create the pyramidality.

Agreed. Namedropping something like shoelace isn't going to help anyone. If you need to be reminded that shoelace is the fastest way, you probably don't know shoelace in the first place. I think, in this one example, shoelace might be a little hard to explain in the midst of a tossup - there'd be too many wasted words. I don't know right now how I would do such a shoelace clue. In general, yes, theorems can't just be namedropped.

[pray for elves]

What also makes that name-drop of the shoelace theorem stupid is that the "shoelace theorem" is not its actual name.

[Sen. Estes Kefauver (D-TN)]

Do expand please!
I'd never heard of the concept until now since we're having this discussion.

[Saiem]

What is the official name of the shoelace theorem? The only thing I could think of is pick's theorem.

Other awesome names for math things:
Chicken McNugget Theorem (real cool/simple concept that could be used in brain bowl)
Simon's Favorite Factoring Trick (this guy didn't come up with it. However, he definitely has popularized it to the point where he has become associated with the concept. It is often addressed as SFFT in many contests.)

[Susan]

It's not a math thing, but there's that most delicious of microscopy problems, the "coffee-stain effect" (also called the doughnut effect). It's when you dry out a drop of solution and the solute tends to deposit in a ring around the edge of the droplet rather than depositing evenly across the area covered by the droplet.

[pray for elves]

What is the official name of the shoelace theorem? The only thing I could think of is pick's theorem.

Right you are, the official name is Pick's Theorem.

[Saiem]

You know... I'm beginning to think that they are not one and the same.

The shoelace theorem is derived from the box product of a bunch of vectors. It shouldn't really be called a theorem.. so much as a method.

basically any time you're given a bunch of points take the determinant of this matrix:

x1 y1 1
x2 y2 1
x3 y3 1

and then absolute value it and divide by two. that'll give you the area of the triangle formed by the three points. If you don't divide by two, it'll give you the area of a parallelogram formed by the three points (i.e. two vectors centered at the third point... its tricky stuff. look it up in a book... box product)

I think we've talked about this enough.

Chicken McNugget Theorem
http://en.wikipedia.org/wiki/Chicken_McNugget_Theorem

Sample tossup question: You live on the planet Zebulon 5. The currency there is very simple. The currency is divided into two types, 5 cent coins and 7 cent coins. You want to know what is the greatest value of money you cannot attain with this money. For example having 8 cents is not possible on Zebulon 5. For Ten Points, What is the greatest amount of money unattainable on Zebulon 5?

If you keep the numbers small enough.. you can just figure it out. In this case it would be 5*7 - 5 - 7=23 cents.

[Sen. Estes Kefauver (D-TN)]

You live on the planet Zebulon 5. The currency there is very simple.

From a totally non-math stylistic point of view, I think that kind of useless information introducing a question does nothing but give the math players even less time to do their problem. Either cut that crap altogether, find a more useful clue to use up that space instead, or else if you absoultely must, at least put that stuff later so the math people actually have as much time as possible to work.

[Saiem]

Hmm, fair enough. It just helps to have a setting.

I just don't see how I can give much more insight into the problem without flat out saying "Hey, use this really cool formula that you should know."

If I were house-writing a tournament, I'd definitely let a question like that slide since there isn't much substance to the question. It does definitely make sense to move clues like that to the end, so you can continue to work.

[setht]

I know I'm getting into this thread rather late; my apologies if you already figured out whatever you wanted to figure out, or if the thread should have been left undisturbed. Also, this post has turned into a monster, so I apologize for that, too.

Let me state up front that I did not play quizbowl in high school, and that I have never had much connection with high school quizbowl. I can't respond to the poll because I have almost no idea what the status quo is for computation questions in high school quizbowl.

That being said, here are some thoughts on the little exposure I've had to math questions (computational and non-computational) in high school quizbowl and some responses to previous posts in the thread.

I think math, including computational math, deserves a place in high school quizbowl, but people should try to bring the questions more in line with the rest of the distribution.

My own high school experience featured 4 years of math with an emphasis on calculation--most of the time our homework and test questions involved some amount of calculation. My teachers didn't spend time (in high school) teaching us how to add or multiply numbers, but they did expect us to carry out all the calculations associated with geometric/algebraic series, finding determinants, etc. I don't see why questions asking for similar calculations shouldn't appear. Jerry's right that math and science students spend lots of time applying and understanding concepts, but I think that also happens with good calculation questions--that is, the actual calculation step of a good calculation question should be fairly easy, with the difficulty placed on figuring out the relevant calculation.

I think the available answer- and clue-spaces for non-computational math questions are small enough to preclude having large numbers of good non-computational math questions in any question set. Again, I don't know what the status quo is, but if we're talking about something like 2/2 math questions per packet, I don't think it's possible for multiple tournaments to fill that requirement using strictly non-computational questions without lots of repeats between tournaments and/or going into overly obscure material. Several people have called for a Heron's Formula tossup as an example of a good topic in non-computational math, but I don't think there's much in the way of good clues for such a question. I could be wrong, but in my high school math classes no one spent any time connecting Heron's Formula with almost anything else. I think I could come up with 2 or 3 clues for a Heron's Formula tossup based solely on what I learned about it in high school. A quick look at Wikipedia suggests that many such tossups would include clues on Brahmagupta's formula, Bretschneider's formula, and/or Tartaglia's formula, all of which seem like useless clues for almost all high schoolers. There are good candidates for non-computational math questions, but I don't think there are all that many; in particular, not everything with a name that gets mentioned in high school math classes falls in that category.

Several people have given candidates for good, pyramidal computation questions, and discussed what can be done to write questions in that style. One of the things I have seen in some high school question sets, and in some of these examples, is that many of these questions have useless text giving irrelevant background information or waste time naming not very well-known formulas without focusing on the relevant details (namely, how one applies these formulas). I would say that these characteristics are not at all in accord with the principles of pyramidal question-writing. In addition, many of these questions don't take into account the rapid nature of quizbowl questions--asking people to apply Heron's formula to calculate an area given the side lengths of a triangle might be a fine clue, except for the fact that no one is going to do that in less than 5 seconds, by which time the reader has probably already given a much juicier clue.

Another issue that people have mentioned is that many calculation questions feel like nearly-one-clue questions. I've also seen this in some high school question sets: questions will often state a problem, give a somewhat inefficient method of solution, give a more efficient method, then finish by restating the problem. I find these questions aesthetically unpleasing, but I guess they can be written pyramidally with useful clues--they often don't seem like they're written that way, but I think it's possible.

I think there's another route to consider. Here's a sample tossup I just whipped up:

This number equals the order of the Klein group and the derivative of tangent of x at x equals pi over 3 radians. It also equals the highest polynomial order for which a general solution in radicals exists, and the base-11 logarithm of 14,641, which can be found by noting that the first several rows of Pascal’s triangle give successive powers of 11. The determinant of a matrix with entries 0, 2, -2 and 3 is, FTP, this smallest composite number whose square is 16.
ANSWER: 4


This question gives a bunch of different, entirely unrelated ways of arriving at the same number. In this particular example, some of the methods are computational, some are not--I like having a mix of both, but not every question needs this. I've probably screwed up on ordering clues (there might be too much of a drop-off from the first sentence to the second, for one thing) and setting up my pyramid, but I think the general idea can work: give lots of clues, arranged from least well known (I don't think that many high schoolers learn about the Klein group; if no one does, then this is an empty clue and should be cut in favor of something more appropriate) to most well known. Doing this allows the question writer to work in, say, calculus techniques without completely shafting any players that haven't had calculus--you just give one or two calculus-based clues, then move on to stuff that people learn earlier in high school. You can even occasionally incorporate some early collegiate-level material. You can also avoid having to spend time explaining how to apply less well-known formulas, leaving more space for other clues. In fact, you can set things up so you never suggest a technique at all--let the players decide how they want to attack any particular clue.

Looking over at SaiemGilani's sample Chicken McNugget theorem question, I think that question could benefit from a similar treatment: just say something like, "what's the smallest dollar value that can't be represented using special coins of value $5 and $7," and leave space for other clues (e.g., the number of problems proposed by Hilbert, the number of positive non-trivial divisors of 6^4, etc.).

I think this type of question also lends itself well to game flow--you don't sit on one word problem the whole time, and you can easily set things up so you don't have to give 10 seconds for extra computation at the end. I'd say that this sort of question corresponds much more closely to typical pyramidal questions in other categories. Much of the time, I think the players who get these questions will not be the ones that can crunch lots of numbers the fastest, but rather the first to recognize and understand a particular clue.

Does this type of math question seem more palatable? I don't think "computation questions are usually written poorly" is an argument against including them (and trying to find a way to write them well), but if no one can come up with a good way of writing them for quizbowl then perhaps they should be de-emphasized.

-Seth

[Stained Diviner]

While your example does bring computational questions in line with noncomputational questions, it does raise some problems.

For one thing, it could be frustrating to play on. If you hear the tangent derivative clue but need to use the quotient rule to take advantage of it, do you focus on working it out or on listening to the next few clues? The same type of decision might have to be made at a few points in your question. (Should I start multiplying 11s? Should I write down the matrix?) Such strategic decisions could have as much of an impact on who answers the question as pure knowledge or skill.

The questions you are advocating are also difficult to write. It is difficult to write a wide variety of computational calculus questions that can be read by a moderator who might be weak in math, have simple answers, and can be worked out without a calculator in under 30 seconds. If a high percentage of math questions include some computational calculus, you run into the same problem that you have with noncomputational math.

I think those are the two reasons to keep computational questions as single problems, perhaps with a hint or two thrown in.

[theMoMA]

I've always enjoyed the type of question that Seth wrote there, even though there haven't been many of them. If I recall, Leo wrote one for PACE a few years ago. That seems like a good way to bring math--even semi-computational math--up to par with the rest of the distribution in terms of convertability and general feel.

[setht]

While your example does bring computational questions in line with noncomputational questions, it does raise some problems.

For one thing, it could be frustrating to play on. If you hear the tangent derivative clue but need to use the quotient rule to take advantage of it, do you focus on working it out or on listening to the next few clues? The same type of decision might have to be made at a few points in your question. (Should I start multiplying 11s? Should I write down the matrix?) Such strategic decisions could have as much of an impact on who answers the question as pure knowledge or skill.

The questions you are advocating are also difficult to write. It is difficult to write a wide variety of computational calculus questions that can be read by a moderator who might be weak in math, have simple answers, and can be worked out without a calculator in under 30 seconds. If a high percentage of math questions include some computational calculus, you run into the same problem that you have with noncomputational math.

I think those are the two reasons to keep computational questions as single problems, perhaps with a hint or two thrown in.

I thought d(tangent x)/dx = secant^2(x) was the sort of thing people memorized in AP Calc, but perhaps that clue's too hard/takes too long. If so, cut it and replace it with another one--this is easy to do in a pyramidal question with lots of different clues, not so easy in a pyramidal question that consists entirely of one word problem. I really don't think the powers of 11 clue should take anyone more than 3 seconds to figure out, if they're going to figure it out at all: you can note using binomial expansion that (10+1)^n has n in the 10's place for the first several n, you can do the same thing with Pascal's triangle, or you could just think, "hey, 14,641 is somewhat more than 10,000 = 10^4, so this should be 11^4." The Pascal's triangle clue can probably be cut, again allowing more space for other clues (perhaps the tangent derivative clue can be kept, and expanded by adding "or secant squared of pi over 3 radians." Perhaps the question could add a clue on map coloring, or "the index of the first partial sum of the harmonic series that is greater than 2"--there are tons of possibilities.

In any case, I think this objection might apply to a few of the clues in the example, but not to the idea of pyramidal math questions with some computational clues. My idea is that there can be some computational clues that take, say, ~3 seconds to compute if you know what to do. If you do know what to do, you can buzz in and then use your 3 seconds to finish computing. If you don't, you wait for the next clue. As long as you keep your clues to relatively short computations, I see no reason to worry about people making bad decisions about which clues to focus on.

I don't think "it's hard to write good questions" is a worthy objection to advocating good questions. If these questions are not good, they're not worth writing regardless of how hard it is to write them. If they are good, they're worth writing. Writers and editors should try to write good questions; if that means pyramidal questions with lots of clues, they should give it their best shot. If they find that they can't handle it, they should see if they can get help from people that can handle it. Failing that, they should be prepared for complaints about question quality. I don't see any reason to treat math questions differently from questions in other topics in this respect.

I don't think it'd be that hard to write a wide variety of computational questions (they don't need to be computational calculus questions, I'm not sure where that came from). For one thing, I'm not imagining that there will be only one logarithm-based clue in the whole set, or only one "take this simple derivative" clue.

I'm having trouble seeing how the example I posted could give readers trouble, and I would imagine most questions shouldn't give readers problems if the writers just write things out (rather than writing "it's the only real root of the polynomial x^3 - x^2 + x -1 = 0," write "it's the only real root of the polynomial x cubed minus x squared plus x minus 1 equals 0"). Put in pronunciation guides if you think they're needed.

I'm also not sure what you're referring to when you say, "If a high percentage of math questions include some computational calculus, you run into the same problem that you have with noncomputational math." What is the problem with noncomputational math? Small answer space? I don't see how including computational calculus will particularly restrict the answer space.

My impression is that most people take the "math computation" approach to writing math computation questions--a single word problem which requires a significant amount of time to complete. I think it should be possible to take more of a "pyramidal-style quizbowl" approach to writing these questions. I think this would be a good thing for tournaments aiming for pyramidal-style quizbowl. I gather there are high school tournaments out there that don't shoot for pyramidal-style questions, in which case there's no reason to try for that approach in the math computation questions.

-Seth

[grapesmoker]

I thought d(tangent x)/dx = secant^2(x) was the sort of thing people memorized in AP Calc

I think the kind of question Seth whipped up is probably a more correct way to have computation-style questions in high school. I realize I'm not the target audience for such questions, but it would certainly take me longer to come up with the above identity than it would to remember the Klein 4-group, though I'm guessing high school players who are taking calculus are much more likely to be fast on that clue. David Reinstein is right that there arises a dilemma between trying to compute the answer to a clue and waiting for the next one, but I still think that this kind of question works better in the context of quizbowl than what are essentially one-clue computational tossups.

[Stained Diviner]

The New Trier Varsity might be a good place to experiment with Seth's idea and see what happens. I'll see if the head editors are interested.

[Sen. Estes Kefauver (D-TN)]

It wouldn't if they really want it to be ACF Fall style, since ACF generally precludes computation.

[leapfrog314]

ACF generally precludes computation.

We're from Illinois, though, so we might get shot if we have no computation. We're not sure how we're going to handle math yet. And frankly, we have some other tournaments to take care of before we can start thinking too hard about the NT set.