Monday, July 7, 2008

Figure Not Drawn To Scale! An SAT-Type Geometry/Summer Diversion

In the circle at the left, O is the center, A, B and C are on the circle and OABC is a parallelogram. If AB = 6, what is the length of segment AC (not drawn)?

(A) 3√2 (B) 3√3 (C) 6 (D) 6√2 (E) 6√3


Is this an appropriate standardized test question?

Are you an opponent of multiple choice (aka, "multiple guess") questions. Why?

We can also say much about the issue of drawing figures that do not appear to be what they are? Is it just the testmaker's way of misleading or trapping students or is there a valid purpose to this?

Your thoughts about this problem...


Eric Jablow said...

This actually requires students to know an important fact about angles inscribed in circles. The one everyone remembers is that an angle inscribed in a semicircle must be a right angle: If AC is a diameter, and B is on the circle between A and C, then angle ABC is 90 degrees.

The more general result is (in the original picture) angle ABC is half of angle AOC (with the non-obvious orientation). So, ABC = 1/2 (360 - AOC). But ABC is AOC, and so both angles are 120 degrees. The computation is now obvious (which means that I'm being lazy).

YzW731 said...

I tried that problem and I think multiple guess is pretty nice - a break from a lot of free response students always get.

That picture looked pretty annoying - i had to redraw it two times *laughs* In the end I just assumed a bunch of 30 60 90 triangles existed and based the answer on what looked right. Is the answer E) 6 root 3?

Dave Marain said...

Yes, you got it! I think it's important for educators to see how students often make correct intuitive guesses for these kinds of questions. Making assumptions without formal verification is also part of being "test-wise" even if mathematicians and math educators cringe at this!

Eric's inscribed angles argument is very nice however I'd be curious about how many students would recognize that triangle OAB is equilateral since AB = OC (opp sides of a parallelogram) and OA, OB, and OC are all radii!

There is the test-taking strategy component to this discussion, then there is the issue of what important mathematics students can learn from this type of problem.

Keep your thoughts coming!!

jd2718 said...

I had Dave's response - recognize radii, and then segments equal to radii.

I would rather see this as a discussion question than a test question.

The amount of thought required seems relatively high for a multiple choice item, and a short answer variant would be really harsh on partial question. This is, more or less, an all or nothing question.


Florian said...

Nice question!

I think that this problem can be solved using very basic knowledge about triangles and parallelograms and pythagoras only:

AB=6 => OC=6 (because ABCO is a parallelogram)
OC=6 => OA=OB=6 (because O is the center and A and C lie on the circle)

Each side in ABO and BCO equals 6. Which means we can compute the height of the two equilateral triangles easily:

h = sqrt(6*6-3*3) = 3*sqrt(3)

OB is the line where the bases of the two triangles are connected and if we add the heights of those triangles with respect to OB we get the distance AC = 2h=6*sqrt(3).

I don't see a reason why this type of question or drawing could be a bad idea for a test. After all students are required to think aren't they?

PhD in Math? Pfft! What a waste! said...

Hi there!

Interesting post, to say the least. I think in answering these questions why must first ask: what is the goal of these standardized exams? Initially, when I got first intimately acquainted with the American K-12 education system at the age of 15, I was shocked by the emphasis on standardization. It seems that recently this emphasis has only grown (especially here in California, plus such useless money wasters as the "no child left behind program," but that alone is worth of a post).

These tests, needless to say, will not succeed in filtering out the bright minds with potential for further development or, much more so, encourage these minds academically. Standardization of any shape or form will have the same advert effect. In addition, given that most teachers (especially in public schools) are pushed to pull the test scores up (via "preparing" their students to pass such standard exams, in some cases giving out answers--I've witnessed this myself), the vicious cycle continues.

I have seen bright students with great potential (in mathematics per se) fail miserably, while students with enough will power to power memorization succeed greatly - they shall make fine clerks, administrators and politicians later, and lets face it, we need them, while brighter minds are caught in the gears of standardization and are twisted into "I'm just not good enough" cubicle workers.

Needless to say, the above perception is only a gross simplification of the problem that's been growing through the public education system like cancer. Here's a much nicer, deeper and more intelligent (than this comment) reflection on this problem:

PhD in Math? Pfft! What a waste! said...

I would also like to add that I agree with Florian: nice problem indeed. My post was targeting standardization. Suffice it to say, I am against multiple choice questions (these I also encountered for the first time in America - I am comparing with the Russian education system; it seems, however, that Russia's education system, despite desperate pleas from many faculty has been progressively westernized with respect to standard examination... well, I presume it is the cheapest way to deal with the growing body of student population).

Florian said...

PHDiM? made some great comments.
Of course this type of problem
cannot be the only one for students.
During my exchangestudent year in the
US I was amazed how easily I could
get through in MC tests by reasoning
out bad answers. In many cases it
was not necessary to solve the problem

I personaly think a balanced approach of MC and open problems could combine the best of two worlds. MC questions can be used as a warmup in a test to check basic understanding followed by the bigger points free problems.

The problem with standarized testing seems to be that the variety of questions is not large enough allowing students to study problem cases instead of problem solving. If the pool of questions were bigger (and the material base the same) woudn't this be a solution?

Alex said...

Re: the standardised test... I can't really comment, as I'm not familiar with your standards.

Multiple choice I like. Sure, you can reason away wrong answers - but that simply means there's another way for ingenious people to solve the problem, and I see that as a good thing.

Multiple choice papers also make it easy to analyse the results. With carefully chosen wrong options, they can be one of the best ways to see what a large group of students is doing wrong. Here's an example: "What is 3 plus 4 times 6? Is it 30, 42, or 27?"

Note the question isn't any easier or harder for being multiple choice - but it's certainly easier to tabulate the results, which could matter if you've got a sample size of a hundred or more.

Misleading figures - take it question-by-question. Here, the key observation is that IF OABC is a parallelogram, then OAB is isosceles. Drawing the figure accurately might have given that away.

Oh, and the problem? I like it. As with many in geometry, there's more than one solution. I'd pitch it to the brighter kids as extension... for them I feel the difficulty works well. There's too much thinking ahead requred for weaker students, IMO.

Hypatia said...

I agree with all of the posters with some qualifications.

[1] I like the problem, but not necessarily the answer choices. The first three answers were obviously wrong. I solved the problem by surmising that the parallelogram was a rhombus. Could it also be a square? No - OB was a radius and therefore 6, not 6* sqrt(2). So answer d is incorrect. By process of elimination, e had to be the correct answer without any calculations whateoever. Now if answer b were eliminated and a final choice e being "none of these answers", I might actually have been forced to do a little work.
[2] I agree with Jonathan when he suggested it was a good discussion question. Several people solved the problem in different ways. A classroom discussion sharing these ways, or perhaps seeking other ways, would be wonderful. It might be interesting to further the discussion by asking for the area of OABC. Actually I like that question better than the original with at least three plausible choices or the 'none of these'and two plausible. -- It's also a great queestion for reviewing all the properties of special quadrilaterals.
[3] Most high school math teachers don't assess their students using the MC format -although there are some who use the 'book' exam :( If some MC questions are included on an exam, we usually ask 'why' an answer was chosen. Since our students are assessed in such a variety of ways, I'm not really that opposed to the format, if the questions and answer choices are good. It's just that that isn't the case on most state exams and the standards are minimum. It shouldn't be difficult for all students to be proficient. The fact that so many of them do so poorly means that we (collectively speaking) are doing something very, very wrong.
[4] I also agree with 'phd in math'. I, too, have seen very accomplished math students who have done misearbly on standardized tests. I'll save that story for another day because this post is so long.(Sorry Dave!) I will add though that I can probably cite more examples than he can as I have been teaching math longer than he is old. (It pains me to admit that!)

Dave Marain said...

Thank you for your thoughtful reply. I'm not that impressed by the distractors among those choices either and I wrote them! I spent far more time developing the problem which turned out to be fairly challenging for the SAT groups with whom I'm working with this summer. For the most part, these are motivated higher-ability students but, when faced with questions that require deeper reasoning, they struggle like most would.

This question did discriminate fairly well -- close to 20% answered it correctly and these were mainly the highest scoring students.

Some thoughts about the choices...

(1) On a real SAT, there are often 2-3 choices that are easily eliminated as being 'out of range'. Despite this many students did not do this.
(2) I would have preferred to make this a "grid-in" question, however, the radical prevented this. Answers to grid-in problems have traditionally been restricted to rational numbers on the SAT.
(3) I asked the few students who got the question right how they did it. Most replied something like this: "I re-drew the diagram, assumed there were equilateral triangles or 30-60-90 triangles and went with the 'radical 3' answer at the end since it had to be more than 6." None went through the process we're describing in the complete solutions offered thus far in these comments. BUT, of course, when faced with 2-3 minutes to solve one of these "test-wise" students often are skillful at eliminating choices and making educated guesses.

Most importantly, they are confident risk-takers which separate them from equally or more capable students who are not as confident. This is unfortunately the nature of standardized testing -- always has been, always will be.

(4) (E) None of the preceding answers is correct
ETS has moved away from using this as a choice although it still occasionally appears. This is why i didn't use it. However, I could have produced more quality distractors if I had put more thought into it.

Yes, the quality of "state tests" is usually not very high, however, I cannot say that for the SAT whose questions are screened, refined and edited by many.

I've devoted more time in this reply to the issues of testing than to the mathematics underlying this problem, but that's what happens when one opens up Pandora's Box as I did! I appreciate all of the comments made about this problem. I enjoyed writing it and the class discussion that resulted from this question was worth its weight in gold. I did feel it reviewed so many important ideas as well as test-taking strategies.

Hopefully you will enjoy more of these as I write them this summer.
Thanks again to Alex, Florian, 'Phd in math', Hypatia, Eric, jd2718 and yzw who took the time to enrich this discussion. There is so much more i want to say about this problem but I'll stop here.