The June 2008 Math B exam is available as a PDF from JMAP.

I already posted three bad problems, and then three more. There is ongoing math teacher discussion of individual questions at a Math A/B listserve run by the Association of Mathematics Teachers of New York State)

Here’s the last two that really bother me are 33 and 34, the 6 point free response questions, most valuable on the exam:

#33 Solve for x: $log_3(x^2 - 4) - log_3(x + 2) = 2$

So here’s the deal: $log_3\frac{x^2 - 4}{x + 2} = 2$
$\frac{x^2 - 4}{x + 2} = 9$
$x^2 - 4 = 9(x + 2)$

That’s what the State expected (then solve the quadratic, reject the bad root. Talk about failure to anticipate!

Most kids seem to have done this, instead:

(more common answer, and the worst problem, below the fold)

$log_3\frac{x^2 - 4}{x + 2} = 2$
$log_3\frac{(x + 2)(x - 2)}{x + 2} = 2$
$log_3(x-2) = 2$
$x - 2 = 9$

And the state was not prepared for this. The only guideline that comes close calls for a 50% deduction (3 of 6 points) Clearly the illegal cancellation is a mistake, unless the kid restricts x. But just as clearly, deducting 1 of 6 points would be appropriate. So what’s the state do? In panic, clearly, they say to mark it completely right…

Helps my students, but doesn’t make me happy.

#34. This is the worst.

Gerardo and Bennie are pushing a box. Gerardo pushes with a force of 50 pounds in an easterly direction, and Bennie pushes with a force of 39 pounds in a northeasterly direction. The resultant force forms an angle of 32° with the 39-pound force.

Find the angle between the 50-pound force and the 39-pound force, to the nearest tenth of a degree.

Find the magnitude of the resultant force, to the nearest pound.

So, let’s hear it for meaning what they mean and not what they say. “Easterly” means due east… 0° in polar coordinates. “Northeasterly”? Guess what? it doesn’t mean 45° . Any student taking physics resolves the vectors into components, using the regular meanings of the words, and gets lost.

Bravo, SED. Your rubric fails to specify how many points to deduct from the child who does not misread the way you intended.

1. June 23, 2008 pm30 12:34 pm 12:34 pm

First question: I really don’t see a problem with your “wrong” method at all. log_3 (x+2) is identical to the given LHS except where it’s undefined – and frankly I don’t think a student at this level should be penalised for not stating “except where x=2”. After all, if x=2 you’ve already written down a divide-by-zero in the “correct” method without making any assertions about *it* being a formal divide that you don’t really mean.

If, on the other hand, you think the question was specifically intended to check that they’re rejecting invalid solutions at the end, then yes, it was broken. But I don’t think that’s clear.

For the last question: pleasant change from my argumentativeness so far :). This question is completely broken. It’s possible to deduce what they were trying to ask, but they shouldn’t have to. I absolutely agree that this is the worst of the lot.

2. June 23, 2008 pm30 1:21 pm 1:21 pm

They pointed this anticipating that there were both a log part and a quadratic part to the problem. They did not notice that cancellation was possible.

Here’s the scoring guide:

(6) 11 and appropriate work is shown

(5) Appropriate work is shown, but one computational error is made. or

(5) The given equation is solved correctly for x, but the extraneous root is not rejected.

(4) Appropriate work is shown, but two or more computational errors are made.

(3) Appropriate work is shown, but one conceptual error is made. or

(3) The equation $x^2 - 9x -22 = 0$ is written, but no further correct work is shown.

(2) Appropriate work is shown, but one conceptual and one computational error are made. or

(2) The equation $\frac{x^2 - 4}{x + 2} = 9$ is written, but no further correct work is shown.

(1) The equation $log_3 (x - 2) = 2$ is written, but no further work is shown. or

(1) 11, but no work is shown.

(0) A zero response is completely incorrect, irrelevant, or incoherent, or is a correct response that was obtained by an obviously incorrect procedure.

3. June 23, 2008 pm30 3:32 pm 3:32 pm

So you’re saying it’s too many points for the difficulty level?

This seems in a way like the converse of the other question, where you were upset that a question with a valid longer method was worth fewer marks. Here the problem is that a valid shorter method is worth too many marks?

4. June 23, 2008 pm30 3:40 pm 3:40 pm

Googling around, I see that this exam must be passed for graduation? – what’s the pass mark?

I guess, looking at the rubric, that significant partial credit should be easy to obtain (seeing method marks awarded for “one conceptual error”, etc, which here would get you nothing). That probably ties into your pass marks being quite high by our standards.

5. June 23, 2008 pm30 4:06 pm 4:06 pm

B is not a graduation requirement – but it is needed for an “advanced” diploma.

The log question was fine, but the authors did not think it through, and so the key didn’t work. It would have given 3 of 6 (one conceptual error) for taking a shortcut but forgetting to restrict x. When they realized, they overreacted by awarding full credit. Five out of six would have been just fine; at this level kids know the danger of division by 0.

The other question is a bit similar in that the authors failed to anticipate what students would do. I’m not saying every goofy approach needs to be taken into account, but major variations should be. They pointed it as a procedural question without setting it up as one.

In neither case are the questions bad – but in both cases there are problems with scoring.

Eventually the student receives a raw score (out of 88) that is converted to a scaled score (not a straight percent). 46 points is passing for this June exam. The number varies a bit each time (another bone to pick…)

June 24, 2008 am30 12:57 am 12:57 am

can you explain what you mean by 46 points is passing for this June exam?

7. June 24, 2008 am30 1:16 am 1:16 am

46 after scaling or before?

I may be being obtuse, but I still don’t understand why you want to penalise the cancellation for the log question at all – I think they’re quite right not to. The division by zero does not occur (as if it did, the original equation is undefined and so clearly unsatisfied). Further, in the solution you like, you’ve written out that potentially-0/0 division, just not cancelled through it – you not only divide by zero but multiply by it again! (Of course you could avoid that by a different route of log manipulation, but none of this is necessary.)

The short solution is, IMO, completely correct.

June 25, 2008 am30 1:14 am 1:14 am

This question is terrible. I would have done like the rest of the kids did [and I, uhh, have a fair bit more education].

9. June 26, 2008 am30 12:12 am 12:12 am

I think I understand why dr rick wants to say that this is completely correct solution, and I would almost agree with him. Almost. Namely, they didn’t actually say that x can’t be -2, because it’s not in the domain, and therefore they aren’t dividing by 0, so there is no problem (I do agree that there is no problem). In other words, do we know that they didn’t say anything because they KNEW there wasn’t a problem, or because they DIDN’T KNOW that there could be a problem. I would have liked if they stated the domain in the beginning of the problem, in which case there would be no reason to doubt their work. So, I would take a point off for not giving me the domain.

10. June 26, 2008 pm30 2:43 pm 2:43 pm

@e,
yup. just write down the restriction, and then cancel away.

11. June 26, 2008 pm30 4:19 pm 4:19 pm

I still say it’s not necessary. Why is it in the denominator? Because it’s the argument of a log (being subtracted). Why is it ok to put it in the denominator, and cancel it? Because, as the argument of a log, it cannot be zero! The difficulty is guaranteed not to arise by the very nature of the technique and so need not be mentioned.

Your method – multiplying through by something which then turns out to be a spurious root – INTRODUCES a problem which they then have to solve by specifically noting the rested domain. To make them introduce a statement in their shorter better method to solve a problem that isn’t there is perverse.

Similarly, if the problem were “solve $\frac{x^2-4}{x+2}=9$”, nobody would suggest that cancelling through should be penalised unless you note that x must not be -2. We know that already – it’s in a denominator! Now, if you choose to multiply up and solve it quadratically, yes, you have to note x is not -2 after all, but if you don’t introduce the difficulty you don’t need to cater for it.