This semester I have decided to give students a simple question or two at the beginning of every calculus class that will trap them into making the most common errors that we all know...e.g. the freshman dream. There is a list/partial classification of such errors here. They generally involve crimes with square roots, fractions, function composition, lines (thinking they have written the equation for a tangent line that is not a straight line at all, for example), exponents, logarithms and strange other beliefs such as all functions....especially trig functions...being linear.
I always have imagined that someone would have constructed a nice list of simple "trap" problems that instructors could use to construct the kinds of short quizzes that I'm looking to give, but surprisingly there isn't a really good repository of quick, but good, problems that aim to isolate the most common student crimes.
As a specific restriction, for example, I'd really like to avoid problems that just say "simplify" or "rationalize", because these commands reinforce the cookbook thinking that got students into this mess in the first place. I would, however, like the problems to be short...i.e. take a competent student around 5 minutes to complete.
Question: Where is a good source of short problems (not of "simplify" or "rationalize" type) that are "traps" to elicit common algebra/trig/precalculus student blunders? I'm interested in online sources, or your favorite problems listed as answers.
Edit: I think it is important to point out, due to the valid concern expressed by Daniel R. Collins and echoed by others, that I am asking for these sorts of questions precisely because telling students about pitfalls seems to inevitably lead to them falling in precisely the pits you told them not to fall in. The philosophy of the question is that questions and exercises answered by the students are just that...questions and exercises answered by the students. Certainly nobody is claiming that if a student solves an exercise that elicits a particularly pernicious pitfall that the student will suffer a spectacular fall from grace and will then be unable to avoid making the mistake! The idea is that problems that show these mistakes to their would-be solver will be memorable in the right way.
This said, if indeed tricky problems like this do lead to "black hole errors" then this would explain the shenanigans I keep seeing on people's papers...