As you probably know, the same sorts of errors appear year after year in the Improving Sentences and Identifying Sentence Errors sub-sections of the SAT Writing Test. Some might say ETS is striving for reliability, but the beret-wearing inner writer in me says they just lack imagination.
Many of these errors, unfortunately, require students to read and carefully consider all options before identifying the error and selecting the right answer, but at least one sort of commonly (not to say "universally," in test after test after test) appearing error is easy to spot and correct; I have trained even the most grammatically / stylistically challenged students to correctly answer such questions in 5 seconds at most, freeing up precious time to spend on the more nuanced items.
The error is the dangling modifier.
In case you're a little rusty, or have sensibly been spending your time thinking about almost anything other than esoteric grammar errors, dangling modifiers are participial phrases (the first word will be a participle, ending in -ing or -ed) that begin sentences without being directly followed by a comma and the noun or pronoun they modify. All phrases function like another part of speech (noun, adjective, adverb), and participial phrases function like adjectives. For syntactical sense, then, they need to be followed by the pronoun or noun these "adjectives" modify. If not, we have the dangling modifier.
Here are some examples:
a) Walking the dog, he had to stop frequently to urinate on fire-hydrants.
b) Smashed on the tennis court, I found my father's watch.
Uh-oh! Option a), as written, seems to suggest that whoever is walking the dog is the one urinating on fire-hydrants. Option b) suggests that I was lying smashed on the the tennis court when I found my father's watch. I suppose both are technically possible (while in college, maybe?), but probably neither is what the writer meant!
a) Walking the dog, he had to stop frequently for his four-legged friend to urinate on fire-hydrants.
b) I found my father's watch smashed on the tennis court.
Note that both require substantial revision, a) because I inserted "four-legged friend" in order to avoid either repeating "dog" or creating an ambiguous pronoun reference with "him," and b) because, although all the words are the same, I transferred the participial phrase to the end of the sentence, immediately following the noun it modifies. In my experience as a classroom teacher, students sometimes struggle with these significant modifications--they know what the sentence intends (which, if you think about it, is probably why we have dangling modifiers in the first place) but not how to fix it.
Fortunately, on the SAT they never have to. Either they just have to spot the error or pick from some pre-fabricated sentences with the error corrected. Also, while there is no rule requiring participial phrases to begin sentences, that is almost always where dangling modifiers appear and is always where they appear on the SAT.
So here is how I train them:
1) Give a mini-lesson (<20 minutes) just like the one described above, and make it comical;
1) Get them in the habit of looking for participial phrases at the beginning of the provided sentence, which are easy to spot because they begin with "an -ing or -ed form of a verb" and end with a comma;
2) Get them to check that immediately following the comma there is a noun or pronoun "doing" or "being" the -ing or -ed form of the verb;
3) If there is no appropriate noun or pronoun, skip the rest of the sentence and quickly scan the provided multiple-choice answers for the one that either identifies (in the Identifying Sentence Errors sub-section) or corrects (in the Improving Sentences sub-section) the dangling modifier.
Start to finish, 5 seconds. Test after test after test.
Notice that I avoided using the words "participle" and "participial"--it's at your discretion, of course, but unless teaching verbal phrases is part of your grammar curriculum or central to your sense of self-worth, these words are unnecessarily polysyllabic (as are, as it happens, "unnecessarily polysyllabic"), confusing because of their similarity to each other, and off-putting, so just a distraction. You don't need them to spot or correct the error, so I usually just say "-ed or -ing form of a verb."
Dangling modifier, btw, is also a mouthful. In my classes, some of which were in urban environments, we would create fake "sign" to flash when we noticed a particular error: in this case, the first three fingers of one hand, hanging downward to represent an m and kind of swinging forward and backward, as dangling does. I guess. Caveat emptor, better check with your local administrator before using this technique, lest you are required to check with your union rep afterward: kids can get so excited with their new gestures that, in the hallway, they appear to be flashing gang-sign. Not me, but this guy I know... he learned that the hard way.
Good luck! It's really, really easy.