Here are two introductory phrases that do not refer to the nouns or pronouns following them -- which in these cases are the subjects of the sentences.
Subjects are in blue, verbs in red, problem words in purple. The first example is from a letter offering customers a Platinum Visa card:
As a valued Brand Insurance customer, we are pleased to offer you …
The company making the offer has confused itself with the customer to whom the offer is being made. This is a very common syntactical problem. There are two fixes:
Fix #1 -- Change 'we' to 'you', and re-write the remainder of the sentence to suit:
As a valued Brand Insurance customer, you are invited to accept …
Fix #2 -- Move the introductory phrase or clause to follow the noun or pronoun to which it refers. In this location it needs to be as short as possible, so I'll delete the brand name.
We are pleased to offer you, as a valued customer, …
The next example comes from a Forbes online story:
After all, prior to filing its Initial Public Offering (IPO), I argued that Groupon lacks a competitive advantage …
This journalist may very well have argued that Groupon lacked a competitive advantage, but he never filed Groupon's IPO forms. In this case, the best fix is to move the misleading phrase to immediately before 'Groupon', because that's what the phrase is talking about.
After all, I argued that prior to filing its Initial Public Offering (IPO), Groupon lacked a competitive advantage …
Note that I changed 'lacks' to 'lacked', to suit the past-tense verb, 'argued'. Sticklers would also insert a comma after 'that'. I would join them, except for the comma that already lies after 'all'. Two comma in five words seems a bit chunky, but ... it's up to you.
The video tutorial presents all of this with sound, colour and movment.
Many grammarians identify the participles in introductory phrases or clauses as "dangling" when they are followed by an illogical noun or pronoun. This is usually the subject of the sentence. Here's an absurd example, with the dangling participle in purple and the pronoun subject in blue:
Swarming aggressively, I was attacked by bees.
The participle 'swarming' is said to be dangling because in syntax it modifies the first noun or pronoun following the introductory phrase, which is 'I'; but in logic it modifies 'bees'.
I think it's misleading to call these things dangling participles. That implies that the participle is in the wrong place. But as you can see, the problem is with the noun or pronoun rather than the participle. And the best fix is almost always to leave the participle where it is and to re-write the sentence with the logical noun or pronoun as the subject, followed by an active verb (in red) and its object (in green), like this:
Swarming aggressively, the bees attacked me.
This is pretty obvious in such a simple and outrageous example. But here's the same problem in a sentence you could see in any business document or email:
Listening to the market, the price was dropped by 15 percent.
Well, no … the price is an inanimate abstraction; it cannot listen to real life. And there's another problem usually associated with these misbegotten subjects -- passive verbs.
The solution to both problems, as in the swarming bees story, is to change the subject to the logical noun or pronoun, followed by an active verb (red) and its object (green):
Listening to the market, we dropped the price by 15 percent.
So let's stop blaming dangling participles when the real problem is illogical subjects. That term focuses the student's mind on the noun or pronoun, where the real trouble lies.
Watch the video for a more entertaining presentation of the same stuff.
The "Dangling Participle" rule is simple: Write participles and prepositions before their objects, never after.
Most editors, and even many English teachers, are happy to let them slide these days. But I think you should correct danglers, because they go "clank!" in your reader's inner ear. The trick is how to fix them without resorting to old-fashioned or pompous language.
Prepositions like 'with', 'at', 'by' and 'on' are often left swinging in the breeze. Here's an example from an 'FAQ' page on the web. I've highlighted the key words.
How many computers can I install my single-user licence on?
Danglers are a syntax error. The preposition appears after its object, instead of before it, which is standard in English. For the old-fashioned fix, just move the preposition ahead of its object:
On how many computers can I install my single-user licence? (10 words)
While grammatically correct, this does sound a bit stuffy. But put the phrase after the clause, and we have the same thing in more modern syntax:
I can install my single-user licence on how many computers? (10 words)
Only now we've lost what was best about the original sentence. The question is 'How many?', which is strongest at the head of the sentence. To keep this emphasis, while losing the dangler, make 'computers' the subject of the sentence, like this: (Subject in blue, verb in red.)
How many computers are covered by my single-user licence? (9 words)
Oops -- this requires a passive verb. But with 'licence' as the subject, we can make it active, like this:
How many computers does my single-user licence cover? (8 words)
The video shows this in animated form, with a bit more detail.