These are sometimes called hanging participles but since a) the people who make the mistake probably don't know what participles are and b) some of them aren't participles, I call them misaligned subjects.
Here is my favourite example, which I heard on a BBC television report many years ago when the Queen visited some ancient ruins in Jordan.
Two and a half thousand years old, the Queen seemed to enjoy her visit to the ancient city of Petra.I think you see the problem. The queen is a ripe old age, but not as old as that... In terms of syntax, what has happened is that the first phrase has no subject and therefore must borrow the subject of the main verb ("seemed to enjoy").
Equally wrong would be this rephrasing to include a participle:
Dating back two and a half thousand years, the queen seemed to enjoy the ancient ruins at Petra.No, you need to rephrase completely. For example:
The queen seemed to anjoy the ancient ruins at Petra, which date back two and a half thousand years.Or
The queen seemed to enjoy the two and a half thousand year old ruins at Petra. (Except I frankly get a bit boggled by hyphens and can't decide where to put them. Anyway, it really doesn't sound nice, does it?)It's this type of thing which old-fashioned grammar teaching gives us. It's not about rule-following. It's just about the ability to say exactly what you mean to say and what you think you are saying.