There was a recent post on FB that was based on assuming that a preposition had been left out of a prepositional phrase. There is really no such thing as a prepositional phrase without the preposition. The preposition cannot be “assumed.” It is either there, or it is not. If it is there, there is a prepositional phrase; if it is not there, there is no prepositional phrase.
The sentence under consideration was something like
…And approximately one year we were there with him.
And the discussion was whether there is a comma needed after “year,” based on the rule that three words do not take a comma and four words do when they are at the beginning of a sentence — which is another issue.
In this sentence, “year” with its adjective modifier “one” is called an adverbial objective. It is a noun that answers an adverb question — “how long?”
…We spent three days.
…It took five hours.
…They spent a thousand dollars.
In the original sentence, “approximately” is an adverb that modifies/limits the adjective “one.” These adverbial objectives can be anywhere in the sentence, as can most adverbs, without being considered to be “out of place.”
So the introductory element is short and just modifies and does not take a comma.
Further, I do not believe it is a hard-and-fast rule to count the number of words for prepositional phrases at the beginning of a sentence.
…on that very day
…during the uncomplicated negotiations
These are each four words. There does not need to be a comma after the first one at the beginning of a sentence; the second one would most likely take a comma. And part of a discussion for another day is the kind of prepositional phrases that always take a comma at the beginning of the sentence.
I see the makings of a good class on introductory elements!