A phrase is a group of two or more grammatically linked words without a subject and predicate -- a group of grammatically-linked words with a subject and predicate is called a clause.
The group "teacher both students and" is not a phrase because the words have no grammatical relationship to one another. Similarly, the group "bay the across" is not a phrase.
In both cases, the words need to be rearranged in order to create phrases. The group "both teachers and students" and the group "across the bay" are both phrases.
You use a phrase to add information to a sentence and it can perform the functions of a subject, an object, a subject complement or object complement, a verb, an adjective, or an adverb.
The highlighted words in each of the following sentences make up a phrase:
She bought some spinach when she went to the corner store.
Lightning flashed brightly in the night sky.
They heard high pitched cries in the middle of the night.
In early October, Giselle planted twenty tulip bulbs; unfortunately, squirrels ate the bulbs and none bloomed.
Small children often insist that they can do it by themselves.