If the modifying phrase is headed by an adverb then its form is an adverbial phrase:
Adverbial phraseNote that walking is a present participle.
While walking to class, I ran into an old friend.
=> Here 'while' modifies 'walking to class'.
Walking to class, I ran into an old friend.
=> Here 'walking to class' modifies 'I'.
While I was walking into class, I ran into an old friend.
=> Here 'while I was walking to class' functions as a dependent clause.
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