the chioldren were running
In this sentence, "were running" is the verb phrase, in the past continuous tense.
"The children were running to the bus stop." The verb phrase is "were running."
The gerund phrase in this sentence is "children singing."
"children singing" is the gerund phrase in this sentence. It serves as the object of the preposition "of".
A participial phrase functions as an adjective and starts with a participle (a verb form ending in -ing, -ed, etc.). For example, in the sentence "Running quickly, the athlete crossed the finish line," the participial phrase is "Running quickly."
Yes, "all the way" and "there" together form a prepositional phrase in the sentence.
Children singing
children singing
running on the track
for joggingThe gerund is jogging, in the prepositional phrase "for jogging." The gerund is a noun here.
I'm not sure I understand your question. What phrase are you asking about, "children for school?" If that is what you mean, it could be correct depending on how it is used in the sentence. For example, a sentence such as "We must prepare the children for school." would be correct. If you make your question a bit clearer I could help more.
The gerund phrase in the sentence is "children singing." It functions as the subject complement that describes the favorite sound.