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A group of words made up of a preposition, its object, and any of the object's modifiers.
A verb form that functions as a substantive while retaining certain verbal characteristics, such as modification by adverbs, and that in English may be preceded by to, as in To go willingly is to show strength or We want him to work harder, or may also occur without to, as in She had them read the letter or We may finish today. See Usage Note at split infinitive.
INfinitive: tp run, to walk, to fly
Gerund: I am walkING, he is runnING, it is flyING
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so what does it mean to "Write five sentences, each containing one or more infinitive phrases." ..... i write them but what does it mean by "each containing one or more infinitive phrases."
Let's clarify gerund phrases, first of all.
In these sentences --
I am walking.
He is running.
It is flying.
These -ing forms are NOT gerunds. In each of these sentences, the verb phrase comprises two words: some form of to be plus the present participle (-ing form). This is the present progressive -- it's a verb tense. So "am walking" is the present progressive verb form of the verb "to walk."
A gerund is any single-word -ing form (or any phrase beginning with the -ing form) where that word (or that phrase) functions as a noun does -- that is, as a subject, as an object, and so on.
So in these two sentences, the word "walking" is functioning as a gerund --
Walking is fun. ["walking" is the subject of the sentence]
I like walking. ["walking" is the object of the verb]
You can also expand "walking" out into a phrase --
Walking in the morning is fun.
I like walking in the morning.
You now have gerund phrases.
You can also form gerunds or gerund phrases by using the -ing form of the word (or a phrase beginning with the -ing form), as nitprin noted, as the object of a preposition.
So, for example --
I keep in shape by running every day.
The phrase "by running every day" is a gerund phrase. It's a gerund phrase because the phrase is functioning as the object of the preposition, and that's a noun function.
To summarize --
If the -ing form of a word (or a phrase beginning with the -ing form of the word) is FUNCTIONING like a noun, then it's a gerund (or a gerund phrase).
Does that make sense?
As for your last question, about the sentences containing infinitive phrases, do you understand what an infinitive phrase is? Your five sentences must each have within it at least one such phrase.
What are your five sentences? Have you written them? If so, post them, and let's go through them.