1. In grade seven, I became the captain of my school and held fund-raising events for people in need.

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What is the part of speech of 'for people in need'? Is it an adjective phrase or an adverbial phrase?

2. They will hold a party for their teacher's birthday.
[What about 'for their teacher's birthday'? Does it modify the verb 'hold' or 'party'?]

1. What is "for people in need" modifying?

If it's modifying/describing a noun, then it's an adjective phrase.

The best thing to remember when you're dealing with these is that phrases like this are almost always modifying the word they come right after (or sometimes right before)

What words are being modified by these phrases? What words are right next to (in front of, most likely) the phrases?