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Well, if you are doing any sort of writing at all, you will be using nouns, verbs, adjectives, and adverbs because those are the various (main) classes that the words you are using fall into. Are you supposed to be identifying all the words you use as falling into one or the other of these categories?
How would the nouns, verbs, adjectives and adverbs need to be used?
Every time you write a sentence, you are using some combination of these words. Your subject is likely to be a noun or noun phrase (group of related words centering around a noun). The verb is the pivot point. What follows that verb is likely to be another noun or an adjective or an adjective and a noun. Adverbs are likely to be sprinkled in here and there.
Nouns, verbs, adjectives, and adverbs are the basic building blocks of your sentences. They are the basic words that convey meaning in the sentence. Then there are other more restricted classes of words -- prepositions, for example, or conjunctions -- that are used as connectors in the sentence.
But you cannot write *without* using nouns, verbs, adjectives, and adverbs. Perhaps your teacher wants you to begin using them consciously? Or to identify them in a passage that you write?
What is the exact assignment? Is it written down somewhere?