Notes:

忙 máng is an adjectival verb, as the predicate of a sentence it describes the subject just like an adjective. Various intensifiers can be placed before the verb to modify the degree of the adjective verb, for instance, 太 tài too and 很 hěn very.

Notes:

A little more about the adverb 很 hěn very:

很 hěn is an adverbial adjunct, which means that it is used to modify a verb, or in this case, an adjective acting as a verb. In a lot of sentences with adjectival predicates, 很 hěn does not necessarily indicate an increased degree of the verb or adjective, but rather fulfills a purely grammatical function.

In other words, a lot of times 很 hěn doesn't really mean anything, it is just put in the sentence because otherwise it will sound weird.

Notes:

Notice that when Matt addresses Teacher Zhang he uses her professional title of "teacher" in place of a personal pronoun or name. This is a way of conveying respect.

Notes:

死了 sǐ le is a conventional phrase, which means that the 死 sǐ and 了 le always go together. The phrase emphasizes the degree of the adjective.