Wow! “The”, “an”, and “a”. Three words and we have a whole class of the English language. Part of a larger class of words referred to as Determiners, Articles can also be described as Adjectives that give a certain amount of description to the noun they come before.
The distinctiveness that articles give to their associative nouns has to do with either the definite quality they (actually just one article) provides the noun, or the indefinite quality they provide the noun. The Definite Article “the” designates a definite quality. When one says, “the cat”, or “the baseball”, they are referring to a thing specific; they know which one they are talking about, and to some degree, so does the listener. “The baseball game was exciting.”, is a clear statement in English that indicates that both speaker and listener know which baseball game is being referred to in the conversation. (See…. In saying “the conversation” both I and you know that I am referring to the preceding statement, “The baseball game was exciting.”) If one says, “Give me the pencil.”, the sentence context is implicit that the listener knows which pencil the speaker is talking about; even if that information isn’t revealed until the speaker points at the pencil. “The” is a very important word indeed!
The Indefinite Articles, “a” and “an”, are no less important. Both describe or assign an indefinite character to the noun that follows. In traveling linguistically from, “the baseball”, to “a baseball”, one moves from the concrete to the abstract. Now, like some Platonian discussion of Forms, all the listener can conjure up is either the ideal baseball in all its perfection or some derivative thereof that suits the context of a conversation being had. It might be any baseball or some variant that suits the listener’s imagination. As abstract as this all sounds, this quality of noun speaks volumes when used in context; as it informs the listeners of those things that are still in need of proper context and identification.
When to use “a”, and “an”, is relatively simple. Use “a” before words that start with consonants, and use “an” with words that start with vowels. Are there exceptions (is water wet)? I could go on about certain words starting with the letter “h”, that requests the use of the article “an”, but I’m not going to. It’s not critical to the formation of proper usage of articles for a student. They will learn them through use of the English language within the community they have settled into. Choices here, as to use “an” or not, are highly dialectical or cultural. It doesn’t help to only confuse your student when they hear someone use the word “an” improperly (at least according to your tutoring).
Rather, you will want to spend your time discussing when to use one of these three articles according to the type of noun they are associative to. Here we get into Countable and Noncountable Nouns, as well as a good amount of geography.