Learn English – How to say that you are available on weekday except one day (e.g., Tuesday)

John: What did you do yesterday?

Sam (a prosecutor): I was hearing a [or: the?] reason a murderer killed the victim being justified.

In short, the reason sounds better, because in life as we and John know it, there is usually only one reason given by, or on behalf of, a murderer. This is true, whether John has heard about this specific case, this specific trial, this specific murderer, or not. This is just the way life, and most murder trials work. (Although, Prosecutor Sam is being a little over-zealous for his cause here, because in the USA, a person is innocent until proven guilty; so a person on trial for murder, is an accused murderer, not a murder--unless he has murdered previously, but still regarding this newest trial, he is an accused murderer: still Sam could, off the record, call the person a "murderer.")

In rare cases, more than one (thus a) reason could be given, as in when people are trying to guess "the reason." But there is usually only one reason, and this one reason can include multiple facts, strands, or sub-reasons, but it can be summarized as "the reason." And this is "the reason' that Jill, the accused murderer's defense attorney is going to try to justify. Not one among many, but just this one.

If this murderer did indeed have more than one reason, then a could be used.

But it is not true that "a" and "the" are entirely governed by whether the listener already knows it or not. This is a false thesis. Read more of Abbott.

What usually governs the use of a and the is the assumption that the speaker makes about the listener. If the speaker assumes the listener can identify which referent he is talking about, then the speaker will use the. Else, he will use a. But notice even this is just a guideline. Speakers do not have to abide by this rule, and this rule does not cover the corollary, the use of a. In fact, a speaker can use a even if he knows that the listener can identify the referent. Let's go over this using the sentence a about the house.

If say to you

Our friend Jason bought the house with a big back yard.

this usually means I would expect you to be able identify which house I'm talking about. Either you are familiar with this house, because you've seen it or we've discussed it before, or something along those lines. But, usually, if I don't assume you can identify which house I'm talking about, I would use a house with a big back yard.

However, if we are limiting the discourse context to one sentence, I would probably say

Our friend Jason bought the house with the big back yard.

Notice I say both the house and the big back yard. This is because, given the discourse context of only one sentence, the house I expect you to be able to identify is identifiable because of its big back yard. This is the only characteristic of the house that distinguishes it from the other houses that Jason was considering buying. In other words, among the houses that Jason was serious about buying, only one of them had a big back yard.

If I say

Jason bought a house with a big back yard.

Taking this sentence as our only context, I would normally say it if I assumed you could not identify which house I'm talking about.

However analyzing the whys and wherefores of individual sentences shorn from any context is extremely problematic especially when taking about ten use of articles. Also important are speaker's intention, what the speaker assumes assumes, and any other context. Thus

If we are standing in front of the house that Jason bought, and it has a big back yard and I say

Well, Jason bought a house with a big back yard (just like he said he would); just look how big that back yard is!

this doesn't mean I don't expect you to be able to identify which house Jason bought. We both know which house it is: we're standing in front of it, talking about it! In this case, the noun phrase a house with a big back yard refers to a specific house, namely the one we are standing in front of and talking about. But, grammatically I am not identifying it as a definite house; for that, I would have to use the definite article.

Both "I won't be needing that" and "I don't need that" are grammatical.

The problem with your teacher's logic is that "need" is not a modal in either of those. Two easy-to-detect signs of modality are:

• They have the syntactic properties associated with auxiliary verbs in English, principally that they can undergo subject–auxiliary inversion (in questions, for example) and can be negated by the appending of not after the verb.
• They do not inflect (in the modern language) except insofar as some of them come in present–past (present–preterite) pairs. They do not add the ending -(e)s in the third-person singular (the present-tense modals therefore follow the preterite-present paradigm).
  _Wikipedia

For the first point I listed, it should be pretty obvious that not does not follow need in "I don't need". For the second point, since people say "he needs...", we know it's not modal in this sense either.

"Need" specifically is a semi-modal. Your sentence needs modification were it to use need as a modal:

I need not have help.

According to the statistics here, the modal form of need is not used very often in American English (which is why I think it sounds old fashioned). A COCA search for BE needing (capitalizing "be" means it matches all forms of the verb, such as be, were, 're, etc.) returns 356 results, so I think it's safe to say that educated speakers do use "will not be needing" and similar (I would).