"Pleading the fifth" is a reference to the Fifth Amendment to the US Constitution. It protects a person from being compelled to testify in court when their testimony would incriminate themselves. It does not apply in family relationships, nor in police interrogation (as incorrectly instructed by your teacher). In the context of custodial interrogation, the US Constitution protect the right to remain silent. But, that rule doesn't protect me from you your mother either.
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Asserting your Fifth Amendment rights to your mother was inappropriate and will get you nowhere. Fifth Amendment rights are only appropriate during interrogation by the police or during court proceedings or in certain other criminal situations. Your teacher was correct. Hopefully you will never be in a situation where you must clean your Fifth Amendment rights.
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Context is key! The assertion of your fifth amendment right not to incriminate yourself via your own statement(s) applies to custodial interrogations by the police. Anything that a person says after a voluntary and knowing waiver of his/her 5th amendment rights can(is available to the prosecutor) be used in court proceedings against the defendant.
The concept of remaining silent to protect one's innocence has become an important part of the American culture. As such it is cited(used) in the colloquial as a catch phrase to make it clear to the interagator, in your case MOM, that you chose not to talk. The rub here is that mom's although not police have ways to make you talk that are not protected by the U.S. Constitution and are worse(LOL) than jail or prison.
This answer/response is based on the information provided in your inquiry and requires a much more complete context than is available in this public forum. Please do NOT use this answer/response to say or do anything regarding your situation. BEFORE you say or do anything consult with an experienced local Federal and/or state criminal defense attorney in your jurisdiction who will listen to you and your concerns.
No, the fifth amendment applies in court proceedings and congressional hearings as well as during police interrogations (to be technical about it, in state court proceedings and local police interrogations it is the Fourteenth Amendment, not the Fifth, that you are asserting) and it would also apply in certain other Government-related situations. For example, you do not have to declare criminal activity on your income tax return. Thus, while you might have to report illegal income from burglaries, you do not have to write "Burglar" on your Form 1040 where it asks for your income.. All of these situations, however, share a common thread. It is the Government, in one form or another, that is asking the question. That is not the case in the family situation that you describe. Constitutional rights lie against the Government, not against individuals. So there is, I am afraid, no constitutional issue in your case and your assertion of the Fifth/Fourteenth Amendment privilege will be denied. Having ruled correctly on the substance of the constitutional issue, albeit on incorrect grounds, your mother may now ask you about your girlfriend. Whether you tell her anything or not is up to you, but it is not a constitutional issue.
I note, by the way that "she yelled at me and maid me answer." Unless you can establish that her yelling at you went so far as to render your answer involuntary, the fact that you answered at all could be taken as a waiver or forfeiture of any constitutional privilege that might have existed.