Which one of these sentences is correct when I want to mean I registered for a class? Or can either of them be used interchangeably?
I joined a dance class.
I signed up for a dance class.
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This really depends on whether there is an acceptance process or not. To sign up means to formally state your wish to join the class. To join means to actually start attending lessons. If acceptance is automatic, then you can probably use these terms interchangeably.
However, acceptance may not be automatic. Maybe it's an advanced class that requires an audition. Maybe the class is oversubscribed and the organizer will be picking people randomly from the sign-up list. Maybe the class hasn't actually been scheduled yet and there's a chance that it will be scheduled at time when you are not available.
In all those cases, there's going to be a time gap between signing up and joining; and maybe you'll even find out that you can't join the class you signed up for.
One can sign up by adding one's name to a list on a bulletin board, or on a clipboard or electronically or by (in an institution) telling the admin person you want to take the class.
You sign up for a class that you want to take. You sign up to take it.
Once you have signed up. and are taking the class, you can use the verb join to describe being in the class.
Patsy: How long have you been taking that ballet class?
Annie: I joined the class in January.
Patsy: Was it easy to get into that advanced class?
Annie: No, I signed up and did not have to audition for it.
To join means to become part of something larger than yourself where there are members or participants in a larger entity.
In that sense, you can join many things: clubs, organizations, institutions, classes, courses, groups, churches, etc.
Teachers often say to distracted students, something like this:
"Mr. Smith, would you care to join us?" (pay attention to what is being said)
If you run into someone in a restaurant who is with others, that person might say to you:
"Would you like to join us?" = become part of our "group" at a table