american english - Usage of "homework," "schoolwork," and "assignment" in AmEng for schoolwork given to students to do at home

As far as AmEng goes, is there any difference in using either homework, schoolwork, or assignment to call schoolwork given to students to be done at home? Can these be used just about interchangeably?

ASSIGNMENT vs. HOMEWORK

1. assignment

An assignment is a task that someone is given to do, usually as part of their job.

My first major assignment as a reporter was to cover a large-scale riot.

An assignment is also a piece of academic work given to students.

The course has heavy reading assignments.

When class begins, he gives us an assignment and we have seven minutes to work at it.

In American English, an assignment is also a piece of work given to students to do at home. (emphasis is mine.)

2. homework

Work given to schoolchildren to do at home is also called homework.

He never did any homework.

HOUSEWORK vs. HOMEWORK

1. homework

Homework is work that school pupils (Chiefly BrEng)/students (Chiefly AmEng) are given to do at home.

Have you done your English homework?

2. housework

Housework is work such as cleaning or washing that is done in a house.

She relied on him to do most of the housework.

Collins COBUILD English Usage

SCHOOLWORK

: work that is done in classes in a school or given to students to do at home.

Merriam-Webster

EDIT

Point to all of this is, if some native speakers of AmEng actually use exclusively "assignment" (or schoolwork) for "homework" in the sense "schoolwork done at home," wouldn't the reason for that be that "homework" might have a quite different meaning in their vernacular, e.g. "housework" or "work done on the home"?

Homeworker in the US is a person who works on homes or a person who works in homes (maid). It is not a "clear" phrase at all because it tends to have ambiguous or no/multiple meanings. It is certainly something that I would expect to hear from a foreigner in the US. (emphasis is mine.) RyeBread - ELU

Please, consider the following example:

I'll help you with your homework/schoolwork/assignments once I'm done with the housework. (standard AmEng)

-and-

I'll help you with your schoolwork/assignments once I'm done with the homework. (nonstandard regional AmEng?)

Answer

I've always used these terms in a different way that @James McLeod is describing them. Perhaps they are regional differences.

Assignment: One singular task or project; large tasks may be broken up into smaller assignments. I may get one or more assignment per class. Whatever it is, it is thought of as a singular 'unit of work' when submitted or completed.
Complex, multi-part, or really large assignments are generally referred to as projects.

Homework: The collection of all the assignments I have to do at home

Schoolwork: The collection of all the assignments I have to do for school, regardless of the location of where it is done. Can be a synonym for homework, but less frequently used as such.

(Further clarification after reading some comments): An assignment can also be done IN school. Assignment just means 'one task'. Which is why a lot of people do use homework assignment. They're specifying that this assignment is to be done at home (or at least, not during that class)

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Not sure about other parts of the country, but I have never heard of homework used to describe not-related-to-school work done on or in the home.

Yes, Housework means work done for the house/home. This typically means tasks like cooking, cleaning, laundry, yard work, and small house repairs (when done on one's own home, this term is not used if professionals are called in).

I have never known a native speaker to confuse housework and homework, but if someone learning AmE were to mix the terms up, I do believe they would still be understood by most people.