I can't say what is normal. Private schools do tend to load them up on homework. They think it makes them "better" schools and many actually advertise the homework load as if it is a selling point.
That's my experience anyway.
I also think that if your daughter just had a few worksheets to do each night, it wouldn't seem so overwhelming to either of you (vs. having to log into a website and click around to various ones).
Does the school ofter "after-care"? Some schools do and they often are a little more helpful when it comes to getting the homework knocked out. She could at least read for 10 minutes.
Depending upon your car ride, she could read to you in the car and have that already done before you arrive home.
I would also recommend that you review the spelling words for a few minutes on the weekend and then "quiz" her verbally in the car. Copy the list down yourself and keep it on a sticky note so that when you stop at a light you can glance at it and call out one or two words.
These seem small and maybe not how you want your car ride home to go. But it can save you 20 minutes at home on the back end. Neither of you "loses" time... you are trapped in the car for the duration of the ride whether you call out spelling words or she reads aloud or not.
Is there any chance that her after school care (sitter?) has a laptop she can use to log into her math site for 15 minutes? If it is a busy place, then that isn't likely to work, because ALL the kids will want to play on the computer. But if it is small enough, perhaps it is a possibility.
I would also recommend you figure out a schedule for yourselves if you aren't already following one.
M, T, Th, she logs in and does math.
M, T, W she does one each of the 3 homework assignments of her choosing (that she has to do 3 of)
W, Th, F, Sat Sun she reads.
And practice spelling for 5-10 minutes in the car, then have her copy the list once at home on paper.
Done.
I know, sounds so much easier than the reality of making it happen.
I would talk to the teacher and find out what they consider the priorities to be if she is unable to get to everything. Which item would she LEAST be bothered by if it didn't happen. Strive to get them all, but if sometimes one thing has to slide, be sure it is that thing.
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And yes, there is child care that makes sure that homework needs are addressed during the time they are there. The martial arts studio our kids go to also offers an after-school program and they pick the kids up from school, provide a snack, then like another poster said, they give them a work area and supervision/help. When the homework is done, there is a martial arts class. Then the kids have free time until the parents pick them up... usually by 6:00 pm.