"I have a 6 months old daughter and she is a very goodbaby. She has been going to the same daycare provider since she was 3 months old. We both have had no problems. Recently though, she has been having tantrums during the day at daycare, but not at home. She will scream if her sitter will put her down to play and go into another room. Our sitter has other children there as well and they all play with her all of the time. But now that it is summer, they want to go outside. So she is left in the house with the sitter and another little baby a few months younger. My daughter is already crawling and sitting up on her own. So the sitter rotates her activities when she starts to get fussy. The swing to the bouncy chair, to the floor, and then the jumper and so on…all day. But I came early and I could hear her through the door. She has such a fit if she gets no attention. But at home, she is the only child and she doesn’t do that at all, and we do not pick her up all the time. We want to teach her to be more independent and to occupy herself to an extent. We keep her on a very good schedule. Bath time at the same time every night, bed time and bottle at the same time, etc. What can we do to teach her that she can not get away with this kind of behavior at day care? Is it ok to let her cry as long as we know she is not hungry or have a dirty diaper?"
There is no way you can teach a baby to be independent. Kids will separate when they're ready to separate, and if you try to force it too soon by setting rigid limits too soon it'll backfire. A baby needs a certain amount of closeness and nurturing and attention to be able to separate, and if the baby doesn't get what she needs she'll be more clingy in an effort to get that attention. Trying to teach a kid to be independent is like trying to teach a kid to walk earlier--you can make yourself crazy doing all kinds of goofy exercises, but kids walk when they walk.
A 6-month-old is not "getting away" with anything--babies that age absolutely do not have that level of thought yet. And there's also not a chance under the sun that you can teach a 6-month-old to entertain herself. Yes, some babies do on their own, but if your girl is one who likes to be interacting with other people, then good luck trying to change her personality. Most of the development books will tell you that you can't reliably expect a baby to completely occupy herself (meaning that you can be in another room and she can't see you) until the baby is at least a year old.
So I think you'll have an easier time with your daughter if you adjust your expectations. Pick her up when she cries, snuggle her, love her up as much as you can right now. The more attention you give her now, the easier it will be for her to separate when it's the right time for it. I know it's grueling to be carrying around a baby so much, but in the long run it'll make her more secure (and let her be independent more quickly) and you'll know you gave her everything she needed when she was at the tiny baby stage. It's definitely worth it to invest in a sling or Ergo or other body carrier so you can carry her around with you but still be hands-free to get some stuff done. As she gets older she'll start to want to be held and carried less and less as part of her normal development.
I wouldn't be surprised if carrying her around more at home would help her calm down at daycare, too. If she's getting enough attention at home then she won't need it so much at daycare. (If she's not getting it at daycare and you're not giving her a lot of attention at home then she's going to have a really tough time separating the way she's supposed to.) If she's working on new skills developmentally she could be going through a massive cranky phase. Not being able to go out with the bigger kids will just make her feel more cranky. Probably the easiest thing for your daycare provider to do would be to get a clean sheet and put it down on the grass outside near where the big kids are playing and plop your daughter down on it (assuming there's decent shade). That way your daughter can practice her next thing (pulling up and then walking, I'd guess) while she's outside being happy around the other kids.
It sounds like your daughter is very social, and needs a lot of contact and attention from people of all ages. Trust me when I say that at this time next year she will be running away from you as fast as she can. Don't let her babyhood pass you by by pushing her to separate before it's developmentally appropriate. It will only end up frustrating all of you anyway.