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There's no hard-and-fast time frame for IF and when a skater has to start competing. She's definitely at a level where she can start taking private lessons.
I have a student around your DD's level that I actually DON'T want in competitions. S/he's been passed up through the levels too easily, without real strength or mastery of the elements. S/he's painfully shy and afraid of attention, so to combine that with weak skills at a crowded competition would be a recipe for failure. I'd rather spend the lesson time improving his/her basic skating, honestly. I think it's a better investment for the parents - I don't spend my skating parents' money pointlessly.
When a skater starts working on a program for a competition or a skating show, in either a group or private lesson, new skills aren't introduced. The coach will use the highest skills the skater has completed to make the program, so they'll get a workout, but no new skills will really be gained. When I used to coach the show groups, I made that clear to the parents. Our entire lesson would be spent learning choreography and using existing skills with maybe one cute spin position or a simple jump, but nothing that would help them move up a skill level. I always suggested adding a second group lesson so that the skater was still working on skills, otherwise they would plateau until after the show was over. Just a word to the wise.
However, there are kids who love the limelight and a show or competition program is perfect. It lets them grow as a performer and have a lot of fun.
Most Basic Skills competitions require the skater to compete at their highest level completed as of a certain deadline date, usually a month before the event. Check the registration form. I would suggest that your daughter compete at Basic 8 since that's the last level she completely passed. However, let your new coach determine what level she should compete at because the standards for competitions will always be more demanding than the group lesson program evaluations. They're really just to organize the next round of classes.
Our rink has early-morning and late-afternoon freestyle sessions. Most of the skaters taking lessons in the mornings are older and the younger skaters come in the afternoons. It's because of school and lifestyle schedules, mostly. So while you may have been at the rink for group lessons on one day, that doesn't mean that the coaches don't have any younger private students. Some coaches specialize in different skater groups, but it's usually based on skating level, not age. There are some coaches who prefer adult students, or serious competitors. They'll tell you honestly and refer you to someone else.
My personal opinion is that Basic 6 or so is a good jumping-off point to start privates because the higher Basic levels are real foundation skills that are best learned properly. One-on-one attention will always be better than a group setting, especially if the skater is a little shy or gets overlooked easily.
Talk to your daughter about the group lesson instructor she's liked in the past year. The rink can give you their contact information. Ask if they're accepting new students at your DD's level, find out what time slots they have available, their rates for lessons AND what is needed for the competition. You probably would want to be clear that you're going to use the competition as a "trial" basis or that you'll think about continuing lessons after the event, but you weren't able to commit at this time. That's fine and honest in making it clear that your daughter wants to try this competition, but you weren't planning on taking privates forever.
Ask the coach if there's enough time to properly prepare for the competition. If the deadline's next week, you're pushing it - more time beforehand would be better. I usually suggest two months of lessons before the competition, to ensure that the skater has enough time to prepare and practice. That cuts down on the nerves and stress.
If your DD ASKED about the competition, you should probably try to make it happen before that interest wanes.