If your 11 year-old sticks with the flute I would recommend eventually buying him/her a personal instrument, and make it a good middle-of-the-road quality instrument. If you don't have a musical education to know the difference, enlist the help of the kid's teacher for recommendations. In my opinion (played piano since 5, flute since 5th grade, French horn since 10th) once you have surpassed the beginner level, it does make a difference to have a better instrument. They sound better, they play more easily, they teach the kid's ear what the sound is supposed to be like. On some instruments a higher-quality instrument may have features included that a beginner one doesn't. For example, a higher-quality horn will be double rather than single, on a flute you'll have the B-foot and open holes, both of which take some more skill and practice to master.
And for god's sake if you have a piano at home, get that thing tuned regularly!!