IME 6-7 and 9-10 (Y2 and Y5) are both ages at which DCs want to be more grown up than they are ready to handle.
This has a nasty tendency to show it's self as assertive stroppyness and toddler tantrums when they don't get there own way. DNiece (7) had a beauty last time we saw her.
What can you do? Firm boundaries help. We had a lot of "go to your room, until you want to be nice". By 6 DD2 knew full well that backchat, disobedience and baiting your big sister were not nice.
(And yes it would have been more PC to say when you want to behave nicely, but I'm crap at this modern positive parenting stuff).
Once you have been firm, ignore the tantrum, whining and cring that follows. Do not try and reason with the DC until they calm down same definitely goes for 9year olds). They may look big, but they still get into toddler type adrenaline fueled tizzes they can't control.
And finally as Redface says, individual attention, days out, evenings, DD2 liked to choose a board game to play with me when DH took DD1 to a late night dance class.
Going to see an elderly DF with a dog. DD2 loved walking and playing with the dog. DD1 is scared of dogs, so again this was something special for her.
If you can let them choose what to eat, where to eat, a simple choice of MC Donald's or KFC, Even Sainsbury's or Tesco's to do the food shopping or which of local towns to visit for new school trousers.
If your life is controlled by school rules six hours a day, you need a bit of choice. What small DC decide are important choices may seem very petty to adults, but become very important to small DCs. The more you can give them choice the easier it is for you both not to feel bad when mum's opinion has to prevail.