So Hagrid's wand is broken when he is expelled but he still uses the pieces to do magic. This doesn't make sense because when Ron's wand broke, every spell backfired. So how is Hagrid doing it?

This would've been the answer I would've given.

oh, so weak spells is equivalent to backfiring spells? i guess it just depends on the type of wand maybe as to how it will work. thanks

maybe but Hagrid technically only had one năm thêm of schooling than Ron had at the point that his wand was broken

Yeah, that's what I'm thinking as well. Dumbledore seemed to know that Hagrid was still using magic, even after his wand was snapped. Dumbledore definitely had something to do with it.

i dunno, i'm still inclined to believe that Mrinmayee is closer to the truth. if bạn look back to PS in the Keeper of the Keys chapter, it says that Hagrid actually wanted to turn Dudley into a pig but only managed to give him a pig's tail. He can make fires easily enough but transfiguration would require a proper wand. actually maybe you're right because Hagrid only had 3 years of schooling so transfiguration would have been difficult for him, especially on that scale.