My DD has recently discovered the joys of feedi...

My DD has recently discovered the joys of feeding herself grated food. So far she's had carrot, cheese and apple. I was wondering if anyone could tell me the best way to store it? Can I freeze carrot and apple grated? Will it defrost ok or will it go yuck? (cheese isn't a problem as we buy it pre-grated so there's always some available.) We're finding its a great snack for her but I don't want to have to grate it every time I give it to her.