When my kids were younger, we would go to picnics and it would always end up that one of the 4 would end up with their food on them, the paper plates would give a way. So I started looking for anything that would work to hold their paper plates, in a pinch.
I found that a lid from a plastic gallon of ice cream was my answer. It fits the paper plates perfectly and the kids didn't wear their dinner.
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A Frisbee will add support to paper plates when the plate is place inside the Frisbee.