Put a colander or strainer in a bowl. Pour cherries into the strainer, saving the cherry juice in the bowl.
Put one whole cherry in each paper cup. Put the remaining cherries on a cutting board. With a sharp knife, carefully cut cherries into small pieces. Have an adult show you how to use the knife.
Put chopped cherries, lemonade concentrate, water and the juice your saved from the jar of cherries in the container of an electric blender or food processor. Cover blender or processor. Puree or blend until smooth. Do not put a spoon or spatula in the blender while it is running and keep your hands clear of the working parts.
Fill paper cups with equal amounts of cherry mixture. Freeze 30 to 40 minutes, or until very slushy.
Place popsicle sticks in the center of each cup. Freeze 1 hour longer, or until firm. To serve, peel off paper cups.