Put the watermelon and pineapple in plastic containers and place in the freezer until completely frozen, at least 4 hours. Also, place the rum in the freezer for at least 4 hours.
Combine 2 cups of the watermelon, 1/4 cup plus 2 tablespoons of the pineapple, 1 cup rum, 1/2 cup Ginger Syrup, and 1/4 cup lemon juice in a blender. Blend on high speed until well combined. Add 2 cups ice and blend until smooth. Pour into a pitcher. Repeat with the remaining ingredients. Serve in margarita glasses or other wide-mouthed, stemmed glasses. Garnish with watermelon wedges.
Makes about 2 quarts, 8 servings
Combine the ingredients in a large saucepan over low heat. Simmer for 10 minutes, remove from the heat, and allow to cool. Once cool, strain out the ginger.
Makes about 1 quart