Puree fresh cucumber chunks with a small amount of water in a blender or Magic Bullet until well blended.
Pour cucumber pulp into a metal sieve (or coffee filter lined colander) with a bowl underneath to catch the cucumber juice.
Let drain for 10-15 minutes.
(You'll want to use 1 to 1 1/4 cups of cucumber juice in this recipe.)
Meanwhile, combine 1 cup sugar and 1 cup lemon juice in a 2 quart pitcher.
Add 5 1/2 cups water and stir well.
(Discard cucumber pulp or set aside for another use.)
Stir in 1 cup cucumber juice.
Taste test and add more water if needed.
You may add extra cucumber juice if desired.
Muddle a sprig of mint in each serving glass.
Add ice and pour cucumber lemonade over all.
Garnish with a sprig of fresh mint and cucumber and lemon slices.