Combine sugar and 1 cup of the water in a medium (2- or 3-quart) saucepan.
Bring to a boil over medium heat. Stir until all the sugar is dissolved. Remove from heat and cool to room temperature. Refrigerate this mixture (simple syrup) while you make the rest of the lemonade.
Juice the lemons over a strainer, until you have 1 cup of fresh lemon juice.
Add the lemon juice to the remaining 5 cups of cold water. Stir in the refrigerated simple syrup (sugar and water mixture).
Garnish with lemon slices or fresh mint, if desired.