First, the child must learn how to properly handle age-appropriate scissors. Teach him to always hold the scissors with his thumb toward the ceiling with his dominant hand. Let the child decide which hand feels more comfortable and buy left-handed scissors if the need arises. Learning to hold the paper and the scissors at the same time may overwhelm the child, so it's more manageable for him to cut while an adult holds the paper steady. Encourage the child to use only his dominant hand rather than both. It may also help to start on smaller scraps of paper rather than a full sheet.
Use a green piece of construction paper and have the child snip a fringe to create grass for a jungle or farm scene. Creating the fringe requires only one cut at a time and helps the child become accustomed to opening and closing the scissors without making the learning curve too steep. You may also have her cut corners from other colors of paper and use those small cuttings to create animals for the scene. Encouraging children to play with different sizes, shapes and colors builds confidence in their abilities and provides additional learning opportunities.
Fold several pieces of printer paper in half and draw half of a shape on each. Have the child cut along the lines you have drawn and open the papers to reveal the shapes. Create hearts, squares, circles and more. Have the child identify the shapes and talk about symmetry. Creating something through cutting gives the child a strong sense of accomplishment. Once the child gets the hang of this, give him a folded piece of blank paper and have him cut a shape the best he can.
Standard coffee filters make the project of creating snowflakes easy to manage and very inexpensive. Take a white coffee filter and fold it to form a half circle. Then fold it in thirds to create a small cone shape. Have the child snip various sized shapes in the cone, being sure to snip the tip off, and then open to reveal the finished snowflakes.