Collage is the easiest and most straightforward of cutting and gluing activities. Gather a pile of old magazines, the more colorful the better. National Geographic is good for this activity and is available nearly everywhere in charity shops or at yard sales. Encourage children to consider how and why they are juxtaposing images, and help them to build a basis of conscious composition in their art. Guiding children as they use materials and images is a good way to get them interested in reading and in the world around them. Don't hesitate to postpone the activity in favor of reading an article if the child shows an interest.
Combining cutting and gluing with a child's imagination and a pile of paper and cardboard can lead to wonderful costumes. Kids can make armor and monsters from cardboard boxes and fantasy outfits from modified paper. By cutting newspaper into thin strips and gluing them together to make a fringe, you can make anything from a fringed jacket to a sheepdog costume. Cardboard tubes are good for robot costumes as arms and legs, with sturdy cardboard boxes for head and torso.
This is a project that children will want to show their friends. Cut a strip of paper about 2 inches wide and about 20 inches long. Lay it out on a flat surface. Pick up both ends, and bring them together to make a loop. Turn one end of the strip over, and glue the piece of paper into a loop so that it has half a turn in it. This is a Mobius Strip. Now, run a pencil all the way down the center of the strip, until you come back to where your pencil line started. You will find that the pencil line covers both sides of the loop because it only has one side. Amaze the children by cutting down the center of the loop with a pair of scissors. Cut all the way around until you have two loops. But, in fact you won't have two loops, you will only have one because it's a Mobius Strip.