Go to the Making Learning Fun website (see Resources) and print out the engineer hat template onto thin, bendable, white cardboard. Give your child a blue pencil, or crayon, and tell him to color in the hat with blue stripes and leave the "Engineer" badge white. Cut out the hat with scissors and then cut a 2 1/2-cm-wide piece of cardboard that will fit around the back of your child's head to hold the engineer hat in place. Stick this strap to either side of the colored-in template with sticky tape where the band juts out slightly. Alternatively, draw the front outline of an engineer hat onto a blank sheet of cardboard and have your child color that. Draw an engineer badge on the middle of your hat and extend a strap that runs between the cap and peak either side of the outline for you to attach the long 2 1/2-cm-wide strap of cardboard to.
Find a cheap painter's cap. You may even find one free if you ask family and friends. With your child, paint or draw blue stripes onto the white cap. Leave white gaps of the same width as the blue stripes. Use waterproof child paints or a felt tip pen for the blue stripes.
Buy a white plastic bowl that will fit your child's head at a garage sale or thrift shop. Place the bowl upside down on white construction paper and draw a third of the way around it with a pencil. Remove the bowl and draw a cap peak shape out from the arc you have already drawn. Cut this out and stick it to the bottom of the upside down bowl with sticky tape so that it sticks out at a 90-degree angle. Have your child help paint the hat with blue stripes.
Inflate a balloon to roughly the size of your child's head and tie it off. Get your child to rip up a newspaper into small pieces. Help your child dip the newspaper pieces in white art glue and cover the top half of the balloon with it. When the layer of papier-mache is about 1 mm thick, let it dry then add another layer. When the second layer is dry, pop the balloon and trim around the bottom of the remaining papier-mache bowl shape. Cut a brim shape out of cardboard and stick this to the bottom of the upside-down bowl shape with sticky tape so that it juts out at a right angle. With your child paint the whole structure white and when it dries cover it with blue painted stripes.