When handling broken pieces of black fiberglass tissue, it's crucial to take safety precautions. Wear protective gear such as gloves, long sleeves, and a dust mask to prevent skin contact and inhalation of fibers. Use a broom and dustpan or a vacuum cleaner with a HEPA filter to collect the broken pieces. Avoid sweeping with a regular broom as it can release more fibers into the air. Dispose of the collected material in a sealed plastic bag, marking it as hazardous waste. Check local regulations for proper disposal methods, as fiberglass may require special disposal procedures.