Foam board is a type of lightweight panel most often used as a form of backing for projects that require something lighter, cheaper or more flexible than classic panelling materials. The core will typically be composed of polystyrene (more commonly known as styrofoam) and surrounded on either side by another type of plastic or paper.
When discarding foam board, you should consider the materials it is composed of and their properties. There is an easy way to figure out whether or not what you have is recyclable. Just look for the recycling trademark logo on the product or packaging and locate the number in the centre. For most foam board, the number will be 6--signifying polystyrene plastic, or styrofoam.
Once you know the recycling number, go to the website for your local recycling centre or give them a call to determine whether or not they accept materials with the corresponding code. In rural areas or smaller cities, the option to recycle certain materials may not be available. In these cases, the option to reuse your foam board is available.
Styrofoam has been suggested as useful material for insulation and can also serve as packaging if broken down (such as the plastic peanuts you often see in shipping boxes). Also, there are mail-in programs that accept these types of materials to be converted into recycled packaging materials.