Does anyone know what this is?

[paul monfils]

You are correct. It does fit together with other similar pieces. Good observation! This is one of the 8 shell plates that join together to form the shell of a chiton. Chitons are a separate class of mollusks distinct from the bivalves and gastropods. They usually live attached to rocks or other hard surfaces, since their shell covers only the dorsal aspect of their body, and the soft tissues of their underside are exposed if they fall off the rocks. Some of them can curl up for protection if this happens however. The 8 shell plates are held together by a muscular band called the girdle, which encircles the outer edge of the animal. In some species it is smooth, on others bumpy or scaly or spiny. When the animal dies, this band of tissue decomposes, releasing the shell plates. They often wash up on the beach because chitons usually live in fairly shallow water.

http://www.snorkelstj.com/chitons.html