What were these built for? How were they used?
These tiles reflect a long-standing tradition of ceramics in the southwestern pueblos dating from 7500 B.C.E. to the present day. Historically all pottery production was for Native use but the Hopi began making tiles for decoration in the 19th century, like the ones you see in that drawer.
The designs on these tiles mirrored the abstracted motifs on their pottery. By the early 20th century, merchants and collectors passing through the region created a market for portable Native tokens. Small bowls and decorative tiles using traditional Hopi designs filled this market nicely.
How do you mend a crack like that?
Ceramics are actually quite easy to mend! A variety of glues can be used, the jagged surfaces of breaks provide a lot of surface area for adhesive.