The first pot shard is made up of two different pots. The two halves were stitched together from found pieces. You can see how the curve of the pot is very different on the two halves. These assembled pieces are fairly common.
The second pot looks Mexican. Possibly Oaxacan native. This style has been made there for a very long time. You can get an idea of how modern it is by flicking the rim with your fingernail, if it goes *thud* it could be pre 20th century, if it has a ringing sound it's modern.
The third one appears to be a Hohokam corrugated pot. Could be anywhere from about 1200 AD to 1840's. There were many other tribes that made this type of pot, usually for bulk dry goods storage. This one looks a lot like Hohokam pots I've handled but this style of pot has been made in China, Africa, Indonesia, South and North America throughout history. If your grandmother lived in the Cave Creek area and the pot was found there it's more than likely Hohokam.
Value would be in the eye of the beholder. Not knowing where or when they were found pretty much puts them in the class with all other clay pots. If there were some culturally identifying designs or a document about where they were found the value would be considerably enhanced.
You might consider calling the Heard Museum in Phoenix. They used to have a program to help identify objects from peoples personal collections.