I would sure like to see a picture if you have one.

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The flat pad is 9" long and 6" wide -- one of the smaller ones from the parent plant.

Whoa! None of those are Opuntia ficus-indica. The middle one is an Opuntia, but a different one. I'm pretty sure that the other pieces are monstrose cholla segments (Cylindropuntia fulgida). I used to pick pieces like those up in alleys when I lived in Tempe. The only thing that makes me wonder is that the chollas I found were usually still somewhat spiny.

Bingo! pasimachus it looks like they are cylindropuntia fulgida var. mammillata. The C. fulgida are way more spiny than the ones I found. But pictures I found on-line of the var. mammillata are an exact match. Thank you for your input. I always like to know what I am growing!