So I did some experimenting. The floor of the shower has the rougher surface that the soap scum clings to and it ends up being the hardest part of the shower to clean. Here is what I did. I divided the shower into three sections and cleaned one with baking soda; one with baking soda and salt; and one with Oxy clean - the dry powder.
I was looking for a way to clean the shower without too harsh chemicals but it also had to be easy on my arm. In other words, it should cut through the scum and get the job done without my having to ice my shoulder.
The winner was the baking soda and salt. Baking soda is already a natural safe scrub in itself but alone it took too much scrubbing. The addition of some table salt gave the baking soda more scrubbing power. As to the Oxy Clean, the baking soda did better than it did.
So from now on it is just a sprinkle of baking soda and a sprinkle of salt. It's easy on the environment, easy on my arm and best of it all - it is cheap.