If you look at the first photo you will see that our sliding door from our house is not too far from the pool. All of our cats are strictly indoors, but some of them want to explore outside. To get out, many times they will dart out of the door in almost a blind run and nearly end up in the pool. Many pets will "panic swim" if they end up in the pool. This is where they flap their feet basically trying to walk on the water rather than swim through it. Often times they get worn out and drown.
I've got six cats and Sassy is the only one that will dart out of the door and is the only one I feel needs swimming lessons. She also happens to be a very adventurous kitty who I was sure wouldn't have problems once she got use to the water.
Based on some comments I will add a couple pieces of information here.
1. The pool has a solar heater and was around 90 degrees F.
2. The chlorine levels in the pool are maintained to just above drinking water.
3. Sassy had been around the pool for many months prior to this and so wasn't really afraid of it anymore.
4. I changed to title to better reflect what is actually happening. I'm not teaching the cat to swim, but simply giving her a safe and calm environment to develop what should come naturally to her.
Step 1: Safely get the cat into the water.
Hold the cat comfortably in your arms and do whatever you can to keep the kitty calm. Walk backwards in to the water so the cat cannot see you entering the pool.