Just thought I'd tell you, you should do 100% water changes rather than 50% a week. It's not quite enough on an unfiltered tank because there's nothing to convert the ammonia your betta will make. If it's a 1G bowl, you should change the water totally every 4 days or so. If it's a 2G, change it about once a week. There's not really a point in small water changes for a betta unless their tank is filtered.
And don't worry about shock from all-new water, it doesn't seem to bother them. Just get a little cup, scoop the fish into it, rinse the bowl out with hot water then fill it with room-temperature water (or as close as you can get by feel). Float the cup in the bowl until the temps are equal (doesn't take long with the small amount of water in the cup) and pour him back in.
Hope this helps!