The thing about unraveling sweaters is that the process leaves "sweater poop" all over the floor in your knitting space. This becomes socially unacceptable if you do some of your unraveling while watching your sons' swim practice. It also takes time to unravel and roll into balls.
I had this great cotton sweater that I wanted to use the yarn from in my mother's day dish towels. I didn't see myself having the time to unravel and knit. I decided to get the unraveling started. This means getting past all the weird, impossible to unravel parts that exist in the neck of some sweaters. Once I was able to get a solid strand of yarn unraveling smoothly, I began to knit while I unraveled rather than rolling the unraveled yarn into a ball first. To keep the sweater poop at bay, I kept the piece that was unraveling as I was knitting in a bag. This worked out very well. The bag is a sweater litter box keeping my floor sweater poop free. (I'd show you but that would ruin the surprise so you'll just have to wait!)