The threads get messy while cutting and crocheting, but the fraying stops once the project is worked up. If you wanted to spend the time trimming all the bits from the finished project, you could... But just like cutting and joining the "right" way, that's not for me. (And wow, that would take forever! You might as well spend the time making bias strips instead.) I think the threads give it extra style, like the fraying bottoms of cut-offs that haven't been hemmed.
So, here's my work in all it's glory; I won't hide the mess from you. The ends of the strips are sloppy, threads are everywhere, and I don't even cut the material straight. But the joins are strong and smooth, plus one simple extra step will help you avoid tails popping out. It may not be as pretty as working with bias strips, but it gets the job done more quickly.
The one thing I actually do that isn't sloppy is wind the material up as I go; otherwise, I'm always left with piles of tangles. You don't have to wind your fabric yarn, but I do recommend it if you'll be making all the material before you begin crocheting.
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Is it the best way? No. But it's the way that gets it done for me. I'm not good at sewing, and I have a ton of material to cut up with no help. So when I have time, I sit on the floor with a scrap piece of plywood and my rotary cutter, and cut... And cut, and cut, and cut. It took me a whole day to cut a king-sized fitted sheet into strips. Then it took me more than a day to join them all. And I still have two more sheets to cut up for the project. No, it's not the "best" way, but if I do it any other way, I'll never get it done.
It always seems like such a great idea when I'm looking at my old sheets and wondering what I should do with them... "Yay, let's recycle and crochet something!" But after hours on the floor cutting and joining and I'm still only finished with one sheet, all I can think of is getting back to