Most production shops have a few pieces of corrugated board, cut from a shipping carton, that work very well. The really fancy ones have a cutout so they'll fit over a 6" Kurt vise, and a welding magnet or two hold the cutout in position.
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If you work on shield positioning, you can direct some chips at the people you don't like, while avoiding your boss (or the instructor)
I use a piece of 1/2" thick plywood held on edge in a 2x4 base. I'll snap a photo later. It keeps most of the chips in one area so clean up is faster.
Shields can be attached directly to lathe tooling, like the photo below, or held between the tool & the quick change block. You'll want one of these if you ever have to turn or bore brass for bushings - it sprays off the tool like (very hot) water.
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