A Sleeve Protector is an object that is applied to the wrist and/or elbows to keep the sleeves of a shirt from getting dirty while doing desk work. Granny herself even used it to aid her in putting down a fight between Black☆Star & Kilik.
As the trio prepare to do work, Anya questions Granny on what was on her sleeve. Granny explained what the sleeve protector was and why it was used. It was then she overheard Kilik and Black Star argue on who's taking a quest. She promptly puts one on and secures it on her elbow. She then intervenes, knocking both of the students out with one move.