Hi Melissa
It is funny how people get a little persnickety about certain details. We all do! It sometimes is even worth it. Other times it is wasted energy.
I want to give you a different spin on it. If you were at a regular class of yoga you would turn your body to the side and step out along the length of your mat whenever you did a warrior pose. In fact you would step out so that at all times your base would be your mat.
If you have a knee problem or a hip problem then a slippery floor could be so dangerous. It seems you are aware of the danger.
It could be fixed by a technique modification. I would have to drill down a little more to work out the best out of the few approaches I could offer you.
By the way I don’t consider you to be cheating. When you do step out along your mat instead of across it, you are actually invited to explore your relationship with the mat. Here’s something else, at my home studio we actually perform the whole standing series with the towel across the mat. We have carpet and we find that the room will positively stink if we sweat on the floor, but with the towel cross ways the carpet stays fresh and clean and we take our sweat away from the room and we wash it away soon after. I am not against using a towel the other way. I am against towel adjusting for particular poses, or fixing or smoothing out a towel because it bunches up.
Your teacher’s admonishments are not appropriate especially since you slip on the rubbery floor. I could ask you whether others have a problem slipping as well. Your state being ruined is your own responsibility however and as hard as it is to totally ignore that, you really have to. You are not being disrespectful, I should think it is the other way around. In my opinion if she can’t offer you the tools to fix that (technical points, modifications to your own technique or even using a prop) then she should leave you alone.
It could be that she is noticing you changing your towel just before triangle. Why don’t you start and then keep your towel that way from the beginning of class and only change it when you get to the floor? There are no rules about having to do it a certain way.
Please go and look at all the other tips on triangle because there are some valuable tips on how not to slip. One of those is here: Slipping and Sliding Feet and another is here: What Should I Try To Be Accomplishing In This Pose. It is possible that I can give you another tip but I want you to see if these (and those in the other threads) work first.
It is possible that for the moment you could benefit from not lunging so far down (see second link for details) by not stepping out as far which means thigh not quite parallel to ground. Keep the shin vertical. If this still doesn’t give you a feeling of safety then get back to me. I have other great tips to share.
Namaste
Gabrielle