Do you not get paid by the hour? We have a calendar on our fridge and our nanny writes down when she arrives (sometimes it varies, although it's generally the same time) and when she leaves (same thing). Then at the end of the pay period we look to see if she worked the 8 am - 6 pm every day that we expected or if she came in or left earlier or later on any day. No discussion, no argument, it's just taken care of. 30 minutes late is unacceptable. You should absolutely be compensated for that time.