Can you become pregnant on birth control

I've been with my boyfriend for six months now and we've been having sex for the past two months. To be safe I wanted to go on birth control. I told my doctor that I have cramps and that I'm sexually active. But only told my mom the cramps part. I have a gynocologist appointment next week, the doctor will be putting me on birth control. However do some birth controls not project against getting pregnant(the real reason I'm getting it) I know it sounds crazy but I want to be able to have unprotected SAFE sex. I still however only having unprotected sexual during my non fertile periods. Also my boyfriend and I always have sex while I'm on my period. Is it possible for me to become pregnant while taking the sugar pills during my period?