How to solve math problems without a calculator?

I'm a high school student and I have gotten used to using calculators real bad. I'm in algebra two and my teacher this year is strict compared to my other math teachers (which lets the class use calculators). My teacher this year does not allow us to use calculators and I feel like I will do bad on my tests/quizzes.  I'm not good at memorizing numbers and such so that means that I don't completely know my times and division table. However, I know how to set up algebra equations. I've just gotten to dependant on calculators. How can I make sure I pass my math class this school year?

I've noticed that I mostly have a difficult time with high numbers. Example: 8*6, 8*8, 9*7. Things like that