To multiply 2-digit numbers, have kids begin by splitting up the numbers into expanded notation - hundreds, tens and ones. For example, to do 38 x 7, think of 38 as 30 and 8. Now multiply those numbers by 7. So, 30 x 7 = 210 and 8 x 7 = 56. 210 + 56 = 266, therefore 38 x 7 = 266. The picture above shows this thinking symbolically.
Check out my YouTube video below for a full explanation of this method, as well as the Partial Quotients method of dividing large numbers, which I explain below.
To begin the Partial Quotients method of division, I like to start with a quick list of multiplication facts, which will lessen the mental load I'm carrying while I do the division. So if I'm dividing by 3, I make a list of the 3x facts that can help me - specifically, 3x1, 3x2, 3x5 and 3x10. The example in the picture above shows the facts of 42 that I can draw on to solve 687÷42. Next I think about how many groups of 42 I can take from 687. I use the facts I recorded to check. I see that I can take 10 groups away, so I record that, and take the 10 groups away. Now I still have 267, so I can take 5 groups of 42 away from that, which is 210. I still have 57, so I take one more group of 42 away and I have 15 left over. That means I have taken away 16 groups of 42 and there is a remainder of 15. Therefor, 687÷42=16 r15. Easy peasy.
As I mentioned with the multiplication strategy above, introduce this method before you show students the traditional method of "long division." If you introduce the standard algorithm too early, there is the danger of kids learning the "dance steps" without understanding the concept. This is setting them up for failure down the road. I encourage you to watch the video, as the method is easier to understand when you see it in action. Good luck!