I wanted to put together all the methods of doing (long) mathematics, so that I have one place to go for reference. I was going to try to write something up, but there are some fantastic resources online.
Firstly times tables tricks
Traditional (add a zero to each subsequent line of multiplication)
Box method (boxes showing separated units, tens and hundreds, multiplied together then the parts added. This relies on knowing what to do with all the zeros when when multiplying hundreds by hundreds, etc)
Napier method (commonly used, boxes with diagonal lines)
Russian Peasant (I think this is magic, really! Relies on dividing the left side in half and doubling the right side... )
Super clear explanation here:
Chunking (also called Partial Quotent Method)