This one is one of the most space-consuming, but is also one of the easiest, as it only requires you to know your tables up to 9x9. This makes it especially useful for KS2 or less-able KS3 students (age 9+)
The rest is adding.
Step 1: The Grid Method.
To multiply two numbers together, the numbers are first broken down into their component place-value chunks.
For instance, let us multiply 47 by 68.
"47" is actually "40 + 7" and "68" is "60 + 8".
These numbers are written into a grid, as in the illustration below:
Step 2: Multiply the Rows and the Columns.
Ignoring the zeros, multiply the digits at the top of the columns with those at the left of the rows.
4x6 = 24
4x8 = 32
7x6 = 42
7x8 = 56
Now we add the zeros back on - 24 gains a zero from the 40, and one from the 60, so becomes 2400.
Similarly, 32 becomes 320 and 42 becomes 420.
Step 3: Adding.
All you have to do now is add up the four numbers in the grid. Remember to be careful about place value, and align them up to the right.
Step 4: You Want More??
It is possible to use it to multiply more than two numbers, but you need to work them out as you go along (for example, 23x46x17 would need you to work out 23x46 and then multiply that result by 17).
You are not just limited to two-digit numbers - here are a pair of three-digit numbers worked out on a scrap of paper.