A lot of the Trachtenberg Speed Math System involves multiplication. In the Trachtenberg system, multiplication is split into several sections.
The first section is Basic Multiplication which involves multiplying a number by the numbers zero to twelve. For someone who is poor in math or a child who has not yet learnt their multiplication tables then this is the section to start with. The Trachtenberg system of Basic Multiplication does not require the multiplication tables as the answer is calculated by mainly using addition and subtraction and following a set of rules.
The commutative property of multiplication means that the order you multiply is not important so for those who need to learn the multiplication tables, like the one pictured above, you only need to learn the numbers I have left in. The numbers I have removed are those that are repeated where the order of the two factors is swapped. Also, you may notice that when going down the table when you get to the square of the number you turn right to continue the series for that digit. I have highlighted the series for multiplying by 5 in yellow. So there are not as many numbers to learn in this table as you may have first thought.
The Basic Multiplication enables those who have not yet mastered the multiplication table above to be able to not only multiply any pair of numbers in the table but also multiply even much larger numbers by zero to twelve. However, eventually, you will need to know the tables to be able to move onto direct multiplication.
The next section is referred to as direct multiplication and can be used when multiplying a number by another number. It is the same as the multiplication that you would normally do but instead of writing the numbers one above the other and requiring possibly several lines for the answer, depending on how big the multiplier is, the direct multiplication method has the equation written on one line and the answer is written directly below the multiplicand.
What makes the direct method faster than the standard way to multiply is that although you still do the same amount of calculations, most are done mentally without writing anything down except the answer. The layout of the equation is also important and is what makes following the method easier than it would be than if you wrote the numbers being multiplied one above the other.
Although you can mentally do most of the work, in use it does become difficult to use once the multiplier is larger than 4 or 5 digits as the numbers you may need to mentally add together get rather large. This was something Jakow Trachtenberg also realized and he looked for a better way.
The last section is referred to as speed multiplication, the "two-finger" method or "unit-tens" multiplication, which was the result of Jakow looking for an improvement to the direct multiplication. This method is a little more advanced than the direct method but it does greatly reduce the size of the numbers you need to mentally add together even when multiplying by multipliers of more than 5 digits in length.
You can learn about both the Direct multiplication and Two Finger Multiplication methods in How to Multiply Two-Digit Numbers.