Both commutative and distributive
Multiplication is distributive over addition
$$a\times(b+c) = a\times b + a\times c$$for example, $$3\times(2+4) = 3\times 2 + 3\times 4$$
$3\times(2+4)$ $=$ $3\times2+3\times4$
Note that $5\times 30 = 150$ follows from associativity
\begin{align} 5 \times 30 &= 5 \times (3 \times 10) \\ &= (5 \times 3) \times 10 \\ &= 15 \times 10 = 150 \end{align}I ran a Google search for "multiplication tricks" and found this page.
All the tricks there can be explained using the algebraic properties of multiplication.
Trick: Double, then double again.
Example: $4\times9$
Explanation: $4\times n = (2\times2)\times n = 2\times(2\times n)$
Trick: Multiply by 10, then cut in half.
Example: $5\times9$
Explanation: $$5\times n = \left(\frac{1}{2}\times10\right)\times n = \frac{1}{2}\times(10\times n)$$
Trick: Multiply by 10, then subtract the number itself.
Example: $9\times13$
Explanation: $9\times n = (10-1)\times n = 10\times n - n$