Early in math education, children are taught the order of operations. The basic rule of order of operations states that multiplication and/or division comes before addition and/or subtraction when solving an equation. Knowing this helps students know how to approach solving an equation. Multiplication also has its own rules of thumb that can help students simplify the equation-solving process.
The commutative property dictates that one number ("x") multiplied by another number ("y") will equal the same regardless of the order of the numbers. For example, 3 * 5 = 15 and 5 * 3 = 15. This is true even if there is a larger string of numbers. For example, 3 * 2 * 5 = 30 and 2 * 5 * 3 = 30. In variables, the commutative property can be written as: x * y = y * x.
The associative property pertains to numbers being multiplied to numbers set inside a parenthesis. Using variables, the property states that (x * y) * z = x * (y * z). For example: (2 * 1) * 3 = 6 and 2 * (1 * 3) = 6. The only thing that the parenthesis change in this type of equation is what order things are multiplied in. The numbers inside get multiplied together first, then the outside number. But, as the commutative property showed, the order of multiplication doesn't change the outcome.
Identity Property and Zero Product Property
Perhaps the easiest property to remember, the identity property merely states that any number multiplied by 1 equals the number: x * 1 = x. For example, 3 * 1 = 3. Whatever the number is, including negative numbers or fractions, multiplying it by 1 just results in the same exact number.
Conversely, multiplying any number by 0 just results in 0: x * 0 = 0. For example, 3 * 0 = 0. This is called the zero product property.
The distributive property is the hardest multiplication property because it also involves addition. Using variables, it states: x * (y + z) = xy + zy. For example: 3 * (2 + 5) = 21 and 3 * 2 + 5 * 3 = 21. Remember that the order of operations states that multiplication gets done before adding. If it makes it easier for you to visualize, place the multiplied portions of the equation in parenthesis, like so: (3 * 2) + (5 * 3) = 21.