Multiplication - Properties of Multiplication

Commutative Property (Order Property): This property states that changing the order of the factors does not change the product. For example, $4 \times 5 = 20$ and $5 \times 4 = 20$. Visually, if you have an array with 4 rows and 5 dots in each row, it contains the same total number of dots as an array with 5 rows and 4 dots in each row.

Associative Property (Grouping Property): When multiplying three or more numbers, the product remains the same regardless of how the numbers are grouped using parentheses. For example, $(2 \times 3) \times 4 = 6 \times 4 = 24$, and $2 \times (3 \times 4) = 2 \times 12 = 24$. Visually, this is like calculating the total items in multiple identical boxes by either grouping the boxes first or the items inside first.

Multiplicative Identity Property (Property of 1): Any number multiplied by $1$ results in the number itself. For instance, $15 \times 1 = 15$. Visually, this is equivalent to having 1 group containing 15 items or 15 groups containing 1 item each.

Zero Property of Multiplication: The product of any number and zero is always zero. For example, $9 \times 0 = 0$ and $0 \times 125 = 0$. Visually, imagine having 10 empty baskets; since every basket has 0 items, the total number of items is 0.

Distributive Property of Multiplication over Addition: This property allows you to split a large multiplication problem into two smaller, easier problems. It states that $a \times (b + c) = (a \times b) + (a \times c)$. Visually, you can find the area of a large rectangle by splitting it into two smaller rectangles and adding their areas together.

Multiplication Terms: It is important to know the names of the parts of a multiplication sentence. In $8 \times 2 = 16$, the number $8$ is the Multiplicand (the number being multiplied), $2$ is the Multiplier (the number of times it is multiplied), and $16$ is the Product (the result). Both $8$ and $2$ are also known as Factors.