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, if you have an array of stars with $5$ rows and $3$ columns ($5 \times 3 = 15$), it contains the same number of stars as an array with $3$ rows and $5$ columns ($3 \times 5 = 15$).

Associative Property (Grouping Property): When multiplying three or more numbers, the way in which the numbers are grouped does not change the final result. Imagine three boxes, each containing $2$ sets of $4$ marbles; whether you calculate $(3 \times 2) \times 4$ or $3 \times (2 \times 4)$, the total count of $24$ marbles remains the same.

Multiplicative Identity Property (Property of 1): Any number multiplied by $1$ gives the number itself as the product. Visually, if you have $7$ baskets and each basket has $1$ apple, you have a total of $7$ apples ($7 \times 1 = 7$).

Multiplicative Property of Zero: Any number multiplied by $0$ is always $0$. If you have $5$ empty plates (plates with $0$ cookies), the total number of cookies is $0$ ($5 \times 0 = 0$).

Distributive Property of Multiplication over Addition: This property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products together. For example, to find the area of a large rectangle split into two smaller ones, you can calculate the total width first or find the two smaller areas and add them: $5 \times (10 + 2) = (5 \times 10) + (5 \times 2)$.

Terms of Multiplication: In a multiplication sentence like $12 \times 3 = 36$, the numbers $12$ and $3$ are called 'factors', and the result $36$ is called the 'product'. You can visualize this as 'Group Size' $\times$ 'Number of Groups' $=$ 'Total'.

Multiplication by 10, 100, and 1000: When a number is multiplied by $10$, $100$, or $1000$, we simply write the number and place one, two, or three zeros respectively to its right. For instance, $45 \times 100$ results in $45$ followed by two zeros, which is $4500$.