Division - Division as Equal Sharing and Grouping

Division as Equal Sharing: This concept involves distributing a total quantity of items equally into a specific number of groups to find out how many items each group receives. Visually, imagine you have $12$ apples and $3$ baskets; you place one apple into each basket, then a second, and then a third, until all apples are gone. Each basket ends up with $4$ apples, which means $12 \div 3 = 4$.

Division as Equal Grouping: This process involves taking a total quantity and splitting it into groups of a fixed size to find out how many groups can be formed. For example, if you have $20$ beads and you put $5$ beads into each bag, you will see $4$ bags filled. This can be visualized as circling sets of $5$ objects from a large pile until no objects are left.

The Terms of Division: A division sentence has three main parts. The Dividend is the total number being divided ($15$), the Divisor is the number of groups or the size of each group ($3$), and the Quotient is the result or answer ($5$). In a long division visual format, the Dividend sits inside a 'house' or bracket, the Divisor sits outside to the left, and the Quotient is written on top.

Division as Repeated Subtraction: Division can be understood as subtracting the same number over and over until you reach zero. For example, $12 \div 4$ means $12 - 4 = 8$, $8 - 4 = 4$, and $4 - 4 = 0$. Since we subtracted $4$ exactly $3$ times, the answer is $3$. Visually, this is represented on a number line as a series of equal jumps backward from a starting number towards $0$.

Relationship between Multiplication and Division: Multiplication and division are inverse (opposite) operations. This can be visualized using a rectangular array of dots. If an array has $4$ rows and $6$ columns ($4 \times 6 = 24$ total dots), you can see that $24$ dots divided into $4$ rows results in $6$ dots per row ($24 \div 4 = 6$). Every multiplication fact helps you solve two division facts.

Properties of Division: There are three special rules to remember. First, any number divided by $1$ is the number itself ($7 \div 1 = 7$). Second, any number divided by itself is $1$ ($9 \div 9 = 1$). Third, zero divided by any number is $0$ ($0 \div 5 = 0$). Note that you can never divide a number by $0$.

Remainders in Division: Sometimes, when sharing or grouping, some items are left over because they cannot form a full group. These leftovers are called the Remainder. Visually, if you try to share $7$ candies between $2$ children, each gets $3$ candies and $1$ candy is left over. This is written as $7 \div 2 = 3$ remainder $1$.