Referring to a mathematical operation that yields the same result regardless of the grouping of the elements.
The Four Arithmetic Operations
Addition is the most basic operation of arithmetic. In its simplest form, addition combines two quantities into a single quantity, or sum. For example, say you have a group of 2 boxes and another group of 3 boxes. If you combine both groups together, you now have one group of 5 boxes. To represent this idea in mathematical terms:
Subtraction is the opposite of addition. Instead of adding quantities together, we are removing one quantity from another to find the difference between the two. Continuing the previous example, say you start with a group of 5 boxes. If you then remove 3 boxes from that group, you are left with 2 boxes. In mathematical terms:
Multiplication also combines multiple quantities into a single quantity, called the product. In fact, multiplication can be thought of as a consolidation of many additions. Specifically, the product of $x$ and $y$ is the result of $x$ added together $y$ times. For example, one way of counting four groups of two boxes is to add the groups together:
However, another way to count the boxes is to multiply the quantities:
$2 \cdot 4 = 8$
Note that both methods give you the same result—8—but in many cases, particularly when you have large quantities or many groups, multiplying can be much faster.
Division is the inverse of multiplication. Rather than multiplying quantities together to result in a larger value, you are splitting a quantity into a smaller value, called the quotient. Again, to return to the box example, splitting up a group of 8 boxes into 4 equal groups results in 4 groups of 2 boxes:
The commutative property describes equations in which the order of the numbers involved does not affect the result. Addition and multiplication are commutative operations:
$5 \cdot 2=2 \cdot 5=10$
Subtraction and division, however, are not commutative.
The associative property describes equations in which the grouping of the numbers involved does not affect the result. As with the commutative property, addition and multiplication are associative operations:
$(4 \cdot 1) \cdot 2=4 \cdot (1 \cdot 2)=8$
Once again, subtraction and division are not associative.
The distributive property can be used when the sum of two quantities is then multiplied by a third quantity.