Associative Property of Addition

The associative property of addition shows us that when adding multiple values together, the outcome will always be the same. You can group numbers together in parenthesis, and it will still end up having the same outcome. For example, ${\displaystyle 3+(4+5)=(3+4)+5}$. The order remains the same, but the grouping has changed. The result, however, is consistent.

Associative Property of Multiplication

The associative properties work for both addition and multiplication. Think of grouping ${\displaystyle 3\times 3}$ together. You end up with 9. What about ${\displaystyle 3\times (3\times 4)}$? If we change the grouping here, the result will be the same. Try to visualize why this is. When multiplying, you're often building objects in rows and columns.

For example, a 2 inch ${\displaystyle \times }$ 4 inch block will be 2 inches across, and 4 downward. If you had a block 4 inches across and 2 inches downward, it would be the same size overall. When you put objects in parenthesis, remember to do those operations first. You may end up with certain "sides" of the operation or object being larger or smaller, but the total area will always have the same outcome.

Commutative Property of Addition

The commutative property of addition shows that no matter what order numbers are in when we add them, the result will always be the same. For example, ${\displaystyle x+y=y+x}$, just how ${\displaystyle 3+4=4+3}$. Even though the order in which we added has shifted, the result doesn't change, and the statements on both sides of the equal sign remain true.

Addition Property of Equality

The addition property of equality states that if two variables or numbers are equal to each other on each side of the equation, and the operation they go through is alike, the resulting sum will be the same. If for example, both ${\displaystyle x}$ and y are 6, and you add ${\displaystyle z}$ to each of them, that is,

if ${\displaystyle x=y}$ then ${\displaystyle x+z=y+z}$

or if ${\displaystyle 6=6}$ then ${\displaystyle 6+z=6+z}$

Subtraction Property of Equality

The subtraction property of equality states that if two variables or numbers are equal to each other on each side of the equation, and the operation they go through is alike, the resulting difference will be the same. If for example, both ${\displaystyle x}$ and y are 6, and you subtract ${\displaystyle z}$ from each of them, that is,

if ${\displaystyle x=y}$ then ${\displaystyle x-z=y-z}$

or if ${\displaystyle 6=6}$ then ${\displaystyle 6-z=6-z}$