Properties of Addition and Multiplication: Practice

The same two numbers are added, but in the opposite order, and the total does not change.

$6 + 9 = 15 = 9 + 6$

Changing the order of the inputs is the commutative property.

The numbers $2$, $5$, and $8$ stay in the same order, but the parentheses move, so the grouping changes.

$(2 + 5) + 8 = 15 = 2 + (5 + 8)$

Changing which pair you add first is the associative property.

One is the multiplicative identity, so multiplying by it leaves the number unchanged.

$58 \times 1 = 58$

The additive identity is the number that leaves any value unchanged when added.

$a + 0 = a$

That number is $0$.

The factors $3$, $4$, and $5$ keep their order; only the grouping changes.

$(3 \times 4) \times 5 = 60 = 3 \times (4 \times 5)$

Changing the grouping is the associative property.

Addition is commutative, so the order of the two numbers does not matter.

$2 + 7 = 9 = 7 + 2$

Subtraction and division are not commutative, and $6 \times 0 = 0$ (not $6$), so only this equation holds.

Multiplying by $1$ leaves the number unchanged.

$14 \times 1 = 14$

This is the multiplicative identity property.

Commutativity swaps the order of the same two numbers.

$8 + 12 = 12 + 8$

So the blank is $8$.

Multiplying any number by $0$ gives $0$.

$135 \times 0 = 0$

This is the zero property of multiplication, not the identity property (which uses $1$).

Only addition and multiplication are commutative. A quick check shows the others fail.

$5 - 3 = 2, \qquad 3 - 5 \ne 2$

Reversing $5 - 3$ asks you to take $5$ from just $3$, which cannot give $2$, so subtraction is not commutative. That leaves "addition is commutative" as the true statement.

The commutative property reverses the order of the same two numbers.

$12 + 7 = 19 = 7 + 12$

The other equations show the identity, identity, and zero properties in turn.

The zero property states that any number times $0$ is $0$.

$5 \times 0 = 0$

The first equation is the additive identity, and the second is the multiplicative identity.