Algebra 1 Properties

Does commutative work with:

• Addition?

• Multiplication?

• Subtraction?

• Division?

❌ Does NOT work with subtraction or division.

What does commutative mean?

Commutative = Change the ORDER

Changing the order of numbers does not change the answer for addition or multiplication.

This is an example of:
a + b = b + a

Example:
3 + 7 = 7 + 3

Commutative Property - This illustrates how the order of addition does not affect the sum.

This is an example of:
ab = ba

Example:

4 × 5 = 5 × 4

Commutative Property - This illustrates how the order of multiplication does not affect the product.

What does associative mean?

What changes?

Associative = Change the GROUPING

Changing the grouping using parentheses does not change the answer.

Remember: Associative changes GROUPING, not order.

This is an example of:

Addition:
(a + b) + c = a + (b + c)

Example:
(2 + 3) + 4 = 2 + (3 + 4)

5 + 4 = 2 + 7

9 = 9

Associative Property - This demonstrates how changing the grouping of addends does not alter the sum.

This is an example of:

Multiplication:
(ab)c = a(bc)

Associative Property - This illustrates that changing the grouping of factors in multiplication does not affect the product.

Distributive Property

What do you do when a number is outside parentheses?

Distribute = Multiply the outside number by EVERY term inside.

a(b + c) = ab + ac

This is an exampe of:

Example:

3(x + 4)

= 3x + 12

The 3 is multiplied by both x and 4.

Distributive Property - The Distributive Property states that multiplying a number by a sum is the same as multiplying that number by each addend in the sum individually and then adding the results.

This is an example of:

With subtraction:

5(x − 2)

= 5x − 10

Distributive Property when using subtraction. This shows that a number is multiplied by each term inside the parentheses, including negatives.

Identity Property

What numbers can you use without changing the original number?

An identity leaves a number unchanged.

Additive Identity

Adding 0 changes nothing.

a + 0 = a

This is an example of:
8 + 0 = 8

Addition Identity Property

+ 0 → stays the same

Identity Property

What numbers can you use without changing the original number?

An identity leaves a number unchanged.

Multiplicative Identity

Multiplying by 1 changes nothing.

a × 1 = a

This is an example of:

8 × 1 = 8

Multiplication Identity Property

× 1 → stays the same

Inverse Property

What happens when you combine a number with its opposite or reciprocal?

An inverse brings a number back to an identity.

Additive Inverse

A number plus its opposite equals 0.

a + (−a) = 0

Example:
7 + (−7) = 0

Inverse Property

What happens when you combine a number with its opposite or reciprocal?

Multiplicative Inverse

A number multiplied by its reciprocal equals 1.

a × 1/a = 1

Example:
5 × 1/5 = 1

Multiplication Property of Zero

What happens when you multiply any number by 0?

Any number multiplied by 0 equals 0.

a × 0 = 0

Anything × 0 = 0

This is an example of:

12 × 0 = 0

Multiplication Property of Zero

This is an example of:

−5 × 0 = 0

Multiplication Property of Zero

This is an example of:

x × 0 = 0

Multiplication Property of Zero

Reflexive Property

What is a number or quantity always equal to?

A quantity is always equal to itself.

a = a

This is an example of:

5 = 5

Reflective Property

This is an example of:

x = x

Reflective Property

This is an example of:

AB = AB

Reflexive Property

Symmetric Property

If a = b, what can you conclude?

You can flip the equation.

If: a = b

Then: b = a

Example: If x = 10, then 10 = x.

This is an example of:

If x = 10, then 10 = x.

Symmetric Property of Equality

Transitive Property

If a = b and b = c, what can you conclude?

If two quantities are equal to the same quantity, they are equal to each other.

If: a = b and b = c

Then: a = c

Example:

If x = 5 and 5 = y then x = y.

Substitution Property

If a = b, what can you do?

You can replace one quantity with an equal quantity.

If: a = b

then a can replace b in an expression.

Example:

If x = 4: 2x + 3

Replace x with 4:

2(4) + 3 = 11

Commutative

Commutative → ORDER

Change the order.

3 + 5 = 5 + 3

Associative

Associative → GROUP

Change the grouping.

(3 + 5) + 2 = 3 + (5 + 2)

Distributive

Distributive → MULTIPLY

Multiply by every term.

3(x + 2) = 3x + 6

What is the difference between Identity and Inverse?

IDENTITY

Leaves the number unchanged.

+ 0 → stays the same

× 1 → stays the same

INVERSE

Combines with a number to produce an identity.

Opposite → 0

Reciprocal → 1

What is the difference between Identity and Inverse?

Identity = Same

Inverse = Undo

Reflexive

Reflexive → SELF

a = a

Symmetric

Symmetric → FLIP

a = b → b = a

Transitive

Transitive → CHAIN

a = b and b = c → a = c

Substitution

Substitution → REPLACE

If a = b, replace a with b.

Name the property:

4 + 9 = 9 + 4

Commutative Property

The order changed.

4 + 9 became 9 + 4

Name the property:

(2 × 3) × 4 = 2 × (3 × 4)

Associative Property

The grouping changed.

(2 × 3) × 4 became 2 × (3 × 4)

Name the property:

6(x + 2) = 6x + 12

Distributive Property

The 6 was multiplied by each term inside the parentheses.

6(x) + 6(2) = 6x + 12

Name the property:

x + 0 = x

Additive Identity Property

Adding 0 leaves the number unchanged.

Name the property:

x + (−x) = 0

Additive Inverse Property

A number and its opposite add to 0.

Name the property:

x × 0 = 0

Multiplication Property of Zero

Any number multiplied by 0 equals 0.

Name the property:

If x = 7, then 7 = x

Symmetric Property of Equality

The equation is flipped.

x = 7 → 7 = x

Name the property:

If x = y and y = z, then x = z.

Transitive Property of Equality

The equalities form a chain: x = y = z

Therefore: x = z

Name the property:

If x = 5, then: 3x + 2 can become: 3(5) + 2

Substitution Property of Equality

Because x and 5 are equal, 5 can replace x.