a = a

If something is equal to its identical twin, you have used the

Reflexive Property

a = b & b = a

If something flipped sides of the equal sign, you have used the

Symmetric Property

a = b, c = b so     a = c

If two items are equal to a third item, the two are equal, you have used the

Transitive Property 

a+b = b+a

If you reversed the order of addition or multiplication you have used the

Commutative Property

a+(b+c) = (a+b)+c

If you changed a grouping rearranged parenthesis, but kept everything else in the same order, you have used the

Associative Property

If a=b then a+c = b+c

If you added the same nonzero # to both sides, have used the

Addition Property

If a = b ac = bc

If you multiplied the same nonzero # to both sides you have used the

Multiplication Property

a + 0 = a

If you added 0 to get the same # back, you used the

Additive Identity

a _* 1 = a

If you multiplied by 1 to get the same# back, then you have used the

Multiplicative Identity

a + (-a) = 0

If you added opposite #’s and ended with 0, you have used the

Property of Opposites

b _* ¹/_b=1

If you multiplied by a reciprocal to get 1, you have used the

Property of Reciprocals

a(b+c) = ab+ac  qr+rs = (q+s)r

If you multiplied a # into or pulled a # out of parenthesis, you have used the

Distributive Property

a _* 0 = 0

If you multiplied by 0 and got 0, you have used the

Multiplication Property of 0

w _* (-1) = -w

If you multiplied by (-1) and got the opposite of what you started with, you have used the

Multiplicative Property of (-1)

If you have stated that a<b, a=b or a>b, you have used the

Comparison Property

a < b, c is +, then   ac < bc

If you multiplied an inequality by a positive # and maintained the inequality, then you have used the

1^st Multiplication Property of Order

 a < b, c is –, then   ac > bc

If you multiplied as inequality by a negative # and reversed the inequality, then you have used the

2^nd Multiplication Property of Order

a+c = b+c then     a = b

If you cancelled the same quantity from both sides of an equation (by subtracting) you have used the

Cancellation Property of Addition

ac = bc so a = b

If you cancelled the same nonzero quantity from both sides of an equation (by division) you have used the

Cancellation Property of Multiplication

ab = 0 iff a = 0 or    b = 0

If a product is zero, so you know that one of the factors has to be zero, you have used the

Zero Product Property