This is the list of properties that students need to know how to use. These mostly apply to algebraic expressions (8.14), multi-step equations (8.17), and multi-step inequalities (8.18). You'll also use them for some other units (e.g., volume & surface area, Pythagorean Theorem, etc.).
These properties apply to expressions, equations, and inequalities. There are other properties that don't apply to expressions because they require doing something to both sides of an equation or inequality.
Commutative (change order)
Addition: a + b = b + a
Multiplication: a*b = b*a
Associative Property (grouping symbols)
Addition: (a + b) + c = a + (b + c)
Multiplication: (a*b)*c = a*(bc)
Distributive property:
a*(b + c) = ab + bc
a*(b – c) = ab – ac
Identity (number stays the same):
Addition: a + 0 = 0 + a = a
Multiplicative: 1*a = a*1 = a
Inverses (numbers that combine with other numbers and result in "identity elements" of 0 for addition and 1 for multiplication):
Addition: a + (–a) = 0 or (–a) + a = 0
Multiplication: a * 1/a = 1 or 1/a * 1 = 1
Multiplicative Property of zero
a*0 = 0 or 0*a = 0
Substitution property:
Given x = 3, then 3 can be substituted for x in an expression, equation, or inequality.
Example: "2x + 4" becomes "2(3) + 4"
These properties apply to equations and inequalities but not expressions:
Addition property (add something to both sides):
Given a = b, you can also do a + c = b + c
Given a > b, you can also do a + c > b + c
Subtraction property: (subtract something from both sides)
Given a = b, you can also do a – c = b – c
Given a > b, you can also do a – c > b – c
Multiplication property (multiply both sides by the same factor)
Given a = b, you can also do a*c = b*c
Given a > b, you can also do a*c > b*c
Division property (divide both sides by the same factor):
Given a = b, you can also do a/c = b/c
Given a > b, you can also do a/c > b/c
When solving the Pythagorean Theorem, you can also take the square root of both sides. This is not always going to be a perfect square. Example:
a^2 = 9
√ (a^2) = √ (9)
a = 3
The direction of the inequalitysign changes when multiplying or dividing by a negative number.
Multiplication Example:
8 > 4
8(–1) > 4(–1)
–8 > –4 This is not true
–8 < –4 Switch the direction of the inequality sign & now it’s true.
Division Example:
8 > 4
8/(–2) > 4/(–2)
–4 > –2 This is not true
–4 < –2 Switch the direction of the inequality sign & now it’s true.
BrainPop
Virtual Nerd
Substitution Property (for variables)
Videos on Properties (Art of Problem Solving)
Lessons:
Topics:
2.K.1-5 Properties
3.N.1-11 Properties
4th Grade
4.B.6 properties of addition
4.D.24 properties of multiplication
4.D.25 distributive property (find missing factor)
4.D.26 multiply with distributive property
4.E.5 properties of division
4.E.9 divide using distributive property
5th Grade
5.B.6 properties of addition
5.B.7 adding using properties
6.Y.11-16 properties
7.R.10-13 properties
8th Grade
8.V.12-14 properties
8.W.5 properties of equality
8.W.New! Solve equations with the distributive property
A.H.1-5 Properties