1. Properties
Identity, Equality, Commutative, Associative, and Distributive Properties

2. Why Oh Why?!?! Properties lay the firm foundation for Algebra
Provides you with accurate information for proofs in upper level math courses
Helps you perform algebraic manipulation with confidence

3. Additive and Multiplicative
Identity Properties
Additive and Multiplicative

4. Additive Identity (Add ID)
Adding _____ to any number or variable does not change that number or variable.
Look for the addition of zero in any part of the expression or equation.
Examples:
ex. 5+0=5 ex. b+0=b
0+5= b=b
!!Notice that the addition of zero can come first or second!!

5. Multiplicative Identity (Mult. ID)
Multiplying by _____ does not change that number or variable.
Look for any part of the equation or expression being multiplied by one.
Examples:
ex. 3*1=3 ex. 5(1)=5 ex. 1a=a
1*3= (5)= a1=a
!!Notice the various notations for multiplication!!

6. Multiplicative Property of Zero (Mult. 0)
Any number or variable multiplied by zero equals zero.
Look for one side being zero and the other side having multiplication of zero in it.
Examples:
ex. 36*41*0= ex. ac(4*8*0*9b)=0
!!Notice that the *0 can come anywhere in the equation!!

7. Multiplicative Inverses (Mult. Inv)
Are two numbers whose product is one.
Another name for these is reciprocals.
Look for one side being 1 and the other side having reciprocals multiplied together.
Examples:
ex. 5*(1/5)=1 ex. (3/4)(4/3)=1 ex. m(1/m)=1
(1/5)*5= (4/3)(3/4)= (1/m)m=1
The only number that does not have a reciprocal is zero.

8. Reflexive, Symmetric, Transitive, and Substitution
Equality Properties
Reflexive, Symmetric, Transitive, and Substitution

9. Reflexive Property (reflection)
Any quantity is equal to itself (duh)
If a variable is on both sides, it holds the same value on both sides.
Examples:
ex. 2+6=2+6
ex. 5*x=5*x
!!Notice that the simplified expressions gives you the same value on both sides!!

10. Symmetric Property (like a palindrome)
If one quantity is equal to a second, then the second is equal to the first. (WOW)
Examples:
ex. If b=3, then 3=b
ex. If 10=7+3, then 7+3=10.
(They are exactly the same forwards and backwards, like a palindrome.)

11. Transitive Property
Root word transit which means to move across or through.
Takes out the “middle man”.
Examples:
ex. If a=b and b=c, then a=c.
ex. 5+7=8+4 and 8+4=12, so 5+7=12

12. Substitution Property
Most commonly used in simplification.
States that any quantity can replace its equal in any situation.
Example:
ex. If n=17, then in 4+n, n can be replaced with 17. So 4+17=21
ex. (8+3)4 = (11)4 since 8+3=11

13. Ok, Now what??
Here is an example of what these properties allow us to do:
Evaluate. Justify the steps using properties.
6(12-48÷4)
6(12-12) by Substitution
6(0) by Substitution
0 by Mult. 0

14. You Try one:
Evaluate. Justify your steps using the Properties. (remember PEMDAS) Ex. ¼(12-8)+3(15÷5-2) ¼(4)+3(3-2) by Substitution ¼(4)+3(1) by Substitution 1+3(1) by Mult. Inv 1+3 by Mult. Id 4 by Substitution

15. Test Yourself Identify the property then find the value of n.
1. 13n= =n ½n=1
Mult Add Id Mult Inv
Quick Quiz:

16. Commutative, Associative, and Distributive Properties

17. Commutative Property of Addition (Comm. +)
Commute-which means to move or travel between.
The order in which you add the numbers or variables, does not change the final answer.
Allows you to move the addends around the addition symbol.

18. Commutative Property
Examples: Ex. 2+3=3+2 Ex. a+6=6+a Ex =13 5= =13 !!Using this property will help you do mental math more easily!! Ex Commute the numbers Simplify 92

19. Commutative Property of Multiplication (Comm. *)
The order in which you multiply your factors does not change your final product.
Allows you to move your factors around the multiplication symbol.
Examples:
Ex. 2*3=3*2=6 Ex. 2x*3=2*3*x=6x
!!Notice how it helps you simplify things that look complex!!

20. Associative Property of Addition (Assoc +)
Associate-which means to group together or hang around.
The way the numbers or variables are being grouped in an addition problem can be switched.
Allows you to switch the numbers or variables inside the ( ).
Examples: (2+3)+4 = 2+(3+4)
= 2+7
= 9

21. Associative Property of Multiplication (Assoc. *)
The way the numbers or variables are grouped in a multiplication problem can be switched.
Allows you to switch the numbers or variables inside the ( ) in a multiplication problem.
Examples:
Ex.2*3*4=3*4*2 Ex. 5*a*2=2*5*a

22. Let’s watch again… Ok, see if this clears things up a bit.

23. Distributive Property
Distribute-to handout or share
Has an associated group inside ( ) with a number on the outside that needs to be “handed out” to the inside using mult.
Example:
Ex. 2(x+1) Ex. (y+4)5 Ex. -2(m+1)
Ex. 3(3-x) Ex. (y-7)(-2) Ex.

24. Here is how we use it Rewriting numeric expressions: Ex. (3+2)4
You create equivalent expressions when you rewrite. Try another one.
Ex. -5(6-3)
Use it to simplify expressions like: 5(x2+x+3) and -4(x2-2x+2).

25. Like Terms
Like terms have the same variable letter with the same exponent.
Are these like terms? 2x and x?
X3 and x2? X and x2?
Try this: 3x+4x+x
Ex. x+y+2y ex. y-2y+3 ex. 3y+4x-2xy

26. Last one…