Pre Algebra Period 1

EQUATIONS & INEQUALITIES: Properties 2-1




WHAT ARE PROPERTIES? (Why are they your friends?)

You can count on properties. They always work.
There are 0 COUNTEREXAMPLES!

COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)
Because you can count on them, you can use them to JUSTIFY what you do.

JUSTIFY = a reason for doing what you did


PROPERTIES ARE EXCEPTIONS TO AUNT SALLY:

Commutative (order) Property
a + b = b + a

ab = ba

3 + 5 = 5 + 3

3 (5) = 5 (3)

(you can HEAR the change in order!)

Aunt Sally says that you always need to go left to right, but Commutative says not necessary if you have all multiplication or all addition.



Associative (groupings) Property
a + b + c = a + (b + c)

abc = a(bc)

(3 + 2) + 8 = 3 + (2 + 8)

(Why would you want to? Sometimes it's easier!)

[57 x 5] (2) = (57) [ 5 (2) ]

(you can't hear this property! but you can SEE it!)

Aunt Sally says you must always do parentheses first, but Associative says that you can actually take the parentheses away, put parentheses in, or change where the parentheses are if you have all multiplication or all addition.


These properties give you a choice when it's all multiplication OR all addition

There are no counterexamples for these two operations.


BUT THEY DO NOT WORK FOR SUBTRACTION OR DIVISION
(lots of counterexamples!
10 - 2 does not equal 2 - 10

15 ÷ 5 does not equal 5 ÷ 15)


SO WHY SHOULD YOU CARE????

Because it makes the math easier sometimes!

Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???

Commutative allows you to choose!


ANOTHER EXAMPLE: [(543)(5)](2)

Aunt Sally would say you must do the 543 by the 5 first since it's in [ ]
But our friend the Associative Property allows us to simply move the [ ]

[(543)(5)](2) = (543)[(5)(2)]
which is so much easier to multiply in your head!!!



TWO MORE FRIENDS: THE IDENTITY PROPERTIES OF

ADDITION AND MULTIPLICATION
F
For addition, we know that adding zero to anything will not change the IDENTITY of what you started with: a + 0 = a
(what you started with)
0 is known as the ADDITIVE IDENTITY.


For multiplication, we know that multiplying 1 by anything will not change the IDENTITY of what you started with:
(1)(a) = a
(what you started with)
1 is known as the MULTIPLICATIVE IDENTITY.

Sometimes 1 is "incognito" (disguised!)

We use this concept all the time to get EQUIVALENT FRACTIONS.

Say we have 3/4 but we want the denominator to be 12

We multiply both the numerator and the denominator by 3 and get 9/12

We actually used the MULTIPLICATIVE IDENTITY of 1, but it was disguised as 3/3

ANYTHING OVER ITSELF = 1
(except zero because dividing by zero is UNDEFINED!)

a + b - c = 1
a + b - c


We also use this property to SIMPLIFY fractions.
We simplify all the parts on the top and the bottom that equal 1 (your parents would say that we are reducing the fraction)


6abc/2a  =  3bc  
2a/2a = 1 and that's why we can divide the fraction by it.
                    

10a/5 =     2a                                                                                              
AGAIN, WE LOVE THESE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!