Pre Algebra Period 2 (Monday)

Properties of Numbers 2-1
WHAT ARE PROPERTIES? (Why are they your friends?)
You can count on properties. They always work. There are no COUNTEREXAMPLES!

COUNTEREXAMPLE = an example that shows that something does not work
(counters what you have said)

An example from Math: You can't switch the order of subtraction because it's not the same value.
10 - 8 DOES NOT EQUAL 8 - 10
That's a COUNTEREXAMPLE to saying that you can switch subtraction
(We'll say that it's a COUNTEREXAMPLE to the existence of a
COMMUTATIVE PROPERTY OF SUBTRACTION
so that property does not exist!)

An example from Math: You switch the order of addition to make the adding easier.
20 + 547 + 80 = 20 + 80 + 547
(both equal 647, but the right side is much easier!)
What allowed you to switch the order?
A property called the Commutative Property of Addition says you can!
You'll always get the same value!

Now Aunt Sally doesn't like some of the properties because they allow us to do things that are exception to the Order of Operations!

Commutative Property
You can switch the order of all addition or all multiplication
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 Property
You can group all addition or all multiplication any way you want
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

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/a + b - c = 1


We also use this property to SIMPLIFY fractions.
We "simplify" all the parts on the top (the numerator) and the bottom (the denominator) that equal 1
(your parents would say that we are reducing the fraction)
6abc/10a =3bc/5 since both the numerator and denominator can be divided by
2a/2a



WE LOVE PROPERTIES BECAUSE THEY MAKE OUR LIFE EASIER!
AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!!