Algebra/Algebra Honors: Properties of Numbers 1-3

CHAPTER 1-3: Properties (2 days)

Re-introducing you to lots of old friends today!

WHAT ARE PROPERTIES?  They are characteristics of math operations that can be identified

WHY ARE THEY YOUR FRIENDS? (BFFs or Best Friends Forever) You can count on properties.

They always work. There are NO COUNTER EXAMPLES!

THEY ALLOW YOU TO WRITE EQUIVALENT EXPRESSIONS FOR AN EXPRESSION AND THE NEW EXPRESSION MAY BE EASIER TO USE!!

COUNTEREXAMPLE = an example that shows that something does NOT WORK
(counters what you have said)

PROPERTIES ARE THE EXCEPTIONS TO AUNT SALLY

Some 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)

JUSTIFYING

Because you can ALWAYS count on PROPERTIES, you can use them to JUSTIFY what you do mathematically.
JUSTIFY = giving a reason for doing what you did, and those reasons are your BFFS, the properties!

There are 2 parts to justifying:

1) First of all, what did you change OR if you’re looking at what someone else did, WHAT CHANGED?

(Did the order change? Did the (  ) change? Has anything been simplified?)

2) What allowed you (or them) to make that change?

(Commutative? Associative? Distributive?)

Example: You’re given (565)(5)(2) but you change it to:  (5)(2)(565) and get quickly (10)(565) = 5650

JUSTIFY! (what did you do to find the answer)

1) You changed the ORDER

2) Commutative Property of Multiplication allows you to change the order

Example: You’re given (565)(5)(2) but you change it to:  (565)[(5)(2)] and get quickly (565)(10) = 5650

JUSTIFY! (what did you do to find the answer)

1) You put in a set of [  ]  

2) Associative Property of Multiplication allows you to either add are take away a set of parentheses

WHY DOES AUNT SALLY DISLIKE PROPERTIES INTENSELY???
BECAUSE PROPERTIES ARE EXCEPTIONS TO HER RULES (ORDER OF OPERATIONS OR PEMDAS)!!!

She’s happy though that sometimes your justification can be ORDER OF OPERATIONS (in other words you just simplified  or did the math in the proper order of PEMDAS)

AGAIN WHY DO WE LOVE PROPERTIES???

WHY SHOULD YOU CARE????
 Because they make the math easier sometimes!
BUT AUNT SALLY HATES THEM BECAUSE THEY ALLOW US TO BREAK HER RULES!!

I.      COMMUTATIVE PROPERTY
PROPERTIES ARE OUR FRIENDS! (mathematically speaking)
YOU CAN ALWAYS DEPEND ON THEM --- THEY HAVE NO COUNTEREXAMPLES!

COMMUTATIVE PROPERTY (works for all multiplication or all addition)
You can SWITCH THE ORDER and still get the same sum or product.
This is the property YOU CAN HEAR because you've switched the order.
a + b = b + a OR ab = ba
Therefore, we say that both sides of the equations have EQUIVALENT (=) EXPRESSIONS
SO WHY SHOULD YOU CARE????
Because it makes the math easier sometimes!
Which would you rather multiply:
(2)(543)(5) OR (2)(5)(543) ???

II.   ASSOCIATIVE PROPERTY
ANOTHER FRIEND!
This friend allows you to GROUP all multiplication or all addition ANYWAY YOU CHOOSE!
a + (b + c) = (a + b) + c
a(bc) = (ab)c
Why? TO MAKE THE MATH EASIER OF COURSE!
This is the property that YOU CAN SEE instead of hearing because you use ( ) but DON'T CHANGE THE ORDER AS IT IS GIVEN.
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: 

III.      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 = 1
a + b - c

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

                              6abc = 3bc since both the numerator and denominator can be divided by 2a.

                               2a

IV.  PROPERTIES OF EQUALITY
(these are also called AXIOMS)

REFLEXIVE:
a = a
3 = 3
In words: It looks exactly the same on both sides! (like reflecting in a mirror)
This seems ridiculous, but in Geometry it's used all the time.
I'll show you that in class.

SYMMETRIC:
a = b then b = a
3 + 5 = 8 then 8 = 3 + 5

In words: You can switch the sides of an equation.
We use this all the time to switch the sides if the variable ends up on the right side:
12 = 5y -3
The Symmetric property allows us to switch sides:
5y - 3 = 12

TRANSITIVE:

                        If a = b and b = c then a = c

                        3 + 5 = 8, and 2 + 6 = 8 then 3 + 5 = 2 + 6
                        In words: If 2 things both equal a third thing, then we can just say that the first 2 things are equal.
                        If Jane is 14 years old and Bobby is 14 years old, then we can say that Jane and Bobby are the same age
                        It's like cutting out the "middle man"!

I've got a pattern that will help you recognize the difference between these 3 properties specifically.

If you put these 3 properties in order alphabetically, they'll be in order this way:
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations

SO REMEMBER THIS: R,S,T…1,2,3!

V.      SUBSTITUTION:

Very similar to Transitive

If a = b, then a may be SUBSTITUTED in for b in any other expression.

If 3 + 5 = 8 then 3 + 5 may be substituted for 8 in any other expression:

50 + 8 = 58

50 + (3 + 5) also = 58

In Algebra, we use substitution all the time to substitute a value in for a variable:

3 + n if n = 10

3 + 10 would be an equivalent expression because n = 10 so we can replace n with 10 in the original expression

VI.  INVERSE PROPERTIES:

ADDITIVE INVERSE:

Adding opposites signs of the same term = 0.

This "friend" saves us time when adding a lot of integers together (THAT’S WHY WE SAY “YAY”!)...always look for opposites FIRST and cross them out!

a + (-a) = 0

MULTIPLICATIVE INVERSE:

Multiplying by the reciprocal of a term = 1.

(a)(1/a) = 1

(4/5)(5/4) = 1

(-2)(-1/2) = 1

This friend helps because you can make math easier with fractions by allowing you to cross cancel!

Both inverses are used in equation balancing.