Algebra (Period 1)

REVIEW OF PROPERTIES:


1. Commutative: switch order of all + or all x (Ch 1-2)
C+ or Cx

2. Associative: change groupings using ( ) of all + or all x (Ch 1-4)
A+ or Ax
3. Identity: + 0 or x 1 does NOT change the identity of what you have
x + 0 = x and 1(x) = x
[We use multiplicative identity to make equivalent fractions! When you change 2/3 to 10/15 you are multiplying by 1...but 1 = 5/5] (Ch 1-2)
ID+ or IDx
 
4. Distributive: Distribute (give by multiplying) the outside term to EACH term inside the (   ).  The full name is the Distributive Property of Multiplication with respect to Addition and Distributive Property of Multiplication with respect to Subtraction but we just call it the Distributive Property(Ch 1-5)
DP
 
5.  Inverse:
Additive Inverse: adding opposites signs of the same term = 0....This "friend" saves us time when adding a lot of integers together...always look for opposites FIRST and cross them out! a + (-a) = 0

Multiplicative inverse: multiplying reciprocal by a term = 1 (Ch 2-10)
x (1/x) = 1
 
6. Multiplicative property of 0: When you multiply anything by 0 = 0  (Ch 2-10, p. 105 in Try This i.)
 
7. Multiplicative property of -1: When you multiply anything by -1, you get the opposite sign.  (Ch 2-8)
 -1(a) = -a

8. Inverse property of a sum: When you multiply a SUM by -1, DISTRIBUTE the negative to EACH TERM in the (   ).   (Ch 2-8) On Practice Test we'll do #3a, b, c, and d.
 
WE did several including
[6(x + 4) -12] - [-5(x -8) + 11] why does that expression become
6x + 24 - 12 -[-5x +40 + 11] ?
and then
6x + 12 + 5x - 51
which simplifies to
11x - 39.

-3(2x + 3y -1) - 4x -(-2y) -12
-6x -9y + 3 - 4x +2y -12
-10x -7y - 9

Sometimes students memorize that Associative Property is the one that uses (  ), but Distributive Property also uses (  ) and sometimes you can have Commutative Property with (  ).
SO HOW CAN YOU TELL THE DIFFERENCE???
 
There are 2 types of Properties:
Axioms and Theorems

Axioms = properties we accept as obvious and so we don't need to prove them

Theorems = properties that need to be proved USING THE AXIOMS WE ACCEPT AS FACT!



EXAMPLES OF AXIOMS: 
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse



EXAMPLE OF A THEOREM: 
Distributive Property in REVERSE (a + b)c = ac + bc
 
WE WILL LEARN A FEW MORE (on your handout and also in Ch 2-10):


PROPERTIES OF EQUALITY (these are 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.




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:
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.



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
 
Properties with JUSTIFYING
JUSTIFYING (a proof)
When you do the steps of a problem, you should be able to give the reason why it's ok that you did what you did.
In arithmetic, the reason is generally O³.
But as problems become more complicated, you use our properties to make the math easier! :)
If you were late for math class, I could ask you to justify yourself. What would I be asking you to do??? Give me a good reason why you're late
(Ex. you were in counseling office and you have a note)

In math, the properties are our friends because they are the reasons why we do mathematically what we do....
The properties are our JUSTIFICATIONS.

A really easy example of justifying:
Given:  (436)(5)(2)
You change it to: (5)(2)(436)
JUSTIFY: Commutative of multiplication (Cx)
Why did you change it? The math was easier! :)
(WE LOVE THESE PROPERTIES!)

Same really easy example, but you do something different:
A really easy example of justifying:
Given:  (436)(5)(2)
You change it to:  (436)[(5)(2)]
JUSTIFY: Associative of multiplication (Ax)
Why did you change it? The math was easier! :)

(WE LOVE THESE PROPERTIES!)

EQUATION BALANCING PROPERTIES OF EQUALITY:

There are 4 of these.

Whatever YOU DO TO BALANCE an equation, that operation is the property of equality that was used.
 
So if you have x + 3 = 10, you used the SUBTRACTION PROPERTY OF EQUALITY because you need to SUBTRACT 3 from each side equally.
abbreviate with -prop=
If you have x - 3 = 10, you used the ADDITION PROPERTY OF EQUALITY because you need to ADD 3 from each side equally.
abbreviate with +prop=

If you have 3x = 10, you used the DIVISION PROPERTY OF EQUALITY because you need to DIVIDE each side equally by 3.
abbreviate with ÷prop=

If you have x/3 = 10, you used the MULTIPLICATION PROPERTY OF EQUALITY because you need to MULTIPLY each side equally by 3.
abbreviate with ×prop=
 
SOMETIMES, WE SAY THERE ARE ONLY 2 BALANCING PROPERTIES OF EQUALITY
CAN YOU GUESS WHICH 2 ARE "DROPPED OUT"?

Since we say we never subtract and we really never divide, it's those 2.
GOING BACK TO OUR PREVIOUS EXAMPLES:

If you have x + 3 = 10, you could say that we ADDED -3 to each side equally; therefore, we used the ADDITION (not subtraction) PROPERTY.

If you have 3x = 10, you could say that we MULTIPLIED each side equally by 1/3;
therefore, we used the MULTIPLICATION (not division) PROPERTY. 

(We always multiply by the MULTIPLICATIVE INVERSE).