Algebra Honors (Periods 6 & 7)

Using Several Transformations 3-3

a+b - b = a
and ab ÷ b = a
Use inverse operations
We can use those to solve equations with more than one step
5n - 9 = 71
by adding 9 to both sides of the equation ( using the +prop=)
we get 5n = 80
By then dividing BOTH sides by 5 using the ÷prop=
we arrive at n = 16
Using set notation {16}

(½)x + 3 = 9
By subtracting 3 from both sides using -prop=
we get ½x = 6
By multiplying BOTH sides by the reciprocal of ½ which is 2 (using xprop=)
we get
x = 12

w-5 = 2
9

multiply BOTH sides by 9 using xprop=
we get
w-5 = 18
adding 5 to both sides using +prop=
w = 23
{23}

32 = 7a + 9a
Combine Like terms FIRST
32 = 16a
2 = a {2}

4(y +8) - 7 =15
You must use the Distributive Property first
4y + 32 - 7 = 15
combine like terms on the left side
4y + 25 = 15
subtract 25 from both sides
4y = -10
divide by 4
y = -10/4
simplify BUT leave as an improper fraction
y = -5/2
Using set notation {-5/2}


But what about
-4(m +12) = 36
Yes you could do the Distributive Property BUT... why not divide both sides by -4 FIRST
-4(m + 12)= 36/-4
-4

m + 12 = -9
subtract 12 from both sides using the -prop=
and get
m = -21
{-21}
(c + 3) -2c -(1-3c) = 2

BEFORE WE COMBINE terms let's look at
-(1-3c) = -1 + 3c that is the inverse of a sum!!

c + 3 -2c -1 + 3c = 2
combining like terms we get
2c + 2 = 2
2c = 0
c = 0
{0}

Math 6A (periods 1 & 2)

Order of Operations 1-5

O³ because Order Of Operations

Grouping symbols ( ) hugs!! so important in middle school...and in math!!
[ ] brackets and { }


These Grouping symbols show which operations are performed first.
Even the Fraction bar is a grouping symbol!!

8 + 3 - 9⋅ 2 ÷ 3

Notice there are NO grouping symbols. WE then tried to come up with a number and in several of our classes we have over 6 different answers... we can't have that happening!!

So follow these rules:
When there are NO grouping symbols
1) Do ALL of the multiplication or division in ORDER from Left to Right
2) THEN do ALL of the addition or subtraction in ORDER from left to right!!

We tried it again... with all sorts of beeps.... and discovered the funnel process

8 + 3 - 9⋅ 2 ÷ 3
8 + 3 - 18 ÷3
8 + 3 -6
11-6
5

WE then tried it on
72 -24 ÷ 3
72 - 8
64

and then

8 + 6 ⋅ 12 ÷ 4 + 6
8 + 72 ÷ 4 + 6
8 + 18 + 6
26 + 6
32

The last one we did in class was
126 - 9÷ 3 ⋅ 28
126 -3 ⋅ 28
126 - 84
42


if you have a series of grouping symbols... do the inside ones first.. called nesting

{ [ ( )]}

(14 + 77) ÷ 7
91 ÷ 7
13

Make sure to show your division as a side bar. Be proud of your work!!
[3 + (4 ×5)]×10
[3 + 20]×10
[23]10
230

P--> Please --> Parentheses ( )
E --> Excuse --> Exponents e²
M --> My ----> Multiplication
D --> Dear ---> division
A--> Aunt --> Addition
S-->Sally --> Subtraction

PEMDAS but we talked about how it should really be
P
E
MD
AS

or
P
E
DM
SA

Because it is multiplication or division, whichever comes first!!
and then addition or subtraction, whichever comes first


After we have checked grouping symbols...
do all the multiplication and division in order from left to right
then,
do all the addition and subtraction in order from left to right

97 -7 5÷ 5×3 + 68
as we scan... or used our typewriter skills...(where are your sound effects???)
we discover we must divide first
funneling down, we get
97 -15× 3 + 68
97 -45 +68
52 + 68
120

what if
t = 12
w = 10
x = 9
y = 4
z = 3

tx- wz
(12)(9) - (10(3)
108 - 30
78

x(y + w)
9(4 +10)
9(14)
126

Math 7 ( Period 4)

Justifying the properties

What are properties- They are characteristics of math operations that can be identified

Why are they your friends... your BFF  you can count on properties. They  always  work. there are no (0) COUNTEREXMPLES

COUNTEREXAMPLE- an example that shows that something does NOT work.

It counters what you said!

Because you can ALWAYS count on properties, you can use the to JUSTIFY what you do mathematically.

JUSTIFY = giving a reason for doing what you did and you’re reasons are your BFF’s the properties

There are 2 parts of 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 the change?

(Commutative ?  Associative ? Distributive?)

Example  you’re given (5)(565)(2) but you change it to:

(5)(2)(565)

(10)(565)

5650

JUSTIFY what you did... What did you do to get the solution?

1) You changed the order

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

This time you are given

(565)(5)(2) but you change it to  (565)[(5)(2)] and quickly get 565(10) = 5650

Justify... ( What did you do to get your answer?)

1) you put in a set of [  ]

2) Associative Property of Multiplication allows you to either add or take away a set of ( ) and [  ] are part of the group in ( )  

Why do you think Aunt Sally would dislike these properties intensely?

Because properties are the 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!

Aunt Sally hates them because they allow us to break her rules!

Algebra Honors (Periods 6 & 7)

Transforming Equations: All Four Op's 3-1 & 3-2

Addition Property of Equality
for all real numbers, a, b, and c
and given a= b
then
a + c = b + c
and c + a = c + b
we abbreviated it as
+prop=
Make sure to read this as the Addition Property of Equality
Now
a -c = b - c
is the Subtraction Property of Equality
-prop=
But realize with a tiny change
a + -c = b + -c you have the +prop=

x - 8 = 17
If we add 8 to both sides ---> That's using the +prop=
x -8 + (+8) = 17 + (+8)
x = 25


x + 5 = 9
If we subtract 5 from both sides--> we are using the -prop=
x +5 (-5) = 9 + (-5)
x = 4

Multiplication Property of Equality
ac = bc
and ca = cb
and we write it
xprop=

Division Property of Equality
where c ≠ 0
a/c = b/c
and we write that
÷prop=

(-2/3)t = 8
If we use the Multiplication Property of Equality and
multiply BOTH SIDES by the reciprocal of -2/3
we have,
(-3/2)(2/3)t = 8(-3/2)
t = -12

⎮m⎮ =6
2
using the Multiplication Property of Equality (xprop=)

2⎮m⎮ =6(2)
2
⎮m⎮ =12
so m = -12 and 12
and using the set notation

Math 7( Period 4)

Distributive Property 1.8

Another Friend
You can make equivalent (equal expressions) using the Distributive Property... this property allows you to multiply each term inside the (  ) instead of doing the (  ) first.
Distributing literally means giving... so think of DP as giving the multiplication to each term inside the (  )

It's Halloween and you are giving candy to each child who comes in the door.  Every ghost, vampire, and princess wants the candy... you can’t just stop somewhere in the middle

c( g + p + v) =  cg + cp + cv

a( b + c + d) = ab + ac + ad

a(b – c- d) = ab – ac – ad

How is this property different from Order of Operations?

Aunt Sally or PEMDAS) says Please ( ) first... do the operations inside the parenthesis first but the DP allows you to multiply the number/variable outside the parenthesis by each term inside to get the same results.

REMEMBER  you must have the following characteristics for the DP:

1.   a number of variable outside the (  ) that is multiplied

2.   terms inside that are separated by  +  or  -

2(3 + 5)

Aunt Sally  ( order of operations) says 2 (8) = 16

Distributive property says 2 (3) + 2(5) = 6 + 10 = 16

Both are correct—just different ways to arrive at the same answer. Now in this case—Aunt Sally was smarter—using her method we arrived at the solutions faster... right?

So when does the DP help the most?

When variables are present

2( y + 5)

Now Aunt Sally can’t do ANYTHING... She is stuck

But the DP can help you simplify to 2y + 2(5) which equals 2y + 10

The DP is also great for mental math

15(23)

with this you just need to multiply out but using the distributive property you can change 23 into 20 + 3

15( 20 + 3) you probably can multiply 15(20) in your head thinking well 15 (2) is 30 so 15(20) must be 300 and 15(3) is 45 so I add 300 + 45 ... and I can do that in my head  345

15(20+3) = 15(20) + 15(3) = 300 + 45 = 345

You can also use the DP Backwards  = FACTORING the GCF in Algebra

You are given

11(24) + 11 (76)

What do you normally do?

Instead look at this expression... you notice that 11 is a COMMON FACTOR in each term....

The DP allows you to go BACKWARDS BEFORE the 11 was distributed to both the 24 and 76.

Think: What would that look like????

Se up a pair of parentheses

(            )

Place the common factor in front

11(            )

Place the two other factors inside with the SAME operation given

11(  24 + 76 )

BUT WHY WOULD YOU WANT TO DO THIS????

Because it easier to do the math this way using Order of Operations

look

11(  24 + 76 )

 = 11(100) = 1100

So the DP can also be stated this way

ab + ac = a( b + c)

Algebra Honors (Periods 6 & 7)

More on Properties 
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

Examples of Axioms 
Commutative, Associative, Identity, Distributive, Additive Inverse, Multiplicative Inverse

Example of Theorem:
The DP in reverse (a + b) c = ac + bc

Property of Negative 1 and the Inverse Property of a Sum

Both of these properties have to do with multiplying  -1 to another factor

Property of Negative 1

When you multiple any term by -1  you always get the opposite sign

-1(3a) = -3a  and   -1(-3a) = 3a

Both of these examples can be written with just the negative sign—without a 1

-(3a) = -3a  and –(-3a) = 3a

Inverse Property of a SUM

When you multiply by -1 by more than 1 term—you always get the opposite sign of each term

-1(3a + 5) = -3a – 5

-1(-3a – 5) = 3a + 5

Both of the above examples can be rewritten with just a negative sign ( without the 1)

Remember when you see a negative sign it is just a shortened for of -1.

-(3a + 5) = -3a – 5  and  -(-3a -5) = 3a + 5 

Properties of Equality (these are axioms)

Reflexive
a= a
 it looks the same – identical--on both sides !

Symmetric
a= b then b = a
3 + 5 = 8 the 8 = 3 + 5  You can switch the sides of an equation.

Transitive

a = b  b = c then a = c

3 + 5 = 8 and 2 + 6 = 8, then 3 + 5 = 2 + 6

If two things both equal a third, then we can say that they equal each other as well.

The Reflexive Property only has ONE equation

The Symmetric Property only has TWO equations

The Transitive Property only has THREE Equations