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

Algebra Honors (Periods 6 & 7)

Dividing Real Numbers 2-9
Quotient a divided by b or a ÷b should be written as a fraction
a/b
Definition of division --> Which we abbreviated as Def (÷)
a÷ b = a ⋅(1/b)
to divide by a non zero number, multiply by its reciprocal

a+b
c
= a/c + b/c
where c ≠ 0
and

a-b
c
= a/c - b/c

Justifying leads us to the following:
(a+b)/c
= (a+b)(1/c) --> def (÷)
=(a ⋅ 1/c) + (b ⋅ 1/c) --> DP
=a/c + b/c --> def (÷)

Math 6A( Periods 1 & 2)

 The Distributive Property 1-4

The Distributive Property of Multiplication with Respect to Addition 

DP+

For any whole numbers a, b, and c

 a(b + c) = ab + ac

(b+c)a = ba + ca

13(15) … well you could just multiply it out but using the distributive property you might be able to do this in your head…

13(15) = 13( 10+5) = 13(10) + 13(5) = 130 + 65 = 195

9(11)  now we know this but let’s see what happens if we try using the distributive property

9(11) = 9(10 + 1) = 90 + 9 = 99  YAY!!!

We could have completed this another way…

9(11) = (10-1)(11) = 10(11) – 1(11) = 110 – 11= 99

This last example is really  an example of the

Distributive Property of Multiplication with Respect to Subtraction or

DP-

For any whole numbers a, b, and c

 a(b - c) = ab- ac

(b - c)a = ba - ca

What should we do with the following:

(117∙ 4) – (17∙ 4)

Last year you would multiply  (117∙ 4) and then multiply (17∙ 4) and then take the difference… but that is a long process

What happens if we use the distributive property?

(117∙ 4) – (17∙ 4) = (117-17)∙4= (100)4= 400

WOW—that was so much easier

More Examples of the Distributive Property:
(7∙ 9) + (13 ∙9)

They both share a 9 so pull the 9 out

(7 + 13)9

(20)9 = 180

WOW.. look at that!

But… what happens with

57 ∙ 9

There are a number of different ways we might set this up

(50 + 7)(9) = 50(9) + 7(9)

Or

57(10-1) = 57(10) – 57(1)

Or

(60-3)(9) = 60(9) – 3(9)

We could even set it up with the following but this would not help us to solve it easier…

(37 + 20)(9)… remember our discussion in class—it was to make the multiplication easier

The results for all of the above is 513

What about 25 ∙ 38

We could do 25(40 -2) = 25(40) – 25(2)

Now that’s easy…. 1000 – 50 = 950

25(30 + 8) also works but probably isn’t  quite as easy

25(30) + 25(8) = 750 + 200 = 950

101(34) = (100 + 1)(34) = 100(34) + 1(34) = 3434

(89∙ 5) – (19∙ 5)

(89-19)(5)

(70)(5) =350

So much easier than multiplying each separately!

BUT… what happens if you have

4(17 + 42)

Well we would first add the two numbers in the (  )

That gives us 4(59)… but that really did not make it any easier..

So now look at

4(59) and ask yourself what can you do with that number

4(60-1) = 4(60) – 4(1) = 240 -4 = 236!!

YAY!!

Math 7 ( Period 4)

 The Commutative & Associative Properties 1.7

What are properties... They are characteristic of math operations that can be identified

Why are they your friends?  (BFF) you can count on properties they always work they will NOT let you down... there are NO COUNTER EXAMPLES..

 Properties are the exceptions to Aunt Sally

These properties give you a choice when
it is ALL MULTIPLICATION or
it is ALL ADDITION

(they DO not work for subtraction or division)  lots of counterexamples for those...

 10 – 2 does not equal 2- 10

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 that you can hear because you switched the order

 a+ b = b + a  or ab= ba 

Commutative Big C looks like an ear... 

Why care?  it makes the math easier ... sometimes

 Which  would you rather multiply

(2)(543) (5)  or (2)(5)(543)

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

again.. to make the math easier

This is the property that you can SEE instead of hear because  you use (parentheses) but you don’t change the ORDER GIVEN

[(543)(5)](2)  what would Aunt Sally say you MUST do? 

but our friend the associative property allows use to move the brackets...

(543)[(5)(2)] which is so much easier to multiply—even in your head...

Algebra Honors (Periods 6 & 7)

Problem Solving Consecutive Integers 2-7


Consecutive- list in natural order from least to greatest
-2, -1, 0, 1 ..

an integer n
the next three numbers are n + 1, n+ 2, and n+ 3
the number that receded n is n-1

Let's create the following equation: The sum of four consecutive integers is 66
Let n be the first integer
n + (n + 1) + (n + 2) + (n + 3) = 66
4n + 6 = 66
n = 15
Now, if the question asked what was the first integer the solution would be {15}
but if the question asks you to state all the integers the solution would be
{15, 16, 17, 18}

If x = 15, how would you write 14 and 16 in terms of x?
14 --> x -1
16 --> x + 1

if m is odd, m + 1 is even

How do you write 4 consecutive even integers starting with x ?
x, x + 2, x + 4, x + 6

How do you write 4 consecutive odd integers, starting with x ?
x , x + 2, x + 4, x + 6


In the above cases, you defined what x was-- in the first case x was even... so you need to add multiples of 2 to continue getting an even number... AND in the 2nd case you defined that this time x was odd.. and again you need to add multiples of 2 to continue getting an odd number.


The sum of 2 consecutive integers is 43. Let x = 1st integer
x + ( x + 1) = 43

The sum of three consecutive odd integers is 40 more than the smallest.
What are the integers?

(Hint: Reread the word problem carefully!!)

Let x represent the 1st odd integer
x + (x + 2) + (x + 4) = x + 40
3x + 6 = x + 40
2x = 34
x = 17
so
{17, 19, 21} is the solution.. and if you substitute in to check, you will see you have solved the question!!

However, in our book, it asks this question a little differently, It says with the domain for the smallest given as {13, 17, 25}.
In this case you could work backwards and substitute each of the three integers to see if it works.

The Reciprocal of Real Numbers 2-8



Two numbers whose product is 1 are called reciprocals or multiplicative inverses
5 and 1/5 are reciprocals because 5⋅1/5 = 1
4/5 and 5/4 are reciprocals 4/5⋅5/4 = 1
and so are (-1.25) and (-0.8) because (-1.25) ⋅ (-0.8) = 1
look closely at (-1.25) and (-0.8) and you realize that they are really -5/4 ⋅ -4/5 = 1

If a is any real number, its reciprocal is 1/a

Property of Reciprocals
for any non zero real number a,
a⋅1/a = 1 and 1/a⋅ a = 1
-a and -1/a are also reciprocals
Notice, we discussed in class that
1/-a = -1/a = -(1/a) those are equivalent

(ab)(1/a⋅1/b) = a (1/a)⋅b(1/b) = a/a⋅b/b= 1⋅1 = 1


so try the following
1/3(42m -3v)
1/3(42m) - (1/3)(3v)
14m - v

Make sure to distribute the negative to each term

How about
-1/20( 5z -4w) - 6(-1/30w - 1/24z)

Again make sure to distribute the negative to each term!!
(-1/4)z + (1/5)w + (1/5)w + (1/4)z
or
-z/4 + w/5 + w/5 + z/4
= 2w/5 also written as (2/5)w

Math 6A (Periods 1 & 2)

Inverse Operations 1-3

Inverse Operations undo each other
Addition & subtraction are inverse operations
Multiplication &division are inverse operations

We can use these relationships between inverse operations to simplify some numerical and variable expressions

17- 8 + 8 instead of doing 17 - 8 and then adding 8 again simply cancel the + and the -
That is, 17 - 8 + 8 = 17

9 - x + x = 9 because adding x and subtracting x are inverse operations and they undo each other...

108 ÷ 6 × 6 ... the long way would be to divide 108 by 6 first. That's 18 and then multiply 18 by 6 and get 108.. but why do that? Using the inverse operations you realize that multiplication and division undo each other so 108 ÷ 6 × 6 is simply 108

3n ÷ 3 = 3

Subtraction Property of Zero
For any whole number a
a-0 = a and a-a = 0

In Algebra
a- a = 0 is called the Property of Opposites ( The Prop of Opp's)

Division Property of One
For any whole number a, except 0
a ÷ 1 = a and a ÷ a = 1

Division Property of Zero
For any whole number a, except 0
0 ÷ a = 0


Remember we discussed that we can never divide any number by 0
We said that any number
a ÷ 0 was said to be 'undefined.'

Math 7 (Period 4)

 Powers & Exponents 1.3
A Power has two parts:

BASE= the large number on the bottom ( base)

EXPONENT = the tiny number raise up

EXPONENTIAL NOTATION = the ay you write the expression with the exponent

EXPANDED NOTATION = showing the multiplication the appropriate number of times reflected by the exponent

In your book this is called “ repeated multiplication”

STANDARD NOTATION = the actual value ( what you would expect) you find this by evaluating the power—doing the repeated multiplication

3⁴ is exponential notation where 3 is the base and 4 is the exponent

(3)(3)(3)(3)  is expanded notation

81 is standard notation

The entire expression is called     3 to the power of 4

(4y)³ = (4y)(4y)(4y) = 64y³

Evaluating a Power

Evaluating is plugging and chugging when a variable is raised to a power

x⁴ when x = 3   3⁴ = 81

Some special powers to know

There are two special powers that have ‘nicknames”

3² can be read  “ 3 to the second power” but its nickname is 3 squared

This is based on finding the area of a square

If one side of the square is 3 then the area would be ( 3)(3) which is 3²

Likewise 3³ can be read 3 to the third power.... but its nickname is “ 3 cubed”

This name is based on finding the volume of a cube.

(3)(3)(3) = 3³

Math 7 (Period 4)

Expressions & Variables 1-2

(we are skipping 1.1 for now—will pick it up  with a unit on data and stats)

LOTS of VOCAB

term: a number and or variable added or subtracted to another number and or variable

factors : a number and or variable multiplied by another number and or variable

addition words: sum, total plus, more than greater than addends  altogether both

subtraction words: difference   minus, less than, less minuend subtrahend

multiplication words: product, times, factors, each

division words: quotient, dividend divisor , shared equally

expression: at least one term ( with NO equal sign)

equation: equal expressions ( it must have an = sign)

numerical: all numbers ( called constants)

variable: a letter ( or symbol) used to represent a number

algebraic (or variable): has at least one variable

simplify a numerical expression: do order of operations you will get a simpler expression

evaluate- given the value of the variable you “plug and chug” in an algebraic expression and simplify using operations

You should be able to write a numerical expression in words:

 3 + 7 is The sum of three and seven

You should be able to write a numerical equation in words:

 3 + 7 = 10

The sum of three and seven is ten

Evaluating VARIABLE ( or algebraic) expressions

Evaluate 5x + y when x = 1.2 and y = 0.8

5(1.2) + 0.8 = 6.8 

Math 6A (Periods 1 & 2)

Properties of Addition &Multiplication 1-2


1, 2, 3, 4,5 .... are counting numbers or natural numbers.
If we add 0... we have 0,1,2,3,4,... and we have the set of whole numbers

Some important properties are found using whole numbers:

Commutative Property of Addition (C+)
for any whole numbers a and b
a + b = b + a

Commutative Property of Multiplication (Cx)
for any whole numbers a and b
ab = ba

You HEAR the differences.. remember we made the big C into an ear... or at least we attempted to!!

However look at the following
(6 +5) + 7 = 11 + 7 = 18
but so does
6 + (5 + 7) = 6 + 12 = 18

Associative Property of Addition ( A+)
For any whole numbers a, b, and c
(a + b) + c = a + ( b + c)

Associative Property of Multiplication ( Ax)
For any whole numbers a, b, and c
(ab)c = a(bc)

You SEE the difference with the hugs ( ).. hugs are really important in life.. and in math!!

Addition Property of Zero from the book
also known as the Identity Property of Addition ( ID+)
for any whole number a
a + 0 = a and 0 + a = a

Multiplication Property of One
also know as the Identity Property of Multiplication (IDx)
For any whole number a
a(1) = a and 1(a) = a

Multiplication Property of Zero  MPZ
for any whole number a
0(a) = 0 and 0(a) = 0

So how do we justify?

16 + 19 + 34 --- given
16 + 34 + 19 ---C+
(16 + 34) + 19--- A+
50 + 19---simplify
69--- simplify


2 (14)(15)--- given
2 (15) (14) --- Cx
[2(15)] (14) --- Ax
30 (14)--- simplify
420--- simplify

how about the following. What would make it easier to simplify???
8⋅ 14⋅ 25 ⋅5 --- given
14 ⋅ 8 ⋅ 5⋅ 25 --- Cx
14 ⋅ (8 ⋅ 5)⋅ 25 --->Ax
14 ⋅ 40⋅ 25---simplify
14 ⋅ (40⋅ 25) --- Ax
14 ⋅ 1000 --- simplify
14000 --- simplify

Make sure you justify each step of tonight's homework!!

Algebra Honors ( Periods 6 & 7)

Chapter 2 Highlights Continued

Substitution Principle-- an expression may be replaced by another expression that has the same value.

JUSTIFY THE FOLLOWING:
b + (a -b)
= b + [a +(-b)] Definition of subtraction ( we used def-)
= b + [-b + a] C+
=(b + -b) + a A+
= 0 + a Property of Opposites ( How about Prop of Op for this one!!)
= a Id+

Check out the 5 step plan on Page 27... use the index in your textbook to find many concepts and explanations...

Multiplication Property of One a(1) = a and 1(a) = a IDx 1 is the identity element for multiplication

Multiplication Property of Zero A90) = 0 and 0(a) = 0 Ox

Multiplicative Prop of -1
a(-1) = -a and (-1)a = -a

Justify the following
a(-1) + a
= a(-1) + a(1) ID x
= a(-1 +1) DP
=a(0) Prop of Op
=0 Ox

Property of Opposites in Products or the Prop of Opposites in Products or just POP
(-a)b= -ab
a(-b) = -ab
(-a)(-b) = ab

JUSTIFY
a(b-c)
=a[b + (-c)] Def-=ab + a(-c) DP
ab + (-ac) POP
ab -ac Def -

-3(7c + d) - 2(10d-c)
-21c -3d -20d + 2c
-19c -23d

Math 6A (Periods 1 & 2)

Mathematical Expressions 1-1

A variable is a symbol used to represent one or more numbers. The numbers are called the values of the variable.


An expression, such as 3 x n, that involves a variable is called a variable expression.


Expressions, such as 3 x 2, that name a certain number are called numerical expressions


When we write a product that involves a variable, we usually omit the multiplication symbol (whether that be written as x or as ∙ or even with parentheses). Thus, 3 x n is written as 3n
and 2 x a x b is written as 2ab

In numerical expressions for products a multiplication symbol must be used to avoid confusion.

9 x 7 may be written as 9 ∙ 7 or even 9(7)

When a mathematical sentence uses an equal sign, it is called an equation. An equation tells us that two expressions name the same number. The expression to the left of the equals sign is called the left side of the equation and the expression to the right of the equals sign is called the right side.
expression = expression

When a number is substituted for a variable in the variable expression and the indicated operation is carried out, we say that the variable expression has been evaluated. For example, if n has the value 6 in the variable expression 3 x n, then 3 x n has the value 3 x 6, or 18
Example: Evaluate the expression 6a when the variable has the following values:
6a; 2, 4, 6, 8
You would substitute in each value for the variable a
6(2) = 12
6(4) = 24
6(6) = 36
6(8) = 48

148 ÷ 4 =
148/4
37


if m = 3 and n = 18
n ÷ m
substitute in
n/m or 18/3 = 6


If y = 18 and x = 8
4y ÷ 3x immediately set this up as
4y/3x

Now substitute in your values
4(18) / 3(8)
72/24 = 3

Algebra Honors (Periods 6 & 7)

Chapter 1 Highlights
Variable→ symbol used to represent one or more numbers
Variable expression contains a variable
Numerical expression names a particular number
Simplifying the expression→Replacing a numerical expression by the simplest name for its value
Grouping symbols→ parentheses, brackets, fraction bar, etc used to enclose an expression that should be simplified first.
Equation→ two numerical or variable expressions that are equal. Represented by an equals sign placed between the two sides of the equation.
Open Sentences→ contain variables, such as 5x – 1 =9
or y + 2 = 2 + y

The given set of numbers that a variable represents is called the domain of the variable.
Use brackets { } to show a set of numbers. A short way to write “the set whose members are 1, 2, and 3” is {1, 2, 3}
Any value of a variable that turns an open sentence into a true sentence is a solution or root of the sentence.
The set of all solutions of an open sentence is called the solution set of the sentence.
Some equations have only one solution, and some equation have no solutions. The sentence y + 2 = 2 + y has an infinite number. The solution set is the set of all numbers. If, however, you are asked to solve of the domain {0,1,2,3}, you state that the solution set is the domain itself {0, 1, 2, 3}.

Another way to express this is

Read the above as
“y belongs to the set whose members are 0, 1, 2, 3”

Real Numbers
Natural Numbers or Counting Numbers→ 1, 2, 3, 4, 5….
Whole Numbers (Natural Numbers + 0) → 0, 1,2, 3, 4, 5 …
Integers (Natural Numbers , their opposites, + 0) → …-4, -3, -2, -1, 0, 1, 2, 3, 4, …
Rational Numbers (quotient of two integers)
So any number that is either positive, negative, or zero is called a real number.

Each number is a pair, such as 4 and -4 is called the opposite of the other number. The opposite of a is written –a.
The numerals -4(lowered minus sign) and ^-4 (raised minus sign) name the same number.
Caution –a read “the opposite of a” is NOT necessarily a negative number.
For example, if a= -2 then –a would be –(-2) = 2

Absolute value is a distance concept. It my be thought of as the distance between the graph of a number and the origin on a number line. The graphs of -4 and 4 are both 4 units from the origin.