Algebra Period 3

REVIEW OF SIMPLE EQUATIONS 3-1 through 3-3

1 and 2-step equations
What is the goal? ... to determine the value of the variable
What does isolate the variable mean? ... getting the variable alone on one side of the equation
How? Use inverse operations ( the opposites) to 'get rid' of everything on the side with the variable.
FOCUS? ... on the variable
Addition/ Subtraction Properties of Equality
For all rational numbers a, b, and c if you begin with a= b then
a+c = b+c
as long as you add/subtract the same value to BOTH SIDES you are still equal or in balance
Sometimes, you combine these two properties becasue we say you never subtract-- you always ADD the Opposite

Multiplication/ DIvision Proeprties of Equality
For all rational numbers a, b, and c if you begin with a = b then
ac= bc
or a/c = b/c

Again, as long as you multiply/ divide the same valu to BOTH SIDES you are still equal or in balance

ALso works for (3/4) x = 12 because you multiply by the reciprocal....
You are using the Multiplication Property of Equality with the Multiplicative Inverse Property
that's 2 friends at once!!
(3/4)x = 12
(4/3)(3/4)x = 12 (4/3)
x = 16

Using both properties together 2-step equations

Use the Addition Property of Equality first ( get rid of addition or subtraction)
Use the Multiplication Property of Equality next ( get rid of mult/division)

Collecting Terms- sometimes you need to collect LIKE TERMS on the SMAE SIDE of the equation before balancing

FORMAL CHECK- 3 steps
1) REWRITE the original equation
2) SUBSTITUTE your solution and put a "?" over the equal sign
3) DO THE MATH and check it!!


2- Steps with distributive Property
USUALLY you want to distribut first!!
(Unless the factor outside the ( ) can be dividied out of BOTH SIDES PERFECTLY!!

5y -2(2y +8) = 16
5y -4y -16 = 16 {distibute accurately}
y - 16 = 16 { collect like terms}
y = 32 { solve by adding 16 to both sides!!}

-3(4 +3x) = -9
Wait you could distribute but if you divide both sides by -3
{the -3 goes into BOTH SIDES PERFECTLY}
you have 4 + 3x = 3
3x = -1 { subtract 4 from both sides}
x = -1/3 { Divide both sides by 3}

What if you have variables on BOTH SIDES of the equation?

Simplify each side of the equation first
THEN use the additive inverse property to move the variables to the other side, Usually we try to move the smaller coefficient to the larger because somethimes that avoids negative coefficients. But it isnot always the case.

3y - 10 - y = -10y + 12
combine like terms
2y - 10 = -10y +12
In this case I would add 10y to both sides to get rid of a negative coefficient
12y - 10 = 12
Now add 10 to both sides
12y = 22
Now divide both sides by 12
y = 22/12 or y = 11/6

Pre Algebra Period 2( Week of October 6)

Two Step Equations 7-1

Use the Addition Property of Equality or the Subtraction Property of Equality to get rid of addition or subtraction
+prop =
-prop =

Then use the Multiplication Property of Equality or the Division Property of Equality to get rid of multiplication or division
xprop =
÷prop=

This is the opposite of Aunt Sally because for solving equations you must work backwards-- or go in reverse!!

3x + 7 = 28
Subtract seven from both sides
-7 = -7 you subtract 7 form both sides so you used the -prop =
3x = 21
NOW Divide both sides by 3
3x/3 = 21/3
x = 7

CHECKING a 2 step equation

You need 3 steps to check also

1) rewrite the equation
3x + 7 = 28
2) substitute your answers and put a "?" over the "="
3) DO THE MATH!! use order of operations
3(7) + 7 ? 28
21 + 7 ? 28
28 = 28

Math 6 Honors Periods 1, 6 & 7

One Step Equations with all 4 Operations 2-4 and 2-5

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one?
that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality
We used:
" +prop= " to represent Addition Property of Equality
" -prop= " to represent Subtraction Property of Equality
" xprop= " to represent Multiplication Property of Equality
" ÷prop= " to represent Division Property of Equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of x = ____and box this in.

FORMAL CHECK OF YOUR ANSWER

There are three (3) steps to a formal check:

1. REWRITE the original equation from the original source-- this is just in case you find you copied the problem wrong!!

2. SUBSTITUTE your answer where the variable is and QUESTION your answer by placing a "?" over the =.

3. DO THE MATH-- that is check your answer by doing the math and finally putting a check mark at the end.

Pre Algebra Period 2 (Monday)

One Step Equations with all 4 Operations 2-5 and 2-6

GOAL: You use the INVERSE operation to ISOLATE the variable on one side of the equation

Here are the steps and justifications (reasons)
1. focus on the side where the variable is and focus specifically on what is in the way of the variable being by itself ( isolated)
2. What is the operation the variable is doing with that number in its way?
3. Get rid of that number by using the opposite ( inverse) operation
*Use + if there is a subtraction problem
*Use - if there is an addition problem
*Use x if there is a division problem
*Use ÷ if there is a multiplication problem

GOLDEN RULE OF EQUATIONS; DO UNTO ONE SIDE OF THE EQUATION WHATEVER YOU DO TO THE OTHER!!

4. Justification: You have just used one of the PROPERTIES OF EQUALITY
which one? that's easy-- Whatever operation YOU USED to balance both sides that's the property of equality

5. You should now have the variable all alone ( isolated) on one side of the equal sign.

6. Justification: Why is the variable alone?
For + and - equations you used the Identity Property of Addition (ID+) which simply means that you don't bring down the ZERO because you add zero to anything-- it doesn't change anything... [Note: there is no ID of subtraction]

For x and ÷ equations, you used the Identity Property of Multiplication (IDx) which simply means that you don't bring down the ONE because when you multiply by one it doesn't change anything [NOTE: there is no ID of division]

7. Put answer in the final form of

x = and box this in.

FORMAL CHECK OF YOUR ANSWER

There are thre (3) steps to a formal check:

1. REWRITE the original equation from the original source-- this is just in case you find you copied the problem wrong!!

2. SUBSTITUTE your answer where the variable is and QUESTION your answer by placing a "?" over the =.

3. DO THE MATH-- that is check your answer by doing the math and finally putting a check mark at the end.

Math 6 Honors Periods 1, 6 & 7

Writing Inequalities 2-3

2 < 7 and 7 > 2 are two inequalities that state the relationship between the numbers 2 and 7

2 < 7 reads 2 is the less than 7
7 > 2 reads 7 is greater than 2
The symbols < and > are called inequality symbols.

The point of the number line that is paired with a number is called the graph of that number.
Check out the graph on page 39 of our textbook.

Looking at the graph of numbers, we see that the larger number will be to the right of the smaller number.

A number n is between 6 and 12 would be 6 < n < 12 or 12 > n > 6

Pre Algebra Period 2 (Thursday)

Simplifying Variable Expressions 2-3
Review of Algebraic terminology:
In the expression, 3y + 5
3 is the coefficient (number attached to variable –
remember, "co" means to go along with)
y is the variable
5 is the constant (number not attached to variable)
terms are separated by ADDITION ONLY!

COMBINING LIKE TERMS:
1) Same variable (or no variable)
2) Same power

You can combine by addition or subtraction LIKE TERMS.
You cannot combine UNLIKE TERMS.

EX: 3a + 4a = 7a
but
3a + 4b = 3a + 4b
3a + 4a² = 3a + 4a²

YOU SHOULD ALWAYS COMBINE LIKE TERMS BEFORE YOU EVALUATE!
IT'S MUCH SIMPLER!
-25a + 5a - (-10a) when a = -14
First combine like terms: -10a
Then plug in for a = -14: -10(-14) = 140

Algebra Period 3 (Wednesday)

Number properties and Proofs 2-10
MORE NEW FRIENDS! (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 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


PROPERTIES OF EQUALITY
(these are AXIOMS)
” Prop = “
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.
The Reflexive Property only has ONE equation
The Symmetric Property only has TWO equations
The Transitive Property only has THREE equations

Algebra Period 3 (Monday)

Using the Distributive Property 2-7
Hanging out with an old friend:
THE DISTRIBUTIVE PROPERTY WITH NEGATIVES
The distributive property works the same when there is subtraction in the ( )
a(b - c) = ab - ac

Inverse of a Sum 2-8
Property of -1:
For any rational number a,
(-1) a = -a
In words: MULTIPLYING BY -1 changes a term to its OPPOSITE SIGN

INVERSE OF A SUM PROPERTY:
DISTRIBUTING THE NEGATIVE SIGN
incognito, it's simply distributing -1
TO EACH ADDEND INSIDE THE PARENTHESES

EXAMPLE: -(3 + x) = -1(3 + x) = (-1)(3) + (-1)(x) = -3 + -x or -3 - x
ALL THAT HAPPENED WAS THAT EACH SIGN CHANGED TO ITS OPPOSITE!
-(a + b) = -a - b

Of course they get MUCH HARDER (but the principle is the same!)
[5(x + 2) - 3y] - [3(y + 2) - 7(x - 3)]

Distribute and simplify inside each [ ] first
[5x + 10 - 3y] - [3y + 6 - 7x + 21]

Now, the subtraction sign between them is really a -1 being distributed!
"Double check" to see this (change the subtraction to adding a negative):
[5x + 10 - 3y] + - 1[3y + 6 - 7x + 21]

Distribute the -1 to all the terms in the 2nd [ ]
[5x + 10 - 3y] + -3y + -6 + 7x + - 21

Simplify by combining like terms:
12x - 6y -17

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!!!

Math 6 H Periods 1, 6 & 7 (Monday)

Writing Mathematical Expressions 2-1

Make sure to glue the 'pink 1/2 sheet' of math word phrases that we associate with each of the four basic operations -- into your spiral notebook (SN)

We can use the same mathematical expression to translate many different word phrases
Five less than a number n
The number n decreased by five
The difference when five is subtracted from a number n

All three of those phrases can be translated into the variable expression

n-5

The quotient of a number y divided by ten becomes y/10. It may look like only a fraction to you-- but if you read y/10 as always " y divided by 10" you have used the proper math language.

Twelve more than three times a number m

Wait-- where are you starting from... in this case you are adding 12 to 3m so you must write

3m + 12


Not all word phrases translate directly into mathematical expressions. Sometimes we need to interpret a situation.. we might need to use relationships between to help create our word phrase.

In writing a variable expression for the number of hours in w workdays, if each workday consists of 8 hours...

First set up a T chart- as discussed in class
put the unknown on the left side of the T chart... The unknown is always the one that reads like " w workdays"

so in this case

w workdays on the left side and under it you put
1
2
3

On the right side put the other variable-- in this case hours
under hours put the corresponding facts you know-- the relationship between workdays and hours as given in this case
hours
8
16
24

all of those would be on the right side of the T chart.

Now look at the relationships and ask yourself--
What do you do to the left side to get the right side?

and in this case

What do you do to 1 to get 8?
What do you do to 2 to get 16?
What do you do to 3 to get 24?

Do you see the pattern?

For each of those the answer is "Multiply by 8" so
what do you do to w-- The answer is Multiply b 8
so the mathematical expression in this case is "8w."

What about writing an expression for
The number of feet in i inches

i inches is the unknown... so that goes on the left side of the T chart... with feet on the right

i inches ___feet
12...............1
24...............2
36...............3

I filled in three known relationships between inches and feet Now, ask your self those questions again...
What do you do to the left side to get the right side?

and in this case

What do you do to 12 to get 1?
What do you do to 24 to get 2?
What do you do to 36 to get 3?

In each of these, the answer is divide by 12
so What do you do to i? the answer is divide by 12
i inches ___feet
12...............1
24...............2
36...............3
i................i/12



and it is written i/12


Some everyday words we use to so relationships with numbers:
consecutive whole numbers are whole numbers that increase by 1 for example 4, 5, 6
A preceding whole number is the whole number that is 1 less and the next whole number is the whole number that is 1 greater.

Math 6 Honors Periods 1, 6 & 7

Order of Operations 1-5
PEMDAS in my math class... PEMDAS in my math class... just follow the song and make sure you use the rules

Always perform the operation enclosed in the inner pair of group symbols ( such as parentheses and bracket [ ]) FIRST.
Then do all the exponents
Do all multiplication and division in order from left to right
Do all addition and subtractions in order from left to right.

A Problem Solving Model 1-6

make sure to glue in the "Plan for Solving Word Problems" into your spiral notebook

Read the problem carefully. Make sure that you understand what it says. You may need to read it more than once... That's okay!! It is great to re read things!!

Use questions like these in planning the solution:
What is asked for?
What facts are given?
Are there enough facts? Are there some unnecessary facts?
Determine which operation or operations can be used to solve the problem

Carry out the operations CAREFULLY!!
Check your results with the facts given in the problem.

Math 6 Honors Periods 1, 6 & 7

The Distributive Property 1-4

Distributive Property of Multiplication
(with Respect to Addition)

For any whole numbers a, b, and c,

a x (b + c) = (a x b) + (a x c)
and (b + c ) x a = (b x a ) + (c x a)

or written without the multiplication operator symbol
a(b + c) = ab + ac
(b+c)a = ba + bc
Remember that the parentheses indicate which operation to do first.


How could we simplify the following using the distributive property?

a. 13 x 15
13(10 + 5) = 13(10) + 13(5)


b. (11 x 4) + (11 x 6)
11( 4 + 6) = 11(10)

Distributive Property of Multiplication
(with Respect to Subtraction)

For any whole numbers a, b, and c,

a x (b - c) = (a x b) - (a x c)
and (b - c ) x a = (b x a ) - (c x a)

or written without the multiplication operator symbol

a(b -c) = ab - ac
(b-c)a = ba - ca which is better written ab- ac

Since multiplication is distributive with respect to both addition and subtraction, we refer to both properties as the distributive property.

Pre Algebra Period 2 (Monday)

CHAPTER 1-9: MULTIPLYING AND DIVIDING INTEGERS

They have the SAME rules!!!!

Math Book Rules:
If you have 2 signs that are the SAME -
answer is POSITIVE
If you have 2 signs that are DIFFERENT -
answer is NEGATIVE

Good Guy/Bad Guy Rules:
GOOD thing happens to GOOD person= GOOD = POSITIVE

BAD thing happens to BAD person = GOOD = POSITIVE
(they got what they deserved!)

GOOD thing happens to a BAD person = BAD = NEGATIVE
(we hate when good things happen to people who don't deserve it!)

BAD thing happens to a GOOD person = BAD = NEGATIVE
(we hate when that happens because it's so UNFAIR!)

Finger Rules:
If your index finger represents the negative sign, then if you have 2 negatives, you have the index fingers of both your left hand and your right hand and they make a plus sign!
If you just have one negative, it just stays negative because you don't have another finger to cross it!
If you have more than 2 negatives, you just keep using your index fingers to determine the sign.
We did this in class. It's fun! But once you get it, you won't need to keep doing it! (unless you want to keep having fun!!!)

WHAT IF THERE IS MORE THAN 2 SIGNS?
Use Aunt Sally Rules and go left to right
or
BE A SIGN COUNTER:
an ODD number of NEGATIVES = NEGATIVE
an EVEN number of NEGATIVES = POSITIVE

EXAMPLE: (-2) (-5) (-3) = -30 (odd number of negatives)
(2) (-5) (-3) = +30 (even number of negatives)
(2) (5) (-3) = -30 (odd number of negatives)

Averages ( or the Mean): Adding up all the numbers and dividing by the number of numbers

Algebra Period 3 (Monday)

CHAPTER 2-1: ABSOLUTE VALUE
Note:
It is difficult to show the symbol for absolute value here so l n l should be read as “the absolute value of n”

l n l = 5 has 2 possible answers: {-5, 5}
l n l= -5 is impossible! It's the null set and that symbol is either { } or a 0 with a slanted line through it

FORMAL DEFINITION OF ABSOLUTE VALUE:
absolute value of n is n if n was a positive number or zero
absolute value is the opposite of n if n was a negative number
(since absolute value is always positive)

The absolute value is a distance concept- the absolute value is the distance a number is from zero on a number line.

2 words often misunderstood:
withdrawing money is actually considered negative, while depositing is considered positive
The focus is not the money in your wallet, but the money in your bank account!

CHAPTER 2-2: RATIONAL NUMBERS:
Counting Numbers = natural numbers = 1, 2, 3, 4 …..
Whole Numbers = the natural numbers + 0… so 0, 1, 2, 3, 4, ….
Integers = the whole numbers and their opposites… so …-4, -3, -2, -1, 0, 1, 2, 3, 4….
Rational number = any number that can be expressed as the ratio (fraction) of two integers
a/b, where a and b are both integers and b cannot be zero
b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!
Proof of this was done in class

There are positive and negative rational numbers.
One way to put them in order from least to greatest is to simply place them on the REAL number line. Numbers will be least to greatest if read from the left to the right.
If you have both fractions and decimals, it's often easier to change the fractions into decimals by simply dividing the numerator by the denominator.
REMEMBER: When numbers are NEGATIVE, the closer they are to zero, the bigger they are.
Example: -1/2 is greater than -3/4
-2.3 is greater than -2.5

You can also find the absolute value of rational numbers -
They are ALWAYS POSITIVE unless you're talking about zero (which is neutral)