Math 6 Honors (Period 6 and 7)

Polygons 4-5

A polygon is a closed figure formed by joining segments ( the sides of the polygon) at their endpoints ( the vertices of the polygon). Polygons are named for the number of sides they have.
Triangle- 3 sides
Quadrilateral- 4 sides
Pentagon- 5 sides
Hexagon- 6 sides
Octagon- 8 sides
Decagon- 10 sides

A polygon is regular if ALL of its sides are congruent and ALL of its angles are congruent.

To name the polygon we name its consecutive vertices IN ORDER.

A diagonal of a polygon is a segment joining two nonconsecutive vertices.
Look at the quadrilateral on Page 123 and notice the two segments that represent the diagonals of the quadrilateral PQRS.

Certain quadrilaterals have special names.
A parallelogram has it opposite sides parallel and congruent.

A trapezoid has just one pair of parallel sides.

Certain parallelograms also have special names

A rhombus ( rhombii plural) has all it sides congruent.
A square has congruent sides and congruent angles
A rectangle has all its angles congruent.

Thus a square is a rectangle.. but a rectangle isn't necessarily a square!!

TH\he perimeter of a figure is the distance (think fence) around it. Thus the perimeter of a polygon is the sum of the lengths of its sides.

Always label your perimeters. If the figure provides a specific measure, such as meters (m), centimeters (cm), feet (ft), inches (in.)-- make sure to use that label.

If no unit of measure is given, always include "units"




Algebra (Period 1)

Factoring by Group 6-6

First a review:
Checklist of how to factor thus far-->

1. Look for a GCF of all terms

2. Binomials - look for difference of two squares

both perfect squares - double hug - one positive, one negative - square roots of both terms

3. Trinomials - look for Trinomial Square (factors as a binomial squared)

first and last must be perfect squares - middle must be double the product of the two square roots

SINGLE hug - square roots of both terms - sign is middle sign

4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

5. Trinomials - last sign negative - double hug with different signs, putting middle sign in first hug - factors that multiply to last and subtract to middle - middle sign will always be with the bigger factor



REMEMBER: 
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

(we're skipping 6-5 and then going back to it)
When you have 4 TERMS IN YOUR POLYNOMIAL!


You put the polynomial in 2 sets of 2 by using ( )

Then you factor out the GCF for each set of 2 terms individually


DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?

Of course not!
But for this section of the math book, it will!

What happens if it doesn't work? The polynomial may just not be factorable!

MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ - 9x² + 4x - 6
First notice there is NO GCF of all the terms!!

Factoring by grouping says if there is no GCF of the 4 terms, look and see if there is a GCF of just 2 terms at a time!!

Put ( ) around the first 2 terms and another ( ) around the 2nd set of terms.
(6x³ - 9x²) + (4x - 6)
Factor out the GCF from each set of two terms
3x²(2x - 3) + 2(2x - 3)
Look for a COMMON factor to factor out between the two sets
In this case its (2x - 3)
Pull out
(2x - 3)(3x + 2)
and check to make sure you cannot continue to factor!!

Try these:

x³ + x² + 2x - 2
First... is there a GCF? No

okay
now set up in 2 groups of TWO

(x³ + x²) + (2x - 2)
x² (x + 1) + 2 (x-1)
wait... they are NOT the same...
cannot be factored.. not factorable!!

2x² - 4x + xz - 2z
Is there a GCF? NO

(2x² - 4x) + (xz - 2z)
2x(x -2) + z(x-2)
(x-2)(2x + z)

24x³ + 27x² - 8x - 9
Is there a GCF? NO

(24x³ + 27x²) + (-8x - 9)

3x²(8x +9) -1(8x + 9)
(8x + 9)(3x² - 1)

c⁶ -c⁴ - c² + 1
(c⁶ -c⁴) + (-c² + 1)
c⁴(c² -1) -1c² -1)

Look carefully at that results.. why did the second term become -1?

(c² -1)(c⁴ -1)
and ask yourself... are you finished factoring? ...
NO
I see The difference of Two Squares...


(c + 1)(c-1)(c²+1)(c² -1)
Now are you finished?
No.. I still see the difference of Two Squares... bring everything down...
(c + 1)(c-1)(c²+1)(c + 1)(c-1)
and now write it in the correct order
(c² + 1)(c + 1)(c + 1)(c-1)(c-1)


4y⁵ + 6y⁴ +6y³ +9y²
First thing-- Is there a GCF? YES

pull out a y² and you are left with
y²(4y³ + 6y² + 6y + 9)
Now put those 4 terms in 2 groups of two!!.. use brackets..

y²[(4y³ + 6y²) + 6y + 9)]
Look for a GCF in each of the hugs!!

y²[2y²(2y +3) + 3(2y+3)]

What do each of them have in common? What do they share? 2y + 3
when you put that in the first set of hugs... what's left?

y²(2y+3)(2y²+3)

Pre Algebra (Period 2 & 4)

Simplifying Fractions 4-4

Equivalent Fractions - Just multiply the numerator and the denominator by the same number and you will get an equivalent (equal) fraction to the one you started with.


GOLDEN RULE OF FRACTIONS = Do unto the numerator as you do unto the denominator


Simplifying fractions (your parents call this "reducing")

2 good ways:

(1) Just divide both the numerator and denominator by the GCF

(2) Another way: Rewrite the numerator and denominator in prime factorization form. Then simply cross out each common factor on the top and bottom
(they cross out because it's 1)

You'll be left with the simplified fraction every time!!!!
                                           


THE GCF METHOD:

One of the reasons we learn the GCF is because it's the FASTEST WAY TO SIMPLIFY FRACTIONS IN ONE STEP!!!

Just divide both the numerator and denominator by the GCF

THE PROBLEM WITH THE METHOD:
If you're not comfortable finding the GCF, you're pretty much sunk with this method! :(

THE BEST REASON TO USE THIS METHOD (other than it's a Calif. STAR Key Standard), it truly is the FASTEST :)

So imagine you have a "GCF Magical Voice" in your head...
The voice tells you the GCF of the numerator and the denominator...
You simply use that GCF to divide both the top and bottom of your fraction and you're done in one step!



THE PRIME FACTORIZATION METHOD:

This is sort of using the GCF "incognito" (in disguise)!

Rewrite the numerator and denominator in prime factorization form.

(Use a Factor Tree or Inverted Division to find the Prime Factorization if necessary).

Then simply cross out each common factor on the top and bottom.

(You're actually using the ID Property of Multiplication because 
each "crossout" is really a quotient of 1!)

You'll be left with the simplified fraction every time!!!!

If you actually multiplied together all your cross-outs, you'd get the GCF...
so you're using the GCF without even computing it!

THE PROBLEM WITH THIS METHOD:
You may think it's a lot of work


THE BEST REASON TO USE THIS METHOD: Although it takes time, everyone can do a Factor Tree or Inverted Division and create the Prime Factorization...
You'll never get the wrong answer with this one!



THE CROSS OUT METHOD:

You simply think of the first number that comes to your mind that "goz-into" both the numerator and the denominator and keep going until it's simplified.

If it's even, most people start with dividing it in half....and then in half again, etc.

This probably takes the longest, but in practice, most people use this method!

THE PROBLEM WITH THIS METHOD: You may think that a fraction is simplified, but you've missed a factor...this especially happens when the number is odd and you're always used to using 2 to divide the top and the bottom!

THE BEST REASON TO USE THIS METHOD: No one ever forgets how to do this method...it just comes naturally and there are no "precise" steps to do!
 
EXAMPLE: Simplify by each method:
36/
54
 
GCF METHOD:

The GCF is 18:

36 ÷ 18 = 2

54 ÷ 18 = 3



PRIME FACTORIZATION METHOD:

36 = 2 x 2 x 3 x 3

54 = 2 x 3 x 3 x 3

Two of the 3s cross out and one of the 2s

You are now left with:

2/
3

That's it!!!!!!!!!



CROSS OUT METHOD:
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 3 = 2

54 ÷ 2 = 27 ÷ 3 = 9 ÷ 3 = 3


so 36/54 = 2/3
Do the same thing with variables!

Pre Algebra (Period 2 & 4)

Prime Factorization & GCF 4-3

Greatest Common Factor - think of it backwards to understand it!

Factor = must be a number that goes into the numbers

Common = must be a number that goes into BOTH the numbers

Greatest = must be the biggest number that goes into BOTH the numbers

There are several ways to find it.

1) List all the factors of each number and circle the biggest one that is common to both
 (takes too long!!)
2) Circle the common factors in the prime factorizations of each number and multiply

3) list the factors in a table and bring down the factors whose column is filled.
Then multiply.


EXAMPLE:
Find the GCF of 36, 45 and 54


LIST ALL THE FACTORS OF EACH NUMBER:

1, 2, 3, 4, 6, 9, 12, 18, 36

1, 3, 5, 9, 15, 45

1, 2, 3, 6, 9, 18, 27, 54

The GCF is 9


FIND THE PRIME FACTORIZATIONS ON A FACTOR TREE OR INVERTED DIVISION AND MULTIPLY THE COMMON FACTORS:

36 = 2 x 2 x 3 x 3

45 = 3 x 3 x 5

54 = 2 x 3 x 3 x 3

GCF = 3 x 3 = 9

PUT THE PRIME FACTORIZATIONS IN A BOX WITH COLUMNS:
   as shown in class                                     

DO THE SAME THING WITH VARIABLES:
The GCF of the variables is the most of each variable that each term has in common.
EXAMPLE:
Find the GCF of a²b³c⁴   ac³d   a³c²f
The COMMON variables are a and c
How many of each variable is COMMON to all 3 terms:
They each have 1 a (although the first term has 2 and the 3rd term has 3)
They each have 2 c's (although the 1st term has 4 and the 2nd has 3)
GCF = ac²
Again, the GCF of variables is simply the lowest power of common variables
You should look for a special case of GCFs:
When one number goes into the other number(s), the smaller number is always the GCF.
Example: The GCF of 50 and 100 is 50
50 is the biggest factor that goes into both 50 and 100!

Algebra (Period 1)

Factoring x² +bx + c or Factoring Trinomials 6-4

You are reversing it back to BEFORE it was FOILed.
Always check your factoring by FOILing or BOXing back!!


Factoring Trinomials with a:
PLUS sign as the second sign
x² + bx + c
Following these steps:

1. set up your hugs ( )( )
2. When the last sign is positive the BOTH signs in each of the ( )( ) are the SAME!!
3. How do you know what those two signs are? It is whatever the sign is of the 2nd term of the trinomial. Put that sign in BOTH parentheses.
4. to factor (unFOIL), you will need to find two factors that
MULTIPLY to the LAST term and
ADD to the MIDDLE term

you can set up a box with

___ X ___ =
___ + ___ =

and fill in the blanks.
I suggest you make a T-chart with all the factors of the last term-- using your divisibility rules!!
Example:
x² +8x + 15
Follow the steps
( )( )
Think: last term sign is + so both signs are the same
Think: first sign (sign of the 2nd term) is + so both signs are positive
put + into the ( )( )
( + )( + )
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x + )(x + )
Now to get to the L in FOIL you need two factors whose product is 15. This is easy but using a T chart
15
1 I 15
3 I 5

you see 1 X 15 or 3 X 5 are possibilities

BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to 8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
3 X 5 = 15
3 + 5 = 8
(x + 5) (x +3)

At this point it does not matter which factor you put into the first ( ) because they are the SAME sign but I always tend to put the LARGER number in the first ( ) because of other rules -- which you will learn later this week)

Next example
x² - 8x + 15
Follow the steps
( ) ( )
THINK: Last sign is + so the signs are the same
THINK: First sign ( 2nd term) is NEGATIVE so BOTH signs are NEGATIVE
( - ) ( - )

Again,
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x - )(x - )

Now to get the "L" in FOIL, you need two factors whose product is 15
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to -8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
-3 X -5 = 15
-3 + -5 = -8
(x - 5)(x - 3)
(It doesn't matter which is first because they're the same sign!)
Now FOIL to see if we're right!

Last example:
x² - 8xy + 15y²
Same problem as the one above except now there are two variables. Simply use the same steps above and include the y

(x -5y)(x -3y)


Factoring Trinomials with a:
NEGATIVE sign as the second sign

x² + bx - c
We will use the same method as yesterday to factor these basic trinomials!
1.
Set up your (       )(       )

2. Look at the SECOND or last sign

If it's negative, then the signs in the (   ) are DIFFERENT
Why?
Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs.
Remember that the last term is the product of the two LAST terms in FOILing.


3. Now look at the sign of the second term.

It tells you "Who wins," meaning which sign must have the larger absolute value. 
Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing.
Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the sign of the larger absolute value.
( This is just integer rules!!)
Put that sign in the first parentheses and always put the bigger number in the first parentheses.
4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term.
(yesterday the factors needed to ADD to the middle)

Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!

This is still an educated guess and check!

To help you do this, I suggest to set it up like this:

____ x ____ = ____
____
 - ____ = ____

Again, setting up a T-chart with all the factors also helps you visualize the two numbers you are looking for!!



EXAMPLE:

x² + 2x - 15
( ) ( )
THINK: LAST sign is - so the signs are DIFFERENT
THINK: First sign is + so the POSITIVE WINS!!
( + ) ( - )
You know the "F" in FOIL means that both the fist terms must be x so
(x + )(x - )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
(Don't forget the sign1!!)
Several possibilities like 1 X -15 or 15 X -1 or 3 X -5 or 5 X -3
BUT since the POSITIVE must win , according to the middle term of the example (+2x)
you know that the bigger factor must be positive ( so it can win!!)
Therefore your choices are POSITIVE 15 X NEGATIVE 1 or POSITIVE 5 X NEGATIVE 3
BUT, you also need them to ADD to the I and O in FOIL so pick the two factors that also ADD to POSITIVE 2
Since -3 + 5 = +2 these must be the two factors that will work
I set it up like this:

____ x ____ = 15

____ - ____ = 2

so
5 x 3 = 15

5 - 3 = 2

YOU CAN ALSO DO THIS WITH THE APPROPRIATE SIGNS and adding:

+____ x -____ = -15

+____ + -____ = + 2

so

5 x (-3) = -15

5 +( -3) = 2


THIS IS WHERE IT DOES MATTER WHICH NUMBER YOU DO HAVE WITH THE SIGN BECASUE THE + MUST WIN!!
( x + 5 )( x - 3 )



NEXT EXAMPLE:

x² - 2x -15

(      )(      )

THINK: Last sign is - so signs are DIFFERENT!

THINK: First sign is - so NEGATIVE MUST WIN

(    -   )(    +   )

You know the the "F" in FOIL means that both first terms must be x

( x - )( x + )

Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15

Like 1 and 15, or 3 and 5

But you also need to add to the I and O in FOIL which means that the two factors
 must add to NEGATIVE 2

Since 3 + -5 = -2, this must be the two factors that will work:

( x - 5 )( x + 3)

It matters which number you have with which sign because the negatives must win!

That's why I always put the sign of the middle term in the first parentheses.

That way, I always know to put the larger number in the first parentheses, so that sign will win.

Now FOIL to see if we're right!


LAST EXAMPLE:

x² - 2xy -15y²
Same problem as the one before, except now there are 2 variables!

Simply use the same factorization and include the y

( x - 5y )( x + 3y)


ALWAYS CHECK BY FOILing or BOXing Back!!

Pre Algebra (Period 2 & 4)

Prime Factorization & GCF 4-3


Prime Factorization
Now that you know what a number divides by, 
you can get its prime factorization


Prime = a number with exactly 2 factors (itself and 1)


Composite = a number with at least 3 factors


1 and 0 are neither composite nor prime


How do you find all the prime factors of a number?

You learned Factor Trees in previous years and Factor Trees are a good strategy

Make sure you take all the bottom factors only!


EXAMPLE: 54




The prime factorization of 54 = 2x3x3x3 or 2(3³)


Make sure you list the factors from least to greatest!

There are multiple Factor Trees based on what you decide to use as your first 2 factors. But each one will get you to the same "bottom" of the tree.

The prime factorization of 54 IS ALWAYS = 2x3x3x3 or 2(3³) no matter how you start your Factor Tree

You can also use something called INVERTED DIVISION.

This is similar to Factor Trees but you must begin with the SMALLEST PRIME FACTOR that goes into the number and keep using it until it no longer works.

Then you go on to the next higher prime factor, etc.

EXAMPLE: For 54 again, but this time using Inverted Division.


THERE IS ONLY ONE CORRECT FORM OF INVERTED DIVISION!
BUT... the factors will always be in order if you follow the rules!!

Pre Algebra (Period 2 & 4)

Exponents 4-2

2⁶ represents 2⋅2⋅2⋅2⋅2⋅2 = 64

the 2 in 2⁶ is the base
the 6 in 2⁶ is the exponent
and together 2⁶ it is the power!!
The base is used as a factor 6 times to produce 64

There are 3 forms here:
2⁶ is in exponential notation
2⋅2⋅2⋅2⋅2⋅2 is in expanded notation
64 is in standard notation

You must use integer rules-- even when raising a negative number to a power!!

odd power = negative
even power = positive
(-5)³ = (-5)(-5)(-5) = -125

(-5)⁴ = (-5)(-5)(-5)(-5) = 625


an odd number of negative signs or an odd power---> it's negative
an even number of negative signs or an even power ---> it's positive

(-2)³ = ( -2)(-2)(-2) = -8
(-2)⁴ = (-2)(-2)(-2)(-2) = 16

If there is a negative BUT NO parenthesis it is ALWAYS negative

-2⁵
is read as the opposite of 2⁵

Look at what the exponent is touching.. and in this case it is just the 2!!
You can also think of -2⁵ as (-1)-2⁵

so this is -1⋅2⋅2⋅2⋅2⋅2 = -32

-2⁴ = -2⋅2⋅2⋅2 or -1⋅2⋅2⋅2⋅2 = -16
BUT REMEMBER
(-2)⁴ = (-2)(-2)(-2)(-2) = +16


THis works with variables as well. WHen you substitute in for variables put the number in parenthesis!! Hugs are important in life-- and equally important in math!!
For example Solve for x
x⁴ - 10 when x = -2
(-2)⁴ - 10
+16 - 10 = 6
But what happens when the expression is
-x⁴ - 10? Still solve for x , when x = -2
This time
the exponent is touching just the x
-(-2)⁴ - 10
You must use PEMDAS and do the exponent first!!
(-2)⁴ = 16 so substitute that back in
-(16) - 10
becomes
-16 -10 = -26

What about 4(2y -3)² when y = 5
substitute in
4[2(5) -3]²
4(10-3)²
4(7)²
4(49) =196


How about
-2x³ + 4y . When x = -2 and y = 3

-2(-2)³ + 4(3)
-2(-8) + 12
16 + 12 = 28

Algebra (Period 1)

Trinomial Squares 6-3

This is a special product that we learned in Chapter 5 when we did FOILing.

FOIL:
(a + 3)² (called a binomial squared)

(a + 3)(a + 3) = a² + 3a + 3a + 9
a² + 6a + 9 (called a trinomial square)



Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
   
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:

1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)

2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?

4) Is the middle term double the product of the square roots of the first and last terms?


IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
 
TO FACTOR A TRINOMIAL SQUARE: a²
 + 6a + 9
1) (     )²

2) Put the sign of the middle term in the (   +   )²

3) Find the square root of the first term and the last term and place in the parentheses:  
(3 + a)²

4) Check by FOILing back. (or using the BOX method)

x² - 14x + 49
(x -7)²

16x² - 56xy + 49y²
ask yourself those important questions. They are all YES... so
(4x - 7y)²

x² -4xy + 4y²
(x - 2y)²
y⁶ + 16y³ + 64
(y³ + 8)²

9a⁸ - 30a⁴b + 25b²
(3a⁴ -5b)²

What about
2x² -40x + 200
What's the first thing you must look for? ALWAYS!!! the GCF
2(x² -20x + 100) now it is a trinomial square
2(x -10)²

similarly with
2x² -4x + 2
2(x² -2x + 1)... again NOW it is a trinomial square
2(x -1)²

18x³ + 12x² + 2x
What's the GCF? 2x

2x(9x² + 6x + 1)
2x(3x + 1)²

But look at
(a +4)² - 2(a +4) + 1
How can that be a trinomial square?
Well, think about the following
A² + 2AB + B² and
A² - 2AB + B²
represent generic trinomial squares
so if we let (a + 4) represent A... it works
[(a +4) -1]²
but we can simplify that to
(a +3)²

You could multiply everything out and then combine like terms,
that is,
(a +4)² - 2(a +4) + 1 becomes
a² -8a + 16 -2a -8 + 1 which then simplifies
a² -6a + 9 which is definitely a trinomial square
(a-3)²
but that's what we got using a simpler method!!

Try
(y+3)² + 2(y+3) + 1
It is a trinomial square in the form
A² + 2AB + B²
so [(y + 3) +1]²
which becomes (y +4)²

Math 6 Honors (Period 6 and 7)

Triangles 4-4
A triangle is the figure formed when three points, not on a line are jointed by segments.

Triangle ABC ΔABC
Each of the Points A, B, C is called a vertex
(plural: Vertices) of ΔABC

Each of the angles angle A. angle B. and angle C is called an angle of ΔABC

In any triangle-
The sum of the lengths of any two sides is greater than the length of the third side
The sum of the measures of the angles is 180

There are several ways to name triangles. One way is by angles

Acute Triangle
3 acute angles

Right Triangle
1 right angle

Obtuse Triangle
1 obtuse angle

Triangles can be classified by their sides

Scalene Triangle
no 2 sides congruent

Isosceles Triangle
at least 2 sides congruent

Equilateral Triangle
all 3 sides congruent


The longest side of a triangle is opposite the largest angle and the shortest side is opposite the smallest angle. Two angles are congruent if and only if the sides opposite them are congruent.

Algebra (Period 1)

Difference of Two Squares 6-2

Again, remember that FACTORING just UNDOES multiplication.

In this case, the multiplication that you'll be UNDOING is FOILING.

FOIL:
(a + b)(a - b)

You will get:
a² - b²
This is the DIFFERENCE (subtraction) of TWO SQUARES.

Now FACTOR:
a² - b²
You undo the FOILING and get:
(a + b)(a - b)


REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!

You will get ADDITIVE INVERSES which will become ZERO



HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:

1) Is it a binomial?

2) Is it a difference?

3) Are both terms perfect squares?


IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!

HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:

1) Double hug  (    )(    )

2) Find square root of each term (sq rt sqrt)(sq rt sq rt)

3) Make one sign positive and one sign negative.
              
(sq rt + sqrt)(sq rt -sq rt)


                           
Of course, they get more complicated! 
We can combine pulling out the GCF with this!

ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!

EXAMPLE:

27y² - 48y⁴ 

First, look for a GCF that can be pulled out.

The GCF = 3y²

Factor out the GCF (look at Chapter 6-1):
3y²(9 - 16y² )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:


3y²(3 - 4y)(3 + 4y)


CALLED FACTORING COMPLETELY BECAUSE
 YOU CANNOT FACTOR FURTHER!


Always check your factoring by distributing or FOILing back!


THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!

a² + b² CANNOT BE FACTORED!!!!!


BUT - b² + a²
= + a² -b²
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)

Algebra (Period 1)

Factoring Polynomials 6-1


Chapter 5 was a very important building block of Algebra but

CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!



REMEMBER THIS KEY CONCEPT:
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL …SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL.)


Factoring is a skill that you must understand to be successful in higher level math!!!

We did a simple version of this back in Chapter 1 and you had a RACE on it!


Factoring is simply UNDOING multiplying


Say you multiplied 5 by 10 and got 50

How would you undo it?
DIVIDE by 5!


So FACTORING uses the concept of DIVIDING.

You're actually undoing the DISTRIBUTIVE PROPERTY.

How?
You look for the most of every common factor....the GCF!

Then you pull out the GCF (divide it out of) from each term,
placing the GCF in front of ( )


EXAMPLE:

FIRST, DISTRIBUTE:
2m²n (2n2 + n + 3)
=
4m²n³ + 2m²n² + 6m²n


Now, pretend you don't want the 2m²n to be distributed anymore...

What should you end up with once you UNDO the Distributive Property?

2m²n (2n² + n + 3)


That's exactly what you started with!

So is it that easy?

Well yes... and no...

Yes because that is the answer
…and
…
No because it was only that easy because I gave you how it started!

You won't know how it started in a real problem!



THIS IS AN EXAMPLE OF THE WAY YOU WOULD USUALLY SEE IT.

The question would say:
FACTOR:
 4m²n³ + 2m²n² + 6m²n


Step 1: What does each term have in common (what is the GCF) ?

They each can be divided by 2m²n


Step 2: Put the GCF in front of a set of ( ) and divide each term by the GCF
2m²n ( 4m²n/2m²n + 2m²n²/2m²n + 6m²n/2m²n)


Step 3: SIMPLIFY and you'll get:

2m²n (2n² + n + 3)



Step 4: Check your answer!!!!!

Always check your factoring of the GCF by distributing back!


(incognito, it should be the same thing)

Another check is to make sure you have factored out the entire GCF.

Look inside the parentheses and ask yourself if there is still any factors in common between the terms.
If there is, then you haven't factored out the GREATEST Common Factor.
For example, let's say in the prior example that you only factored out 2mn instead of 2m²n. You would have:
2mn ( 4m²n³/2mn + 2m²n²/2mn + 6m²n/2mn)


= 2mn(2mn²+ mn + 3m)


If you just check by distributing back, the problem will check.

BUT ...
Look inside the ( ) and notice that each term still has a common factor of m!
So this would not be the fully factored form!
So make sure you always look inside the ( )!!!

RELATIVELY PRIME TERMS -

TERMS WITH NO COMMON FACTORS

THAT MEANS THAT THEY CANNOT BE FACTORED
(GCF = 1)

We say they are "not factorable"

Math 6 Honors (Period 6 and 7)

Angles and Angle Measure 4-3
An angle is a figure formed by two rays with the same endpoints. The common endpoint is called the vertex. The rays are called the sides.
We may name an angle by giving its vertex letter if this is the only angle with that vertex, or my listing letters for points on the two sides with the vertex letter in the middle. We use the symbol from the textbook.

To measure segments we use a rule to mark off unit lengths. To measure angles, we use a protractor that is marked off in units of angle measure called degrees.

To use a protractor, place its center point at the vertex of the angle to be measured and one of its zero points on the side.


We often label angels with their measures. When angles have equal measures we can write m angle A = m angle B
We say that angle A and angle B are congruent angles


If two lines intersect so that the angles they form are all congruent, the lines are perpendicular. We use the symbol that looks like an upside down capital T to mean “is perpendicular to.”



Angles formed by perpendicular lines each have measure of 90° . A 90° angle is called a right angle. A small square is often used to indicate a right angle in a diagram

An acute angle is an angle with measure less than 90°. An obtuse angle has measure between 90° and 180°



Two angles are complementary if the sum of the measures is 90°
Two angles are supplementary if the sum of their measures is 180°

Algebra (Period 1)

Multiplying Polynomials 5-11

To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.

The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.

When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!


2 ways:

1) box method

2)column method
(double or triple distributive with columns to combine like terms)




If you have a trinomial times a binomial, it's easier to use the Commutative Property

and make it a binomial times a trinomial:

(x² + x - 1) (x - 1)

switch it to

(x - 1) (x² + x - 1)

Remember the following rules
(A + B)(A + B) = (A + B) ² = A² + 2AB + B²

(A - B)(A - B) = (A - B)² = A² -2AB + B²

(A + B)(A - B)= A²- B²
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms

You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!

Math 6 Honors (Period 6 and 7)

Points, Lines, Planes 4-1
We can describe but CANNOT DEFINE point, line or plane in Geometry

We use a single small dot to represent a point and in class we labeled with a P and we called it Point P
A straight line in Geometry is usually just called a line...
Two points DETERMINE exactly ONE LINE

we connected Point P with Point Q and created Line PQ
We placed this type of arrow ↔ over PQ to show a line
↔
PQ

that would represent the line PQ but we found we could write
↔
QP
and mean the SAME line!!

You can name ANY line with ANY TWO points that fall on that line!! USE only TWO points to name a line!!


Three or more points on the same line are called collinear. Notice the word "line" in collinear.
collinear

Points NOT on a same line are called noncollinear.

We have a RAY if we have an endpoint and it extends through other points. We name the rame by Naming the endpoint FIRST
→
PQ is RAY PQ and it begins at P and goes through Q. It is NOT the SAME as

→
QP which is Ray QP, which begins at Q and goes through P

Segments are parts of lines with TWO ENDPOINTS.
⎯
PQ is a segment with endpoints Point P and POint Q

We then looked at the drawing from page 105 and the class named all of the names for the line in the drawing, all of the rays that existed in the figure as well as the segments. We found that there were just 3 collinear points: A, X, B but that we could name 3 sets of non collinear points
X, Y, B and A, X, Y, AND A, B, Y

Three non-collinear points determine a flat surface called a plane.
We name a Plane by using three of its non collinear points!! Plane ABC was out example.

Lines in the same plane that do not intersect are PARALLEL lines. Two segments or rays are parallel if they are parts of parallel lines.
↔
AB is parallel to

↔
CD

may be written
↔ ↔
ABllCD
using two straight lines to indicate parallel

Parallel lines DO NOT intersect.

Intersecting lines intersect in a single point!!



Planes that do not intersect are called parallel planes... we looked around the room and found examples of parts of planes.. noticing which ones were parallel!! (the floor and ceiling were a great example)
Then we drew the box from PAge 106 and identified parallel segments and lines from that box.

Two non parallel lines that do not intersect are called SKEW LINES.

Pre Algebra (Period 2 & 4)

Graphing & Writing Inequalities Review

Some words to really know:
At least--> means greater than or EQUAL TO

I want at least $150 to go shopping for holiday gifts!!
m ≥ 150


AT MOST --> less than or EQuAL TO

You might say that you want at most 15 minutes of homework tonight. That means you will take 15 minutes but you sure would like less....
m ≤ 15

Review of some of the most missed problems from our recent races:
y is at most 5
y ≤ 5

x is no more than 7
x ≤ 7

y is at least 20

y ≥ 20

y is less than 8
y < 8. Now 5 < y reads " Five is less than than y" --> if that is true then y must be greater than 5 or

y > 5 so now you can graph easily because you know exactly which way the arrow ( solution set) must be pointing... in the same direction as the inequality symbol. THis ONLY works when you have the variable on the left!! as in y > 5.
ALWAYS CHANGE so the variable is on the left!!

Solve one step inequalities just like you do for equations.. just make sure to end with the variable on the left-- to be able to graph easily

y - 7 > -3
add 7 to BOTH Sides
y > 4
Now you can graph easily... It's an open dot on your graph Because it DOES NOT inlcude 4 as part of the solution

FORMAL CHECK
(1) REWRITE the inequality
(2) Substitute in for the variable--using ANY number that fits the solution (except the boundary number) In this example y > 4,
4 is the boundary number. We showed in class why you can't use the boundary number so if y > 4 then 5, 10, 100, etc would work...
In class we used 10
10 -7 > -3 (Remember to put a "?" over the inequality)
(3) REALLY DO THE MATH. that is, in this case,
do 10-7.
Well 10-7 = 3 so
3 > -3 That's true-- it works!!

If you get something that is not true, chances are you did your problem wrong. re work the problem and see if you can discover your error!!

WHen dividing or multiplying by a negative coefficient (that's the number attached to the variable) you must remember to
REWRITE the inequality AS YOU "FLIP THE SWITCH" (change the inequality symbol)
y/-5 ≥ -3
We need to multiply both sides by a -5 so we must REWRITE and FLIP
(-5)(y/-5) ≤ (-3)(-5)
y ≤ 15

FORMAL CHECK
(1) REWRITE THE INEQUALITY
(2) Substitute in one of the SOLUTIONS-- any one that really fits and is easy to work with!!
(3) DO THE MATH-- look at your results-- does it make sense?... don't just put a happy face!!

(1) y/-5 ≥ -3
well, we found out the y ≤ 15 that means any number less than 15 could be used to check I would pick something that was easy to work with, something that was divisible by 5... like 0, or 5 or even 10
we used 0 in class
(2) 0/-5 ≥ -3
(3) 0 ≥ -3 That's true!! IT WORKED

But what if I used 5?
well step 2 would be
(2) 5/-5 ≥ -3
(3) -1 ≥ -3 and that's still true !!


5y ≤ -35
I just divide by 5. I don't need to switch anything because I am NOT dividing by a negative number
5y/5 ≤ -35/5
y ≤ -7

But what about
-15 ≤ -3m

In this case I must rewrite as I "FLIP THE SWITCH" (Remember the poster from class)
-15/-3 ≥ -3m/-3
5 ≥ m

Before we continue we need to realize that if five is greater than or equal to m... that means m must be less than or equal to 5 or m ≤ 5. Now its easy to graph.

FORMAL CHECK
-15 ≤ -3m
-15 ≤ -3(4)
-15 ≤ -12 which is true!!

In class we used a number that was not correct to show what happens.

When working with Inequalities with 2 steps and variables on both sides:
(1) Do Distributive Property (DP) carefully-- if necessary
(2) combine like terms on each side
(3) "Jump" the variables to one side of the equation
(4) Add or subtract
(5) Multiply or Divide-- being careful if have negative coefficients to rewrite the inequality and "flip the Switch"
(6) MAKE SURE the variable is on the LEFT Side

Algebra (Period 1)

Multiplying Binomials: Special Products 5-10

LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!


Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart. 

When you do, you actually don't need to show any work because you do it in your head! 

(That should make a lot of you happy! :)



DIFFERENCE OF TWO SQUARES:

You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS

(x + 6)(x - 6) =
x² - 6x + 6x - 36 =
x² - 36


This will happen every time!
 
The middle terms are additive inverses so they become zero.


You're left with a difference (subtraction) of two terms that are squared.



SQUARING A BINOMIAL:

When you multiply one binomial by itself (squaring it), you end up with:

First term squared + twice the product of both terms + last term squared


(x + 6)² =
(x + 6)(x + 6) =
x² + 2(6x) + 6² =
x² + 12x + 36


If you foiled you would have:

x²  + 6x + 6x + 36


CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP??? 

WHY???


Another example with subtraction in the middle:

(x - 6)² =
(x - 6)(x - 6) =
x² + 2(-6x) + 6² =
x² - 12x + 36


If you foiled you would have:

x² - 6x - 6x + 36


CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP??? 
WHY???



PLEASE NOTE:

NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE 
MULTIPLICATIONS FASTER!

IF YOU EVER FORGET THEM, JUST FOIL!


(but you will need to recognize them for factoring in Chapter 6)

Algebra (Period 1)

Addition of Polynomials 5-7
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power

You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) + (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:


Subtraction of Polynomials 5-8
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)

EXAMPLE OF COLUMN FORM:
(5x⁴ - 3x² - (-4x) + 3) - (-10x⁴ + 3x³- 3x² - x + 3)
Rewrite in column form, lining up like terms:
5x⁴ - 3x² - (-4x) + 3
- ( -10x⁴ + 3x³- 3x² - x + 3)
-----------------------------------

For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x⁴ + 0x³ - 3x² -(-4x) + 3
-(-10x⁴ +3x³- 3x² - x + 3)
-----------------------------------

DISTRIBUTE THE NEGATIVE, THEN ADD:
5x⁴ + 0x³ - 3x² - (-4x) + 3
+10x⁴ -3x³ +3x² + x - 3
-----------------------------------
15x⁴ - 3x³ + 5 x

Pre Algebra (Period 2 & 4)

Solving 1 Step Inequalities with Addition & Subtraction 2-9


These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same:

ISOLATE THE VARIABLE

You sitll balance with the INVERSE OPERATION

Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



Solving 1 Step Inequalities with Multiplication & Division 2-10

These are solved just like equations except you have a greater than, less than, greater than or equal, less than or equal sign instead of the equal sign.


Your goal is still the same: ISOLATE THE VARIABLE


You sitll balance with the INVERSE OPERATION


Your justifications are still almost the same:
IDENTITY AND PROPERTIES OF INEQUALITY



THERE IS ONE MAJOR EXCEPTION!!!!

When you multiply or divide by a NEGATIVE, 
the symbol CHANGES DIRECTION!


-3y > 9

You need to divide both sides by NEGATIVE 3 so the symbol will switch from > to < in the solution
 y < -3 is the answer

 If you want to understand why:
 3 < 10 correct? 
Now multiply both sides by -1 (mult prop of equality)
 You get -3 < -10, but THAT'S NOT TRUE!!! 
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10

Algebra (Period 1)

Polynomials 5-5

Polynomials = SUM of monomials

Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰

1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Some TERMS YOU NEED TO KNOW

Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term




More on Polynomials 5-6

Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest powerr first ( actually NEVER used in practice)

Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Pre Algebra (Period 2 & 4)

Inequalities & Their Graphs 2-8

What most real life situations call for

How often to you think...I need at least $20 (not exactly $20)
....
I want at most 20 minutes of homework (not exactly 20 minutes)?


Inequalities have many answers (most of the time an infinite number!)

Examples: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)

n ≥ 3 means still means that every real number greater than 3 is a solution, 
but now 3 is also a solution


n < 3 means that every real number less than 3 is a solution! (but NOT 3)
 n ≤ 3 means still means that every real number less than 3 is a solution,   but now 3 is also a solution

 GRAPHING INEQUALITIES:
 First, graphing an equation's solution is easy
 1) Say you found out that y = 5, you would just put a dot on 5 on the number line
 2) But now you have the y ≥ 5
 You still put the dot but now also darken in an arrow going to the right showing all those numbers are also solutions
 I say: the equal sign part of it is a crayon...Pick up the crayon and color in the circle
 3) Finally, you find in another example that y > 5

You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!

I say: there isn't a crayon so you can't color in the circle.
Therefore it stays open!
 

TRANSLATING WORDS:

Some key words to know:


AT LEAST means greater than or equal that is...
≥

AT MOST means less than or equal. THat is ≤


I need at least $20 to go to the mall means I must have $20, but I'd like to have even more!


I want at most 15 minutes of homework means that I can have 15 minutes, but I'm hoping for even less!

Pre Algebra (Period 2 & 4)

Adding & Subtracting Decimals Equations 3-5

Most important... LINE UP THE DECIMALS!!
We did a number of examples from the textbook... check out this section!!


Make sure you use a SIDE BAR for your calculations!!


Multiplying & Dividing Decimal Equations 3-6


For multiplication, count the decimals and place the decimal from the right!!
THese are difficult to show here as well...but

( m/7.2) = -12.5

multiply both sides by -7.2

(-7.2)(m/-7.2) = -12.5(-7.2)
m = 90

Make sure to use a side bar and multiply carefully!!

s/2.5 = 5

multiply both sides by 2.5

(2.5)(s/2.5) = 5(2.5)
3 = 12.5


4.5 = m/-3.3

multiply both sides by -3.3
Notice your answer will be a negative!!

(-3.3) (4.5) = [m/-3.3](-3.3)

-14.85 = m

... and I used a sidebar in class to show the multiplication!!

For division

0.9r = -5.4
Divide both sides by 0.9

.9r = -5.4
0.9 0.9

r = -6

use a sidebar to divide carefully

divide both sides by -0.9
-0.9n = -81.81

-0.9n/-0.9 = -81.81/-0.9

notice that your solution will be a positive!!
and use a side bar to divide carefully

n = + 90.9

Algebra (Period 1)

Scientific Notation 5-4



You've had this since 6th grade!
It is just a bit more complicated


You restate very big or very small numbers using powers of 10 in exponential form


Move the decimal so the number fits in this range:

less than 10 and greater than or equal to 1
That is 1 ≤ n < 10 Scientific notation is the product of two factors: one of the factors is a number (1)1 ≤ n < 10 and the other factor (2) is a power of ten 
Count the number of places you moved the decimal and make that your exponent 
Very big numbers - exponent is positive 
Very small numbers (decimals) - exponent is negative (just like a fraction!) 

Remember that STANDARD notation is what you expect (the normal number)
 
When you multiply or divide scientific notations, use the power rules! 
 Just be careful that if your answer does not fit the scientific notation range, that you restate it. (2.2 x 10 ^-3)(3.0 X 10 ⁵)
(2.2)(3.0) X 10 ^-310 ⁵

6.6 X 10 ²

(6.1 x 10 ⁹)(2.5 x 10^-4)
(6.1)(2.5) x 10 ⁹⁺⁴
15.25 x 10 ⁵
But that isn't in scientific notation so change just 15.25 into proper scientific notation first
that is
15.25 = 1.525 x 10¹
put that back in your work
1.525 x 10¹ ⋅ 10⁵
1.525 X 10⁶


(2.5 x 10 ^-3)(4.0 X 10 ^-8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 ^-3)(10 ^-8)

(2.5)(4)= 10
so initially you have 10 X 10 ^-11
but 10 does not fall into the required 1 ≤ n < 10 so changed 10 into scientific notation or 1.0 X 10¹
so you have 1.0 X 10¹ X 10 ^-11
or 1 X 10 ^-10

(5.4 X 10^-6)(5.1 X 10 ^-8)

(5.4)(5.1) = 27.54
so
27.54 X 10 ^-6+-8
27.54 X 10 ^-14
but again 27.54 isn't in scientific notation... changed just that number into scientific notation... FIRST
2.754 X 10¹
put it back into your work
2.754 x 10¹ ⋅ 10^-14
2.754 X 10 ^{1 + -14}
2.754 X 10 ^-13


What about
9.0 X 10 ⁸
3.0 X 10 ²

do your factors separately just like you did with multiplication
That is,
9.0 = 3
3.0


and
10⁸
10²

= 10 ^8-2
=10⁶
so the answer is
3 x 10 ⁶

6.9 X 10 ⁴
3.0 X 10 ⁸

2.3 X 10 ^4-8
2.3 X 10 ^-4

2.7 X 10 ¹²
9.0 X 10 ¹²
Divide 2.7 by 9
you get
0.3
10^{12 -12} = 10 ⁰
but wait 0.3 isn't in the correct for so change that..
0.3= 3.0 x 10^-1
so the answer is
3.0 X 10 ^-1





6.0 x 10 ⁷
3.0 X 10²




First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 ^{7 -2} or 10⁵
so 2.0 X 10⁵

4.2 X 10⁵
2.1 X 10³

= 2 X 10²

2.5 X 10 ^-7
5.0 X 10 ⁶
first 2.5/5 = 0.5 and 10^-7/10⁶ = 10^-7-6 = 10^-13
so initially you have
0.5 x 10^-13 but 0.5 is not within 1 ≤ n < 10 so change 0.5 to scientific notation 5.0 X 10 ^-1 and mult that by 10^-13
5 X 10 ^-14


How about this harder one:

(3.6 X 10⁶)(4 X 10 ^-3
(4.8 x 10 ^-2)(1.2 X 10 ⁶)

put all your factors together and see if you can simplify before you begin...

(3.6)(4)
(4.8)(1.2)

You can simplify
(3.6)
(4.8)


isn't that 3/4
and then your 4's simplify
left with
3.o
1.2
which simplifies to 2.5 ... so going back to the original problem you now have
2.5 x 10^6-3
10 ^{-2 + 6}
which
becomes
2.5 X 10 ^3-4 or
2.5 X 10 ^-1

Math 6 Honors (Period 6 and 7)

Multiplying Decimals 3-8

According to our textbook:
Place the decimal point in the product so that the number of places to the right of the decimal point in the product is the sum of the number of places to the right of the decimal point in the factors!!

You do NOT need to line up the decimal point when you are multiplying.

11.32 X 8.73

11.32
8.73

98.8236

estimate and you get 11 X 9 = 99

What would you do if you had
(19.81 x 5.1) + (19.81 X 4.9)

Wait... wait... remember the Distributive Property????
Look you can use it to make this problem soooooo much easier
19.81(5.1 _ 4.9)
19.81 (10) = 198.1


How about 50(.25) + 50(.75)
This is one you can even do in your head because
50(0.25 + 0.75) = 50 (1) = 50






Dividing Decimals 3-9

According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.

Check out our textbook for some examples!!

When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.

Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!

the quotient is 0.3431 which rounds to 0.343


To divide one decimal by another

Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number


Divide the new dividend by the new divisor

Check by multiplying the quotient and the divisor.

Algebra (Period 1)

POWER TO ANOTHER POWER


MULTIPLY the POWERS
(m⁵)³ = m¹⁵


To check, EXPAND it out:
(m⁵)(m⁵)(m⁵) = m¹⁵



PRODUCT TO A POWER

DISTRIBUTE the power to EACH FACTOR
(m⁵n⁴)³ = m¹⁵n¹²



RAISING A QUOTIENT TO A POWER:


DISTRIBUTE THE POWER to the numerator and the denominator

(m²/n⁶)³ = m⁶/n¹⁸

Math 6 Honors (Period 6 and 7)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the nth power of ten, move the decimal point n places to the right.

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of two factors

(1) a number greater than or equal to 1, but less than 10,

AND

(2) a power of ten

are said to be written in scientific notation.
We can write 'a number greater than or equal to 1, but less than 10' as an mathematical inequality 1 ≤ n < 10 To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point. 4,592,000,000 in scientific notation First move the decimal point to the left to get a number between 1 and 10 4,592,000,000 the first factor in scientific notation becomes 4.592 Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation



4.592 x 10⁹ (Yes, you get to use the × symbol for multiplication .. but only for this!!


When we move a decimal point to the left, we are actually dividing by a power of ten.


Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros


3.1 ÷ 10⁴ = 0.00031

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

Math 6 Honors (Period 6 and 7)

Adding & Subtracting Decimals 3-6

RULES TO FOLLOW:
1. Write the given numbers one above the other (STACK'EM) with the decimal points lined up!!

2. Add zeros to get the same number of decimal places and then add/sub just like whole numbers.

3. Place the decimal point in your sum/ difference in the position right under the decimal points in the given numbers.

6.47 + 340.8 + 73.523

STACK THEM!!
The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. You can see if your answer is reasonable by using estimation.

Algebra (Period 1)

Exponents: 5-1
POWER RULES:


MULTIPLYING Powers with LIKE BASES:

Simply ADD THE POWERS

m⁵m³ = m⁸

You can check this by EXPANDING:
(mmmmm)(mmm) = m⁸



DIVIDING Powers with LIKE BASES:

Simply SUBTRACT the POWERS

m⁸/m⁵ = m³     


Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm = mmm

ZERO POWERS:

Anything to the zero power = 1


(except zero to the zero power is undefined)


Proof of this was given in class:

1 = mmmmmmmm/mmmmmmmm
= m⁸/m⁸
= m⁰ (by power rules for division)
       


By the transitive property of equality : 1 = m⁰


Review the odd/even rule

IF THERE IS A NEGATIVE INSIDE PARENTHESES:

Odd number of negative signs or odd power = negative

Even number of negative signs or even power = positive


EXAMPLES:
(-2)⁵ = -32

(-2)⁴ = +16



IF THERE IS A NEGATIVE BUT NO PARENTHESES:

ALWAYS NEGATIVE!!!!

-2⁵ = -32

-2⁴ = -16

JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS


They're in the wrong place in the fraction

m³/m⁵ = m^-2
        

m³/m⁵ = mmm/ mmmmm
= 1/mm


Again, by transitive property of equality:

m³/m⁵ = m^-2 = 1/m²

Remember the rule of powers with (  ) 
When there is a product inside the (  ), then everything inside is to the power!

If there are no (  ), then only the variable/number right next to the power is raised to that power.

3x^-2 does not equal (3x)^-2
The first is 3/x² and the second is 1/9x²

RESTATE A FRACTION INTO A NEGATIVE POWER:

1) Restate the denominator into a power

2) Move to the numerator by turning the power negative


EXAMPLE: 
1/32
 = 1/(2)⁵ = (2)^-5

Math 6 Honors (Period 6 and 7)

Rounding 3-5 (continued)

Round the following number to the designated place value:
509.690285

tenths: 509.690285
You underline the place value you are rounding to and look directly to the right. If it is 0-4 you round down; if it is 5-9 you round up 1.
so here we round to
509.7

hundredths
509.690285
becomes 509.69

hundred-thousandths
509.690285
becomes
509.69029

tens
509.690285
becomes
510

(a) What is the least whole number that satisfies the following condition?

(b) What is the greatest whole number that satisfies the following condition?
A whole number rounded to the nearest ten is 520.
Well, 515, 516, 517, 58, 519, 520, 521, 522, 523, 524 all would round to 520
so

(a) 515
(b) 524

A whole number rounded to the nearest ten is 650
(a) 645
(b) 654

A whole number rounded to the nearest hundred is 1200
(a) 1150
(b) 1249
How about these...
(a) What is the least possible amount of money that satisfies the following condition?
(b) What is the greatest possible amount?

A sum of money, rounded to the nearest dollar is $57
(a) $56.50
9b) $57.49

A sum of money rounded to the nearest ten dollars $4980
(a) $4975
(b) $4984.99

Pre Algebra (Period 2 & 4)

Decimal Review

First up: Place value

The root word in decimal is "decem" which means 10.

Each time you move in the decimal system you are multiplying or dividing the place value by 10.
If you move to the LEFT, place value is MULTIPLIED by 10. If you move to the RIGHT, place value is DIVIDED by 10 (or multiplied by 1/10...remember that dividing by 10 is the same as multiplying by its reciprocal...says our BFF, the Multiplicative Inverse Property)

The middle of the decimal system is the ones place (not the decimal!)

Place value names mirror each other to the left and to the right of the ones.
EXAMPLE: Take 1 and multiply by 10 and you get the TENS.

Take 1 and divide by 10 (or multiply by 1/10) and you get the TENTHS.

Another pattern in PLACE VALUE NAMES:

They are related to how many groups of "000" after 1,000
1 million = 1,000,000 

million is from "mille" which meant 1000 and 1 million is 1000 times 1000 or 1000¹ bigger than 1000
1 billion = 1,000,000,000 as in 1000 times 1000 times 1000 or 1000² bigger than 1000
One more example: 
1 trillion = 1,000,000,000,000 as in 1000³bigger than 1000

SO TO MAKE THIS EASIER TO UNDERSTAND:

EACH NEW PLACE VALUE NAME IS COUNTING THE GROUPS OF  ZEROS (000) AFTER 1000!

1 QUADrillion...QUAD means 4 so 4 groups of zeros after 1000:
1,000,000,000,000,000

AND OTHER THE OTHER SIDE OF THE DECIMAL, THE NAMES ARE THE SAME EXCEPT THEY HAVE A "TH"

What you need to do on the other side is to remember that the ones is still the starting point
on the left side of the decimal point though. So if you're looking for 1 millionth, you count 6
places in total = .000001
You see that there are only 5 zeros, not 6

1 millionth = 1 times 1/1,000,000 = .000001

Instead of writing all these zeros, we can use exponents.

Exponents for powers of 10 count the number of zeros.
So 1,000,000 = 106
and one millionth = 1/1,000,000 = 10-6

ROUNDING: 
Round 0 - 4 down and 5 - 9 up

EXAMPLE: 0.34 rounded to the tenths is 0.3
0.35 rounded to the tenths is 0.4
Notice you don't add zeros at the end
If there is a 9 in the next place, then think of rounding up as adding 1, that will make the number go up to "10"
EXAMPLE: 0.96 rounded to the tenths is 1.0

Think of it as adding 1 in the tenths column which would carry to the ones.
Now there is a zero in the tenths to show the reader of the number that you rounded to the tenths and not the ones.

EXAMPLE: 24,300 rounded to the thousands is 24,000

Now you need zeros to the decimal point or you'll lose the place value!

EXAMPLE: 29,600 rounded to the thousands is 30,000
Again, if it's a 9, it will need to go up to the next column...
Think of it as adding 1

Math 6H (Period 6 & 7) Yosemite Week

Comparing Decimals 3-4

Printed Notes will be handed out in class on Monday, November 1st.

We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.


In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:


1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68. 2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 . 3. 0.83 and 0.833 To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833 0.83 = 0.830 Then compare 0.830 and 0.833 since 0 <3 Then 0.830 < 0.833. Write in order from least to greatest 4.164, 4.16, 4.163, 4.1 First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100 The order of the numbers from least to greatest is 4.1, 4.16, 4.163, 4.164
Rounding 3-5

A method for rounding may be stated as follows:

Find the place to which you wish to round, mark it with an underline ___

Look at the digit to the right.

If the digit to the right is 5 or greater, add 1 to the marked digit.

If the digit to the right is less than 5, leave the marked digit unchanged.

Replace each digit to the right of the marked place with a 0

Round 32,567 to (a) the nearest ten thousand,
(b) the nearest thousand,
(c) the nearest hundred, and
(d) the nearest ten

(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros

30,000

(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros

33,000

(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros

32,600

(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero

32,570

A similar method of rounding can be used with decimals.
The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.

Round 4.8637 to
(a) the nearest thousandth,
(b) the nearest hundredth,
(c) the nearest tenth, and
(d) the nearest unit

a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7

4.864

b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7

4.86

c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7

4.9

d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7

5

Math 6H (Period 6 & 7)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.

Remember how we proved that any number to he zero power was equal to 1
or a⁰ = 1

Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3
or 3⁴⁺² = 3 ⁶
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract

3⁴ /3² = 3^4-2 =3²

What would happen if we had
3² / 3⁴ ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3

Which would be
1
3²

or 1/3²

but you can write that as 3^-2
We just subtract-- using the same rule.


Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN

SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point

Math 6H (Period 6 & 7)

The Decimal System 3-2

Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)

Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?

Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT
Think of the comma as a 'signal' for you to mention a place name category!!

In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."

The expanded notation for 420,346 is given by

4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)

Using exponents the expanded notation may be given as
4(10⁵) + 2(10⁴)+0(10³)+3(10²)+4(10¹)+6(10⁰)

What hmmm.. how is (10⁰) = 1
any number to the zero power is 1.


From yesterday we noticed that
10² ⋅ 10³ = 10⁵
and
10⁶ ⋅ 10³ =10⁹
so we wrote a rule for any exponent values a and b
10^a ⋅ 10^b = 10^a+b
Then I asked what would be a rule for

10⁸
10⁵
and
10⁶
10⁴
Students decided that since we added the exponents when multiplying we could subtract the exponents when dividing!!

so could we write a rule for any exponent values a and b?
YES
10^a
10^b
= 10^a-b
but then
10⁵
10⁵
would be
10 ^5-5 which is 10⁰
BUT WAIT... isn't any number divided by itself equal to ONE. That is,
10⁵
10⁵
= 1
So
10⁰ = 1

279,043
is read as two hundred seventy-nine thousand, forty-three.
and in expanded notation becomes:
2(100,000) + 7(10,000) + 9(1000) + 0(100) + 4(10) + 3(1)

or with exponents:
2(10⁵) + 7(10⁴) + 9(10³) + 0(10²) + 4(10¹) + 3(10⁰)
Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???

Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.

so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value

10t + 2 becomes the correct expression

What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 in front of the x for the ones' digit? No it is... invisible!!

Some of you asked that I put the Magic Number Trick that we did online... try this one on your parents...

Have your 'victim' follow these directions:
Choose any four-digit number
Write the thousand's digit
Write the thousands' and hundreds' digits
Write the thousands' hundred's and tens' digits.
Add these numbers

Multiply by 9

Find the sum of the digits of the original number
add this sum to the previous results.

Your 'victim's' answers should always be the same as their original number chosen.
Example
I used in class 1492
Write the thousand's digit-------> 1
Write the thousands' and hundreds' digits ---> 14
Write the thousands' hundred's and tens' digits--->149
Add these numbers 1 + 14 + 149 = 164
Multiply by 9 or 164(9)= 1476

Find the sum of the digits of the original number 1 + 4 + 9 + 2 = 16
add this sum to the previous results. 1476 + 16 = 1492

Voila!! It worked!!
Email me your parent's comments or post them here!!

Pre Algebra (Period 2 & 4)

Solving Multi-Step Equations 7-2


2 STEP EQUATIONS WITH SOME MORE STEPS BEFORE BALANCING



1) Sometimes equations have LIKE TERMS on the SAME SIDE.


The easiest first step is to COMBINE these before starting to balance:


9x -2x = -42
combine like terms ( those terms on the same side of the = sign)
7x = -42
now divide by 7
7x/7 = -42/7
x= -6


4a + 1 - a = 19

ADD THE 4a to the -1a FIRST

3a + 1 = 19


NOW SOLVE AS A 2 STEP EQUATION
3a + 1 = 19
-1 = -1
3a = 18
then
3a/3 = 18/3

a = 6



2) Sometimes equations have DISTRIBUTIVE PROPERTY on one side:

Usually, you want to do DISTRIBUTE FIRST!


UNLESS THE FACTOR OUTSIDE THE ( ) CAN BE DIVIDED
OUT OF BOTH SIDES PERFECTLY!!!!
(meaning it's compatible!)
In the following example you must distribute -2(2y + 8)


EXAMPLE: 
5y - 2(2y + 8) = 16

5y - 4y - 16 = 16 [distribute]

y - 16 = 16 [collect like terms]

y = 32 [solve by adding 16 to both sides]



But, with the following example you can divide both sides by 3 BEFORE you distribute
EXAMPLE:
3(4 + x) = 9

4 + x = 3 [Don't distribute! Divide both sides by 3.
The 3 goes into both sides perfectly!

x = -1 [Subtract 4 from both sides]

38 = -3(4y +2) + y
38 = -12y + -6 + y
38 = -11y - 6
+6 +6
44 = -11y
so then
44/-11 = -11y/-11


-4 = y

Useful Steps for Solving A Multi-Step Equation

Step 1: Use DP if necessary
Step 2: Combine like terms
Step 3: Undo Addition or subtraction using +prop= or -props=
Step 4: Undo Multiplication or division using xprop+ or ÷prop=

Algebra (Period 1)

WORD PROBLEMS--REVIEW Continued

Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.

GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches

AGE PROBLEMS:
Mary and Sam's ages sum to 50

Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.


Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.

x = one integer and y = other integer x + y = 26

x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22

Algebraic Inequalities:

TRANSLATING WORDS:

Some key words to know:

AT LEAST means greater than or equal

AT MOST means less than or equal

I need at least $200 to go to the mall means I must have $200, but I'd like to have even more!

I want at most 15 minutes of homework means that I can have 15 minutes, 
but I'm hoping for even less!
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.

For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.

n + (n + 2) > 50
n > 24

The smallest odd integers in the set are 25 and 27

For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.

n + (n + 2) < 50
n < 24

The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)

Pre Algebra (Period 2 & 4)

Solving Two-Step Equations 7-1
TWO STEP EQUATIONS:

1. Use the ADDITION PROPERTY OF EQUALITY (first) 
(get rid of addition or subtraction)

Addition property of equality abbreviated as +prop=

Subtraction property of equality abbreviated as -prop=


2. Use the MULTIPLICATION PROPERTY OF EQUALITY (second) 
(get rid of multiplication/division)

Multiplication property of equality abbreviated as ×prop=
Division property of equality abbreviated as ÷ prop=

DID YOU NOTICE THAT THIS IS THE OPPOSITE OF AUNT SALLY?


That's because solving equations is actually doing order of operations backwards!



EXAMPLE:

3x + 7 = 28
         
-7  -7     

SUBTRACT 7 FROM BOTH SIDES

3x = 21

NOW DIVIDE BOTH SIDES BY 3


3x = 21

3     3



x = 7


CHECKING A 2 STEP EQUATION:

You'll need 2 steps to check, too!

(1) Re write the equation

3x + 7 = 28

(2) SUBSTITUTE YOUR ANSWER AND PUT A ? OVER THE =



3(7) + 7 = 28

USE ORDER OF OPERATIONS - MULTIPLY FIRST

(3) Do the Math!! Really do the Math!!
21 + 7 = 28

28 = 28


-d/7 + 21 = 0

(okay addition or subtraction first so... since + 21 O will need to subtract 21_
-d/7 + 21 = 0
-21 -21
-d/7 + 0 = -21

-d/7 = -21
Now since it is minus d divided by 7 I need to multiply both sides by 7
but if I only use 7 I will still have a negative d and I want to end up with just d... so what if we multiply both sides by -7

(-7)(-d/7) = -21(-7)
d = 147

9 - 3p = -27
most students think that the - sign is attached to the 0 but IT IS NOT!!
this is 9 minus 3p so
9- 3p = -27
you need to subtract 0 from both sides to clear the +9 in the original equation
9 -3p = -27
-9 -9
-3p = -36

Wait how did I get -36... well we have -27 - 9 on the right side and using the integer rules... same sign rule... just add them and use their sign...

now
-3p = -36
If I divide both sides by 3 I still would have a negative on the left with the variable so I must divide both sides by -3
-3p = -36
-3 -3

p = 12

Algebra (Period 1)

WORD PROBLEMS--REVIEW
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!

CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times you can translate words into Algebra word by word just like you translate English to Spanish or French.

5 more than a number

n + 5
the product of 5 and a number

5n

the quotient of 5 and a number

5/n

the difference of a number and 5

n - 5

NOTE: Because multiplication & addition are both commutative, when solving for a solution the order will not matter for the solution BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words... AND, FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER....GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS EXCEPT (counterexample!)...

5 less THAN a number
or

5 subtracted FROM a number
Both of these are: n - 5

The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5

For word problems like someone's age or the amount of money you have, you also should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.

EXAMPLE: Tom is 3 years older than 5 times the age of Julie

Translating: 3 + 5J

Does that make sense? Is Tom a lot older than Julie or is Julie older?

Try any age for Julie. Say she is 4 years old.

3 + 5(4) = 23

In your check, Tom is 23.
Is Tom 3 years older than 5 times Julie's age?

YES!
You're algebra is correct!



STRATEGY #2: DRAWING A PICTURE

I have 5 times the number of quarters as I have dimes.

Let’s say I first translate to: 5Q = D
I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes

Does this make sense? That would mean I have a lot more dimes than quarters.

The original problem says I have a lot more quarters!

My algebra is WRONG!
I need to switch the variables.

5D = Q

I check: If I assume that I have 20 quarters, then 5D = 20
D = 4

Does this make sense? YES!
I have 20 quarters and only 4 dimes.

Sometimes it helps to make a quick picture.

Imagine 2 piles of coins.

The pile of quarters is 5 times as high as the pile of dimes.

You can clearly see that you would need to multiply the number of dimes
to make that pile the same height as the number of quarters!



STRATEGY #3: MAKE A T-CHART
-- this is my favorite!!
To translate known relationships to algebra, it often helps to make a T-Chart.

You always put the unknown variable on the LEFT side and what you know on the right.

Fill in the chart with at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?

Then, you use that mathematical relationship with a variable.


EXAMPLE: The number of hours in d days

Your unknown is d days so that goes on the left side:

d days l number of hours

1 ------l------24

2 ------l------48

3 ------l------72

Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each 
You must MULTIPLY the left column BY 24 to get to the right column

The last line of the chart will then use your variable d
d days number of hours

d days l number of hours

1 ------l------24

2 ------l------48

3 ------l------72

d ------l-----24d



EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:

h hours l number of days

24------l------- 1

48------l------- 2

72 ------l-------3
(Why did I start with 24 and not 1 hour this time?)
 Now look at the relationship between the left column and the right column. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" 
You must DIVIDE the left column BY 24 to get to the right column

The last line of the chart will then use your variable h
h hours number of days

h hours l number of days

24------l------- 1

48------l------- 2

72 ------l-------3
h -------l-------h/24



Some interesting translations used all the time in Algebra:

The next consecutive number after n: n + 1

Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6


The next EVEN consecutive number after n: n + 2

Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14


The next ODD consecutive number after n: n + 2

Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11