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!