Math 6H (Period 3, 6 & 7)

Quotients of Integers 11-6

We all remember 2⋅ 5 = 10
and we know corresponding information
10÷ 5 = 2

The quotient of two positive OR two negative integers is POSITIVE!!

The quotient of a positive AND a negative integer is NEGATIVE!!

The same rules of multiplication apply to division. The same life story!! :-)

Although we talked about the sum of two integers always being an integer and
the difference of two integers always being an integers...

the QUOTIENT of two integers is NOT ALWAYS an integer!!

10/4 = 2 1/2 --> which is NOT an integer!!

We also reviewed:

10/0 --> is undefined!!
whereas,
0/10 = 0

98/-14 = -7

Simplify the following:
6 × 8 + -3 × 5
7 × 5 + -3 × 8

Make sure you perform operations using PEMDAS
(also called Aunt Sally's rules or even Order of Operation O³)

6 × 8 + -3 × 5
7 × 5 + -3 × 8

= 3

Pre Algebra Period 1

CHAPTER 5-3 ADDING AND SUBTRACTING FRACTIONS

(positive only)

1) Check for a common denominator

2) If you don't have one, use the LCM as the LCD (least common denominator)

3) Use equivalent fractions to restate to the common denominator

4) When subtracting, be sure to BORROW IF YOU NEED TO!



Adding or subtracting VARIABLES:

If the variable is just in the numerator,
just follow the normal method of finding a common denominator

EXAMPLE:
x/2 + 3x/5 
The common denominator is 10

(5)x/10 + 3x(2)/10      
  (
5x + 6x)/ 
10    = 11/10  




If the variable is in the denominator, but they are the same variables, just add/subtract normally because you have a common denominator

EXAMPLE:
3/y + 7/y = 10
/y  

If in the denominator, but the denominators are not the same, multiply the two denominators together and use equivalent fractions just as you would do with number denominators.


This time you will have to multiply by a variable.
EXAMPLE:
3/y - 7/10 
 
The LCD is 10y:

[(10)3 - (y) 7 ]/
(10) y   =

(30 - 7y)/10y

 
There's also a trick to this that I'll show you in class!
You simply multiply the 2nd denominator by the 1st numerator.
Then multiply the 1st denominator by the 2nd numerator.
The denominator is the product of both denominators.
Only works if there are just 2 terms!!!

NEGATIVE FRACTIONS


1) Double check any subtractions just as you would for integer problems

2) Place the bigger fraction on the top (no matter the sign)

3) Restate to common denominators if needed

4) Borrow if the fraction below is larger than the fraction above

5) Make sure your answers have consistent signs - In other words, if you have a negative fraction, make sure your whole number part is also negative

OR

1) You can simply make all mixed numbers into improper fractions first

2) Find a common denominator

3) Use integer rules with the numerators

4) Restate back into mixed numbers if required

EXAMPLE using both methods:


5 2/3 - 10 1/4


KEEP THEM AS MIXED NUMBERS:
First of all, you know the final answer will be NEGATIVE so make sure it is!
STACK THEM!! With the WINNER on TOP

10 1/4
5 2/3


Find a common denominator:

10 3/12

5 8/12


Borrow because the bottom number is bigger than the top number

9 15/12

5 8/12

Use integer rules to add or subtract:

-4 7/12


Check: If so, you've got the answer:
(if not, you needed to borrow and you forgot to!)
-4 7/12



Pre Algebra Period 1

Multiplying & Dividing Fractions 5-4

This is the easiest fraction skill because you don't need a common denominator. 

MULTIPLICATION

1. Turn any mixed number into an IMPROPER FRACTION

2. Simplify if possible (it makes the math easier!)

3. Multiply numerator x numerator and denominator x denominator

4. Simplify if necessary



DIVISION:

WE NEVER DIVIDE, WE FLIP THE SECOND FRACTION AND MULTIPLY!

(never, ever touch the first fraction!)

1. Turn any mixed number into an IMPROPER FRACTION

2. FLIP THE SECOND FRACTION AND CHANGE TO MULTIPLICATION

3. simplify if possible (it makes the math easier!)

4. Multiply numerator x numerator and denominator x denominator

5. Simplify if necessary



NEGATIVES:

Follow your integer rules (FINGER RULES) to determine the sign



VARIABLES:
You can cross cancel them as well!

Math 6H (Period 3, 6 & 7)

Products with One Negative Factor 11-4


3 ⋅ -2 = -6
Its really repeated addition
or
-2 + -2 + -2 which we learned a few sections ago was equal to -6.

The product of a positive integer and a negative integer is a negative integer.

The product of ZERO and any integer is ALWAYS ZERO!!
a⋅0 = 0

Math imitates life...and Karma(?)
What was the story I told in class... it applies to
Multiplication & Division ...
+ ⋅ + = +
- ⋅ + = -
+ ⋅ - = -
-⋅ - = +

Products with Several Negative Factors 11-5

The product of -1 and any integer equals the opposite of that integer.
(-1)(a) = -a

The product of two negative integers is a positive integer

For a product with NO ZERO factors:
-->if the number of NEGATIVE factors is odd, the product is negative
-->if the number of NEGATIVE factors is even, then the product is positive

Every integer and its opposite have equal squares!!

Remember-- if its all multiplication use the Associative & Commutative Properties of Multiplication to make your work EASIER!!

Algebra Period 4

ZERO PRODUCTS PROPERTY: 6-8

A new friend that you can ALWAYS count on!

For any two rational numbers a and b, if ab = 0,
then either a = 0, b = 0 or both equal zero
HOW DOES OUR NEW FRIEND HELP US?
Try to solve x² + 10x + 24 = 0
You'll -24 from each side: x² + 10x  = -24
Now what????
It's a quadratic (x² term).
You can't isolate x because there's another term with x² in it.
We need the Zero Products Property to solve quadratics.


Using this property, we can solve quadratic equations by factoring the equation and setting each factor to zero and solving.

EXAMPLE: (5x + 1)(x - 7) = 0

Using the Zero Products Property, we know that either 5x - 1 must equal zero
or x - 7 must equal zero.
(That's the only way the product could be zero.)

Set both equal to zero and then solve:

if 5x + 1 = 0 then x must be -1/5

if x - 7 = 0 then x must be 7

If you substitute these answers for x back in the original equation, they will both end up as 0 = 0

(IN OTHER WORDS, THE SOLUTIONS WORK!)


In the first example, the equation was already factored.
Now you will first need to factor the equation (if possible), and then set each factor equal to zero to solve.
MAKE SURE YOU MOVE EVERYTHING TO ONE SIDE OF THE EQUATION!!!!!!
YOU MUST HAVE ZERO ON ONE SIDE OF THE EQUATION TO USE THE ZERO PRODUCTS PROPERTY!

EXAMPLE: x² = 16

First move the 16 to the left side: x²2 - 16 = 0

Now factor as the difference of two squares: (x + 4)(x - 4) = 0

Set each factor equal to zero and solve: x + 4 = 0 or x - 4 = 0

x = -4 or x = 4
Why do you have to have zero on one side? Just think about the simple equation
ab = 24.
Can you solve this? a could be 1 and then b would be 24. But a could be 3 and then b would be 8. Or a could be 4 and be would be 6. Or a could be 5/7 and then be would be 168/5. There are infinite possibilities!
The only way you can find 2 answers definitively is if the product is 0.
If ab = 0, then for sure either a = 0 or b = 0.
Otherwise the product could not be 0! 



NEW ALGEBRA TERM TO LEARN!

root: any solution that turns the equation into the value of zero is called a root of the polynomial, or a ZERO of the polynomial because when you're graphing the polynomial the y value is zero at this point

So if the directions say "Find the roots of......." it just means get zero on one side of the equation, factor, set each factor equal to zero, and solve.

SUMMARY OF SOLVING QUADRATICS: 
solutions = roots = zeros of the polynomial
1. Get zero on one side of the equation
2. Factor

3. Set each piece (factor) equal to zero

4. Solve as one or two step

5. Check by substituting in the ORIGINAL equation

Pre Algebra Period 1

How to Change repeating decimals to fractions

Step 1: set up "n = the repeating decimal
Step 2: determine how many numbers are under the bar
Step 3: use that number as a power of 10
Step 4: multiply both sides of the equation in Step 1 by that power of 10
Step 5: Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation
Step 6: Solve as a 1 or 2-step equation
Step 7: simplify, if necessary

REMEMBER-- SOMETIMES YOU NEED TO GET THE DECIMAL OUT OF THE NUMERATOR--> MULTIPLY BOTH THE NUMERATOR AND THE DENOMINATOR BY A POWER OF 10


example:
change .416666... into a fraction
let n = .416666...
notice there is one 1 number that is repeating so it is only 10 to the 1st power
multiply both sides by 10
10n = 4.16666....
now subtract the 1st equation

10n = 4.16666...
-n = 0.41666...
9n = 3.75

divide both sides by 9
9n = 3.75
9 9

or 9n/9 = 3.75/9

move the decimal 2 places in both the numerator and denominator
n = 375/900
simplify
n = 5/12

LEARN THE MOST COMMON ONES BY HEARS
Need to know:
1/9 family
1/11 family
Easy trick to recall
If the repeating decimal starts repeating right after the decimal, you can easily get the fraction just by putting the same number of 9's as the length of the repeating decimal.. then simplify:
Examples:
.4444... = 4/9
.454545.... = 45/99 = 5/11
.162162162... = 162/999 = 18/111

Math 6H ( Periods 3, 6, & 7)

Subtracting Integers 11-3

The life story about someone who was so negative-- you wanted to take a little negativity away but since you can't do that you add a little positiveness-- works in math as well!!
Instead of subtracting ... "ADD THE OPPOSITE!"
We proved it in class with our little red and yellow tiles... If you need to review, make your own out of red and yellow paper-- or whatever colors you want!!

In life-- to take away a little negative-- add some positive

To take away an integer... add its opposite.

Rule from our textbook
for all integers a and b

a - b = a + (the opposite of b) or
a - b = a + (-b)

Instead of subtracting.. "ADD THE OPPOSITE"
make sure you do the check, check.. you need to have two check marks.. one changing the subtraction to addition and the other changing the sign of the 2nd number to its opposite.

5 - ^- 2=
5 + ⁺ 2= 7

-2 - 5 =
- 2 + ^- 5 =
before I give you the answer look... we are looking at
-2 + ^-5
We are adding two negatives.. so we are back to the rules from Section 11-2...
when adding the same sign just add the number and use their sign so
- 2 + ^- 5 = -7

But what about -2 - ^-5 ?
adding the opposite, we get
-2 + ⁺ 5 .
Now, the signs are different so the rule from Section 11-2 is
ask yourself... "Who wins?" and "By how much?"

Okay, here the positive wins so I know the answer will be +
and by how much means.. to take the difference
5-2 = 3
so -2 + ⁺ 5 = +3
Do you need to put the + sign? No, but I like to in the beginning to show that I checked WHO WON!!

What about
-120 - ^-48?
add the opposite
-120 + ⁺ 48 follows the
Different signs rule... so
ask yourself
Who wins? answer: the negative.. so I know the answer will be negative..
And "By How Much?" take the difference 120-48 = 72
so
-120 + ⁺ 48 = -72

NEVER EVER CHANGE THE FIRST NUMBER'S SIGN!!

WALK THE LINE, the number line-- that is!!
Remember... Attitude is such a little thing... but it makes a BIG difference!!
Always start with a positive attitude!!
When you walk the line, Which way are you always facing when you start???

Always attempt to get everything into addition so we can follow the rules of Section 11-2 Adding Integers.
1) SAME SIGN rule---> just add the numbers and use their sign
-4 + -5 = -9
2) DIFFERENT SIGNS rule
ask yourself those 2 important questions
a) Who wins? (answer is either negative or positive)
b) By how much? (Take the difference)

Page 376 answers to # 2-18 (evens)
2. -9
4. -5
6. 22
8. -9
10. -74
12. -32
14. -160
16. 498
18. -284

Algebra Period 4

Factoring ax² +bx + c Section: 6-5

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 pos, one neg - 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

6. 4 terms - Factor by grouping - make sure in descending order - pair off the first 2 terms and the last 2 terms -
make sure there's a PLUS sign in between the 2 pairs -
factor out the GCF of each pair - if the binomial left in the (  ) is the same, it's factorable.


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

Factoring ax²+ bx + c  Sections 6-5 or
FACTORING TRINOMIALS WITH A COEFFICIENT ON THE 1ST TERM.  
We'll use FACTORING BY GROUPING (Chapter 6-6)

I call this "Xbox 360" and you'll see why in class!
When you have a trinomial with a coefficient on the first term, factoring becomes more difficult.

A good method to factor is to use factoring by grouping.

1) Multiply the first term's coefficient by the last term's coefficient

2) By guess and check, find 2 factors that multiply to the product you got in #1 and add to your middle term
(just like how you did it for trinomials without a coefficient on the first term)
I like to set up a T Chart and go from there!!

3) Rewrite the original trinomial as a 4 term polynomial using the first term, the two factors you found, and the the last term.
IT SHOULD SIMPLIFY TO THE ORIGINAL PROBLEM!

4) Factor by grouping

AGAIN, NOT ALL TRINOMIALS ARE FACTORABLE!


EXAMPLE: 15n² - 19n - 10

1) Multiply 15 by 10 = 150

2) Find 2 factors that multiply to 150 and add to -19 (which means subtract to 19)

Think: 15 and 10? NO

30 and 5? NO

25 and 6? YES! 

Since the negatives must win, it must be -25 and +6 = -19!


3) Rewrite 15n² - 19n - 10 as a 4 term polynomial:
15n² - 25n + 6n - 10


4) Factor by grouping: (15n² - 25n) + (6n - 10)

5) Pull out the GCF: 5n(3n - 5) + 2(3n - 5)

(3n - 5)(5n + 2)


6) Check to make sure you can't factor any more.

7) FOIL to make sure you did not make a silly mistake!