Math 6 Honors Periods 6 & 7 (Friday)

Quotients of Integers

One of last night's homework problems could have been
-7 X 4 = -28 if that is true then,

-28 ÷ 4 = -7

The rules that apply for multiplication also apply for division.

The quotient of two positive integers is positive

The quotient of two negative integers is positive

The quotient of a positive integer and a negative integers is negative.

Notice: the quotient of two integers need not be an integer. For example, there is no integer n such that 10/4 = n because there is no integer n such that n X 4 = 10

Actually the quotient 10/-4 = -2.5

Remember that we never divide by 0. 10/0 is UNDEFINED

Let n be a positive integer and –n be its opposite

n ÷ -1 = -n

-n ÷ -1 = n

-n ÷ -n = 1

-n ÷ n = -1

n² ÷ n = n

Math 6 Honors Periods 6 & 7 (Thursday)

Products With One or More Negative Factors 11-4 & 11-5

The product of a positive integer and a negative integer is a negative integer.
3 X -2 = -6
Multiplication is just repeated addition so this is really -2 + -2 + -2 = -6

The product of ZERO and any integer is always ZERO!!


4(-5X3) = 4(-15) = -60

-12(3 X 5) = -180

[25 - (6 X -2)](-3) = -111


The product of -1 and any integer equals the opposite of that integer.

-1 X 7 = -1

-1 X -8 = 8

The product of two negative integers is a positive integer

-3 X 4 X -2 = 24

-5 X 2 X -3 X -4 = -120

For a product with NO zero factors:
1. If the number of negative factors is ODD, the product is negative.

2. If the number of negative factors is EVEN, the product is positive.

-6 X -3 = 18

8 X -2 X -1 = 18

-2 X - 7 X 11 X 0 = 0

What happens in life -- happens in math:

Something good happens to someone good---- that's good

Something bad happens to someone good --- that's bad

Something good happens to someone bad--- that's bad

Something bad happens to someone bad ( really bad of course)-- that's good!!

Otherwise know as

positive X positive = positive

negative X positive = negative

positive X negative = negative

negative X negative = positive

Algebra Period 3

We won't be covering Chapter 7-1 in class.
This is simple Pre-Algebra!
I have included a review below:

Review of x y Coordinate Plane Graphing from Pre-Algebra (ch 7-1 in your book)

Cartesian plane: Named after French mathematician Descartes.
plane: a two dimensional (across and up/down) flat surface that extends infinitely in all directions.

quadrant: 2 perpendicular lines called axes split the plane into 4 regions....quad means 4
quadrant names: begin in the top right (where you normally write your name!) and go counterclockwise in a big "C" (remember it for "C"oordinate)
They are named I, II, III, IV in Roman Numerals

The axes are NOT part of any quadrant. A point on the x-axis or the y-axis is not in a quadrant since it is on the boundary between quadrants.

coordinate - A coordinate is the position of a point in the Cartesian plane
coordinate = "co" means goes along with (COefficient, COworker, CO-president, CO-champions)

"ordinate" means in order

So coordinate means numbers that go along with each other in a certain order
The numbers are the x and y values and the order is that the x always comes first

Also called an ordered pair (x y "ordered" and they are a "pair" of numbers)
Ordered pairs are recognized by the use of ( x , y ) format

origin = (0, 0) the center of the graph (its beginning or origin)
When you count the coordinate' s position, you count from the origin.

x comes before y in the alphabet so the order is (x, y) ....
always go right or left first, then up or down

the x axis is the horizontal axis (goes across)
Remember that because the number line also is horizontal and you learn that first
(the pattern to remember is x is always first and the number line is before going up and down)

NOW LET'S GET TO WHAT YOU ACTUALLY DO!!!
1) Count your x value:
positive x, count right from origin (positive numbers are to the right of zero on number line)
negative x, count left from origin
2) Count your y value:
positive y value, count up from where your x value was (up is the positive direction)
negative y value, count down from where your x value was (down is the negative direction)

EXAMPLE:
(3, 5) Count 3 to the right from the origin, then 5 up
(3, -5) Still count 2 to the right, but now count 5 down
(-3, 5) Count 3 to the left from the origin, then count 5 up
(-3, -5) Again count 3 to the left, but now count 5 down

BUT WHAT HAPPENS WHEN
ONE OF THE VALUES IS ZERO?
If the y value is zero it means that you move right or left, but don't go up or down:
SO YOUR POINT WILL BE ON THE x AXIS........x axis is where y = 0
Example: (3, 0) is a point on the x axis, 3 places to the RIGHT
Example: (-3, 0) is a point on the x axis, 3 places to the LEFT

If the x value is zero it means that you don't move right or left, you just go up or down.
SO YOUR POINT WILL BE ON THE y AXIS...........y axis is where x = 0
Example: (0, 3) is a point on the y axis, 3 places UP
Example: (0, -3) is a point on the y axis, 3 places DOWN

Pre Algebra Periods 1, 2, & 4

Chapter 4-9: Scientific notation
You've had this since 6th grade!
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

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.

ORDERING SCIENTIFIC NOTATION NUMBERS:
As long as numbers are in scientific notation,

they are easy to put in order from least to greatest!

1) If they are all different powers, simply order them by powers
2) If they have the same power, simply order them using your decimal ordering skills.

EXAMPLE 1: Order 3.7 x 10⁸, 4.3 x 10^-2, 9.3 x 10⁵, and 8.7 x 10^-5

8.7 x 10^-5, 4.3 x 10^-2, 9.3 x 10⁵, 3.7 x 10⁸

EXAMPLE 2: 3.7 x 10⁸, 4.3 x 10⁸, 9.3 x 10⁸, and 8.7 x 10⁸

3.7 x 10⁸, 4.3 x 10⁸, 8.7 x 10⁸, 9.3 x 10⁸

Algebra Period 3

Reviewed final checklist of how to factor:

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
2. 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
3. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

4. 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
5. Trinomial with "a" coefficient - Use XBox - multiply first to last to get new product - then find factors that multiply to that new produce and either add or subtract to the middle term (use trinomial rules above) - replace middle term with these two factors and place appropriate signs so they will add to the original middle term - proceed as if you have factoring by grouping (see 6 below)
6. 4 term polynomial - factor by grouping - pair of the first 2 terms and then the second 2 terms by placing parentheses around them - make sure you always have a plus sign between the 2 pairs ( you may need to double check) - factor out the GCF of each pair - if it factors, there should now be a new GCF - factor that out in front parentheses and place what ever is left in the second parentheses

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