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

Equivalent Fractions 6-2

Fractions can be pictured on the number line.

If you multiply the numerator and the denominator of a fraction but the same nonzero number the resulting fraction is equivalent to the original fraction

1/2 (3/3) = 3/6

Properties

For any whole numbers a, b, and c, with b≠ 0 and c≠ 0

a/b = a(c)/b(c) and a/b = (a÷c) / (b÷c)

A fraction is in lowest terms if its numerator and denominator are relatively prime If their Greatest Common Factor (GCF) is 1



Fractions and Mixed Numbers 6-3

You know that ½ + ½ + ½ = 3/2
A fraction such as 3/2 (whose numerator is greater than or equal to its denominator) is called an improper fraction. Every improper fraction is greater than or equal to 1. A proper fraction is a fraction whose numerator is less than its denominator.

Proper fractions
1/4 2/3 5/9 10/12

Improper fractions
5/2, 8/3. 11/9 , 18/15

You can express any improper fractions as the sum of a whole number and a proper fraction

3/2 = 1 + ½ or 1½

A number such as 1½ (that is expressed as the sum of a whole number and a fraction) is called a mixed number.

If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simplest form.

To change an improper fraction to a mixed number in simple form, we divide the numerator by the denominator and express the remainder as a fraction

14/3 = 14 ÷ 3 = 4 2/3

30/4 = 30 ÷ 4 = 7 2/4 = 7 ½

To change a mixed number to an improper fraction, rewrite the whole number part as a fraction with the same denominator as the fractional part

1 3/8 = 8/8 + 3/8 = 11/8

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

Fractions 6-1

Can you think of some familiar expressions that include fractions?

Notice that the symbol ¼ can mean several things:

 It means one divided by four
 It represents one out of four equal parts
 It is a number that has a position on a number line.

If an object is divided into 8 equal parts, each part is one eight of the whole.
1/8 means 1 divided by 8 or 1 ÷ 8.

If an object is divided into eight parts and three of these parts are being considered, then the fraction that represents the parts is 3/8

A fraction consists of two numbers.
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.

Properties

For any whole numbers a, b, and c, with b ≠ 0

(a/b)(c) = ac/b

Examples:

2/7 + 2/ 7 + 2/7 = 6/7 which is really (3) 2/7 = 6/7

Algebra Period 3 (Wednesday)

How do you determine whether a given number is a solution?
Plug it in, plug it in, plug it in! Do this carefully. Use ( ) when you plug in a value for x and for y.

How do you find a solution to an equation yourself?
Plug in for x and find y!
You can use ANY number for x
Then plug in your number and find y

How can you graph a linear equation?
Make an x/y table of values and then graph the coordinates.
You only need 3 coordinates to make a good line!
(The 3rd coordinate serves as a "check" for the other two...in case you made a mistake!)
I always try x = zero and y = zero first because it's usually easy. Then pick another easy x value!
If this doesn't work well (you get a fraction as an answer and that's not easy to graph),
then try setting x equal to 1, then 2, then 3

Linear equations 7-3
What do they look like ( and what is not a linear equation?)
The variable is to the 1 power - like x, or y, or a, or b
What is not a linear equation? the variable is not to the 1 power - like x2, x3, etc, or 1/x (x-1)

2 ways to graph:
1) 3 points using a table (like Ch 7-2)
EXAMPLE: 2x - 3y = -6
x y
0 2
3 4
-3 0

2) 2 points using the y and x intercepts (where the line intersects the y and x axis)
Standard form of a linear equation: Ax +By = C
A, B and C should not be fractions
A should be positive (y will be positive or negative)
We won't be using this form to look at the slope of the line!
This is a good format for finding the x and y intercepts!

If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.

EXAMPLE: 2x - 3y = -6
If x = 0, y = 2
If y = 0, x = -3

Special linear equations:
Ones that are parallel to either the x or the y axis:
Lines parallel to the y axis are vertical lines:
They end up as the form x = with no y variable in the equation at all!
EXAMPLE: x = 4 ends up as a vertical line at x = 4
Still don't get this???
Pick of few points with the x value of 4:
(4, 0) (4, 2) (4, -3)
Graph those and join them in a line.
What do you get???
A vertical line!

Lines parallel to the x axis are horizontal lines:
They end up as the form y = with no x variable in the equation at all!
EXAMPLE: y = 4 ends up as a horizontal line at y = 4
Still don't get this???
Pick of few points with the y value of 4:
(0, 4) (2, 4) (-3, 4)
Graph those and join them in a line.
What do you get???
A horizontal line!

Algebra Period 3 (Tuesday)

Graphing Ordered Pairs 7-1
Review of x y Coordinate Plane Graphing from Pre-Algebra

Cartesian plane: Named after French mathematician Descartes.
PLANE: a two dimensional (across and up/down) flat surface that extends infinitely in all directions. It’s 2-D

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

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

Algebra Period 3 (Review)

FACTORING CHECKLIST
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
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 the 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 T chart - 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!!!!

Math 6 H Periods 1, 6 & 7

Graphs of Equations 11-9

An equation in two variables can produce many ordered pairs.
y = 2 – x

If we give x the value of 3, for example a corresponding value of y is determined.

y = 2- 3
y = -1
We describe this correspondence by the ordered pair (3, -1). We can created a table that shows several other ordered pairs produced by y = 2 –x

x ---> 2 - x = y ---> Ordered pair (x, y)
-1 ---> 2- -1 = 3 ---> (-1, 3)
0 ---> 2 – 0 = 2---> (0, 2)
1 ---> 2 - 1 = 1---> (1, 1)
2 ---> 2 - 2 = 0---> (2, 0)
3 ---> 2 - 3 = -1---> (3, -1)
4 ---> 2 - 4 = -2---> (4, -2)

Let’s plot those ordered pairs. The diagram suggests that if we were able to graph all of the ordered pairs produced by y = 2-x we would obtain a line. This line is the graph of the equation y = 2-x



Graph the equation y = 2x – 3

First make a table of ordered pairs and then graph the ordered pairs on a coordinate plane.

x---> 2x-3 = y---> ordered pair
-1---> 2(-1) – 3 = -5---> (-1, -5)
0---> 2(0) – 3 = -3---> (0, -3)
1---> 2(1) – 3 = -1---> (1, -1)
2---> 2(2) – 3 = 1---> (2, 1)



In the set of ordered pairs, for each value of x there is exactly one value of y. A set of ordered pairs such as this, in which no two ordered pairs have the same first component is called a function. For example, we can say that y = 2x -3 defines y as a function of x

Math 6 H Periods 1, 6 & 7

Graphs of Ordered Pairs 11-8


The location of a desk in a classroom can be described as “ second row, third desk” If we write (2, 3) to represent this location, the order of the numbers is important since (3, 2) would represent “third row, second desk) a pair of numbers whose order is important is called an ordered pair. Remember when you first walked into our classroom on the very first day—You had to find your seat – and it was an ordered pair!!

We graph a number as a point on a number line. We graph an ordered pair of numbers as a point on a plane marked with two perpendicular number lines called axes. The first number of an ordered pair is associated with the horizontal number line—called the x- axis and the second number with the vertical number line, called the y axis. The axes meet in a point, called the origin (0, 0).

Open your books to Page 389 to review - we are looking at the top graph on that page. To locate the ordered pair (3,2) start at the origin—where is that?—go 3 units to the right and then 2 units up. The numbers 3 and 2 are the coordinates of the graph (3, 2) the plane itself is called a coordinate plane.
Remember (x, y) it is x first, then y. It is Horizontal first, then Vertical, Across and then Up/Down its easy to remember… each is alphabetically in order..

Pre Algebra Period 2

REVIEW

ADDING OR SUBTRACTING FRACTIONS 5-3
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 smaller 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 create an answer box right now—and put the negative sign in it!!
Put bigger absolute value on top and take their difference
10 1/4
5 2/3
Find a common denominator:
10 3/12
5 8/12
Borrow because the bottom number is smaller than the top number
9 15/12
5 8/12
Use integer rules to add or subtract:
4 7/12
Put this in your answer box—that you already created with the negative sign and you have
-4 7/12

RESTATE THEM INTO IMPROPER FRACTIONS:
5 2/3 - 10 1/4
17/3 - 41/4

The common denominator is 12
(4)17/(4)(3) - (3)41/(3)(4)

68/12 - 123/12

Subtract using integer rules
-55/12

Restate into a mixed number if required
-4 7/12


EQUATIONS USING ADDING AND SUBTRACTING FRACTIONS 5-7
Same as using adding and subtracting with integers
Use the OPPOSITE (inverse) OPERATION (sign)
If there are fractions on both sides, remember to find a COMMON DENOMINATOR.

EXAMPLE - with a common denominator:
y - 1/8 = 5/8
ADD 1/8 to both sides and you get
y = 5/8 + 1/8 = 6/8
SIMPLIFY to get y = 3/4

EXAMPLE - with different denominators:
y + 4 5/12 = 5 3/8
SUBTRACT 4 5/12 from both sides and get
y = 5 3/8 - 4 5/12
Draw a vertical line separating the whole numbers from the fractions
FIND A COMMON DENOMINATOR which is 24
5 3/8 9/24
-4 5/12 10/24

NOW YOU'LL NEED TO BORROW
4 33/24
-4 10/24
23/24
RESTATE WITH THE VARIABLE FOR YOUR ANSWER y = 23/24

EXAMPLE: When you'll have to double check and put the larger ( absolute value) number on top
y + 2 1/6 = 1 3/8
SUBTRACT 2 1/6 FROM EACH SIDE
y = 1 3/8 - 2 1/6
DOUBLE CHECK , LOOK AND SEE THE SIGNS ARE DIFFERENT—so you will need a SIDE BAR to do your work. PUT THE WINNER ON TOP In this case, PUT 2 1/6 ON TOP (larger absolute value)… Which also means you need to create an answer box with the variable, an equal sign, and the sign of the winner. so on your page you would have in a box y = - just waiting for your results…
- 2 1/6
+1 3/8
Draw a VERTICAL line separating the whole numbers from the fractions
Now you need a COMMON DENOMINATOR:
- 2 1/6 4/24
+1 3/8 9/24

YOU WILL NEED TO BORROW SINCE 9 IS BIGGER THAN 4
At this point I never worry about the signs, I just take their difference—knowing that I have created that answer box with the sign of the answer. ( see above)
2 1/6 4/24
-1 3/8 9/24

1 28/23
1 9/24
19/24
but you put this back in the answer box which is just waiting for your answer and you have y = -19/24
YOU CAN ALSO CHANGE THEM BOTH INTO IMPROPER FRACTIONS FIRST!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
-13/6 -13/6
y = 33/24 - 52/24

y = -19/24
ONE OTHER METHOD (that they use in Algebra next year):
GET RID OF ALL DENOMINATORS BY MULTIPLYING BY THE LCD!!!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
24(y + 13/6) = 24(11/8)
24y + 52 = 33
-52 -52
24y = -19

y = -19/24




MULTIPLYING AND DIVIDING FRACTIONS 5-4
YOU DO NOT NEED A COMMON DENOMINATOR!!!!

MULTIPLICATION
1. Turn any mixed number into an IMPROPER FRACTION
2. Simplify (Some teachers call this : Cross Cancel) 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. Cross cancel 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 to determine the sign

VARIABLES:
You can cross cancel them as well!


EQUATIONS USING MULTIPLYING AND DIVIDING FRACTIONS 5-8
SOLVING ONE STEPS WITH FRACTIONS THAT ARE MULTIPLIED
SIMPLY MULTIPLY BY THEIR RECIPROCAL
3/5 y = 20
(5/3)(3/5)y = 20(5/3)
y = 100/3
Make mixed if required: 33 1/3

MIXED NUMBERS?
TURN THEM INTO IMPROPER, THEN MULTIPLY BY RECIPROCAL
4 2/3 y = 2 1/6
14/3 y = 13/6
(3/14)(14/3) y = (13/6)(3/14)
y = 13/28

Pre Algebra Period 2 (Tuesday)

Chapter 5-2 FRACTIONS = DECIMALS

How to change a fraction to a decimal
1. Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 =
3 divided by 4 =
0.75

If the quotient starts repeating, then put a bar over the number(s) that repeat.

OR

2. Use equivalent fractions (SOMETIMES WORKS!)

Works if the denominator can be easily made into a power of 10
SAME EXAMPLE: 3/4
but this time you will multiply by 25/25 to get
75/100 = 0.75

3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = 0.5

IF IT'S A MIXED NUMBER, JUST ADD THE WHOLE NUMBER AT THE END!
EXAMPLE: 8 3/4
For the fraction: 3 divided by 4 = .75
Add the whole number:
8.75

IF THE MIXED NUMBER OR FRACTION IS NEGATIVE, SO IS THE DECIMAL!


Changing decimals to fractions.

CHANGING TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
I learned this catchy phrase READ IT WRITE IT REDUCE!!
But we no longer say "reduce". We now say "simplify"

EXAMPLE:
Change .24 to a fraction
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25

EXAMPLE with whole number:
Change 7.24 to a fraction
The 7 is the whole number in the mixed number so you just put the 7 at the end
1) READ IT: 24 hundredths
2) WRITE IT: 24/100
3) SIMPLIFY: 24/100 = 6/25
4) (Now is the time to put the whole number back!!) 7 6/25


HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS:
Repeating decimals (we'll use algebra!)
This involves algebra and takes some work, so MEMORIZE THE FOLLOWING:
1/3 = .333 . . . and 2/3 = .666. . .
Also showed the 1/9 family pattern which is the numerator with a bar
Another great time-saving pattern - 1/11 family: Multiply the numerator by 9 and put a bar over it

To get an exact answer when doing math operations with repeating decimals,
make all repeating decimals into their fraction equivalents and do the operations with fractions!

To change repeating decimals to fractions, follow these steps:
Let n = the repeating decimal
Multiply both sides by a power of 10 equal to the number of places that repeat under the bar
Subtract n on the left side and the repeating decimal equal to n on the right side
Solve as a one-step equation
Multiply both numerator and denominator by a power of 10 if necessary
to get the decimal out of the numerator.
Simplify

EXAMPLE: Restate .41666 . . . into a fraction
The repeating portion is .6 or one place so multiply both sides by 10
n = .41666 . . .

10n = 4.1666 . . .
- n = -.4166 . . .

9n = 3.75 so divide both sides by 9 to get

9n/9 = 3.75/9

n = 375/900

n = 5/12

MULTIPLICATION OF MONOMIALS AND BINOMIALS 5-9
Multiplying a monomial by a polynomial is just the Distributive Property
You'll need to remember your power rules because you'll be MULTIPLYING SAME BASES
and then combining only LIKE TERMS.
EXAMPLE:
3x³ ( 2x² - 3x + 10)
3x³(2x²) + 3x³(- 3x) + 3x³(10)
6x⁵ - 9 x⁵ + 30x³
(no like terms to combine in this one!)
When you multiply a binomial by another binomial, I used to think of it as DOUBLE Distributive Property...See if you can figure out why!

Multiplying a binomial by another binomial:
FOILing binomials:
This is a memory device so you won't forget to multiply any of the factors!
First - first term in each binomial (terms on the left)
Outside - two outside terms in each binomial ( the first one in the
left parentheses and the second one in the right parentheses)
Inside - two inside terms (the terms right next to each other in the
different parentheses - the second one in the first parentheses
and the first one in the second parentheses)
Last - second term in each binomial (terms on the right)

I know this may seem overwhelming when you first see it,
but after you practice it, it does make sense.
It helps a lot of students to not forget any of the 4 multiplications!
EXAMPLE: (5x + 6)(3x - 7)
F O I L
FIRST OUTSIDE INSIDE LAST
= (5x)(3x) + (5x)(-7) + (6)(3x) + (6)(-7)
Simplify: 15x² + (-35x) + 18x - 42
Combine like terms: 15x² + -17x - 42
By the way, you can also do this in different orders --- as long as you multiply each term in one parentheses by each term in the other parentheses.
There are other ways to do the same problem:

Showed you the "box" method that allows you to use any order and protects you from ever missing one of the four multiplications. ( I can't get a table embedded here so make sure to look at your notes about the box method-- or email me.
5x +6
3x 15x² 18x
-7 -35x -42



15x²+ 18 x – 35x -42 =
15x² – 17x – 42
Also showed you "unibrow" method where you use arrows to show which terms you are multiplying. THIS IS HOW I LEARNED AND SEE IF YOU CAN FIGURE OUT WHY I USED TO THINK OF IT AS DOUBLE DISTRIBUTIVE PROPERTY!
I can't really show this on this website so watch carefully in class!

Math 6 Honors Periods 6 & 7 (Tuesday)

Adding Integers 11-2

Rules: The sum of two positive integers is a positive integer.
The sum of two negative integers is a negative integer.

So- if the two numbers have the same sign, use their sign and just add the numbers.

-15 + -13 = - 28

-10 + -4 = -14

Rules: The sum of a positive integer and a negative integer is :

POSITIVE… IF the positive number has a greater absolute value

NEGATIVE… IF the negative number has a greater absolute value

ZERO… IF both numbers have the same absolute value

Think of a game between two teams-
The POSITIVE TEAM vs. The NEGATIVE TEAM.
30 + -16 … ask yourself the all important question…
“WHO WINS? in this case the positive and then ask
“BY HOW MUCH?” take the difference 14

14 + - 52…
“WHO WINS?” the negative…
“BY HOW MUCH?”
38 so 14 + (-52) = -38

Although Aunt Sally says you must do ( ) first, when you are ONLY adding you can use the Commutative and Associative Properties to aid you!!

(-2 + 3) + - 6 you can work this 2 ways

(-2 + 3) + - 6 = 1 + -6 = -5 or
using all the properties that work for whole numbers
Commutative and Associative Properties of Addition
can change expression to (-2 + -6) + 3 or -8 + 3 = -5 you still arrive at the same solution.

You want to use these properties when you are adding more than 2 integers.
First look for zero pairs—you can cross them out right away!!
3 + (-3) = 0
-9 + 9 = 0

Then you can use C(+) to move the integers around to make it easier to add them together rather than adding them in the original order. In addition, you can use A(+) to group your positive and negative numbers in ways that make it easier to add as well.

One surefire way is to add all the positives up… and then add all the negatives up. At this point ask yourself that all important question…
WHO WINS? … use the winner’s sign..
and then ask yourself..
BY HOW MUCH?

example:

-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13

Taking a good scan of the numbers, do you see any zero pairs?
YES—so cross them out and you are left with

-4 + 5 + (-4) + 13
add your positives 5 + 13 = 18
add your negatives and use their sign – 4 + -4 = -8

Okay, Who wins? the positive
By how much? 10

so
-4 + 27 +(-6) + 5 + (-4) + (6) + (-27) + 13 = 10

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

Negative Numbers 11-1

On a horizontal number line we use negative numbers for the coordinates of points to the left of zero. We denote the number called ‘negative four’ by the symbol -4. The symbol -4 is normally read ‘ negative 4’ but we can also say ‘ the opposite of 4.’

The graphs of 4 and -4 are the same distance from 0—but in opposite directions. Thus they are opposites. -4 is the opposite of 4.

The opposite of 0 is 0

Absolute Value is a distance concept. Absolute value is the distance of a number from 0 on a number line. The absolute value of a number can NEVER be negative!!

Counting (also known as Natural) numbers: 1, 2, 3, 4, ….
Whole numbers 0, 1, 2, 3, 4….
Integers are natural numbers and their opposites AND zero
…-4, -3, -2, -1, 0, 1, 2, 3, 4….

The opposite of 0 is 0.

The integer 0 is neither positive nor negative.

The farther we go to the right on a number line--- the bigger the number. We can compare two integers by looking at their position on a number line.

if x < 0 what do we know? x is negative number
if x > 0, what do we know? x is a positive number

We have been practicing representing integers by their graphs, that is, by points on a number line. Make sure that your number line includes arrows at both ends and a line indicating where zero falls on your number line. The graph of a number MUST have a closed dot right on the number line at that specific number. Please see our testbook page 366 for an accurate example.

Algebra Period 3 (Review)

REVIEW:WEEK BEFORE WINTER BREAK
Sections 5-5 to 5-8 Polynomials

Polynomial = sum of monomials

Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial

Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)

Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x² - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y

Constant = the number that is not attached to any variable

Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM

Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!


Sections 5-7 ADDING POLYNOMIALS
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:

Section 5-8: SUBTRACTING POLYNOMIALS
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

Algebra Period 3 (Review)

REVIEW:
EXPONENTS SECTIONS 5-1 TO 5-3
SCIENTIFIC NOTATION SECTION 5-4

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

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
cancel out


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 = m^8-8 = m⁰
(by power rules for division)
By the transitive property of equality : 1 = m⁰

NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!

EXAMPLE:
m³ / m⁵ = m^-2
m³ / m⁵ = mmm/mmmmm = 1/mm
Again, by transitive property of equality:
m³ / m⁵ = m^-2 = 1/m²

EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER

EXAMPLE:
(-2)^-5 = 1/(-2)⁵ = 1/-32 OR -1/32

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

More with Exponents 5-2 &
Multiplying and Dividing Monomials 5-3
A monomial is an expression that is either a numeral, a variable, or a product of numerals and variables with whole number exponents.
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 = (m²)³/(n⁶)³ = m⁶/n¹⁸




Scientific Notation 5-4
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 That is, 1 ≤ n < 10
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 is your answer does not fit the scientific notation range, that you restate it.

TRY THIS LINK THAT TAKES YOU FROM HUGE POWERS TO
LITTLE TINY POWERS (NEGATIVE POWERS OR DECIMALS)

Try this link to practice scientific notation!

Try this link to practice multiplication of scientific notation!

Try this link to practice division!

Try this link to practice harder problems!