Algebra (Period 1)

SLOPE
: 7-4

First, let's talk about what the word "slope" means in the real world:


You can think of the slope of a line as the slope of a ski mountain -


When you're climbing up, it's positive


When you're sliding down, it's negative
 (if you're looking at the mountain from left to right)


The steeper the mountain, the higher the slope value


(A slope of 6 would be an expert slope because it
is much steeper than a slope of 2 which would be an intermediate's slope)


"Bunny slopes" for beginners will be lower numbers,
 generally fractional slopes (like 1/2 or 2/3)



A good benchmark to know is a slope of 1 or -1 is a 45 degree angle


You can also think of slope as rise/run - read this "rise over run"


Rise is how tall the mountain is (the y value)


Run is how wide the mountain is (the x value)


A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)


Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)


You can think of slope as a calculation using 2 coordinates.

We use m to denote slope:


m = rise/run

change in y
change in x

Rise = Change in y value = Difference in y value = y₂ - y₁
Run = Change in x value = Difference in x value = x₂ - x₁
THE LITTLE 1 AND 2 SUBSCRIPTS JUST STAND FOR Y VALUE OF THE FIRST POINT AND Y VALUE OF THE 2ND POINT.

To calculate slope you need 2 coordinates.

It doesn't matter which one you start with.

Just be consistent!

If you start with the y value of the FIRST point, make sure you start with the x value of the FIRST point!

If you start with the y value of the SECOND point, make sure you start with the x value of the SECOND point!



You can count the slope of a line:


1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)


2) Now count how far over the point is across - You'll need to either go right or left.


Make this the denominator of your slope (because the x value is the denominator of slope)


If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)


If you went to the LEFT, the value is NEGATIVE



Special slopes:


Horizontal lines in the form of y = have slopes of zero (they're flat!)


Vertical lines in the form of x = have no slope or undefined because the denominator is zero

Math 6 Honors (Period 6 and 7)

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 and 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 the answer is -38


(-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

Pre Algebra (Period 2 & 4)

Adding/ Subtracting Fractions (Negative) 5-3


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


2) As yourself "Who Wins?" Place the "winners sign" in the answer box you've created

3) Stack them...Place the winner on the top (no matter the sign)
 Take the difference

3) Restate to common denominators if needed


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

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





EXAMPLE :


5 2/3 - 10 1/4



METHOD 1: KEEP THEM AS MIXED NUMBERS:


First of all, you know the final answer will be NEGATIVE so make sure it is!

Put bigger absolute value on top:
 and take the difference
10 1/4
-
5 2/3


Find a common denominator:


10 3/12
-5 8/12


Borrow because the top fraction portion is smaller than the bottom
9 15/12
5 8/12

Use integer rules to add or subtract:
 - 4 7/12


Check

Algebra (Period 1)

LINEAR EQUATIONS: 7-3

What do they look like (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)



Two 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

So the 3 points of (0,2) (3,4) and (-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
 where 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 = constant 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 = constant 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!

Pre Algebra (Period 2 & 4)

Multiplying & Dividing Fractions 5-4

(skipped 5-3 negative fractions until tomorrow)


EASIEST FRACTION SKILL BECAUSE YOU DON'T NEED A COMMON DENOMINATOR!!!!



MULTIPLICATION

1. Turn any mixed number into an IMPROPER FRACTION

2. Simplify or 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 the 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!

Algebra (Period 1)

GRAPHING LINEAR EQUATIONS: 7-2

How do you determine whether a given number is a solution?

Plug it in, plug it in, plug it in!



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

Math 6 Honors (Period 6 and 7)

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 textbook page 366 for an accurate example.

Algebra (Period 1)

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.


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

Pre Algebra (Period 2 & 4)

Adding & Subtracting Fractions 5-3

(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    = 11x/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 showed 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