Algebra Period 3 (Tues/Wed)

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)

VISUALIZE THE FOLLOWING 2 MOUNTAINS TO HELP YOU UNDERSTAND:
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)
It has a much longer time to slowly reach the 1000 foot top of the mountain!

You can think of slope as a calculation using 2 coordinates:
Rise/Run
=Change in y value/Change in x value
=Difference in y value/ Difference in x value
= y₂ - y₁/ x₂ - x₁
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 one point, make sure you start with the same x!

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


Slope Intercept Form 7 -5

Finding the slope-intercept form of a line:
y = mx + b
where m = slope and b = y intercept
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign
(I explained this when we did Chapter 7-3 to easily find 3 coordinates in your T Chart. We just didn't call it slope intercept form at that time!)
It helps to solve the equation for y before you pick your x values, but you don't have to.

EXAMPLE from above: 2x - 3y = -6
Solve the above equation for y.
Subtract 2x from each side:
-3y = -2x - 6
Divide each side by -3:
y = 2/3 x + 2
Now pick your x values, put them on the left side of the T chart, then solve for y.
Instead of picking 0, 1, 2, it makes sense to pick x values that are multiples of 3.
Why? Because you will need to multiply the x value by 2/3 and this will keep the y value an integer:
x y
0 2
3 4
-3 0
Now graph these coordinates and join as your line

Restate Standard Form to Slope Intercept Form:
Example: 3x + 4y = 10 is the STANDARD FORM of a line
Solve for y
first subtracting 3x from both sides:
4y = -3x + 10
Now divide both sides by 4:
y = -3/4 x + 10/4 or y = -3/4 x + 5/2
The slope is the coefficient of the x
m = -3/4 (so you're sliding down at a little less than a 45 degree angle)
The y intercept is the constant
b = 5/2 (so the line crosses the y axis at 2 1/2.)

Graph when line is in Slope Intercept Form:
If you have the slope-intercept form of the equation, it's really easy to graph the line:
1) Graph the intercept on the y axis
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 3x - 2
1) Put a dot at (0, -2)
2) From (0, -2) count up 3 and over to the right 1 to find the next coordinate (1, 1)
Remember, slope is y over x, so the numerator is the y change and the denominator is the x change
If it's positive, you're counting up (positive) and to the right (positive)
or you can count down and to the left because 2 negatives make a positive.
If it's negative, you're counting down (negative) and to the right (positive)
or you can count up and to the left because you would have a positive and negative = negative

If you're given the slope and the y intercept,
you can write the equation of any line!
Just use: y = mx + b
EXAMPLE: m = -2/3 and b = -12
The line would be y = -2/3 x - 12

Math 6 H Periods 1, 6 & 7 (Tues & Wed.)

Comparing Fractions 6-4

When 2 fractions have equal denominators-- it is easy to tell which of the fractions are greater. Compare their numerators.

5/11 < 7/11 because 5 < 7

If the fractions have different denominators, find a common denominator. Using the lCM of the denominators-- the LCD-- is a surefire way of determinng the relationship between fractions.
Which is greater 5/6 or 7/9?
Since the LCM (6,9) = 18
Using equivalent fractions
5/6 = 15/18
and 7/9 = 14/18
so 5/6 > 7/9

( See Section 6-2 notes if you need to review equivalent fractions)

Another way is to use cross products to compare.

What if you needed to name a fraction between two other fractions?

FOr instance, between 7/15 and 12/ 25

FInd the LCM (15, 25) using the methods taught from chapter 5
LCM ( 15, 25) = 75
finding equivalent fractions for
7/15 = 35/75
12/25 = 36/75
If you want a fraction between, simply double the denominators and then double the numerators
35/75 = 70/150
36/75 = 72/150
So 71/150 would be a fraction that is between the two given fractions.

Algebra Period 3 ( Review)

1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug
Usually, we just try 0, 1, 2 first

2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug
This works really well when the line is in STANDARD form and the coefficients are factors
of the constant on the other side of the equation.

3) We know how to GRAPH a line by using slope-intercept form... y = mx + b
We isolate y on one side
We read the y intercept (the b -> the constant on the other side)
We graph that value on the y axis
We COUNT to the next point by reading the slope, the coefficient of the x
The slope should be read what happens to the y value (+up or -down) and then what
happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1