Math 6 Honors (Period 6 and 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 1)

Find the Equation of a Line
: 7-6

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:


We isolate y on one side

We read the y intercept (the b which is 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 as the y value (+up or -down)
and then the x value (+ right or - left)


If the slope is not a fraction, make it a fraction by putting the integer over 1



TODAY WE'RE NOT LEARNING TO GRAPH ANOTHER WAY...WE'RE LEARNING HOW TO FIND THE EQUATION OF A LINE

We usually use the slope-intercept form of the line as our "template"

We know that y = mx + b so


we can substitute in what we know (what the problem gives us)
 and solve for whatever we're missing!


[The problem will give you enough information to know the slope or know the y intercept! Then you'll solve for the missing item.]



It helps to memorize this little rhyme


Oh mystery line,
What could you be?

If I could just find you,

y = mx + b


If I could find m

And I could find b.

Then I could put it together

And I would see:

y = mx + b


The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:


1) Do I have the slope (m)? If not, find it by using the slope formula or counting it if you have the graph


2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.


3) Don't forget to put it all together in one equation at the end!



THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:


First case:
You're given the slope and the y intercept
 (easiest case) 
m = 3/2 and b = -7/5


Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5



Second case:
You're given a point and the slope and need to find the intercept (b)


(3, 1) is a point on the line and m = 2

Plug in the point and the slope and solve for b


1 = 2(3) + b
1 = 6 + b


b = -5


Now put it altogether with the given slope and the intercept you just found:


y = 2x -5



Third case:
You're given a point and the y intercept and need to find the slope


(3, 1) is still a point on the line, but this time you know b = 2


Plug in the point and the y intercept and solve for slope


1 = 3m + 2
-1 = 3m
m = -1/3


Now put it altogether with the given intercept and the slope you just found:


y = -1/3 x + 2



Fourth case:
You're given 2 points and need to find the slope and the intercept

(1 , 3) and (-2 , -3) are 2 points on the line


You need to first find the slope using the formula:

m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2


Now plug the slope in with one (you pick the easiest!) of the points and find the intercept, b


3 = 2(1) + b

3 = 2 + b
b = 1


Finally, put it all together:
 y = 2x + 1



Fifth case:
You have a graph of a line and need to determine the equation


Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)


(If the y intercept is not an integer, then follow fourth case completely)


Put the information together in y = mx + b form

Algebra (Period 1)

Equations and Slope 7-5



Chapter 7-5: SLOPE INTERCEPT FORM



WE HAVE COVERED GRAPHING
 THE FOLLOWING 2 WAYS:


3 random x points using a table (Ch 7-2)


Most pick 0, 1, 2 and just plug and chug to find the y value


This works well when you don't end up with fraction answers!



EXAMPLE: -3y =-2x - 6


x y

0 2

3 4

-3 0
Then we learned a second way to graph:


Graph the 2 special points that are the y and x intercepts (Ch 7-3)


x intercept: where the line intersects the x axis


y intercept: where the line intersects the y axis


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 so the y intercept is (0, 2)


If y = 0, x = -3 so the x intercept is (-3, 0)


x y

0 2

-3 0


The x intercept is the one that has the x value (y is 0)


The y intercept is the one that has the y value (x is 0)



TODAY, WE WILL COVER THE MOST USED FORM OF A LINEAR EQUATION:

THE SLOPE INTERCEPT FORM




You must restate the equation to get it into the following format:


y = mx + b


where m = slope and b = y intercept (where the line hits the y axis)
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign!
Look at the example I gave you above:

-3y =-2x - 6
If you tried the x points 0, 1, 2, only 0 gave an integer for y.
All the other y values were fractions.
The slope intercept form provides the explanation on why.
RESTATE -3y =-2x - 6 INTO SLOPE INTERCEPT FORM:
Divide both sides by -3:
y = 2/3 x + 2


Look at the coefficient for x. WHAT X VALUES WILL GIVE YOU INTEGER ANSWERS FOR Y????
They need to be multiples of 3!



Now look at the graph of y = 2/3 x + 2
Notice that the + 2 at the end is the y intercept! (without doing any work!)
Use the counting method for slope on your graph.
You should have counted: UP 2, RIGHT 3
The slope therefore is 2/3.

Look at the equation.

It told you 2/3 without any work!



Graph when line is in Slope Intercept Form:


So 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 (that's the +/- constant at the end)


2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 2/3 x + 2
) Put a dot at (0, 2)

3) From (0,2) count up 2 and over to the right 3 to find the next coordinate (3,4)


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 (negative) and to the left (negative) 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 (positive) and to the left (negative) 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



Restate Standard Form to Slope Intercept Form:


Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(x and y on the same side, x is positive, no fractions, constant alone)

Restate into Slope Intercept Form:


Solve for y

first subtract 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)
Notice that the "b" is a fraction.

When this happens, the slope intercept form may not be the best form to graph the line.

You must start at 2 1/2 on the y axis and count down 3, right 4. THAT'S HARD!
Neither 3 nor 4 is a factor of the constant 10 so intercepts will be fractions.

We need the x term to end up with 1/2 so that when we add that to the b of 5/2, we'll get an integer!
So let's make x = 2 because that will cross cancel with the -3/4 slope to halves:
y = (-3/4)(2) + 5/2 = -3/2 + 5/2 = 2/2 = 1
So we found a coordinate that has just integers: (2, 1)


Now graph that point.


Now count the slope from that point (instead of from the y intercept)!

Math 6 Honors (Period 6 and 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