Pre Algebra Period 2 (Tuesday)

Absolute Value 1-4
ABSOLUTE VALUE IS ALWAYS POSITIVE!
(except zero because zero has no sign)
Absolute value is a DISTANCE concept and that is why it can't be negative.

On the other hand, every integer has an ADDITIVE INVERSE which will be its OPPOSITE SIGN (except zero which has no sign - neutral)

Adding Integers 1-5
Three ways to understand adding integers:
1. positive negative sketch - make positive numbers positive signs and make negative numbers negative signs. Then match up all the positives with the negatives by box them in. Whatever is left, is the answer and the sign of the answer.

2. number line - draw the arrows and see where you end up

3. Who wins and by how much?
Different signs: Put the winner on top and take the difference
Take the DIFFERENCE (subtract) and keep the BIGGER (absolute value)number's SIGN
I say: 2 teams came to play: the positive team and the negative team
If you have 10 + (-15) then the positive team scored 10 while the negative team scored 15
Who won?
The negative team (so keep that sign)
By how much?
5 points
Answer: -5

Same sign: Just add and keep the sign you have
I say: only one team came to play so, of course, they won!
You would just add up the scores because all the players are on the same team!
Example: -5 + (-10), so just the negative team came to play
Therefore, the negative team won and you just add up their scores
Answer: -15

Additive inverses: the sum of additive inverses (same number with opposite signs) is always zero.
I say: it's a tie and no one wins! The answer would be zero!
Example: -5 + 5 = 0
-x + x = 0

Again, the rules are:
SAME sign: Just ADD them and KEEP the sign
DIFFERENT signs: Take the difference (SUBTRACT) and take the BIGGER number's SIGN.

MORE THAN 2 ADDENDS:
When adding a lot of addends, here's a good strategy:
1) SEE IF ANY ARE ADDITIVE INVERSES AND JUST CROSS THEM OUT BECAUSE ADDING INVERSES ALWAYS GIVES YOU ZERO!
What justifies crossing them out? The Identity Property of Addition
2) Add the positives to the positives
Add the negatives to the negatives
What justifies changing the order and grouping the addends this way?
The Commutative (order) and Associative (grouping) Properties of Addition

3) Finally, add the positive sum to the negative sum at the end and
see who wins and by how much
Usually, you will make less silly mistakes this way than just going left to right!

Adding integers with variable expressions:
Just substitute in for the variable, putting the substituted number into
( ),then evaluate using the integer rules.
(I say plug and chug!)
y + 5 where y = -12
(-12) + 5 = -7

Algebra Period 3 (Tuesday)

CHAPTER 1-5: FACTORING (part 2 starting on p. 25)
First, I'll review what Like Terms are and clear up the difference between
"combining" vs multiplying/dividing terms
LIKE TERMS:
1. Same variable
2. Same exponent
Constants are like terms because they all have no variables
(you can actually say that they have a variable raised to the 0 power,
which is 1 times the coefficient)
Example: 3x + 4 can be thought of as 3x + 4x0.= 3x + 4(1).
You can only combine (add or subtract) like terms.

BUT YOU CAN MULTIPLY/DIVIDE UNLIKE TERMS!
3a(7y) = 21ay
BUT
3a + 7y cannot be simplified

48xy/6x = 8y
BUT
48xy - 6x cannot be simplified

FACTORING
THE MAJOR "WATERSHED" MOMENT IN ALGEBRA I!
Undoing the Distributive Property
5 ( 6y + 11 )
This simplifies to 30y + 55

Now you can WORK BACKWARDS to the way the expression began before you used the distributive property.
In 7th grade, we called this using the distributive property backwards.
Now, in Algebra, we call this FACTORING.
FACTORING IS A KEY CONCEPT TO UNDERSTAND IN ALGEBRA.
WITHOUT IT, YOU WILL STRUGGLE THE ENTIRE YEAR!

Let's use the same problem as above and work backwards (FACTOR)
EXAMPLE: FACTOR 30y + 55

THINK: What do 30y and 55 have in common (What is their GCF?)
They both divide by 5

FACTOR out the GCF of 5
(factoring out is really dividing each term by the 5)

MIDDLE STEP TO UNDERSTAND FACTORING:
5 ( 30y + 55 )
5 5
we haven't changed the value because we've divided each piece by 5 and multiplied by 5 as well
(multiply a number by 5 and then divide it by 5 and you'll see have your starting number!)

FINAL FACTORED FORM: 5 ( 6y + 11 )
Wait a minute...Isn't that what I started with before I used the Distributive Property????
Of course it is! This is the key concept in factoring!!!!!

FACTORING DOES NOT CHANGE THE VALUE OF WHAT YOU STARTED WITH!
It's incognito the same thing!
Therefore, you can always check your factored form by multiplying back to the original simplified form.
If it doesn't get back to that,
YOU HAVEN'T FACTORED PROPERLY!

We say that the factored form and simplified form are EQUIVALENT EXPRESSIONS..
(I say they are INCOGNITO the same thing!)

Pre Algebra Period 2 (Monday)

Variables & Expressions 1-1 (continued)

You can translate words into Algebra word by word
just like you translate English to Spanish or French.
5 more than a number
5 + n
the product of 5 and a number
5n
the quotient of 5 and a number
5/n
the difference of a number and 5
n - 5
NOTE: BECAUSE MULTIPLICATION & ADDITION IS COMMUTATIVE, YOU DON'T NEED TO WORRY ABOUT THE ORDER
BUT
FOR SUBTRACTION AND DIVISION, YOU MUST BE CAREFUL ABOUT THE ORDER
GENERALLY, THE ORDER FOLLOWS THE ORDER OF THE WORDS
EXCEPT (counterexample!)
5 less THAN a number
or
5 subtracted FROM a number
Both of these are: n - 5
The order SWITCHES form the words because the words state that you have a number that is more than you want it to be so you need to take away 5 from it.
For word problems like someone's age or the amount of money you have, you should always check your algebraic expression by substituting actual numbers to see if your expression makes sense.
EXAMPLE: Tom is 3 years older than 5 times the age of Julie
Translating: T = 3 + 5J
Does that make sense? Is Tom a lot older than Julie or is Julie older?
Try any age for Julie. Say she is 4 years old.
T = 3 + 5(4) = 23
In your check, Tom is 23. Is Tom 3 years older than 5 times Julie's age?


STRATEGY #2: MAKE A T-CHART
To translate known relationships to algebra, it often helps to make a T-Chart.
You always put the unknown variable on the LEFT side and what you know on the right.
Fill in the chart with 3 lines of numbers and look for the relationship between the 2 columns.
Then, you use that mathematical relationship with a variable.

EXAMPLE: The number of hours in d days
Your unknown is d days so that goes on the left side:

d days number of hours
1 24
2 48
3 72

Now look at the relationship between the left column and the right column.
You must MULTIPLY the left column BY 24 to get to the right column
The last line of the chart will then use your variable d

d days number of hours
1 24
2 48
3 72
d 24d

EXAMPLE: The number of days in h hours (The flip of the first example)
Your unknown is h hours so that goes on the left side:

h hours number of days
24 1
48 2
72 3



(Why did I start with 24 and not 1 hour this time?)


Now look at the relationship between the left column and the right column.
You must DIVIDE the left column BY 24 to get to the right column
The last line of the chart will then use your variable h

h hours number of days
24 1
48 2
72 3
h h/24

Algebra Period 3 (Monday)

Solving Equations: An Intro 1- 7

Equations use an = sign to show that two expressions are equivalent
Equations can be true, false, or open
If an equation is NUMERICAL (numbers), it must be either TRUE or FALSE:
3 + 5 = 8 is TRUE
3 + 5 = 9 is FALSE
If an equation is ALGEBRAIC (variables), it is OPEN for discussion
(it all depends on what the variable is!)
y + 4 = 10 is OPEN

SOLUTION = answer
REPLACEMENT SET: The set of answers that you have to choose from for the answer.
SOLUTION SET: All the answers that make a statement true.
{ } set symbol
EXAMPLE: Solve y + 4 = 10 for the replacement set {2, 6, 10}
Substitute each element of the replacement set into the equation to see which one(s) work
Only 6 works, so only 6 is a true solution
The solution set is {6}
Note that if the replacement set was {2, 7, 10} for y + 4 = 10, then the solution set would be the NULL SET which means the solution does not exist in the replacement set.
The symbol for the null set is a zero with a line through it.

Equations must always be in balance.
GOLDEN RULE: WHATEVER YOU DO TO ONE SIDE OF AN EQUATION, YOU MUST DO TO THE OTHER SIDE or you'll be out of balance.
One way they might ask this question on the STAR Test is to ask:
What was done to the first equation to get the second?
EXAMPLE: 3x - 5 = 12 and 9x - 15 = 36
The first equation was multiplied by 3 to get to the second equation so they are equivalent.

Using Formulas 1-9
What's the difference between an ordinary equation and a FORMula?
A FORMula has a KNOWN form, meaning it's a known relationship.
The area of a rectangle is known to be length times width (or base times height)
Contrast that to 2y + 7 = what??? This is not a known relationship!
Once you know a formula, you plug in the variables that you know (are "given")
and solve for the variable that you don't know.
MAKE SURE THE DIMENSIONS (labels) are consistent!
Example...if the length is given in feet and the width is given in inches, make them both inches
(usually, it's easier to change to the smaller label)