Experimenting with Zoho

Experimenting with Zoho writer...
MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ + 9x² + 4x - 6

NOTICE THAT THERE IS NO GCF OF ALL 4 TERMS!

Factoring by grouping says if there is no GCF of all 4 terms, look and see if there is a GCF of just 2 terms at a time.
Put ( ) around the first 2 terms and the 2nd 2 terms:
(6x³ + 9x² ) + ( 4x - 6)
FACTOR OUT THE GCF FROM EACH SET OF TWO TERMS:

3x² (2x - 3) + 2(2x - 3)

Math 6H ( Periods 3, 6, & 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 1

Comparing and Ordering Fractions 

5-1 (cont'd)

Ordering or comparing fractions:(last method)

4: Make them into decimals because DECIMALS = FRACTION posers!

(or decimals are just fraction wannabes!)
Today, we will change fractions to decimals and decimals to fractions.

How to change a fraction to a decimal
1.
Divide (ALWAYS WORKS!)

EXAMPLE: 3/4 = 3 divided by 4 = .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: but this time you will multiply by 25/25 to get 75/100 = .75



3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = .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 TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
Read it, Write it, 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) 7 6/25



HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS---> we will do that in 2010!!

Math 6H ( Periods 3, 6, & 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.

Pre Algebra Period 1

Comparing and Ordering Fractions

 5-1
HOW TO FIND THE LEAST COMMON MULTIPLE:

LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!

Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12...     multiples of 5: 5, 10, 15, 20...)

Common = must be a number that ALL THE NUMBERS go into 

Least = must be the SMALLEST number that ALL THE NUMBERS go into

There are the same ways to find it as the GCF:

1) List multiples of each number and circle the smallest one that is common to all the numbers
---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply

3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.

THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.

WHY DO WE NEED EVERY FACTOR THIS TIME?

Because it's a multiple of all your numbers!

Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.

For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36

LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3

36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)

LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ???? 
1944!!!! 
Would you rather use 108 or 1944???


LCM WITH A COLUMN APPROACH:
You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column.
If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY!

                                    SAME EXAMPLE: 54, 36
                                     
36 = 2 x 2 x 3 x 3
                                     
54 = 2 x       3 x 3 x 3
                                   
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.

If you do it in exponential form, you take the HIGHEST POWER of each column!

Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!

CHECKING THE LCM TO MAKE SURE IT WORKS:

Algebra Period 4

Factoring by Group 6-6

First a review:
Checklist of how to factor thus far-->

1. 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

3. 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 middle sign

4. Trinomials - last sign positive - double hug with same sign as middle term - factors that multiply to last and add to middle

5. 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



REMEMBER: 
FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!

(we're skipping 6-5 and then going back to it)
When you have 4 TERMS IN YOUR POLYNOMIAL!


You put the polynomial in 2 sets of 2 by using ( )

Then you factor out the GCF for each set of 2 terms individually


DOES THIS ALWAYS WORK FOR EVERY 4 TERM POLYNOMIAL?

Of course not!
But for this section of the math book, it will!

What happens if it doesn't work? The polynomial may just not be factorable!

MAKE SURE IT'S IN DESCENDING ORDER FIRST!!!!

EXAMPLE: 6x³ - 9x² + 4x - 6
First notice there is NO GCF of all the terms!!

Factoring by grouping says if there is no GCF of the 4 terms, look and see if there is a GCF of just 2 terms at a time!!

Put ( ) around the first 2 terms and another ( ) around the 2nd set of terms.
(6x³ - 9x²) + (4x - 6)
Factor out the GCF from each set of two terms
3x²(2x - 3) + 2(2x - 3)
Look for a COMMON factor to factor out between the two sets
In this case its (2x - 3)
Pull out
(2x - 3)(3x + 2)
and check to make sure you cannot continue to factor!!

Math 6H ( Periods 3, 6, & 7)

Least Common Multiple 5-6

Also check out December 4, 2009 posting of LCM!! Here is a review of that lesson...
Let’s look at the nonzero multiples of 8 and 12—listed in order
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72…

Multiples of 12: 12, 24, 36, 48, 60, 72, ….

The numbers 24, 48, and 72, ... are called common multiples of 8 and 12. The least of these multiples is 24 and is therefore called the least common multiple.

LCM(8, 12) = 24

To find the LCM of two whole numbers, we can write out lists of multiples of the two numbers.

Or, we can use prime factorization

Lets find LCM(12, 15)

12 = 2²∙3
15 = 3∙5

The LCM will be made up of the greatest power of each factor

LCM will be 2²∙3∙5 = 60
The book has a third option or method
you can check out, if you’d like

Let’s find LCM (54, 60)


54= 2∙3∙3∙3 = 2∙3³
60 = 2∙2∙3∙5 = 2²∙3∙5

The greatest power of 2 that occurs in either prime factorization is 2²
The greatest power of 3 that occurs in either prime factorization is 3³
The greatest power of 5 that occurs in either prime factorization is 5
Therefore, LCM(54,60) is 2²∙3³∙5 = 540


REMEMBER:
The GCF (greatest common factor) is a factor. The GCF of two numbers will be either the smaller of the two or smaller than both

The LCM (least common multiple) is a multiple. The LCM of the two numbers will be the largest of the two or larger than both.

Algebra Period 4

Factoring x² +bx + c or Factoring Trinomials 6-4

You are reversing it back to BEFORE it was FOILed.
Always check your factoring by FOILing or BOXing back!!


Factoring Trinomials with a:
PLUS sign as the second sign
x² + bx + c
Following these steps:

1. set up your hugs ( )( )
2. When the last sign is positive the BOTH signs in each of the ( )( ) are the SAME!!
3. How do you know what those two signs are? It is whatever the sign is of the 2nd term of the trinomial. Put that sign in BOTH parentheses.
4. to factor (unFOIL), you will need to find two factors that
MULTIPLY to the LAST term and
ADD to the MIDDLE term

you can set up a box with

___ X ___ =
___ + ___ =

and fill in the blanks.
I suggest you make a T-chart with all the factors of the last term-- using your divisibility rules!!
Example:
x² +8x + 15
Follow the steps
( )( )
Think: last term sign is + so both signs are the same
Think: first sign (sign of the 2nd term) is + so both signs are positive
put + into the ( )( )
( + )( + )
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x + )(x + )
Now to get to the L in FOIL you need two factors whose product is 15. This is easy but using a T chart
15
1 I 15
3 I 5

you see 1 X 15 or 3 X 5 are possibilities

BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to 8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
3 X 5 = 15
3 + 5 = 8
(x + 5) (x +3)

At this point it does not matter which factor you put into the first ( ) because they are the SAME sign but I always tend to put the LARGER number in the first ( ) because of other rules -- which you will learn later this week)

Next example
x² - 8x + 15
Follow the steps
( ) ( )
THINK: Last sign is + so the signs are the same
THINK: First sign ( 2nd term) is NEGATIVE so BOTH signs are NEGATIVE
( - ) ( - )

Again,
You already know the "F" in FOIL means that both first terms must be x --> so put those terms in
(x - )(x - )

Now to get the "L" in FOIL, you need two factors whose product is 15
BUT, you also need two numbers to add to the I and O of FOIL which means that the two numbers must add up to -8 ( the middle term)
Since 3 + 5 = 8, they must be the two factors that will work
-3 X -5 = 15
-3 + -5 = -8
(x - 5)(x - 3)
(It doesn't matter which is first because they're the same sign!)
Now FOIL to see if we're right!

Last example:
x² - 8xy + 15y²
Same problem as the one above except now there are two variables. Simply use the same steps above and include the y

(x -5y)(x -3y)


Factoring Trinomials with a:
NEGATIVE sign as the second sign

x² + bx - c
We will use the same method as yesterday to factor these basic trinomials!
1.
Set up your (       )(       )

2. Look at the SECOND or last sign

If it's negative, then the signs in the (   ) are DIFFERENT
Why?
Because when you multiply integers and get a NEGATIVE product, the only way that will happen is if they are DIFFERENT signs.
Remember that the last term is the product of the two LAST terms in FOILing.


3. Now look at the sign of the second term.

It tells you "Who wins," meaning which sign must have the larger absolute value. 
Remember that the middle term is the SUM of the "O" and the "I" terms when FOILing.
Because these two terms have DIFFERENT signs, when you add them, you actually "subtract" and take the sign of the larger absolute value.
( This is just integer rules!!)
Put that sign in the first parentheses and always put the bigger number in the first parentheses.
4. To UNFOIL (factor), you will need to find 2 FACTORS that MULTIPLY to the last term, but SUBTRACT to the middle term.
(yesterday the factors needed to ADD to the middle)

Or you can still say you're adding, but since they are DIFFERENT signs, you will end up subtracting!

This is still an educated guess and check!

To help you do this, I suggest to set it up like this:

____ x ____ = ____
____
 - ____ = ____

Again, setting up a T-chart with all the factors also helps you visualize the two numbers you are looking for!!



EXAMPLE:

x² + 2x - 15
( ) ( )
THINK: LAST sign is - so the signs are DIFFERENT
THINK: First sign is + so the POSITIVE WINS!!
( + ) ( - )
You know the "F" in FOIL means that both the fist terms must be x so
(x + )(x - )
Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15
(Don't forget the sign1!!)
Several possibilities like 1 X -15 or 15 X -1 or 3 X -5 or 5 X -3
BUT since the POSITIVE must win , according to the middle term of the example (+2x)
you know that the bigger factor must be positive ( so it can win!!)
Therefore your choices are POSITIVE 15 X NEGATIVE 1 or POSITIVE 5 X NEGATIVE 3
BUT, you also need them to ADD to the I and O in FOIL so pick the two factors that also ADD to POSITIVE 2
Since -3 + 5 = +2 these must be the two factors that will work
I set it up like this:

____ x ____ = 15

____ - ____ = 2

so
5 x 3 = 15

5 - 3 = 2

YOU CAN ALSO DO THIS WITH THE APPROPRIATE SIGNS and adding:

+____ x -____ = -15

+____ + -____ = + 2

so

5 x (-3) = -15

5 +( -3) = 2


THIS IS WHERE IT DOES MATTER WHICH NUMBER YOU DO HAVE WITH THE SIGN BECASUE THE + MUST WIN!!
( x + 5 )( x - 3 )



NEXT EXAMPLE:

x² - 2x -15

(      )(      )

THINK: Last sign is - so signs are DIFFERENT!

THINK: First sign is - so NEGATIVE MUST WIN

(    -   )(    +   )

You know the the "F" in FOIL means that both first terms must be x

( x - )( x + )

Now to get the "L" in FOIL, you need 2 factors whose product is NEGATIVE 15

Like 1 and 15, or 3 and 5

But you also need to add to the I and O in FOIL which means that the two factors
 must add to NEGATIVE 2

Since 3 + -5 = -2, this must be the two factors that will work:

( x - 5 )( x + 3)

It matters which number you have with which sign because the negatives must win!

That's why I always put the sign of the middle term in the first parentheses.

That way, I always know to put the larger number in the first parentheses, so that sign will win.

Now FOIL to see if we're right!


LAST EXAMPLE:

x² - 2xy -15y²
Same problem as the one before, except now there are 2 variables!

Simply use the same factorization and include the y

( x - 5y )( x + 3y)


ALWAYS CHECK BY FOILing or BOXing Back!!

Pre Algebra Period 1

Rational Numbers 4-6

THE NUMBER SYSTEMS

Natural (counting): 1, 2, 3, ....

Whole: 0, 1, 2, 3, ...

Integers: ...-3, -2, -1, 0, 1, 2, 3, ...

Rational: All numbers that can be written as a fraction or RATIO (decimals-terminating and repeating, fractions, mixed numbers, integers)

Irrational: Decimals that never terminate or repeat - like pi and square root of 2


Real numbers: All the number systems together



Complex or Imaginary Numbers: we don't study these until Algebra II, but they are stated as

a + bi where i is equal to the square root of -1



Again, Rational number = any number that can be expressed as the ratio (fraction) of two integers

a/b, where a and b are both integers and b cannot be zero

b cannot be zero because you cannot divide by zero.....IT'S UNDEFINED!


There are positive and negative rational numbers.

Algebra Period 4

Trinomial Squares 6-3

This is a special product that we learned in Chapter 5 when we did FOILing.

FOIL:
(3 + a)² (called a binomial squared)

(3 + a)(3 + a) = 9 + 3a + 3a + a²
=
9 + 6a + a² (called a trinomial square)



Again, you see that the middle term is DOUBLE the product of the two terms in the binomial, and the first and last terms are simply the squares of each term in the binomial.
   
HOW TO RECOGNIZE THAT IT IS A BINOMIAL SQUARED:

1) Is it a trinomial? (if it's a binomial, it cannot be a binomial squared - it may be diff of 2 squares)

2) Are the first and last terms POSITIVE?
3) Are the first and last terms perfect squares?

4) Is the middle term double the product of the square roots of the first and last terms?


IF YES TO ALL OF THESE QUESTIONS, THEN YOU HAVE A TRINOMIAL SQUARE
 
TO FACTOR A TRINOMIAL SQUARE: 9 + 6a + a²

1) (     )²

2) Put the sign of the middle term in the (   +   )²

3) Find the square root of the first term and the last term and place in the parentheses:   (3 + a)²

4) Check by FOILing back.

Math 6H ( Periods 3, 6, & 7)

Greatest Common Factor 5-5

Also check out December 1, 2009 posting of GCF. Here is a review of that lesson...


When the factors in the numbers 30 and 42 are listed, the numbers 1, 2, 3, and 6 appear in both lists
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

These numbers are called common factors of 30 and 42. The number 6 is the greatest of these numbers and is therefore called the greatest common factor of the two numbers.
We write
GCF(30,42) = 6
to denote the greatest common factor of 30 and 42

Find GCF(54, 72)
List the factors of each number

54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

the common factors are 1, 2, 3, 6, 9, and 18

the greatest number in both lists is 18. Therefore,

GCF(54, 72) = 18

Another way to find the GCF of two numbers is to use prime factorization.
Find GCF (54, 72)
First find the prime factorization of 54 and of 75

54 = 2∙3∙3∙3∙3
72 = 2∙2∙2∙3∙3

Find the greatest power of 2 that occurs in both prime factorization. 2
Find the greatest power of 3 that occurs in both prime factorization 3²
Therefore GCF(54,72) = 2∙3² = 18

Try GCF(45, 60) using the prime factorization method

The number 1 is a common factor of any two whole numbers.
If 1 is the GCF, then the two numbers are said to be relatively prime.
Show that 15 and 16 are relatively prime
List the factors of each number
Factors of 15 = 1, 3, 5, 15
Factors of 16: 1, 2, 4, 8, 16
Since the GCF(15, 16) = 1, the two numbers are relatively prime

Pre Algebra Period 1

Simplifying Fractions 4-4

Equivalent Fractions - Just multiply the numerator and the denominator by the same number and you will get an equivalent (equal) fraction to the one you started with.


GOLDEN RULE OF FRACTIONS = Do unto the numerator as you do unto the denominator


Simplifying fractions (your parents call this "reducing")

2 good ways:

(1) Just divide both the numerator and denominator by the GCF

(2) Another way: Rewrite the numerator and denominator in prime factorization form. Then simply cross out each common factor on the top and bottom
(they cross out because it's 1)

You'll be left with the simplified fraction every time!!!!
                                           


THE GCF METHOD:

One of the reasons we learn the GCF is because it's the FASTEST WAY TO SIMPLIFY FRACTIONS IN ONE STEP!!!

Just divide both the numerator and denominator by the GCF

THE PROBLEM WITH THE METHOD:
If you're not comfortable finding the GCF, you're pretty much sunk with this method! :(

THE BEST REASON TO USE THIS METHOD (other than it's a Calif. STAR Key Standard), it truly is the FASTEST :)

So imagine you have a "GCF Magical Voice" in your head...
The voice tells you the GCF of the numerator and the denominator...
You simply use that GCF to divide both the top and bottom of your fraction and you're done in one step!



THE PRIME FACTORIZATION METHOD:

This is sort of using the GCF "incognito" (in disguise)!

Rewrite the numerator and denominator in prime factorization form.

(Use a Factor Tree or Inverted Division to find the Prime Factorization if necessary).

Then simply cross out each common factor on the top and bottom.

(You're actually using the ID Property of Multiplication because 
each "crossout" is really a quotient of 1!)

You'll be left with the simplified fraction every time!!!!

If you actually multiplied together all your cross-outs, you'd get the GCF...
so you're using the GCF without even computing it!

THE PROBLEM WITH THIS METHOD:
You may think it's a lot of work


THE BEST REASON TO USE THIS METHOD: Although it takes time, everyone can do a Factor Tree or Inverted Division and create the Prime Factorization...
You'll never get the wrong answer with this one!



THE CROSS OUT METHOD:

You simply think of the first number that comes to your mind that "goz-into" both the numerator and the denominator and keep going until it's simplified.

If it's even, most people start with dividing it in half....and then in half again, etc.

This probably takes the longest, but in practice, most people use this method!

THE PROBLEM WITH THIS METHOD: You may think that a fraction is simplified, but you've missed a factor...this especially happens when the number is odd and you're always used to using 2 to divide the top and the bottom!

THE BEST REASON TO USE THIS METHOD: No one ever forgets how to do this method...it just comes naturally and there are no "precise" steps to do!
 
EXAMPLE: Simplify by each method:
36/
54
 
GCF METHOD:

The GCF is 18:

36 ÷ 18 = 2

54 ÷ 18 = 3



PRIME FACTORIZATION METHOD:

36 = 2 x 2 x 3 x 3

54 = 2 x 3 x 3 x 3

Two of the 3s cross out and one of the 2s

You are now left with:

2/
3

That's it!!!!!!!!!



CROSS OUT METHOD:
36 ÷ 2 = 18 ÷ 3 = 6 ÷ 3 = 2

54 ÷ 2 = 27 ÷ 3 = 9 ÷ 3 = 3


so 36/54 = 2/3
Do the same thing with variables!

Algebra Period 4

Difference of Two Squares 6-2

Again, remember that FACTORING just UNDOES multiplication.

In this case, the multiplication that you'll be UNDOING is FOILING.

FOIL:
(a + b)(a - b)

You will get:
a² - b²
This is the DIFFERENCE (subtraction) of TWO SQUARES.

Now FACTOR:
a² - b²
You undo the FOILING and get:
(a + b)(a - b)


REMEMBER:
You must have two different signs because that's how the MIDDLE TERM disappears!

You will get ADDITIVE INVERSES which will become ZERO



HOW TO RECOGNIZE THE DIFFERENCE OF TWO SQUARES:

1) Is it a binomial?

2) Is it a difference?

3) Are both terms perfect squares?


IF YES TO ALL 3 QUESTIONS, THEN YOU HAVE A DIFFERENCE OF 2 SQUARES!!

HOW TO FACTOR THE DIFFERENCE OF 2 SQUARES:

1) Double hug  (    )(    )

2) Find square root of each term (sq rt sqrt)(sq rt sq rt)

3) Make one sign positive and one sign negative.
              
(sq rt + sqrt)(sq rt -sq rt)


                           
Of course, they get more complicated! 
We can combine pulling out the GCF with this!

ALWAYS LOOK FOR A GCF TO PULL OUT FIRST!!!!!!

EXAMPLE:

27y² - 48y⁴ 

First, look for a GCF that can be pulled out.

The GCF = 3y²

Factor out the GCF (look at Chapter 6-1):
3y²(9 - 16y² )
NOW YOU HAVE A DIFFERENCE OF TWO SQUARES TO FACTOR:


3y²(3 - 4y)(3 + 4y)


CALLED FACTORING COMPLETELY BECAUSE
 YOU CANNOT FACTOR FURTHER!


Always check your factoring by distributing or FOILing back!


THERE IS NO SUCH THING AS THE SUM OF TWO SQUARES!

a² + b² CANNOT BE FACTORED!!!!!


BUT - b² + a²
= + a² -b²
= (a + b)(a - b)
BECAUSE IT'S JUST SWITCHED (COMMUTATIVE)

Algebra Period 4

Factoring Polynomials 6-1

Chapter 5 was a very important chapter that you cannot survive without...

CHAPTER 6 IS EVEN MORE IMPORTANT FOR HIGH SCHOOL!!!



As I stated-- these chapters are the Meat & Potatoes of Algebra!!

REMEMBER:

FACTORING WILL NEVER CHANGE THE ORIGINAL VALUE OF THE POLYNOMIAL SO YOU SHOULD ALWAYS CHECK BY MULTIPLYING BACK!!!!
(You'll either distribute or FOIL...that's what you learned how to do in Chapter 5!)


CHAPTER 6-1: FACTORING THE GCF

Factoring is a skill that you must understand to be successful in higher level math!!!

We did a simple version of this back in Chapter 1!


Factoring is simply UNDOING multiplying

Say you multiplied 5 by 10 and got 50

How would you undo it?
DIVIDE by 5!

So FACTORING uses the concept of DIVIDING.

You're actually undoing the DISTRIBUTIVE PROPERTY.

How?

You look for the most of every common factor....the GCF!

Then you pull out the GCF (divide it out of) from each term, 
Placing the GCF in front of ( )


EXAMPLE:

FIRST,
DISTRIBUTE:

2m²n(2n² + n + 3)


4m² n³ + 2m² n² + 6m² n



Now, pretend you don't want the 2m²n distributed anymore...
What should you end up with once you UNDO the Distributive Property?

2m² n (2n² + n + 3)



That's exactly what you started with!

So is it that easy?

Well yes... and no...

Yes because that is the answer
 and

No because it was only that easy because I gave you how it started!

You won't know how it started in a real problem!
THIS IS AN EXAMPLE THE WAY YOU WOULD USUALLY SEE IT.

The question would say:
FACTOR: 
4m² n³ + 2m² n² + 6m² n



Step 1: What does each term have in common (what is the GCF) ?

They each can be divided by 2m² n


Step 2: Put the GCF in front of a set of (   ) and divide each term by the GCF
2m² n [ (4m² n³)/2m² n + (2m² n²)/2m² n + (6m²n/2m² n]

Step 3: SIMPLIFY and you'll get:

2m²n (2n² + n + 3)



Step 4: Check your answer!!!!!

Always check your factoring of the GCF by distributing back!

(incognito, it should be the same thing)



RELATIVELY PRIME TERMS ARE 
TERMS WITH NO COMMON FACTORS
 THAT MEANS THAT THEY CANNOT BE FACTORED (GCF = 1)
We say they are "not factorable"


Math 6H ( Periods 3, 6, & 7)

Prime Numbers and Composite Numbers 5-4

A prime number is a positive integer greater than 1 with exactly two factors, 1 and the number itself. The numbers 2, 3, 5, 7 are examples of prime numbers

A composite number is a positive integer greater than 1 with more than two factors. The numbers 4, 6, 8, 9, and 10 are examples of composite numbers.

Since 1 has exactly 1 factor, it is neither prime nor composite.

About 230 BCE Erathosthenes, a Greek Mathematician suggested a way to find prime numbers—up to a specific number. The method is called the Sieve of Eratosthenes because it picks out the prime numbers as a strainer, or sieve, picks out solid particles from a liquid.

You may factor a number into prime factors by using either of the following methods
➢ Inverted short division
➢ Factor tree

Both were shown in class.

Could you start the factor tree differently? If so, would you end up with the same answer?


The prime factors of 42 are the same in either factor tree, except for their order.

Every composite number greater than 1 can be written as a product of prime factors in exactly one way, except for the order of the factors.

When we write 42 as 2 ∙ 3 ∙ 7 this product is called the prime factorization of 42

Notice the order in which prime factorization is written.

Let’s try finding the prime factorization of 60

The prime factorization of 60 = 2 ∙ 2∙ 3 ∙ 5 or 2²∙ 3∙ 5

Pre Algebra Period 1

Prime Factorization & GCF 4-3
Greatest Common Factor - think of it backwards to understand it!

Factor = must be a number that goes into the numbers

Common = must be a number that goes into BOTH the numbers

Greatest = must be the biggest number that goes into BOTH the numbers

There are several ways to find it.

1) List all the factors of each number and circle the biggest one that is common to both
 (takes too long!!)
2) Circle the common factors in the prime factorizations of each number and multiply

3) list the factors in a table and bring down the factors whose column is filled.
Then multiply.


EXAMPLE:
Find the GCF of 36, 45 and 54


LIST ALL THE FACTORS OF EACH NUMBER:

1, 2, 3, 4, 6, 9, 12, 18, 36

1, 3, 5, 9, 15, 45

1, 2, 3, 6, 9, 18, 27, 54

The GCF is 9


FIND THE PRIME FACTORIZATIONS ON A FACTOR TREE OR INVERTED DIVISION AND MULTIPLY THE COMMON FACTORS:

36 = 2 x 2 x 3 x 3

45 = 3 x 3 x 5

54 = 2 x 3 x 3 x 3

GCF = 3 x 3 = 9

PUT THE PRIME FACTORIZATIONS IN A BOX WITH COLUMNS:
   as shown in class                                     

DO THE SAME THING WITH VARIABLES:
The GCF of the variables is the most of each variable that each term has in common.
EXAMPLE:
Find the GCF of a²b³c⁴   ac³d   a³c²f
The COMMON variables are a and c
How many of each variable is COMMON to all 3 terms:
They each have 1 a (although the first term has 2 and the 3rd term has 3)
They each have 2 c's (although the 1st term has 4 and the 2nd has 3)
GCF = ac²
Again, the GCF of variables is simply the lowest power of common variables
You should look for a special case of GCFs:
When one number goes into the other number(s), the smaller number is always the GCF.
Example: The GCF of 50 and 100 is 50
50 is the biggest factor that goes into both 50 and 100!

Math 6H ( Periods 3, 6, & 7)

Square Numbers & Square Roots (Continued) 5-3

Before we began---We reviewed the terminology for various numbers using 75 as our example
75---the standard form of the number
3⋅ 5⋅ 5 is it written in expanded prime factorization
3⋅ 5² is it's exponential prime factorization

If you have a prime number all three are exactly the same
For example
41 --- is the standard form of the number
but 41 is also the expanded prime factorization
and 41 ( or 41¹) is it written in exponential prime factorization.
[You can leave off the exponent 1 for the power of 1 because it is truly invisible!!]


Knowing the SQ's & SQRT's of numbers up to 20 is critical for finding the approximate square root of any non perfect square.
That is,
You know the √36 = 6
and √25 = 5
But what would be
√28 ?

There are several strategies to use.. one involved dividing and taking the average. This is from the yellow worksheet handed out on Friday 12/4
You know that √28 is less than 6 but more than 5
so you start with 5
divide 28 by 5
28 ÷ 5 = 5.6
Now take the average of 5 and 5.6 or
5 + 5.6
2

which equals 5.3

Divide 28 by 5.3 now
28 ÷ 5.3 = equals 5.28 or rounded 5.3
Since the divisor (5.3) and the quotient (5.3) match STOP...and say

√28≈ 5.3

That's fairly complicated... in class we showed some neat tricks about the relationship between the perfect squares...
Using the same number √28

You still think of the perfect square below and above

Stack them
36
√28
25

then take the square roots for those perfect squares

√36 = 6
√28 ≈
√25 = 5

√36 = 6
√28 ≈ 5
√25 = 5

Now, find the difference between your two perfect squares
36 -25 = 11
We discovered that the difference will always be the sum of the square roots of the perfect squares!! WOW!!

That number becomes your denominator

Now, find the difference between your number (in this case 28) and the lower of the two perfect squares (25) so 28-25 = 3
That number becomes the numerator
so you have

3/11

Now, we haven't learned about fractions but you can estimate ( since this is all about estimating... anyway)
and you know

3/10
That can be written as .3
so adding that to your estimate
we can safely estimate
√28 ≈ 5.3

Try finding the √110... with this method..

Math 6H ( Periods 3, 6, & 7)

Least Common Multiple 5-6

We looked at the first few non-zero multiples of 8 and 12
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
Multiples of 12: 12, 24, 36, 48, 60, 72, 84 ...

We noticed 24, 48, and 72 appear in both lists. These numbers are called common multiples of 8 and 12 and the LEAST of the multiples is 24...
It is called... the least common multiple.
We write the least common multiple of 8 and 12 as
LCM(8,12) = 24

To find the LCM of two whole numbers you could write out the lists of multiples-- and that works relatively easily with small numbers... but there are more efficient ways to find the least common multiple of two whole numbers.

1. Write out the first few multiples of the larger of the two numbers and test each multiple for divisibility by the smaller number. The first multiple of the larger number that is divisible by the smaller number is the LCM

2. You can use prime factorization to find the LCM. The LCM is EVERY factor to its GREATEST power!!

LCM(54, 60)
54 = 2⋅ 3⋅ 3⋅ 3 = 2⋅ 3³
60 = 2⋅ 2⋅ 3⋅ 5 = 2²⋅ 3⋅ 5
So the greatest power of 2 is 2²
The greatest power of 3 is just 3
and the greatest pwoer of 5 is just 5
so the product of 2²⋅ 3⋅ 5 will be the LCM
LCM(54, 60) = 540

3. You may use the BOX method as shown in class... unfortunately it does not show well here. Remember you need to create a L. The numbers on the side of the box represent the GCF!! You need to multiple them with the last row of factors.
See me before or after class if you want any review!!


We reviewed the concept of relatively prime and noticed that any two prime numbers are relatively prime. We also noticed that if two numbers are relatively prime-- neither of them must be prime....

We also found out that if one number is a factor of a second number, the GCF of the two numbers is the first number AND... if one whole number is a factor of a second whole number the LCM of the two numbers is the second number!!
GCF(12,24) = 12
LCM(12,24) = 24

WOW!!

If two whole numbers are relatively prime---
their GCF = 1
and their LCM is their product!!
GCF(8,9) =1
GCF(8,9) = 72

WOW!!

LCM & GCF Story PRoblems
1) Read the problem
2) Re-read the problem!!
3) Figure out what is being asked for!!
4) find the "magic " word... to help you determine if you are finding GCF or LCM
5) When in doubt... draw it out!!

Algebra Period 4

Multiplying Polynomials 5-11

To multiply two polynomials, multiply each term of one polynomial by every term of the other. THEN ADD the results.

The textbook shows a column approach, please see page 249 for instructions.
In class we used the BOX method... and then combined terms.

When you multiply a trinomial by a binomial or two trinomials, it gets really tricky!


2 ways:

1) box method

2)column method
(double or triple distributive with columns to combine like terms)




If you have a trinomial times a binomial, it's easier to use the Commutative Property

and make it a binomial times a trinomial:

(x² + x - 1) (x - 1)

switch it to

(x - 1) (x² + x - 1)

Remember the following rules
(A + B)(A + B) = (A + B) ² = A² + 2AB + B²

(A - B)(A - B) = (A - B)² = A² -2AB + B²

(A + B)(A - B)= A²- B²
You can use FOIL to multiply two binomials
remember FOIL is First Terms, Outside Terms, Inside Terms, Last Terms

You can always use FOIL-- or the BOX method but knowing these rules will make computation quicker if you know the above rules!!

Math 6H ( Periods 3, 6, & 7)

Greatest Common Factor 5-5

If we list the factors of 30 and 42, we notice
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

We notice that 1, 2, 3, and 6 are all COMMON factors of these two numbers. The number 6 is the greatest of these and therefore is called the
GREATEST COMMON FACTOR of the two numbers. We write
GCF(30,42) = 6

Although listing the factors of two numbers and then comparing their common factors is one way to determine the greatest common factor, using prime factorization is another easy way to find the GCF

Find GCF(54, 72)
54 = 2 ⋅ 3 ⋅ 3 ⋅ 3
72 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3
Find the greatest power of 2 that occurs IN BOTH prime factorization. The greatest power of 2 that occurs in both is just 2 ¹
Find the greatest power of 3 that occurs IN BOTH prime factorizations. The greatest power of 3 that occurs in both is 3²
Therefore
GCF(54, 72) = 2 ⋅ 3² = 18

In class we circled the common factors and realized that
GCF(54, 72) = 2 ⋅ 3 ⋅ 3 = 18


Fin the GCF( 45, 60)
45 = 3 ⋅ 3⋅ 5
60 = 2⋅ 2⋅ 3⋅ 5
Since 2 is NOT a factor of 45-- there is NO greatest power of 2 that occurs in both prime factorizations.
The greatest power of 3 is just 3¹
and the greatest power of 5 is just 5¹
Therefore,
GCF(45,60) = 3⋅ 5 = 15

The number 1 is a common factor of any two whole numbers!! If 1 is the GCF , then the two numbers are said to be RELATIVELY PRIME. Two numbers can be relatively prime even if one or both of them are composite.

Show that 15 and 16 are relatively prime
List the factors of each number
FACTORS of 15: 1, 3, 5, 15
FACTORS of 16: 1, 2, 4, 8, 16

Since the GCF(15,16) = 1. The two numbers are relatively prime!!

Math 6H ( Periods 3, 6, & 7)

Prime Numbers & Composite Numbers 5-4

A prime number is one that has only two factors: 1 and the number itself, such as 2, 3, 5, 7, 11, 13...
A counting number that has more than two factors is called a composite number, such as 4, 6, 8, 9, 10...

Since one has exactly ONE factor, it is NEITHER PRIME NOR COMPOSITE!!
Zero is also NEITHER PRIME NOR COMPOSITE!!

Every counting number greater than 1 has at least one prime factor -- which may be the number itself.
You can factor a number into PRIME FACTORS by using a factor tree or the inverted division, as shown in class.

Using the inverted division, you also start with the smallest prime number that is a factor... and work down
give the prime factors of 42
2⎣42
3⎣21
7

When we write 42 as 2⋅3⋅7 this product of prime factors is called the prime factorization of 42.

Two is the only even prime number because all the other even numbers have two as a factor.

Explain how you know that each of the following numbers must be composite...
111; 111,111; 111,111,111; and so on....
Using your divisibility rules you notice that the sums of the digits are multiples of 3.

List all the possible digits that can be the last digit of a prime number that is greater than 10.
1, 3, 7, 9.

Choose any six digit number such that the last three digits are a repeat of the first three digits. For example
652,652. You will find that 7, 11, and 13 are all factors of that number... no matter what number you choose... why is that???? email me your response.

Algebra Period 4

Multiplying Binomials: Special Products 5-10

LEARN TO RECOGNIZE SOME SPECIAL PRODUCTS - IT MAKES IT EASIER!


Remember: You can FOIL these just like the other products until you remember these special patterns....but when we get to factoring next week, it will really help you to know these patterns by heart. 

When you do, you actually don't need to show any work because you do it in your head! 

(That should make a lot of you happy! :)



DIFFERENCE OF TWO SQUARES:

You will notice that the two factors are IDENTICAL except they have DIFFERENT SIGNS

(x + 6)(x - 6) =
x² - 6x + 6x - 36 =
x² - 36


This will happen every time!
 
The middle terms are additive inverses so they become zero.


You're left with a difference (subtraction) of two terms that are squared.



SQUARING A BINOMIAL:

When you multiply one binomial by itself (squaring it), you end up with:

First term squared + twice the product of both terms + last term squared


(x + 6)² =
(x + 6)(x + 6) =
x² + 2(6x) + 6² =
x² + 12x + 36


If you foiled you would have:

x²  + 6x + 6x + 36


CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP??? 

WHY???


Another example with subtraction in the middle:

(x - 6)² =
(x - 6)(x - 6) =
x² + 2(-6x) + 6² =
x² - 12x + 36


If you foiled you would have:

x² - 6x - 6x + 36


CAN YOU SEE THAT THE 2 MIDDLE TERMS ARE JUST DOUBLING UP??? 
WHY???



PLEASE NOTE:

NOTICING THESE SPECIAL PRODUCTS HELPS YOU DO THESE 
MULTIPLICATIONS FASTER!

IF YOU EVER FORGET THEM, JUST FOIL!


(but you will need to recognize them for factoring in Chapter 6)

Math 6H ( Periods 3, 6, & 7)

Square Numbers and Square Roots 5-3

Numbers such as 1, 4, 9, 16, 25, 36, 49... are called square numbers or PERFECT SQUARES.

One of two EQUAL factors of a square is called the square root of the number. To denote a square root of a number we use a radical sign (looks like a check mark with an extension) See our textbook page 157.

Although we use a radical sign to denote cube roots, fourth roots and more, without a small number on the radical sign, we have come to call that the square root.
SQRT = stands for square root, since this blog will not let me use the proper symbol) √ is the closest to the symbol

so the SQRT of 25 is 5. Actually 5 is the principal square root. Since 5 X 5 = 25
There is another root because
(-5)(-5) = 25 but in this class we are primarily interested in the principal square root or the positive square root.

Evaluate the following:
SQRT 36 + SQRT 64 = 6 + 8 = 14
SQRT 100 = 10
Is it true that SQRT 36 + SQRT 64 = SQRT 100? No
You cannot add square roots in that manner.
However look at the following:
Evaluate
SQRT 225 = 15
(SQRT 9)(SQRT 25)= (3)(5) = 15
so
SQRT 225 = (SQRT 9)(SQRT 25)

Also notice that the SQRT 1600 = 40
But notice that SQRT 1600 = SQRT (16)(100) = 4(10) = 40

Try this:
Take an odd perfect square, such as 9. Square the largest whole number that is less than half of it. ( For 9 this would be 4). If you add this square to the original number what kind of number do you get? Try it with other odd perfect squares...

In this case, 9 + 16 = 25... hmmm... what's 25???

Math 6H ( Periods 3, 6, & 7)

Tests for Divisibility 5-2

It is important to learn the following divisibility rules:
A number is divisibility by:

2 ... if the ones digit of the number is even
3 ... if the sum of the digits is divisible by three ( add the digits together)
4 ... if the number formed by the last two digits is divisible by by four ( Just LOOK at the last two numbers-- DON"T ADD them!!)
5 ... if the ones digits of the number is a 5 or a 0
6 ... if the number is divisible by both 2 and 3... (or if it is even and divisible by 3)
8 ... if the number formed by the last three digits is divisible by 8. (Like FOUR, just look at the last three digits-- divide them by 8)
9 ... if the sum of the digits is divisible by 9
10 ... if the ones digits of the number is a 0.

You will not need to know the divisibility rules for 7 or 11 but they are interesting...

You can test for divisibility by 7
Let's start with a number 959
Step 1: drop the one's digit so we have 95
Step 2: Subtract twice the ones' digit ( that you dropped) in this case we dropped a 9
so we double that and subtract 18 from 95
or 95-18 = 77. If the results, in the case, 77, is divisible by 7 --- so is the original number 959.
Step 3: If the number you get is still to big.. continue the process until you can determine if your number is divisible by 7.


To test for divisibility by 11
add the alternative digits beginning with the first
so let's try the following
4,378,396
Step 1: Add the alternate digits beginning with the 1st 4 + 7+ 3 + 6 = 20
Step 2: Add alternate digits beginning with the 2nd 3 + 8 + 9 = 20

Step 3: If the difference of the sums is divisible by 11 so is the original number.
In this case, 20-20 = 0 and 0/11= 0 so
4,378,396 is divisible by 11.


A good test for divisibility by 25 would be if the last two digits represent a multiple of 25.

A perfect number is one that is the SUM of all its factors except itself. The smallest perfect number is 6, since 6 = 1 + 2+ 3
The next perfect number is 28 since
28 = 1 + 2 + 4 + 7 + 14
What is the next perfect number?

Algebra Period 4

Polynomials 5-5

Polynomials = SUM of monomials

Monomials must have variables with whole number powers. Review section 5-3 for detail on monomials!!
no variables in the denominator, no roots of numbers!!
so 1/x is not a monomial
neither is x ^1/2

constants have whole number power of zero..
7 is really 7x⁰

1 term = monomial
2 terms = binomial
3 terms = trinomial

TERMS are separated by addition
( if see subtraction-- THINK: add the opposite!!)

Coefficient - number attached to the variable ( it can be a fraction)
3x² - 10x
the coefficients are 3 and -10. Make sure to attach the negative sign to the coefficient -- and ADD the OPPOSITE

y/6 is really (1/6)y so the coefficient is 1/6
if you have -x/3 that is really (-1/3)x so the coefficient is -1/3


Constant = the number that is not attached to ANY variable

Some TERMS YOU NEED TO KNOW

Degree of a term = SUM of the exponents of all its variables
-6x⁴ : the degree is 4
8x² : the degree is 2
-2x : the degree is 1
9 : the degree is 0 ( think 9 is really 9x⁰

Degree of a polynomial - HIGHEST degree of any of its terms
so
-6x⁴ + 8x² + -2x + 9
The degree of the polynomial is : 4

Leading term = term with the HIGHEST degree
Leading coefficient- the coefficient of the leading term




More on Polynomials 5-6

Descending order- write the variables with the highest power first ( This is the way it is usually written)

Ascending order- write the variables with the lowest pwoer first ( actually NEVER used in practice)

Evaluating a polynomial- this is what we have been doing all year... plug it in, plug it in!!
Remember to ALWAYS put the number you substitute in parentheses!!

2x²y + 5xy - 4, where x = -4 and y = 5
Substitute carefully:

2(-4)²(5) + 5(-4)(5) - 4
= 2(16)(5) +(-20)(5) - 4
= 160 +(-100) -4
= 160 -104
= 56

Math 6H ( Periods 3, 6, & 7)

Finding Factors and Multiples 5-1


You know that 60 can be written as the product of 5 and 12. 5 and 12 are called whole number factors of 6-. A number is said to be divisible by its whole numbered factors.

To find out if a smaller whole number is a factor of a larger whole number, you divide the larger number by the smaller.--- if the remainder is 0, the smaller number IS a factor of the larger number.

We set up T charts to find al the factors of numbers.
For example. Find all the factors of 24
24
1--24
2--12
3--8
4--6

We you go down the left side and back up the right you have
1, 2, 3, 4, 6, 8, 12, 24
all the factors of 24 in order!!!

A multiple of a whole number is the product of that whole number and ANY whole number. You can find the multiples of given whole numbers by multiplying that number by 0, 1, 2, 3, 4, ...and so on
The first four multiples of 7 are
0, 7, 14, 21
because 0(7) = 0 ; 1(7) = 7 ; 2(7) = 14; 3(7) = 21

If you were to ask for the first four NON-ZERO Multiples of 7
the answer would be 7, 14, 21, 28


Generally, any number is a multiple of each of its factors. That is, 21 is a multiple of 7 and it is a multiple of 3!!

Any multiple of 2 is called an EVEN number
A whole number that is NOT an even number is called an ODD number
Since 0 is a multiple of 2 .. that is 0 = 0(2) 0 is an EVEN number







The word factor is derived from the Latin word for "maker" the same root for factory and manufacture. When multiplied together factors 'make' a number.
factor X factor = product.

Algebra Period 4

Scientific Notation 5-4



You've had this since 6th grade!
It is just a bit more complicated
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
Scientific notation is the product of two factors
one of the factors is a number that is 1 ≤ n < 10 and the other factor is a power of ten

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 if your answer does not fit the scientific notation range, that you restate it.

(2.5 x 10 ^-3)(4.0 X 10 ^-8)
multiply the first factors or (2.5)(4.) and multiply the powers of ten
(10 ^-3)(10 ^-8)

(2.5)(4)= 10
so initially you have 10 X 10 ^-11
but 10 does not fall into the required 1 ≤ n < 10
so changed 10 into scientific notation or 1.0 X 10¹
so you have 1.0 X 10¹ X 10 ^-11
or 1 X 10 ^-10

What about
6.0 x 10 ⁷
3.0 X 10²
First divide 6.0/3.0 = 2.0
and use the exponent rules of 10 ^{7 -2} or 10⁵
so 2.0 X 10⁵

4.2 X 10⁵
2.1 X 10³

= 2 X 10²

2.5 X 10 ^-7
5.0 X 10 ⁶
first 2.5/5 = 0.5 and 10^-7/10⁶ = 10^-7-6 = 10^-13
so initially you have
0.5 x 10^-13 but 0.5 is not within 1 ≤ n < 10
so change 0.5 to scientific notation
5.0 X 10 ^-1 and mult that by 10^-13
5 X 10 ^-14

Math 6H ( Periods 3, 6, & 7)

Dividing Decimals 3-9


According to our textbook-
In using the division process to divide a decimal by a counting number, place the decimal point in the quotient directly over the decimal point in the dividend.

Check out our textbook for some examples!!

When a division does not terminate-- or does not come out evenly-- we usually round to a specified number of decimal places. This is done by adding zeros to the end of the dividend, which as you know, does NOT change the value of the decimal. We then divide ONE place beyond the specified number of places.

Divide 2.745 by 8 to the nearest thousandths.
See the set up in our textbook on page 89. Notice that they have added a zero and the end of the dividend ( 2.745 becomes 2.7450) because you want to round to the thousandths and we need to go ONE place additional.
DIVIDE carefully!!

the quotient is 0.3431 which rounds to 0.343


To divide one decimal by another

Multiply the dividend and the divisor by a power of ten that makes the DIVISOR a counting number


Divide the new dividend by the new divisor

Check by multiplying the quotient and the divisor.

Tickets to the school play cost $5.25 each. THe total receipts were $651. How many tickets were sold?

Divide $651 by $5.25

that is, divide 651 by 5.25

or divide 65100 by 525.

124

So A total of 124 tickets were sold.


Check the textbook for the details to the division-- as it is impossible to do here!!

Algebra Period 4

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. IF the monomial is just a numeral we call is a constant.

Some examples: 4x³, -7ab, y (1/2)x⁵ and 2x⁴y

The following are NOT monomials
1/y, x^1/2, x² +4, y² + 2y + 4

Multiplying monomials - use the properties of rational numbers and the properties of exponents:
(3x)(4x) = 12x²

(3x²)(-x) = (3x²)(-1x)
= (3)(-1)(x²)(x)
=-3x³

(-3a)(4a²)(-a⁴ = (-3)(4)(-1)(a)(a²)(a)
=12a⁷

Dividing is similar-- Use the properties of rational numbers and the properties of exponents:

x⁵/x² = x^5-2 = x³

4x²/5x⁷ = (4/5)x^2-7 = (4/5)x^-5 OR
(4/5x⁵)

12m⁵/4m³ = 3m²
BUT
4m⁵/12m³ = m²/3

Be careful Keep the RULES Separate!! Don't mix up your exponent RULES with what you know previously for rational numbers!!

Math 6H ( Periods 3, 6, & 7)

Multiplying or Dividing by a Power of Ten 3-7

We have learned that in a decimal or a whole number each place value is ten times the place value to its right.

10 ∙ 1 = 10
10 ∙ 10 = 100
10 ∙ 100 = 1000

10 ∙ 0.1 = 1
10 ∙ 0.01 = 0.1
10 ∙ 0.001 = 0.01

Notice that multiplying by ten has resulted in the decimal point being moved one place to the right and in zeros being inserted or dropped.

Multiplying by ten moves the decimal point one place to the right

10 ∙ 762 = 7620

10 ∙ 4.931 = 49.31

At the beginning of this chapter you learned about powers of ten

10⁴ = 10 ∙10 ∙ 10 ∙10 = 10,000

We can see that multiplying by a power of 10 is the same as multiplying by 10 repeatedly.

2.64874 ∙10⁴ = 26,387.4

Notice that we have moved the decimal point four places to the right.

Rule

To multiply a number by the nth power of ten, move the decimal point n places to the right.

Powers of ten provide a convenient way to write very large numbers. Numbers that are expressed as products of a number greater than or equal to 1, but less than 10, AND a power of ten are said to be written in scientific notation.

To write a number in scientific notation we move the decimal point to the left until the resulting number is between 1 and 10. We then multiply this number by the power of 10, whose exponent is equal to the number of places we moved the decimal point.

4,592,000,000 in scientific notation

First move the decimal point to the left to get a number between 1 and 10

4,592,000,000 the first factor in scientific notation becomes 4.592

Since the decimal point was moved 9 places, we multiply 4.592 by 10⁹ to express the number in scientific notation

4.592 x 10⁹

When we move a decimal point to the left, we are actually dividing by a power of ten.

Notice that in dividing by a power of 10 we move the decimal point to the left the same number of places as the exponent. Sometimes we may have to add zeros

3.1 ÷ 10⁴ = 0.00031

Rule

To divide a number by the nth power of ten, move the decimal point n places to the left, adding zeros as necessary.

Algebra Period 4

Exponents: 5-1
POWER RULES:


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 = mmm

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
= m⁰ (by power rules for division)
       


By the transitive property of equality : 1 = m⁰


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

JUST REMEMBER
NEGATIVE POWERS MEANS THE NUMBERS ARE FRACTIONS


They're in the wrong place in the fraction

m³/m⁵ = m^-2
        

m³/m⁵ = mmm/ mmmmm
= 1/mm


Again, by transitive property of equality:

m³/m⁵ = m^-2 = 1/m²

Remember the rule of powers with (  ) 
When there is a product inside the (  ), then everything inside is to the power!

If there are no (  ), then only the variable/number right next to the power is raised to that power.

3x^-2 does not equal (3x)^-2
The first is 3/x² and the second is 1/9x²

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

Math 6H ( Periods 3, 6, & 7) Yosemite Week

Comparing Decimals 3-4

We have used number lines to compare whole numbers. Number lines can be used to show comparisons of decimals. As with whole numbers, a larger number is graphed to the right of a smaller number.

In order to compare decimals, we compare the digits in the place farthest to the left where the decimals have different digits.

Compare the following:

1. 0.64 and 0.68 since 4 < 8 then 0.64 < 0.68.

2. 2.58 and 2.62 since 5 < 6 then 2.58 < 2.62 .

3. 0.83 and 0.833

To make it easier to compare, first express 0.83 to the same number of decimal places as 0.833

0.83 = 0.830 Then compare

0.830 and 0.833 since 0 <3 Then 0.830 < 0.833.

Write in order from least to greatest

4.164, 4.16, 4.163, 4.1

First, express each number to the same number of decimal places Then compare. 4.164, 4.160, 4.163, 4.100

The order of the numbers from least to greatest is

4.1, 4.16, 4.163, 4.164


Rounding 3-5

A method for rounding may be stated as follows: Find the place to which you wish to round, mark it with an underline ___ Look at the digit to the right. If the digit to the right is 5 or greater, add 1 to the marked digit. If the digit to the right is less than 5, leave the marked digit unchanged. Replace each digit to the right of the marked place with a 0

Round 32,567 to (a) the nearest ten thousand, (b) the nearest thousand, (c) the nearest hundred, and (d) the nearest ten

(a)32, 567: since 2 is less than 5, we leave the 3 unchanged, and replace 2, 5, 6, and 7 with zeros

30,000

(b) 32,567: since the digit to the right of 2 is 5, we add a 1 to 2 and get 3 and we replace 5, 6, and 7 with zeros

33,000

(c)32,567: since 6 is greater than 5, we add 1 to 5 and replace 6 and 7 with zeros

32,600

(d) 32,567: since 7 is greater than 5, we add 1 to 6 and replace 7 with a zero

32,570

A similar method of rounding can be used with decimals. The difference between the two methods is that when rounding decimals, we do not have to replace the dropped digits with zeros.

Round 4.8637 to (a) the nearest thousandth, (b) the nearest hundredth, (c) the nearest tenth, and (d) the nearest unit

a. 4.8637: Since 7 is greater than 5, we add 1 to 3 --get 4 & drop the 7

4.864

b. 4.8637: Since 3 is less than 5, we leave 6 unchanged and drop 3 & 7

4.86

c. 4.8637: Since 6 is greater than 5, we add 1 to 8 and drop 6,3, &7

4.9

d. 4.8637: Since 8 is greater than 5, we add 1 to 4 and drop 8, 6, 3, & 7

5

Math 6H ( Periods 3, 6, & 7)

Decimals 3-3

Although decimals ( termed decimal fractions) had been used for centuries, Simon Stevin in the 16th century began using them on a daily basis and he helped establish their use in the fields of sciences and engineering.

Note that
1/10 = 1/10¹
1/100 = 1/10²
1/1000 = 1/10³

We also know that
1/10= 0/1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
and so on... these strings of digits are called decimals.

Remember how we proved that any number to he zero power was equal to 1
or a⁰ = 1

Refer back to your notes or to the blog a few days ago...
we also showed how
What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3
or 3⁴⁺² = 3 ⁶
We just add the exponents if the bases are the same!!
When we divide by the same base we just subtract

3⁴ /3² = 3^4-2 =3²

What would happen if we had
3² / 3⁴ ?
Let's look at what we would actually have
3⋅3
3⋅3⋅3⋅3

Which would be
1
3²

or 1/3²

but you can write that as 3^-2
We just subtract-- using the same rule.


Now let's get back to our decimal lesson and apply that to decimals -- and the Powers of TEN

SO 1/10 = 1/10¹= 0.01 and it is equal to 10^-1
Notice that 10^-1 is NOT a negative number-- it is a small number
and 10^-21 is not a negative number it is a VERY TINY number

AS with whole numbers, decimals use place values. These place values are to the RIGHT of the decimal point.
We need to be able to write decimals in words as well as expanded notation.
In class we used 0.6394 as our example

zero and six thousand three hundred ninety-four ten-thousandths.

Notice how this number when written in words begins...with "ZERO AND"
Why do we need to do that?

Also notice that there is a hyphen between ten and thousandths in ten-thousandths. It is critical to understand when you must place a hyphen.
We read the entire number to the right of the decimal point as if it represented a whole number, and then we give the place value of the digit farthest to the right.

So, although 0.400 is equivalent to 0.4
we must read 0.400 as "zero and four hundred thousandths."

Now look at the following words
"zero and four hundred-thousandths." What is the subtle difference between those two phrases above?
There is a hyphen in the last phrase-- which means that the hundred and the thousandths are attached and represent a place value so

zero and four hundred-thousandths is 0.00004 while
zero and four hundred thousandths is 0.400

Carefully see the distinction!!

Getting back to our 0.6394

to write it in decimals sums and then in exponents:
0 + 0.6 + 0.03 + 0.009 + 0.0004

0 + 6(0.1) + 3(0.01) +9(0.001) + 4(0.0001)

0(10⁰) + 6(10^-1)+ 3(10^-2)+ 9(10^-3)+ 4(10^-4)

14.35 is read as fourteen AND thirty-five hundredths.
When reading numbers, only use the AND to indicate the decimal point

Algebra Period 4

Word Problems.. Continuted...

Generally, you see these type of word problems for geometry, age problems, and finding two integers that have some sort of relationship to each other.

GEOMETRY:
The length of a rectangle is twice its width. The perimeter is 48 inches. What is the length and width? Let w = width and l = length of the rectangle Using P = 2l + 2w
2l + 2w = 48 and What else do we know? l = 2w
We can't solve either of these 2 equations because they each have 2 different variables. But...we can substitute in for one of the variables and "get rid of it"! :)
2l + 2w = 48 Substitute 2w in for length: ( because we know l = 2w)
2(2w) + 2w = 48
6w = 48
w = 8 inches
l = 2w so l = 2(8) = 16 inches

AGE PROBLEMS:
Mary and Sam's ages sum to 50

Mary's age is 10 less than twice Sam's age. Find their ages
Let M = Mary's age and S = Sam's age
M = 2S - 10
M + S = 50
Substitute (2S - 10) for Mary's age so you'll only have 1 variable:
(2S - 10) + S = 50
3S - 10 = 50
3S = 60
Sam (S) = 20 years old
Mary = 2S - 10 = 2(20) - 10 = 30 years old.


Finding 2 integers:
The sum of 2 integers is 26.
One integer is 10 more than 3 times the other. Find the 2 integers.

x = one integer and y = other integer x + y = 26

x = 3y + 10 Substitute in for x:
(3y + 10) + y = 26 4y + 10 = 26
4y = 16 y = 4
x = 3(4) + 10 = 22

Algebraic Inequalities:

TRANSLATING WORDS:

Some key words to know:

AT LEAST means greater than or equal

AT MOST means less than or equal

I need at least $200 to go to the mall means I must have $200, but I'd like to have even more!

I want at most 15 minutes of homework means that I can have 15 minutes, 
but I'm hoping for even less! 
Because the answers in an inequality are infinite, the word problems usually are worded to either ask for the least or the greatest answers in the solution set.
 
For example if the problem asks for 2 consecutive odd integers that sum to at least 50, it will say: "give the smallest" in the solution set.

n + (n + 2) > 50
n > 24

The smallest odd integers in the set are 25 and 27

For example if the problem asks for 2 consecutive odd integers that sum to at most 50, it will say: "give the greatest" in the solution set.

n + (n + 2) < 50
n < 24

The greatest odd integer in the set is 23, so the integers are 23 and 25 (n + 2)

Math 6H ( Periods 3, 6, & 7)

The Decimal System 3-2

Our system of numbers uses the following ten digits:
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9

Whole numbers greater than 9 can actually be represented as sums. For example
386 = 300 + 80 + 6
or
3(100) + 8(10) + 6(1)

Notice that each place value is ten times the value of the place value to its RIGHT!!
The number 10 is called the BASE of this system of writing numbers.
The system itself is called the DECIMAL SYSTEM from the Latin word decem-- which means ten
Think December- but why is that month the 12th month? hmmm.. Did anyone know from class?

Look at the chart given to you in class and notice the place names for the first several numbers.
To make numbers with MORE THAN four digits easier to read, commas are used to separate the digits into groups of three-- starting from the RIGHT

In words the number 420,346 is written as
"four hundred twenty thousand, three hundred forty-six."

The expanded notation for 420,346 is given by

4(100,000) + 2(10,00,000) + 0(1000) + 3(100) + 4(10) + 6(1)

Using exponents the expanded notation may be given as
4(10⁵) + 2(10⁴)+0(10³)+3(10²)+4(10¹)+6(10⁰)

What hmmm.. how is (10⁰) = 1


Writing a variable expression to represent two or three digit numbers requires you to think of the value of each place.
For example,
The ten's digit is t and the ones' digit is 2
you can't just put t + 2 WHY???

Let's say you are thinking of the number 12 when we said the expression
"The ten's digit is t and the ones' digit is 2"
If you said t + 2 you would get 1+2
and that = 3
It isn't the two digit number we wanted--- 12.

so what is the place value of the 1?
It is really in the ten's place or written as 1(10)
To write the variable expression we must in include the value

10t + 2 becomes the correct expression

What about the ten's digit is 5: the ones' digit is x? 5t + x. Do I need to put a 1 infront of the x for the ones' digit? No it is... invisible!!

Math 6H ( Periods 3, 6, & 7)

Exponents and Powers of Ten 3-1
When two or more numbers are multiplied together--each of the numbers is called a factor of the product.

A product in which each factor is the SAME is called a power of that factor.

2 X 2 X 2 X 2 = 16. 16 is called the fourth power of 2 and we can write this as
2⁴ = 16

The small numeral (in this case the 4) is called the exponent and represents the number of times 2 is a factor of 16.
The number two, in this case, is called the base.

When you are asked to evaluate... simplify... solve... find the answer
That is,
Evaluate
4³ = 4 X 4 X 4 = 16 X 4 = 64

The second and third powers of a numeral have special names.
The second power is called the square of the number and the third power is called the cube.

We read 12² as "twelve squared" and to evaluate it
12² = 12 X 12 = 144

Powers of TEN are important in our number system.
Make sure to check out the blue sheet and glue it into your spiral notebook
First Power: 10¹ but the exponent is invisible = 10
Second Power: 10² = 10 X 10 = 100
Third Power 10³ = 10 X 10 X 10 = 1000
Fourth Power 10⁴ =10 X 10 X 10 X 10 = 10,000
Fifth Power 10⁵ = 10 X 10 X 10 X 10 X 10 = 100,000

Take a look at this list carefully and you will probably see a pattern that we can turn into a general rule:

The exponent in a POWER of TEN is the same as the number of ZEROS when the number is written out.

The number of ZEROS in the product of POWERS OF TEN is the sum of the numbers of ZEROS in the factors.

For example Multiply.
100 X 1000
Since there are 2 Zeros in 100 and 3 zeros in 1000,
the product will have 2 + 3 , or 5 zeroes.
100 X 1000 = 100,000

When you need to multiply other bases:

first multiply each
For example

3⁴ X 2 ³ would be
(3 X 3 X 3X 3) X ( 2 X 2 X 2)
= 81 X 8 = 648

What happens when you multiply the same bases?
3⁴ ⋅ 3² = 3⋅3⋅3⋅3⋅3⋅3 or 3 ⁶
We just add the exponents if the bases are the same!!

Well then, what about (3⁴)² ?
Wait.. look carefully isn't that saying 3⁴ Squared?
That would be (3⁴)(3⁴), right?
.. and looking at the rule above all we have to do here is then add those bases or 4 + 4 = 8 so the answer would be 3⁸.
OR
we could have made each (3⁴) = (3⋅3⋅3⋅3)
so (3⁴)² would be 3⋅3⋅3⋅3⋅3⋅3⋅3⋅3 or still 3⁸
But wait... isn't that multiplying the two powers? So when raising a power to a power-- you multiply!!
(3⁴)² = 3⁸

1 to any more is still just 1
1⁵ = 1

0 to any power is still 0!!

Evaluate if a = 3 and b = 5
Just substitute in... but use hugs () we all love our hugs!!
a³ + b²
would be (3)³ + (5) ²
= 27 + 25 = 52




Check out this great Video on the Powers of Ten

Algebra Period 4

WORD PROBLEMS--REVIEW
Writing algebraic expressions will NOT have an equal sign and you will NOT be able to solve them!

CHAPTER 1-6: WRITING ALGEBRAIC EXPRESSIONS
STRATEGY #1: TRANSLATE WORD BY WORD
Many times 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 are both commutative, when solving for a solution the order will not matter BUT I require that you translate accurately-- similarly to when you speak another language you are required to learn the proper order of words. AND, 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. If you aren’t sure about the order with these, I suggest that you try plugging in an actually number and see what you would do with the phrase.
For example, if the phrase was
“5 subtracted from 12”
you would immediately know to write
12-5

For word problems like someone's age or the amount of money you have, you also 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?

YES!
You're algebra is correct!



STRATEGY #2: DRAWING A PICTURE

I have 5 times the number of quarters as I have dimes.

Let’s say I first translate to: 5Q = D
I
check: If I assume that I have 20 quarters, then 5(20) = 100 dimes

Does this make sense? That would mean I have a lot more dimes than quarters.

The original problem says I have a lot more quarters!

My algebra is WRONG!
I need to switch the variables.

5D = Q

I check: If I assume that I have 20 quarters, then 5D = 20
D = 4

Does this make sense? YES!
I have 20 quarters and only 4 dimes.

Sometimes it helps to make a quick picture.

Imagine 2 piles of coins.

The pile of quarters is 5 times as high as the pile of dimes.

You can clearly see that you would need to multiply the number of dimes
to make that pile the same height as the number of quarters!



STRATEGY #3: MAKE A T-CHART
-- this is my favorite!!
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 at least 3 lines of numbers and look for the relationship between the 2 columns.
Ask yourself what do you do to the left side to get to the right?

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 l number of hours

1 ------l------24

2 ------l------48

3 ------l------72

Now look at the relationship between the left column and the right column. What do I do to 1 to get 24? What do I do to 2 to get 48? What do I do to 3 to get 72? For each 
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

d days l number of hours

1 ------l------24

2 ------l------48

3 ------l------72

d ------l-----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 l number of days

24------l------- 1

48------l------- 2

72 ------l-------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. or ask yourself "What do I do to 24 to get 1? What do I do to 48 to get 2? What do I do to 72 to get 3?" 
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

h hours l number of days

24------l------- 1

48------l------- 2

72 ------l-------3
h -------l-------h/24



Some interesting translations used all the time in Algebra:

The next consecutive number after n: n + 1

Does it work? Try it with any number: if you have 5, then 5 + 1 will give you 6


The next EVEN consecutive number after n: n + 2

Does it work? Try it with any EVEN number: if you have 12, then 12 + 2 will give you 14


The next ODD consecutive number after n: n + 2

Does it work? Try it with any number: if you have 9, then 9 + 2 will give you 11

Math 6H ( Periods 3, 6, & 7)

Check out this great Video about the Powers of Ten

Math 6H Period 3, 6 & 7

Solving Equations & Inequalities 2-4 & 2-5

Inequalities Continued

We know about > greater and as well as < less than
so now we look at
≥ which means " greater than or equal to" and
≤ which means " less than or equal to"

This time the boundary point ( or endpoint) is included in the solution set.
The good news is that we still solve these inequalities the same way in which we solved equations-- using the properties of equality.
w/4 ≥ 3
we multiply both sides by 4/1
(4/1)(w/4) ≥ 3(4/1) by the X prop =
1w ≥ 12
w ≥ 12 by the ID(x)

Which means that any number greater than 12 is part of the solution AND 12 is also part of that solution

Take the following:
b - 3 ≤ 150 ( we need to add 3 to both sides of the equation)
+3 = +3 using the + prop =
b + 0 ≤ 153
or b ≤ 153 using the ID (+)

Problem Solving: Using Mathematical Expression 2-6

Seventeen less than a number is fifty six

I suggest lining up and placing the "equal sign" right under the word is
then complete the right side = 56
after that take your time translating the left
Start with a "let statement."
A "let statement" tells your reader what variable you are going to use to represent the number in your equation.
So in this case Let b = the number
it becomes
b- 17 = 56
Now solve as we have been practicing for a couple of weeks.
b - 17 = 56
+17 = +17 using the +prop=
b + 0 = 73
b = 73 by the ID(+)

How could we check?
A FORMAL CHECK involves three steps:
1) Re write the equation ( from the original source)
2) substitute your solution or... "plug it in, plug it in...."
3) DO the MATH!! actually do the math to check!!

so to check the above
b - 17 = 56
substitute 73 and put a "?" above the equal sign...

73 - 17 ?=? 56
Now really do the math!! Use a side bar to DO the MATH!!
That is, what is 73- 17? it is 56
so 56 = 56


Practice some of the class exercised on Page 50, Just practice setting up the equations from the verbal sentences.