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!!