Algebra (Period 1)

PRODUCT OF RADICALS 11-4
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)

√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7

HELPFUL HINT:
If you are multiplying √250 • √50,
I would suggest that you don't multiply 250 x 50 too quickly!
Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!
So why not factor each factor first?!!!

Using my example above:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5

Dividing & Simplifying 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.

HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)

EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4


√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6


EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3

Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3

Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3

Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!

AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!

RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)

If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.

RATIONALIZING THE DENOMINATOR EXAMPLE:
√7/√ 3

There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.

You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3

Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!

Algebra (Period 1)

SIMPLIFYING RADICALS 11-3

SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect squares left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: square root of 50 = SQRT(25 * 2)
Next, simplify the SQRT of the perfect square and leave the nonperfect factor under the radical:
SQRT(25 * 2) = SQRT(25) *SQRT(2) = 5 SQRT( 2 )
***Imagine the square root sign top extends over all numbers****
√(25 * 2) = √25 * √2 = 5√2

HELPFUL HINTS:
When you are factoring the radicand, you're looking for the LARGEST PERFECT SQUARE that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
You can even use all the perfect squares that you have memorized... it really helps// just remember when you pull out a number... you are pulling out the number which multiples by itself to get that perfect square... NOT the perfect square itself...


Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors
(remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!
Example: SQRT 250
Prime factorization = 2 * 5 * 5 * 5
Circle the first two 5's
5 * 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical
Final answer: SQRT 250 = 5 SQRT 10
√250 = 5√10


VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: SQRT (x¹⁰ ) = x⁵ or
√x¹⁰ = x⁵
We saw this already in factoring!!!


SIMPLIFYING NONPERFECT VARIABLES:
If the variable has an even power:
EXAMPLE: The square root of 75x¹⁰ = SQRT [(25)(3)(x10 )] =
SQRT [ (5)(5)](3)(x⁵x⁵)

***Imagine the square root sign top extends over all numbers****
√75x¹⁰ = √[(25)(3)(x¹⁰ )] = √[ (5)(5)](3)(x⁵x⁵)


Simplified, you can pull out a factor of 5 and x⁵
Final simplified radical = 5x⁵ SQRT(3)
or
5x⁵√3



If the variable has an odd power:
If you have an odd power variable, simply express it as the even power one below that odd power times that variable to the 1 power:
Example: x⁵ = x⁴ x
so if you have the SQRT( x⁵ ) = SQRT (x⁴ x) = x²SQRT(x)

√x⁵ =√x⁴ x = x²√x



FACTORING A GCF FIRST, THEN FINDING A BINOMIAL SQUARED:
Sometimes you will need to factor what's under the radical before you start to simplify
Example: SQRT(3x² + 12x + 12) or

***Imagine the square root sign top extends over all numbers****
√3x² + 12x + 12

First factor out a 3:

SQRT [3 (x² + 4x + 4) ] or
***Keep imagining that line across all the numbers under the radical √ ****
√3(x² + 4x + 4)

Now factor the trinomial: SQRT [3 (x + 2) (x + 2) ] or

√3(x + 2)²

The (x + 2)² is a perfect square so SQRT [3 (x + 2) (x + 2) ] = (x + 2) SQRT( 3 )

or (x+2)√3

you need to include the HUGS around the (x +2) because you want that binomial to be multiplied by the √3 .. not just the 2.

Algebra (Period 1)

Radical Expressions 11-2


If an expression under the square root sign is NEGATIVE, it does not exist in the REAL numbers!


There is no number that you can square and get a NEGATIVE PRODUCT



VARIABLES UNDER THE SQUARE ROOT SIGN: 
If you have a variable under the square root sign,
you need to determine what values of the variable will keep the radicand greater than or equal zero


The square root of x then is only real when x is greater than or equal to zero


The SQRT (x + 2) is only real when x + 2 is greater than or equal to 0

Set x + 2 greater than or equal to 0 and solve as an inequality!

You will find that x must be greater than or equal to -2



SPECIAL CASE!!!! a variable squared plus a positive integer under radical: 
If you're trying to find the principal square root of x² (or any variable squared) plus a positive integer, then all numbers will work because a squared number will always end up either positive or zero!


Example: √(x² + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.


Then you're just adding another positive number.



If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.

EXAMPLE: √(x² - 10)

x² must be equal or greater than 10, so x must be at least the square root of 10

(the square root of 10 squared is 10)



ANOTHER SPECIAL CASE!!!!!!!!!!

ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.


Example: The square root of x² is the absolute value (positive) of x SHown here as: ( I x I )


Why? 
Because it is assumed that you're finding the PRINCIPAL (positive) square root.


EXAMPLE:
x = -3
 √x² = √(-3)² = √9 = 3 (not -3)

so you have to put absolute value signs around the answer


IF THERE IS A VARIABLE SQUARED
(see p. 489 #17-30)


TRINOMIALS UNDER THE RADICAL:

What do you think you would do if you saw x² + 10x + 25 under the radical sign????


FACTOR IT! 


IT MAY BE A PERFECT SQUARE (a binomial squared!)



EXAMPLE:
√( x² + 10x + 25) factors to
√(x + 5)² = I x + 5 I

Determine the values for the variable that will make each expression a real number
√m(m+3)

you know that m(m+3) ≥ 0 so m ≥0 OR m ≤ -3

√x²(x-3)

again set x²(x-3) ≥ 0 and you discover x = 0 or x ≥3

Given a and c, what must be true of b to make

√b² -4ac
a real number?

a = -3 and c = 2
substitute in and we have

√b² -4(-3)(2)

√b² +24
b can be any real number!!

But what if we have
a = 2 and c = 8
√b² -4ac
substitute in
√b² -4(2)(8)
√b² -64
b is either
b≤ -8 or b ≥ 8


Determine whether each of the follow statements is sometimes, always or never true:

√a² + b ² is a real number ---> ALWAYS

√3 - t is a real number for t ≥ 3 ---> SOMETIMES

√a² - b ² is a real number ---> SOMETIMES

√a² + 2ab + b ² is a real number ---> ALWAYS


For a polynomial in the form ax² + bx + c = 0 to have real solutions,
√b² -4ac must be a real number. Which of the following polynomials have real solutions?

x² - 12x + 3 = 0 is real because √b² -4ac = √12² -4(1)(3) =
√144 -12 √132 and that's real

x² + 5x + 7 = 0 is NOT REAL because √5² -4(1)(7) = √25 -28 = √-3 which is NOT REAL

Algebra (Period 1)

Real Numbers 11-1

...
as opposed to IMAGINARY numbers! : )
(Seriously!)

√ is the symbol for square root


MAIN CONCEPT:
Square rooting "undoes" squaring!

It's the inverse operation!!!

Just as subtraction undoes addition

Just as division undoes multiplication


If you square a square root:

(√243)² = 243 (what you started with)

If you square root something squared:

√243² = 243 (what you started with)

If you multiply a square root by the same square root:

(√243)(√243) = 243 (what you started with)



IN SUMMARY:
(√243)²= √243² = (√243)(√243) = 243





1)RADICAL sign: The root sign, which looks like a check mark.

If there is no little number on the radical, you assume it's the square root

But many times there will be a number there and then you are finding the root that the number says.

For example, if there is a 3 in the "check mark," you are finding the cubed root.

One more example:
The square root of 64 is 8.
The cubed root of 64 is 3.
The 6th root of 64 is 2.



2)RADICAND : Whatever is under the RADICAL sign

In the example above, 64 was the radicand in every case.



3) ROOT (the answer): the number/variable that was squared (cubed, raised to a power)
to get the RADICAND (whatever is under the radical sign)

In the example above, the roots were 8, 3, and 2.



4) SQUARE ROOTS: (What we primarily cover in Algebra I) The number that is squared to get to the radicand.


Every POSITIVE number has 2 square roots - one positive and one negative.


Example: The square root of 25 means what number squared = 25


Answer: Either positive 5 squared OR negative 5 squared = 25



5) PRINCIPAL SQUARE ROOT: The positive square root.


Generally, the first section just asks for the principal square root unless there is a negative sign in front of the radical sign.



6) ± sign in front of the root denotes both the positive and negative roots at one time!


Example: √ 25 = ±5



7) ORDER OF OPERATIONS with RADICALS: Radicals function like parentheses when there is an operation under the radical.

In other words, if there is addition under the radical, you must do that first (like you would do parentheses first) before finding the root.


EXAMPLE: √ (36 + 64) = 10 not 14!!!!


First add 36 + 64 = 100

Then find √100 = 10



Radicals by themselves function as exponents in order of operations
(that makes sense because they undo exponents). 

Actually, roots are FRACTIONAL EXPONENTS!
 
Square roots = 1/2 power,

Cubed roots = 1/3 power,

Fourth roots = 1/4 power, etc.

So √25 = 25^½ = 5



EXAMPLE: 3 + 4√25
 you would do powers first...in this case square root of 25 first!

3 + 4(5)
Now do the multiplication

3 + 20

Now do the addition
23



8) THE SQUARE ROOT OF ANYTHING SQUARED IS ITSELF!!!

EXAMPLE: √5² = 5

√(a -7)² = a - 7



RATIONAL SQUARE ROOTS:
Square roots of perfect squares are RATIONAL



REVIEW OF NUMBER SYSTEMS:

Rational numbers are decimals that either terminate or repeat
 which means they can be restated into a RATIO a/b of two integers a and b where b is not zero.


Natural numbers: 1, 2, 3, ... are RATIOnal because you can put them over 1

Whole numbers: 0, 1, 2, 3,....are RATIOnal because you can put them over 1

Integers: ....-3, -2, -1, 0, 1, 2, 3,....are RATIOnal because you can put them over 1


Rational numbers = natural, whole, integers PLUS all the bits and pieces in between that can be expressed as repeating or terminating decimals: 2/3, .6, -3.2, -10.7 bar, etc.


Real numbers: all of these!

In Algebra II you will find out that there are Imaginary Numbers!

Square roots of NEGATIVE numbers are IMAGINARY



IRRATIONAL SQUARE ROOTS:
Square roots of a nonperfect squares are IRRATIONAL -
They cannot be stated as the ratio of two integers -

As decimals, they never terminate and never repeat -
you round them and use approximately sign.


MOST FAMOUS OF ALL IRRATIONAL NUMBERS IS PI!


There is another group of irrational numbers: 

Square roots of not perfect squares


Square roots are MOSTLY IRRATIONAL!


There are fewer perfect squares than not perfect!


Here are some perfect squares: 0, 4, 9, 16, 25, 36, etc.

PERFECT SQUARES CAN ALSO BE TERMINATING DECIMALS!

EXAMPLE: √(.04) is rational because it is ±0.2



But all the square roots in between these perfect squares are IRRATIONAL

For example, the square root of 2, the square root of 3, the square root of 5, etc.


You can estimate irrational square roots.

For example, the square root of 50 is close to 7 because the square root of 49 is 7. 
You can estimate that the square root of 50 is 7.1 and then square 7.1 to see what you get. 

If that's too much, try 7.05 and square that.

This works much better with a calculator! 

And obviously, a calculator will give you irrational square roots to whatever place your calculator goes to.

Remember: These will never end or repeat
(even though your calculator only shows a certain number of places physically!)

In class we went through a method of finding a good approximate to any square using the perfect square above it and below it!!

Pi Day

HAPPY PI DAY


HAPPY PI DAY! WHAT AN IRRATIONAL DAY!!!!
BE IRRATIONAL.. AND TRANSCENDENTAL
... here are the facts:
Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle.
Pi is an IRRATIONAL number. That means it has no pattern and never terminates.
It CANNOT be written as a fraction with an integer in the numerator and denominator.

We use 3.14 and 22/7 as APPROXIMATIONS of pi.

These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers.

You will be able to discover many interesting facts about pi—even finding it on your own.

Here's some links to click on if you're interested in the mystery of Pi:




History of pi
Find your birthday in pi
The first ten thousand digits of pi


Here is a rap I really like...


Lose Yourself in the Digits of pi


... and of course... here are the songs we sang

Happy Pi Day (to the tune of “Happy Birthday”)

Happy PI day to you
Happy PI day to you
Happy PI day everybody
Happy PI day to you



Oh, Number Pi (to the tune of “Oh, Christmas Tree”)

Oh, Number Pi
Oh, Number Pi
Your digits are unending.
Oh Number Pi
Oh Number Pi
No pattern are you sending
You’re three point one four one five nine
And even more if we had time,
Oh, number Pi
Oh, number Pi
For circle lengths unbending.

Oh, number Pi
Oh, number Pi
You are a number very sweet
Oh, number Pi
Oh, number Pi
Your uses are so very neat.
There’s 2 Pi r and Pi r squared
A half a circle and you’re there,
Oh, number Pi
Oh, number Pi
We know that Pi’s a tasty treat




Pi Day Song (to the tune of “Jingle Bells”)

Pi day songs
All day long
Oh what fun it is
To sing a jolly pi day song
In a fun math class
Like this

Verse;
Circles in the snow
Around and round we go
How far did we have to run?
Diameter times pi! (Refrain)

We wish you a Happy Pi Day (to the tune of “We wish you a Merry Xmas”)

We wish you a happy Pi Day
We wish you a happy Pi Day
We wish you a happy Pi Day
To you and to all

Pi numbers for you
For you and for all
Pi numbers in the month of March
So three point one four!!

Pre Algebra (Period 2 & 4)

Fractions, Decimals & Percents 6-5

Fractions = Decimals = Percents!!
A percent is a ratio that compares a number to 100.
Therefore you can write a percent as a fraction with a denominator of 100

Percent means "per hundred" or "out of a hundred"
THe symbol % comes from the fraction bar and a denominator of 100

TO CHANGE ANY % TO A FRACTION:

simply get rid of the % sign and put the number over a denominator of 100
Simplify
5% = 5/100 = 1/20

TO CHANGE ANY % TO A DECIMAL

Simply get rid of the % and move the decimal point over to the LEFT-TWO PLACES
( its running away from the %... it is so sad that the % sign went away!!)
5% = 0.05

TO CHANGE ANY DECIMAL TO A %
Simply put the % to the right of the number and move the decimal point to the RIGHT TWO PLACES ( the decimal is so happy to see the percent sign -- it runs toward it!!)

0.245 = 24.5%

TO CHANGE ANY FRACTION TO A %
Simply change it to a decimal first, then follow as above

20/50 = 2/5 = 0.4 = 40%
But this one is a good one to think of changing 20/50 = 40/100 and then easily seeing 40%

If the denominator will go into 100 easily, you can use equivalent fractions!!(as in this case)

If you ever forget how many places to move the decimal, just look at the % sign-- It tells you 2 places ( 2 zeros)

You need to know a few percentages
10%--> just move the decimal point one place to the LEFT. That's it
$75.00 with a 10% discount
The discount would be $7.50

On $80.00 the 10% discount would be $8.00 and a 5% would be half of that or $4.00

You can find %20 easily as well-- just double 10%

Know
50% --> 1/2
25% --. 1/4
75% --> 3/4

Math 6 Honors (Period 6 and 7)

Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions


Or compare 5/6 and 7/9

again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9

Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35

The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!

and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!

What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them

What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!

If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!

If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.


Ordering or comparing fractions:

Different ways:

I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?

When the numerator is close to the denominator, the fraction is approaching 1

(Ex: 9/11)

When you double the numerator and it's close to the denominator, the fraction is close to 1/2
(Ex: 4/9)

When the numerator is very far from the denominator, the fraction is approaching zero
(Ex: 1/8)

Also, if one number is improper or mixed number and other is a proper fraction,
then obviously the number greater than 1 will be bigger!

II. LCD - give them all the same denominator using the LCM as the LCD


III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions

IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)

Math 6 Honors (Period 6 and 7)

Fractions & Mixed Numbers 6-3

1/2 + 1/2 + 1/2 = 3/2

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1

1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions


5/2, 8/3, 18/15, 12/5 are all improper fractions


You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number

If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.


To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2


To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.

or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!

2 5/6 =

(2 x 6) + 5
6
=17/6


Practice these:

785 ÷ 3

852÷ 5

3751÷ 16

98001÷231


post your answers below in the comments for extra credit !!

Math 6 Honors (Period 6 and 7)

Fractions 6-1

The symbol 1/4 can mean several things:
1) It means one divided by four
2) It represents one out of four equal parts
3) It is a number that has a position on a number line.



1/8 means 1 divided by 8 or 1 ÷ 8
A fraction consists of two numbers
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
we noted that we could abbreviate ...

denominator as denom with a line above it

and numerator as numer

we found that you could add

1/3 + 1/3 + 1/3 = 3/3 = 1
or 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1
we also noted that 8 X 1/8 = 8/8 = 1

We also noticed that 2/7 X 3 = 6/7


So we discussed the properties
For any whole numbers a, b,and c with b not equal to zero

1/b + 1/b + 1/b ... + 1/b = b/b = 1 for b numbers added together

and we noticed that b X 1/b = b/b = 1
we also noticed that
(a/b) X c = ac/b

We talked about the parking lot problem on Page 180

A count of cars and trucks was taken at a parking lot on several different days. For each count, give the fraction of the total vehicles represented by
(a) cars

(b) trucks

Given: 8 cars and 7 trucks
We noticed that you needed to find the total vehicles or 8 + 7 = 15 vehicles
so

(a) fraction represented by cars is 8/15
(b) fraction represented by trucks is 7/15


What if the given was: 15 trucks and 32 vehicles
This time we need to find how many were cars. so 32 -15 = 17 so 17 cars

(a) fraction represented by cars is 17/32
(b) fraction represented by trucks is 15/32

Equivalent Fractions 6-2


We drew the four number lines from Page 182 and noticed that 1/2, 2/4, 3/6, and 4/8 all were at the midpoints of the segment from 0 to 1. They all denoted the same number and are called equivalent fractions.

If you multiply the numerator and the denominator by the same number the results will be a fraction that is equivalent to the original fraction

1/2 = 1 x 3/2 x 3 = 3/6

It works for division as well
4/8 = 4 ÷ 4 / 4 ÷ 8 = 1/2

So we can generalize and see the following properties
For any whole numbers a, b, c, with b not equal to zero and c not equal to zero

a/b = a x c/ b x c and
a/b = a ÷ c / b ÷c


Find a fraction equivalent to 2/3 with a denominator of 12
we want a number such that 2/3 = n/12
You could look at this and say
" What do I do to 3 to get it to be 12?
Multiply by 4
so you multiply 2 by 4 and get 8 so
8/12 is an equivalent fraction


A fraction is in lowest terms if its numerator and denominator are relatively prime-- That is if their GCF is 1

3/4, 2/7, and 3/5 are in lowest terms.
They are simplified
You can write a fraction in lowest terms by dividing the numerator and denominator by their GCF.


Write 12/18 is lowest terms
The GCF (12 and 18) = 6

so 12/18 = 12÷ 6 / 18 ÷ 6 = 2/3

Find two fractions with the same denominator that are equivalent to 7/8 and 5/12
This time you need to find the least common multiple of the denominators!! or the LCD
Using the box method from Chapter 5, we find that the LCM (8, 12 ) = 24

7/8 = 7 X 3 / 8 X 3 = 21/24
and
5/12 = 5 X 2 / 12 X 2 = 10/24


When finding equations such as
3/5 = n/15 we noticed we could multiply the numerator of the first fraction by the denominator of the second fraction and set that equal to the denominator of the first fraction times the numerator of the second... or

3(15) = 5n now we have a one step equation

If we divide both sides by 5 we can isolate the variable n and solve...
3(15)/ 5 = n
9 = n

We found we could generalize

If a/b = c/d then ad = bc

Math 6 Honors (Period 6 and 7)

The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!1
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 -x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!
Graphing Inequalities

You will need to look at the graphs in your textbook. .. page 397

Whenever we graph relations that are inequalities we must be aware of all the facts that can influence your work. You need to ask yourself, "What kind of numbers is the solution supposed to be?"
When you graphed inequalities such as
-3 < x < 2 where x was an integer we used a point on the number line for each integer that could be a solution to that inequality. To show every number in x < 2 we would use a number line and place an Open Dot at 2 indicating that 2 was NOT part of the solution and then draw a darkened ray away from 2 indicating 1, 0, -1, -2... were all part of the solution.
To show that this line has infinite solutions in that direction, you MUST place an arrow at the end of that darkened ray.

If the inequality was a " less than or equal to" " ≤" you would use a Closed Dot at 2 to indicate that 2 was part of the solution.

We can now graph inequalities such as y ≥ x + 2
first you find the BOUNDARY LINE which is just y = x + 2 and you can use the 3 column table as we have done before or use a T chart as shown in class.

Remember you only need 2 points to determine a line---> but 3 points will help you make sure you have 3 correct points on the line!!

I am going to try to set up a T chart using "I" to separate the x and y
X I Y
-2 I 0
-1 I 1
0 I 2
1 I 3
2 I 4

(Note: it doesn't line up well here.. but hopefully you get the idea)

Plot those points on the graph and you have what appears to be a straight line. Since we are graphing y ≥ x + 2 we ARE including the line so we draw a solid line.

But.. what points are included?
Well, we know that (-2,0) works but we also see if we plug into our inequality that (-2,1) and (-2,2) work as well.

We need to shade the part above the line to indicate all those points are part of the solution as well.

Three set method for graphing an inequality

(1) Determine the boundary line. Draw it--
use a solid line if the boundary line is part of the graph (≤ or ≥)
use a dashed line if the boundary line is NOT part of the graph (< or >)

(2) Shaded either the part above the boundary line or the part below the boundary line.
If the inequality reads y > or y ≥ shade ABOVE the line.
If the inequality reads y < or y ≤ shade BELOW the line

(3) Always CHECK- choose a point you think works within the shaded region and see if it does work.. or use (0,0) and determine if it is part of the solution or not!!

Math 6 Honors (Period 6 and 7)

Graphs of Equations 11-9

An equation in two variables y = x + 2
produces an infinite number of ordered pairs
If we give x the value of 3, a corresponding value of y is determined
y = (3) + 2 = 5
The ordered pair is (3, 5)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -2
y = (-2) + 2 = 0 ( -2, 0)
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
I like to remember ordered pairs---> ( ordered, pairs)

y = 2x -3
in the future you will see it written as
f(x) = 2x -3
so if x = 3
f(3) = 2(3) -3 = 6-3 = 3
so f(3) = 3
if x = 5
f(5)= 2(5) - 3 = 10 -3 = 7
so f(5) = 7

We used a three column chart to compute our ordered pairs.
Please refer to the blue sheet glued into your spiral notebook for the examples we completed in class-- if you were absent, please come in one morning and I will review that chart with you.

Pre Algebra (Period 2 & 4)

Proportions 6-2

A proportion = 2 equal ratios (2 equivalent fractions)

Solve using equivalent fractions or

Cross multiplication and then a one-step equation

(see if you can simplify the fractions before multiplying)

Example: Solve the proportion for y:


4/3 = y
/21

EQUIVALENT FRACTION APPROACH:
Multiply both top and bottom by 7, y = 28



CROSS PRODUCTS APPROACH:

You'll get 3y = (21)(4)
Now divide each side by 3.
Do this before multiplying on the right side!
Why? Because a lot of the time you'll be able to simplify and keep the numbers smaller!


3y/3 = (21)(4) /3

See how the 3 cross cancels into the 21?
so y = 28


ALWAYS SIMPLIFY THE FRACTIONS FIRST!

WORD PROBLEMS WITH PROPORTIONS:


It's all about setting up the LABELS first!


label A _____________ = ____________ label A

label B xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxlabel B

Math 6 Honors (Period 6 and 7)

Graphs of Equations 11-9

An equation in two variables y = x + 2
produces an infinite number of ordered pairs
If we give x the value of 3, a corresponding value of y is determined
y = (3) + 2 = 5
The ordered pair is (3, 5)
If we let x = 4
y = (4) + 2 = 6
and we get the ordered pair (4, 6)
What happens if x = 0
y = (0) + 2 = 2 ( 0, 2)
or x = -2
y = (-2) + 2 = 0 ( -2, 0)
For each value of x there is EXACTLY 1 value of y.
set of ordered pairs in which no two ordered pairs have the same x is called a FUNCTION
I like to remember ordered pairs---> ( ordered, pairs)

y = 2x -3
in the future you will see it written as
f(x) = 2x -3
so if x = 3
f(3) = 2(3) -3 = 6-3 = 3
so f(3) = 3
if x = 5
f(5)= 2(5) - 3 = 10 -3 = 7
so f(5) = 7

We used a three column chart to compute our ordered pairs.
Please refer to the blue sheet glued into your spiral notebook for the examples we completed in class-- if you were absent, please come in one morning and I will review that chart with you.


The following equations create curves that are called PARABOLAS!! Notice the difference in these equations from our previous equations
y = x² +1
when we create your three column table using integers from -2 to 2
we notice
y = (-2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)
y = (-1)² +1 = 1 + 1 = 2 ordered pair (-1, 2)
y = (0)² +1 = 0 + 1 = 1 ordered pair (0, 1)
y = (1)² +1 = 1 + 1 = 2 ordered pair (1, 2)
y = (2)² +1 = 4 + 1 = 5 ordered pair (-2, 5)

When you graph this... you get a "U" shaped graph.

Remember linear equations LINEar equations are lines!1
and look like y = x + 2

PARABOLAS have the form y = x² or y = -x²

Let's try
y = 2 -x²
With our 3 column table
for values of x from -2 to 2
we find
y = 2 -(-2)² = 2 -(4) = -2 and the ordered pair is (-2,-2)
y = 2 -(-1)² = 2 - (1) = 1 and the ordered pair is ( -1, 1)
y = 2 -(0)² = 2 - 0 = 2 and the ordered pair is (0, 2)
y = 2 -(1)² = 2 -1 = 1 and the ordered pair is (1, 1)
y = 2 -(2)² = 2 - (4) = -2 and the ordered pair is (2, -2)

When you graph these ordered points you find you have an upside down U
hmmm... y = -x² results in a sad face parabola
and y = x² results in a happy face parabola!!

Math 6 Honors (Period 6 and 7)

Graphs of Ordered Pairs 11-8

A PAIR of numbers whose ORDER is important is called an
ordered pair!!
(ordered, pair)

(2,3) is not the same as (3,2)

The two perpendicular lines are called axes.
The x-axis deals with the 1st number of the ordered pair and the y-axis deals with the 2nd number of the ordered pair.
The AXES meet at a point called the Origin (0,0)
The plane is called the coordinate plane

There are 4 quadrants, Use Roman Numerals to name them!!
Quadrant I ---> both the x and y coordinates are positive
(x,y) (+,+)

Quadrant II --> the x coordinate is negative but the y is positive
(-x,y) (-,+)

Quadrant III --. both the x and y coordinates are negative
(-x,-y) (-,-)

Quadrant IV --> the x coordinate is positive but the y coordinate is negative
(x,-y) (+,-)