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
Wednesday, December 16, 2009
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!!
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!!
Tuesday, December 15, 2009
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.
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:
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!!
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!!
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