Graphs of Ordered Pairs 11-8
The location of a desk in a classroom can be described as “ second row, third desk” If we write (2, 3) to represent this location, the order of the numbers is important since (3, 2) would represent “third row, second desk) a pair of numbers whose order is important is called an ordered pair. Remember when you first walked into our classroom on the very first day—You had to find your seat – and it was an ordered pair!!
We graph a number as a point on a number line. We graph an ordered pair of numbers as a point on a plane marked with two perpendicular number lines called axes. The first number of an ordered pair is associated with the horizontal number line—called the x- axis and the second number with the vertical number line, called the y axis. The axes meet in a point, called the origin (0, 0).
Open your books to Page 389 to review - we are looking at the top graph on that page. To locate the ordered pair (3,2) start at the origin—where is that?—go 3 units to the right and then 2 units up. The numbers 3 and 2 are the coordinates of the graph (3, 2) the plane itself is called a coordinate plane.
Remember (x, y) it is x first, then y. It is Horizontal first, then Vertical, Across and then Up/Down its easy to remember… each is alphabetically in order..
Sunday, February 1, 2009
Wednesday, January 21, 2009
Pre Algebra Period 2
REVIEW
ADDING OR SUBTRACTING FRACTIONS 5-3
NEGATIVE FRACTIONS
1) Double check any subtractions just as you would for integer problems
2) Place the bigger fraction on the top (no matter the sign)
3) Restate to common denominators if needed
4) Borrow if the fraction below is smaller than the fraction above
5) Make sure your answers have consistent signs - In other words, if you have a negative fraction, make sure your whole number part is also negative
OR
1) You can simply make all mixed numbers into improper fractions first
2) Find a common denominator
3) Use integer rules with the numerators
4) Restate back into mixed numbers if required
EXAMPLE using both methods:
5 2/3 - 10 1/4
KEEP THEM AS MIXED NUMBERS:
First of all, you know the final answer will be NEGATIVE so create an answer box right now—and put the negative sign in it!!
Put bigger absolute value on top and take their difference
10 1/4
5 2/3
Find a common denominator:
10 3/12
5 8/12
Borrow because the bottom number is smaller than the top number
9 15/12
5 8/12
Use integer rules to add or subtract:
4 7/12
Put this in your answer box—that you already created with the negative sign and you have
-4 7/12
RESTATE THEM INTO IMPROPER FRACTIONS:
5 2/3 - 10 1/4
17/3 - 41/4
The common denominator is 12
(4)17/(4)(3) - (3)41/(3)(4)
68/12 - 123/12
Subtract using integer rules
-55/12
Restate into a mixed number if required
-4 7/12
EQUATIONS USING ADDING AND SUBTRACTING FRACTIONS 5-7
Same as using adding and subtracting with integers
Use the OPPOSITE (inverse) OPERATION (sign)
If there are fractions on both sides, remember to find a COMMON DENOMINATOR.
EXAMPLE - with a common denominator:
y - 1/8 = 5/8
ADD 1/8 to both sides and you get
y = 5/8 + 1/8 = 6/8
SIMPLIFY to get y = 3/4
EXAMPLE - with different denominators:
y + 4 5/12 = 5 3/8
SUBTRACT 4 5/12 from both sides and get
y = 5 3/8 - 4 5/12
Draw a vertical line separating the whole numbers from the fractions
FIND A COMMON DENOMINATOR which is 24
5 3/8 9/24
-4 5/12 10/24
NOW YOU'LL NEED TO BORROW
4 33/24
-4 10/24
23/24
RESTATE WITH THE VARIABLE FOR YOUR ANSWER y = 23/24
EXAMPLE: When you'll have to double check and put the larger ( absolute value) number on top
y + 2 1/6 = 1 3/8
SUBTRACT 2 1/6 FROM EACH SIDE
y = 1 3/8 - 2 1/6
DOUBLE CHECK , LOOK AND SEE THE SIGNS ARE DIFFERENT—so you will need a SIDE BAR to do your work. PUT THE WINNER ON TOP In this case, PUT 2 1/6 ON TOP (larger absolute value)… Which also means you need to create an answer box with the variable, an equal sign, and the sign of the winner. so on your page you would have in a box y = - just waiting for your results…
- 2 1/6
+1 3/8
Draw a VERTICAL line separating the whole numbers from the fractions
Now you need a COMMON DENOMINATOR:
- 2 1/6 4/24
+1 3/8 9/24
YOU WILL NEED TO BORROW SINCE 9 IS BIGGER THAN 4
At this point I never worry about the signs, I just take their difference—knowing that I have created that answer box with the sign of the answer. ( see above)
2 1/6 4/24
-1 3/8 9/24
1 28/23
1 9/24
19/24
but you put this back in the answer box which is just waiting for your answer and you have y = -19/24
YOU CAN ALSO CHANGE THEM BOTH INTO IMPROPER FRACTIONS FIRST!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
-13/6 -13/6
y = 33/24 - 52/24
y = -19/24
ONE OTHER METHOD (that they use in Algebra next year):
GET RID OF ALL DENOMINATORS BY MULTIPLYING BY THE LCD!!!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
24(y + 13/6) = 24(11/8)
24y + 52 = 33
-52 -52
24y = -19
y = -19/24
MULTIPLYING AND DIVIDING FRACTIONS 5-4
YOU DO NOT NEED A COMMON DENOMINATOR!!!!
MULTIPLICATION
1. Turn any mixed number into an IMPROPER FRACTION
2. Simplify (Some teachers call this : Cross Cancel) if possible (it makes the math easier!)
3. Multiply numerator x numerator and denominator x denominator
4. Simplify if necessary
DIVISION:
WE NEVER DIVIDE, WE FLIP THE SECOND FRACTION AND MULTIPLY!
(never, ever touch the first fraction!)
1. Turn any mixed number into an IMPROPER FRACTION
2. FLIP THE SECOND FRACTION AND CHANGE TO MULTIPLICATION
3. Cross cancel if possible (it makes the math easier!)
4. Multiply numerator x numerator and denominator x denominator
5. Simplify if necessary
NEGATIVES:
Follow your integer rules to determine the sign
VARIABLES:
You can cross cancel them as well!
EQUATIONS USING MULTIPLYING AND DIVIDING FRACTIONS 5-8
SOLVING ONE STEPS WITH FRACTIONS THAT ARE MULTIPLIED
SIMPLY MULTIPLY BY THEIR RECIPROCAL
3/5 y = 20
(5/3)(3/5)y = 20(5/3)
y = 100/3
Make mixed if required: 33 1/3
MIXED NUMBERS?
TURN THEM INTO IMPROPER, THEN MULTIPLY BY RECIPROCAL
4 2/3 y = 2 1/6
14/3 y = 13/6
(3/14)(14/3) y = (13/6)(3/14)
y = 13/28
ADDING OR SUBTRACTING FRACTIONS 5-3
NEGATIVE FRACTIONS
1) Double check any subtractions just as you would for integer problems
2) Place the bigger fraction on the top (no matter the sign)
3) Restate to common denominators if needed
4) Borrow if the fraction below is smaller than the fraction above
5) Make sure your answers have consistent signs - In other words, if you have a negative fraction, make sure your whole number part is also negative
OR
1) You can simply make all mixed numbers into improper fractions first
2) Find a common denominator
3) Use integer rules with the numerators
4) Restate back into mixed numbers if required
EXAMPLE using both methods:
5 2/3 - 10 1/4
KEEP THEM AS MIXED NUMBERS:
First of all, you know the final answer will be NEGATIVE so create an answer box right now—and put the negative sign in it!!
Put bigger absolute value on top and take their difference
10 1/4
5 2/3
Find a common denominator:
10 3/12
5 8/12
Borrow because the bottom number is smaller than the top number
9 15/12
5 8/12
Use integer rules to add or subtract:
4 7/12
Put this in your answer box—that you already created with the negative sign and you have
-4 7/12
RESTATE THEM INTO IMPROPER FRACTIONS:
5 2/3 - 10 1/4
17/3 - 41/4
The common denominator is 12
(4)17/(4)(3) - (3)41/(3)(4)
68/12 - 123/12
Subtract using integer rules
-55/12
Restate into a mixed number if required
-4 7/12
EQUATIONS USING ADDING AND SUBTRACTING FRACTIONS 5-7
Same as using adding and subtracting with integers
Use the OPPOSITE (inverse) OPERATION (sign)
If there are fractions on both sides, remember to find a COMMON DENOMINATOR.
EXAMPLE - with a common denominator:
y - 1/8 = 5/8
ADD 1/8 to both sides and you get
y = 5/8 + 1/8 = 6/8
SIMPLIFY to get y = 3/4
EXAMPLE - with different denominators:
y + 4 5/12 = 5 3/8
SUBTRACT 4 5/12 from both sides and get
y = 5 3/8 - 4 5/12
Draw a vertical line separating the whole numbers from the fractions
FIND A COMMON DENOMINATOR which is 24
5 3/8 9/24
-4 5/12 10/24
NOW YOU'LL NEED TO BORROW
4 33/24
-4 10/24
23/24
RESTATE WITH THE VARIABLE FOR YOUR ANSWER y = 23/24
EXAMPLE: When you'll have to double check and put the larger ( absolute value) number on top
y + 2 1/6 = 1 3/8
SUBTRACT 2 1/6 FROM EACH SIDE
y = 1 3/8 - 2 1/6
DOUBLE CHECK , LOOK AND SEE THE SIGNS ARE DIFFERENT—so you will need a SIDE BAR to do your work. PUT THE WINNER ON TOP In this case, PUT 2 1/6 ON TOP (larger absolute value)… Which also means you need to create an answer box with the variable, an equal sign, and the sign of the winner. so on your page you would have in a box y = - just waiting for your results…
- 2 1/6
+1 3/8
Draw a VERTICAL line separating the whole numbers from the fractions
Now you need a COMMON DENOMINATOR:
- 2 1/6 4/24
+1 3/8 9/24
YOU WILL NEED TO BORROW SINCE 9 IS BIGGER THAN 4
At this point I never worry about the signs, I just take their difference—knowing that I have created that answer box with the sign of the answer. ( see above)
2 1/6 4/24
-1 3/8 9/24
1 28/23
1 9/24
19/24
but you put this back in the answer box which is just waiting for your answer and you have y = -19/24
YOU CAN ALSO CHANGE THEM BOTH INTO IMPROPER FRACTIONS FIRST!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
-13/6 -13/6
y = 33/24 - 52/24
y = -19/24
ONE OTHER METHOD (that they use in Algebra next year):
GET RID OF ALL DENOMINATORS BY MULTIPLYING BY THE LCD!!!
y + 2 1/6 = 1 3/8
y + 13/6 = 11/8
24(y + 13/6) = 24(11/8)
24y + 52 = 33
-52 -52
24y = -19
y = -19/24
MULTIPLYING AND DIVIDING FRACTIONS 5-4
YOU DO NOT NEED A COMMON DENOMINATOR!!!!
MULTIPLICATION
1. Turn any mixed number into an IMPROPER FRACTION
2. Simplify (Some teachers call this : Cross Cancel) if possible (it makes the math easier!)
3. Multiply numerator x numerator and denominator x denominator
4. Simplify if necessary
DIVISION:
WE NEVER DIVIDE, WE FLIP THE SECOND FRACTION AND MULTIPLY!
(never, ever touch the first fraction!)
1. Turn any mixed number into an IMPROPER FRACTION
2. FLIP THE SECOND FRACTION AND CHANGE TO MULTIPLICATION
3. Cross cancel if possible (it makes the math easier!)
4. Multiply numerator x numerator and denominator x denominator
5. Simplify if necessary
NEGATIVES:
Follow your integer rules to determine the sign
VARIABLES:
You can cross cancel them as well!
EQUATIONS USING MULTIPLYING AND DIVIDING FRACTIONS 5-8
SOLVING ONE STEPS WITH FRACTIONS THAT ARE MULTIPLIED
SIMPLY MULTIPLY BY THEIR RECIPROCAL
3/5 y = 20
(5/3)(3/5)y = 20(5/3)
y = 100/3
Make mixed if required: 33 1/3
MIXED NUMBERS?
TURN THEM INTO IMPROPER, THEN MULTIPLY BY RECIPROCAL
4 2/3 y = 2 1/6
14/3 y = 13/6
(3/14)(14/3) y = (13/6)(3/14)
y = 13/28
Wednesday, January 7, 2009
Pre Algebra Period 2 (Tuesday)
Chapter 5-2 FRACTIONS = DECIMALS
How to change a fraction to a decimal
1. Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 =
3 divided by 4 =
0.75
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: 3/4
but this time you will multiply by 25/25 to get
75/100 = 0.75
3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = 0.5
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 decimals to fractions.
CHANGING TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
I learned this catchy phrase READ IT WRITE IT REDUCE!!
But we no longer say "reduce". We now say "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) (Now is the time to put the whole number back!!) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS:
Repeating decimals (we'll use algebra!)
This involves algebra and takes some work, so MEMORIZE THE FOLLOWING:
1/3 = .333 . . . and 2/3 = .666. . .
Also showed the 1/9 family pattern which is the numerator with a bar
Another great time-saving pattern - 1/11 family: Multiply the numerator by 9 and put a bar over it
To get an exact answer when doing math operations with repeating decimals,
make all repeating decimals into their fraction equivalents and do the operations with fractions!
To change repeating decimals to fractions, follow these steps:
Let n = the repeating decimal
Multiply both sides by a power of 10 equal to the number of places that repeat under the bar
Subtract n on the left side and the repeating decimal equal to n on the right side
Solve as a one-step equation
Multiply both numerator and denominator by a power of 10 if necessary
to get the decimal out of the numerator.
Simplify
EXAMPLE: Restate .41666 . . . into a fraction
The repeating portion is .6 or one place so multiply both sides by 10
n = .41666 . . .
10n = 4.1666 . . .
- n = -.4166 . . .
9n = 3.75 so divide both sides by 9 to get
9n/9 = 3.75/9
n = 375/900
n = 5/12
How to change a fraction to a decimal
1. Divide (ALWAYS WORKS!)
EXAMPLE: 3/4 =
3 divided by 4 =
0.75
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: 3/4
but this time you will multiply by 25/25 to get
75/100 = 0.75
3. MEMORY! Some equivalencies you should just know!
EXAMPLE: 1/2 = 0.5
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 decimals to fractions.
CHANGING TERMINATING DECIMALS TO FRACTIONS:
EASY!!!
I learned this catchy phrase READ IT WRITE IT REDUCE!!
But we no longer say "reduce". We now say "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) (Now is the time to put the whole number back!!) 7 6/25
HOW TO CHANGE REPEATING DECIMALS TO FRACTIONS:
Repeating decimals (we'll use algebra!)
This involves algebra and takes some work, so MEMORIZE THE FOLLOWING:
1/3 = .333 . . . and 2/3 = .666. . .
Also showed the 1/9 family pattern which is the numerator with a bar
Another great time-saving pattern - 1/11 family: Multiply the numerator by 9 and put a bar over it
To get an exact answer when doing math operations with repeating decimals,
make all repeating decimals into their fraction equivalents and do the operations with fractions!
To change repeating decimals to fractions, follow these steps:
Let n = the repeating decimal
Multiply both sides by a power of 10 equal to the number of places that repeat under the bar
Subtract n on the left side and the repeating decimal equal to n on the right side
Solve as a one-step equation
Multiply both numerator and denominator by a power of 10 if necessary
to get the decimal out of the numerator.
Simplify
EXAMPLE: Restate .41666 . . . into a fraction
The repeating portion is .6 or one place so multiply both sides by 10
n = .41666 . . .
10n = 4.1666 . . .
- n = -.4166 . . .
9n = 3.75 so divide both sides by 9 to get
9n/9 = 3.75/9
n = 375/900
n = 5/12
MULTIPLICATION OF MONOMIALS AND BINOMIALS 5-9
Multiplying a monomial by a polynomial is just the Distributive Property
You'll need to remember your power rules because you'll be MULTIPLYING SAME BASES
and then combining only LIKE TERMS.
EXAMPLE:
3x3 ( 2x2 - 3x + 10)
3x3(2x2) + 3x3(- 3x) + 3x3(10)
6x5 - 9 x5 + 30x3
(no like terms to combine in this one!)
When you multiply a binomial by another binomial, I used to think of it as DOUBLE Distributive Property...See if you can figure out why!
Multiplying a binomial by another binomial:
FOILing binomials:
This is a memory device so you won't forget to multiply any of the factors!
First - first term in each binomial (terms on the left)
Outside - two outside terms in each binomial ( the first one in the
left parentheses and the second one in the right parentheses)
Inside - two inside terms (the terms right next to each other in the
different parentheses - the second one in the first parentheses
and the first one in the second parentheses)
Last - second term in each binomial (terms on the right)
I know this may seem overwhelming when you first see it,
but after you practice it, it does make sense.
It helps a lot of students to not forget any of the 4 multiplications!
EXAMPLE: (5x + 6)(3x - 7)
F O I L
FIRST OUTSIDE INSIDE LAST
= (5x)(3x) + (5x)(-7) + (6)(3x) + (6)(-7)
Simplify: 15x2 + (-35x) + 18x - 42
Combine like terms: 15x2 + -17x - 42
By the way, you can also do this in different orders --- as long as you multiply each term in one parentheses by each term in the other parentheses.
There are other ways to do the same problem:
Showed you the "box" method that allows you to use any order and protects you from ever missing one of the four multiplications. ( I can't get a table embedded here so make sure to look at your notes about the box method-- or email me.
5x +6
3x 15x2 18x
-7 -35x -42
15x2+ 18 x – 35x -42 =
15x2 – 17x – 42
Also showed you "unibrow" method where you use arrows to show which terms you are multiplying. THIS IS HOW I LEARNED AND SEE IF YOU CAN FIGURE OUT WHY I USED TO THINK OF IT AS DOUBLE DISTRIBUTIVE PROPERTY!
I can't really show this on this website so watch carefully in class!
Multiplying a monomial by a polynomial is just the Distributive Property
You'll need to remember your power rules because you'll be MULTIPLYING SAME BASES
and then combining only LIKE TERMS.
EXAMPLE:
3x3 ( 2x2 - 3x + 10)
3x3(2x2) + 3x3(- 3x) + 3x3(10)
6x5 - 9 x5 + 30x3
(no like terms to combine in this one!)
When you multiply a binomial by another binomial, I used to think of it as DOUBLE Distributive Property...See if you can figure out why!
Multiplying a binomial by another binomial:
FOILing binomials:
This is a memory device so you won't forget to multiply any of the factors!
First - first term in each binomial (terms on the left)
Outside - two outside terms in each binomial ( the first one in the
left parentheses and the second one in the right parentheses)
Inside - two inside terms (the terms right next to each other in the
different parentheses - the second one in the first parentheses
and the first one in the second parentheses)
Last - second term in each binomial (terms on the right)
I know this may seem overwhelming when you first see it,
but after you practice it, it does make sense.
It helps a lot of students to not forget any of the 4 multiplications!
EXAMPLE: (5x + 6)(3x - 7)
F O I L
FIRST OUTSIDE INSIDE LAST
= (5x)(3x) + (5x)(-7) + (6)(3x) + (6)(-7)
Simplify: 15x2 + (-35x) + 18x - 42
Combine like terms: 15x2 + -17x - 42
By the way, you can also do this in different orders --- as long as you multiply each term in one parentheses by each term in the other parentheses.
There are other ways to do the same problem:
Showed you the "box" method that allows you to use any order and protects you from ever missing one of the four multiplications. ( I can't get a table embedded here so make sure to look at your notes about the box method-- or email me.
5x +6
3x 15x2 18x
-7 -35x -42
15x2+ 18 x – 35x -42 =
15x2 – 17x – 42
Also showed you "unibrow" method where you use arrows to show which terms you are multiplying. THIS IS HOW I LEARNED AND SEE IF YOU CAN FIGURE OUT WHY I USED TO THINK OF IT AS DOUBLE DISTRIBUTIVE PROPERTY!
I can't really show this on this website so watch carefully in class!
Math 6 Honors Periods 6 & 7 (Tuesday)
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 vs. 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 14 + (-52) = -38
Although Aunt Sally says you must do ( ) first, when you are ONLY adding you can use the Commutative and Associative Properties to aid you!!
(-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
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 vs. 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 14 + (-52) = -38
Although Aunt Sally says you must do ( ) first, when you are ONLY adding you can use the Commutative and Associative Properties to aid you!!
(-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
Sunday, January 4, 2009
Math 6 H Periods 1, 6 & 7 (Review)
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 testbook 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 testbook page 366 for an accurate example.
Algebra Period 3 (Review)
REVIEW:WEEK BEFORE WINTER BREAK
Sections 5-5 to 5-8 Polynomials
Polynomial = sum of monomials
Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial
Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)
Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x2 - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y
Constant = the number that is not attached to any variable
Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM
Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!
Sections 5-7 ADDING POLYNOMIALS
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Section 5-8: SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
Sections 5-5 to 5-8 Polynomials
Polynomial = sum of monomials
Monomials must have variables with WHOLE NUMBER powers
(constants have whole number powers because you can say it has a variable to the zero)
(no variables in the denominator and no roots of numbers)
1 Term = monomial
2 terms = binomial
3 terms = trinomial
Terms are separated by addition
(and subtraction...although we never subtract...we add the opposite.)
Coefficient = Number attached to variable (can be a fraction!)
The sign of the coefficient should be looked at AFTER you change any subtractions to addition
For example: 3x2 - 10x
The coefficients are 3 and -10
Also, if you have y/6, you really have 1y/6 (ID prop of multiplication)
so the coefficient of y/6 is 1/6 and can be written as (1/6) y
if you have -y/6, you really have -1y/6 (ID prop of multiplication)
so the coefficient of -y/6 is -1/6 and can be written as (-1/6) y
Constant = the number that is not attached to any variable
Degree of a term = sum of the exponents of all its VARIABLES
Degree of a polynomial = HIGHEST degree of any of its terms
Leading term = term with the HIGHEST degree
Leading coefficient = the coefficient of the LEADING TERM
Sections 5-6
DESCENDING ORDER - Write the variables with the highest power first
ASCENDING ORDER - Write the variable with the lowest power first
(this order is actually never used in practice!)
EVALUATING A POLYNOMIAL - WE'VE BEEN DOING THIS ALL YEAR! Plug it in, plug it in, plug it in! Then use Aunt Sally!
Remember to ALWAYS put the number you substitute in parentheses!!!
Sections 5-7 ADDING POLYNOMIALS
This is nothing more than combining LIKE TERMS
LIKE TERMS = same variable AND same power
You can either do this using 3 different strategies:
1. Simply do it in your head, but keep track by crossing out the terms as you use them.
2. Rewrite putting the like terms together (commutative and associative property)
3. Rewrite in COLUMN form, putting like terms on top of each other like you do when adding a column of numbers.
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) + (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
Section 5-8: SUBTRACTING POLYNOMIALS
You can use the ADDITIVE INVERSE PROPERTY with polynomials!
Subtracting is simply adding the opposite so.............
DISTRIBUTE THE NEGATIVE SIGN TO EACH TERM!!
(Change all the signs of the second polynomial!)
After you change all the signs, use one of your ADDING POLYNOMIAL strategies!
(see the 3 strategies listed above under Chapter 5-7)
EXAMPLE OF COLUMN FORM:
(5x4 - 3x2 - (-4x) + 3) - (-10x4 + 3x3- 3x2 - x + 3)
Rewrite in column form, lining up like terms:
5x4 - 3x2 - (-4x) + 3
- ( -10x4 + 3x3- 3x2 - x + 3)
-----------------------------------
For the sake of showing you here, I have added ZERO Terms to line up columns
+ 5x4 + 0x3 - 3x2 -(-4x) + 3
-(-10x4 +3x3- 3x2 - x + 3)
-----------------------------------
DISTRIBUTE THE NEGATIVE, THEN ADD:
5x4 + 0x3 - 3x2 - (-4x) + 3
+10x4 -3x3 +3x2 + x - 3
-----------------------------------
15x4 - 3x3 + 5 x
Algebra Period 3 (Review)
REVIEW:
EXPONENTS SECTIONS 5-1 TO 5-3
SCIENTIFIC NOTATION SECTION 5-4
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
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
cancel out
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 = m8-8 = m0
(by power rules for division)
By the transitive property of equality : 1 = m0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m3 / m5 = m-2
m3 / m5 = mmm/mmmmm = 1/mm
Again, by transitive property of equality:
m3 / m5 = m-2 = 1/m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)-5 = 1/(-2)5 = 1/-32 OR -1/32
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
More with Exponents 5-2 &
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.
POWER TO ANOTHER POWER
MULTIPLY the POWERS
(m5)3 = m15
To check, EXPAND it out:
(m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR
(m5n4)3 = m15n12
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator
(m2/n6)3 = (m2)3/(n6)3 = m6/n18
Scientific Notation 5-4
You've had this since 6th grade!
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
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 is your answer does not fit the scientific notation range, that you restate it.
TRY THIS LINK THAT TAKES YOU FROM HUGE POWERS TO
LITTLE TINY POWERS (NEGATIVE POWERS OR DECIMALS)
Try this link to practice scientific notation!
Try this link to practice multiplication of scientific notation!
Try this link to practice division!
Try this link to practice harder problems!
EXPONENTS SECTIONS 5-1 TO 5-3
SCIENTIFIC NOTATION SECTION 5-4
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
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
cancel out
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 = m8-8 = m0
(by power rules for division)
By the transitive property of equality : 1 = m0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m3 / m5 = m-2
m3 / m5 = mmm/mmmmm = 1/mm
Again, by transitive property of equality:
m3 / m5 = m-2 = 1/m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)-5 = 1/(-2)5 = 1/-32 OR -1/32
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
More with Exponents 5-2 &
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.
POWER TO ANOTHER POWER
MULTIPLY the POWERS
(m5)3 = m15
To check, EXPAND it out:
(m5)(m5)(m5) = m15
PRODUCT TO A POWER
DISTRIBUTE the power to EACH FACTOR
(m5n4)3 = m15n12
RAISING A QUOTIENT TO A POWER:
DISTRIBUTE THE POWER to the numerator and the denominator
(m2/n6)3 = (m2)3/(n6)3 = m6/n18
Scientific Notation 5-4
You've had this since 6th grade!
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
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 is your answer does not fit the scientific notation range, that you restate it.
TRY THIS LINK THAT TAKES YOU FROM HUGE POWERS TO
LITTLE TINY POWERS (NEGATIVE POWERS OR DECIMALS)
Try this link to practice scientific notation!
Try this link to practice multiplication of scientific notation!
Try this link to practice division!
Try this link to practice harder problems!
Monday, December 1, 2008
Algebra Period 3
Exponents: 5-1
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!!!!
-2^5 = -32
-2^4 = -16
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m^5m^3 = m^8
You can check this by EXPANDING:
(mmmmm)(mmm) = m^8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m^8/ m^5 = m^3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
By the transitive property of equality : 1 = m^0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m^3/m^5 = m^-2
m^3/m^5 = m^-2
mmm/mmmmm 1/mm
Again, by transitive property of equality:
m^3/m^5 = m^-2 = 1/m^2
m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)^-5 = 1/(-2)^5 = 1/-32 OR -1/32
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
More on Exponents: 5-2
POWER to another POWER
Multiply the POWERS
(m^5)^3 = m^15
to check EXPAND it out (m^5) (m^5) (m^5) = m^15
PRODUCT to a POWER
DISTRIBUTE the power to EACH FACTOR
(m^5n^4)^3 = m^15n^12
RAISING a QUOTIENT to a POWER
distribute the power to the numerator AND to the denominator
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!!!!
-2^5 = -32
-2^4 = -16
MULTIPLYING Powers with LIKE BASES:
Simply ADD THE POWERS
m^5m^3 = m^8
You can check this by EXPANDING:
(mmmmm)(mmm) = m^8
DIVIDING Powers with LIKE BASES:
Simply SUBTRACT the POWERS
m^8/ m^5 = m^3
Again, you can check this by EXPANDING:
mmmmmmmm/mmmmm
ZERO POWERS:
Anything to the zero power = 1
(except zero to the zero power is undefined)
Proof of this was given in class:
By the transitive property of equality : 1 = m^0
NEGATIVE POWERS = FRACTIONS
They're in the wrong place in the fraction!
NEGATIVE POWERS ARE NOT NEGATIVE NUMBERS!
THEY HAPPEN WHEN THERE IS A DIVISION OF LIKE BASES WHERE THE POWER ON THE TOP IS SMALLER THAN THE POWER ON THE BOTTOM!
WHEN YOU USE THE POWER RULES, YOU WILL SUBTRACT A BIGGER NUMBER FROM A SMALLER NUMBER AND THAT WILL CREATE A NEGATIVE POWER!
EXAMPLE:
m^3/m^5 = m^-2
m^3/m^5 = m^-2
mmm/mmmmm 1/mm
Again, by transitive property of equality:
m^3/m^5 = m^-2 = 1/m^2
m2
EXPRESS NEGATIVE POWERS WITHOUT EXPONENTS:
1) MOVE TO DENOMINATOR
2) EXPAND THE POWER
EXAMPLE:
(-2)^-5 = 1/(-2)^5 = 1/-32 OR -1/32
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
More on Exponents: 5-2
POWER to another POWER
Multiply the POWERS
(m^5)^3 = m^15
to check EXPAND it out (m^5) (m^5) (m^5) = m^15
PRODUCT to a POWER
DISTRIBUTE the power to EACH FACTOR
(m^5n^4)^3 = m^15n^12
RAISING a QUOTIENT to a POWER
distribute the power to the numerator AND to the denominator
Math 6 H Periods 1, 6 & 7
Review of Sections 5.1- 5.3
You know that 60 can be written as the product of 5 and 12. 5 and 12 are whole number FACTORS of 60.
A number is said to be divisible by its whole number factors if the remainder is 0.
A multiple of a whole number is the product of that number and any other whole number ( including zero-- because zero is a whole number)
You can find the multiples of any given whole number by multiplying that number by 0, 1, 2, 3, 4...
Any multiple of 2 is called an even number. A whole number that is not an even number is called an odd number.
Zero is an even number!!
Tests for Divisibility
A Number is Divisible by:
2 if is even
3 if the SUM of its digits is a multiple of 3
4 if the last two digits in the number represent a multiple of 4
5 if its last digit is either a 0 or a 5
6 if the number is divisible by 2 AND 3
8 if the last three digits in the number represent a multiple of 8
9 if the SUM of its digits is a multiple of 9
0 if the last digit is a 0.
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 + 4. What would be the next perfect number?
Try this and let me know for extra credit... email me your response.
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25 are called perfect squares or square numbers.
1 = 1 times 1
4 = 2 times 2
9 = 3 times 3 and so on
One of the two equal factors of a square number is called the square root of that number. To denote the square root of a number we use a radical sign. √ it looks something like a check mark with a line extending. See our textbook for better examples. ( Page 157)
We can use our knowledge of perfect squares to find the lengths of sides of squares. For example IF the Area of a square is 49 cm squared. We can find the length of each side using the formula for the area of a square. A = lw but in this case it is also A = s^2, which is read "Area equals side squared."
so we know 49 = s^2. so the square root (SQRT) of 49... hmmm.. we know 7 times 7 = 49 so s = 7 cm. Notice the label isn't squared- we are talking about linear measurements with each side-- only the AREA is squared.
Positive SQ RT are also called principal square roots.
negative square roots exist also. For example the SQRT of 81 is 9 but it could also be -9 because (-9) times (-9) also = 81
We are focusing on positive SQ RTs in this chapter.
Memorize the Perfect Squares and corresponding SQ RTs up to 30.
KNOW the rule for squaring any number that ends in 5.
See me if you forgotten the rule!!
Check out these websites for how to calculate a SQ RT without a calculator
http://mathforum.org/library/drmath/sets/mid_square_roots.html
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
http://www.nist.gov/dads/HTML/squareRoot.html
You know that 60 can be written as the product of 5 and 12. 5 and 12 are whole number FACTORS of 60.
A number is said to be divisible by its whole number factors if the remainder is 0.
A multiple of a whole number is the product of that number and any other whole number ( including zero-- because zero is a whole number)
You can find the multiples of any given whole number by multiplying that number by 0, 1, 2, 3, 4...
Any multiple of 2 is called an even number. A whole number that is not an even number is called an odd number.
Zero is an even number!!
Tests for Divisibility
A Number is Divisible by:
2 if is even
3 if the SUM of its digits is a multiple of 3
4 if the last two digits in the number represent a multiple of 4
5 if its last digit is either a 0 or a 5
6 if the number is divisible by 2 AND 3
8 if the last three digits in the number represent a multiple of 8
9 if the SUM of its digits is a multiple of 9
0 if the last digit is a 0.
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 + 4. What would be the next perfect number?
Try this and let me know for extra credit... email me your response.
Square Numbers and Square Roots 5-3
Numbers such as 1, 4, 9, 16, 25 are called perfect squares or square numbers.
1 = 1 times 1
4 = 2 times 2
9 = 3 times 3 and so on
One of the two equal factors of a square number is called the square root of that number. To denote the square root of a number we use a radical sign. √ it looks something like a check mark with a line extending. See our textbook for better examples. ( Page 157)
We can use our knowledge of perfect squares to find the lengths of sides of squares. For example IF the Area of a square is 49 cm squared. We can find the length of each side using the formula for the area of a square. A = lw but in this case it is also A = s^2, which is read "Area equals side squared."
so we know 49 = s^2. so the square root (SQRT) of 49... hmmm.. we know 7 times 7 = 49 so s = 7 cm. Notice the label isn't squared- we are talking about linear measurements with each side-- only the AREA is squared.
Positive SQ RT are also called principal square roots.
negative square roots exist also. For example the SQRT of 81 is 9 but it could also be -9 because (-9) times (-9) also = 81
We are focusing on positive SQ RTs in this chapter.
Memorize the Perfect Squares and corresponding SQ RTs up to 30.
KNOW the rule for squaring any number that ends in 5.
See me if you forgotten the rule!!
Check out these websites for how to calculate a SQ RT without a calculator
http://mathforum.org/library/drmath/sets/mid_square_roots.html
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
http://www.nist.gov/dads/HTML/squareRoot.html
Tuesday, November 11, 2008
Algebra Period 3
CHAPTER 4-1
Introduction to INEQUALITIES and graphing them:
Writing an inequality
Graphing an inequality: open dot is < or >
Closed dot mean “less than or EQUAL” or “ greater than or EQUAL”
(think of the = sign as a crayon that you can use to COLOR IN THE DOT!)
The closed dot INCLUDES the number as part of the solution set.
For example, x ≥ -5 is any number greater than -5 but it also includes -5.
What’s the difference of inequalities from equations? Inequalities have many answers (most of the time an infinite number!)
Example: n > 3 means that every real number greater than 3 is a solution! (but NOT 3)
n ≥ 3 means still means that every real number greater than 3 is a solution, but now 3 is also a solution
GRAPHING INEQUALITIES:
First, graphing an equation's solution is easy
1) Say you found out that y = 5, you would just put a dot on 5 on the number line
2) But now you have the y ≥ 5
You still put the dot but now also darken in an arrow going to the right
showing all those numbers are also solutions
3) Finally, you find in another example that y > 5
You still have the arrow pointing right, but now you OPEN THE DOT on the 5 to show that 5 IS NOT A SOLUTION!
TRANSLATING WORDS:
Some key words to know: so make FLASH CARDS or do whatever you do to MEMORIZE these words!!
AT LEAST means greater than or equal
AT MOST means less than or equal
I need at least $20 to go to the mall means I must have $20, 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 sure hoping for even less!
CHAPTER 4-2 Solving Inequalities with adding and subtracting:
Simply use the Additive Inverse Property as if you were balancing an equation!
The only difference is that now you have more than one possible answer.
Example: 5y + 4 > 29
You would -4 from each side, then divide by 5 on each side and get:
y > 5
Your answer is infinite! Any real number bigger than 5 will work!
See Graphing above to review how to graph the solution!!
CHAPTER 4-3 Solving inequalities with multiplication or division:
Again, you will use your equation skills,
but this time with the Multiplicative Inverse Property.
ONE MAJOR DIFFERENCE FROM EQUATIONS!
When you multiply or divide by a NEGATIVE to BALANCE,
you must SWITCH the inequality SYMBOL!
(Does not apply to adding or subtracting negatives.)
EXAMPLE: -3y > 9
You need to divide both sides by NEGATIVE 3 so the symbol
will switch from > to < in the solution
y < -3 is the answer
If you want to understand why:
3 < 10
Now multiply both sides by -1 (multiplication prop of equality)
You get -3 < -10, but THAT'S NOT TRUE!!!
You have to SWITCH THE SYMBOL to make the answer true: -3 > -10
REMEMBER: when you MULTIPLY or DIVIDE by a NEGATIVE,
the symbol SWITCHES
Chapter 4-4; DOING 2 STEPS WITH INEQUALITY SIGNS -
Same as equations except make sure you switch the symbol
if you multiply or divide by a negative!
Always finish with the variable on the left.
Check with whatever solution is easiest in the solution set!
With two steps:
Before you start, you may want to clear fractions or decimals, but if you don't mind using them, just get started with the checklist below. If you want to clear them, you should do that right after you distribute (between steps 1 and 2 below)
1. Do distributive property first (if necessary)
2 Combine like terms on each side of the wall (equal sign)
3. Jump the variables to one side of the wall (get all the variables on one side of the equation) by using the Additive Inverse Property (add or subtract using the opposite sign of the variable term)
4. Add or subtract
5. Multiply or divide
6. Make sure the variable is on the LEFT side when finished.
By putting the variable on the left side of the inequality, you will be able to graph the solutions much easier. The arrow you will draw will follow the same direction as the inequality sign.
Tuesday, October 28, 2008
Pre Algebra Period 2
Statistics Unit
There are many ways to ORGANIZE and REPRESENT data.
The whole point is to make the data more UNDERSTANDABLE.
There are 3 ways to express the CENTRAL TENDENCY of data:
Mean or average (add up all the values and divide by the number of items)
Mode or the most often seen value
Median or middle of data from least to greatest (if even number of items, average the middle two)
The Range can be expressed in two ways:
Showing the lowest value to the highest value
Calculating the difference between those two values (highest - lowest)
Tally Tables help you put COUNT the data and begin to ORGANIZE it.
You can tally numeric data (like grades) or non-numeric data (like names for the giraffe)
Frequency Tables take the tallies and SUMMARIZE them in an easy to read table.
Can be numeric or non-numeric (like the Tally Table)
Line Plots are graphs that make the Frequency Table summary more VISUAL.
It's really easy to see the MODE in the Line Plot.
For numeric data, you can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
Can be numeric or non-numeric (Like Tally and Frequency Tallies)
Stem and Leaf Plots are graphs that CLUSTER the data in INTERVALS.
This helps make the graph more UNDERSTANDABLE because you can see PATTERNS.
Data must be NUMERIC.
You can determine the MEDIAN easily because it's in order from
LEAST TO GREATEST
You can also see the MODE but not as quickly as the Line Plot because it will be in an interval with other data items. On the other hand, if you're looking for the interval that has the most data (not the individual item), the Stem and Leaf will show the "Interval Mode" very quickly.
You can read every data item on the Stem and Leaf.
Scatter Plots are graphs that plot TWO DIFFERENT TYPES OF DATA against each other, one one the x axis and one on the y axis to determine if there is a CORRELATION between the two types of data. ("co" meaning together and "relation" meaning a pattern)
If the two types of data have a correlation, you can draw a LINE OF BEST FIT through the center of the data points.
POSITIVE CORRELATION: the data moves in the SAME DIRECTION
(the more you study, the higher your grade....or the less you study, the lower your grade)
The line of best fit looks like it is going UP from left to right
NEGATIVE CORRELATION: the data moves in OPPOSITE DIRECTIONS.
(the higher the temperature, the lower the number of people using their heat....or the lower the temperature, the higher the number of people using their heat)
The line of best fit looks like it is going DOWN from left to right
NO CORRELATION: the two types of data has no relationship and so the data points are scattered everywhere with no pattern
(temperature changes should have no relationship with how well a student does)
TODAY:
LINE GRAPHS: p. 98-99
Data that takes place OVER TIME is represented well on a Line Graph.
For example, a student's math grade over the course of a year, or over middle school, or even over a single month. The length of the time interval depends on what you're trying to show.
If you have more than one set of data over the same time period,
you can make MULTIPLE LINES on the graph.
For example, a student's math grade, along with his science grade.
BAR GRAPHS:
Data over time can also be shown as a Bar Graph, but Bar Graphs are especially useful to show data that compares either non-numeric data or comparing multiple types of the same data at the same time.
Non-numeric data: Really a Line Plot but without the "X's" Allows you to represent very large data sets easily because you can label the y axis (vertical axis) in any way that you need to.
Numeric data: Going back to a student's grades on the Line Graph...Say you wanted to compare multiple students' math grades and science grades in 8th grade, you could make a DOUBLE BAR GRAPH with one bar color representing math grades and another bar color representing their science grades
HISTOGRAMS
Isn't this just a Bar Graph???
Yes! It's a specific type of bar graph that shows the FREQUENCY of data items
Isn't that what a Line Plot and a Stem and Leaf Plot show???
Yes!
If you made Bar Graph out of a Line Plot, it would be called a HISTOGRAM
or
If you turn a Stem and Leaf Plot on its side, outline the data listed in the leaves and then erase the data numbers and instead color in the bars, you will also have a HISTOGRAM.
What you lose:
The actual data items (you'll still know 12 students scored in the 80's, but not exactly what score in the 80's that each of them received)
You cannot determine the exact MODE or MEDIAN, but can make an analysis of the intervals.
What you gain:
For large amount of data items, a HISTOGRAM is much less cluttered than either a Stem and Leaf or Line Plot, making the patterns easier to understand.
For large amount of data items, because you can adjust the labels on the y axis, you can fit all the data in in whatever
Math 6 Honors Periods 6 & 7 (Tuesday )
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 = 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 = 2.63874 = 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 umber 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 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 = 0003.1 = 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.
Math 6 Honors Periods 1, 6 & 7 ( Yosemite Week in review)
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
Adding and Subtracting Decimals 3-6
Decimals may be added or subtracted using the same rules as whole numbers
Write the given numbers one above the other with the decimal points in line.
Annex zeros to get the same number of decimal places and then add or subtract as if the numbers were whole numbers.
Place a decimal point in the number for the sum or difference in position under the decimal points in the given numbers.
Add 6.47 + 340.8 + 73.523
| STEP 1 | STEP 2 | STEP 3 |
| 0006.47 | 006.470 | 006.470 |
| 0340.8 | 340.800 | 340.800 |
| + 073.523 | + 73.52 | + 73.523 |
| | 420 793 | 420.793 |
The use of rounded numbers to get an approximate answer is called estimation. We use estimates to check actual answers. Use estimates as a habit to check if your answer is reasonable. To find an estimate, first round the highest place value of the smallest number, then compare.
Add 8.574 + 81.03 + 59.432. Then estimate to check your answer. What is the highest place value of the smallest number? 9 + 81 + 59 = 149
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