RADICAL EXPRESSIONS 11-2
If an expression under the SQ RT 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 SQ RT 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 square root of (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: √(x 2 + 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 ( lxl )
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 = l x + 5l
Thursday, February 18, 2010
Algebra (Period 4)
REAL NUMBERS 11-1
as opposed to IMAGINARY numbers! (Seriously!)
√ is the symbol for radicals. We use this symbol (without any small number on the radical) to represent square root (At times in the post I may need to use SQ RT to refer to square root)
Square rooting "undoes" squaring!
It's the inverse operation: Just as subtraction undoes addition and Just as division undoes multiplication
EXAMPLES:
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 = 25½ = 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!
(PI day is on a Sunday this year, BUT we will celebrate it either Friday before or the Monday after... we will be approximating pi day...))
There is another group of irrational numbers: Square roots of nonperfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than nonperfect!
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 ± .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 SQRT of 50 is close to 7 because the square root of 49 is 7.
You can estimate that the SQ RT 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!)
Another way to calculate the square root.
Take the perfect square above and below.
Find the difference between the two perfect squares ( it will always bee their sum)
Use that number as the denominator.
Now take the difference between your number and the bottom perfect quare. That difference will be the numerator. Now, if necessary estimate-- using benchmarks to change your number into a decimal.
For instance, what is the square root so 181
Set up your problem like this
198
181
169
because 196 and 169 are your perfect squares between 181
Then you know
√196 = 14
181
√169 = 13
put the approximate sign after your number within a square root sign and place the bottom perfect square and a decimal point after that because you know your number falls between the two square roots., that is,
√196 = 14
√181 ≈ 13.
√169 = 13
Now, 196 - 169 = 27 but it is also try that 14 + 13 = 27... either way you use that number as the denominator.
The numerator is the difference between the number you are using and th bottom perfect square... in this case 181- 169 = 12
now 12/27 becomes 4/9 which we know is .44444... so
√181 ≈ 13.44
as opposed to IMAGINARY numbers! (Seriously!)
√ is the symbol for radicals. We use this symbol (without any small number on the radical) to represent square root (At times in the post I may need to use SQ RT to refer to square root)
Square rooting "undoes" squaring!
It's the inverse operation: Just as subtraction undoes addition and Just as division undoes multiplication
EXAMPLES:
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 = 25½ = 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!
(PI day is on a Sunday this year, BUT we will celebrate it either Friday before or the Monday after... we will be approximating pi day...))
There is another group of irrational numbers: Square roots of nonperfect squares
Square roots are MOSTLY IRRATIONAL!
There are fewer perfect squares than nonperfect!
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 ± .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 SQRT of 50 is close to 7 because the square root of 49 is 7.
You can estimate that the SQ RT 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!)
Another way to calculate the square root.
Take the perfect square above and below.
Find the difference between the two perfect squares ( it will always bee their sum)
Use that number as the denominator.
Now take the difference between your number and the bottom perfect quare. That difference will be the numerator. Now, if necessary estimate-- using benchmarks to change your number into a decimal.
For instance, what is the square root so 181
Set up your problem like this
198
181
169
because 196 and 169 are your perfect squares between 181
Then you know
√196 = 14
181
√169 = 13
put the approximate sign after your number within a square root sign and place the bottom perfect square and a decimal point after that because you know your number falls between the two square roots., that is,
√196 = 14
√181 ≈ 13.
√169 = 13
Now, 196 - 169 = 27 but it is also try that 14 + 13 = 27... either way you use that number as the denominator.
The numerator is the difference between the number you are using and th bottom perfect square... in this case 181- 169 = 12
now 12/27 becomes 4/9 which we know is .44444... so
√181 ≈ 13.44
Thursday, February 11, 2010
Algebra (Period 4)
Fitting Equations to Data:7-7
(word problems)
Many real world relationships are LINEAR (or approximate a linear relationship), meaning they can be graphed with a LINE.
For example, if a candy bar costs $1.50, then 2 bars cost $3.00 etc
Think of the number of candy bars as the x value because that's the independent variable...
you decide how many candy bars you're going to buy.
The y value is the dependent variable.
How much money you owe at the register is the y value!
x y
1 $1.50
2 $3.00
3 $4.50
and so on....
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price!!! ($1.50)
Think of it as the change in y ($ you owe)
over the change in x (# of candy bars)
The money you owe goes up $1.50 every time you buy 1 more candy bar
This is a POSITIVE slope of 1.50!
So another meaning of a positive slope in the real world is that two types of data are changing at a constant rate IN THE SAME DIRECTION (called a POSITIVE CORRELATION in statistics)
You can reverse both directions and have the same relationship just as we did when we counted the slope going both up and down:
Every time you decrease your purchase by 1 bar, your purchase price also decreases by $1.50.
The data is still going in the SAME DIRECTION (now both going down!)
How do you find the equation of this linear relationship?
You use the Slope Intercept or Point Slope formulas of a line!
(It's just a real life mystery line!!! :)
SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
You can pick any point, which means any number of candy bars with its associated cost.
I used 1 candy bar (the x value) with its cost of $1.50:
y - 1.50 = 1.50(x - 1)
Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:
The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything,
But often, the y intercept value will be a number.
For example, think of cell phone use.
Say you are charged $.10 per minute of use, but your monthly charge is $25.
Even if you don't use your cell phone, you still owe $25!
The linear equation would be:
y = .10x + 25 where y = what you owe
and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)
NEGATIVE SLOPES IN THE REAL WORLD:
Let's think of a real life example that will give us a NEGATIVE SLOPE...
Often, the more you buy, the smaller the unit price per item.
This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.
SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!
DATA THAT APPROXIMATES A LINEAR EQUATION:
Finally, sometimes real world data can be APPROXIMATED as a linear relationship.
In other words, it may not be exact, but a good way to understand the data is to look at it that way.
Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.
(word problems)
Many real world relationships are LINEAR (or approximate a linear relationship), meaning they can be graphed with a LINE.
For example, if a candy bar costs $1.50, then 2 bars cost $3.00 etc
Think of the number of candy bars as the x value because that's the independent variable...
you decide how many candy bars you're going to buy.
The y value is the dependent variable.
How much money you owe at the register is the y value!
x y
1 $1.50
2 $3.00
3 $4.50
and so on....
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price!!! ($1.50)
Think of it as the change in y ($ you owe)
over the change in x (# of candy bars)
The money you owe goes up $1.50 every time you buy 1 more candy bar
This is a POSITIVE slope of 1.50!
So another meaning of a positive slope in the real world is that two types of data are changing at a constant rate IN THE SAME DIRECTION (called a POSITIVE CORRELATION in statistics)
You can reverse both directions and have the same relationship just as we did when we counted the slope going both up and down:
Every time you decrease your purchase by 1 bar, your purchase price also decreases by $1.50.
The data is still going in the SAME DIRECTION (now both going down!)
How do you find the equation of this linear relationship?
You use the Slope Intercept or Point Slope formulas of a line!
(It's just a real life mystery line!!! :)
SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
You can pick any point, which means any number of candy bars with its associated cost.
I used 1 candy bar (the x value) with its cost of $1.50:
y - 1.50 = 1.50(x - 1)
Y INTERCEPTS IN THE REAL WORLD THAT ARE NOT ZERO:
The scenario above has a y intercept value of 0 because you don't owe anything if you don't buy anything,
But often, the y intercept value will be a number.
For example, think of cell phone use.
Say you are charged $.10 per minute of use, but your monthly charge is $25.
Even if you don't use your cell phone, you still owe $25!
The linear equation would be:
y = .10x + 25 where y = what you owe
and x = number of minutes used
Plug in 0 for x, number of minutes, and you still owe $25 (y value)
NEGATIVE SLOPES IN THE REAL WORLD:
Let's think of a real life example that will give us a NEGATIVE SLOPE...
Often, the more you buy, the smaller the unit price per item.
This happens with copying or buying things like invitations.
Companies give you a "break" if you buy more.
SEE THE EXAMPLE ON PAGE 333 IN YOUR BOOK!
DATA THAT APPROXIMATES A LINEAR EQUATION:
Finally, sometimes real world data can be APPROXIMATED as a linear relationship.
In other words, it may not be exact, but a good way to understand the data is to look at it that way.
Think SCATTER PLOTS with POSITIVE or NEGATIVE CORRELATIONS.
Tuesday, February 9, 2010
Math 6H ( Periods 3, 6, & 7)
Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 2 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 36/99 = .36363636...
we talked about the 1/7th family.
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 2 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 36/99 = .36363636...
we talked about the 1/7th family.
Monday, February 8, 2010
Math 6H ( Periods 3, 6, & 7)
Word Problems 6-3 & 6-4
Read-- and read read your word problems. Really determine what the question is asking...
When in doubt, draw it out!!
Sally & Bill prepared 40 sandwiches for a picnic with 14 friends. If the sandwiches were divided equally among all of them, how many sandwiches would each person receive?
Wait, think... there are 16 total people because you must count Sally & Bill
so 40/16 = 2 1/2 sandwiches
A class has 48 minutes available to hear reports for 15 students. If the time is to be divided equally, how long may each student take?
48/ 15 = 3 1/5 minutes
An assembly period has 50 minutes for speeches by 4 candidates for class president. If two minutes are allowed for each introduction and the remaining time is to be divide equally, how long may each candidate speak?
4 introductions so 2 X 4 = 8 50 -8 = 42 minutes to be divided equally
42/ 4 = 10 1/2 minutes each
A fence is to be put up along a side of a lot measuring 114 feet. Nineteen fence posts are to be used-- one at each end and 17 equally spaced between ends. What is the distance between centers of two consecutive fence posts?
This is a Must DRAW OUT
I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I
count the spaces between and you will find that there are 18 spaces between the posts so 114/18 = 6 1/3 ft
A play consists of 3 acts of equal length with 2 twenty minute intermissions between acts. (this means there is one between the first and the second act and another between the second and the third act) If the whole production takes 3 hours, how long is each act in minutes?
When in doubt-- draw it out
l_______________l_____l_______________l_____l_______________l
First change 3 hours to 180 minutes
then deduct the two twenty minute intermissions 180 - 40 = 140
140 minutes left for the three equal acts
140/3 = 46 2/3 min
An hour TV program consists of four segments of equal length. There are 3 on-minute commercials between EACH segment. There is also a 2 minute intro and a 2 minute closing. How long is each segment?
When in doubt-- draw it out
l______l__________l__l__l__l__________l__l__l__l__________l______l
there are 4 minutes for into and closing and 3(3) minutes of commercials 4 + 9 = 13 minutes
60 -13 = 47 47/ 4 = 11 3/4 minutes
A table top is in the shape of a square 45 inches on each side. A chessboard is inlaid so that a border of 3 inches exists on all four sides. How wide is each of the 8 rows of squares of chessboard?
When in doubt... draw it out
You will see that there is a border of 3 inches all around so you need to deduct 3 inches from one side and 3 inches from the other or 6 inches
45-6 = 39. Now the chessboard has 8 rows so divide
39/8 = 4 7/8 inches
The Bears won 11 out of 16 games. The Eagles won 17 out of 24 games. Which team won a greater fraction of its games?
Compare
11/16 or 17/24
WAIT... not those two fractions again!!
The Eagles...
Did you have to change to a common denominator??? I hope not!!
There were 10 sunny days out of 28 in February and 12 sunny days out of 30 in November. (Anyone remember the poem...or the way to check with your knuckles???)
Which month had the higher fraction of sunny days?
10/28 12/30 I might simplify first but I can tell by comparing that November had more sunny days
What method did you use???
Read-- and read read your word problems. Really determine what the question is asking...
When in doubt, draw it out!!
Sally & Bill prepared 40 sandwiches for a picnic with 14 friends. If the sandwiches were divided equally among all of them, how many sandwiches would each person receive?
Wait, think... there are 16 total people because you must count Sally & Bill
so 40/16 = 2 1/2 sandwiches
A class has 48 minutes available to hear reports for 15 students. If the time is to be divided equally, how long may each student take?
48/ 15 = 3 1/5 minutes
An assembly period has 50 minutes for speeches by 4 candidates for class president. If two minutes are allowed for each introduction and the remaining time is to be divide equally, how long may each candidate speak?
4 introductions so 2 X 4 = 8 50 -8 = 42 minutes to be divided equally
42/ 4 = 10 1/2 minutes each
A fence is to be put up along a side of a lot measuring 114 feet. Nineteen fence posts are to be used-- one at each end and 17 equally spaced between ends. What is the distance between centers of two consecutive fence posts?
This is a Must DRAW OUT
I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I
count the spaces between and you will find that there are 18 spaces between the posts so 114/18 = 6 1/3 ft
A play consists of 3 acts of equal length with 2 twenty minute intermissions between acts. (this means there is one between the first and the second act and another between the second and the third act) If the whole production takes 3 hours, how long is each act in minutes?
When in doubt-- draw it out
l_______________l_____l_______________l_____l_______________l
First change 3 hours to 180 minutes
then deduct the two twenty minute intermissions 180 - 40 = 140
140 minutes left for the three equal acts
140/3 = 46 2/3 min
An hour TV program consists of four segments of equal length. There are 3 on-minute commercials between EACH segment. There is also a 2 minute intro and a 2 minute closing. How long is each segment?
When in doubt-- draw it out
l______l__________l__l__l__l__________l__l__l__l__________l______l
there are 4 minutes for into and closing and 3(3) minutes of commercials 4 + 9 = 13 minutes
60 -13 = 47 47/ 4 = 11 3/4 minutes
A table top is in the shape of a square 45 inches on each side. A chessboard is inlaid so that a border of 3 inches exists on all four sides. How wide is each of the 8 rows of squares of chessboard?
When in doubt... draw it out
You will see that there is a border of 3 inches all around so you need to deduct 3 inches from one side and 3 inches from the other or 6 inches
45-6 = 39. Now the chessboard has 8 rows so divide
39/8 = 4 7/8 inches
The Bears won 11 out of 16 games. The Eagles won 17 out of 24 games. Which team won a greater fraction of its games?
Compare
11/16 or 17/24
WAIT... not those two fractions again!!
The Eagles...
Did you have to change to a common denominator??? I hope not!!
There were 10 sunny days out of 28 in February and 12 sunny days out of 30 in November. (Anyone remember the poem...or the way to check with your knuckles???)
Which month had the higher fraction of sunny days?
10/28 12/30 I might simplify first but I can tell by comparing that November had more sunny days
What method did you use???
Thursday, February 4, 2010
Math 6H ( Periods 3, 6, & 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!!)
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!!)
Wednesday, February 3, 2010
Algebra Period 4
Find the Equation of a Line
: 7-6
1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug
Usually, we just try 0, 1, 2 first
2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug
This works really well when the line is in STANDARD form and the coefficients are factors of the constant on the other side of the equation.
3) We know how to GRAPH a line by using slope-intercept form:
We isolate y on one side
We read the y intercept (the b which is the constant on the other side)
We graph that value on the y axis
We COUNT to the next point by reading the slope, the coefficient of the x
The slope should be read as the y value (+up or -down)
and then the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1
TODAY WE'RE NOT LEARNING TO GRAPH ANOTHER WAY...WE'RE LEARNING HOW TO FIND THE EQUATION OF A LINE
We usually use the slope-intercept form of the line as our "template"
We know that y = mx + b so
we can substitute in what we know (what the problem gives us) and solve for whatever we're missing!
[The problem will give you enough information to know the slope or know the y intercept! Then you'll solve for the missing item.]
It helps to memorize this little rhyme
Oh mystery line, What could you be?
If I could just find you,
y = mx + b
If I could find m
And I could find b.
Then I could put it together
And I would see:
y = mx + b
The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula or counting it if you have the graph
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.
3) Don't forget to put it all together in one equation at the end!
THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:
First case:
You're given the slope and the y intercept (easiest case) m = 3/2 and b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5
Second case:
You're given a point and the slope and need to find the intercept (b)
(3, 1) is a point on the line and m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5
Third case:
You're given a point and the y intercept and need to find the slope
(3, 1) is still a point on the line, but this time you know b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2
Fourth case:
You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3) are 2 points on the line
You need to first find the slope using the formula:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one (you pick the easiest!) of the points and find the intercept, b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together: y = 2x + 1
Fifth case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form
1) We know how to GRAPH a line by 3 points where we decide what to plug in and chug
Usually, we just try 0, 1, 2 first
2) We know how to GRAPH a line by intercepts...we plug in zero for y and x and chug
This works really well when the line is in STANDARD form and the coefficients are factors of the constant on the other side of the equation.
3) We know how to GRAPH a line by using slope-intercept form:
We isolate y on one side
We read the y intercept (the b which is the constant on the other side)
We graph that value on the y axis
We COUNT to the next point by reading the slope, the coefficient of the x
The slope should be read as the y value (+up or -down)
and then the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1
TODAY WE'RE NOT LEARNING TO GRAPH ANOTHER WAY...WE'RE LEARNING HOW TO FIND THE EQUATION OF A LINE
We usually use the slope-intercept form of the line as our "template"
We know that y = mx + b so
we can substitute in what we know (what the problem gives us) and solve for whatever we're missing!
[The problem will give you enough information to know the slope or know the y intercept! Then you'll solve for the missing item.]
It helps to memorize this little rhyme
Oh mystery line, What could you be?
If I could just find you,
y = mx + b
If I could find m
And I could find b.
Then I could put it together
And I would see:
y = mx + b
The rhyme has 3 steps and usually you will have 3 steps or questions to ask yourself:
1) Do I have the slope (m)? If not, find it by using the slope formula or counting it if you have the graph
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope and solving for b or if you have the graph, just read it on the y axis.
3) Don't forget to put it all together in one equation at the end!
THERE ARE 5 CASES THAT YOUR BOOK INCLUDES:
First case:
You're given the slope and the y intercept (easiest case) m = 3/2 and b = -7/5
Just plug in to the generic slope intercept equation: y = 3/2 x - 7/5
Second case:
You're given a point and the slope and need to find the intercept (b)
(3, 1) is a point on the line and m = 2
Plug in the point and the slope and solve for b
1 = 2(3) + b
1 = 6 + b
b = -5
Now put it altogether with the given slope and the intercept you just found:
y = 2x -5
Third case:
You're given a point and the y intercept and need to find the slope
(3, 1) is still a point on the line, but this time you know b = 2
Plug in the point and the y intercept and solve for slope
1 = 3m + 2
-1 = 3m
m = -1/3
Now put it altogether with the given intercept and the slope you just found:
y = -1/3 x + 2
Fourth case:
You're given 2 points and need to find the slope and the intercept
(1 , 3) and (-2 , -3) are 2 points on the line
You need to first find the slope using the formula:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one (you pick the easiest!) of the points and find the intercept, b
3 = 2(1) + b
3 = 2 + b
b = 1
Finally, put it all together: y = 2x + 1
Fifth case:
You have a graph of a line and need to determine the equation
Look at the graph and find 2 easy points to use to find the slope (make sure they are integers!)
(If the y intercept is not an integer, then follow fourth case completely)
Put the information together in y = mx + b form
Math 6H (Period 3, 6 & 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 !!
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 !!
Tuesday, February 2, 2010
Math 6H (Period 3, 6, & 7)
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
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
Algebra (Period 4)
Equations and Slope 7-5
Chapter 7-5: SLOPE INTERCEPT FORM
WE HAVE COVERED GRAPHING THE FOLLOWING 2 WAYS:
3 random x points using a table (Ch 7-2)
Most pick 0, 1, 2 and just plug and chug to find the y value
This works well when you don't end up with fraction answers!
EXAMPLE: -3y =-2x - 6
x y
0 2
3 4
-3 0
Then we learned a second way to graph:
Graph the 2 special points that are the y and x intercepts (Ch 7-3)
x intercept: where the line intersects the x axis
y intercept: where the line intersects the y axis
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2 so the y intercept is (0, 2)
If y = 0, x = -3 so the x intercept is (-3, 0)
x y
0 2
-3 0
The x intercept is the one that has the x value (y is 0)
The y intercept is the one that has the y value (x is 0)
TODAY, WE WILL COVER THE MOST USED FORM OF A LINEAR EQUATION:
THE SLOPE INTERCEPT FORM
You must restate the equation to get it into the following format:
y = mx + b
where m = slope and b = y intercept (where the line hits the y axis)
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign!
Look at the example I gave you above:
-3y =-2x - 6
If you tried the x points 0, 1, 2, only 0 gave an integer for y.
All the other y values were fractions.
The slope intercept form provides the explanation on why.
RESTATE -3y =-2x - 6 INTO SLOPE INTERCEPT FORM:
Divide both sides by -3:
y = 2/3 x + 2
Look at the coefficient for x. WHAT X VALUES WILL GIVE YOU INTEGER ANSWERS FOR Y????
They need to be multiples of 3!
Now look at the graph of y = 2/3 x + 2
Notice that the + 2 at the end is the y intercept! (without doing any work!)
Use the counting method for slope on your graph.
You should have counted: UP 2, RIGHT 3
The slope therefore is 2/3.
Look at the equation.
It told you 2/3 without any work!
Graph when line is in Slope Intercept Form:
So if you have the slope-intercept form of the equation, it's really easy to graph the line:
1) Graph the intercept on the y axis (that's the +/- constant at the end)
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 2/3 x + 2 ) Put a dot at (0, 2)
3) From (0,2) count up 2 and over to the right 3 to find the next coordinate (3,4)
Remember, slope is y over x, so the numerator is the y change and the denominator is the x change!
If it's positive, you're counting up (positive) and to the right (positive) or you can count down (negative) and to the left (negative) because 2 negatives make a positive.
If it's negative, you're counting down (negative) and to the right (positive) or you can count up (positive) and to the left (negative) because you would have a positive and negative = negative
If you're given the slope and the y intercept, you can write the equation of any line!
Just use: y = mx + b
EXAMPLE: m = -2/3 and b = -12
The line would be y = -2/3 x - 12
Restate Standard Form to Slope Intercept Form:
Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(x and y on the same side, x is positive, no fractions, constant alone)
Restate into Slope Intercept Form:
Solve for y
first subtract 3x from both sides: 4y = -3x + 10
Now divide both sides by 4:
y = -3/4 x + 10/4 or
y = -3/4 x + 5/2
The slope is the coefficient of the x m = -3/4 (so you're sliding down at a little less than a 45 degree angle)
The y intercept is the constant b = 5/2 (so the line crosses the y axis at 2 1/2)
Notice that the "b" is a fraction.
When this happens, the slope intercept form may not be the best form to graph the line.
You must start at 2 1/2 on the y axis and count down 3, right 4. THAT'S HARD!
Neither 3 nor 4 is a factor of the constant 10 so intercepts will be fractions.
We need the x term to end up with 1/2 so that when we add that to the b of 5/2, we'll get an integer!
So let's make x = 2 because that will cross cancel with the -3/4 slope to halves:
y = (-3/4)(2) + 5/2 = -3/2 + 5/2 = 2/2 = 1
So we found a coordinate that has just integers: (2, 1)
Now graph that point.
Now count the slope from that point (instead of from the y intercept)!
Chapter 7-5: SLOPE INTERCEPT FORM
WE HAVE COVERED GRAPHING THE FOLLOWING 2 WAYS:
3 random x points using a table (Ch 7-2)
Most pick 0, 1, 2 and just plug and chug to find the y value
This works well when you don't end up with fraction answers!
EXAMPLE: -3y =-2x - 6
x y
0 2
3 4
-3 0
Then we learned a second way to graph:
Graph the 2 special points that are the y and x intercepts (Ch 7-3)
x intercept: where the line intersects the x axis
y intercept: where the line intersects the y axis
If it's in standard form, this way works great if both the x and y coefficients are factors of the constant on the other side of the equal sign.
EXAMPLE: 2x - 3y = -6
If x = 0, y = 2 so the y intercept is (0, 2)
If y = 0, x = -3 so the x intercept is (-3, 0)
x y
0 2
-3 0
The x intercept is the one that has the x value (y is 0)
The y intercept is the one that has the y value (x is 0)
TODAY, WE WILL COVER THE MOST USED FORM OF A LINEAR EQUATION:
THE SLOPE INTERCEPT FORM
You must restate the equation to get it into the following format:
y = mx + b
where m = slope and b = y intercept (where the line hits the y axis)
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign!
Look at the example I gave you above:
-3y =-2x - 6
If you tried the x points 0, 1, 2, only 0 gave an integer for y.
All the other y values were fractions.
The slope intercept form provides the explanation on why.
RESTATE -3y =-2x - 6 INTO SLOPE INTERCEPT FORM:
Divide both sides by -3:
y = 2/3 x + 2
Look at the coefficient for x. WHAT X VALUES WILL GIVE YOU INTEGER ANSWERS FOR Y????
They need to be multiples of 3!
Now look at the graph of y = 2/3 x + 2
Notice that the + 2 at the end is the y intercept! (without doing any work!)
Use the counting method for slope on your graph.
You should have counted: UP 2, RIGHT 3
The slope therefore is 2/3.
Look at the equation.
It told you 2/3 without any work!
Graph when line is in Slope Intercept Form:
So if you have the slope-intercept form of the equation, it's really easy to graph the line:
1) Graph the intercept on the y axis (that's the +/- constant at the end)
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 2/3 x + 2 ) Put a dot at (0, 2)
3) From (0,2) count up 2 and over to the right 3 to find the next coordinate (3,4)
Remember, slope is y over x, so the numerator is the y change and the denominator is the x change!
If it's positive, you're counting up (positive) and to the right (positive) or you can count down (negative) and to the left (negative) because 2 negatives make a positive.
If it's negative, you're counting down (negative) and to the right (positive) or you can count up (positive) and to the left (negative) because you would have a positive and negative = negative
If you're given the slope and the y intercept, you can write the equation of any line!
Just use: y = mx + b
EXAMPLE: m = -2/3 and b = -12
The line would be y = -2/3 x - 12
Restate Standard Form to Slope Intercept Form:
Another example: 3x + 4y = 10 is the STANDARD FORM of a line
(x and y on the same side, x is positive, no fractions, constant alone)
Restate into Slope Intercept Form:
Solve for y
first subtract 3x from both sides: 4y = -3x + 10
Now divide both sides by 4:
y = -3/4 x + 10/4 or
y = -3/4 x + 5/2
The slope is the coefficient of the x m = -3/4 (so you're sliding down at a little less than a 45 degree angle)
The y intercept is the constant b = 5/2 (so the line crosses the y axis at 2 1/2)
Notice that the "b" is a fraction.
When this happens, the slope intercept form may not be the best form to graph the line.
You must start at 2 1/2 on the y axis and count down 3, right 4. THAT'S HARD!
Neither 3 nor 4 is a factor of the constant 10 so intercepts will be fractions.
We need the x term to end up with 1/2 so that when we add that to the b of 5/2, we'll get an integer!
So let's make x = 2 because that will cross cancel with the -3/4 slope to halves:
y = (-3/4)(2) + 5/2 = -3/2 + 5/2 = 2/2 = 1
So we found a coordinate that has just integers: (2, 1)
Now graph that point.
Now count the slope from that point (instead of from the y intercept)!
Pre Algebra (Period 1)
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
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
Monday, February 1, 2010
Math 6H ( Period 3, 6, & 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
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
Pre Algebra ( Period 1)
RATIOS AND RATES: 6-1
Ratios = fractions with meaning (it's all about the labels!)
3 ways to write a ratio:
EXAMPLE: 16 girls and 14 boys at a party
16 girls to 14 boys or
16 girls: 14 boys or
16 girls/14 boys
You can simplify this just like a fraction:
8 girls to 7 boys
In fact, anything you can do with a fraction, you can do with a ratio!
Rates = ratios with 2 DIFFERENT LABELS
Miles per gallon, miles per hour
10 out of 16 girls went to my party is not a rate.
Not 2 different labels! (but it is a ratio)
Unit rates = rates with a denominator of 1
(SO MUST HAVE 2 DIFFERENT LABELS)
I drive 150 miles in 3 hours is a rate
To change it to a UNIT RATE, simply DIVIDE the numerator by the denominator
150 miles/3 hours = 50 miles per hour
NOW IT'S A UNIT RATE
People focus on MPGs these days when they buy cars!
A Honda Civic = 40 mpg while a Hummer = 8 mpg
A special unit rate called the UNIT PRICE:
I USE UNIT RATES ALL THE TIME WHEN I TRY TO DECIDE WHETHER IT'S WORTH GOING TO COSTCO INSTEAD OF PAVILIONS
Goldfish = $7.99 at Pavilions for 33.5 oz and $10.99 at Costco for 48 oz.
If you divide $/oz you get a unit rate know as UNIT PRICE MONEY MUST BE THE NUMERATOR!!!!
IN CLASS: Chapter 6-2: Proportions
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!
Ratios = fractions with meaning (it's all about the labels!)
3 ways to write a ratio:
EXAMPLE: 16 girls and 14 boys at a party
16 girls to 14 boys or
16 girls: 14 boys or
16 girls/14 boys
You can simplify this just like a fraction:
8 girls to 7 boys
In fact, anything you can do with a fraction, you can do with a ratio!
Rates = ratios with 2 DIFFERENT LABELS
Miles per gallon, miles per hour
10 out of 16 girls went to my party is not a rate.
Not 2 different labels! (but it is a ratio)
Unit rates = rates with a denominator of 1
(SO MUST HAVE 2 DIFFERENT LABELS)
I drive 150 miles in 3 hours is a rate
To change it to a UNIT RATE, simply DIVIDE the numerator by the denominator
150 miles/3 hours = 50 miles per hour
NOW IT'S A UNIT RATE
People focus on MPGs these days when they buy cars!
A Honda Civic = 40 mpg while a Hummer = 8 mpg
A special unit rate called the UNIT PRICE:
I USE UNIT RATES ALL THE TIME WHEN I TRY TO DECIDE WHETHER IT'S WORTH GOING TO COSTCO INSTEAD OF PAVILIONS
Goldfish = $7.99 at Pavilions for 33.5 oz and $10.99 at Costco for 48 oz.
If you divide $/oz you get a unit rate know as UNIT PRICE MONEY MUST BE THE NUMERATOR!!!!
IN CLASS: Chapter 6-2: Proportions
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!
Algebra Period 4
SLOPE
: 7-4
First, let's talk about what the word "slope" means in the real world:
You can think of the slope of a line as the slope of a ski mountain -
When you're climbing up, it's positive
When you're sliding down, it's negative (if you're looking at the mountain from left to right)
The steeper the mountain, the higher the slope value
(A slope of 6 would be an expert slope because it is much steeper than a slope of 2 which would be an intermediate's slope)
"Bunny slopes" for beginners will be lower numbers, generally fractional slopes (like 1/2 or 2/3)
A good benchmark to know is a slope of 1 or -1 is a 45 degree angle
You can also think of slope as rise/run - read this "rise over run"
Rise is how tall the mountain is (the y value)
Run is how wide the mountain is (the x value)
A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)
Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)
You can think of slope as a calculation using 2 coordinates.
We use m to denote slope:
m = rise/run
change in y
change in x
Rise = Change in y value = Difference in y value = y2 - y1
Run = Change in x value = Difference in x value = x2 - x1
THE LITTLE 1 AND 2 SUBSCRIPTS JUST STAND FOR Y VALUE OF THE FIRST POINT AND Y VALUE OF THE 2ND POINT.
To calculate slope you need 2 coordinates.
It doesn't matter which one you start with.
Just be consistent!
If you start with the y value of the FIRST point, make sure you start with the x value of the FIRST point!
If you start with the y value of the SECOND point, make sure you start with the x value of the SECOND point!
You can count the slope of a line:
1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)
2) Now count how far over the point is across - You'll need to either go right or left.
Make this the denominator of your slope (because the x value is the denominator of slope)
If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)
If you went to the LEFT, the value is NEGATIVE
Special slopes:
Horizontal lines in the form of y = have slopes of zero (they're flat!)
Vertical lines in the form of x = have no slope or undefined because the denominator is zero
First, let's talk about what the word "slope" means in the real world:
You can think of the slope of a line as the slope of a ski mountain -
When you're climbing up, it's positive
When you're sliding down, it's negative (if you're looking at the mountain from left to right)
The steeper the mountain, the higher the slope value
(A slope of 6 would be an expert slope because it is much steeper than a slope of 2 which would be an intermediate's slope)
"Bunny slopes" for beginners will be lower numbers, generally fractional slopes (like 1/2 or 2/3)
A good benchmark to know is a slope of 1 or -1 is a 45 degree angle
You can also think of slope as rise/run - read this "rise over run"
Rise is how tall the mountain is (the y value)
Run is how wide the mountain is (the x value)
A 1000 foot high mountain (the rise) is very steep if it's only 200 feet wide (the run) (slope = 5)
Another mountain that is also 1000 feet high is not very steep if it is 2000 feet wide (slope = 1/2)
You can think of slope as a calculation using 2 coordinates.
We use m to denote slope:
m = rise/run
change in y
change in x
Rise = Change in y value = Difference in y value = y2 - y1
Run = Change in x value = Difference in x value = x2 - x1
THE LITTLE 1 AND 2 SUBSCRIPTS JUST STAND FOR Y VALUE OF THE FIRST POINT AND Y VALUE OF THE 2ND POINT.
To calculate slope you need 2 coordinates.
It doesn't matter which one you start with.
Just be consistent!
If you start with the y value of the FIRST point, make sure you start with the x value of the FIRST point!
If you start with the y value of the SECOND point, make sure you start with the x value of the SECOND point!
You can count the slope of a line:
1) Beginning with one point, count up to another point; however far that is, make that the numerator of your slope (because the y value of slope is the numerator)
2) Now count how far over the point is across - You'll need to either go right or left.
Make this the denominator of your slope (because the x value is the denominator of slope)
If you went to the RIGHT, the value is POSITIVE (x values going to the right or positive)
If you went to the LEFT, the value is NEGATIVE
Special slopes:
Horizontal lines in the form of y = have slopes of zero (they're flat!)
Vertical lines in the form of x = have no slope or undefined because the denominator is zero
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