FINDING SQUARE ROOTS
Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!
Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.
Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:
1. Guess
2. Divide
3. Average.
... and then just keep repeating steps 2 and 3.
First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.
In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465
We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225
Method 2- Estimating your Square root
This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 132 and 142 So our SQ RT must also be between 13 and 14.
196
183
169
Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14
Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5
What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.
Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.
If you have any questions or comments about these two ways, post a comment here.
Monday, March 16, 2009
Algebra Period 3
DIVIDING RADICALS 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3 = √3
√16 4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25= 5
√36 6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27
√ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3
√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7
√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 = √7 • √3 = √(7• 3) = √21
√ 3 √3 √ 3 3 3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Approximating Pi Day
Since Pi Day fell on a Saturday this year, we approximated Pi Day and celebrated on Monday instead... So What did we learn about pi?
Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle. Pi is an IRRATIONAL number. That means it has no pattern and never terminates. It CANNOT be written as a fraction with an integer in the numerator and denominator. We use 3.14 and 22/7 as APPROXIMATIONS of pi. These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers. You will be able to discover many interesting facts about pi—even finding it on your own.
After measuring 3 different circular objects and completing the table...
Think about your results and answer these questions:
How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large?
Compare your values to 3.14 Were your calculations greater or less than the 3.14 values of pi?
Are you values of pi consistent?Explain why or why not?
What reasons do you think would account for these differences?
On your computer, read about the history of pi at this website
http://ualr.edu/lasmoller/pi.html
Record 5 new or interesting facts about pi that you learned
Finding Yourself in Pi
Pi is an irrational number. That means that it is a non-terminating, non-repeating decimal. Since the number order keeps changing you will eventually find any group of numbers in a sequence, somewhere in the never-ending list. Through the influence of high speed computers that can process large amounts of data, we can examine this aspect of pi much more easily. There are even websites that will instantly search pi for any string of numbers.
Your assignment is to locate a specific string of numbers in pi. Pick a string of 7 or more numbers that is meaningful or significant to you, such as your phone number or birthday (for example: if your birthday is February 9, 1993, then your number string would be 02091993). On your computer, use the Pi Searcher at http://www.angio.net/pi/piquery to find where in pi your number string occurs. Record the string of numbers and what position it holds in the list of numbers.
There are a number of great Pi day songs.. but I think the best is from Fort Vancouver High School .. Check out this video and rap they created...
Today we celebrate Π (Pi), a very cool number. Π is a comparison between the measurements of the circumference to the diameter of a circle—any circle. Pi is an IRRATIONAL number. That means it has no pattern and never terminates. It CANNOT be written as a fraction with an integer in the numerator and denominator. We use 3.14 and 22/7 as APPROXIMATIONS of pi. These are not the exact values. The only symbol that tells the exact value is Π. Pi is a ratio, a comparison between two numbers. You will be able to discover many interesting facts about pi—even finding it on your own.
After measuring 3 different circular objects and completing the table...
Think about your results and answer these questions:
How does the measurement of the circumference compare to the measurement of the diameter? Is it twice as large? Is it three times as large or more than three times as large?
Compare your values to 3.14 Were your calculations greater or less than the 3.14 values of pi?
Are you values of pi consistent?Explain why or why not?
What reasons do you think would account for these differences?
On your computer, read about the history of pi at this website
http://ualr.edu/lasmoller/pi.html
Record 5 new or interesting facts about pi that you learned
Finding Yourself in Pi
Pi is an irrational number. That means that it is a non-terminating, non-repeating decimal. Since the number order keeps changing you will eventually find any group of numbers in a sequence, somewhere in the never-ending list. Through the influence of high speed computers that can process large amounts of data, we can examine this aspect of pi much more easily. There are even websites that will instantly search pi for any string of numbers.
Your assignment is to locate a specific string of numbers in pi. Pick a string of 7 or more numbers that is meaningful or significant to you, such as your phone number or birthday (for example: if your birthday is February 9, 1993, then your number string would be 02091993). On your computer, use the Pi Searcher at http://www.angio.net/pi/piquery to find where in pi your number string occurs. Record the string of numbers and what position it holds in the list of numbers.
There are a number of great Pi day songs.. but I think the best is from Fort Vancouver High School .. Check out this video and rap they created...
Wednesday, March 11, 2009
Algebra Period 3 (Wednesday)
PRODUCT OF RADICALS 11-4
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7
HELPFUL HINT:
If you are multiplying √250 • √50,
I would suggest that you don't multiply 250 x 50 too quickly!
Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!
So why not factor each factor first?!!!
Using my example above:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5
Dividing & Simplifying 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7/√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Basically, a radical is similar to a variable in that you can always multiply them,
but only add or subtract them if they are the exact same radicand (like terms)
√2 •√14 = √28
√28 can be simplified to 2√7, read 2 "rad" 7
HELPFUL HINT:
If you are multiplying √250 • √50,
I would suggest that you don't multiply 250 x 50 too quickly!
Instead, factor 250 and factor 50
Then use the circling pairs method
This will actually save time generally (if you are not allowed to use a calculator!) because you won't end up with a humongous number that you will have to then simplify!
The way you would simplify is then to factor this big number!!!
So why not factor each factor first?!!!
Using my example above:
√250 • √50
Factor each number first:
√(2 x 5 x 5 x 5) •√(2 x 5 x 5)
Combine under one radical sign in PAIRS:
√(2 x 2)(5 x 5)(5 x 5)(5)
Simplify by taking one of each pair out of the radical
(2 x 5 x 5)√5
Multiply all the perfect roots that you took out of the radical:
50√5
Dividing & Simplifying 11-5
Just as you can multiply radicals, you can also divide them by either
1) separating the numerator from the denominator,
or
2) simplifying the entire fraction underneath the radical.
HOW DO YOU KNOW WHICH METHOD TO USE?
Try both and see which one works best! (Examples below)
EXAMPLE OF TAKING THE QUOTIENT UNDER THE RADICAL APART:
Take apart fractions where either the numerator, the denominator, or both are perfect squares!
√(3/16)
Notice that the denominator is a perfect square so it makes sense to look at the denominator separately from the numerator:
√3/√16 = √3/4
√ (25/36)
Notice that both the numerator and denominator are perfect squares so it makes sense to simplify them apart:
√25/√36 = 5/6
EXAMPLE OF SIMPLIFYING THE FRACTION
UNDER THE RADICAL FIRST:
Sometimes, the fraction under the radical will simplify.
If this is true, always do that first!
EXAMPLE: √(27/3)
27/3 simplifies to 9:
√9 = 3
Notice that if you took this fraction apart first and
then tried to find the square root of each part, it's much more complicated:
√(27/3)
Separate the numerator from the denominator:
√27 / √ 3
Factor 27:
√(3x3)x3
√3
Simplify the numerator in pairs:
3√3/√3
Cross cancel if possible:
3
You get the same answer, but with lots more steps!!!
AGAIN, SO HOW DO YOU KNOW WHICH TO DO????
Check both ways and see which works best!!!!!
RATIONALIZING THE DENOMINATOR
THE RULE: Simplified form has
NO RADICALS IN THE DENOMINATOR.
(and you cannot change this rule even if you don't like it or think it makes sense!!!)
If you end up with a radical there, you must get rid of it by squaring whatever is under the radical.
Squaring it will result in the denominator becoming whatever was under the radical sign.
But you cannot do something to the denominator without doing the same thing to the numerator
(golden rule of fractions), so you must multiply the numerator by whatever you multiplied the denominator by.
RATIONALIZING THE DENOMINATOR EXAMPLE:
√7/√ 3
There is nothing you can simplify, whether you put it together or take it apart!
But you can't leave it this way because the rule is that
you can't leave the √3 in the denominator.
You need to multiply both numerator and denominator by √3 to get it out of there:
√7 /√ 3 = √7 • √3 /√ 3•√ 3 = √(7• 3) /3 = √21 /3
Note that you cannot cross cancel the 3 in denominator with 21 in numerator
because one is a square root and the other is not (they are unlike terms!)
The √21 is not 21!
It's irrational and approximately 4.58
You can't cross cancel 4.58 with 3 in the denominator!
Math 6 H Periods 1, 6 & 7
Addition and Subtraction of Fractions 7-1
Most of you already know how to add and subtract fractions, although some of you may need just a little review.
5/9 + 2/9 = 7/9
and that
7/9 – 2/9 = 5/9
The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!
In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.
The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.
LCD ( a/b, c/d) = LCM(b, d) where band d both cannot be equal to 0
For example LCD ( 3/4, 5/6) = LCM(4,6) =12
3/4 = 9/12 and 5/6 = 10/12
Let’s do:
7/15 + 8/9
First find the LCD
LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!
15 = 3• 5
9 = 32
So LCM(15,9) = [every factor to its greatest power] 32•5 = 45
Then find equivalent factions with a LCD of 45, and add
7/15 = 21/45
8/9 = 40/45
21/45 + 40/45 = 61/45 = 1 16/45
Addition and Subtraction of Mixed Numbers 7-2
To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.
STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9
If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.
5 3/10 + 7 7/15
Stack
5 3/10
+7 7/15
Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…
9 5/9 - 4 13/15
Most of you already know how to add and subtract fractions, although some of you may need just a little review.
5/9 + 2/9 = 7/9
and that
7/9 – 2/9 = 5/9
The properties of addition and subtraction of whole numbers also apply to fractions.
If the denominators are the same— add or subtract the numerators AND use the numerator!!
In order to add two fractions with different denominators, we first find two fractions, with a common denominator, equivalent to the given fractions. Then add these two fractions.
The most convenient denominator to use as a common denominator is the least common denominator of LCD, of the two fractions. That is, the least common multiple of the two denominators.
LCD ( a/b, c/d) = LCM(b, d) where band d both cannot be equal to 0
For example LCD ( 3/4, 5/6) = LCM(4,6) =12
3/4 = 9/12 and 5/6 = 10/12
Let’s do:
7/15 + 8/9
First find the LCD
LCM(15, 9) Do your factor trees or inverted division – or just by knowing!!
15 = 3• 5
9 = 32
So LCM(15,9) = [every factor to its greatest power] 32•5 = 45
Then find equivalent factions with a LCD of 45, and add
7/15 = 21/45
8/9 = 40/45
21/45 + 40/45 = 61/45 = 1 16/45
Addition and Subtraction of Mixed Numbers 7-2
To add or subtract mixed numbers we could first change the mixed numbers to improper fractions and then use the method from 7-1 .
1 4/9 + 3 1/9 = 13/9 + 28/9 = 41/9 = 4 5/9 but that was 5th grade….
In the second method, and the one I prefer, you work separately with the fractional and whole number parts of the given mixed numbers.
STACK THEM!!
3 4/9
1 7/9
4 11/9 = 5 2/9
If the fractional parts of the given mixed numbers have different denominators, we find equivalent mixed numbers whose fractional parts have the same denominator, usually the LCD.
5 3/10 + 7 7/15
Stack
5 3/10
+7 7/15
Draw a line separating the fractional part from the whole numbers Find the LCM of the denominators the LCD and add…
9 5/9 - 4 13/15
Algebra Period 3
SIMPLIFYING RADICALS 11-3
SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect square left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: √50 = √(25 * 2)
Next, simplify the sqrt of the perfect square and leave the nonperfect factor under the radical:
√(25 * 2) = √25 * √2 = 5√2
We usually read the answer as "5 rad 2"
HELPFUL HINTS:
When you are factoring the radicand,
you're looking for the LARGEST PERFECT SQUARE
that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.
Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors (remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!
Example: √ 250
Prime factorization = 2 x 5 x 5 x 5
Circle the first two 5's
5 x 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical
Final answer: √ 250 = 5√(2 x 5) = 5 √10
VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: √x10 = x5
We saw this already in factoring!!!
EXAMPLE: √75x10
√ [(25)(3)(x10 )]
or
√[(5)(5)](3)(x5x5)
Simplified, you can pull out a factor of 5 and x5
5x5√3
If the variable has an odd power:
If you have an odd power variable, simply express it as the
(even power one below)(variable to the 1 power)
EXAMPLE: √x5
5 is an odd power, so go down to the next even power (4)
√ (x4 x)
Now you can find the square root of the even power and the 1 power is just left under the √
x2√x
YOU SHOULD NEVER HAVE A VARIABLE OF MORE THAN THE 1 POWER
UNDER THE SQUARE ROOT SIGN!
IF YOU DO,
YOU HAVE NOT SIMPLIFIED ALL THE WAY!
FACTORING A GCF FIRST,
THEN FINDING A BINOMIAL SQUARED:
Sometimes you will need to factor what's under the radical before you start to simplify
Example: √(3x2 + 12x + 12)
First factor out a 3:
√[3 (x2 + 4x + 4) ]
Now factor the trinomial:
√ [3 (x + 2) (x + 2) ]
The (x + 2)2 is a perfect square so
√ [3(x + 2)2] = (x + 2) √ 3
SIMPLIFYING NONPERFECT NUMBERS UNDER THE RADICAL:
A simplified radical expression is one where there is no perfect square left under the radical sign
You can factor the expression under the radical to find any perfect squares in the number:
EXAMPLE: √50 = √(25 * 2)
Next, simplify the sqrt of the perfect square and leave the nonperfect factor under the radical:
√(25 * 2) = √25 * √2 = 5√2
We usually read the answer as "5 rad 2"
HELPFUL HINTS:
When you are factoring the radicand,
you're looking for the LARGEST PERFECT SQUARE
that is a FACTOR of the radicand.
So start with:
Does 4 go into it?
Does 9 go into it?
Does 16 go into it?
Does 25 go into it?
etc.
Another method: Inverted Division or Factor Trees
Factor the radicand completely into its prime factors (remember this from Pre-Algebra?)
Find the prime factorization either way in order from least to greatest.
Circle factors in PAIRS
Every time you have a pair, you have a factor that is squared!
Then, you can take that factor out of the radical sign.
Remember that you are just taking one of those factors out!
Example: √ 250
Prime factorization = 2 x 5 x 5 x 5
Circle the first two 5's
5 x 5 is 25 and so you can take the square root of 25 = 5 out of the radicand
Everything else is not in a pair (squared) so it must remain under the radical
Final answer: √ 250 = 5√(2 x 5) = 5 √10
VARIABLES UNDER THE SQUARE ROOT SIGN:
An even power of a variable just needs to be divided by two to find its square root
EXAMPLE: √x10 = x5
We saw this already in factoring!!!
EXAMPLE: √75x10
√ [(25)(3)(x10 )]
or
√[(5)(5)](3)(x5x5)
Simplified, you can pull out a factor of 5 and x5
5x5√3
If the variable has an odd power:
If you have an odd power variable, simply express it as the
(even power one below)(variable to the 1 power)
EXAMPLE: √x5
5 is an odd power, so go down to the next even power (4)
√ (x4 x)
Now you can find the square root of the even power and the 1 power is just left under the √
x2√x
YOU SHOULD NEVER HAVE A VARIABLE OF MORE THAN THE 1 POWER
UNDER THE SQUARE ROOT SIGN!
IF YOU DO,
YOU HAVE NOT SIMPLIFIED ALL THE WAY!
FACTORING A GCF FIRST,
THEN FINDING A BINOMIAL SQUARED:
Sometimes you will need to factor what's under the radical before you start to simplify
Example: √(3x2 + 12x + 12)
First factor out a 3:
√[3 (x2 + 4x + 4) ]
Now factor the trinomial:
√ [3 (x + 2) (x + 2) ]
The (x + 2)2 is a perfect square so
√ [3(x + 2)2] = (x + 2) √ 3
Sunday, March 8, 2009
Algebra Period 3 (Monday)
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 = 251/2 = 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 THIS SATURDAY, BUT WE'LL CELEBRATE IT THIS NEXT MONDAY-- as we approximate 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!)
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: √(x2 + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.
If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x2 - 10)
x2 must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)
ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x2 is the absolute value (positive) of x ( IxI )
Why?
Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE:
x = -3
√x2 = √(-3)2 = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED
(see p. 489 #17-30)
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)
EXAMPLE:
√( x2 + 10x + 25) factors to √(x + 5)2 = I x + 5I
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 = 251/2 = 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 THIS SATURDAY, BUT WE'LL CELEBRATE IT THIS NEXT MONDAY-- as we approximate 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!)
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: √(x2 + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.
If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x2 - 10)
x2 must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)
ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x2 is the absolute value (positive) of x ( IxI )
Why?
Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE:
x = -3
√x2 = √(-3)2 = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED
(see p. 489 #17-30)
TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x2 + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)
EXAMPLE:
√( x2 + 10x + 25) factors to √(x + 5)2 = I x + 5I
Monday, March 2, 2009
Math 6 H Periods 1, 6 & 7 (Monday)
Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction into a decimal.
The first one, we try to find an equivalent fraction whose denominator is a power of 10.
13/25 is a great example because we can easily change the denominator into 100 : multiplying 24 by 4.
So
13/25 ( 4/4) = 52/100 = .52
In the second method of changing a fraction into a decimal, we divide the numerator by the denominator.
Change 3/8
When the remainder is 0, as above, the decimal is referred to as a terminating decimal. By examining the denominator of 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 of then 2 or 5, the decimal representation will terminate.. (This is so since the fraction can be written as an equivalent fraction whose denominator is a power of ten)
7/40
40 = 23 ∙ 5; since the only prime factors of the denominator are 2 and 5, the fraction can be expressed as a terminating decimal
5/12
12 = 22 ∙ 3 since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal
9/12 = 3/4
4 = 22. Since 4 has no prime factors other than 2, this fraction can be expressed as a terminating decimal
Now, what happens if the denominator of a fraction has prime factors other than 2 or 5
Change 15/22 to a decimal I know that this cannot be expressed as a terminating decimal because the denominator (22) has the prime factorization of 2 ∙ 11.
divide carefully and you will get 0.6818181….
Notice the pattern of repeating remainders of 18 and 4. They produce a repeating block of digits 81, in the quotient.
we write 15/22 = 0.681818181…. or 0.681 with a bar over the 81 where the bar, also know as the vinculum, means that the block 81 repeats without ending.
a decimal such as 0.681 , in which a block of digits continues to repeat indefinitely is called a repeating decimal.
Property
Every fraction can be expressed as either a terminating decimal or a repeating decimal..
Changing a Decimal to a Fraction 6-6
As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.
To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.
Change 0.385 to a fraction in lowest terms
.385 = 385/1000 = 77/200
Change 3.64 to a mixed number in simple form
3.64 = 3 64/100 = 3 16/25
To change a repeating decimal into a fraction follow these examples
th__
0.54
tththththh__
Let n = 0.54 = 0.54545454….
[How many numbers are under the vinculum?] 2
Multiple both sides by 102
So then, 100n = 54.54545454…
100n = 54.54545454…
n = .54545454….
We can subtract n from 100n to get 99n
100n = 54.54545454…
- n = .54545454….
99n = 54
Divide both sides by 99
99n = 54
99 99
n = 54/99 = 6/11
Let’s try
th___
0.243
theitheith___
Let n = 0.243 = .243243243243….
How many numbers are under the vinculum? 3
So multiply both sides by 103
1000n = 243.243243243243….
1000n = 243.243243…
n = 243.243243
999n = 243
Divide both sides by 999
n = 243/999 = 27/111 = 9/37
Let’s try one that is a bit more complicated
thethehtett__
Change 0.318 the vinculym is over just the 18.
[Notice this isn’t 0.318 nor is it 0.318
the__
0.318 so that means it is 03.1818181818....
How many numbers are under the vinculum? 2
So, we multiply by 102
Let n = 0.318181818…
100n = 31.818181818…
n= .318181818…
99n = 31.500000…
Divide both sides by 99
99n/99 = 31.5/99
n = 31.5/99 but that isn’t a proper fraction. What can I do to change this?
Multiply by 10
315/990 = 63/198 = 7/22
HERE ARE SOME STEPS TO FOLLOW:
Step 1 set up “ n= the repeating decimal” n = .515151…
Step 2 determine how many numbers are under the bar in this case = 2
Step 3 Use that number as a power of 10 102 = 100
Step 4 Multiply both sides of the equation in step 1 by that
power of 10 100n = 51.515151…
Step 5 Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation 100n = 51.5151…
- 00n= 51.5151…
Step 6 Solve as a 1-step equation 99n = 51 so n = 51/99
Step 7 Simplify 51/99 = 17/33
**** REMEMBER- sometimes you need to get the decimal out of the numerator—so multiply by a power of 10
There are two methods that can be used to change a fraction into a decimal.
The first one, we try to find an equivalent fraction whose denominator is a power of 10.
13/25 is a great example because we can easily change the denominator into 100 : multiplying 24 by 4.
So
13/25 ( 4/4) = 52/100 = .52
In the second method of changing a fraction into a decimal, we divide the numerator by the denominator.
Change 3/8
When the remainder is 0, as above, the decimal is referred to as a terminating decimal. By examining the denominator of 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 of then 2 or 5, the decimal representation will terminate.. (This is so since the fraction can be written as an equivalent fraction whose denominator is a power of ten)
7/40
40 = 23 ∙ 5; since the only prime factors of the denominator are 2 and 5, the fraction can be expressed as a terminating decimal
5/12
12 = 22 ∙ 3 since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal
9/12 = 3/4
4 = 22. Since 4 has no prime factors other than 2, this fraction can be expressed as a terminating decimal
Now, what happens if the denominator of a fraction has prime factors other than 2 or 5
Change 15/22 to a decimal I know that this cannot be expressed as a terminating decimal because the denominator (22) has the prime factorization of 2 ∙ 11.
divide carefully and you will get 0.6818181….
Notice the pattern of repeating remainders of 18 and 4. They produce a repeating block of digits 81, in the quotient.
we write 15/22 = 0.681818181…. or 0.681 with a bar over the 81 where the bar, also know as the vinculum, means that the block 81 repeats without ending.
a decimal such as 0.681 , in which a block of digits continues to repeat indefinitely is called a repeating decimal.
Property
Every fraction can be expressed as either a terminating decimal or a repeating decimal..
Changing a Decimal to a Fraction 6-6
As we have seen, every fraction is equal to either a terminating decimal or a repeating decimal. It is also true that every terminating or repeating decimal is equal to a fraction.
To change a terminating decimal to a fraction in lowest terms, we write the decimal as a fraction whose denominator is a power of 10. We then write this fraction in lowest terms.
Change 0.385 to a fraction in lowest terms
.385 = 385/1000 = 77/200
Change 3.64 to a mixed number in simple form
3.64 = 3 64/100 = 3 16/25
To change a repeating decimal into a fraction follow these examples
th__
0.54
tththththh__
Let n = 0.54 = 0.54545454….
[How many numbers are under the vinculum?] 2
Multiple both sides by 102
So then, 100n = 54.54545454…
100n = 54.54545454…
n = .54545454….
We can subtract n from 100n to get 99n
100n = 54.54545454…
- n = .54545454….
99n = 54
Divide both sides by 99
99n = 54
99 99
n = 54/99 = 6/11
Let’s try
th___
0.243
theitheith___
Let n = 0.243 = .243243243243….
How many numbers are under the vinculum? 3
So multiply both sides by 103
1000n = 243.243243243243….
1000n = 243.243243…
n = 243.243243
999n = 243
Divide both sides by 999
n = 243/999 = 27/111 = 9/37
Let’s try one that is a bit more complicated
thethehtett__
Change 0.318 the vinculym is over just the 18.
[Notice this isn’t 0.318 nor is it 0.318
the__
0.318 so that means it is 03.1818181818....
How many numbers are under the vinculum? 2
So, we multiply by 102
Let n = 0.318181818…
100n = 31.818181818…
n= .318181818…
99n = 31.500000…
Divide both sides by 99
99n/99 = 31.5/99
n = 31.5/99 but that isn’t a proper fraction. What can I do to change this?
Multiply by 10
315/990 = 63/198 = 7/22
HERE ARE SOME STEPS TO FOLLOW:
Step 1 set up “ n= the repeating decimal” n = .515151…
Step 2 determine how many numbers are under the bar in this case = 2
Step 3 Use that number as a power of 10 102 = 100
Step 4 Multiply both sides of the equation in step 1 by that
power of 10 100n = 51.515151…
Step 5 Rewrite the equations so that you subtract the 1st equation FROM the 2nd equation 100n = 51.5151…
- 00n= 51.5151…
Step 6 Solve as a 1-step equation 99n = 51 so n = 51/99
Step 7 Simplify 51/99 = 17/33
**** REMEMBER- sometimes you need to get the decimal out of the numerator—so multiply by a power of 10
Sunday, March 1, 2009
Math 6 H Periods 1, 6 & 7
I found this very old clip of teaching fractions-- I thought you'd enjoy seeing what they taught students about fractions and decimals-- even before my time!!
Algebra Period 3
COMPARING THE 3 FORMS OF LINEAR EQUATIONS
Standard Form
Ax + By = C
3x + 4y = 10 is the STANDARD FORM of a line
x and y are on the same side of the equation and both coefficients are integers.
This format works especially well when the coefficients are both factors of the constant.
Use the x and y intercepts to graph.
Slope Intercept Form
y = mx + b
y = -3/4 x + 5/2 is SLOPE INTERCEPT FORM of the same line
y is isolated on one side, x term is first, then the constant on the other side of the equation
The coefficient of the x term is the slope.
The constant is the y intercept.
Graph the y intercept, then count the slope to another point.
For graphing, it doesn't work well if the y intercept is a fraction!
Point Slope Form
y - y1 = m(x - x1)
y - 3 = 3/4(x - 4)
Works really well if you have the slope and a point on the line.
You can use this method, and then simplify to the point slope form.
FITTING EQUATIONS TO DATA: 7-7
(word problems)
Many real world relationships are LINEAR, meaning they can be graphed with a LINE.
For example, if 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 what you decide
(how many candy bars you're going to buy)
The result will be how much money you owe at the register (the y value)
x y
1 --> $1.50
2 --> $3.00
3 --> $4.50
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price = 1.50
Think of it as the change in y ($ 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 is two types of data that GO IN THE SAME DIRECTION
You can reverse both directions as well:
You don't have enough money for 3 candy bars, so you decrease your purchase by 1 bar
Then your purchase price also decreases by $1.50
The data is still going in the SAME DIRECTION (both now going down!)
If we now what to find the equation of this linear relationship, simply use the slope intercept or point slope formulas of a line!
SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
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)
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!
(we'll go over this one in class)
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.
2 SPECIAL LINES AND THEIR SLOPES: 7-8
PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2
PERPENDICULAR LINES:
2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS and RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4
THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!
EXAMPLE:
Your mystery line has a point of (2, -5) and is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line ( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.
EXAMPLE:
Your mystery line has a point of (2, -5) and is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line
(since the given line's slope is 2, the mystery line's slope is -1/2)
Substitute the slope and the point given on the mystery line and solve for b.
Standard Form
Ax + By = C
3x + 4y = 10 is the STANDARD FORM of a line
x and y are on the same side of the equation and both coefficients are integers.
This format works especially well when the coefficients are both factors of the constant.
Use the x and y intercepts to graph.
Slope Intercept Form
y = mx + b
y = -3/4 x + 5/2 is SLOPE INTERCEPT FORM of the same line
y is isolated on one side, x term is first, then the constant on the other side of the equation
The coefficient of the x term is the slope.
The constant is the y intercept.
Graph the y intercept, then count the slope to another point.
For graphing, it doesn't work well if the y intercept is a fraction!
Point Slope Form
y - y1 = m(x - x1)
y - 3 = 3/4(x - 4)
Works really well if you have the slope and a point on the line.
You can use this method, and then simplify to the point slope form.
FITTING EQUATIONS TO DATA: 7-7
(word problems)
Many real world relationships are LINEAR, meaning they can be graphed with a LINE.
For example, if 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 what you decide
(how many candy bars you're going to buy)
The result will be how much money you owe at the register (the y value)
x y
1 --> $1.50
2 --> $3.00
3 --> $4.50
If you graph this, you'll get a line because the price is constant.
That means that the slope is constant.
What is the slope????
The price = 1.50
Think of it as the change in y ($ 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 is two types of data that GO IN THE SAME DIRECTION
You can reverse both directions as well:
You don't have enough money for 3 candy bars, so you decrease your purchase by 1 bar
Then your purchase price also decreases by $1.50
The data is still going in the SAME DIRECTION (both now going down!)
If we now what to find the equation of this linear relationship, simply use the slope intercept or point slope formulas of a line!
SLOPE INTERCEPT:
y = 1.5x
(the y intercept is 0 because at 0 candy bars, you owe 0)
POINT SLOPE:
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)
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!
(we'll go over this one in class)
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.
2 SPECIAL LINES AND THEIR SLOPES: 7-8
PARALLEL LINES:
2 lines that are parallel to each have the SAME SLOPE!
y = 2x - 10 and y = 2x + 3/4 are parallel because they both have a slope of 2
PERPENDICULAR LINES:
2 lines that are perpendicular to each other have SLOPES that are:
OPPOSITE SIGNS and RECIPROCALS
y = 2x - 10 is perpendicular to y = -1/2 x + 3/4
THIS IS ANOTHER TWIST TO OUR MYSTERY LINE PUZZLE!!!
If you know that the mystery line is parallel or perpendicular to another given line
then you know the mystery line's slope!!!
EXAMPLE:
Your mystery line has a point of (2, -5) and is PARALLEL to line y = 2x + 3/4
So you know the mystery line's slope because it is the same as the given line ( m = 2)
Substitute the slope and the point given on the mystery line and solve for b.
EXAMPLE:
Your mystery line has a point of (2, -5) and is PERPENDICULAR to line y = 2x + 3/4
So you know the mystery line's slope because it is the opposite sign reciprocal of the given line
(since the given line's slope is 2, the mystery line's slope is -1/2)
Substitute the slope and the point given on the mystery line and solve for b.
Algebra Period 3 (Thursday)
Finding the Equation of a Line: 7-6
Review:
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 -> 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 what happens to the y value (+up or -down) and then what happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1
Oh mystery line,
What can you be?
If I could only find you,
y = mx + b
So first I find m
Then I find b.
Now put it all together
And you've found me!
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
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
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 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) 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 and the y intercept and need to find the slope
(3, 1) 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)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one 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
ANOTHER WAY TO FIND THE EQUATION
WITH 1 POINT & SLOPE: ( and it is MY favorite)
POINT SLOPE FORM OF THE EQUATION
You know one point and the slope. This is the same case as the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form.
Most people use the slope intercept form for all cases.
Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y1 = m (x - x1)
Using the same example from the second case above: (3 , 1) and m = 2
y - 1 = 2 (x - 3)
What you have now is point slope form of the line
If you simplify this, you will get the slope intercept form of the line!
y - 1 = 2 (x - 3)
y - 1 = 2x - 6
y = 2x - 5
If you're trying to link the slope-intercept form to the point slope form of the same line:
The point-slope version eliminates one step from using the slope intercept form.
In the slope-intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found.
In point-slope form, once you plug in the point and slope, you just simplify and the equation is already done!
Review:
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 -> 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 what happens to the y value (+up or -down) and then what happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1
Oh mystery line,
What can you be?
If I could only find you,
y = mx + b
So first I find m
Then I find b.
Now put it all together
And you've found me!
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
2) Do I have the y intercept (b)? If not, find it by plugging in a point and the slope
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 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) 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 and the y intercept and need to find the slope
(3, 1) 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)
You need to first find the slope:
m = change in y / change in x = 3 - (-3)/ 1 - (-2) = 6/3 = 2
Now plug the slope in with one 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
ANOTHER WAY TO FIND THE EQUATION
WITH 1 POINT & SLOPE: ( and it is MY favorite)
POINT SLOPE FORM OF THE EQUATION
You know one point and the slope. This is the same case as the SECOND CASE, but there is a ANOTHER WAY to solve it other using slope intercept form.
Most people use the slope intercept form for all cases.
Point-slope form of a line: You know one point and the slope. Use the following formula:
y - y1 = m (x - x1)
Using the same example from the second case above: (3 , 1) and m = 2
y - 1 = 2 (x - 3)
What you have now is point slope form of the line
If you simplify this, you will get the slope intercept form of the line!
y - 1 = 2 (x - 3)
y - 1 = 2x - 6
y = 2x - 5
If you're trying to link the slope-intercept form to the point slope form of the same line:
The point-slope version eliminates one step from using the slope intercept form.
In the slope-intercept form, you plug in the point and slope, solve for b, and then rewrite the equation using the intercept that you found.
In point-slope form, once you plug in the point and slope, you just simplify and the equation is already done!
Friday, February 27, 2009
Algebra Period 3 (Tues/Wed)
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)
VISUALIZE THE FOLLOWING 2 MOUNTAINS TO HELP YOU UNDERSTAND:
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)
It has a much longer time to slowly reach the 1000 foot top of the mountain!
You can think of slope as a calculation using 2 coordinates:
Rise/Run
=Change in y value/Change in x value
=Difference in y value/ Difference in x value
= y2 - y1/ x2 - x1
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 one point, make sure you start with the same x!
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
Slope Intercept Form 7 -5
Finding the slope-intercept form of a line:
y = mx + b
where m = slope and b = y intercept
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign
(I explained this when we did Chapter 7-3 to easily find 3 coordinates in your T Chart. We just didn't call it slope intercept form at that time!)
It helps to solve the equation for y before you pick your x values, but you don't have to.
EXAMPLE from above: 2x - 3y = -6
Solve the above equation for y.
Subtract 2x from each side:
-3y = -2x - 6
Divide each side by -3:
y = 2/3 x + 2
Now pick your x values, put them on the left side of the T chart, then solve for y.
Instead of picking 0, 1, 2, it makes sense to pick x values that are multiples of 3.
Why? Because you will need to multiply the x value by 2/3 and this will keep the y value an integer:
x y
0 2
3 4
-3 0
Now graph these coordinates and join as your line
Restate Standard Form to Slope Intercept Form:
Example: 3x + 4y = 10 is the STANDARD FORM of a line
Solve for y
first subtracting 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.)
Graph when line is in Slope Intercept Form:
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
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 3x - 2
1) Put a dot at (0, -2)
2) From (0, -2) count up 3 and over to the right 1 to find the next coordinate (1, 1)
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 and to the left 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 and to the left 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
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)
VISUALIZE THE FOLLOWING 2 MOUNTAINS TO HELP YOU UNDERSTAND:
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)
It has a much longer time to slowly reach the 1000 foot top of the mountain!
You can think of slope as a calculation using 2 coordinates:
Rise/Run
=Change in y value/Change in x value
=Difference in y value/ Difference in x value
= y2 - y1/ x2 - x1
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 one point, make sure you start with the same x!
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
Slope Intercept Form 7 -5
Finding the slope-intercept form of a line:
y = mx + b
where m = slope and b = y intercept
All you do is solve the equation for "y" meaning isolate the y on one side of the equal sign
(I explained this when we did Chapter 7-3 to easily find 3 coordinates in your T Chart. We just didn't call it slope intercept form at that time!)
It helps to solve the equation for y before you pick your x values, but you don't have to.
EXAMPLE from above: 2x - 3y = -6
Solve the above equation for y.
Subtract 2x from each side:
-3y = -2x - 6
Divide each side by -3:
y = 2/3 x + 2
Now pick your x values, put them on the left side of the T chart, then solve for y.
Instead of picking 0, 1, 2, it makes sense to pick x values that are multiples of 3.
Why? Because you will need to multiply the x value by 2/3 and this will keep the y value an integer:
x y
0 2
3 4
-3 0
Now graph these coordinates and join as your line
Restate Standard Form to Slope Intercept Form:
Example: 3x + 4y = 10 is the STANDARD FORM of a line
Solve for y
first subtracting 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.)
Graph when line is in Slope Intercept Form:
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
2) "Count" the next point by using the slope or x coefficient as a FRACTION
For the equation y = 3x - 2
1) Put a dot at (0, -2)
2) From (0, -2) count up 3 and over to the right 1 to find the next coordinate (1, 1)
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 and to the left 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 and to the left 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
Math 6 H Periods 1, 6 & 7 (Tues & Wed.)
Comparing Fractions 6-4
When 2 fractions have equal denominators-- it is easy to tell which of the fractions are greater. Compare their numerators.
5/11 < 7/11 because 5 < 7
If the fractions have different denominators, find a common denominator. Using the lCM of the denominators-- the LCD-- is a surefire way of determinng the relationship between fractions.
Which is greater 5/6 or 7/9?
Since the LCM (6,9) = 18
Using equivalent fractions
5/6 = 15/18
and 7/9 = 14/18
so 5/6 > 7/9
( See Section 6-2 notes if you need to review equivalent fractions)
Another way is to use cross products to compare.
What if you needed to name a fraction between two other fractions?
FOr instance, between 7/15 and 12/ 25
FInd the LCM (15, 25) using the methods taught from chapter 5
LCM ( 15, 25) = 75
finding equivalent fractions for
7/15 = 35/75
12/25 = 36/75
If you want a fraction between, simply double the denominators and then double the numerators
35/75 = 70/150
36/75 = 72/150
So 71/150 would be a fraction that is between the two given fractions.
When 2 fractions have equal denominators-- it is easy to tell which of the fractions are greater. Compare their numerators.
5/11 < 7/11 because 5 < 7
If the fractions have different denominators, find a common denominator. Using the lCM of the denominators-- the LCD-- is a surefire way of determinng the relationship between fractions.
Which is greater 5/6 or 7/9?
Since the LCM (6,9) = 18
Using equivalent fractions
5/6 = 15/18
and 7/9 = 14/18
so 5/6 > 7/9
( See Section 6-2 notes if you need to review equivalent fractions)
Another way is to use cross products to compare.
What if you needed to name a fraction between two other fractions?
FOr instance, between 7/15 and 12/ 25
FInd the LCM (15, 25) using the methods taught from chapter 5
LCM ( 15, 25) = 75
finding equivalent fractions for
7/15 = 35/75
12/25 = 36/75
If you want a fraction between, simply double the denominators and then double the numerators
35/75 = 70/150
36/75 = 72/150
So 71/150 would be a fraction that is between the two given fractions.
Sunday, February 22, 2009
Algebra Period 3 ( Review)
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
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... y = mx + b
We isolate y on one side
We read the y intercept (the b -> 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 what happens to the y value (+up or -down) and then what
happens to the x value (+ right or - left)
If the slope is not a fraction, make it a fraction by putting the integer over 1
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