ADDING AND SUBTRACTING RADICALS 11-6
Radicals function like variables, so you can only COMBINE LIKE RADICALS!
You cannot add √2 to √3!!!!
However, you may add 3√2 to 5√2 and get 8√2:
(3 + 5)√2 = 8√2
Make sure you simplify all radical expressions before trying to combine them!
Sometimes, it looks like they are not like radicands, but then after simplifying they are.
EXAMPLE #34 from p. 505:
√( x2y) + √( 4x2y) + √(9y) - √( y3)
Simplify each term first!!!!!!!!
x√y + 2x√y + 3√y - y√ y
NOW THEY ARE ALL LIKE TERMS BECAUSE ALL HAVE SQRT y!
(x + 2x + 3 - y)√y
(3x - y + 3)√y (final simplified answer)
Friday, February 26, 2010
Math 6H ( Periods 3, 6, & 7)
Addition and Subtraction of Mixed Numbers 7-2
Although the textbook states that to add or subtract mixed numbers you can first change the mixed numbers to improper fractions and then use the methods of Section 7-1, I want you to practice the methods taught in class this week.
I want you to stack you mixed numbers, separating the fractional parts from the whole numbers. Proceed to add ( or subtract) the fractional parts, using LCD-- where necessary. At times you will need to borrow from the whole number to be able to subtract properly.
3 4/9 + 1 7/9
stack
3 4/9
+1 7/9
4 11/9 = 5 2/9
9 3/7 - 4 5/7
9 3/7
-4 5/7
need to change to
8 10/7
-4 5/7
4 5/7
Again, 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
Although the textbook states that to add or subtract mixed numbers you can first change the mixed numbers to improper fractions and then use the methods of Section 7-1, I want you to practice the methods taught in class this week.
I want you to stack you mixed numbers, separating the fractional parts from the whole numbers. Proceed to add ( or subtract) the fractional parts, using LCD-- where necessary. At times you will need to borrow from the whole number to be able to subtract properly.
3 4/9 + 1 7/9
stack
3 4/9
+1 7/9
4 11/9 = 5 2/9
9 3/7 - 4 5/7
9 3/7
-4 5/7
need to change to
8 10/7
-4 5/7
4 5/7
Again, 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
Wednesday, February 24, 2010
Algebra Period 4
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.
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.
Tuesday, February 23, 2010
Algebra Period 4
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!
Sunday, February 21, 2010
Algebra (Period 4)
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 6H ( Periods 3, 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
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