Addition & 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
Wednesday, January 25, 2012
Tuesday, January 24, 2012
Algebra Honors (Period 6 & 7)
Simple Radical Equations 11-10
Solving equations involving radicals are solved by isolating the radical on one side of the equals sign and then squaring both sides of the equation.
140 = √2(9.8)d all under the √
140 = √19.6d
(140)2 = (√19.6d)2
19600 = 19.6d
1000=d
The solution set is {1000}
Solve
√(5x+1) + 2 = 6
√(5x+1) = 4
[√(5x+1)]2 = (4)2
5x + 1 = 16
5x = 15
x = 3
The solution set is {3}
When you square both sides of an equation, the new equation may NOT be equivalent to the original equation Therefore, you must CHECK EVERY POSSIBLE ROOT IN THE ORIGINAL EQUATION to see whether it is indeeed a root.
Solve
√(11x2 -63) - 2x = 0
√(11x2 -63) = 2x
√(11x2 -63)2 = (2x)2
11x2 -63 = 4x2
7x2 = 63
x2 = 9
x = ± 3
Now we need to check for BOTH + 3 and - 3
Rewrite the original equation
√(11x2 -63) - 2x = 0
√(11(3)2 -63) - 2(3) = 0
√99-63 - 6 = 0
√36 - 6 = 0
6-6 = 0
That's true
Now for x = -3
√(11(-3)2 -63) - 2(-3) = 0
√(99 -63) + 6 = 0
√36 + 6 = 0
12 ≠ 0
So -3 is NOT a solution
Solving equations involving radicals are solved by isolating the radical on one side of the equals sign and then squaring both sides of the equation.
140 = √2(9.8)d all under the √
140 = √19.6d
(140)2 = (√19.6d)2
19600 = 19.6d
1000=d
The solution set is {1000}
Solve
√(5x+1) + 2 = 6
√(5x+1) = 4
[√(5x+1)]2 = (4)2
5x + 1 = 16
5x = 15
x = 3
The solution set is {3}
When you square both sides of an equation, the new equation may NOT be equivalent to the original equation Therefore, you must CHECK EVERY POSSIBLE ROOT IN THE ORIGINAL EQUATION to see whether it is indeeed a root.
Solve
√(11x2 -63) - 2x = 0
√(11x2 -63) = 2x
√(11x2 -63)2 = (2x)2
11x2 -63 = 4x2
7x2 = 63
x2 = 9
x = ± 3
Now we need to check for BOTH + 3 and - 3
Rewrite the original equation
√(11x2 -63) - 2x = 0
√(11(3)2 -63) - 2(3) = 0
√99-63 - 6 = 0
√36 - 6 = 0
6-6 = 0
That's true
Now for x = -3
√(11(-3)2 -63) - 2(-3) = 0
√(99 -63) + 6 = 0
√36 + 6 = 0
12 ≠ 0
So -3 is NOT a solution
Math 6 Honors ( Periods 1, 2, & 3)
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
13/12 - 5/12 = 8/12 = 2/3
and that
7/9 – 2/9 = 5/9
13/12 - 5/12 = 8/12 = 2/3
a/c + b/c = (a +b)/c where c does not equal 0
a/c - b/c = (a -b)/c
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 b and 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
5/6- 11/24
Stack them and use the LCD
5/6 = 20/24
-11/24 = -11/24
9/24 = 3/8
7/12 + 4/9 + 3/4
several strategies ca be used. You can find the LCD for all three you can use the C+ and the A+
and change it to
(7/12 + 3/4) + 4/9
then add the first two factions
7/12 + 3/4 becomes 7/12 + 9/12 = 16/12 = 4/3
then add 4/3 + 4/9
change 4/3 to 12/9
12/9 + 4/9 = 16/9 = 1 7/9
What about 17/10 - ( 3/5 + 5/6)
You must do the parenthesis first
so 3/5 + 5/6
3/5 = 18/30
5/6 = 25/30
43/30
Now you have
17/10 - 43/30
stack those
17/10 = 51/30
51/30
-43/30
8/30 = 4/15
n + 1/2 = 5/6
you need to isolate the variable so add 1/2 to BOTH sides of the equation
n + 1/2 = 5/6
-1/2 = -1/2 Change 1/2 to 3/6 and subtract carefully
n = 2/6
n = 1/3
make sure to box your answer
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
13/12 - 5/12 = 8/12 = 2/3
and that
7/9 – 2/9 = 5/9
13/12 - 5/12 = 8/12 = 2/3
a/c + b/c = (a +b)/c where c does not equal 0
a/c - b/c = (a -b)/c
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 b and 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
5/6- 11/24
Stack them and use the LCD
5/6 = 20/24
-11/24 = -11/24
9/24 = 3/8
7/12 + 4/9 + 3/4
several strategies ca be used. You can find the LCD for all three you can use the C+ and the A+
and change it to
(7/12 + 3/4) + 4/9
then add the first two factions
7/12 + 3/4 becomes 7/12 + 9/12 = 16/12 = 4/3
then add 4/3 + 4/9
change 4/3 to 12/9
12/9 + 4/9 = 16/9 = 1 7/9
What about 17/10 - ( 3/5 + 5/6)
You must do the parenthesis first
so 3/5 + 5/6
3/5 = 18/30
5/6 = 25/30
43/30
Now you have
17/10 - 43/30
stack those
17/10 = 51/30
51/30
-43/30
8/30 = 4/15
n + 1/2 = 5/6
you need to isolate the variable so add 1/2 to BOTH sides of the equation
n + 1/2 = 5/6
-1/2 = -1/2 Change 1/2 to 3/6 and subtract carefully
n = 2/6
n = 1/3
make sure to box your answer
Monday, January 23, 2012
Algebra Honors (Period 6 & 7)
Multiplication of Binomials Containing Radicals 11-9
Chapter 5 taught us how to multiply binomials-- we can use those methods when multiplying binomials that contain square root radicals.
(6 + √11)(6 - √11)
The pattern is
(a +b)(a -b) = a2 - b2
so using that we get
62 - (√11)2
36 - 11 = 25
Simplify (3 + √5)2
The pattern here is
(a + b)2 = a2 + 2ab + b2
so ( 3 + √5)2 =
32 + 2[(3)(√5)] + (√5)2 =
9 + 6√5 + 5 =
14 + 6√5
Simplify (2 √3 - 5√7)2
The pattern here is (a - b)2 = a2 - 2ab + b2
(2 √3 - 5√7)2 =
(2 √3)2 -2[(2)(5)(√3)(√7)] +(5√7)2 =
4(3) -20√21 +25(7) =
12 -20√21+ 175 =
187 -20√21
If both b and d are nonnegative, then the binomials
a√b + c√d AND a√b - c√d are called conjugates of one another. COnjugates differ ONLY in the sign of one term
if a, b, c, and d are all integers then the product (a√b + c√d)(a√b - c√d) will be an integer... see the first example!!
Conjugates can be used to rationalize binomial denominators that contain radicals.. getting rid of the radicals in the denominator
Rationalize
3/(5- 2√7)
3/(5- 2√7) = [ 3/(5- 2√7)] × [((5+ 2√7)/(5+2√7)]
This doesn't show well here hopefully you can remember what was done in class...
= 3(5 +2√7)/25-(2√7)2 =
(15+6√7)/25-28 =
(15+6√7)/-3 =
15/-3 +6√7/-3 =
-5 -2√7
√√
Chapter 5 taught us how to multiply binomials-- we can use those methods when multiplying binomials that contain square root radicals.
(6 + √11)(6 - √11)
The pattern is
(a +b)(a -b) = a2 - b2
so using that we get
62 - (√11)2
36 - 11 = 25
Simplify (3 + √5)2
The pattern here is
(a + b)2 = a2 + 2ab + b2
so ( 3 + √5)2 =
32 + 2[(3)(√5)] + (√5)2 =
9 + 6√5 + 5 =
14 + 6√5
Simplify (2 √3 - 5√7)2
The pattern here is (a - b)2 = a2 - 2ab + b2
(2 √3 - 5√7)2 =
(2 √3)2 -2[(2)(5)(√3)(√7)] +(5√7)2 =
4(3) -20√21 +25(7) =
12 -20√21+ 175 =
187 -20√21
If both b and d are nonnegative, then the binomials
a√b + c√d AND a√b - c√d are called conjugates of one another. COnjugates differ ONLY in the sign of one term
if a, b, c, and d are all integers then the product (a√b + c√d)(a√b - c√d) will be an integer... see the first example!!
Conjugates can be used to rationalize binomial denominators that contain radicals.. getting rid of the radicals in the denominator
Rationalize
3/(5- 2√7)
3/(5- 2√7) = [ 3/(5- 2√7)] × [((5+ 2√7)/(5+2√7)]
This doesn't show well here hopefully you can remember what was done in class...
= 3(5 +2√7)/25-(2√7)2 =
(15+6√7)/25-28 =
(15+6√7)/-3 =
15/-3 +6√7/-3 =
-5 -2√7
√√
Tuesday, January 17, 2012
Algebra Honors (Period 6 & 7)
Adding and Subtracting Radicals 11-8
You can use the Distributive Property to simplify the sums of radicals like
4√7 + 5√7 = (4 +5)√7 = 9√7
UNLIKE radicands CANNOT be combined.
3√6-2√13+5√6
combine the LIKE radicands
8√6 -2√13
Express each radical in simplest form and you can sometimes combine terms in sums and differences of radicals
7√3- 4√6 + 2√48 -6√54 =
7√3 - 4√6 + 2√(16⋅3) -6√(9⋅6) =
7√3 - 4√6 + 8√3 - 18√6 =
15√3 -22√6
To simplify sums or differences of square root radicals
Express each radical in simplest form
Use the DP to add or subtract radicals with LIKE RADICANDS
You can use the Distributive Property to simplify the sums of radicals like
4√7 + 5√7 = (4 +5)√7 = 9√7
UNLIKE radicands CANNOT be combined.
3√6-2√13+5√6
combine the LIKE radicands
8√6 -2√13
Express each radical in simplest form and you can sometimes combine terms in sums and differences of radicals
7√3- 4√6 + 2√48 -6√54 =
7√3 - 4√6 + 2√(16⋅3) -6√(9⋅6) =
7√3 - 4√6 + 8√3 - 18√6 =
15√3 -22√6
To simplify sums or differences of square root radicals
Express each radical in simplest form
Use the DP to add or subtract radicals with LIKE RADICANDS
Tuesday, January 10, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Changing a Fraction to a Decimal 6-5
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 72/99 = .72727272...
There are two methods that can be used to change a fraction to a decimal.
1) find an equivalent fraction whose denominator is a power of 10. ( this method does not always work but when it does it becomes really easy to change to a decimal)
13/25 multiply the numerator and the denominator by 4 to get 52/100 and then just close your eyes and see Chapter 3... and 0.52
2) divide the numerator by the denominator. It's a great way to determine your score out of 100 and then figure out your percent.
If you got 67/75 on the last test
divide 67 by 75 carefully 0.8933333... you earned a B+
take 3/8 and divide 3 by 8 8 goes into 3 0.375 times
so 3/8 = 0.375
If the numerator is smaller than the denominator we know our number must be between 0 and 1--> it must be a decimal.
When the remainder is 0 as in the case of dividing 3 by 8, it is called a terminating decimal.
By examining a fraction in lowest terms, we can determine whether the fraction can be expressed as a terminating decimal.
If the denominator has no prime factors other than 2 or 5, the decimal representation will terminate.
7/40
looking at 40 we notice the prime factorization ( oh no, it's Chapter 5)
40 = 23·5 Since the only prime factors are 2 and 5
7/40 must terminate.
What about 5/12 ?
12 = 22· 3 Since 3 is a prime factor of the denominator, the fraction cannot be expressed as a terminating decimal.
What about 9/12 ? At first it looks the same as the one above, but look carefully and realize 9/ 12 = 3/4
Since 4 = 22 , 4 has no other prime factors except 2, this can be expressed as a terminating decimal.
Let's look at 15/22
Since 22 has the prime factor of 11 we know that this will not terminate. In fact when you divide 15 by 22 you end up with 0.681818181...
We write this as ( Please check page 196) Notice that the bar is only over the 81 and represents a block numbers that continues to repeat indefinitely and is called a repeating decimal.
EVERY FRACTION CAN BE EXPRESSED AS EITHER A TERMINATING DECIMAL OR A REPEATING DECIMAL.
Let's look at 4/9 = 0.4444444....
5/9 =
7/9 =
31/99 = 0.3131313131...
8/ 11 = 72/99 = .72727272...
Monday, January 9, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Comparing Fractions 6-4
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions
Or compare 5/6 and 7/9
again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9
Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35
The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!
and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!
What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them
What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!
There were 10 sunny days in February ( 28 total days)
12 sunny days in November (30 total days)
Which month has higher fraction of sunny days?
10/28 or 12/30
First simplify each fraction... then compare
10/28 = 5/14 and 12/30 = 2/5
Now use one oc the methods to compare
Cross products works really well and you discover
NOVEMBER is the month with the higher fraction of sunny days
The Bears won 11 out of 16 games.
The Eagles won 17 out of 24 games.
Which team won a great fraction of their games?
11/16 or 17/24
Again, I would use cross multiplication and we discovered that THE EAGLES won a greater fraction of their games.
If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!
If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.
Ordering or comparing fractions:
Different ways:
I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?
When the numerator is close to the denominator, the fraction is approaching 1
(Ex: 9/11)
When you double the numerator and it's close to the denominator, the fraction is close to 1/2 (Ex: 4/9)
When the numerator is very far from the denominator, the fraction is approaching zero (Ex: 1/8)
Also, if one number is improper or mixed number and other is a proper fraction, then obviously the number greater than 1 will be bigger!
II. LCD - give them all the same denominator using the LCM as the LCD
III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions
IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)
When two fractions have equal denominators it is easy to tell which of the fractions is greater.
We simply compare their numerators.
3/11 < 5/11 since 3< 5 If the fractions have different denominators, there are a variety of methods to consider. We could find a common denominator, which we will need to do when we add or subtract fractions... but when comparing let's try other methods... Take 2/3 and 4/5 Comparing Fractions
Or compare 5/6 and 7/9
again this time you would multiply
5(9) = 45 and 7(6) = 42
so 5/6 > 7/9
Also if the numerator is the same
2/3, 2/7, 2/9, 2/11, 2/21, 2/35
The larger the denominator the smaller the fractions so to list in order from least to greatest start with the largest number in the denominator!!
and if you have fractions with the numerator just one away from the denominator
such as 3/4, 5/6, 7/8, 9/10, 23/24, 45/46
the smallest fraction will be the one with the smallest numbers
3/4 is the smallest fraction and that list is in order from least to greatest!!
What if you need to name a fraction between two fraction 1/6 and 3/8
you could find the LCD
1/6 = 4/24 and
3/8 = 9/24
so you could state
5/24, 6/24 ( but that is really 1/4), 7/24, or 8/24 ( but that is really 1/3.
There are actually an infinite number of fractions... these are only 4 of them
What if you need to find a fraction between 3/7 and 4/7
sometimes you need to change the denominators just to realize that there really are other fractions between
for instance, 3/7 = 6/14 and 4/7 = 8/ 14 so doesn't 7/14 ( or actually 1/2) work!!
... and that's just one of the fractions!!
There were 10 sunny days in February ( 28 total days)
12 sunny days in November (30 total days)
Which month has higher fraction of sunny days?
10/28 or 12/30
First simplify each fraction... then compare
10/28 = 5/14 and 12/30 = 2/5
Now use one oc the methods to compare
Cross products works really well and you discover
NOVEMBER is the month with the higher fraction of sunny days
The Bears won 11 out of 16 games.
The Eagles won 17 out of 24 games.
Which team won a great fraction of their games?
11/16 or 17/24
Again, I would use cross multiplication and we discovered that THE EAGLES won a greater fraction of their games.
If n > 0
Then
if a < b a/n < b/n Think about this one!! Plug in some numbers and see what happens and if a < b, then n/a > n/b
Again, plug in some numbers and see what happens!!
If a/b and c/d are fractions and if ad > bc, which fraction is greater,
a/b or c/d ?
Post your answer below in the comments for extra credit. Make sure to give your reasoning for your answer.
Ordering or comparing fractions:
Different ways:
I. Benchmarks - 0, 1/4, 1/2, 3/4, and 1 (using your gut feeling)
How do you figure out which benchmark to use?
When the numerator is close to the denominator, the fraction is approaching 1
(Ex: 9/11)
When you double the numerator and it's close to the denominator, the fraction is close to 1/2 (Ex: 4/9)
When the numerator is very far from the denominator, the fraction is approaching zero (Ex: 1/8)
Also, if one number is improper or mixed number and other is a proper fraction, then obviously the number greater than 1 will be bigger!
II. LCD - give them all the same denominator using the LCM as the LCD
III. Use cross multiplication when comparing two... do it several times when comparing a list of fractions
IV. Change them to decimals ( works well if you are great at decimals-- but I want you to become GREAT at fractions!!)
Friday, January 6, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Fractions & Mixed Numbers 6-3
1/2 + 1/2 + 1/2 = 3/2
A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1
1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions
5/2, 8/3, 18/15, 12/5 are all improper fractions
You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number
If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.
To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2
To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.
or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!
2 5/6 =
(2 x 6) + 5
6
=17/6
Practice these:
785 ÷ 3
852÷ 5
3751÷ 16
98001÷231
post your answers below in the comments for extra credit !!
1/2 + 1/2 + 1/2 = 3/2
A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
Every improper fractions is greater than 1
A proper fraction is a fraction whose numerator is less than its denominator.
Thus, a proper fraction is always between 0 and 1
1/4, 2/3, 5/9. 10/12 17/18 are all proper fractions
5/2, 8/3, 18/15, 12/5 are all improper fractions
You can express any improper fraction as the sum of a whole number and a fraction
a number such as 1 1/2 is called a mixed number
If the fractional part of a mixed number is a proper fraction in lowest terms, the mixed number is said to be in simple form.
To change an improper fraction into a mixed number in simple form, divide the numerator by the denominator and express the remainder as a fraction.
14/3 = 4 2/3
30/4 = 7 2/4 = 7 1/2
To change a mixed number to an improper fraction rewrite the whole number part as a fraction with the same denominator as the fraction part and add together.
or multiply the denominator by the whole number part and add the fractional part to that...
In class I showed the circle shortcut. If you were absent, check with a friend or ask me in class!!
2 5/6 =
(2 x 6) + 5
6
=17/6
Practice these:
785 ÷ 3
852÷ 5
3751÷ 16
98001÷231
post your answers below in the comments for extra credit !!
Thursday, January 5, 2012
Wednesday, January 4, 2012
Math 6 Honors ( Periods 1, 2, & 3)
Equivalent Fractions 6-2
We drew the four number lines from Page 182 and noticed that 1/2, 2/4, 3/6, and 4/8 all were at the midpoints of the segment from 0 to 1. They all denoted the same number and are called equivalent fractions.
If you multiply the numerator and the denominator by the same number the results will be a fraction that is equivalent to the original fraction
1/2 = 1 x 3/2 x 3 = 3/6
It works for division as well
4/8 = 4 ÷ 4 / 4 ÷ 8 = 1/2
So we can generalize and see the following properties
For any whole numbers a, b, c, with b not equal to zero and c not equal to zero
a/b = a x c/ b x c and
a/b = a ÷ c / b ÷c
Find a fraction equivalent to 2/3 with a denominator of 12
we want a number such that 2/3 = n/12
You could look at this and say
" What do I do to 3 to get it to be 12?
Multiply by 4
so you multiply 2 by 4 and get 8 so
8/12 is an equivalent fraction
A fraction is in lowest terms if its numerator and denominator are relatively prime-- That is if their GCF is 1
3/4, 2/7, and 3/5 are in lowest terms.
They are simplified
You can write a fraction in lowest terms by dividing the numerator and denominator by their GCF.
Write 12/18 is lowest terms
The GCF (12 and 18) = 6
so 12/18 = 12÷ 6 / 18 ÷ 6 = 2/3
Find two fractions with the same denominator that are equivalent to 7/8 and 5/12
This time you need to find the least common multiple of the denominators!! or the LCD
Using the box method from Chapter 5, we find that the LCM (8, 12 ) = 24
7/8 = 7 X 3 / 8 X 3 = 21/24
and
5/12 = 5 X 2 / 12 X 2 = 10/24
When finding equations such as
3/5 = n/15 we noticed we could multiply the numerator of the first fraction by the denominator of the second fraction and set that equal to the denominator of the first fraction times the numerator of the second... or
3(15) = 5n now we have a one step equation
If we divide both sides by 5 we can isolate the variable n and solve...
3(15)/ 5 = n
9 = n
We found we could generalize
If a/b = c/d then ad = bc
We drew the four number lines from Page 182 and noticed that 1/2, 2/4, 3/6, and 4/8 all were at the midpoints of the segment from 0 to 1. They all denoted the same number and are called equivalent fractions.
If you multiply the numerator and the denominator by the same number the results will be a fraction that is equivalent to the original fraction
1/2 = 1 x 3/2 x 3 = 3/6
It works for division as well
4/8 = 4 ÷ 4 / 4 ÷ 8 = 1/2
So we can generalize and see the following properties
For any whole numbers a, b, c, with b not equal to zero and c not equal to zero
a/b = a x c/ b x c and
a/b = a ÷ c / b ÷c
Find a fraction equivalent to 2/3 with a denominator of 12
we want a number such that 2/3 = n/12
You could look at this and say
" What do I do to 3 to get it to be 12?
Multiply by 4
so you multiply 2 by 4 and get 8 so
8/12 is an equivalent fraction
A fraction is in lowest terms if its numerator and denominator are relatively prime-- That is if their GCF is 1
3/4, 2/7, and 3/5 are in lowest terms.
They are simplified
You can write a fraction in lowest terms by dividing the numerator and denominator by their GCF.
Write 12/18 is lowest terms
The GCF (12 and 18) = 6
so 12/18 = 12÷ 6 / 18 ÷ 6 = 2/3
Find two fractions with the same denominator that are equivalent to 7/8 and 5/12
This time you need to find the least common multiple of the denominators!! or the LCD
Using the box method from Chapter 5, we find that the LCM (8, 12 ) = 24
7/8 = 7 X 3 / 8 X 3 = 21/24
and
5/12 = 5 X 2 / 12 X 2 = 10/24
When finding equations such as
3/5 = n/15 we noticed we could multiply the numerator of the first fraction by the denominator of the second fraction and set that equal to the denominator of the first fraction times the numerator of the second... or
3(15) = 5n now we have a one step equation
If we divide both sides by 5 we can isolate the variable n and solve...
3(15)/ 5 = n
9 = n
We found we could generalize
If a/b = c/d then ad = bc
Algebra Honors (Period 6 & 7)
Rational Square Roots 11-3
You know that subtraction undoes addition, and that division by a nonzero number undoes multiplication, Similarly squaring a number can be undone by finding a square root.
If a2 = b then a is a square root of b
Notice that 72= 49 and so does (-7)2 = 49
So 7 and -7 are square roots of 49
the radical symbol √ is used to write the principal or positive square root of a positive number.
is read “The positive square root of 49 equals 7
A negative square root is associated with the symbol - √
is read “The negative square root of 49 equals -7”
Let’s use ± to indicate both the positive and negative square root
so ±√49 means the positive or negative square root of 49 or ±7
Let’s look at √49 the number written beneath the radical sign (such as 49) is called the radicand.
For all positive real numbers a:
Every positive real number a has two square roots
The symbol √a denotes the principal square root of a
Zero has only one square root—itself.
Because the square of every real number is either positive or zero—NEGATIVE NUMBERS DO NOT HAVE SQUARE ROOTS IN THE SET OF REAL NUMBERS.
does not have a solution in the set of real numbers!!
Notice that SQRT(4•25)=SQRT(100) = 10 and
that SQRT (4) • SQRT(25) = 2 • 5 = 10
so
Product Property of Square Roots
For any nonnegative real numbers a and b,
SQRT(AB) = SQRT(A) •SQRT(B)
Find:
Let’s say you forgot your perfect squares—OH MY!!
but looking at 225, using your skills from previous years you realize 225= 9 • 25 so
SQRT 225= SQ•5 =15RT(9 •25) = 3•5 =15
What about SQRT 2304
If you cannot see any perfect squares that divide the radicand—begin by factoring it!!
Then see if you have any perfect squares. USE INVERTED DIVISION!!
use inverted division along with divisibility rules to find perfect squares
Look for the largest perfect square factors and you discover that
SQRT (2304) = SQRT(22•32•82) = 2•3•8= 48
You know that subtraction undoes addition, and that division by a nonzero number undoes multiplication, Similarly squaring a number can be undone by finding a square root.
If a2 = b then a is a square root of b
Notice that 72= 49 and so does (-7)2 = 49
So 7 and -7 are square roots of 49
the radical symbol √ is used to write the principal or positive square root of a positive number.
is read “The positive square root of 49 equals 7
A negative square root is associated with the symbol - √
is read “The negative square root of 49 equals -7”
Let’s use ± to indicate both the positive and negative square root
so ±√49 means the positive or negative square root of 49 or ±7
Let’s look at √49 the number written beneath the radical sign (such as 49) is called the radicand.
For all positive real numbers a:
Every positive real number a has two square roots
The symbol √a denotes the principal square root of a
Zero has only one square root—itself.
Because the square of every real number is either positive or zero—NEGATIVE NUMBERS DO NOT HAVE SQUARE ROOTS IN THE SET OF REAL NUMBERS.
does not have a solution in the set of real numbers!!
Notice that SQRT(4•25)=SQRT(100) = 10 and
that SQRT (4) • SQRT(25) = 2 • 5 = 10
so
Product Property of Square Roots
For any nonnegative real numbers a and b,
SQRT(AB) = SQRT(A) •SQRT(B)
Find:
Let’s say you forgot your perfect squares—OH MY!!
but looking at 225, using your skills from previous years you realize 225= 9 • 25 so
SQRT 225= SQ•5 =15RT(9 •25) = 3•5 =15
What about SQRT 2304
If you cannot see any perfect squares that divide the radicand—begin by factoring it!!
Then see if you have any perfect squares. USE INVERTED DIVISION!!
use inverted division along with divisibility rules to find perfect squares
Look for the largest perfect square factors and you discover that
SQRT (2304) = SQRT(22•32•82) = 2•3•8= 48
Tuesday, January 3, 2012
Algebra Honors (Period 6 & 7)
Decimal Forms of Rational Numbers 11-2
Any common fraction can be written as a decimal by dividing the numerator by the denominator. If the remainder is zero, the decimal is called a terminating, or ending, or finite decimal.
3/8
Actually this is one of the fractions you need to know by heart !
If you don’t a remainder of zero when dividing the numerator by the denominator, continue to divide until the remainder begins to repeat.
5/6
7/11
3 2/7
The decimal quotient above are nonterminating, nonending, or infinite. The dots indicate that the decimals continue without end.
They are also called repeating or periodic because the same digit or block of digits repeats unendingly. A bar (vinculum) is used to indicate the block of digits that repeat.
What ones do you need to know by heat… same from 6th grade
1/3 family, 1/11 family, and let’s look at the 1/7 family (my favorite)
Let’s look at this algebraically… when you divide a positive integer n by a positive integer d, the remainder r at each step must be zero or a positive integer less than d. For example, if the divisor is 6, the reminders will be 0, 1, 2, 3, 4, or 5 and the division will terminate or begin repeating within 5 steps after only zeros remain to be brought down.
For every integer n and every positive integer d, the decimal form of the rational number n/d either terminates or eventually repeats in a block of fewer than d digits.
Any common fraction can be written as a decimal by dividing the numerator by the denominator. If the remainder is zero, the decimal is called a terminating, or ending, or finite decimal.
3/8
Actually this is one of the fractions you need to know by heart !
If you don’t a remainder of zero when dividing the numerator by the denominator, continue to divide until the remainder begins to repeat.
5/6
7/11
3 2/7
The decimal quotient above are nonterminating, nonending, or infinite. The dots indicate that the decimals continue without end.
They are also called repeating or periodic because the same digit or block of digits repeats unendingly. A bar (vinculum) is used to indicate the block of digits that repeat.
What ones do you need to know by heat… same from 6th grade
1/3 family, 1/11 family, and let’s look at the 1/7 family (my favorite)
Let’s look at this algebraically… when you divide a positive integer n by a positive integer d, the remainder r at each step must be zero or a positive integer less than d. For example, if the divisor is 6, the reminders will be 0, 1, 2, 3, 4, or 5 and the division will terminate or begin repeating within 5 steps after only zeros remain to be brought down.
For every integer n and every positive integer d, the decimal form of the rational number n/d either terminates or eventually repeats in a block of fewer than d digits.
Thursday, December 15, 2011
Algebra Honors (Period 6 & 7)
Properties of Rational Numbers 11-1
A real number that can be expressed as the quotient of two integers is called a rational number
A rational number can be written as a quotient of integers in an unlimited number of ways.
3 = 3/1= 6/2 = 12/4 = -15/-5
To determine which of two rational numbers is greater, you can write them with the same positive denominator and compare the numerators
Which is greater 8/3 or 17/7?
the LCD is 21
8/3 = 56/21
17/7 = 51/21
so 8/3 > 17/7
For all integers a and b and all positive integers c and d
a/c > b/d if an only if ad > bc
a/c < b/d if and only if ad < bc This method compares the product of the extremes with the product of the means Thus 4/7 > 3/8 because (4)(8) > (3)(7)
The Density Property for Rational Numbers
Between every pair of different rational numbers there is another rational number
The density property implies that it is possible to find an unlimited or endless umber of rational numbers between two given rational numbers.
If a and b are rational numbers and a< b then the number halfway from a to b is
a + (1/2)(b-a);
the number one third of the way from a to b would be
a + (1/3)(b-a) and so on
A real number that can be expressed as the quotient of two integers is called a rational number
A rational number can be written as a quotient of integers in an unlimited number of ways.
3 = 3/1= 6/2 = 12/4 = -15/-5
To determine which of two rational numbers is greater, you can write them with the same positive denominator and compare the numerators
Which is greater 8/3 or 17/7?
the LCD is 21
8/3 = 56/21
17/7 = 51/21
so 8/3 > 17/7
For all integers a and b and all positive integers c and d
a/c > b/d if an only if ad > bc
a/c < b/d if and only if ad < bc This method compares the product of the extremes with the product of the means Thus 4/7 > 3/8 because (4)(8) > (3)(7)
The Density Property for Rational Numbers
Between every pair of different rational numbers there is another rational number
The density property implies that it is possible to find an unlimited or endless umber of rational numbers between two given rational numbers.
If a and b are rational numbers and a< b then the number halfway from a to b is
a + (1/2)(b-a);
the number one third of the way from a to b would be
a + (1/3)(b-a) and so on
Math 6 Honors ( Periods 1, 2, & 3)
Fractions 6-1
The symbol 1/4 can mean several things:
1) It means one divided by four
2) It represents one out of four equal parts
3) It is a number that has a position on a number line.
1/8 means 1 divided by 8 or 1 ÷ 8
A fraction consists of two numbers
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
we noted that we could abbreviate ...
denominator as denom with a line above it
and numerator as numer
we found that you could add
1/3 + 1/3 + 1/3 = 3/3 = 1
or 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1
we also noted that 8 X 1/8 = 8/8 = 1
We also noticed that 2/7 X 3 = 6/7
So we discussed the properties
For any whole numbers a, b,and c with b not equal to zero
1/b + 1/b + 1/b ... + 1/b = b/b = 1 for b numbers added together
and we noticed that b X 1/b = b/b = 1
we also noticed that
(a/b) X c = ac/b
We talked about the parking lot problem on Page 180
A count of cars and trucks was taken at a parking lot on several different days. For each count, give the fraction of the total vehicles represented by
(a) cars
(b) trucks
Given: 8 cars and 7 trucks
We noticed that you needed to find the total vehicles or 8 + 7 = 15 vehicles
so
(a) fraction represented by cars is 8/15
(b) fraction represented by trucks is 7/15
What if the given was: 12 trucks and 15 cars
(a) fraction represented by cars is 15/27
(b) fraction represented by trucks is 12/27
What about
GIVEN:
9cars
35 vehicles
This time we need to find out how many trucks there are
35 -9 = 26
so
(a) 9/35
(b) 26/35
We aren't simplifying YET
The symbol 1/4 can mean several things:
1) It means one divided by four
2) It represents one out of four equal parts
3) It is a number that has a position on a number line.
1/8 means 1 divided by 8 or 1 ÷ 8
A fraction consists of two numbers
The denominator tells the number of equal parts into which the whole has been divided.
The numerator tells how many of these parts are being considered.
we noted that we could abbreviate ...
denominator as denom with a line above it
and numerator as numer
we found that you could add
1/3 + 1/3 + 1/3 = 3/3 = 1
or 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1
we also noted that 8 X 1/8 = 8/8 = 1
We also noticed that 2/7 X 3 = 6/7
So we discussed the properties
For any whole numbers a, b,and c with b not equal to zero
1/b + 1/b + 1/b ... + 1/b = b/b = 1 for b numbers added together
and we noticed that b X 1/b = b/b = 1
we also noticed that
(a/b) X c = ac/b
We talked about the parking lot problem on Page 180
A count of cars and trucks was taken at a parking lot on several different days. For each count, give the fraction of the total vehicles represented by
(a) cars
(b) trucks
Given: 8 cars and 7 trucks
We noticed that you needed to find the total vehicles or 8 + 7 = 15 vehicles
so
(a) fraction represented by cars is 8/15
(b) fraction represented by trucks is 7/15
What if the given was: 12 trucks and 15 cars
(a) fraction represented by cars is 15/27
(b) fraction represented by trucks is 12/27
What about
GIVEN:
9cars
35 vehicles
This time we need to find out how many trucks there are
35 -9 = 26
so
(a) 9/35
(b) 26/35
We aren't simplifying YET
Subscribe to:
Posts (Atom)
