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 !!
Friday, January 6, 2012
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.
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