Download Exponents, Factors, and Fractions

Exponents, Factors, and Fractions
Chapter 3
Exponents and Order of
Operations
Lesson 3-1
Terms
• An exponent tells you how many times a number
is used as a factor
• A base is the number that is multiplied
4
5 4 is the base, 5 is the exponent.
It means 4 x 4 x 4 x 4 x 4
It does NOT MEAN 4 x 5
Simplify Using Order of Operations
• Simplify 34 •(7-2)3
First: Parentheses
Second: Exponents
Third: Multiply
=
=
=
34 • 53
81• 125
10,125
Simplify Powers with Negatives
-54 and (-5)4 are not equivalent. -54 means the
“negative of” 54, so the base is 5, not -5
• -54 = -(5 • 5 • 5 • 5)
= -625
But watch out for: here the base is the number in
the parentheses
(-5)4 = (-5)(-5)(-5)(-5)
= 625
Partner Practice
• Exponents and Order of Operations
Scientific Notation
Lesson 3-2
Definition
• A number in scientific notation is written as
the product of two factors, one greater than
or equal to 1 but less than 10, and the other
as a power of 10
7,500,000,000,000 = 7.5 x 1012
Example
• The moon orbits the earth at a distance of
384,000 km. Write the number in scientific
notation
First: move the decimal point to get a factor
greater than 1 but less than 10
384,000. becomes 3.84000
Then: rewrite as 3.84 x 105 because I moved the
decimal point 5 places
From Scientific Notation to Standard
Form
• Mars is 2.3 x 108 miles from Earth
Write in standard form:
Move the decimal point to the right 8 places
230,000,000
Think, Pair, Share
• http://www.worksheetworks.com/pdf/eaf/9c
dc46f73b98/WorksheetWorks_Scientific_Nota
tion_1.pdf
Divisibility Tests
Lesson 3-3
Definition
A whole number is
divisible by another if
the remainder is zero
Facts/Characteristics
all even numbers are
divisible by 2
Vocabulary
Word
Divisible
48 ÷ 4 = 12
47 ÷ 4 = 11 r 3
Examples
Non-Examples
Divisibility of Whole Numbers
A whole number is divisible by
•
•
•
•
•
2 if it ends in 0, 2, 4, 6 or 8
3 if the sum of its digits is divisible by 3
5 if it ends in 0 or 5
0 if the sum of the digits is divisible by 9
10 if it ends in 0
Prime Factorization
Lesson 3-4
Definition
Factors are numbers
that are multiplied
Facts/Characteristics
When divided there
is no remainder
Vocabulary
Word
3 x 4 = 12
12÷ 4 = 3
Examples
Factor
12 ÷ 5 = 2 r2
Non-Examples
Definition
Facts/Characteristics
A whole number
numbers that can be
greater than 1 with more divided evenly by more
than two factors
than just 1 and itself
Vocabulary
Word
20
1 x 20
2 x 10
4x5
Examples
Composite
number
19
1 x 19
Non-Examples
Definition
A whole number with
exactly two factors, 1
and the number itself
Facts/Characteristics
The whole numbers 0
and 1 are neither prime
nor composite
Vocabulary
Word
19
1 x 19
Examples
Prime
number
20
1 x 20
2 x 10
4x5
Non-Examples
Prime Factorization
What does this mean???
Writing a composite number as a product of
prime numbers gives the prime factorization
….OK, so what does that mean??? And how do I
do it??
Method One: Division Ladder
• Find the prime factorization of 84
Divide 84 by prime numbers starting with 2 and
work your way up.
2) 84
2) 42
3) 21
7
Divide 84 by the prime number 2. Work down.
The result is 42. Since 42 is even, divide by 2, again.
The result is 21. Divide by the prime number 3.
The prime factorization of 42 is
2x2x3x7
Some things to remember…
• When using division ladders, start by first
dividing with the smallest prime number. For
even numbers, that divisor will be 2.
• Keep dividing until you reach a prime number
as the quotient.
Method Two: Factor Trees
• Gets us to the same result, just a different way
84
2
42
2 21
3 7
Circle the prime numbers
Prime factorization of 84: 2 x 2 x 3 x 7
GCF
• The greatest common factor of two or more
numbers is the greatest factor shared by all
the numbers.
• Find the GCF three different ways:
• Use a list of factors
• Use a division ladder
• Use factor trees
Simplifying Fractions
Lesson 3-5
Definition…
• Equivalent fractions are fractions that name
the same amount
• Form equivalent fractions by multiplying the
numerator and denominator by the same nonzero number
Simplest Form
• Fractions are in simplest form when the only
common factor of the numerator and
denominator is 1
• Example: 2 is on simplest form since 2 and 3
3 only have the number 1 as a
common factor
Partner Practice
• Get the fractions into simplest form
Write the Fraction in Simplest Form
Fractions Puzzlers Task Cards
Comparing and Ordering
Fractions
Lesson 3-6
Least Common Denominator
• To compare fractions, start with the LCD
The Least Common Denominator of two or more
fractions is the Least Common Multiple of the
denominators
Ex: compare 3/4 and 5/6
Compare ¾ and 5/6
Step One: Find LCM for 4 and 6
Option 1: Use a List of Multiples
Multiples of 4: 4, 8, 12, 16, 20, 24
Multiples of 6: 6, 12, 18, 24, 30, 36
12 is the least common multiple for 4 and 6
Compare ¾ and 5/6
• Step Two: Write equivalent fractions using
LCM as the common denominator
• 3/4 = 9/12
• 5/6 = 10/12
3/4 < 5/6
How to Find LCM
Find LCM for 8, 10 and 20
Option 2: Use Prime Factorization
Prime Factorization for 8 = 2 x 2 x 2
Prime Factorization for 10 = 2 x 5
Prime Factorization for 20 = 2 x 2 x 2 x 5
Circle each different factor: 2 x 2 x 2 x 5 = 40
LCM for 8, 10 and 20 is 40
Partner Practice
• Use either method for practice
Find the LCM for each pair
What Happens If You Watch TV All Day?
Mixed Numbers and Improper
Fractions
Lesson 3-8
Key Terms
• A proper fraction has a numerator that is less
than its denominator: ⅜
• An improper fraction has a numerator that is
greater than or equal to its denominator: 8/5
• A mixed number is a number written with
both a whole number and a fraction: 2⅔
Write Mixed Numbers as Improper
Fractions
• Write 6⅔ as an improper fraction
• First: Multiply the whole number by the
denominator
(6 x 3 thirds = 18 thirds)
• Second: Add the numerator (18 thirds + 2 thirds =
20 thirds)
Example: 6⅔ = (6 x 3) + 2 = 20
3
3
Write Improper Fractions as Mixed
Numbers
• Write 9
6 as a mixed number
First: divide the numerator by the denominator
Second: write the remainder as a fraction
Third: simplify 1
6)9
-6
3
9 = 1 3/ 6 = 1⅟2
6
Partner Practice
• With your partner, convert Mixed Numbers to
Improper Fractions
Convert Mixed Numbers and Fractions
Cryptic Quiz
Fractions and Decimals
Lesson 3-9
Terminating Decimals
• Write fractions as decimals by dividing the
numerator by the denominator.
• A decimal that stops, or terminates, is a
terminating decimal
• Ex: 5/8 = 5÷ 8 = 0.625
Repeating Decimal
• If when we divide, we discover the same digit
or group of digits in the quotient repeats
without end, that decimal is a repeating
decimal
• Ex: 3/11 = 3 ÷ 11 = 0.272727… = 0.27
Practice
• convert
Rational Numbers
Lesson 3-10
Rational Number
• A rational number is a number that can be
written as a quotient of two integers, where
the divisor is not 0.
Examples: ⅝, 0.46, -6 and 3⅟2
All integers are rational numbers as all integers
can be written with a denominator of 1
Negative Rational Numbers
• Negative Rational Numbers can be written in
three ways
6= 6 =6
7
7 7
Compare Negative Rational Numbers
• Compare -⅟2 and -3/4
• Option 1: Use a number line. Since -3/4 is
farther to the left, it is the smaller number
Compare Negative Rational Numbers
• Compare -⅟2 and -3/4
• Option 2: make equivalent fractions
-⅟2 = - 2/4
Since -3/4 < - 2/4 , then -3/4 < -⅟2
Ordering Rational Numbers
• Order these numbers from greatest to least
⅟4, -0.2, -2/9 , 1.1
Step One: convert the fractions to decimals
by dividing
⅟4 = 0.25 and -2/9 = -0.22
Now order: -0.22 , -0.2, 0.25, 1.1
Partner Practice
http://www.mathworksheetsland.com/6/16expr
at/ip.pdf