Download Number Theory and Fractions

Number Theory and Fractions
Math 6
Chapter 3
Divisibility and Mental Math
Lesson 3-1
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
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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 Numbers and Prime
Factorization
Lesson 3-2
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
Now You Try
• Look at the example you just wrote in your
notes
• Find the Prime Factorization of 15
3)15 15 is not divisible by 2, so divide by 3.
5 The result is 5 and 5 is a prime number
The prime factorization of 15 is 3 x 5
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.
Your turn…
• Look at your notes….
• Find the prime factorization using a division
ladder for the following composite numbers.
)24
)30
)18
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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
Your turn…
• Look at your notes….
• Find the prime factorization using a factor
trees for the following composite numbers.
24
30
18
Continued Practice
• Prime Factorization
• Practice finding the prime factorization.
Do 5 division ladders
Do 5 factor trees
Greatest Common Factor
Lesson 3-3
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
Time to Practice…
• Find the GCF
• Use any one of the three methods
Equivalent Fractions
Lesson 3-4
Definition…
• Equivalent fractions are fractions that name
the same amount
• Form equivalent fractions by multiplying the
numerator and denominator by the same nonzero number
Example
3 x 4 = 12
4 x 4 = 16
12
16 is equivalent to 3
4
6 ÷ 2= 3
10 ÷ 2 = 5
6
10 is equivalent to 3
5
Practice Time
Find the equivalent Fraction
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
Mixed Numbers and Improper
Fractions
Lesson 3-5
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
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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
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Partner Practice
• With your partner, convert Mixed Numbers to
Improper Fractions
Convert Mixed Numbers and Fractions
Least Common Multiple
Lesson 3-6
What is a Multiple?
A multiple of a number is that number x a whole
number. 9 x 4 = 36
36 is a multiple of 9
and 4
How to Find LCM
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
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
Finding LCM
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List the prime factors of each number.
Suppose you want to find the LCM of 18 and 24. List the prime factors of each number:
18 = 2 · 3 · 3
24 = 2 · 2 · 2 · 3
For each prime number listed, underline the most repeated occurrence of this number in any
prime factorization.
The number 2 appears once in the prime factorization of 18 but three times in that of 24, so
underline the three 2s:
18 = 2 · 3 · 3
24 = 2 · 2 · 2 · 3
Similarly, the number 3 appears twice in the prime factorization of 18 but only once in that of 24, so
underline the two 3s:
18 = 2 · 3 · 3
24 = 2 · 2 · 2 · 3
Multiply all the underlined numbers.
Here’s the product:
2 · 2 · 2 · 3 · 3 = 72
So the LCM of 18 and 24 is 72. This checks out because
18 · 4 = 72
24 · 3 = 72
Partner Practice
• Use either method for practice
Find the LCM for each pair
Comparing and Ordering
Fractions
Lesson 3-7
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
Partner Practice
compare fractions
Books Never Written
Fractions and Decimals
Lesson 3-8
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
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