Download Number Theory and Fractions

Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Factors and Prime Factorization
Factors of a product are the numbers that are multiplied to find that
product. A factor is also a whole number that divides the product
with no remainder.
To find all of the factors of 24, make a list of multiplication facts.
1 • 24 = 24
2 • 12 = 24
3 • 8 = 24
4 • 6 = 24
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Write multiplication facts to find the factors of each number.
1. 20
2. 16
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3. 35
4. 31
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A number written as the product of prime factors is called the prime
factorization of the number.
To write the prime factorization of 24, first write it as product of
2 numbers. Then rewrite each factor as the product of 2 numbers
until all of the factors are prime numbers.
24 = 4 • 6
(Write 24 as the product of 2 numbers.)
=2•2•6
(Rewrite 4 as the product of 2 prime numbers.)
=2•2•2•3
(Rewrite 6 as the product of 2 prime numbers.)
So, the prime factorization of 24 is 2 • 2 • 2 • 3 or 23 • 3.
Find the prime factorization of each number.
5. 28
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6. 45
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7. 50
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8. 72
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Greatest Common Factor
The greatest common factor, or GCF, is the largest number that is
the factor of any set of at least two numbers.
You can use prime factorization to find the GCF of two or more
numbers.
To find the GCF of 18 and 24, first write the prime factorization of
each number. Then identify the common prime factors.
18 = 2 • 3 • 3
24 = 2 • 2 • 2 • 3
Next, find the product of the common prime factors.
2•3=6
The GCF of 18 and 24 is 6.
Find the GCF of each set of numbers.
1. 32 and 48
2. 45 and 81
3. 18 and 36
32 = _________________
45 = _________________
18 = _________________
48 = _________________
81 = _________________
36 = _________________
_______________________
_______________________
_______________________
4. 14 and 35
5. 42 and 72
6. 56 and 64
14 = _________________
42 = _________________
56 = _________________
35 = _________________
72 = _________________
64 = _________________
_______________________
_______________________
_______________________
7. 28, 56, and 84
8. 30, 45, and 75
9. 36, 45, and 54
28 = _________________
30 = _________________
36 = _________________
56 = _________________
45 = _________________
45 = _________________
84 = _________________
75 = _________________
54 = _________________
_______________________
_______________________
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Equivalent Expressions
The terms of an expression are the parts of the expression
that are added or subtracted. The expression 28 + 12x has
two terms, 28 and 12x.
Term
Term
28 + 12x
A number that multiplies a variable is a coefficient. In the
term 12x, the number 12 is the coefficient.
Equivalent expressions are expressions that have the
same value for all values of the variables. You can write
equivalent expressions by factoring a given sum of terms
as a product of the GCF and a sum.
Coefficient
Factor 28 + 12x as a product of the GCF and a sum.
28 + 12x
The terms are 28 and 12x.
=2•2•7+2•2•3•x
The GCF of the terms is 2 • 2, or 4.
= 4 • 7 + 4 • 3x
Rewrite each term as a product with the GCF.
= 4(7 + 3x)
Apply the Distributive Property.
Complete to factor the sum of terms as a product of the GCF
and a sum.
1. 20 + 8y
= ____ • ____ • ____ + ____ • ____ • ____ • y
= 4 • _____ + 4 • _____
= 4(__________)
2. 30m + 42
= ____ • ____ • ____ • m + ____ • ____ • ____
= 6 • _____ + 6 • _____
= 6(__________)
Factor the sum of terms as a product of the GCF and a sum.
3. 6c + 10
__________________
6. 7x + 35
__________________
9. 36 + 28z
__________________
4. 12 + 9y
__________________
7. 42k + 21
__________________
10. 32x + 20
__________________
5. 8 + 6n
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8. 18p + 33
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11. 60h + 24
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Decimals and Fractions
You can write decimals as fractions or mixed numbers. A place
value chart will help you read the decimal. Remember the decimal
point is read as the word “and.”
To write 0.47 as a fraction, first think
about the decimal in words.
0.47 is read “forty-seven hundredths.” The place value of the
decimal tells you the denominator is 100.
0.47 = 47
100
To write 8.3 as a mixed number, first
think about the decimal in words.
8.3 is read “eight and three tenths.” The place value of the decimal
tells you the denominator is 10. The decimal point is read as the
word “and.”
8.3 = 8 3
10
Write each decimal as a fraction or mixed number.
1. 0.61
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5. 1.5
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2. 3.43
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6. 0.13
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3. 0.009
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7. 5.002
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4. 4.7
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8. 0.021
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Equivalent Fractions
Fractions that have the same value are equivalent fractions.
3=6
4 8
3 and 6 are equivalent fractions.
4
8
You can use fraction strips to help you find equivalent
fractions.
To solve 2 = ? , first use fraction strips to model the first
3 12
fraction.
Then use 1 fraction pieces to make a length as long as
12
2
the
strip.
3
You need eight 1 pieces, so 2 = 8 .
12
3 12
Find the missing number that makes the fractions equivalent.
1. 3 = ?
4 12
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2. 8 = ?
10 5
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3. 6 = ?
9 3
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4. 5 = ?
6 12
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A fraction is in simplest form when the GCF of the numerator and
the denominator is 1.
To write 4 in simplest form, divide the numerator and the
6
denominator by their GCF, 2.
4÷2=2
6÷2=3
So, 4 in simplest form is 2 .
6
3
Write each fraction in simplest form.
5. 3
9
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6. 12
16
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7. 14
18
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8. 8
20
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Chapter 04: Number Theory and Fractions
Section 06: Mixed Numbers and Improper Fractions Notes
A proper fraction is a fraction whose numerator is less than its
denominator.
2 , 1 , and 2 are examples of proper fractions.
7
3 4
An improper fraction is a fraction whose numerator is greater than or
equal to its denominator.
3 , 8 , and 5 are examples of improper fractions.
2 3
5
Some improper fractions can be written as mixed numbers.
To write 7 as a mixed number, draw circles divided into 1 sections.
4
4
Then shade in 7 of the 1 sections.
4
There is one circle and 3 of a circle shaded.
4
So, 7 = 1 3 .
4
4
Write each improper fraction as a mixed number.
1. 14
3
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2. 11
2
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3. 15
4
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4. 19
6
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Mixed numbers can be written as improper fractions.
To write 2 1 as an improper fraction, draw 3 circles. Divide each
3
circle into 1 sections. Next, shade in 2 whole circles and one 1
3
3
section of the last circle.
Then find the total number of 1 sections that are shaded.
3
Seven 1 sections are shaded, so 2 1 = 7 .
3
3
3
Write each mixed number as an improper fraction.
5. 3 1
4
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6. 5 2
3
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7. 4 1
2
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8. 1 5
6
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
Number Theory and Fractions
Review for Mastery: Comparing and Ordering Fractions
3 and 2 are unlike fractions because they have different
4
3
denominators. To compare these fractions,
graph the fractions on the same number line.
The fraction that is farther to the right on the
number line is greater in value.
9 > 8 , so 3 > 2 .
12 12
4
3
Use the number line to compare the fractions. Write < or >.
1.
2.
1
3
___
5
6
5
10
3.
___
1
4
4.
11
12
___
6
9
4
6
___
1
6
You can use number lines to order fractions from least to greatest.
To order 1 , 5 , and 3 , graph the values on
2 6
4
a number line.
Then list the fractions as they appear
from left to right. From least to greatest
the fractions are 1 , 3 , and 5 .
2 4
6
Use the number line to order the fractions from least to greatest.
5.
6.
7 , 1, 4
12 4 8
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7.
2, 7 , 2
3 12 8
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5 , 1, 5
12 3 6
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8.
1, 3, 3
2 4 9
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Holt McDougal Mathematics
Name _______________________________________ Date __________________ Class __________________
th
6 Grade Chapter 04 Section 06 Notes