Section A
Number Theory
4-1a Divisibility
4-1b Factors and Prime Factorization
4-2 Greatest Common Factor
4-3 Equivalent Expressions
Section A Quiz
Section B
Understanding Fractions
4-4 Decimals and Fractions
4-5 Equivalent Fractions
4-6 Mixed Numbers and Improper Fractions
Section B Quiz
Section C
Introduction to Fraction Operations
4-7 Comparing and Ordering Fractions
4-8 Adding and Subtracting Fractions with Like Denominators
Section C Quiz
Number Theory & Fraction Unit Test
4-1a Divisibility
Vocabulary
________________ - can be divided by a number without leaving a remainder
______________________ - a number greater than 1 that has more than two whole-number factors
______________________ - a whole number greater than 1 that has exactly two factors, 1 and itself
Divisibility Rules
A number is divisible by
2 if the last digit is even (0, 2, 4, 6, 8).
Divisible
3 if the sum of the digits is divisible by 3.
4 if the last two digits form a number divisible by 4.
5 if the last digit is a 0 or 5.
6 if the number is divisible by both 2 and 3.
9 if the sum of the digits is divisible by 9.
10 if the last digit is 0.
Example 1
Checking Divisibility
A) Tell whether 610 is divisible by 2, 3, 4, and 5.
Divisible or Not Divisible
Explain how you know
2
3
4
5
So 610 is divisible by ________________.
B) Tell whether 387 is divisible by 6, 9, 10.
Divisible or Not Divisible
Explain how you know
6
9
10
So 387 is divisible by _________________.
Not Divisible
Example 2
Identifying Prime and Composite Numbers
Tell whether each number is prime or composite.
A) 45
Divisible by: 1, 3, 5, 9, 15, 45
Prime or Composite
B) 13
Divisible by: ____________________
Prime or Composite
Divisible by: ____________________
Prime or Composite
Divisible by: ____________________
Prime or Composite
C) 19
D) 49
Lightly shade in all the prime numbers.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Divisibility Practice #1
Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, and 10
1.
90
2.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
5.
804
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
416
3.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
6.
500
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
308
4.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
7.
972
540
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
8.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
225
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
Tell whether each number is prime or composite
9. 33: _________
10. 69: _________
11. 41: _________
12. 45: _________
13. 58: _________
14. 87: _________
15. 61: _________
16. 53: _________
17. 99: _________
18. Dan counted all the coins in his bank, and he had 72 quarters. Can he exchange the quarters
for an even amount of dollar bills? How do you know?
19. A small town purchased 196 American flags for its Memorial Day Parade. Six locations were
selected to display the flags. Can each location have the same number of flags? Explain.
Divisibility Practice #2
Tell whether each number is divisible by 2, 3, 4, 5, 6, 9, and 10
1.
1,524
2.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
5.
3,174
1,000
3.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
6.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
8,433
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
4,455
4.
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
7.
5,745
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
2,160
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
8.
6,200
2: _______
3: _______
4: _______
5: _______
6: _______
9: _______
10: _______
Write the prime numbers between each set of numbers.
9. 1-10: _____________
10. 21-30: _____________
11. 35-45: _____________
12. 50-60: _____________
13. 65-75: _____________
14. 75-90: _____________
Replace each question mark with a digit that will make the number divisible by 9
15. 35? _____
16. 2,68? _____
17. 7?4 _____
18. 53,?69 _____
19. Several Boy Scouts worked together to create a line of 4,900 dominoes. They each placed
the same amount of dominoes in the line. There were between 50 and 100 Boy Scouts
working on the line. What are all the possible numbers of boy Scouts who could have created
the domino line?
4-1b Factors and Prime Factorization
Vocabulary
___________________ - a number that is multiplied by another number to get a product
__________ ________________ - a number written as the product of its prime factors
Example 1
Finding Factors
List all of the factors of each number.
A)
18
Begin listing factors in pairs
18 = 1 x 18
18 = 2 x 9
18 = 3 x 6
18 = 6 x 3
The factors of 18 are 1, 2, 3, 6, 9, 18.
B)
20
The factors of 20 are ___________________.
C)
13
The factors of 13 are __________________.
D)
36
The factors of 36 are __________________.
1 is a factor
2 is a factor
3 is a factor
4 & 5 are not factors
6 & 3 have already been listed
Example 2
Writing Prime Factorizations
Write the prime factorization of each number.
A) 36 à Choose any two factors to begin. Keep finding factors until each branch ends
at a prime factor.
36
36
B) 54 à Choose any two factors to begin. Keep finding factors until each branch ends
at a prime factor.
54
54
Find the Prime Factorization of the following
30
64
12
630
4-2 Greatest Common Factor
Vocabulary
______________ _____________ _____________ OR ( ___ ___ ___ )
– the largest common factor of two or more given numbers
Example 1
Finding the GCF
Steps:
• Find ALL the factors
• Circle all the common factors
• Find the greatest factor
36
90
GCF = ______
10
16
GCF = ______
5
7
GCF = ______
Find the GCF of each set of numbers.
A)
16 and 24
The GCF of 16 and 24 is _______.
16
24
B)
28 and 42
28
42
24
32
The GCF of 28 and 42 is _______.
C)
12, 24, and 32
Method: Using prime factorization
12
12 =
24 =
32 =
The GCF of 12, 24 and 32 is _______.
Example 2
Problem Solving Application
There are 12 boys and 18 girls in Ms. Ruiz’s science class. The students must form lab groups. Each
group must have the same number of boys and girls. What is the greatest number of groups Ms. Ruiz
can make if every student must be in a group?
Jenna has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the same number
of each color flower in each bouquet. What is the greatest number of bouquets she can make?
4-3 Equivalent Expressions
Vocabulary
________________ - the parts of an expression that are added or subtracted
______________________ - a number that is multiplied by a variable in an algebraic expression
140 + 4x
________________ ______________ - expression that have the same value for all values of the
variables
Example 1: Factoring Numerical Expressions
20 + 56
Step 1: Find the GCF of 20 and 56
GCF = _________
Step 2: Rewrite each term as a product with the GCF
Step 3: Apply the Distributive Property
1. 16 + 12
2. 21 + 9
3. 15 + 20
4. 14 + 18
5. 70 + 42
6. 22 + 99
Example 2: Factoring Algebraic Expressions
15x + 18
Step 1: Find the GCF of 15 and 18
GCF = _________
Step 2: Rewrite each term as a product with the GCF
Step 3: Apply the Distributive Property
7. 6y + 8
8. 25 + 15x
9. 36n + 24
10. 100p + 70
11. 32w + 8
12. 3 + 18x
Example 3: Writing Equivalent Expressions
You can use factors and properties of operations to generate many equivalent expressions. All of the
expressions shown below are equivalent.
52 – 12
52 – 12
52 – 12
= 4 x 13 – 4 x 3
= 2 x 26 – 2 x 6
= 26 x 2 – 6 x 2
= 4(13 – 3)
= 2(26 – 6)
= (26 – 6)2
= 4 x 10
= 2 x 20
= 20 x 2
= 40
= 40
= 40
Write four equivalent expressions for each given expression
8 + 24
40y – 30y
3(1 + 27m)
16 + 30
34 – 4
18x – 12x
Practice: Equivalent Expressions
Factor the sum of terms as a product of the GCF and a sum.
1. 18 + 20
2. 35 + 15
3. 12 + 66
4. 24 + 40
5. 52 + 28
6. 3 + 33
7. 10y + 15
8. 18s + 21
9. 49m + 7
11. 80 + 25z
12. 32b + 48
10. 56 + 24x
Write four equivalent expressions for each given expression.
13. 50 – 10
14. 24y – 8y
15. 4(2 + 7p)
4-4 Decimals and Fractions
Vocabulary
______________ ______________ - a number made up of a whole number that is not zero and a
fraction
______________ ______________ - a decimal number that ends, or terminates
______________ ______________ - a decimal in which one or more digits repeat infinitely
Example 1
Writing Decimals as Fractions or Mixed Numbers
Write the place value of the decimal as the denominator.
Write the digits of the decimal as the numerator.
0.9 =
0.013 =
0.08 =
0.5 =
Write each decimal as a fraction or mixed number.
A)
0.23
B)
0.67
C)
1.7
D)
5.9
Example 2
Writing Fractions as Decimals
Divide the numerator by the denominator. You may need to annex zeros to the dividend.
When you convert a fraction to a decimal, you can end up with a ______________ decimal, or a
______________ decimal.
Terminating Decimal Examples
3
=
4
5
=
8
7
=
25
3
=
10
Repeating Decimal Examples:
To show a repeating decimal, draw a bar over the repeating digits.
4
=
11
7
=
9
Note: Only put bar over part that repeats.
Ex: 0.3333333 =
Ex: 0.437373737 =
Write each fraction or mixed number as a decimal and circle terminating or repeating.
A)
3
4
terminating or repeating
B)
5
2
3
terminating or repeating
C)
3
20
terminating or repeating
D)
6
1
3
terminating or repeating
Example 3
Comparing and Ordering Fractions and Decimals
Order the fractions and decimals from least to greatest.
0.5,
1
, 0.37
5
line up decimals
First rewrite the fraction as a decimal.
1
= 0.2
5
Answer: __________; __________; __________
7
3
, 0.8,
4
10
line up decimals
First rewrite the fraction as a _______________.
3
= ___________
4
7
= ____________
10
Answer: __________; __________; __________
Put the following in order from least to greatest
0.38;
3 1
;
4 5
Answer: __________; __________; __________
4-5 Equivalent Fractions
Vocabulary
______________ ______________ - fractions that name the same amount or part
______________ ______________ - when the numerator and denominator of a fraction have no
common factors other than 1. Also called LOWEST TERMS
Example 1
Finding Equivalent Fractions
Find two equivalent fractions for
_________
=
6
.
8
__________
=
_________
The same area is shaded when the rectangle is divided into 8 parts, 12 parts, and 4 parts.
Example 2
Multiplying and Dividing to Find Equivalent Fractions
Find the missing number that makes the fractions equivalent.
A)
2
=
3 18
B)
3
=
5 20
C)
70 7
=
100
D)
4 80
=
5
Find TWO equivalent fractions for each of the following
3
=
5
1
=
2
4
=
6
10
=
15
Example 3
Writing Fractions in Simplest Form
Greatest Common Factor (GCF) Review
List all of the factors of 24 and 30. Circle all common factors.
24
30
GCF = _______
Examples:
Put the following fractions into lowest terms by dividing the numerator and denominator by
common factors.
6
=
10
12
=
15
7
=
21
Hint: If you divide by the greatest common factor, you will get the fraction in lowest terms
24
=
30
8
=
10
21
=
28
36
=
54
12
=
18
18
=
30
20
=
30
8
=
36
21
=
35
4-6 Mixed Numbers and Improper Fractions
Vocabulary
______________ ______________ - a fraction in which the numerator is greater than or equal to the
denominator
______________ ______________ - a fraction in which the numerator is less than the denominator
Improper and Proper Fractions
Improper Fractions
Ø Numerator equals denominators à fraction is equal to 1
Ø Numerator greater than denominator à fraction is greater
than 1
Proper Fractions
Ø Numerator less than denominator à fraction is less than 1
3
=1
3
102
=1
102
9
>1
5
13
>1
1
2
<1
5
102
<1
351
Writing Mixed Numbers as Improper Fractions
Step 1: multiply denominator and whole number to get new number.
Step 2: add the new number to the numerator à this becomes the new numerator.
Step 3: denominator stays the same.
M.A.D. Wheel
A)
Write
2
1
as an improper fraction. Use multiplication and addition.
5
B)
Write
3
2
as an improper fraction. Use multiplication and addition.
3
Writing Improper Fractions as Mixed Numbers
Astronomy Application
The longest total solar eclipse in the next 200 years will take place in 2186. It will last about
15
15
minutes. Write
as a mixed number.
2
2
METHOD 1: Use a model.
Draw squares divided into half sections. Shade 15 of the half sections.
1
2
1
2
1
2
1
1
2
1
2
2
1
2
1
2
3
1
2
4
1
2
1
2
1
2
5
1
2
6
There are ____ whole squares and ___ half square, or 7
1
2
1
2
1
2
1
2
7
1
squares, shaded.
2
METHOD 2: Use division
Step 1: divide and get a whole number
Step 2: remainder becomes numerator
Step 3: denominator stays the same.
18
=
5
2 15 =
24
=
5
37
=
9
47
=
3
31
=
2
1
2
4-7 Comparing and Ordering Fractions
Vocabulary
_________ ______________ - fractions that have the same denominator
___________ ______________ - fractions with different denominators
______________ ______________ - a denominator that is the same in two or more fractions
Example 1
Comparing Fractions
Compare. Write <, >, =.
A)
1
8
B)
7
10
5
8
1
2
Method 1: Use the least common multiple (LCM) as the common denominator.
8
=
21
5
=
14
8:
3:
LCM = ______
21:
LCM = ______
14:
Method 2: Use cross multiplication
7
10
8
11
Method 3: Rewrite the fractions as decimals. Then compare.
4
5
7
8
Cooking Application
A)
Ray has
2
3
cup of nuts. He needs
cup to make cookies. Does he have enough nuts for the
3
4
recipe?
Compare
B)
2
3
and .
3
4
Rachel and Hannah have
1
2
1
cups of cabbage. They need 1 cups to make pot stickers. Do
3
2
they have enough for the recipe?
Compare
Example 3
A)
Order
1
2
1
and 1 .
3
2
Ordering Fractions
3 3
1
, , and
from least to greatest.
7 4
4
3
= ___
7
3
= ___
4
1
= ___
4
Rename with like denominators
The fractions in order from least to greatest are _____________________________.
B)
Order
4 2
1
, , and
from least to greatest.
5 3
3
4
= ___
5
2
= ___
3
1
= ___
3
Rename with like denominators
The fractions in order from least to greatest are _____________________________.
4-8 Adding and Subtracting with Like Denominators
Example 1
A)
Life Science Application
Sophie plants a young oak tree in her backyard. The distance around the trunk grows at a rate
of
1
inch per month. Use pictures to model how much this distance will increase in two
8
months, then write your answer in simplest form.
=
Ì
1
1
+
8
8
1 1 2
+
=
8 8 8
=
Add the numerators. Keep the same denominators
1
4
Write your answer in simplest form.
The distance around the trunk will increase by _____ inch.
B)
Snow was falling at a rate of
answer in simplest form.
1
inch per hour. How much snow fell after two hours? Write your
4
After two hours _______ inch of snow fell.
Example 2
Subtracting Like Fractions and Mixed Numbers
Subtract. Write each answer in simplest form.
A)
1 -
2
3
___ - ___ = ___
Check
B)
1-
3
5
___ - ___ = ___
Subtract. Write each answer in simplest form.
C)
3
7
1
- 1
12
12
3
7
1
- 1
12
12
2
6
12
2
1
2
Write your answer in lowest terms.
Check:
D)
E)
5
5
12
Subtract the fractions. Then subtract the whole numbers.
- 2
1
12
Example 3
Evaluation Expressions with Fractions
Evaluate each expression for x =
A)
3
. Write each answer in simplest form.
8
5
- x
8
5
- x
8
Write the expression
5
3 2
=
8
8 8
Substitute
=
B)
x +
1
1
8
C)
x +
7
8
Borrowing Practice
A)
8- 2
1
4
B)
6- 2
3
8
1
4
3
8
denominator.
for x and subtract the numerators. Keep the same
Write your answer in simplest form.
Practice
Adding and Subtracting with Like Denominators
Subtract. Write each answer in simplest form.
2. 18 − 10
24
24
1. 1 − 4
7
________________________
4. 8 11 − 5 2
13
13
________________________
5. 5 − 3 1
4
________________________
7. 6 8 − 4 6
9
9
6. 2 − 1 2
7
_______________________
9. 10 − 5 3
5
8. 7 4 − 6 3
11
11
________________________
Evaluate each expression for x =
_______________________
2 . Write each answer in simplest form.
15
12. 13 − x
15
11. x − 1
15
_______________
_______________________
________________________
________________________
10. x + 14
15
3. 2 2 − 1 1
3
3
________________
13. x + 7
15
_______________
_______________
Write each sum or difference in simplest form.
14. 17 − 2
21 21
________________________
17. 27 76 − 14 26
100
100
________________________
15. 13 + 9
32
32
________________________
18. 1 + 4 + 5
15 15 15
________________________
16. 2 + 8
15 15
_______________________
19. 9 + 2 + 5
26
26
26
_______________________
20. Maria has 8 gallons of paint she wants to use in three rooms of her house. She will use 2 1
4
1
gallons in the bedroom and 1 gallons in the bathroom. Use pictures to model how many gallons
4
she will have left to paint the playroom, and then write your answer in simplest form.