7.2
Prime and Composite Numbers
?
Essential Question How can you tell whether a number is prime
or composite?
Texas Essential
Knowledge and Skills
Algebraic Reasoning—5.4.A
Identify prime and composite numbers
MATHEMATICAL PROCESSES
5.1.A Apply mathematics to problems
5.1.D Communicate mathematical ideas and reasoning
How can you tell
whether a number is
prime or composite?
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 7.2 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
prime number
Lesson Opener
Go to Multimedia eGlossary at
thinkcentral.com
Making Connections
Invite children to tell you what they know about multiplication and division.
What whole numbers multiply to 7? (1 and 7) What whole numbers multiply to 8?
(1 and 8, and 2 and 4) State 3 numbers that are divisible by 3. (3, 6, 9, 12, 15, etc.) Is 54
divisible by 3? (Yes, 5 + 4 = 9, which is divisible by 3.)
Using the Digital Lesson
You may wish to have 25 small objects, such as cubes, to represent the cones and
5 segments to represent city blocks. You can place the 25 objects evenly in the space
that represents the city blocks.
Learning Task
What is the problem the children are trying to solve? Connect the story to the problem.
Ask the following questions.
• Have many cones are on the street? (25)
• How many blocks are going to be worked on? (5)
• What is the problem asking? (if the cones can be placed so that there is the same
number on each block)
• What will you be learning about in this lesson? (prime and composite numbers)
Literacy and Mathematics
Choose one or more of the following activities.
• Have students write another story problem similar to the one in the opener.
• Have students draw the scenario in the story problem.
Resources
For the student
For the teacher
Interactive
Student Edition
provides students
with an interactive learning
environment!
Digital Management
Center organizes program
resources by TEKS!
eTeacher
Edition
Math on the Spot
Video Tutor
Online Assessment
System
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 7.2 297A
Name
7.2
?
Unlock the Problem
This activity provides students with one way they can
determine whether a number is prime or composite.
Algebraic
Reasoning—5.4.A
MATHEMATICAL PROCESSES
5.1.A, 5.1.D
Prime and Composite Numbers
Essential Question
How can you tell whether a number is prime
or composite?
Unlock
Unlock the
the Problem
Problem
• Do all composite numbers have the same number
of factors? Explain. No. Although all composite
Students are arranging square tables to make one larger,
rectangular table at a fundraiser for an animal shelter. If the
students want to choose from the greatest number of ways to
arrange the tables, should they use 12 or 13 square tables?
numbers have more than two factors, some will have
more factors than others.
• What are the factors of 12?
1, 2, 3, 4, 6, 12
Use a grid to show all the possible arrangements
of 12 and 13 tables.
• Do all prime numbers have the same number of
factors? Explain. Yes. All prime numbers have exactly
Label each drawing with the factors modeled.
two factors, 1 and the number itself.
ERROR Alert
• How many factors does 1 have? It has one factor,
The same factors in a different
order should be counted only
once. For example, 3 × 4 and
4 × 3 are the same factor pair.
itself.
• Is the number 1 prime? composite? Give a reason
to support each answer. Possible answer: 1 is not
prime because it does not have exactly two factors; 1 is
not composite because it does not have more than
two factors.
Math Talk
Mathematical Processes
Explain how knowing whether
12 and 13 are prime or
composite could have
helped you solve the
problem above.
12 tables.
So, there are more ways to arrange _
• A prime number is a whole number greater than 1 that
has exactly two factors, 1 and itself.
© Houghton Mifflin Harcourt Publishing Company
• A composite number is a whole number greater than 1 that
has more than two factors.
English Language Learners
Leveled Activities
ELPS
Beginning: Activity 19
1.A.1, 3.E, 4.F.7
Intermediate: Activity 26
3.G.1, 4.D, 4.F.2
Advanced: Activity 27
2.I.3, 3.B.3, 4.D
Advanced High: Activity 6
2.I.5, 3.G.2, 4.G.2
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
297 Module 7
1 ,_
2 ,_
3 ,_
4 ,_
6 ,_
12
Factors of 12: _
1 ,_
13
Factors of 13: _
prime
composite number, and 13 is a __
number.
12 is a __
ELL Language Support
Possible explanation: since
13 is prime, it has only
2 factors, so it would have
only one arrangement.
Since 12 is composite, it
would have more than
2 factors, and so it would
have more than one
arrangement.
Module 7 297
Kinesthetic
Small Group
ELPS 1.A.2, 2.C.4, 2.I.5
Strategy: Identify Patterns
Materials: square tiles, grid paper
• Students identify patterns to understand math concepts.
• Have students use color tiles to show all of the rectangles they can
make using 2 tiles. Students draw the rectangles on grid paper.
(A 2 × 1 and a 1 × 2 rectangle are considered the same.)
• Students work through the numbers 3–15 in the same way and record
the dimensions in a chart.
• Have students look for patterns in the chart.
• All the numbers that have only one rectangle are prime numbers. All
the others are composite numbers.
Divisibility You can use divisibility rules to help tell whether a
number is prime or composite. If a number is divisible by any
number other than 1 and itself, then the number is composite.
Math Idea
The number 1 is neither prime
nor composite, since it has only
one factor: 1.
Divisibility
Tell whether 51 is prime or composite.
• Why don’t you have to check for more factors
to tell whether 51 is prime or composite? Possible
Is 51 divisible by 2? No; 51 is not even.
answer: if 51 has more than 2 factors, it is composite. So,
once you find a third factor you can stop. The number of
factors is not what is asked.
Is 51 divisible by 3? Yes; 5 + 1 = 6, and 6 is divisible by 3.
Think: 51 is divisible by a number other than 1 and 51.
51 has more than two factors.
composite
So, 51 is ___
.
Share and Show
Math Talk: Composite; Possible explanation: the
product has more than two factors: 1, the two
prime numbers, and the product itself.
Share
Share and
and Show
Show
Point out to students that they do not need to draw
models or find all the factors of a number to decide
whether it is composite.
Tell whether the number is prime or composite.
1. 11
Think: Does 11 have
2. 73
3. 69
4. 42
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on
the MathBoard.
other factors besides
1 and itself?
prime
prime
composite
Math Talk
Problem
Problem Solving
Solving
3
Mathematical Processes
Analyze Write true or false for each statement.
Explain or give an example to support your answer.
5. The number 1 is not prime.
composite
2
False; 4 has three factors: 1, 2, and 4.
THEN
because it has only one factor, 1.
Differentiate Instruction with
RtI Tier 1 Lesson 46
8. Every multiple of 7 is a composite number.
False; 2 is prime; its only factors are
False; 7 is a multiple of 7, and 7 is
1 and itself.
prime.
Name a 2-digit odd number that is prime. Name a 2-digit
odd number that is composite.
Sample answers: prime: 11, 13, 17; composite: 15, 21, 39
298
© Houghton Mifflin Harcourt Publishing Company
9.
a student misses the checked exercises
IF
6. A composite number cannot have three factors.
True; 1 is neither prime nor composite
7. Only odd numbers are prime numbers.
Quick Check
1
Is the product of two prime
numbers prime or composite?
Explain.
Math Talk
Mathematical Processes
Use Math Talk to focus on students’ understanding
of prime and composite numbers.
COMMON ERRORS
C
E
Error
Students identify a composite number as a
prime number.
Example The factors of 9 are 1 and 9, so 9 is a
prime number.
Enrich
Logical
Individual / Partners
Christian Goldbach was a mathematician who lived in the 1700s. In letters
between mathematician Leonhard Euler and Goldbach, the conjecture
was made that every even number greater than 2 can be expressed as the
sum of two prime numbers. For example, 28 = 23 + 5; 28 = 17 + 11.
Have partners play the following game:
• One partner writes an even number between 20 and 100.
• Students take turns writing the number as a sum of two prime
numbers. They continue until they have found all the different ways to
do so.
• If one partner finds more ways to write the sum than the other partner
does, he or she wins 1 point. Play continues until one partner has
5 points.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Springboard to Learning Tell students to test
consecutive numbers to determine if the number
has more than two factors.
Problem Solving
Problems
Problems 5-8 require students to evaluate the truth
of statements made about prime and composite
numbers.
Go Deeper
Have students write their own true and false
statements about prime and composite numbers.
Lesson 7.2
298
Name
Problem
Problem Solving
Solving
Have students follow the Sieve of Eratosthenes (erah-TOSS-tha-neez) steps to identify all the prime
numbers from 1 to 100. Then ask:
Eratosthenes was a Greek mathematician who lived more than 2,200 years
ago. He invented a method of finding prime numbers, which is now called
the Sieve of Eratosthenes.
• What is the only even prime number? Explain.
10. Multi-Step Follow the steps below to circle all prime numbers
2; every even number greater than 2 is a multiple of 2.
So, every even number greater than 2 has 2, itself, and
1 as factors. Any number with three or more factors is a
composite number.
less than 100. Then list the prime numbers.
STEP 1
STEP 2
STEP 3
STEP 4
Cross out 1, since 1 is not
prime.
Circle 2, since it is prime.
Cross out all other
multiples of 2.
Circle the next number
that is not crossed
out. This number is
prime. Cross out all the
multiples of this number.
Repeat Step 3 until every
number is either circled
or crossed out.
• What number is a factor of both 17 and 91? How
do you know? 1; I know that 1 is a factor of every whole
number.
• Do you need to find all the factors of a number
before you decide if a number is composite?
Why? No. If the number has more than two factors, it is a
composite number.
M
Math
on the Spot
Video Tutor
V
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
2
3
4
5
6
7
8
9
10
So, the prime numbers less
than 100 are
11
12
13
14
15
16
17
18
19
20
2, 3, 5, 7, 11, 13,
21
22
23
24
25
26
27
28
29
30
17, 19, 23, 29, 31,
31
32
33
34
35
36
37
38
39
40
37, 41, 43, 47, 53,
41
42
43
44
45
46
47
48
49
50
59, 61, 67, 71, 73,
51
52
53
54
55
56
57
58
59
60
79, 83, 89, 97.
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
11.
© Houghton Mifflin Harcourt Publishing Company
Through the Math on the Spot Video Tutor,
students will be guided through an interactive
solving of this type of H.O.T. problem. Use this
video to also help students solve the H.O.T.
problem in the Interactive Student Edition. With
these videos and H.O.T. problems, students will
build skills needed in the TEXAS assessment.
1
Explain why the multiples of any number other than 1 are
not prime numbers.
Possible explanation: a multiple of a number has more
factors than just 1 and itself.
Module 7 • Lesson 2 299
3
RtI Tier 1 Lesson 46
2
1
Enrich 44
Name
Name
LESSON
46
5.4.A
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97
A composite number is a whole number greater than 1 that has
more than two factors.
1.
You can use division to find the factors of a number and
tell whether the number is prime or composite.
Tell whether 55 is prime or composite.
Tell whether 61 is prime or composite.
Use division to find all the numbers
that divide into 55 without a
remainder. Those numbers are the
factors of 55.
Use division to find all the numbers
that divide into 61 without a
remainder. Those numbers are the
factors of 61.
1
55 ÷ 5 = 11, so
factors.
5
The factors of 55 are
11 , and
and
and
1
55
are
11
,
are
5
,
© Houghton Mifflin Harcourt Publishing Company
1
and
61
1
and
101
111
121
131
141
151
161
171
181
191
are
61 .
Because 61 has exactly two factors,
61 is a prime number.
Think: Is 44 divisible by
composite
3. 12
any number other
than 1 and 44?
4. 50
2.
2. 53
Think: Does 53 have other
prime
5. 24
composite composite composite
7. 83
prime
Algebraic Reasoning
8. 27
9. 34
102
112
122
132
142
152
162
172
182
192
103
113
123
133
143
153
163
173
183
193
104
114
124
134
144
154
164
174
184
194
105
115
125
135
145
155
165
175
185
195
106
116
126
136
146
156
166
176
186
196
107
117
127
137
147
157
167
177
187
197
108
118
128
138
148
158
168
178
188
198
109
119
129
139
149
159
169
179
189
199
110
120
130
140
150
160
170
180
190
200
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163,
167, 173, 179, 181, 191, 193, 197, 199
Tell whether the number is prime or composite.
1. 44
Find the prime numbers from 101 to 200.
• First draw a line through all the multiples of 2.
• Then draw a line through all the multiples of 3, then all the multiples of 5,
and continue until you have drawn lines through all the multiples of
prime numbers less than 100.
• The remaining numbers are the prime numbers from 101 to 200. List
these below the table.
There are no other numbers that
divide into 61 evenly without a
remainder.
The factors of 61 are
55 .
Because 55 has more than two
factors, 55 is a composite number.
299 Module 7
61 ÷ 1 = 61, so
factors.
Prime Search
All the prime numbers from 1 to 100 are listed below.
A prime number is a whole number greater than 1 that has
exactly two factors, 1 and the number itself.
55 ÷ 1 = 55, so
factors.
Enrich 44
1
Prime and Composite Numbers
OBJECTIVE Determine whether a number is prime or composite.
factors besides
1 and itself?
6. 67
The number 143 has two lines through it,
first as a multiple of 11 and second as a
multiple of 13; so, 143 is the product of
two prime numbers. Find another number
that is the product of two different prime
numbers greater than 7.
Possible answer: 187;
11 3 17 5 187
prime
composite composite composite
Enrich
© Houghton Mifflin Harcourt Publishing Company
Explain how you can
find all the prime numbers from 201 to
1,000.
Possible answer: I can
list all of the numbers
from 201 to 1,000 and
cross out all the multiples
of prime numbers.
10. 78
91
3.
E44
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Fill in the bubble completely to show your answer.
Daily Assessment Task
12. Reasoning Talia’s locker combination consists of three prime numbers.
The sum of these numbers is also a prime number. Which of these might
be her combination?
A
3 - - 8 - - 17
B
2 - - 3 - - 19
C
7 - - 13 - - 3
D
11 - - 2 - - 5
2
1
Can students tell whether a number is prime or
composite?
THEN
IF
NO
•
Soar to Success Math
Warm-Up 31.29
13. A certain number is a whole number. If the number is also composite,
what must be true about the number?
A
It is odd.
B
It has more than two factors.
C
It has exactly two factors.
D
It has two or more digits.
YES
•
TEXAS Test Prep Coach
numbers be?
3, 7, 2
B
6, 6, 0
C
3, 5, 7
D
3, 4, 5
In the Test Prep exercise, if students selected:
B, C, or D
?
TEXAS Test Prep
B
composite.
C
neither prime nor composite.
D
both prime and composite.
© Houghton Mifflin Harcourt Publishing Company
prime.
They do not understand the terms prime
and composite.
Essential Question
Write
Math
How can you tell whether a number is prime or
composite? Possible answer: I can try to find three factors of
15. The number 2 is
A
Enrich 44
Homework and Practice
Lesson 7.2
14. Multi-Step The sum of three prime numbers is 12. What could the
A
•
300
the number. If the number has exactly two factors, I know it
is a prime number. If the number has three or more factors, I
know it is a composite number.
Differentiated Centers Kit
Literature
Eratosthenes and His Sieve
Students read about Eratosthenes
and his contributions to math,
including his sieve for identifying
prime numbers.
Activities
Prime Time
Students complete blue Activity
Card 17 by identifying prime and
composite numbers.
Lesson 7.2
300
5
Algebraic Reasoning—5.4.A
MATHEMATICAL PROCESSES 5.1.A, 5.1.D
Ho mewo rk
and Practice
7.2
Name
Fill in the bubble completely to show your answer.
14. Four boys compare the numbers on their
Prime and Composite Numbers
football jerseys. Parker has the number 55.
Nick has 47. Marshall has 16, and Leon has 9.
Whose jersey has a prime number?
Tell whether the number is prime or composite.
1. 19
2. 81
3. 52
composite
prime
5. 33
TEXAS Test Prep
Lesson
Lesson Check
Check
6. 60
composite
4. 23
composite
prime
7. 31
composite
8. 25
prime
15. Harlan played her favorite game app three
times this morning. In each game, the number
of points she scored was a prime number.
When she adds the points for the three games
together, the sum is also a prime number.
Which of these might be her scores?
A
Parker
B
Nick
A
17, 19, 20
C
Marshall
B
11, 17, 21
D
Leon
C
21, 23, 13
D
29, 19, 23
composite
Write true or false for each statement. Explain or give an example to
support your answer.
9. A prime number is always greater than 1.
True; 1 is not a prime number.
11. Every multiple of 5 is a composite number.
False; 5 is a multiple of 5, and 5 is prime.
Problem Solving
16. Esteban made a list of prime numbers less
10. The number 17 is a prime number.
than 40. He listed the numbers in order from
least to greatest. Which number did Esteban
put on his list after 29?
True; 17 has exactly 2 factors, 1 and 17.
A
41
12. A number can be both prime and composite.
B
23
False; A prime number has exactly two
C
33
factors. A composite number has more
D
31
17. Carolyn found that the difference between two
prime numbers is a composite number. Which
could be the prime numbers?
A
2, 5
B
13, 2
C
23, 19
D
13, 11
than two factors.
pointer lands on three prime numbers. She
says the sum is less than 12. Which could be
the numbers?
Problem
Problem Solving
Solving
13. The students in math class use square tiles to make arrays. Celia says
they can make more arrays with 8 tiles than with 9 tiles because 8 has
more factors. Is Celia correct? Explain.
© Houghton Mifflin Harcourt Publishing Company
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
301-302
Module 7
2, 4, 5
D
with 8 tiles (1 × 8, and 2 × 4) and two arrays with 9 tiles (1 × 9, and 3 × 3).
Homework and Practice
2, 3, 5
B
C
No; Even though 8 has four factors and 9 has three factors, you can make two arrays
Module 7 • Lesson 2
A
301
302
2, 7, 3
0, 9, 1
19. Multi-Step Felix found the sum of two prime
numbers and one composite number to be 45.
The difference between the greatest and
least number is 4. Which could be the three
numbers?
A
13, 17, 15
B
18, 14, 13
C
15, 14, 16
D
11, 19, 12
© Houghton Mifflin Harcourt Publishing Company
18. Multi-Step Trisha spins a spinner. The