7.1
Factors and Divisibility
?
Essential Question How can you tell whether one number is a factor of
another number?
Texas Essential
Knowledge and Skills
How can you tell
whether one number is a
factor of another number ?
Algebraic Reasoning—5.4.A
Identify prime and composite numbers
MATHEMATICAL PROCESSES
5.1.C Select tools, technology, and techniques
5.1.D Communicate mathematical ideas and reasoning
5.1.F Analyze mathematical relationships
Are You Ready?
Access Prior Knowledge
Use the Are You Ready? 7.1 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
composite number, divisible
Lesson Opener
Making Connections
Go to Multimedia eGlossary at
thinkcentral.com
Invite students to tell you what they know about division of whole numbers.
When can you use division to solve a problem? (to find how many equal-numbered
groups of objects there are) What do the numbers in a division problem tell us? (the
total number of objects, the number of groups, and the number of objects in each
group) How are the numbers in a division problem related to multiplication? (Accept all
reasonable answers.)
Using the Digital Lesson
Have students use base-ten blocks to brainstorm how they can group sets of equal
numbers. Ask them to compare their groupings to those of their classmates.
Learning Task
Resources
What is the problem the students are trying to solve? Connect the story to the problem.
• What is the total number of golf balls? (88)
• What does the problem tell you about how to separate the golf balls into the
buckets? (There must be the same number in each bucket.)
• What will the number of buckets used tell you? (how many equal groups there are)
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
Literacy and Mathematics
Choose one or more of the following activities.
• Have students work with a partner to ask and answer questions about the
information given in the problem.
• Have students work in pairs to create a similar problem in which they need to find
how many groups are needed to divide a set of objects into equal numbers.
Math on the Spot
Video Tutor
Online Assessment
System
iTools Virtual
Manipulatives
Soar to Success Math
Online Intervention
Lesson 7.1 291A
Name
7.1
?
Unlock the Problem
Have students read the problem and the caption
beneath the picture of the mosaic. Ask students if
they have ever made a mosaic at school or at home.
Algebraic
Reasoning—5.4.A
MATHEMATICAL PROCESSES
5.1.C, 5.1.D, 5.1.F
Factors and Divisibility
Essential Question
How can you tell whether one number is a factor of
another number?
Unlock
Unlock the
the Problem
Problem
Students in Carlo’s art class painted 32 square tiles
for a mosaic. They will arrange the tiles to make a
rectangle. Can the rectangle have 32 tiles arranged
into 3 equal rows, without gaps or overlaps?
One Way
Discuss how actual tiles could be arranged in a
rectangle. Point out that arranging tiles requires
materials that may not always be available. Drawing
a model is a simpler method to use.
A composite number is a whole number greater than
1 that has more than two factors. You can use models
and division to find factors of composite numbers.
One Way Draw a model.
▲ Mosaics are decorative patterns made
with pieces of glass or other materials.
Think: Try to arrange the tiles into 3 equal rows to make a rectangle.
Another Way
For greater numbers, using tiles or drawing a model
can be time–consuming. Using division is a more
efficient method.
• Is there a remainder when 32 is divided by 3? yes
cannot
A rectangle __
have 32 tiles arranged into 3 equal rows.
• Why does the remainder help you answer the
question? Possible answer: because there is a remainder
Another Way Use division.
Math Talk
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Artville/Getty Images
in the division, I know that 32 tiles cannot be arranged
into 3 rows without gaps or overlaps.
Mathematical Processes
Use Math Talk to help students relate the model
to the division when determining whether one
number is a factor of another number.
English Language Learners
If 3 is a factor of 32, then the unknown factor in 3 × ■ = 32 is a
whole number.
1 0 r2
3)‾
3 2
–_
3
0 2
–_
0
2
whether the unknown
factor is a whole
number.
ELL Language Support
ELPS
Beginning: Activity 20
1.A.1, 3.G.2, 4.C.3
Intermediate: Activity 26
3.G.1, 4.D, 4.F.2
Strategy: Model Language
Advanced: Activity 27
2.I.3, 3.B.3, 4.D
Materials: index cards, markers
Advanced High: Activity 43
4.F.8, 4.G.2, 4.G.4
291 Module 7
Math Talk
Mathematical Processes
is not
a whole number.
The unknown factor in 3 × ■ = 32 __
Leveled Activities
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
Possible answer: the
model shows 3 rows
with 10 tiles and 2 rows
each with an extra tile.
The quotient is 10 with
a remainder of 2.
Think: Divide to see
cannot
So, a rectangle __
have 32 tiles arranged in 3 rows.
Math Idea
A factor of a number divides
the number evenly. This means
the quotient is a whole number
and the remainder is 0.
Explain how the model relates
to the quotient and remainder
for 32 ÷ 3.
Module 7 291
Visual / Auditory
Partners
ELPS 2.C.2, 3.F.1, 3.G.2
•
•
•
•
Students learn the language of divisibility when it is modeled.
Have students write 2-digit numbers on four cards.
Explain the divisibility rules for 2, 3, 5, 6, and 9.
Shuffle the cards and then draw the top card. Tell
36
students by what numbers it is divisible and how
you know.
36 is divisible by 2.
• Draw another card.
• Is this number divisible by 2? How do you know?
• Have partners take turns drawing a new card and asking the
modeled questions to determine divisibility.
Divisibility Rules A number is divisible by another
Divisibility Rules
number if the quotient is a counting number and the
remainder is 0.
Some numbers have a divisibility rule. You can use a
divisibility rule to tell whether one number is a factor
of another.
Is 6 a factor of 72?
Think: If 72 is divisible by 6, then 6 is a factor of 72.
Test for divisibility by 6:
Number
Divisibility Rule
2
The number is even.
3
The sum of the digits
is divisible by 3.
5
The last digit is 0 or 5.
6
The number is even
and divisible by 3.
9
The sum of the digits is
divisible by 9.
yes
Is 72 even? _
What is the sum of the digits of 72?
Divisibility Rules
Read and discuss the divisibility rules. Then ask:
• Suppose that you have to divide two numbers.
Give an example of how divisibility rules can be
used to check your answer. Possible answer: I know
that every even number is divisible by 2. So, if I divide an
even number by 2 and have a remainder in my answer, I
know to recheck my answer.
Possible explanation:
a number is divisible
by each of its factors.
7 +_
2 =_
9
_
Is the sum of the digits divisible by 3?
• Can a number be divisible by more than one
number? Give an example to support your answer.
Yes; possible example: 10 is divisible by 2, by 5, and by 10.
Math Talk
Yes; 9 ÷ 3 = 3, and the remainder is 0.
_______
Mathematical Processes
• Why isn’t there a divisibility rule for 1? All nonzero
How are divisibility and
factors related? Explain.
6 .
72 is divisible by _
numbers are divisible by 1.
• Why isn’t there a divisibility rule for 0? You cannot
So, 6 is a factor of 72.
divide by 0.
Share
Share and
and Show
Show
Share and Show
1. Is 4 a factor of 28? Draw a model to help.
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
Think: Can you make a rectangle with 28 squares in 4 equal rows?
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on
the MathBoard.
Is 5 a factor of the number? Write yes or no.
2. 27
no
_
3. 30
yes
_
4. 36
no
_
5. 53
no
_
292
© Houghton Mifflin Harcourt Publishing Company
is a factor of 28.
4_
3
2
Quick Check
1
a student misses the checked exercises
IF
THEN
Differentiate Instruction with
RtI Tier 1 Lesson 45
COMMON ERRORS
C
Enrich
Logical
Small Group
• Have students find numbers that are divisible by 8 by looking at
multiples of 8. (Students may need to choose greater numbers, like
8 × 56, for the activity.)
• Have students consider the divisibility rules shown in the lesson. Have
students make conjectures of what a divisibility rule for 8 would be.
Encourage them to use the strategy guess, check, and revise. If you can
divide a number by 2 three times, then the number is divisible by 8. For example,
64 ÷ 2 = 32, 32 ÷ 2 = 16, 16 ÷ 2 = 8. So, 64 is divisible by 8.
E
Error
Students look only at the last digit of a
number to decide if it is divisible by 3 or 9.
Example 19 is divisible by 3 and by 9 because the
last digit in the number is divisible by 3 and by 9.
Springboard to Learning To show that 19 is not
divisible by 3 or 9, have students complete the
division problems shown below.
19 ÷ 3 = ■
19 ÷ 9 = ■
Point out that since none of the answers are whole
numbers, 19 is not divisible by 3 or by 9.
• Challenge students to explain why the divisibility rule works. The inverse
of dividing by 2 three times is multiplying by 2 three times. 2 × 2 × 2 is 8. So, you
are basically dividing by 8.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 7.1
292
Name
Problem
Problem Solving
Solving
Problem Solving
List all the factor pairs in the table. Use a model or paper and pencil to help.
Problem
6.
For Problem 10, some students may find it helpful if
you remind them that when one number is divisible
by another number, the division of those numbers
does not produce a remainder.
7.
Factors of 24
Factors of 39
1 ,_
24
24
24 = _
1 ×_
_
_
1 ,_
39
39
39 = _
1 ×_
_
_
2 ,_
12
_
3 ,_
13
39
3 ×_
13 = _
_
_
24
12 = _
2 ×_
_
3 ,_
8
24
8 =_
3 ×_
_
_
4 ,_
6
24
6 =_
4 ×_
_
_
To solve Problem 11, students should infer that
because the correct answer is divisible by 5, two
numbers (78 and 63) can immediately be eliminated
as possible answers.
List all the factor pairs for the number. Make a table to help.
8. 56
Go Deeper
• Ask students to divide 24 by 2 and then divide
the quotient, 12, by 2 again. Is 24 divisible by 4?
Explain. Yes; 24 can be evenly divided by 4.
• Now try it with 46. Divide 46 by 2. Then divide the
quotient by 2. What do you notice? 46 ÷ 2 = 23;
23 ÷ 2 = 11 r1; 46 is not divisible by 4 because there is a
10.
9. 64
1 and 56, 2 and 28, 4 and 14,
1 and 64, 2 and 32, 4 and 16,
7 and 8
8 and 8
What’s the Error? George said if 2 and 4 are factors of
a number, then 8 is a factor of the number. Is he correct? Explain.
No; possible explanation: 2 and 4 are factors of 20, but 8 is not a factor.
Problem
Problem Solving
Solving
remainder.
Stamps Sets
Use the table to solve 11–12.
11. Multi-Step Dirk bought a set of stamps. The number of stamps
• If a number can be divided by 2 twice without a
remainder, then that number is divisible by 4.
in the set he bought is divisible by 2, 3, 5, 6, and 9. Which set is it?
Germany
M
Math
on the Spot
Video Tutor
V
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.
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
© Houghton Mifflin Harcourt Publishing Company
12.
Number of stamps
Germany
90
Sweden
78
Japan
63
Canada
25
Multi-Step Geri wants to put 6 stamps
on some pages in her stamp book and 9 stamps on other
pages. Explain how she could do this with the stamp set
for Sweden.
Possible answer: Geri could break apart 78 into
60 + 18, since 60 is divisible by 6 and 18 is divisible by 9. She could make 10 pages
with 6 stamps each (60 ÷ 6 = 10) and 2 pages with 9 stamps each (18 ÷ 9 = 2).
Module 7 • Lesson 1 293
3
RtI Tier 1 Lesson 45
2
1
Enrich 43
Name
Name
LESSON
45
5.4.A
Invisible Divisible
Use the clues to find all possibilities for the unknown digit in each
number.
1.
Conclusion
39 ÷ 2 19 r1
39 is not divisible by 2.
39 ÷ 3 13 r0
39 is divisible by 3.
The sum of the digits, 3 + 9 = 12, is
divisible by 3, so 39 is divisible by 3.
39 ÷ 5 7 r4
39 is not divisible by 5.
The last digit, 9, is not a 0 or 5, so 39
is not divisible by 5.
39 ÷ 6 6 r3
39 is not divisible by 6.
39 is not divisible by both 2 and 3,
so it is not divisible by 6.
39 ÷ 9 4 r3
39 is not divisible by 9.
The sum of the digits, 3 + 9 = 12,
is not divisible by 9, so 39 is not
divisible by 9.
The number below has 4 as a factor.
What could the unknown digit be?
3,2
6
0, 2, 4, 6, 8
Divisibility Rules
The last digit, 9, is not even, so 39 is
not divisible by 2.
3.
4.
The number below has 5 as a factor.
What could the unknown digit be?
1,9
5
5.
7,71
7.
6.
The number below has 6 as a factor.
What could the unknown digit be?
8.
6,1
5
11
0, 3, 6, 9
The number below has 2 and 9 as
factors. What could the unknown digit
be?
2,3
0, 3, 6, 9
0, 9
The number below has 3 as a factor.
What could the unknown digit be?
4,
0, 6
The number below has 3 and 5 as
factors. What could the unknown digit
be?
1, 3, 5, 7, 9
The number below has 9 as a factor.
What could the unknown digit be?
6,30
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
39 is divisible by 3. So, 3 is a factor of 39.
A number is divisible by each of its factors. A whole number
that has more than two factors is a composite number.
© Houghton Mifflin Harcourt Publishing Company
2.
The number below has 2 as a factor.
What could the unknown digit be?
5,83
Is 39 divisible by 2, 3, 5, 6, or 9?
Result
Enrich 43
1
Factors and Divisibility
OBJECTIVE Determine whether a number is a factor of a given number.
A number is divisible by another number if the quotient is
a counting number and the remainder is 0.
You can decide if a number is divisible by 2, 3, 5, 6, or 9 by
using divisibility rules instead of dividing. Divisibility rules help
you decide if one number is a factor of another.
6
7
Use the chart to tell whether 30 is divisible by each divisor. Explain.
9.
Result Conclusion (yes/no)
1. 30 ÷ 2
2. 30 ÷ 3
3. 30 ÷ 5
4. 30 ÷ 6
5. 30 ÷ 9
15
10
6
5
3 r3
Explanation
yes
30 is an even number.
yes 3 + 0 = 3; 3 is divisible by 3.
yes
The last digit is a 0.
yes 30 is divisible by both 2 and 3.
no 3 + 0 = 3; 3 is not divisible by 9.
Is 4 a factor of the number? Write yes or no.
6. 81
293 Module 7
Country
no
Algebraic Reasoning
7. 24
yes
8. 56
Stretch Your Thinking A number is divisible by 2 if the last
digit is divisible by 2. A number is divisible by 4 if the last two
digits form a number divisible by 4. A number is divisible by 8 if the
last three digits form a number divisible by 8. Describe a possible
pattern in the divisibility rules. Then test each of the following
numbers for divisibility by 8.
3,488
5,614
4,320
3,052
Possible answer: 4 5 2 3 2 and 8 5 2 3 2 3 2. So, for
each successive time 2 is a factor, you need to consider
one more digit; 3,488 and 4,320 are divisible by 8.
yes
89
Enrich
© Houghton Mifflin Harcourt Publishing Company
E43
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Fill in the bubble completely to show your answer.
Daily Assessment Task
13. There are 54 people attending a party. Carla is arranging chairs and
tables. How should she arrange the chairs so that there will be an equal
number of people seated at each table?
A
12 chairs at each table
B
8 chairs at each table
C
9 chairs at each table
D
10 chairs at each table
2
1
Can students tell whether one number is a factor
of another number?
THEN
IF
NO
•
Soar to Success Math
Warm-Up 31.30
14. Analyze Jake organizes 48 marbles into packs. He places the same
number of marbles into each pack. How could he arrange the marbles?
A
8 or 10 in each pack
B
9 or 12 in each pack
C
10 or 12 in each pack
D
8 or 12 in each pack
YES
•
•
Enrich 43
Homework and Practice
Lesson 7.1
15. Multi-Step Megan has 34 rocks in her rock collection. She wants to put
TEXAS Test Prep Coach
5 rocks in some cases and 7 rocks in other cases. How could she arrange
the rocks?
A
Have 6 cases of 5 rocks, and 1 case of 7 rocks.
In the Test Prep exercise, if students selected:
B
Have 4 cases of 5 rocks, and 2 cases of 7 rocks.
A They thought that 80 was divisible by 3.
C
Have 2 cases of 5 rocks, and 4 cases of 7 rocks.
B They thought that 80 was divisible by 6.
D
Have 5 cases of 5 rocks, and 2 cases of 7 rocks.
D They thought that 80 was divisible by 9.
?
TEXAS Test Prep
16. Mrs. Mastrioni bought a set of 80 stamps. She wanted to give all the stamps
A
2 or 3 students.
B
2 or 6 students.
C
2, 4, 5, or 8 students.
D
2, 4, 8, or 9 students.
© Houghton Mifflin Harcourt Publishing Company
to her students as a reward. She could give equal numbers of stamps to
294
Essential Question
Write
Math
How can you tell whether one number is a factor of
another number? Possible answer: I can use a divisibility
rule to check if a number is a factor of another number.
Differentiated Centers Kit
Games
Games
Factor Farm
Students practice determining
factors of whole numbers.
Activities
Flowering Factors
Students complete orange
Activity Card 17 by identifying
the factors of whole numbers.
Lesson 7.1
294
5
Ho mewo rk
and Practice
7.1
Algebraic Reasoning—5.4.A
MATHEMATICAL PROCESSES 5.1.C,
5.1.D, 5.1.F
Name
Fill in the bubble completely to show your answer.
9. Which number is divisible by 2, 3, and 6?
Factors and Divisibility
List all the factor pairs in the table.
1.
Factors of 30
1 ,_
30
1 ×_
30 = _
30
_
_
2 ,_
15
2 ×_
15 = _
30
_
_
TEXAS Test Prep
Lesson
Lesson Check
Check
2.
Factors of 15
A
60
10. Jo Beth organizes 63 party favors into gift bags.
She places the same number of favors in each
bag. How could she arrange the favors?
B
33
A
3 or 6 in each bag
C
46
B
6 or 9 in each bag
D
21
C
7 or 8 in each bag
D
7 or 9 in each bag
1 ,_
15
1 ×_
15 = _
15
_
_
3 ,_
5
3 ×_
15
5 =_
_
_
3 ,_
10
3 ×_
10 = _
30
_
_
11. A package of marbles contains 20 blue
5 ,_
6
5 ×_
6 =_
30
_
_
List all the factor pairs for the number. Make a table to help.
4. 45
1 and 18, 2 and 9, 3 and 6
5. 36
1 and 45, 3 and 15, 5 and 9
6. 63
1 and 36, 2 and 18, 3 and 12, 4 and 9,
© Houghton Mifflin Harcourt Publishing Company
8. A marching band with 75 members makes two
rectangular marching formations. One rectangle
has 6 rows and the other rectangle has 9 rows.
Explain how the members can form the two
rectangles.
Possible answer: Break apart 75 into
so they can plant 4 rows of 14 rose bushes.
30 + 45. They can make 6 rows of 5
members (30 ÷ 6 = 5) and 9 rows of 5
members (45 ÷ 9 = 5).
Module 7 • Lesson 1
Homework and Practice
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
Module 7
A
25
red
B
21
C
green
C
23
D
yellow
D
26
organize. He wants to put 5 books on some
shelves and 6 books on other shelves. How
could he arrange the books?
Yes; Possible explanation: 4 × 14 = 56,
295-296
blue
B
13. Multi-Step The librarian has 94 books to
Problem
Problem Solving
Solving
planting a rose garden. They want to plant
56 rose bushes. Can they arrange the rose bushes
into 4 equal rows? Explain.
A
1 and 63, 3 and 21, 7 and 9
6 and 6
7. Planners for the city’s botanical garden are
math competition will be evenly divided into
teams of 3 students each. How many students
could be participating?
295
296
A
Have 8 shelves of 5 books, and 9 shelves
of 6 books.
B
Have 9 shelves of 5 books, and 8 shelves
of 6 books.
C
Have 5 shelves of 6 books, and 6 shelves
of 5 books.
D
Have 7 shelves of 5 books, and 6 shelves
of 6 books.
14. Multi-Step A group of 72 students went on a
field trip. The students traveled in vans, with
an equal number of girls and boys in each van.
If 42 boys went on the trip, how many vans
were there?
A
9
B
8
C
7
D
6
© Houghton Mifflin Harcourt Publishing Company
3. 18
12. The fifth-grade students participating in the
marbles, 38 red marbles, 48 green marbles,
and 16 yellow marbles. Which color of marbles
can be divided evenly between 6 people?