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```LESSON
5.4
Factors and Multiples
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.OA.B.4 Find all factor pairs for a whole number in the range 1–100.
Recognize that a whole number is a multiple of each of its factors.
Determine whether a given whole number in the range 1–100 is a
multiple of a given one-digit number. Determine whether a given whole number in the
range 1–100 is prime or composite.
MATHEMATICAL PRACTICES
MP6 Attend to precision.
MP7 Look for and make use of structure.
F C R Coherence:
Before
3.OA.C.7 4.OA.B.4 5.OA.B.3
F C R Rigor:
Level 1: Understand Concepts....................Share and Show (
Checked Items)
Level 2: Procedural Skills and Fluency.......On Your Own
Level 3: Applications..................................Think Smarter and Go Deeper
Learning Objective
Understand the relationship between factors
and multiples, and determine whether a number
is a multiple of a given number.
Language Objective
Students list three ways in which factors and
multiples are related.
Materials
MathBoard
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 277H.
Professional Development
Teaching for Depth
Students can use counters to help them determine the
factors of a number. They can model the number by
forming equal groups with the counters. For example,
they can show equal groups of 3 when looking for the
factor 3.
Finding Whether 3 is a Factor of Each Number
3
5
10
12
no
counters
left over
factor
counters
left over
not a
factor
counters
left over
not a
factor
no
counters
left over
factor
Students can look at the modeled number to determine
whether the counters form equal groups with none left
over. If there are no counters left over, the model shows
that the number of counters in each equal group is a
factor of the number. For example, the models for 3 and
12 show that 3 is a factor for those numbers.
The counter model for determining factors also connects
to the concept of ﬁnding multiples. When students
establish the factors of a number, they then also know
that the number is a multiple of those factors. For
example, since 3 and 12 have 3 as a factor, 3 and 12 are
both multiples of 3.
Professional Development Videos
299A
Chapter 5
1 ENGAGE
Daily Routines
Common Core
Problem of the Day 5.4
The table shows the price of a carton of
each item at Jack’s Sporting Goods. Each
carton has 8 balls.
Jack's Sporting Goods
with the Interactive Student Edition
Essential Question
How are factors and multiples related?
Making Connections
• What is a factor? A factor is a number multiplied by another number to
ﬁnd a product.
Item
Price
• Name the factors of 20. 1, 2, 4, 5, 10, 20
Carton of Soccer Balls
\$192
Learning Activity
\$256
Mr. Neal bought a carton of basketballs.
How much did he spend for 2 basketballs?
\$64
Vocabulary common multiple
Interactive Student Edition
Multimedia eGlossary
Fluency Builder
Common Core Fluency
Standard 4.NBT.B.6
What is the problem the students are trying to solve? Connect the
story to the problem. Ask the following questions.
• How can you use division to tell whether a number is a factor of
another number? Possible answer: Divide the number by the possible
factor. If the number divides evenly, then it is a factor.
Literacy and Mathematics
View the lesson opener with the students. Then, choose one or more
of the following activities.
• Have the class make a list of the factors of 36. Then have the class
count by each factor and tell whether they say 36 each time.
• Have students make one list of the numbers they say when they
count by 6s to 72 and another list of the numbers they say when
they count by 8s to 72. Then have them identify the numbers that
are common to both lists.
Division Facts Have students complete these division
facts as quickly as possible.
56 ÷ 7 = 8
24 ÷ 8 = 3
27 ÷ 3 = 9
45 ÷ 9 = 5
30 ÷ 5 = 6
42 ÷ 6 = 7
35 ÷ 7 = 5
20 ÷ 5 = 4
36 ÷ 4 = 9
72 ÷ 9 = 8
48 ÷ 8 = 6
36 ÷ 6 = 6
How are factors and
multiples related?
Lesson 5.4
299B
LESSON
5.4
2 EXPLORE
4.OA.B.4 Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of
each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number.
Determine whether a given whole number in the range 1–100 is prime or composite.
Lesson 5.4
Name
Unlock the Problem
Factors and Multiples
Read and discuss the problem with the class.
Make sure students understand that each row
of the display should have 3 animals and there
should be no animals left over.
MP1 Make sense of problems and
persevere in solving them.
• What must be true about the number of
animals in the sets that Mason could buy?
Unlock
Unlock the
the Problem
Problem
Toy animals are sold in sets of 3, 5, 10, and 12.
Mason wants to make a display with 3 animals
in each row. Which sets could he buy, if he
wants to display all of the animals?
↑
factor factor
• How many animals are sold in each set?
3, 5, 10, or 12
↑
multiple of 3
multiple of 4
One Way Find factors.
MP7 Look for and make use of
structure. Review factors and divisibility with
students. Then have students tell whether 3 is
a factor of each number in the problem.
Tell whether 3 is a factor of each number.
Think: If a number is divisible by 3, then 3 is a factor of the number.
Yes, 3 is divisible by 3.
Is 3 a factor of 3? _____
No, 5 is not divisible by 3.
Is 3 a factor of 5? _____
No, 10 is not divisible by 3.
Is 3 a factor of 10? _____
Another Way
Yes, 12 is divisible by 3.
Is 3 a factor of 12? _____
Before completing the exercise, have a
volunteer write the numbers for skip counting
by 3 from 3 through 12 on the board. 3, 6, 9, 12
• Why could Mason buy sets of 3 or 12 for his
display? Possible answer: those are the numbers that
12 .
3
3 is a factor of _
and _
Another Way Find multiples.
Multiply and make a list.
9 ,_
12 , _
15 ,…
3 ,_
6 ,_
_
1×3
© Houghton Mifflin Harcourt Publishing Company
Use Math Talk to help students
understand the relationship
between factors and multiples.
• Three is a factor of 24. Is it also true that
24 is a multiple of 3? Explain. Possible answer:
3
3 × 4 = 12
↑
One Way
Math
Talk
• How many animals will be in each row?
The product of two numbers is a multiple of
each number. Factors and multiples are related.
Possible answer: the number must have 3 as a factor.
are multiples of 3, so they would make 3 animals in
each row of the display.
Operations and Algebraic
Thinking—4.OA.B.4
MATHEMATICAL PRACTICES
MP6, MP7
Essential Question How are factors and multiples related?
MATHEMATICAL PRACTICES
2×3
3×3
4×3
5×3
3
12 are multiples of 3.
_
and _
3 and _
12 toy animals.
So, Mason could buy sets of _
Math
Talk
Possible explanation: a whole number is
always a multiple of its factors. For example,
3 is a factor of 24, so I know that 24 is a multiple of 3.
MATHEMATICAL PRACTICES 6
Explain how you can use
factors to determine
whether one number is
a multiple of another
number.
Chapter 5 299
since 3 is a factor of 24, 3 can be multiplied by a whole
number to get 24. Any number that is the product of 3
and a counting number is a multiple of 3.
ELL Strategy:
Model Language
Explain the meaning of multiples, and have
students repeat the word.
• Write the multiples of 2 up to 30 on the
board. Say them chorally with students: 2,
4, 6, ...
• The multiples of 2 are the same as skip
counting by 2.
• Student pairs practice counting by naming
multiples of 2. Have each explain the
meaning of multiples.
3
Reteach 5.4
1
Lesson 5.4
Reteach
Name
Differentiated
Di
D
iff
f erentiated
IIn
nst
stru
ruct
ctio
ion
n
Instruction
Enrich 5.4
2
Lesson 5.4
Enrich
Name
Factors and Multiples
Multiple Dates
On January 1, 2011, the Petersons began a new
allowance program for their four children:
You know that 1 3 10 5 10 and 2 3 5 5 10.
So, 1, 2, 5, and 10 are all factors of 10.
Every third day, beginning January 3, Adrian will get his allowance.
Every fourth day, beginning January 4, Beth will get her allowance.
Every fifth day, beginning January 5, Zoe will get her allowance.
Every seventh day, beginning January 7, Eddie will get his allowance.
You can skip count to find multiples of a number:
Count by 1s: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .
Count by 2s: 2, 4, 6, 8, 10, 12, . . .
Count by 5s: 5, 10, 15, 20, 25, . . .
Count by 10s: 10, 20, 30, 40, . . .
1.
What is the first day that Adrian and Beth will get their
allowances on the same day?
2.
What is the first day that Beth and Zoe will get their
allowances on the same day?
3.
What is the first day that Adrian and Eddie will get their
allowances on the same day?
4.
What is the first day that Adrian, Beth, and Zoe will get their
allowances on the same day?
5.
Stretch Your Thinking How many days will it be until
Note that 10 is a multiple of 1, 2, 5, and 10. A number is a multiple of
all of its factors.
January 12
A common multiple is a multiple of two or more numbers. So, 10 is a
common multiple of 1, 2, 5, and 10.
1.
3
2.
January 20
Multiply to list the next five multiples of 3.
,
6 , 9 , 12 , 15 , 18
January 21
Multiply to list the next five multiples of 7.
7 , 14 , 21 , 28 , 35, 42
Is the number a factor of 8? Write yes or no.
3.
2
4.
yes
8
March 1
5.
yes
15
6.
20
no
no
all four children will get their allowances on the same day?
Explain.
420 days; Possible explanation: I multiplied 3 3 4 3
Is the number a multiple of 4? Write yes or no.
7.
2
no
299 Chapter 5
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
8.
12
9.
yes
16
yes
5-11
10.
5 3 7 to find the first common multiple of 3, 4, 5, and 7.
18
no
3 3 4 3 5 3 7 5 3 3 20 3 7 5 60 3 7, or 420.
Reteach
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
5-12
Enrich
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
Common Multiples
Read the definition for common multiple.
Example
Common Multiples A common multiple is a multiple of two or more
numbers.
• How can you tell on which dates Tony and
Amanda work together? Those dates have both a
Example Find common multiples.
Tony works every 3 days and Amanda works every 5 days. If Tony works June
3 and Amanda works June 5, on what days in June will they work together?
circle and a box around them. They are June 15 and 30.
• Why are 15 and 30 common multiples of 3
and 5? because 15 and 30 are multiples of both 3 and 5
Circle multiples of 3. Draw a box around multiples of 5.
June
Sun
Mon
Tue
Wed
Thu
Fri
Sat
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
Math
Talk
Use Math Talk to help students
recognize the relationship
between factors and multiples.
• What is 5 in relationship to 15? Possible
answer: 5 is a factor of 15.
• What is 15 in relationship to 5? Possible
Think: The common multiples have both a circle and a box.
answer: 15 is a multiple of 5.
15 and _
30 .
The common multiples are _
15 and June _
30 .
So, Tony and Amanda will work together on June _
Math
Talk
MATH
BOARD
Identify Relationships
Discuss how factors and
multiples are related. Give
an example.
1. Multiply to list the next five multiples of 4.
4 ,_
8 ,_
12 , _
16 , _
20 , _
24
_
1×4
3. 6
yes
_
4. 16
yes
_
5. 18
no
_
no
_
Is the number a multiple of 6? Write yes or no.
6. 3
7. 6
no
_
Share and Show Possible example: 5 is a factor
of 15. 15 is a multiple of 5.
Is the number a factor of 6? Write yes or no.
2. 3
3 EXPLAIN
MATHEMATICAL PRACTICES 7
8. 16
yes
_
9. 18
no
_
yes
_
© Houghton Mifflin Harcourt Publishing Company
Share
Share and
and Show
Show
MATH
BOARD
The first problem connects to the
learning model. Use the checked exercises
for Quick Check.
Quick Check
Quick Check
If
If
3
2
31
2
1
Rt I
Rt I
a student misses the checked
exercises
300
Then
4_MNLESE342224_C05L04.indd 300
27/02/14 3:12 PM
Visual
Partners
•Present this problem: Rosa makes different
breads on different days. She makes rolls
every third day, loaves every fourth day,
donuts every sixth day, and muffins every
eighth day. If she baked all of the items
today, in how many days will she bake all
the items on the same day again? 24 days
•Have students work in pairs to solve the
problem.
•Have students make up a problem similar to
the problem above. Have pairs of students
Differentiate Instruction with
• Reteach 5.4
• Personal Math Trainer 4.OA.B.4
• RtI Tier 1 Activity (online)
COMMON ERRORS
COMMON ERRORS
Error Students may confuse factors and
multiples.
Example 1 8
is a factor of 6.
3 is a multiple of 6.
Springboard to Learning Have students
use a number line to find multiples of a
number. Multiples of a number are equal to
or greater than the number. Remind students
that factors of a number are always less than
or equal to the number.
Lesson 5.4 300
Name
On
On Your
Own
If students complete the checked exercises
correctly, they may continue with the
remaining exercises.
For Exercise 14, after students list the next nine
multiples of each number, have them circle the
common multiples in each list before writing
the answer on the third line of the exercise.
MP8 Look for and express regularity in
repeated reasoning.
• How are the numbers in each list in
Exercises 15 and 16 related? Possible answer: in
Is the number a multiple of 3? Write yes or no.
10. 4
11. 8
no
_
12. 24
no
_
13. 38
yes
_
no
_
14. List the next nine multiples of each number. Find the common multiples.
4, 6, 8, 10, 12, 14, 16, 18, 20
Multiples of 2: 2, _______
16, 24, 32, 40, 48, 56, 64, 72, 80
Multiples of 8: 8, _______
8, 16
Common multiples: ______
MATHEMATICAL
PRACTICE
8
Generalize Algebra Find the unknown number.
48
15. 12, 24, 36, _
Exercise 15, the numbers are the ﬁrst 3 multiples of 12.
In Exercise 16, they are the ﬁrst 4 multiples of 25.
number in each list? Possible answer: I know that I
125
16. 25, 50, 75, 100, _
Tell whether 20 is a factor or multiple of the number.
Write factor, multiple, or neither.
17. 10
need to write the next multiple in each list.
18. 20
multiple
___
With Exercises 17–19, point out how factors
and multiples are related using more numerical
examples, if students are still having difﬁculty.
SMARTER
factor, multiple
___
neither
___
Write true or false. Explain.
20. Every whole number is a multiple of 1.
DEEPER
19. 30
21. Every whole number is a factor of 1.
True; for every number n, n × 1 = n.
False; 2 is not a factor of 1
SMARTER
Julio wears a blue shirt every
3 days. Larry wears a blue shirt every
4 days. On April 12, both Julio and Larry
wore a blue shirt. What is the next date that
they will both wear a blue shirt?
12
Sun
19
1
26
8
15
22
Sun
29
• What are ﬁve multiples of 1,000?
MP6 Attend to precision.
• Is 25 a multiple or a factor of 50? 25 is a factor
of 50 because 25 × 2 = 50.
SMARTER
For Exercises 20 and 21, students need to
recall that any number multiplied by 1 is
that number. This makes 1 a factor of every
number. Watch for students who confuse
multiples and factors.
For Exercise 22, students may ﬁnd more than
one common multiple of 3 and 4. They must
recognize that only the least of these multiples
will give them the ﬁrst date after April 12 that
Julio and Larry will both wear a blue shirt.
Math on the Spot
Video Tutor
Use this video to help students model and
solve this type of Think Smarter problem.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
301 Chapter 5
© Houghton Mifflin Harcourt Publishing Company
Possible answer: 1,000; 2,000; 3,000; 4,000; 5,000
22.
April 24
13 14April
15 16 17
Mon 21
Tue Wed
Thu 24
Fri
20
22 23
2 3 4 5 6
27 28 29 30 31
9 10 11 12 13
16 17 18 19 20
23September
24 25 26 27
Mon Tue Wed Thu Fri
30
1 2 3
18
Sat
25
7 14
21
28
Sat
4
Chapter 5 • Lesson 4
301
DO NOT EDIT--Changes must be made through "File info"
CorrectionKey=A
MATHEMATICAL PRACTICES
MODEL • REASON • MAKE SENSE
Problem
Problem Solving
Solving •• Applications
Applications
4 ELABORATE
Complete the Venn diagram. Then use it to
solve 23–25.
Factors
of 48
23. What multiples of 4 are not factors of 48?
2
20, 28, 32, 36, 40, 44
48
8
4
3
4, 8, 12, 16, 24, 48
Pose a Problem Look back at
Problem 24. Write a similar problem by changing
the numbers. Then solve.
DEEPER
Check students’ work.
40
12
20
MATHEMATICAL PRACTICES
28
MP4 Model with mathematics. Discuss
each part of the Venn diagram.
• What are the factors of 48? Where are
they shown? 1, 2, 3, 4, 6, 8, 12, 16, 24, 48; on
44
16
32
WRITE
Math
6
Problem Solving • Applications
36
24
1
24. What factors of 48 are multiples of 4?
25.
First Twelve
Multiples of 4
the left side and in the overlapping section
• List 12 multiples of 4 starting with 4.
Where are they shown? 4, 8, 12, 16, 20, 24,
26. Kia paid \$10 for two charms. The price of each charm was a
multiple of \$2. What are the possible prices of the charms?
28, 32, 36, 40, 44, 48; on the right side and in the
overlapping section
Possible answers: \$4 and \$6; \$8 and \$2
27.
• How can you describe the numbers in the
overlapping section? They are factors of 48 and
MATHEMATICAL
PRACTICE
7 Look for Structure The answer is 9, 18, 27, 36, 45.
What is the question?
multiples of 4.
Possible question: What are five multiples of 9?
28.
MP7 Look for and make use of structure.
• In Exercise 27, what is the relationship of
the numbers in the list? Possible answer: the
Math How do you know whether a number is a
WRITE
multiple of another number?
Possible answer: a number is a multiple of
numbers are the first 5 multiples of 9.
another number if it is divisible by the number.
29.
SMARTER
SMARTER
For numbers 29a–29e, select True or False for each
29a.
The number 45 is a multiple of 9.
True
False
29b.
The number 4 is a multiple of 16.
True
False
29c.
The number 28 is a multiple of 4.
True
False
29d.
The number 4 is a factor of 28.
True
False
29e.
The number 32 is a factor of 8.
True
False
© Houghton Mifflin Harcourt Publishing Company
statement.
Students who answer True to number 29b
do not understand the concept of a multiple.
These students may have recognized the pair
of numbers in this problem as part of a basic
fact and incorrectly assumed that must mean
that the number is a multiple of the other.
302
4_MNLESE342224_C05L04.indd 302
27/02/14 2:52 PM
DIFFERENTIATED INSTRUCTION
INDEPENDENT ACTIVITIES
5 EVALUATE Formative
Assessment
Essential Question
Using the Language Objective
Reflect Have students make a list to answer
the Essential Question.
How are factors and multiples related?
Differentiated Centers Kit
Games
Factor Farm
Activities
First One Out
Students complete
orange Activity Card 5
by using multiplication
to find numbers that
match given products.
Students practice
determining factors
of whole numbers.
Games
Possible answer: a factor times a factor is a product. The
product is a multiple of either factor.
Math Journal
WRITE
Math
Write a word problem that can be solved by
finding the numbers that have 4 as a factor.
Lesson 5.4 302
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A
Practice and Homework
Name
Lesson 5.4
Factors and Multiples
COMMON CORE STANDARD—4.OA.B.4
Gain familiarity with factors and multiples.
Is the number a multiple of 8? Write yes or no.
Practice and Homework
1. 4
Think: Since 4 × 2 = 8,
4 is a factor of 8, not a
multiple of 8.
Use the Practice and Homework pages to
provide students with more practice of the
concepts and skills presented in this lesson.
Students master their understanding as they
complete practice items and then challenge
their critical thinking skills with Problem
Solving. Use the Write Math section to
determine student’s understanding of content
for this lesson. Encourage students to use their
Math Journals to record their answers.
2. 8
3. 20
yes
no
4. 40
yes
no
List the next nine multiples of each number.
Find the common multiples.
8, 12, 16, 20, 24, 28, 32, 36, 40
5. Multiples of 4: 4, ___________
14, 21, 28, 35, 42, 49, 56, 63, 70
Multiples of 7: 7, ___________
28
Common multiples: __________
6, 9, 12, 15, 18, 21, 24, 27, 30
6. Multiples of 3: 3, ___________
18, 27, 36, 45, 54, 63, 72, 81, 90
Multiples of 9: 9, ___________
9, 18, 27
Common multiples: __________
Tell whether 24 is a factor or multiple of the number.
Write factor, multiple, or neither.
multiple
7. 6 ___
neither
8. 36 ___
factor
9. 48 ___
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
10. Ken paid \$12 for two magazines. The cost of
PROFESSIONAL
DEVELOPMENT
11. Jodie bought some shirts for \$6 each.
each magazine was a multiple of \$3. What
are the possible prices of the magazines?
Marge bought some shirts for \$8 each.
The girls spent the same amount of money
on shirts. What is the least amount they
could have spent?
\$3 and \$9; \$6 and \$6
12.
\$24
Math Write a word problem that can be solved by
WRITE
finding the numbers that have 4 as a factor.
Check students’ work.
Chapter 5
4_MNLESE342224_C05P04.indd 303
PROFESSIONAL
PROFESSIONAL
DEVELOPMENT
DEVELOPMENT
2/13/14 5:59 PM
CCSS.Math.Practice.MP6 Attend to precision.
Depth of understanding is exhibited when students can use precise and
appropriate mathematical language to describe mathematical concepts
In this lesson, students connect the concepts of factors and multiples,
and precision of language is vital. When having class discussions,
suggest to students that they think about whether they are using the
correct math vocabulary before they respond. Students can reference
the vocabulary definitions if needed.
303 Chapter 5
303
Ask questions such as the following to help students differentiate
factors and multiples:
• Are the factors of a number less than, equal to, or greater than
the number? Can you give an example? The factors are less than
or equal to the number. For example, 1, 2, 4, 5, 10, and 20 are all
factors of 20.
• How can we be sure that 10 is a factor of 20? Multiply 10 by
another number to see if the product is 20. 2 times 10 is equal to
20. So, 10 is a factor of 20.
• How do you find multiples of 20? Multiply 20 by counting
numbers starting with 1, 2, and 3. The answers are the multiples:
20, 40, 60. So, 20 is a multiple of 20.
Lesson Check (4.OA.B.4)
1. Of the numbers listed below, which are
NOT multiples of 4?
2. What number is a common multiple of
5 and 9?
Continue concepts and skills practice with
Lesson Check. Use Spiral Review to engage
students in previously taught concepts and
to promote content retention. Common Core
standards are correlated to each section.
2, 4, 7, 8, 12, 15, 19, 24, 34
2, 7, 15, 19, 34
Spiral Review (4.OA.A.3, 4.NBT.A.2, 4.NBT.B.4, 4.NBT.B.5)
tiles into a rectangular array of 4 rows. How
many tiles will be left over?
2 tiles
5. There are 18 rows of seats in the
auditorium. There are 24 seats in
each row. How many seats are in the
auditorium?
432 seats
4. Jerome added two numbers. The sum
was 83. One of the numbers was 45.
What was the other number?
38
6. The population of Riverdale is 6,735. What
is the value of the 7 in the number 6,735?
© Houghton Mifflin Harcourt Publishing Company
3. Jenny has 50 square tiles. She arranges the
700
FOR MORE PRACTICE
GO TO THE
304
Personal Math Trainer
Lesson 5.4
304
```