LESSON
5.1
Model Factors
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
Learning Objective
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.
Find all the factors of a number by using
models.
MATHEMATICAL PRACTICES
MP1 Make sense of problems and persevere in solving them.
MP4 Model with mathematics.
Students complete a sentence frame, Models
can be used to find factors
by __________.
F C R Coherence:
Standards Across the Grades
Before
Grade 4 After
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
Language Objective
Materials
MathBoard, square tiles, grid paper
(see eTeacher Resources)
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 277H.
About the Math
Professional Development
Why Teach This?
Knowing how to factor is a building block of algebraic
relationships. Students need to internalize the concept of
factors so that algebraic generalizations can be built upon. It
is important for students to understand how to find factors
before they learn how to find the greatest common factor,
which is important in performing operations using fractions
with unlike denominators.
Teaching factors also provides a review of multiplication.
Students will use many multiplication facts as they break
apart numbers into factors.
Factoring can give students insight into numbers; for
example, thinking of 100,000 as 10 × 10 × 10 × 10 × 10
gives some meaning to the number. Factoring is one of
the most important skills required for success in Algebra 1.
Professional Development Videos
279A
Chapter 5
Interactive Student Edition
Personal Math Trainer
Math on the Spot
Animated Math Models
iTools: Counters
HMH Mega Math
1 ENGAGE
Daily Routines
Common Core
Problem of the Day 5.1
with the Interactive Student Edition
Essential Question
How can you use models to find factors?
Sarah has a jar that holds 90 quarters. She
adds 5 quarters a week to the jar. The jar is
full. For how many weeks has Sarah added
quarters to the jar? 18 weeks
Making Connections
Vocabulary factor
• What are the factors in the multiplication sentence 40 = 8 × 5?
8 and 5
™Interactive Student Edition
™Multimedia eGlossary
Encourage students to share what they know about arrays.
• What is factor? A number multiplied by another number to find
a product.
Learning Activity
What is the problem the students are trying to solve? Connect the
story to the problem. Ask the following questions.
Fluency Builder
Common Core Fluency
Standard 4.OA.B.4
Materials 18 two-color counters for each pair
Have students work in pairs. Each student
places some two-color counters on a desk or
table and notes the number of red and the
number of yellow counters. For example,
student 1: 5 red, 4 yellow counters; student
2: 2 red, 7 yellow counters.
Students then work together to find all
possible products using their four numbers
as factor pairs. Possible answer: 5 × 2 = 10,
5 × 4 = 20, 5 × 7 = 35, 4 × 2 = 8, 4 × 7 = 28,
2 × 7 = 14
• Name a multiplication sentence in which the product is 18. Possible
answers: 2 × 9 = 18, 6 × 3 = 18, 1 × 18 = 18
• What would an array look like for each of these multiplication
sentences? Possible answers: 2 rows with 9 tiles in a row; 6 rows with
3 tiles in a row; 1 row with 18 tiles in the row.
Literacy and Mathematics
View the lesson opener with the students. Then, choose one or more
of the following activities.
• Have a number of students act out arranging themselves into an
array. Have the remaining students write a word problem in which
the multiplication equation represented by the array is used to
solve the problem.
• Have students arrange 20 counters in an array and write the
multiplication equation represented by the array. Have students
share the different equations they wrote.
How can you use
models to f ind factors?
Lesson 5.1
279B
LESSON
5.1
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
DO
NOT
must be
made through
"File info"
each
of EDIT--Changes
its factors. Determine
whether
a given whole
number in the range 1–100 is a multiple of a given one-digit number.
CorrectionKey=B
Determine whether a given whole number in the range 1–100 is prime or composite.
Unlock the Problem
Model Factors
Unlock
Unlock the
the Problem
Problem
Read the definition of factor with students.
Activity
A factor is a number multiplied by another number
to find a product. Every whole number greater than
1 has at least two factors, that number and 1.
MP4 Model with mathematics. Guide
students to make arrays using 24 tiles. Begin by
having students place the 24 tiles in one row.
• How can arrays be used to find the factors
of a number? If the array for a number forms a
18 = 1 × 18
↑
factor
16 = 4 × 4
Use all 24 tiles to make as many different arrays as you
can. Record the arrays in the grid, and write the factors
modeled.
Write on the board 8 × 7 = 56.
• Explain that students will draw an array to
model this problem. When we draw an array,
first, you need to identify the two factors.
•Guide students to complete the task using
their language proficiency level:
Beginning: Yes/no: Are 8 and 7
factors of 56?
Intermediate: List the factors of 56.
Advanced: Describe how you would draw
an array to model 8 × 7 = 56.
279 Chapter 5
© Houghton Mifflin Harcourt Publishing Company
2 × 12 = 24
2 ,_
12
Factors: _
1
24 = 24
_
×_
3
8
_
×_
= 24
4 ×_
6
_
= 24
1 ,_
24
Factors: _
3 ,_
8
Factors: _
4 ,_
6
Factors: _
The factors of 24, from least to greatest, are
1 ,_
2 ,_
3 ,_
4 ,_
6 ,_
8 ,_
12 , and _
24 .
_
Two factors that make a product are sometimes called a factor
pair. How many factor pairs does 24 have? Explain.
Math
Talk
4; the factor pairs are 1 and 24, 2 and 12, 3 and 8, 4 and 6.
Yes; possible explanation: you can show the
arrays as 12 rows of 2, 8 rows of 3, 6 rows of 4,
and 24 rows of 1.
MATHEMATICAL PRACTICES 2
Reason Abstractly Can
you arrange the tiles
in each array another
way and show the same
factors? Explain.
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Lesson 5.1
Reteach
Name Scaffold Language
When you are asked to find
factors of a whole number,
only list factors that are also
whole numbers.
Activity Model and record the factors of 24.
Have the students place the 24 tiles in
rows of 2, 3, 4, 5, 6, and so on and record
their arrays in the grid. Remind students
that a rectangle must be formed for the two
numbers to be factors and that a rectangular
array has the same number of tiles in each row
and the same number of tiles in each column.
16 = 2 × 8
Materials ■ square tiles Possible answers are given.
number itself
ELL Strategy:
342 = 1 × 342
factor
16 = 1 × 16
• Every whole number greater than 1 has
at least how many factors? 2; 1 and the
could have the same number of rows as columns. For
example, an array of 16 tiles could have 4 rows of 4.
Both factors are 4.
7=7×1
↑
Many numbers can be broken into factors in different ways.
rectangle, then the number of tiles in each row and in
each column are factors of the product.
Use Math Talk to help students
recognize the Commutative
Property of Multiplication.
The Commutative Property allows students to
model two different arrays for each factor pair.
However, students should keep in mind that
this factor pair should be counted only once.
MP2 Reason abstractly and
quantitatively.
• Could an array model two factors that are
the same? Explain. Possible answer: yes, an array
Operations and Algebraic
Thinking—4.OA.B.4
MATHEMATICAL PRACTICES MP2,
MP4, MP6
Essential Question How can you use models to find factors?
MATHEMATICAL PRACTICES
Math
Talk
Lesson 5.1
Name
Lesson 5.1
Enrich
Name
Festive Factors
Model Factors
Ms. Ramirez is a professional party planner. One of her tasks is
to arrange the seating at tables. Ms. Ramirez likes to have the
same number of party guests seated at each table.
Use tiles to find all the factors of 25. Record the
arrays and write the factors shown.
For each number of guests below, use factors to determine
all the ways Ms. Ramirez can arrange tables and chairs to
have the same number of guests at each table. You do not
have to include the factor 1 and the number itself.
Step 1 Record the array and list the
1 3 25 5 25
MXEFL11AWK4X_RW_C07_L01_Art_01.ai
factors.
Factors: 1 , 25
Think: Every whole number greater than 1
has at least two factors, that number and 1.
Step 2 Make an array to see if 2 is a factor
of 25.
1.
each, 4 tables with 6 chairs each, 6 tables with
So, 2 is not a factor of 25.
MXEFL11AW4X_RW_C07_L01_Art_02
Step 3 Continue making arrays, counting by 1, to find all the other factors of 25.
Is 3 a factor?
4 chairs each, 8 tables with 3 chairs each, 12 tables
Is 4 a factor?
with 2 chairs each
4 rows, 1 tile left
No, 4 is not a factor of 25.
3 rows, 1 tile left
No, 3 is not a factor of 25.
MXEFL11AWK4X_RW_C07_L01_Art_03
24 guests
2 tables with 12 chairs each, 3 tables with 8 chairs
You cannot use all 25 tiles to make an
array that has 2 rows. There is 1 tile left.
Think: An array has the same number of
tiles in every row and the same number of
tiles in every column.
Is 5 a factor?
Differentiated
Instruction
2.
2 tables with 28 chairs each, 4 tables with 14 chairs each,
MXEFL11AWK4X_RW_C07_L01_Art_04
5 rows, all tiles
used.
5 3 5 5 25
7 tables with 8 chairs each, 8 tables with 7 chairs each,
14 tables with 4 chairs each, 28 tables with 2 chairs each
There are the same number of tiles in each
row and column. Yes, 5 is a factor of 25.
If you continue to make arrays up to 24,
you will find there are no additional factors of 25.
3.
So, the factors of 25 are 1, 5, and 25.
MXEFL11AWK4X_RW_C07_L01_Art_05
Two factors that make a product are sometimes called a factor pair.
What are the factor pairs for 25? 1 and 25, 5 and 5
56 guests
Two factors that make a product are sometimes called a factor
pair. Describe how using factor pairs helped you solve the problems.
Possible answer: I know that the factor pairs can
be reversed. For example, 7 and 8 are factors of 56
Use tiles to find all the factors of the product. Record the
arrays and write the factors shown.
and can represent 7 tables with 8 chairs each, or 8
1. 35
tables with 7 chairs each.
Check students’ work.
1, 5, 7, 35
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
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1, 2, 3, 4, 6, 9, 12, 18, 36
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Chapter Resources
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Share
Share and
and Show
Show
3 EXPLAIN
MATH
Share and Show
BOARD
1. Use the arrays to name the factors of 12.
1
12 = 12
_
×_
3
4
_
×_
= 12
2 , 3, _
4 , 6, and _
12 .
The factors of 12 are 1, _
Use tiles to find all the factors of the product. Record the
arrays and write the factors shown. Check students’ work.
1, 5
2. 5: ____
BOARD
The first problem connects to the learning
model. As students work through the
exercises, ask:
• What number is always going to be listed
as a factor? 1
• If the number is even, what other number
do you know is a factor? 2
• When is the array a rectangle that is a
square? Possible answer: if the number of rows and
6
2
_
×_
= 12
Math
Talk
MATH
MATHEMATICAL PRACTICES 6
Use Math Vocabulary
Explain how the numbers
3 and 12 are related. Use
the word factor in your
explanation.
Possible explanation:
3 is a factor of 12.
columns are the same, then the number has 2 ­factors
that are the same.
Math
Talk
Use Math Talk to focus on
students’ understanding of the
vocabulary term factor.
• Of the numbers 3 and 12, which is a factor
of the other? Explain. Possible answer: 3 is a
1, 2, 4, 5, 10, 20
3. 20: ____
factor of 12 because 3 can be multiplied by 4 to get
the product 12.
Use the checked exercises for Quick Check.
© Houghton Mifflin Harcourt Publishing Company
1, 5, 25
4. 25: ____
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Advanced Learners
Visual
Individual
Challenge students to find the factors for greater
numbers.
• Find all the factors of 98. 1, 2, 7, 14, 49, 98
• Have students explain what would happen to
the factor list if 98 was doubled to 196. Possible
answer: I would double each of the original factors. If the
number doubled does not already show up in the list of
factors, it should be included.
1, 2, 4, 7, 14, 28, 49, 98, 196
• Have students find the factors of 130 and 260.
1, 2, 5, 10, 13, 26, 65, 130
1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260
Quick Check
Quick Check
3
2
1
3
2
1
Rt I
Rt I
a student misses the checked
exercises
If
If
Then
Differentiate Instruction with
• Reteach 5.1
• Personal Math Trainer 4.OA.B.4
• RtI Tier 1 Activity (online)
COMMON
COMMON ERRORS
ERRORS
Error Students may assume that all odd
numbers have only 1 set of factors, 1 and
the number.
Example In finding the factors of
15, students may give 1 and 15 as the
only factors.
Springboard to Learning Remind students
that odd numbers can be multiplied by odd
numbers to give another odd number.
Have students skip-count by 3s and 5s to
demonstrate that every other number they
say is odd.
Lesson 5.1 280
Name
On Your Own
On
On Your
Your Own
Own
If students complete the checked exercises
correctly, they may continue. For Exercises 5–8,
have students list factors for each number.
Practice: Copy and Solve Use tiles to find all the factors of the
product. Record the arrays on grid paper and write the factors
shown. Check students’ arrays.
5. 9
4 ELABORATE
9.
MP6 Attend to precision. Have students
read Exercise 9. Ask them to describe how they
would solve the problem. Review the steps to
find factors using tiles.
• How can you find other factor pairs? Possible
1, 2, 3, 6, 9, 18
MATHEMATICAL
PRACTICE
6 Pablo is using 36 tiles to make a patio. Can he
arrange the tiles in another way and show the same factors?
Draw a quick picture and explain.
factors of 36.
For Exercise 10, remind students that
rectangular arrays that show the same two
factors are not different arrays. One array is
the same shape as the other, but it is turned.
281 Chapter 5
8. 18
1, 17
the quick picture shows 4 rows of 9. Both show that 4 and 9 are
SMARTER
For Exercise 11 remind students that 6 can
be broken down into its own factors. These
factors of 6 will also be factors of any number
that 6 is a factor of.
For Exercise 12, help students recognize that
finding the factor pairs for 16 will solve the
problem. Some students may find all the factor
pairs, but omit 1 and 16. Explain that 1 and 16
should be included in their answers because Jean
can buy 1 shirt for $16 or 16 shirts for $1 each.
7. 17
Yes. Possible explanation: Pablo’s tiles show 9 rows of 4, and
answer: make an organized list or draw pictures.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
1, 3, 7, 21
Use the diagram for 9–10.
MATHEMATICAL PRACTICES
10.
SMARTER
How many different rectangular arrays can Pablo
make with all 36 tiles, so none of the arrays show the same factors?
5 rectangular arrays
© Houghton Mifflin Harcourt Publishing Company
Use this video to help students model and
solve this type of Think Smarter problem.
6. 21
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Problem Solving • Applications
Math on the Spot
Video Tutor
1, 3, 9
11. If 6 is a factor of a number, what other numbers must be factors of
the number?
1, 2, and 3
12.
DEEPER
Jean spent $16 on new T-shirts. If each shirt cost the same
whole-dollar amount, how many could she have bought?
1, 2, 4, 8, or 16 T-shirts
Chapter 5 • Lesson 1
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MATHEMATICAL PRACTICES
MODEL • REASON • MAKE SENSE
Unlock the Problem
Unlock
Unlock the
the Problem
Problem
13.
MATHEMATICAL PRACTICES
DEEPER
Carmen has 18 connecting cubes. She wants to model a house
shaped like a rectangle. If the model has a height of one connecting
cube, how many different ways can Carmen model the house using all
18 connecting cubes and none of the models show the same side lengths?
DEEPER
For Exercise 13, encourage students to use
a variety of methods to find the number of
different ways Carmen can construct a house.
Invite volunteers to present their solutions
with models or diagrams.
the number of ways that 18 connecting
a. What do you need to know? _________
cubes can make a rectangle 1 cube high
b. How is finding the number of ways to model a rectangular house
Possible answer: each model will have 18 connecting
related to finding factor pairs? _________
cubes and will be shaped like a rectangle. The length and width of each model will
SMARTER
show a factor pair for 18.
Students who answer Yes to part 14e may
incorrectly be adding the dimensions of
the rectangle rather than finding a pair of
numbers to represent the dimensions of the
rectangle that have a product of 40.
answer:
c. Why is finding the factor pairs only the first step in solving the problem? Possible
___
the problem asks for the number of different ways Carmen can make a model. After
finding the factor pairs, I must count them to solve the problem.
d. Show the steps you used to solve
the problem.
e. Complete the sentences. Factor pairs for
18 are 1 and 18, 2 and 9, 3 and 6.
should demonstrate an understanding of
using the factors of 18 to make different
arrangements of connecting cubes.
14.
5 EVALUATE Formative
Assessment
3
There are _
different ways Carmen
can arrange the cubes to model the
house.
SMARTER
Sarah was organizing vocabulary words using index cards. She arranged
40 index cards in the shape of a rectangle on a poster. For 14a–14e, choose Yes or No to tell
whether a possible arrangement of cards is shown.
14a.
4 rows of 10 cards
Yes
No
14d. 40 rows of 1 card
Yes
No
14b.
6 rows of 8 cards
Yes
No
14e. 35 rows of 5 cards
Yes
No
14c.
20 rows of 2 cards
Yes
No
Essential Question
© Houghton Mifflin Harcourt Publishing Company
Students’ steps will vary. Students
Using the Language Objective
Reflect Have students complete a sentence
frame, Models can be used to find factors
by _____, to answer the Essential Question.
How can you use models to find factors?
Possible answer: I can use square tiles and try to arrange
the required number of tiles into rectangles.
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DIFFERENTIATED INSTRUCTION
INDEPENDENT ACTIVITIES
Math Journal
WRITE
Math
Have students write the answer to the
Essential Question and draw examples to
explain their answer.
Differentiated Centers Kit
Activities
What’s My Fact?
Activities
Flowering Factors
Games
Factor Farm
Games
Students complete
purple Activity
Card 3 by guessing
a factor pair when
given a product.
Students complete
orange Activity
Card 17 by
identifying the
factors of whole
numbers.
Students practice
determining
factors of whole
numbers.
Lesson 5.1 282
Practice and Homework
Lesson 5.1
Name
Model Factors
COMMON CORE STANDARD—4.OA.B.4
Gain familiarity with factors and multiples.
Check students’ work.
Use tiles to find all the factors of the product.
Record the arrays on grid paper and write the factors shown.
1. 15
2. 30
3. 45
Practice and Homework
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.
1 × 15 = 15
1, 2, 3, 5, 6,
3 × 5 = 15
1, 3, 5, 15
10,15, 30
5. 40
6. 36
1, 2, 4, 5, 8,
10, 20 40
1, 3, 5, 9, 15, 45
7. 22
1, 2, 3, 4, 6, 9, 12,
4. 19
1, 19
8. 4
1, 2, 11, 22
1, 2, 4
18, 36
Problem
Problem Solving
Solving
© Houghton Mifflin Harcourt Publishing Company
9. Brooke has to set up 70 chairs in equal rows
11.
10. Eduardo thinks of a number between
for the class talent show. But, there is not
room for more than 20 rows. What are the
possible number of rows that Brooke could
set up?
1 and 20 that has exactly 5 factors.
What number is he thinking of?
1, 2, 5, 7, 10, or 14 rows
______
16
______
Math Have students write the answer to the Essential
WRITE
Question and draw examples to explain their answer.
Check students’ work.
283
© Houghton Mifflin Harcourt Publishing Company
Chapter 5
283 Chapter 5
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CorrectionKey=B
Lesson Check (4.OA.B.4)
1. List all the factors of 24.
2. Natalia has 48 tiles. Write a factor pair for
the number 48.
1, 2, 3, 4, 6, 8, 12, 24
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.
Possible answer: 6 and 8
Spiral Review (4.OA.A.1, 4.NBT.B.5, 4.NBT.B.6)
3. The Pumpkin Patch is open every day. If it
sells 2,750 pounds of pumpkins each day,
about how many pounds does it sell in
7 days?
4. What is the remainder in the division
problem modeled below?
Possible answer: about 21,000 pounds
2
multiplication equation.
6. Channing jogs 10 miles a week. How many
miles will she jog in 52 weeks?
4 × 5 = 20
520 miles
© Houghton Mifflin Harcourt Publishing Company
5. Represent the model shown below using a
FOR MORE PRACTICE
GO TO THE
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Lesson 5.1 284