6.4
Model with Arrays
?
Essential Question How can you use arrays to model multiplication and
find factors?
Texas Essential
Knowledge and Skills
Number and Operations—3.4.D
Determine the total number of objects when equally-sized
groups of objects are combined or arranged in arrays up to
10 by 10
How can you use arrays to
model multiplication and
f ind factors?
3.4.E Represent multiplication facts by using a variety of
approaches such as equal-sized groups and arrays
Algebraic Reasoning—3.5.B Represent and solve
one- and two-step multiplication problems within 100 using
arrays and equations
Also 3.4.K
MATHEMATICAL PROCESSES
3.1.C Select tools, technology, and techniques
3.1.D Communicate mathematical ideas and reasoning
Are You Ready?
Access Prior Knowledge
Lesson Opener
Making Connections
Engage students in a discussion about multiplication. Ask the following questions.
What multiplication sentence can you write for 9 + 9? (2 × 9) What addition
sentence can you write for 4 × 3? (3 + 3 + 3 + 3) How is multiplication like addition?
(Possible answer: You can use the Commutative Property and the Associative Property
to find answers to multiplication and addition problems.)
Using the Digital Lesson
If you want, ask students to look at the image of the leaves lined up in rows and
columns. Ask students to discuss how this model makes it easier to solve the problem.
Learning Task
What is the problem the students are trying to solve? Connect the story to the problem.
• What question are you asked to answer? (Possible answer: how many leaves blew off
the tree)
• What is one way you could find out how many leaves were blown from the tree?
(Possible answer: by finding the sum of all the leaves)
• Suppose the wind blew 15 leaves from the tree. What addition sentence could
you write showing 3 equal groups of leaves? (5 + 5 + 5 = 15) What multiplication
sentence could you write for this addition sentence? (3 × 5 = 15)
Literacy and Mathematics
• Have students work in pairs and create a story problem that involves multiplying
2 whole numbers to find the solution. Have them write the multiplication sentence
as a repeated addition sentence and explain why the 2 equations are equivalent.
• Have students discuss the 4 seasons of a year and name the season when trees
typically drop their leaves. Ask students if they have ever collected fallen leaves.
Use the Are You Ready? 6.4 in the
Assessment Guide to assess students’
understanding of the prerequisite skills
for this lesson.
Vocabulary
array
Go to Multimedia eGlossary at
thinkcentral.com
Materials
square tiles, MathBoard
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 6.4 191A
Name
6.4
?
Unlock the Problem
What do many people grow in gardens? Read the
problem to find out.
Model with Arrays
Essential Question
Algebraic Reasoning—3.5.B
Also 3.4.K
MATHEMATICAL PROCESSES
3.1.C, 3.1.D
How can you use arrays to model multiplication
and find factors?
Unlock
Unlock the
the Problem
Problem
Be sure students understand that rows go across.
Activity 1
Number and
Operations—3.4.D, 3.4.E
Many people grow tomatoes in their gardens. Lee plants 3 rows
of tomato plants with 6 plants in each row. How many tomato
plants are there?
Hands
On
Hands
On
Activity 1
This activity introduces students to an array as
another way to represent multiplication.
Materials ■ square tiles ■ MathBoard
• You make an array by placing the same number of tiles in
each row. Make an array with 3 rows of 6 tiles to show the
tomato plants.
Work through the activity with students.
• How many tiles will you put in each row? 6 tiles
How many rows do you have? 3 rows
• Now draw the array you made.
• What method did you use to find the product?
Possible answer: I skip counted 3 times by the number in
each row.
Check students’ drawings.
• What if Lee planted 7 tomato plants in each row,
instead of 6? How would your array be different?
Each row of the array would have 7 tiles. I’d have to add 1
tile to each row.
No; possible explanation: if you only turn the
tiles, the number stays the same because
you do not add or take away any of the tiles.
• How could you find the new total number of
tiles? Possible answers: I know 3 × 6 = 18. I added 1
more tile to each of 3 rows, so I can add 3 to 18 to get 21;
I can multiply 3 × 7 = 21.
Mathematical Processes
Use Math Talk to focus on students’ understanding
of the proper format of rows and columns. The
first factor tells the number of rows and the second
factor tells the number in each row. So, turning
the array helps introduce the concept of the
Commutative Property of Multiplication.
English Language Learners
Leveled Activities
ELPS
Beginning: Activity 8
2.I.4, 3.G.1, 3.H.3
Intermediate: Activity 3
2.D.2, 2.E.3, 3.F.2
Advanced: Activity 57
2.C.4, 3.D.2, 3.E
Advanced High: Activity 59
1.F, 3.E, 3.H.3, 4.C.3
Go to thinkcentral.com for the ELL Activity
Guide containing these leveled activities.
Multiply.
© Houghton Mifflin Harcourt Publishing Company
Math Talk
• Find the total number of tiles.
3
×
↑
number
of rows
Does the number of tiles
change if you turn the
array to show 6 rows
of 3? Explain.
↑
number
in each row
18 tomato plants.
So, there are _
ELL Language Support
Module 6 191
Visual
Small Group
ELPS 1.A.1, 3.B.3, 3.D.2
Strategy: Identify Relationships
• Students can learn the concept of an array by relating common objects
to arrays.
• Review the definition of array.
July
• Ask students to identify arrays in the
Sunday
Monday
Tuesday Wednesday Thursday
Friday
Saturday
classroom. Possible answer: desks in a row,
1 2 3 4 5
papers on a bulletin board, or the calendar
• Choose one array in the classroom,
and have students describe it using
this model: The calendar has 5 rows of
squares with 7 squares in each row.
191 Module 6
Math Talk
Mathematical Processes
18
6=_
6
13
20
27
7 8
14 15
21 22
28 29
9 10 11 12
16 17 18 19
23 24 25 26
30 31
Hands
On
Activity 2
Materials ■ square tiles ■ MathBoard
Activity 2
Use 8 tiles. Make as many different arrays as you can,
using all 8 tiles. Draw the arrays. The first one is done for you.
A
Hands
On
In this activity, students make arrays to show all the
factors of 8.
B
• What are all the factors of 8? 1, 2, 4, and 8
1 row of 8
1
8 rows of _
1×8=8
8×
C
D
2
4
_
rows of _
2
4
_
rows of _
2
4
_
×_
=8
4
2
_
×_
=8
1
_
• Which number in the multiplication sentences
represents the number of rows? the first number
=8
Share and Show
The first problem connects to the learning model.
Have students use the MathBoard to explain their
thinking.
Use the checked exercises for Quick Check. Students
should show their answers for the Quick Check on the
MathBoard.
4 different arrays using 8 tiles.
You can make _
3
2
Quick Check
1
Share
Share and
and Show
Show
1.
a student misses the checked exercises
IF
Complete. Use the array.
2 rows of _
10 = _
20
_
THEN
2 ×_
10 = _
20
_
Differentiate Instruction with
RtI Tier 1 Lesson 26
2.
3.
3 × 8 = 24
4 × 6 = 24
192
Enrich
COMMON ERRORS
C
E
Error
Students may incorrectly make an
array to solve a problem.
Example 3 × 7 =
There are 24 tiles in the array.
Spatial
Individual
Materials: square tiles, 1-Centimeter Grid Paper (see eTeacher Resources)
• Write the following numbers on the board
8
© Houghton Mifflin Harcourt Publishing Company
Write a multiplication sentence for the array.
12
16
Springboard to Learning Have students circle
the second factor in a multiplication sentence
to remind them that this number represents the
number in each row. Then have them draw this
number of tiles in the first row and use the first
number to determine how many rows to draw.
20
• Have students choose one of the numbers and make as many different
arrays as they can with that number of squares.
• Then have students record the arrays on the grid paper and write the
corresponding multiplication sentence for each array.
Go to Go to thinkcentral.com for additional enrichment
activities in the Enrich Activity Guide.
Lesson 6.4
192
Name
Problem
Problem Solving
Solving
Problem Solving
Mr. Bloom’s Garden
Use the table to solve 4–6.
Have students read Problem 4. Ask them to describe
how they will solve the problem.
4.
For Problem 6, students must analyze the data in the
table and list the vegetables in Mr. Bloom’s garden in
order from greatest number of plants to least number
of plants.
Representations Mr. Bloom grows
vegetables in his garden. Draw an array
and write the multiplication sentence to
show how many corn plants Mr. Bloom
has in his garden. Check students’ drawings.
Vegetable
Planted In
Beans
4 rows of 6
Carrots
2 rows of 8
Corn
5 rows of 9
Beets
4 rows of 7
5 × 9 = 45; 45 corn plants
Problems
5.
Problems 5 and 7 require students to extend their
thinking to find different ways to arrange the plants
in Mr. Bloom’s garden.
Write Math
Could
Mr. Bloom have planted his carrots
in equal rows of 4? If so, how many
rows could he have planted? Explain.
Yes; possible explanation: he has 16 carrot
Write Math
Go Deeper
plants because 2 × 8 = 16. Skip count
Be sure students understand that arrays are made of
rows of objects with the same number of objects in
each row.
by 4s to see if you get 16: 4, 8, 12, 16. You
Show Your Work
count 4 times, so he could plant 4 rows.
The first factor tells the number of rows, and the
second factor tells the number of objects in each row.
6.
• Can you think of any objects around your house
that look like arrays? Possible answers: egg cartons,
Multi-Step List the vegetables in order
by the number of plants. Put the names
in order from greatest to least.
corn 45; beets 28; beans 24; carrots 16
muffin tins, 6-pack of juice, 8-pack of cereal, shoe rack
Have students sketch a picture and write a
multiplication sentence for each real-world example.
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 the H.O.T. problems, students
will build skills needed in the TEXAS assessment.
© Houghton Mifflin Harcourt Publishing Company
M
Math
on the Spot
Video Tutor
V
7.
Mr. Bloom has 12 strawberry
plants. Describe all of the different
arrays that Mr. Bloom could make using
all of his strawberry plants.
2 rows of 6; 6 rows of 2; 1 row of 12;
12 rows of 1; 3 rows of 4; 4 rows of 3
Module 6 • Lesson 4 193
3
RtI Tier 1 Lesson 26
2
1
Math on the Spot videos are in the
Interactive Student Edition and at
thinkcentral.com.
Enrich 29
Name
Name
LESSON
26
3.4.D, 3.4.E
column
row
1.
Yellow mums
2 3 8 array =
16
mums
2.
Purple pansies
6 3 4 array =
24
pansies
3.
Pink begonias
5 3 5 array =
25
begonias
4.
Orange marigolds
3 3 7 array =
21
marigolds
This array has 2 rows and 5 columns.
This array has 5 rows and 2 columns.
Count by 5s.
Count by 2s.
2 rows of 5 are 10.
5 rows of 2 are 10.
5.
White petunias
3 3 4 array =
12
petunias
The multiplication sentence is
2 × 5 = 10.
The multiplication sentence is
5 × 2 = 10.
6.
Blue tulips
3 3 8 array =
24
tulips
Write a multiplication sentence for the array.
2.
1.
© Houghton Mifflin Harcourt Publishing Company
Garden Arrays
Ed’s Garden Nursery is displaying new flowers. Each type of
flower is arranged in an array. Use the clues to label each part of
the flower display. Then find the number of each type of flower.
Color the display.
An array is a set of objects arranged in rows and columns.
Write a multiplication sentence for each array.
Possible sentences
are shown.
CLUES
mums
yellow
marigolds
orange
tulips
blue
3
×
4
=
12
3.
4
×
4
=
7.
×
3
=
15
Number and Operations
begonias
pink
petunias
white
4.
5
pansies
purple
16
8.
193 Module 6
Enrich 29
1
Model with Arrays
OBJECTIVE Use arrays to model products and factors.
3
×
6
=
Check
students’ array.
Add a 2 3 5 array for a red rose garden. Label it and color it red.
Explain how you decided where each type
of flower belonged.
18
Possible answer: I looked at the rows and how many were
in each row and matched it with an array for a flower.
51
Enrich
© Houghton Mifflin Harcourt Publishing Company
E29
Mathematical Processes
Model ¥ Reason ¥ Communicate
Daily
Daily Assessment
Assessment Task
Task
3
Fill in the bubble for the correct answer choice.
8.
9.
Daily Assessment Task
Jacob made this quilt. It has 5 rows of blocks with
6 blocks in each row. Which multiplication sentence
shows the number of blocks in the quilt?
A
4 × 5 = 20
C
5 × 6 = 30
B
6 × 6 = 36
D
5 × 5 = 25
2 rows of 4
C
1 row of 5
B
2 rows of 3
D
3 rows of 3
THEN
IF
NO
•
YES
•
3 × 7 = 21
B
4 × 3 = 12
C
4 × 4 = 16
D
4 × 7 = 28
Enrich 29
Homework and Practice
Lesson 6.4
Multi-Step Ashley and
Daniel each have a garden
with flowers planted in
rows. Which multiplication
sentence shows the number
of flowers in both gardens?
A
Soar to Success Math
Warm-Up 12.35
•
10.
1
Can students use arrays to model multiplication
and find factors?
Apply Karen has 6 animal posters. Which describes
one array Karen could make using all of her posters?
A
2
TEXAS Test Prep Coach
Ashley’s Garden
Test Prep Coach helps teachers to identify common
errors that students can make.
Daniel’s Garden
In the Test Prep exercise, if students selected:
A or C They miscounted the number in each row.
B They counted only 1 row.
TEXAS Test Prep
What multiplication sentence does this array show?
A
2×3=6
C
2 × 5 = 10
B
4×1=4
D
2×4=8
194
© Houghton Mifflin Harcourt Publishing Company
11.
?
Essential Question
Write
Math
How can you use arrays to model multiplication and
find factors? Possible answer: the number of rows is the first
factor, and the number in each row is the second factor.
Differentiated Centers Kit
Games
Games
Multiplication
Bingo
Students practice
multiplication
facts through 10.
Literature
Collections Times Four
Students read the book
and determine how to use
multiplication to find the
total number of objects in
each collection.
Activities
Hurray for Arrays!
Students complete
blue Activity
Card 15 by using
arrays to model
multiplication facts.
Lesson 6.4
194
5
Ho mewo rk
and Practice
6.4
Number and Operations—3.4.D, 3.4.E
Algebraic Reasoning—3.5.B Also 3.4.K
MATHEMATICAL PROCESSES 3.1.C, 3.1.D
Name
TEXAS Test Prep
Lesson
Lesson Check
Check
8.
Ryan makes this array. Which
equation does Ryan’s array show?
9.
Ahmad makes this array. Which
equation does Ahmad’s array show?
Model with Arrays
Write a multiplication sentence for the array.
1.
2.
3 × 5 = 15
4 × 7 = 28
A
3 × 7 = 21
A
2+6=8
B
3 + 7 = 10
B
6+2=8
C
3 × 6 = 18
C
2 × 6 = 12
D
3 × 7 = 20
D
2 × 6 = 26
Use the table to solve 3–5.
Draw an array and write a
multiplication sentence to show
how many science books are on
Anna’s bookshelves.
Anna’s Bookshelves
Kind of
Book
Check students’ drawings.
10.
Number of
Rows
Write a number sentence to show how
many reading books are on Anna’s
bookshelves.
5.
3 rows of 4
A
3 rows of 6
A
28
2 rows of 8
B
2 rows of 9
B
12
Reading
4 rows of 5
C
4 rows of 6
C
57
D
5 rows of 3
D
35
Describe three different arrays that
Anna can make to show how many art
books she has.
12.
8 rows of 2; 4 rows of 4; 1 row of 16
© Houghton Mifflin Harcourt Publishing
Problem
Problem Solving
Solving
6.
Sasha makes an array of tiles. She has
3 rows of 9 tiles. Write a multiplication
sentence to show Sasha’s array.
7.
Dexter makes an array of tiles. He has
5 rows of 6 tiles. Write a multiplication
sentence to show Dexter’s array.
3 × 9 = 27
Module 6 • Lesson 4
Use the Homework and Practice pages to provide
students with more practice on the concepts and
skills of this lesson.
Module 6
195
Multi-Step DeShawn makes an
array using 3 rows of 5 tiles. How
many tiles does Deshawn have if he
adds 2 more rows to the array?
A
15
B
9
C
D
5 × 6 = 30
Homework and Practice
195-196
Keesha makes an array of 5 rows of
7 stickers. How many stickers does
Keesha have?
Art
Possible answers are given.
4 × 5 = 20; 20 books
11.
Science
3 × 4 = 12; 12 books
4.
Becky has 24 stamps. Which
describes one array Becky can
make using all of the stamps?
196
20
25
13.
Multi-Step Ming makes an array
using 2 rows of 7 tiles. She adds
3 more rows to the array. Which
multiplication sentence does
Ming’s array show?
A
2 × 7 = 14
B
3 × 7 = 21
C
5 × 7 = 35
D
9 × 3 = 27
© Houghton Mifflin Harcourt Publishing Company
3.