Multiplication Arrays
Objective To provide opportunities to use arrays, multiplication/
division diagrams, and number models to represent multiplication
d
number stories.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Key Concepts and Skills
Measuring Perimeters of Polygons
• Use basic facts to solve multiplication
number stories. Math Journal 1, p. 82
ruler
Children measure perimeters
of polygons and draw figures
with given perimeters.
[Operations and Computation Goal 3]
• Use arrays to model multiplication. [Operations and Computation Goal 6]
• Write number sentences to model
number stories. [Patterns, Functions, and Algebra Goal 2]
• Model the turn-around rule for multiplication
(Commutative Property of Multiplication)
using an array model. [Patterns, Functions, and Algebra Goal 4]
Key Activities
Curriculum
Focal Points
Math Boxes 4 2
Math Journal 1, p. 83
Children practice and maintain skills
through Math Box problems.
Home Link 4 2
Math Masters, p. 87
Children practice and maintain skills
through Home Link activities.
Children use arrays to represent multiples of
equal groups. Children solve multiplication
number stories using arrays, multiplication/
division diagrams, and number models.
Differentiation Options
READINESS
Building Arrays
Math Masters, pp. 88 and 416
centimeter cubes or other counters 1 six-sided die
Children build arrays.
ENRICHMENT
Exploring Square Numbers
Math Masters, pp. 89 and 416
per group: centimeter cubes or pennies
(or both), tape
Children explore square arrays.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 132
Children add the term array to their Math
Word Banks.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 81. [Operations and Computation Goal 6]
Key Vocabulary
array factor product
Materials
Math Journal 1, p. 81
Home Link 41
transparency of Math Masters, pp. 406 and
419 (optional) per person: 40 counters slate number grid calendar number line
(optional) calculator (optional)
Advance Preparation
Gather a large supply of pennies or other counters. Choose a place for an Arrays Exhibit. See Home
Link 42 for details. Display the Guide to Solving Number Stories on Math Masters, page 406 and a
multiplication/division diagram.
Teacher’s Reference Manual, Grades 1–3 pp. 90–92
248
Unit 4
Multiplication and Division
Interactive
Teacher’s
Lesson Guide
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
Mental Math and
Reflexes
Have children count chorally. They
may use number grids, number lines,
calculators, or calendars as needed.
Skip count forward and backward
by 2s, 5s, and 10s.
Skip count forward and backward
by 6s.
Skip count forward and backward
by 8s.
3.OA.1, 3.OA.3, 3.OA.4, 3.OA.5, 3.OA.7, 3.MD.7a, 3.MD.7b, 3.MD.8
Math Message
Home Link 4 1
Follow-Up
There are 24 trombone
players in a big parade.
Ask several children to share the
multiples of equal groups they found.
They will share some of their stories
later in this lesson.
Use counters to represent trombone
players. Arrange the counters to show
them in equal rows.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Have a volunteer draw an array on the board showing how he
or she arranged the counters. Use circles or Xs to represent
trombone players.
4
24 = 4 × 6
4 × 6 = 24
4
×6
24
6
Ask others to draw different arrangements on the board. Explain
that the pictures show the number of trombone players in arrays.
An array is a group of objects arranged in rows and columns.
Each row has the same number of objects, and each column has
the same number of objects. Arrays provide a way to model
multiplication. Possible arrays: 4-by-6, 6-by-4, 3-by-8, 8-by-3,
2-by-12, 12-by-2, 1-by-24, and 24-by-1
rows
trombones
per row
trombones
in all
4
6
24
1. Draw an array on the board representing 4 rows of trombone
players with 6 players in each row.
2. Display a multiplication/division diagram, and have a
volunteer fill in the known quantities.
3. Write 24 in the bottom right box.
4. Call attention to the fact that the numbers do not have
the same units or labels, as they do in addition and
subtraction stories.
5. Ask a volunteer to write a number model for this array:
4 × 6 = 24.
6. Next, draw an array to represent 6 rows of trombone players
with 4 players in each row.
6
24 = 6 × 4
6 × 4 = 24
6
×4
24
4
rows
trombones
per row
trombones
in all
6
4
24
Lesson 4 2
249
Repeat Steps 1 through 5: Fill in a multiplication/division diagram
and write a number model. Compare the two arrays, emphasizing
that although the total number of players is the same in each
array, the number of rows and the number of players per row are
reversed. Remind children of the turn-around rule for
multiplication (Commutative Property of Multiplication), which
states that two numbers can be multiplied in either order without
changing the product: 4 × 6 = 6 × 4
Solving Multiplication
Number Stories
Using Arrays
WHOLE-CLASS
ACTIVITY
PROBLEM
PRO
PR
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SO
S
SOLVING
OL
O
LV
VING
VIN
IIN
NG
(Math Masters, p. 419)
Adjusting
the Activity
ELL
4 boxes of crayons, 8 crayons per box
Hold up two egg cartons to provide a visual
for 24 eggs in 2 cartons or 2 dozen eggs.
AUDITORY
KINESTHETIC
Children draw arrays on their slates to solve simple multiplication
number stories. Pose multiplication number stories in which
arrays are obvious, such as the following.
TACTILE
VISUAL
6 shoe boxes, 1 pair of shoes per box
5 rows of chairs, 3 chairs per row
5 shelves of books, 8 books per shelf
2 cartons of eggs, 12 eggs per carton
NOTE A factor is a number that is multiplied
by another number. A product is the result of
multiplying factors. It is beneficial to use these
words when discussing multiplication number
stories, but do not insist that children use them.
Teaching Aid Master
Name
Date
Time
Multiplication/Division Diagram
in all
per
in all
per
For each story:
1. Write a brief summary of the story on the board.
2. Fill in a multiplication/division diagram, going over the
number of rows and the number of items per row
(the factors) and the total number of things (the product) as
children tell you what information to enter into the diagram.
To support English language learners, distinguish between the
common meaning and mathematical meaning of product.
3. Ask a volunteer to write a number model for the story on the
board while the other children write number models on their
slates. Children may use either a ? or a letter variable to
represent the unknown quantity.
4. Have children draw arrays on their slates or arrange counters
on their desks to model the story. As they do this, have a
volunteer draw an array on the board or arrange counters on
the overhead projector.
5. Have children solve the problem and share their solution
strategies. Remind them to include the unit in the answer.
in all
per
Math Masters, p. 419
250
Unit 4 Multiplication and Division
6. Ask children to discuss the reasonableness of their answers
and whether their answers make the number model true.
Finally, have children write summary number models on their
slates for the story.
Representing Multiplication
WHOLE-CLASS
ACTIVITY
Student Page
Date
Situations with Arrays
Time
LESSON
Have children share stories from Home Link 4-1 about multiples
of equal groups. Follow steps 1–5 from the previous activity. Make
the point that arrays can be used to represent multiplication
situations, even when the real-life groups do not look like arrays.
Arrays are easy to draw, and they simplify problem solving.
More Multiplication Number Stories
42
(Math Masters, p. 419)
Fill in the multiplication/division diagram. Write a number model.
Make an array with counters. Mark the dots to show the array.
Find the answer. Write the unit with your answer.
1. Mrs. Kwan has 3 boxes of scented markers. Each box has 8 markers.
How many markers does she have?
boxes
markers
per box
markers
in all
3
8
?
3×8=?
Number model:
Answer:
24 markers
(unit)
2. Monica keeps her doll collection in a case with 5 shelves. On each
Solving More
shelf there are 6 dolls. How many dolls are in Monica’s collection?
PARTNER
ACTIVITY
Multiplication
Number Stories
shelves
dolls
per shelf
dolls
in all
5
6
?
PROBLEM
PRO
PR
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VIN
ING
5×6=?
Number model:
30 dolls
Answer:
(unit)
3. During the summer Jack mows lawns. He can mow 4 lawns per day.
How many lawns can he mow in 7 days?
(Math Journal 1, p. 81; Math Masters, p. 406)
Partners or individuals make arrays and use diagrams to solve
multiplication number stories. Children may use counters if they
wish. Circulate and ask guiding questions such as those on the
Guide to Solving Number Stories (Math Masters, page 406).
days
lawns
per day
lawns
in all
7
4
?
7×4=?
Number model:
28 lawns
Answer:
(unit)
Math Journal 1, p. 81
79-101_EMCS_S_SMJ_G3_U04_576353.indd 81
1/26/11 9:11 AM
When reviewing the answers to the journal page, ask follow-up
questions so children can practice more difficult multiplication
facts. For example:
●
Suppose Mrs. Kwan has 7 boxes of markers. How many
markers would she have now? 56 markers
●
If Monica had 60 dolls, how many shelves would she need?
10 shelves
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 81
Use journal page 81 to assess whether children are making progress toward
using arrays and multiples of equal groups to demonstrate the meaning of
multiplication. Children are making adequate progress if they are able to draw
arrays and find the answer for Problems 1, 2, and 3. Some children may be able
to fill in the diagrams and write the number models.
Student Page
Date
Time
LESSON
Measuring Perimeter
42
Measure the perimeter of each shape in inches.
2
1.
1
1
in.
[Operations and Computation Goal 6]
2
2
2.
in.
1
2
150 151
in.
in.
1
in.
1
in.
112
in.
in.
1
6
Perimeter:
3
3.
1
1
1
4.
in.
in.
1
in.
1
in.
in.
1 in.
1 in.
Perimeter: 10 inches
a perimeter of 20 units.
in.
inches
Try This
114
114
in.
114
in.
5. Draw any shape with
7
Perimeter:
inches
in.
Perimeter:
in.
in.
3 _3
4 inches
Sample answer:
What is the area of
your shape? Sample
16
answer:
square units
Math Journal 1, p. 82
79-101_EMCS_S_SMJ_G3_U04_576353.indd 82
1/26/11 9:11 AM
Lesson 4 2
251
Student Page
Date
Time
LESSON
Math Boxes
42
1. Complete the bar graph.
Point Totals
Points
10
2. 10 packs of gum on the shelf in the
candy store. 8 sticks of gum per
pack. How many sticks of gum in
all?
Player A scores
4 points.
Player B scores
8 points.
8
6
packs
Player C scores
3 points.
4
2
0
A
B
C
D
sticks
of gum
in all
8
?
10
Player D scores
9 points.
Answer:
Players
estimate to check
that the answer
makes sense.
(Math Journal 1, p. 82)
4. Solve.
Unit
3×5=
15
15
3 nickels =
= 648 + 209
20
Sample answer:
650 + 200 = 850
20
5. Fill in the empty frames.
Journal page 82 reviews content covered in Unit 3. Problems 1–4
provide children with practice measuring line segments. Children
measure the perimeters of polygons to the nearest _14 inch. Problem
5 involves children drawing a figure with a given perimeter and
then counting unit squares to find the area.
¢
=4×5
¢ = 4 nickels
52 53
56
192
6. 1,798
1
3
2
15
5
6
INDEPENDENT
ACTIVITY
of Polygons
259
estimate:
3
Measuring Perimeters
80 sticks
86 87
3. Solve. Make a ballpark
857
sticks of
gum per
pack
2 Ongoing Learning & Practice
Which digit is in
the tens place?
9
Which digit is in
the hundreds place?
1
Which digit is in
the ones place?
8
Which digit is in the
thousands place?
200 201
Math Boxes 4 2
7
1
(Math Journal 1, p. 83)
18 19
Math Journal 1, p. 83
79-101_EMCS_S_SMJ_G3_U04_576353.indd 83
INDEPENDENT
ACTIVITY
2/4/11 10:24 AM
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 4-4. The skill in Problem 6 previews
Unit 5 content.
Writing/Reasoning Have children write an answer to the
following: Why are estimates helpful in Problem 3?
Sample answer: Estimates help you know whether your
answer makes sense.
Home Link 4 2
INDEPENDENT
ACTIVITY
(Math Masters, p. 87)
Home Connection Children use arrays to solve
multiplication stories. They find pictures from
newspapers, magazines, and other sources that show
arrays of items. You may want to use these pictures to build an
Arrays Exhibit.
Home Link Master
Name
Date
HOME LINK
Time
Arrays
42
Family
Note
Your child is learning how to represent multiplication problems using pictures called arrays.
An array is a group of items arranged in equal rows and equal columns. Help your child use
counters, such as pennies or macaroni, to build the array in each problem. Your
child should record each solution on the dots next to the problem.
64 65
Please return this Home Link to school tomorrow.
For the next few weeks, look for pictures
of items arranged in equal rows and
columns, or arrays. Look in newspapers or
magazines. Have people in your family help
you. Explain that your class is making an
Arrays Exhibit.
This is a 5-by-6 array.
There are 5 rows. There
are 6 dots in each row.
There are 30 dots in all,
since 5 × 6 = 30.
Make an array with counters. Mark the dots to show the array.
1. 4 rows with 6 counters per row
a 4-by-6 array
24
counters
2. 3 rows with 8 counters per row
a 3 × 8 array
24
counters
3. 2 rows with 12 counters per row
a 2 × 12 array
24
counters
Practice
Write these problems on the back of this page. Solve. Show your work.
4. 331 + 27 =
6. 907 - 479 =
358 5. 204
428
= 187 + 17
Unit
Math Masters, p. 87
085-118_EMCS_B_MM_G3_U04_576957.indd 87
252
2/3/11 10:26 AM
Unit 4 Multiplication and Division
NOTE Some teachers include a small section in their Arrays Exhibit titled
Almost an Array. This section may include children’s contributions that are
close but not quite correct. It is important to discuss why such contributions are
considered almost an array but not a true array.
Teaching Master
3 Differentiation Options
Name
Date
LESSON
Time
Building Arrays
42
䉬
Materials
䊐 cm cubes or other counters
䊐 1 six-sided die
READINESS
Building Arrays
INDEPENDENT
ACTIVITY
䊐 centimeter grid paper (Math Masters, p. 416)
1. Roll the die 2 times. The first number tells how many rows to put in
the array. The second number tells how many counters to put in
each row.
5–15 Min
Example
You roll a 1 first and then a 5. You make this figure.
(Math Masters, pp. 88 and 416)
To provide children with experience forming arrays, children
build arrays with counters, following directions on Math Masters,
page 88. They record the arrays on centimeter grid paper, Math
Masters, page 416, and fill in the Array Table on page 88.
2. Draw the arrays you make on centimeter grid paper.
3. Then, fill in the Array Table below.
How many
cubes in each
row?
How many
rows?
How many
cubes in all?
1
2
3
ENRICHMENT
Exploring Square Numbers
SMALL-GROUP
ACTIVITY
4
5
15–30 Min
(Math Masters, pp. 89 and 416)
Algebraic Thinking To further explore arrays, have children build
square arrays with counters for as many numbers as possible,
beginning with 2. They record their arrays on centimeter grid
paper (Math Masters, page 416) and write a number model under
each array. As children build arrays for larger products, they might
need to tape sheets of grid paper together. Have children arrange
the number models and products for each array in order from
smallest to largest. Ask: What patterns do you see? Sample answer:
The differences between consecutive products are odd numbers,
and they increase by 2.
ELL SUPPORT
Building a Math Word Bank
Math Masters, p. 88
SMALL-GROUP
ACTIVITY
5–15 Min
Teaching Master
(Differentiation Handbook, p. 132)
Name
LESSON
To provide language support for multiplication, have children
use the Word Bank template found on Differentiation Handbook,
page 132. Ask children to write the word array, draw a picture
representing the word, and write other related words. See the
Differentiation Handbook for more information.
42
䉬
Date
Time
Square Products
Work in a small group.
Materials
䊐 centimeter grid paper (Math Masters, p. 416)
䊐 centimeter cubes or pennies (or both)
䊐 tape
Directions
1. Each person chooses a different number from 2 to 10.
2. Build an array that shows your number multiplied by itself. Use
centimeter cubes or pennies.
3. Draw each array on centimeter grid paper. Write each number
model under each array.
224
339
4 4 16
4. Continue to build arrays with other numbers. Draw the arrays on
grid paper. You may need to tape pieces of grid paper together for
the larger arrays.
5. Look at the arrays you made. Why do you think the products of the
number models you wrote are called square products?
Sample answer: All of the arrays are squares.
Math Masters, p. 89
Lesson 4 2
253