September 2014
TEACHER’S GUIDE
Vol. 33, No. 1 • ISSN 0732-7773
SEPT
OCT
NOV/DEC
JAN
FEB
MAR
APR
MAY/JUNE
Making Math a Blast!
www.scholastic.com/dynamath
I’m thrilled to be starting a new school year with you on DynaMath!
Your subscription gives you a vital tool for making real-life math
connections, building conceptual understanding, and above all, showing
your students how math is fun!
During my eight years at Scholastic, I’ve always admired DynaMath
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be taking up the torch. We’ve got many fascinating stories to tell this
year. And we’re designing our activities and problem sets to emphasize
critical thinking and a deeper understanding of mathematical practices.
Together, we’ll ensure your students have the tools to succeed at math!
Elizabeth Carney, Editor
FEATURE VIDEO:
Multiplication Diagrams
SKILLS AND STANDARDS
PAGE
SKILL &
ARTICLE TITLE
COMMON CORE STATE
STANDARD
ONLINE MATERIALS
www.scholastic.com/dynamath
4
MULTIPLICATION
Baby Elephant
Rescue
Operations and Algebraic
Thinking: Multiply or divide to
solve word problems involving
multiplicative comparison.
8
EQUATIONS
The Bug Wrangler
Operations and Algebraic
Thinking: Solve word problems
using the four operations.
10
PLACE VALUE
Are We Alone?
Number and Operations in Base
10: Generalize place value
understanding for multi-digit
whole numbers.
12
Skills Sheet: Multiplying With Arrays
RECTANGLE ARRAYS Operations and Algebraic
Thinking: Represent and solve
Meet the Boxtrolls problems involving multiplication. Video: Boxtrolls Trailer
14
POWERS OF 10
Crowd Quake!
DIGITAL ISSUE KEY
Number and Operations in Base
10: Understanding the placevalue system.
Video
Game
Skills
Sheet
Skills Sheet: Word Problem Diagrams
Video: Multiplication Diagrams
Skills Sheet: Writing an Equation
Game: Creepy-Crawly Equations
Skills Sheet: Practicing Place Value
Video: Europa-Jupiter System Mission
Skills Sheet: Understanding Powers of 10
A SUPPLEMENT TO DYNAMATH
DynaMath SEPTEMBER 2014 • T1
LESSON PLANS
page 4 MULTIPLICATION
Baby Elephant Rescue
T4 • DynaMath SEPTEMBER 2014
Application
3. For problem 5, have students share their own
word problem with a partner. Instruct partners
to share whether they feel the problem is
appropriate for the diagram given.
Closure
4. Have students write or discuss what factors,
products, and diagrams are, and explain
how they can help you solve a real-world
multiplication problem.
CONCEPT REINFORCEMENT ONLINE
www.scholastic.com/dynamath
Word
Pr
NAME
__________
__________
__________
_
oblem
Diagr
ams
_____
plate.
___
in each
How
many
group
cupcakes
: _____
_____
_____
does
_____
_____
_____
_____
Ryan
__
and his
people
friend
s will
will get
the gumb share 27
Number
gumb
alls in
alls. Each
of group
all?
s: _____
perso
n will
_____
Number
get 9
___
in each
gumb
alls. How
group
: _____
Total:
many
_____
_____
_____
_____
_____
_____
Miguel
__
each will arrange
row?
24 stick
ers into
Number
6 rows
of group
in his
scrap
s: _____
book.
_____
Number
How
___
many
in each
stickers
group
will be
: _____
Total:
in
_____
_____
_____
_____
_____
_____
__
All rights
reserve
1
TK
d.
In “Baby
Eleph
multiplicat
ion word ant Rescue”
problems. (pages 4-7),
When solvin
you used
diagram
simple
Equal-grou
can help g word proble
diagrams
ms, partic
you solve
number pings proble
called
arrays
ularly
of items
ms have
problems
to help
.
three parts
quickly equal-groupings
you
and accur
: The numb
problems,solve
ately.
er of group
drawing
a
s, numb
Use this
er of items
number
in each
In the word
sente
group,
# of group nce to organ
and total
Put a quest problems
s x # in
ize
each group your inform
ion mark below, identi
ation:
fy
next to
= total
the part the number of
Lulu has
groups,
in which
Lulu have 3 plates
amou
you’re
nt of
missing
in all? of cupcakes.
inform each group, or
There
Number
ation.
total numb
are 4
of group
cupcakes
er of items
s: _____
on each
.
tic Inc.
Number
by Scholas
Total:
page.
©2014
2
ce this
Click on the “Skills Sheet” button
in your digital issue for the “Word
Problem Diagrams” skills sheet.
Students will practice with
different types of multiplication
diagrams.
reprodu
What types of things would you need to know
to find out how much food the workers at
the orphanage need for multiple elephants?
(Possible answers: You need to know how much
one elephant eats and how many elephants there
are in total.)
• Connect to the vocabulary. Discuss what a
factor and a product are. If the technology is
available, project the multiplication diagram on
page 6 of the digital issue. Ask:
Would the number of elephants and how much
they eat be factors in our problem? Explain
your answer.
• As a class or in small groups, ask:
What are some examples of real-life factors
from the article or from your own experiences?
How might you solve for the examples we have
shared? (Possible answers: You could draw a
multiplication diagram or make an array with
manipulatives.)
• Together, look at problem 1. (We suggest
3
sion to
Concept Development
2. A fter reading the article, ask:
• Ask students to share their drawings with their
group, comparing the similarities and differences
between them. Students should understand that
it’s OK for drawings to be different. This enables
us to make connections to our own drawings and
deepen our understanding.
s permis
(either as a whole group, a small group, or with a
partner):
• How would you take care of a baby elephant?
• What kind of food do you think a baby
elephant eats? How much do you think it eats?
teacher
LESSON Engagement
1. Before reading the article, ask students
3 Pints per Feeding
Number of Feedings
XXXOOOOOOOO
3 Pints per
XXX
Feeding OOOOOOOO
XXX
OOOOOOOO
Number of
XXX
Feedings
XXX
XXX
XXX
XXX
grants
Students will draw diagrams to develop conceptual understanding of multiplication and its structure. Students will learn and use precise vocabulary when solving multiplication stories.
tic Inc.
OBJECTIVE
Scholas
2. Reason abstractly and quantitatively.
6. Attend to precision.
7. Look for and make use of structure.
Entin.
MATHEMATICAL PRACTICES STANDARDS
DynaM
ath Septe
mber
2014
• page
1 of 2
by Carli
Operations and Algebraic Thinking:
Multiply or divide to solve word problems involving
multiplicative comparison.
projecting it.) Discuss the diagram. Ask:
What other objects might you draw to represent
the multiplication factors and product for this
problem? (Possible answers: You could draw
circles, stars, X’s, etc.) You want students to
know that the picture’s function is to help you
see the problem.
• Ask students to draw their own versions of
problem 1. Some examples you might see:
Activity
CONTENT STANDARD
LESSON PLANS
page 8 EQUATIONS
The Bug Wrangler
CONTENT STANDARD
Operations and Algebraic Thinking:
Solve word problems using the four operations.
MATHEMATICAL PRACTICES STANDARDS
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
OBJECTIVE
Students will write equations based on
information gathered from word problems.
LESSON
Engagement
1. Before reading the article, ask students (either
as a whole group, a small group, or with a
partner):
Do you think you could do a job that involves
working with and handling insects?
Concept Development
2. After reading the article, give students the
following instructions:
• Explain to your partner what you think an
equation is.
• Next, ask your partner how an equation and
an expression are different. (Possible answers:
An expression has numbers, symbols, and
operators—such as addition and subtraction
signs—to show something’s value. An equation
has an equal sign which shows that two
expressions are equal.)
8 + 2 = 10
● ● ●
● ● ●
● ●
● ●
8 + a = 10
● ● ●
●●
● ● ●
● ●
● ●
●●
● ●
●●
●●
●●
● ●
●●
●●
●●
• Replicate or project the example above and
discuss the similarities and differences between
the two equations. Ask:
What does the equal sign represent? (Note:
We no longer want students to think that the
expression to the right of the equal sign is the
“answer.” It truly represents the equivalence of
two sides of an equation.)
3. Instruct: When you are solving a word
problem, creating an equation from the content
is a great way to help you solve the problem.
Share with a partner techniques you can use
to create an equation from a word problem.
(Possible answers: You could rephrase the
problem a different way, write in your own words
what the question is asking, circle clue words, or
eliminate extra information.)
Application
4. Allow students to practice creating equations
and build a deeper understanding of the equal
sign. (Students may need cubes to help them
conceptually.)
5. Have students work on problems 1 through 4.
Give them time to work independently and then
share their thinking with a partner about which
operation should be used and why.
6. Students may need to collaborate to persevere
through problem 5. Ask students to share their
strategies.
Closure
7. Ask: How can creating an equation from a
word problem help you in your problem solving?
(Possible answers: It helps me organize my
thinking; it is more efficient than drawing every
part of the problem; it makes a number sentence
representing the word problem.)
CONCEPT REINFORCEMENT ONLINE
www.scholastic.com/dynamath
Inside your digital issue, click on the
“Skills Sheet” button for the
“Writing an Equation” skills sheet.
Students will practice writing
number sentences, eliminating
extra information, and deciding
if there is enough information to
solve the problem.
Writing
an
NAME
Equa
tion
__________
__________
__________
_
In “The
Bug Wran
some
case,
not everygler” (page
In fact,
s 8-9),
some
thing
important
times
you wrote
included
to unde a probl
in
to the
equations
question. rstand em doesn’t a word probl
to
have
Use the what the
all the em is nece help you solve
ques
prom
word
pts below tion is askininformatio ssary for you
Alan
probl
n you
takes
to
g and
ems.
to analy
care of
In
to find need to answwrite an
The
ze the
equation.
the
to clean smallest taran 4 tarantulas
follow
er
ing wordkey informatio it. It’s
each
tula is
at
cage.
probl
n that
5 inche the local
How many
ems.
scien
relate
s long;
What
ce muse
s
is the
the
minutes
um. They
word
does longest is
probl
11 inche
it take
each
em askin
Circle
have
Alan
their
the key
g you
to clean s long. It
to find?
informatio
all the takes Alan own cage.
_____
cages
10 minu
Do you
n.
_____
?
tes
_____
have
_____
enough
_____
informatio
_____
If yes,
_____
write
n to solve
_____
and solve
____
the probl
an equa
em?
Melis
tion to
_____
sa had
_____
answer
Man comi
50 comi
___
the word
books
c book
c book
probl
does
s. Of
s. She
em
those
Melis
sa have gave 8 of
, 18
What
her total were Batm
left?
is the
an
comic
word
books comic book
probl
s and
to her
em askin
Circle
13 were
little broth
the key
g you
to find?
er. How Spiderinformatio
_____
many
Do you
n.
_____
comic
_____
have
_____
enough
_____
informatio
_____
If yes,
_____
write
n to solve
_____
and solve
____
the probl
an equa
em?
At Hobs
tion to
_____
_____
answer
There on’s farm,
___
the word
there
are four
are
probl
goats
em
What
living 32 goats, 18
is the
in each
pigs,
word
pen. How and 30
probl
chick
em askin
many
Circle
ens.
female
Nine
the key
g you
of the
pigs are
to find?
informatio
goats
there
_____
Do you
n.
_____
at Hobs are fema
_____
have
on’s farm? les.
_____
enough
_____
informatio
_____
If yes,
_____
write
n to solve
_____
and solve
____
the probl
an equa
em?
tion to
_____
_____
answer
___
the word
probl
em
TK
1
2
3
DynaM
ath Septe
mber
2014
• page
1 of 2
DynaMath SEPTEMBER 2014 • T5
LESSON PLANS
page 10 PLACE VALUE
Closure
4. Project the problem below, asking students to
solve it and justify the answer to a partner.
You won 345,984 dollars! The lottery wrote a
check for you to deposit at the bank. Which
check is written correctly and why?
(either as a whole group, a small group, or with a
partner):
•How far do you think Jupiter is from our
planet? (Allow students to make an educated
guess. The answer is 365 million miles at the
planets’ closest point, since the distance between
them is constantly changing.)
•With your partner, write a number that
represents the distance. (Tell students not to
erase their numbers. They will use them later.)
2400
__________________20______
pay
to the
order of ____________________________________________________________________________________
Your Name
for_________________________________
: 0122105278 :
T6 • DynaMath SEPTEMBER 2014
345,984
______________________________________________________
6724301068 · 2400
2400
__________________20______
pay
to the
order of ____________________________________________________________________________________
Your Name
$
345,984
Three hundred forty-five thousand, nine hundred eighty-four DOLLARS
_____________________________________________________________________________________________________________________
Concept Development
2. Ask: What does it mean to express a number
for_________________________________
: 0122105278 :
______________________________________________________
6724301068 · 2400
2400
__________________20______
pay
to the
order of ____________________________________________________________________________________
Your Name
$
345,984
Three
thousand forty-five thousand, nine hundred eighty-four DOLLARS
_____________________________________________________________________________________________________________________
for_________________________________
: 0122105278 :
______________________________________________________
6724301068 · 2400
CONCEPT REINFORCEMENT ONLINE
www.scholastic.com/dynamath
Worki
ng
NAME
__________
__________
__________
_
With
Place
Value
In “Are
We
stand
ard, expa Alone?”
(page
nded,
and words 10-11), you
learn
form.
ed abou
t
Periods
Thou
sand
S
NS
ns
s
place
value
TK
6/24/14 2:04 PM
and how
to write
numb
Ones
S
Millio
REDS
A digit’
s place
2 4
stand
for “6.” in a numb
6, 3
er deter
It stand
9 4,
mines
s for
Use this
6 millio
its value
1 7
numb
n
. For
5
er sente
exam
ple, a
nce to
“3” in
organ
the millio
ize your
In the
ns place
# of group
informatio
word
probl
of items
doesn’t
sx#
n:
ems
. Put
in each
a ques below, ident
group
tion mark
ify the
= total
numb
next to
the part er of group
In the
numb
s, amou
in whic
er 65,94
nt
h you’r
2, which
e miss of each group
ing infor
digit is
,
in the
matio or total numb
n.
thous
er
ands
place
?.
In the
follow
ing numb
er, circle
the digit
that repre
sents
thirty
millio
The faste
n: 333,3
st roller
reach
33,33
coast
es spee
3.
er in
ds of
149 hours the world
1? _____
per hour. is the Form
_____
ula Ross
_____
In 149,
_____
a
what
in
4? _____
__
the Unite
numb
_____
er is repre
d Arab
_____
Emir
sente
_____
9? _____
d by the ates. It
__
_____
digit.
_____
..
_____
__
1
2
3
DynaM
ath Septe
mber
2014
• page
1 of 2
ONES
In your digital issue, click on the
“Skills Sheet” button for the
“Practicing Place Value” skills
sheet. Students will build fluency
with place value problems.
Checks.indd 1
TENS
in standard form, word form, or expanded form?
With a partner, discuss the forms’ similarities
and differences. (Answers will vary. If the
technology is available, project the place value
chart on page 11 of the digital issue.)
• Have students look at the number they wrote
down before (distance to Jupiter). Ask them to
share what form their number is written in.
•Assuming students wrote the number in
standard form, have them practice writing the
number in expanded form and written form.
•Ask students to break into small groups. Give
each group a different number and have them
model the three numerical forms. Have the
groups share their answers. (Note: Saying very
$
Three hundred forty-five thousand, nine eighty-four
____________________________________________________________________________________________________________________
DOLLARS
HUND
LESSON
Engagement
ave students read the article. Ask students
1. H
5 on their own. Allow them to discuss their
responses with a partner.
HUND
MILLIORED
NS
tudents will deepen their place value
S
understanding by representing values using
standard form, word form, and expanded form.
Students will verbally practice saying the
numbers precisely.
TEN
MILLIO
NS
OBJECTIVE
Application
3. Have students complete problems 1 through
SAND
6. Attend to precision.
8. Look for and express regularity in repeated
reasoning.
MILLIO
MATHEMATICAL PRACTICES STANDARDS
HUND
THOU RED
SAND
S
Number and Operations in Base 10:
Generalize place value understanding for multi-digit
whole numbers.
TEN
THOU
SAND
CONTENT STANDARD
large numbers in standard form, word form, and
expanded form is not easy. This activity will help
students become more confident and precise with
this skill.)
• Have students challenge each other by giving
their partner a number in standard, written, or
expanded form. The partner will check the number
to see if it’s written and then spoken correctly.
THOU
Are We Alone?
ers in
LESSON PLANS
page 12 RECTANGLE ARRAYS
Meet the Boxtrolls
CONTENT STANDARD
Operations and Algebraic Thinking:
Represent and solve problems involving
multiplication.
MATHEMATICAL PRACTICES STANDARDS
1. Make sense of problems and persevere in solving
them.
3. Construct viable arguments and critique the
reasoning of others.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
OBJECTIVE
Students will use rectangular arrays to solve
multiplication problems in context.
LESSON
Engagement
1. Before reading the article, ask students (either
as a whole group, a small group, or with a
partner):
• How many of you have read the book The
Boxtrolls? Have you heard of the movie?
Concept Development
2. After reading the article, ask students:
• Explain to your partner how you use an array
for multiplication. (Possible answer: Arrays show
multiplication [and division] problems visually.)
• Where you have you seen an array in your
daily life? (Possible answers: stacked piles of
bricks, cookies on a cookie sheet, etc.)
• Replicate or project the array below:
Application
3. Together, have students read through the first
problem. Then say:
• Discuss three ways to find the total number of
boxes in an array.
• As a class, allow students to share their
thoughts about the most efficient ways to solve
the problem.
4. Have students work independently or with a
partner to solve for problems 2 through 4. Some
students may need to model with cubes to help
build conceptual understanding.
5. Project the array for problem 5.
• Ask students to come up with different ways
to decompose the array, meaning to break it
into smaller parts. You may have to review (or
introduce) the math term decomposed with your
students.
• Have students use cubes to create the
rectangular array from problem 5. Then ask them
to decompose the array in a way that makes
counting the cubes more efficient. Here is one
way, as shown in problem 5:
4 rows 5
4 rows 5
4 rows 5
4 rows 5
4 columns 4 columns 4 columns 4 columns
Students may have other methods.
Closure
6. Have students share their arrays and
equations with the class or a small group,
comparing how they are similar or different.
Ask: How many rows and columns are in this
array? What’s an equation that shows the
number of cubes in this array?
(4 5 5 = 20 or 5 5 4 = 20)
• Give students cubes to create their own arrays.
Have them write equations based on their arrays
and work with a partner to check that each array
matches the equation.
CONCEPT REINFORCEMENT ONLINE
www.scholastic.com/dynamath
In your digital issue, click on the
“Skills Sheet” button for the
“Multiplying With Arrays” skills
sheet. Students will practice using
arrays to solve multiplication
problems.
Multi
plying
With
Array
s
NAME
__________
__________
__________
_
In “Mee
MULTIPL
and solve t the Boxt
ICATION
use array multiplica rolls” (page
(or math s to multi tion probl s 12-13), you
ems.
ply and
used
Array
Use array rule) says
simple
visua
lly show s are an
diagr
s to answ that you
ams calle
can
the commarrangeme
er the
Miguel
follow multiply numb
utative nt of items d arrays
ing ques
arran
to help
prope
collection
ges the
rty of into equal
tions. ers in any
you write
coins
group
order
multi
into 4
in his
plication.
s.
and
rows
You can
the produ
∏∏∏∏ of 5 coins. coin
ct will This prope
rty
be the
∏
D. Expla
∏∏∏∏
same
in why
.
∏
sente
∏∏∏∏
the
nces
are examtwo array
∏
prope
∏∏∏∏
rty.
ples of s and numb
A. Write
∏
the comm
er
an addit
numb
ion sente
utativ
er of
e
coins
nce to
.
find the
total
Lisette
B. Write
a multi
paper and Annie
total
both
plication
numb
Lisette’s has the most
claim
er of
sente
coins
Paper
stamps that their
nce to
.
find the
xxx
Annie’s on it.
x
Paper
xxx
The comm
x
xxx
xxx
utativ
multi
x
xxx
factors plication e property
Who
says
of
xxx
can be
is corre
the same
changed, that the order
ct? Expla
xx
but the
A. Write .
of
in your x
produ
answer.
the total a number
ct stays
sente
numb
nce that
er of
balls
in this represents
array
How could
:
the comm
some
multi
solve?
utativ
e prope
Provide plication
thinking.
rty
an exam problems
B. In your
easie make
ple to
illustrate r to
this array own draw
your
ing,
based
on the rearrange
the
comm
utativ balls in
Samm
e prope
y break
rty.
equal
parts s a large array
like this:
C. Write
into smal
ler
the total a number
sente
numb
nce that
drew.
er of
balls
in the represents
array
you
Is Samm
of multi y using the
solve? plication to commutativ
Expla
e prope
in your make this
rty
array
answer.
easie
r to
DynaM
1
3
2
4
5
ath Septe
mber
2014
DynaMath SEPTEMBER 2014 • T7
LESSON PLANS
page 14 POWERS OF 10
T8 • DynaMath SEPTEMBER 2014
Closure
4. Have each student research another sound
not mentioned in the story to find out its decibel
level. Then have students share their findings
with the class, pointing out where their sound
would appear on the decibel scale.
CONCEPT REINFORCEMENT ONLINE
www.scholastic.com/dynamath
Unde
NAME
rstand
ing Po
wers
__________
__________
__________
_
In “Crow
d Quak
of differ
ent soun e!” (page
ds. Powe s 14-15
), you
rs of
work
10 are
any numbed with powe
rs of
ers that
For exam
can be 10 to comp
ple, 10,00
written
are
You can
as produ the decib
0 = 10
el level
draw
cts of
x 10 x
numb
10 x 10. s
10 x 10
er disks
x 10
to unde
rstan
d powe
Hundreds
rs of
10 bette
Tens
r.
2
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DynaM
ath Septe
mber
2014
• page
1 of 2
TK
Ones
●
1
of 10
page.
In your digital issue, click on the
“Skills Sheet” button for the
“Understanding Powers of 10”
skills sheet. Students will build
conceptual understanding of how
powers of 10 work.
ce this
• Share your experiences with the highest and
lowest decibel noises you’ve ever heard. (Note:
This is an important discussion to have because
decibels can be an abstract concept for many
children and may challenge their thinking.)
• What data is represented in the left column
and the right column? Compare the two
columns and discuss with a partner.
• What does dB mean? (dB is the abbreviation
for decibel, which measures sound.)
• Help students understand that a sound’s
decibel level tells you how loud a sound is
compared with the quietest sound. Students
should be thinking multiplicatively. For example,
the quietest sound the human ear can hear
measures 0 decibels. A whisper is 1,000 times
(10 5 10 5 10) louder than the quietest sound.
• Explain that a power of 10 says how many
times 10 is multiplied by itself. So, 10 to the
second power would be 10 5 10, or 100. Each
with a partner, allowing for discussion to deepen
their understanding of decibels and multiplicative
reasoning.
reprodu
Concept Development
2.Project the decibel scale on page 15.
Application
3. Have students complete problems 1 through 5
sion to
as a whole group, a small group, or with a
partner):
• How many of you have been to a sporting
event, a concert, or other place where the noise
was very loud?
• Have you ever heard of the term decibel?
s permis
LESSON
Engagement
1. After reading the article, ask students (either
teacher
tudents will use their multiplicative reasoning
S
skills by working with powers of ten to compare
sounds on a decibel scale.
grants
OBJECTIVE
tic Inc.
4. Model with mathematics.
6. Attend to precision.
8. Look for and express regularity in repeated
reasoning.
Scholas
MATHEMATICAL PRACTICES STANDARDS
Ones
Entin.
Number and Operations in Base 10:
Understanding the place value system.
time a product of a power of 10 is once again
multiplied by 10, you add another zero to the
number. So, 100 5 10 = 1,000. That’s because 10
groups of 100 equals 1,000.
• In the projection of page 15­—or within a
student’s print issue­­­—circle the decibel levels
and their corresponding “times as great as the
quietest sound” for a vacuum cleaner, a quiet
place, and a lawn mower. Instruct students to:
Say the value for how many “times as great as
the quietest sound” for each item that’s circled.
(A quiet place is 10 times as loud as the
quietest sound a human ear can hear. Traffic is
10,000,000 [10 million] times as loud as the
quietest sound a human ear can hear. A lawn
mower is 1,000,000,000 [1 billion] times as loud
as the quietest sound a human ear can hear.)
by Carli
CONTENT STANDARD
Activity
Crowd Quake!
GET MORE
SKILLS SHEETS
LIKE THIS
ONLINE:
NAME
_______________________________
MULTIPLICATION
www.scholastic.com
/dynamath
Multiplying With Arrays
In “Meet the Boxtrolls” (pages 12-13), you used simple diagrams called arrays to help you write and solve
multiplication problems. Arrays are an arrangement of items into equal groups. You can use arrays to multiply
and visually show the commutative property of multiplication. This property (or math rule) says that you can
multiply numbers in any order and the product will be the same. Use arrays to answer the following questions.
∏∏∏∏∏
∏∏∏∏∏
∏∏∏∏∏
∏∏∏∏∏
A. Write an addition sentence to find the total
number of coins.
B. Write a multiplication sentence to find the
total number of coins.
2
The commutative property of multiplication
says that the order of factors can be
changed, but the product stays the same.
A. Write a number sentence that represents the
total number of balls in this array:
B. In your own drawing, rearrange the balls in
this array based on the commutative property.
C. Write a number sentence that represents the
total number of balls in the array you drew.
D. Explain why the two arrays and number
sentences are examples of the commutative
property.
3
Lisette and Annie both claim that their
paper has the most stamps on it.
Lisette’s Paper
Annie’s Paper
x x x x
xxx
x x x x
xxx
x x x x
xxx
xxx
Who is correct? Explain your answer.
4
How could the commutative property
make some multiplication problems easier
to solve? Provide an example to illustrate your
thinking.
5
Sammy breaks a large array into smaller
equal parts like this:
Is Sammy using the commutative property
of multiplication to make this array easier to
solve? Explain your answer.
DynaMath SEPTEMBER 2014 • T9
Scholastic Inc. grants teachers permission to reproduce this page. ©2014 by Scholastic Inc. All rights reserved.
1
Miguel arranges the coins in his coin
collection into 4 rows of 5 coins.
ANSWER KEY
Problem of the Day: Our popular monthly
calendar of problems and puzzles is online.
www.scholastic.com/dynamath
PAGE T9:
Multiplying With Arrays
1a. 5 + 5 + 5 + 5 = 20 or 4 + 4 + 4 + 4 + 4 = 20 1b. 4 5 5 = 20 or 5 5 4 = 20
2b.
2c. 2 5 5 = 10 or 5 5 2 = 10
2d. T he order of the factors can change, but the
product does not.
For answers to the student
edition, see page T12.
oth Annie and Lisette have the same
3. B
number of stamps on their papers. Lisette’s
stamps are arranged in a 3 row 5 4 column
array. Annie’s stamps are arranged in a 4 row
5 3 column array. The products are the same.
4. A nswers will vary. One possible answer:
5 5 2 5 75; it is easier to multiply 75 by 10
(the product of 5 and 2) than to multiply
either 5 or 2 by 75.
5. No. Sammy is not changing the order of the factors in the multiplication problem. He’s
making the problem easier by breaking 12 into
3 groups of 4, then adding together the
products of each part.
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SCHOLASTIC DYNAMATH® TEACHER’S GUIDE • VOLUME 33, NO. 1 • September 2014
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REAL WORLD MATH & MONEY LESSONS. SCHOLASTIC.COM/REGIONS
T10 • DynaMath SEPTEMBER 2014
ANSWER KEY
STUDENT EDITION
PAGES 2-3: Numbers in the News
CRUMBLING RUINS: 79 a.d.
SOGGY DOGGIES: 20 shakes
BLOCKS THAT GO VROOM: 500,000 pieces
PAGES 4-7: Baby Elephant Rescue
NOTE: Students may choose to draw pictures,
X’s, O’s, box arrays, or other types of diagrams
for these questions. We chose X’s. Please
accept all reasonable answers.
1a. 3
1b. 8
1c. 3 5 8 = 24
1d. the total number of pints of milk an
elephant drinks per day
2b. 1
2a. x x x x x 2d. 5 tins
2c. 5 3b. 2 5 7 = ? 3a. x x x x x x x
x x x x x x x
3c. 14 tins
4a. x x x x x x x x x x x x
x x x x x x
x x x x x x
x x x x x x
4b. 6 5 ? = 30 (Please also accept 30 ÷ 6 = ?)
4c. 5 groups
5. Answers will vary. Please accept all reasonable answers. Two possibilities we thought of
are:
• One keeper is in charge of feeding 6 baby
elephants. If he feeds each elephant 3
bottles per day, how many total bottles
will he need?
• A keeper is tasked with feeding 20 elephants. He needs 1 new bottle for each
elephant. The bottles come in packs of 3.
The orphanage orders 6 packs. Does the
keeper have enough bottles?
PAGES 8-9: The Bug Wrangler
1. 5 – 2 = 3 kinds of spiders
2. 1 5 4 = 4 drops of water
3. 10,000 – 3,000 = 7,000 locusts
4. 40 ÷ 2 = 20 ants per group
5. 1,500 + (1,500 5 2) = 4,500 beetles. Yes, he
has enough.
.
T12 • DynaMath SEPTEMBER 2014
.
PAGES 10-11: Are We Alone?
1.hundreds 4. 1,321
5. one thousand seventeen
2. 2
3. 400,000 + 10,000 + 6,000 + 900
PAGES 12-13: Meet the Boxtrolls
1a. Answers will vary but could include: You
could count all the boxes individually. You
could multiply the number of rows by the
number of columns (4 5 3). You could divide
the array into two parts and add the parts
together (6 + 6).
1b. Answers will vary, but we found multiplying
the number of rows by the number of
columns to be the most efficient method for
this array. Answer: 12
2a. 9 5 3 = 27
2b.the number of columns (or rows, depending
on the order of the equation in part A)
3. 7 5 5 = 35
4. No, 17 cannot be divided by 2 evenly. There
would be one troll left over.
5.You divided a large array into 4 smaller arrays
(boxes of 16, or 4 5 4). You can multiply
16 5 4 to find the number of boxes in the
whole array. The answer is 64.
PAGES 14-15: Crowd Quake!
1.130 dB
2. MP3 player
3. 80 dB
4. 10,000 times 5. 1,000,000 times
PAGE 16: The Puzzle Page
Math Master
1.D
2. C
3. D
4. B
5. A
Math Blooper
Timmy confused the place value of the 2 in 23. He
meant to multiply 7 5 20 and 7 5 3 and add the
products (161 cups/week). Instead he multiplied
7 5 2 and 7 5 3 and added the products.
Brainteaser
Swap the first penny with the second quarter.