Download Investigating Equal Groups

Investigating Equal
Groups
Q u i c k
R e v i e w
Equal groups have the same number of things in each group.
Here are 5 towers of Snap Cubes.
There are 4 cubes in each tower.
S-
Skip count to find how many cubes.
g
^
16
There are 20 cubes in all.
We write 5 groups of 4 as a multiplication sentence.
4
X
Number of
groups
4
20
A
A
Number of
cubes in each
group
Total number
of cubes
We say: 5 times 4 equals 20.
Try These
Write a multiplication sentence for each picture.
3X2 = 6
4 X 3 = 12
2 X 5 = 10
114
20
Practice
1.
2.
' . ' K ! -\;K\\
i)
I
4)
4
p.'oUUCt.
1
I
1 2
3=
v
8
b) 2 x 4 =
.
e) 2 x 5 =
1
c) 1 < 3=
0
f)
3
5
5 X 1=
Multiply.
6
a< 2 >' 3 =
2
•ii 5 \ 5 =
5
_
b) 3 x 3 -
9
e) 2
2
<
1=
c) 4 v 4 =
f)
,< 2 -
2
1
6
4
: "Me, -i pu iuif- for each multiplication sentence.
Sample Answers
*)
'
1
12
b)
2 x
5=
10
c)
4 x
5 = 20
Strcjtcli Your Thinking
Golf eaa\ come in tubes of 3.
Each box has 4 tubes.
* .a-.v a
< e ,
1
ture and write a muh-oiication sentence to find how
' ha • a box.
!!
1 15
Relating Multiplication
and Repeated Addition
Q u i c k
R e v i e w
You can use repeated addition to think about multiplication.
2+2+2+2=8
4X2-8
You can use a number line to show multiplication as repeated addition.
+2
0
+2
2
1
3
+2
+2
4
Add 2 each time.
4 times 2 equals 8.
Try These
Write an addition sentence and a multiplication sentence for each picture.
3+3+3= 9
3X3 = 9
2.
3.
^
116
^ 0
S
^ 0
2+2+2+2=8
4X2=8
4 + 4 + 4=12
3 X 4 = 12
X
Practice
1. Write an addition sentence for each multiplication senteiv w
a) 3 \ 2 -- 6
2+2+2=6
c) 2 X 4 - 8
b
4+4=8
)
4
N
5
r
S f 5 t 5 + 5 = 20
2 Q
d) 3 X 1 = 3
1+1+1=3
2. Write a multiplication sentence for each addition sentence.
at 1 f 1 + 1 + 1 t 1 = 5 5 X 1 = 5
c) 3 f 3
b) -I i 4 + 4 = 12
2 e 3 - 1 2 4 X 3 = 12
1
d J
3 ,.. 3 ,
f
3*4^=12
2X3 =6
: 6
2. vVr-M •, M ,:X ;•«!. ,r« . " ^ r w i ' «Miit: i multiplication sentence for each number
I me.
-»>
t2
-\2
\'l
f-2
6
-1-2
b)
8
10
f5
,Q
•i - x j
KS
5
i ; s
10
'j r 5 t- 3 - 1;?
>3
X
1
' " . ' W a pie: .
---.r-•-ch multiplication
^» (wu t
i hen write an addition sentence.
•Xi,#ip|o Answers
a) 4 - 4
16
b;
• 3 - 15
•o
4 + 4 + 4 + 4 = 1G
Stretch Yxur
Writ.-* w>*
the i -e
5 +S -
O O
3 + 3 + 3 + 3 + 3 = 15
\\(mkm\i
ii.,._>n -.e-^tence and a multiplication sentence to find
: x? >%• in 5 weeks.
;
:
.
:
- X3
3 y 5 - 2S
+5
f
Q u i c k
Using Arrays to Multiply
R e v i e w
An array shows objects arranged in equal rows.
>
This is a 2-by-5 array.
There are 2 rows of 5 counters.
2 X 5 = 10
O O C
O O C
O O O O
This is a 5-by-5 array.
There are 5 rows of 5 counters.
5 X 5 = 25
J O
o o o o
>
This is a 1 -by-4 array.
There is 1 row of 4 counters.
1X4 = 4
>
This is a 3-by-1 array.
There are 3 rows of 1 counter.
3X1=3
Q Q ( )( )
Try These
Write a multiplication sentence for each array.
3 X 5 = 15
118
2 X4 = 8
4 X 5 = 20
Practice
1. Draw an array to find each product.
O O O O
O O O O
oooo
oooo
oooo
oooo
oooo
a) 4 x 4 =
1
6
._
b) 3
x 4=
o
o
o
o
o
1
c) 5
2
o
o
o
o
o
X 2 = _1<L_
OOO
O O O O O
O O O O O
O O O O O
d)
3 X 5 =
1
5
ooo
ooo
ooo
ooo
O O O O O
O O O O O
_
e)
2 X 5 =
1
0
_
f) 5 X 3 =
I
s
_
2. Use counters, Make an array to find each product.
a) 1 X 2 =
d) 3 X 3 =
2
.
b) 4 X 1 =
.
e) 5 X 4 =
3. There are 4 rows of marchers in the band.
There are 5 marchers in each row.
How many marchers are there in all? _
4
2
c) 2 X 3 =
0
_
4
f)
6
5X5 =
2 ^ 5 = 20
Stretch Your T h i n k i n g
Find different ways of arranging 24 counters in equal rows.
Draw a picture to show each way.
Students' drawings may include arrays of 1 by 24,
2 by 12, 3 by 8, and 4 by 6.
1
19
UNIT
8
Relating Multiplication
Sentences
Q u i c k
R e v i e w
When you multiply 2 numbers, you can switch the order of the numbers
without changing the product.
3 rows of 4
3 X 4 = 12
Turn the array on
its side.
4 rows of 3
4 X 3 = 12
We can show the same idea using equal groups.
2 groups of 5
2 X 5 = 10
5 groups of 2
5 X 2 = 10
Try These
1 . Write 2 multiplication sentences for each array
a) n n n n n
1 2 0
w
3 X 5 = 15
1X4 = 4
5 X 3 = 15
4X1=4
Practice
?
n
,f
is
#
«
:
-
#
~%
•»
\;
#
.;.
;g
^
^
/
4
-
.,.
#
1. Write 2 multiplication sentences and 2 repeated addition sentences for each
array, when you can.
a)
b)
AAA
AAA
4 X 5 = 20
5 X 4 = 20
2X3 =6
5+5+5+5=20
4+4+4+4+4=20
3+3=6
2+2+2=6
3X1=3
1X3=3
O O O O
3X2 =6
ci)
o o o o
2.
2X4=8
4X2=8
4+4=8
2+2+2+2=8
1+1+1=3
Draw an array for each multiplication sentence.
Then write another multiplication sentence for each array.
Sample Answers
a) 2 X 1 = 2
b) 2 X 5 = 10
g
V
OOOOO
OOOOO
1X2 =2
5 X 2 = 10
Stretch Your T h i n k i n g
Draw 20 cookies in an array. Write 2 multiplication sentences for the array.
Sample Answer
OOOOO
OOOOO
OOOOO
OOOOO
4 X 5 = 20
5 X 4 = 20
121
Division as Grouping
Assert*
Q u i c k
R e v i e w
Division can be used to find how many equal groups there are when you
know the size of the groups.
How many equal groups of 2 are there in 10?
>
Start with 10 counters.
>
Divide the 10 counters into groups of 2.
Count the number of groups.
Write the division sentence.
1 0 - 2
=
5
Number of Number in
counters
each group
We say: 10 divided by 2 equals 5.
Number
of groups
Try These
1. Use counters. Find the number of groups.
Write a division sentence.
2.
a) Divide 12 counters into groups of 3. __
12 + 3 = 4
b) Divide 8 counters into groups of 1
8 + 1=8
c) Divide 10 counters into groups of 5
10 + 5 = 2
Use counters. Make equal groups to divide.
a) 15 + 5 =
?
b) 12 + 4 =
W.
c) 8 + 2 =
d) 2 + 1 =
?
e) 6 + 2 =
L
f) 4 + 4 =
1 2 2
Practice
1. Find the number of groups.Then write a division sentence.
a) Make groups of 4.
j
b) Make groups of 3.
c) Make groups of 5.
OyiGfXP
12 + 4 = 3
d) Make groups of 1.
|
10 + 5 = 2
12 + 3 = 4
e) Make groups of 4.
f) Make groups of 2.
oo o
oo
5 + 1=5
000
8+4= 2
6 +2=3
2. Write a division sentence to solve each problem.
a) Ira has 12 plums. He gives 4 plums to each of his friends.
How many people get plums?
1
2
:
4
= 3;Three p e o p ^
b) Suri has 15 photos. She puts 5 photos on each page.
a
e s
How many pages does Suri use? i^Oo."* f J ^ ^ r L H ? ® ! ^ . E 9 :
c) Sahib baked 10 tarts. He put 2 tarts into each bag.
2
How many bags did Sahib use? !9_T... _= 5; Sahib used 5 bags.
Stretch Your T h i n k i n g
The answer is 2 0 + 4 = 5.
What might the problem be?
Sample Answer: There are 20 children in the gym.The teacher wants to
make 4 equal teams. How many children will be on each team?
1 2 3
Division as Sharing
Q u i c k
R e v i e w
Division can be used to find how many are in each group when you
know the number of groups.
12 cookies are shared equally among 3 friends.
How many cookies does each person get?
>
Start with 12 cookies.
>
Divide the 12 cookies into 3 groups.
Count the number of cookies in each group.
>
Write the division sentence.
1 2 - 3
=
4
A
i
k
Number of Number of
cookies
groups
We say: 12 divided by 3 equals 4.
Number of
cookies in
each group
Try These
1. Use counters. Find the number in each group.
Write a division sentence.
a) Divide 20 counters into 4 groups
20 + 4 = 5
b) Divide 16 counters into 4 groups
16 + 4 = 4
c) Divide 3 counters into 3 groups
3+3=1
d) Divide 12 counters into 4 groups.
12 + 4 = 3
1 2 4
Practice
1. Find the number of things in each group.
a) 8 - 4 =
2
d) 1 0 - 2 = _
1
g) 1 0 - 5 =
2
b) 20 - 5 =
4
e) 8 - 2 =
4
h)4-4=
J
1
c) 2 - 2 =
f)
3 - 1=
_ _ J L _
i) 1 5 - 3 =
5
2. Write a division sentence to solve each problem.
a) There are 2 0 people on 4 equal teams. How many people are on each
team?
20 - 4 = 5;There are 5 people on each t e a m .
b) There are 16 muffins in 4 equal-sized tins. How many muffins are in each
j ?
_ 1 6 - 4 = 4; There are 4 muffins in each tin.
t
n
c) There are 25 chairs in 5 equal rows. How many chairs are in each row?
25 - 5 = 5; There are 5 chairs in each row.
d) There are 4 buttons in 2 equal rows. How many buttons are in each row?
4 - 2 = 2; There are 2 buttons in each row.
3. Write an equal sharing problem for 6 - 2 = 3.
Show how to solve the problem using a picture.
Sample Answer: There are 6 marchers in 2 equal rows. How many
marchers are in each row?
¥ V f
A
A
o
"
T
X
A
Q
I
6 - 2 = 3; There are 3 marchers in each row.
Stretch Your T h i n k i n g
There are 12 members in the Boy Scout troop.
They will march in the parade in equal rows.
How many Boy Scouts could be in each row?
3 or 4; Students may aIso answer 2,6, or 12. If so, they are working
beyond grade level.
1 2 5
^
N
T
a
Relating Division and
Repeated Subtraction
° o ,
7
Q u i c k
R e v i e w
You can use repeated subtraction to find 8 - 4 .
Start with 8 counters.
>
Count how many groups of 4 you subtract until no counters remain.
8 subtract 4 is 4,
subtract 4 more
-OOOO
iso.
> O O O O O O O O That's 2 groups.
So, 8 - 4 = 2
>
You can use a number line to show how division is like repeated
subtraction.
_4
_4
^
^
8-4-4 = 0
-M
I I 1 1 1 1 I »
So,8 + 4 = 2
0
4
8
Try These
1. Write a division sentence for each repeated subtraction sentence.
a)
-1
5-1-1-1-1-1=0
5 + 1=5
0
-1
1
-1
2
3
-5
b) 1 0 - 5 - 5 = 0
-1
-1
4
5
-5
10 + 5 = 2
0
-3
c) 6 - 3 - 3 = 0
6+3= 2
0
1 2 6
5
10
-3
Practice
1. Show each division sentence as repeated subtraction on the number line.
a)
12-3=4
10
12
11
b) 8 - 2 = 4
c) 9 - 3 = 3
4
5
2. Write each division sentence as repeated subtraction.
a) !
c) 2-.
• >= 3
15-5-5-5 =0
b
) 4 - M = 4 ±Zlz
\ = 5 20 - 4 - 4 - 4 - 4 - 4 = 0 d) 12 : 4 = 3
e) 25 - 5 = 5 25 - 5 - 5 - 5 - 5 - 5 = 0
f) 5
:
5 ,_ ,
l~
1
~
1
=
0
12-4-4-4=0
5-5 = 0
3. Write a division sentence to solve this problem:
Karl has 20 gerbils. He puts 4 gerbils into each cage.
How many cages does Karl use?
20 - 4 = 5; He uses 5 cages.
Stretch Your Thinking
Find as many ways to put 20 counters into equal groups as you can. Write a
repeated subtraction sentence and a division sentence for each way you find.
2 0 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1- 1- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -
1-1-1=0,
20 - 1 = 20; 2 0 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 - 2 = 0 , 2 0 - 2 = 10;
20 - 4 - 4 - 4 - 4 - 4 = 0, 2 0 - 4 = 5 ; 2 0 - 5 - 5 - 5 - 5 = 0 , 2 0 * 5 = 4 ;
2 0 - 1 0 - 10 = 0 , 2 0 - 10 = 2; 2 0 - 2 0 = 0 , 2 0 - 2 0 = 1
1 2 7
Relating Multiplication
and Division Using
Arrays
8
Q u i c k
R e v i e w
This array has 3 rows of 5.
The multiplication sentence is: 3 X 5
The division sentence is: 15 + 3 = 5
Turn the array to show 5 rows of 3.
The multiplication sentence is: 5 X 3 = 15
The division sentence is: 15 = 5 = 3
These four number sentences
are related sentences.
3X5
5X3
15 =
15 =
XXXXX
XXXXX
X XIX X X
15
XXX
ix x X
X xi X
X Xi
x' X X
1
;
:
= 15
= 15
3=5
5=3
Try These
1. Write a multiplication sentence and a division sentence for each picture.
Sample Answers
a)
XXXX
XXXX
XXXX
XXXX
X
X
X
X
4 X 5 = 20,20 + 4 = 5
1 2 8
b)
XXX X
XXX X
2 X 4 = 8,8 + 2 = 4
Practice
1. Write a multiplication sentence and a division sentence for each picture.
Sample Answers
c)
b)
a)
4 X 4 = 16
3X3 = 9
3 X 5 = 15
16^-4 = 4
9^3 = 3
15 + 3 = 5
2. Write the related sentences for each set of numbers.
a)
1,5,5
1X5 = 5,5X1=5,5 + 5 = 1 5+-1=5
b )
2,3,6
2 X 3 = 6 , 3 X 2 = 6,6 + 3 = 2,6 + 2 = 3
c)
3,5,15 3 X 5 = 15,5 X 3 = 15,15 + 5 = 3,15
f
3= 5
d) 4 . 4 1 6 4 X 4 = 1 6 , 1 6 + - 4 = 4
:
e)
S 7 10 5 X 2 = 1 0 , 2 X 5 = 1 0 , 1 0 - ^ 5 = 2 , 1 0 + 2 = 5
;
;
3. Divide. Use multiplication facts to help you.
a)
12 + 4 =
d)
10-2 =
?
JL_
b) 10 + 5 =
e) 16 + 4
W_
= _JL_
c
)
f)
5
15 + 3 =
4 - 4=
1
.„_
Stretch Your T h i n k i n g
Berta has a collection of antique dolls.
If Berta puts her dolls into groups of 3 or 4, she has 2 dolls left over.
How many dolls might Berta have?
14
1 2 9
Relating ultiplication
arid Division Using
Groups
' " ^ • o ^
_
Review
Q u i c k
Multiplication
•.-''I
<• i v i
1
i
M
!i
1
" M *
and division
i
juon
.i >
1
)
s111Mie11
are
related.
Multiplication: 3
4
12
Division a s sharing: ! 2 :
12
'j
I '
>
•!
4
1
t ! K <ai< e . -n >• *lp v' >u t h i n k a t ) o u t d i v i s i o n .
iiese
i
n H ii I sole ii K H I
• n t e n c c ui<i e d i v i s i o n M - n U ' i i o
b)
OK
c)
2 X 4 = 8
li ••- 2 = 4
6 •:- 3 = 2
8 + 4 = 2
)
-
i»
for ' e d i
2 X 3 = 6
,
-'i
1
•4 X 2 .- 8
• leiaied number
1
f
2
X
5 =
5 X 3
10
= 15
.• - n e -\n - \ f o r e a c h s e t o f n u n i b e r s .
5
X
2
3 X 5
10
10 + 2 = 5
10 + 5 = 2
= 15
1 5 - 5 = 3
15 +
3
= 5
Practice
1. Multiply or divide to solve the riddle.
Match each letter to its answer. Some letters are not used.
Riddle: What goes up a chimney down, but can't go down a chimney up?
9
3X3 =
1
3X5 =
5
(C)
1X4=
4
(B)
5X2=
(R)
20 + 4 =
5
(A)
9 + 3=
1
0
3
4X2 =
8
(Z)
3X4=
1
2
(P)
5X4=
4 + 2 =
2
(U)
4 X 4 =
1
6
(L)
1+ 1=
. J
N
u
M
B
1
(Q)
2
R
E
L
L
15
20
16
16
(M)
0
(E)
(N)
A
2. Write the related number sentences for each set of numbers.
a) 5,5,25
^
2x
b) 2,2,4
c)
5 X^5 = 2 5 , 2 5 + 5 =
:.2ll^
3,3,1
3 X 1 = 3 , 1 X 3 = 3, 3 + 3 = 1,3 + 1 = 3
d) 4,3,12
4 X 3 = 12,3 X 4 = 12,12 + 4 = 3,12 + 3 = 4
3. Pono bought some packages of tennis balls.
Each package holds 3 balls.There are 15 balls altogether.
How many packages did Pono buy?
15 + 3 = 5; He b o u g h t 5 packages.
Stretch Your Thinking
Write as many division sentences as you can that have an answer of 3.
3 + 1 =3
6 + 2=3
9 + 3=3
12 + 4 = 3
15 + 5 = 3
131