Math Gr4 Ch6

Chapter 6
Algebra: Use Multiplication and Division
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6

Lesson 6-1 Multiplication and Division
Expressions
Lesson 6-2 Problem-Solving Strategy: Work
Backward
Lesson 6-3 Order of Operations
Lesson 6-4 Algebra: Solve Equations Mentally
Lesson 6-5 Problem-Solving Investigation:
Choose a Strategy
Lesson 6-6 Algebra: Find a Rule
Lesson 6-7 Balanced Equations

6-1 Multiplication and Division Expressions

Five-Minute Check (over Chapter 5)
Main Idea
California Standards
Example 1
Example 2
Example 3


• I will write and find the value of multiplication
and division expressions.


Standard 4AF1.1 Use letters, boxes, or other
symbols to stand for any number in simple
expressions or equations (e.g., demonstrate an
understanding and the use of the concept of a
variable).


Jake had 4 boxes of apples. There are 6 apples
in each box. Find the value of 4 × n if n = 6.

4×n Write the expression.


4×6 Replace n with 6.

24 Multiply 4 and 6.


Marian has 5 CD cases. Each CD case has 2 CDs
inside. Find the value of 5 × n if n = 2.

A. 7

B. 10

C. 5

D. 2


Find the value of x ÷ (3 × 2) if x = 30.
In Lesson 3-1, you learned that you need to perform
the operations inside parentheses first.

x ÷ (3 × 2) Write the expression.
30 ÷ (3 × 2) Replace x with 30.
30 ÷ 6, or 5 Find (3 × 2) first. Then find 30 6.

Answer: So, the value of x (3 2) if x = 30 is 5.


Find the value of 45 (x × 1) if x = 5.

A. 9

B. 45

C. 5

D. 1


Judy has d dollars to buy bottles of water that cost
$2 each. Write an expression for the number of
bottles of water she can buy.
Words Dollars divided by cost

Variable Let d = dollars.

Expression dollars divided by cost
d ÷ $7

Answer: So the number of bottles of water Judy can
buy is d 2.


Toby has d dollars to spend on discounted books
that cost $3 a piece. Write an expression for the
number of books he can buy.

A. d ÷ 3

B. d – 3

C. d + 3

D. d × 3

6-2 Problem-Solving Strategy: Work Backward

Five-Minute Check (over Lesson 6-1)
Main Idea
Example 1: Problem-Solving Strategy


• I will solve problems by working backward.


Standard 4MR1.1 Analyze problems by
identifying relationships, distinguishing relevant
from irrelevant information, sequencing and
prioritizing information, and observing patterns.


Standard 4NS3.0 Students solve problems
involving addition, subtraction, multiplication, and
division of whole numbers and understand the
relationships among the operations.


Currently, there are 25 students in the chess
club. Last October, 3 students joined. Two
months before that, in August, 8 students
joined. How many students were in the club
originally?


Understand
What facts do you know?
• Currently, there are 25 students in the club.
• 3 students joined in October.
• 8 students joined in August.

What do you need to find?
• The number of students that were in the club
originally.


Plan
Work backward to solve the problem.


Solve
Work backward and use inverse operations. Start
with the end result and subtract the students who
joined the club.

25
– 3
22


Solve

22
– 8
14

Answer: So, there were 14 students in the club
originally.


Check
Look back at the problem. A total of 3 + 8 or 11
students joined the club. So, if there were 14 students
originally, there would be 14 + 11 or 25 students in the
club now.

The answer is correct.

6-3 Order of Operations

Main Idea and Vocabulary
Key Concept: Order of Operations
Example 1
Example 2


• I will use the order of operations to find the value
of expressions.

• order of operations


Standard 4AF1.2 Interpret and evaluate
mathematical expressions that now use
parentheses.

Standard 4AF1.3 Use parentheses to
indicate which operation to perform first when
writing expressions containing more than two
terms and different operations.


Find the value of 12 – (4 + 2) ÷ 3.

12 – (4 + 2) ÷ 3 Write the expression.

12 – 6 ÷3 Parentheses first. (2 + 4) = 6

12 – 2 Multiply and divide from left to right.
6÷3=2

10 Add and subtract from left to right.
12 – 2 = 10


Find the value of 21 (3 + 4) + 5.

A. 16

B. 1

C. 8

D. 12


Find the value of 4x + 3y ÷ 2, when x = 7 and y = 2.

Follow the order of operations.

4x + 3y ÷ 2 = 4 7+3×2÷2 Replace x with 7 and
y with 2.
= 28 + 3 Multiply and divide from
left to right.
= 31 Add.

Answer: 31


Find the value of 3x – 2y + 12 when x = 5 and y = 3.

A. 19

B. 11

C. 21

D. 12

6-4 Algebra: Solve Equations Mentally

Main Idea
Example 1
Example 2
Example 3

Multiplication and Division Equations


• I will solve multiplication and division equations
mentally.


Standard 4AF1.1 Use letters, boxes, or other
symbols to stand for any number in simple
expressions or equations (e.g., demonstrate an
understanding and the use of the concept of a
variable).


Mansis’s Used Car Lot has 8 rows of cars with a
total of 32 cars. Solve 8 × c = 32 to find how many
cars are in each row.


One Way: Use Models
Step 1 Model the
equation.


One Way: Use Models
Step 2 Find the value of c.

8 × c = 32

Answer: So, c = 4.


Another Way: Mental Math

8 × c = 32

8 × 4 = 32 You know that 8 × 4 = 32.

Answer: So, c = 4.


Kyung has just planted a garden. He has a total of
49 vegetables with 7 vegetables in each row. Solve
7 x = 49 to find how many rows of vegetables
there are.
A. 6

B. 7

C. 8

D. 49


Solve 16 ÷ s = 8.

16 ÷ s = 8

16 ÷ 2 = 8

s=2 You know that 16 2 = 8.

Answer: So, the value of s is 2.


Solve 36 p = 6.

A. 6

B. 7

C. 8

D. 9


Six friends went shopping. They each bought the
same number of t-shirts. A total of 24 t-shirts were
bought. Write and solve an equation to find out how
many t-shirts each person bought.
Write the equation.
Words 6 friends bought 24 t-shirts

Variable Let t = the number of t-shirts bought
per person.
Expression 6 t = 24


Solve the equation.

6 t = 24
6 4 = 24
t=4

Answer: So each person bought 4 t-shirts.


Six friends went to a driving range and hit a total of
54 golf balls. If they all hit the same number of golf
balls, how many did each one hit?

A. 7

B. 8

C. 9

D. 10

6-5 Problem-Solving Investigation: Choose a Strategy

Main Idea
Example 1: Problem-Solving Investigation


• I will choose the best strategy to solve a problem.


Standard 4MR1.1 Analyze problems by
identifying relationships, …, and observing
patterns.


4NS3.0 Students solve problems involving
addition, subtraction, multiplication, and division
of whole numbers and understand the
relationships among the operations.


MATT: I take 30-minute guitar
lessons two times a week. There are
four weeks in a month. How many
minutes do I have guitar lessons
each month?

YOUR MISSION: Find how many minutes
Matt has guitar lessons each month.


Understand
What facts do you know?
• Each lesson Matt takes is 30 minutes long.
• He takes lessons two times a week.
• There are four weeks in a month.
What do you need to find?
• Find how many minutes Matt has guitar
lessons each month.


Plan
You can use the four-step plan along with
addition and multiplication to solve the problem.


Solve
Find how many minutes Matt has lessons each
week.

30 lesson 1
+ 30 lesson 2
60 minutes per week


Solve
Find how many minutes Matt has lessons each
week.

60 minutes per week
× 4 weeks per month
240 minutes per month

Answer: So, Matt has lessons 240 minutes
each month.


Check
Matt has lessons 30 + 30 or 60 minutes each
week. This means he has 60 + 60 + 60 + 60
or 240 minutes of lessons each month.

So, the answer is correct.

6-6 Algebra: Find a Rule

Main Idea
Example 1
Example 2
Example 3


• I will find and use a rule to write an equation.


Standard 4AF1.5 Understand that an
equation such as y = 3x + 5 is a prescription
for determining a second number when a
first number is given.


Mike earns $10 when he
babysits for 2 hours. He earns
$20 when he babysits for 4
hours. If he babysits for 6
hours, he earns $30. Write a
rule that describes the money
Mike earns.

Put the information in a table.
Then look for a pattern to
describe the rule.


Pattern: 2 × 5 = 10
4 × 5 = 20
6 × 5 = 30

Rule: Multiply by 5.

Equation: x × 5 = y


Ricardo earns $16 dollars when he mows 2 lawns
of grass. He earns $32 when he mows 4 lawns, and
$48 when he mows 6 lawns. Write a rule that
describes the money Ricardo earns.
A. 8x = y

B. x+y=8

C. 2x + 8 = y

D. x×8=y


Use the equation from
Additional Example 1 to find
how much money Mike earns
for babysitting for 8, 9, or 10
hours.


x×5=y
8 × 5 = $40

40
x×5=y x×5=y
45
9 × 5 = $45 10 × 5 = $50 50

Answer: So, Mike will earn $40, $45, or $50 if he
babysits for 8, 9, or 10 hours.


Use the equation x 8 = y to find how much money
Ricardo earns for mowing 7 or 8 lawns.

A. $49, $64

B. $15, $16

C. $56, $64

D. $63, $72


The cost of admission
into a water park is shown
in the table to the right.
Find a rule that describes
the number pattern. Then
use the rule to write an
equation.


Pattern: 6÷6=1
12 ÷ 6 = 2
18 ÷ 6 = 3

Rule: Divide by 6.

Equation: c ÷ 6 = n


The cost of admission into a basketball game is
shown in the table below. Find a rule that describes
the number pattern. Then use the rule to write an
equation.
A. c÷9=n

B. c+9=n

C. c+n=9

D. c–9=n


Use the equation from
Additional Example 3 to
find how many people
will be admitted to the
park for $24, $30, and
$36.


c÷6=n

24 ÷ 6 = 4

c÷6=n c÷6=n
4
30 ÷ 6 = 5 36 ÷ 6 = 6 5
6

Answer: So, $24, $30, and $36 will by 4, 5, and 6
people tickets.


Use the equation c 9 = n to find how many
people will be admitted to the basketball game
for $45 and $63.

A. 4, 5

B. 5, 6

C. 7,8

D. 5, 7

6-7 Balanced Equations

Main Idea
Example 1
Example 2
Example 3


• I will balance multiplication and division equations.


Standard 4AF2.2 Know and understand
that equals multiplied by equals are equal.


Show that the equality of 6r = 24 does not change
when each side of the equation is divided by 6.

6r = 24 Write the equation.
6r ÷ 6 = 24 ÷ 6 Divide each side by 6.
r=4 So, r = 4.


Check

6r = 24

6 × 4 = 24

24 = 24


Show that the equality of 3y = 9 does not change
when each side of the equation is divided by 3.

A. 3y ÷ 3 = 9 ÷ 3; 6 = 6

B. 3y ÷ 3 = 9 ÷ 3; 3 = 3

C. 3y ÷ 3 = 9; 9 = 9

D. 3y = 9 ÷ 3; 3 = 9


Show that the equality of q ÷ 7 = 4 does not change
when each side of the equation is multiplied by 7.

q÷7=4 Write the equation.
q÷7×7=4×7 Multiply each side by 4.
q = 28 So, q = 28.


Check

q÷7=4
28 ÷ 7 = 4
4=4


Show that the equality v 5 = 5 does not change
when each side of the equation is multiplied by 5.

A. v 5 5 = 5; 10 = 10

B. v 5 5=5 5; 25 = 25

C. v 5 = 5; 5 = 5

D. v 5 5=5 5; 10 = 10


Find the missing number in 5 × 10 × 4 = 50 × .

5 × 10 × 4 = 50 × Write the equation.
5 × 10 × 4 = 50 × You know that 5 × 10 = 50.

Each side of the equation must be multiplied by
the same number to keep the equation balanced.

Answer: So, the missing number is 4.


Find the missing number in 8 5 3 = 40 .

A. 8

B. 5

C. 3

D. 40


Find the missing number in 2 × 12 ÷ 4 = 24 × .

2 × 12 ÷ 4 = 24 × Write the equation.
2 × 12 ÷ 4 = 24 × You know that 2 × 12 = 24.

Each side of the equation must be divided by the
same number to keep the equation balanced.

Answer: So, the missing number is 4.


Find the missing number in 4 11 2 = 44 .

A. 4

B. 11

C. 44

D. 2

6

Five-Minute Checks

Multiplication and Division Equations

6

Lesson 6-1 (over Chapter 5)
Lesson 6-2 (over Lesson 6-1)

6
(over Chapter 5)

Tell whether 13 is composite, prime, or neither.

A. composite

B. prime

C. neither

6
(over Chapter 5)


A. composite

B. prime

C. neither

6
(over Lesson 6-1)

Find the value of each expression if m = 4 and
n = 8.
m 10

A. 18

B. 14

C. 40

D. 80

6
(over Lesson 6-1)

n = 8.
3 (n m)

A. 1.5

B. 6

C. 12

D. 36

6
(over Lesson 6-1)

n = 8.
(12 m) n

A. 6

B. 16

C. 24

D. 64

6
(over Lesson 6-1)

n = 8.
(n m) 2

A. 6

B. 16

C. 24

D. 64

6
(over Lesson 6-2)

Work backward to solve the problem. Lance had 4
granola bars left from his weekend hike. On
Saturday, he ate 2 bars. Before he left for the trip
on Friday, his mother added 5 bars to what he had.
How many bars did he have to start with?
A. 7 bars

B. 5 bars

C. 3 bars

D. 1 bar

6
(over Lesson 6-3)

Find the value of each expression.

4 + (5 2) – 1

A. 6

B. 11

C. 13

D. 14

6
(over Lesson 6-3)


6+6 3

A. 12

B. 15

C. 24

D. 36

6
(over Lesson 6-3)


(17 – 3) – (2 4)

A. 6

B. 7

C. 8

D. 22

6
(over Lesson 6-3)


(21 3) + 3

A. 9

B. 10

C. 21

D. 22

6
(over Lesson 6-4)

Solve each equation mentally.

5 x = 25

A. 4

B. 20

C. 5

D. 6

6
(over Lesson 6-4)


56 m=8

A. 8

B. 48

C. 49

D. 7

6
(over Lesson 6-4)


r 7=3

A. 21

B. 3

C. 24

D. 7

6
(over Lesson 6-4)


k 9 = 36

A. 3

B. 45

C. 4

D. 36

6
(over Lesson 6-5)

Use any strategy to solve. Tell which strategy you
used. Jacobo is 6 years old and his brother is 2
years old. How old will each of them be when
Jacobo is twice his brother’s age?

A. Jacobo will be 12 and his brother will be 6.
B. Jacobo will be 8 and his brother will be 4.
C. Jacobo will be 7 and his brother will be 3.
D. Jacobo will be 10 and his brother will be 6.

6
(over Lesson 6-6)

Find a rule and equation that describes the pattern.
Then use the equation to find the missing number.

A. multiply by 4; x 4 = y;
18
B. add 8; x + 8 = y; 14
C. multiply by 3; x 3 = y;
18
D. multiply by 3; y 3 = x;
18

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Math Gr4 Ch6

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