Math Gr4 Ch6

Chapter 6Algebra: Use Multiplication and Division

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Lesson 6-1 Multiplication and Division Expressions

Lesson 6-2 Problem-Solving Strategy: Work Backward

Lesson 6-3 Order of OperationsLesson 6-4 Algebra: Solve Equations

MentallyLesson 6-5 Problem-Solving Investigation:

Choose a StrategyLesson 6-6 Algebra: Find a RuleLesson 6-7 Balanced Equations

6Algebra: Use Multiplication and Division



Five-Minute Check (over Chapter 5)Main IdeaCalifornia StandardsExample 1Example 2Example 3

6-1 Multiplication and Division Expressions



• 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.



A. 7

B. 10

C. 5

D. 2

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



Find the value of x ÷ (3 × 2) if x = 30.

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.


In Lesson 3-1, you learned that you need to perform the operations inside parentheses first.


A. 9

B. 45

C. 5

D. 1

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



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.

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

Words

Variable

Expression

Dollars

Let d = dollars.

divided by cost

dollars

d

divided by

÷

cost

$7



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




Five-Minute Check (over Lesson 6-1)Main IdeaCalifornia StandardsExample 1: Problem-Solving Strategy

6-2 Problem-Solving Strategy: Work Backward



• 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.


22

25– 3


Solve

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


14

22– 8


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.




Five-Minute Check (over Lesson 6-2)Main Idea and VocabularyCalifornia StandardsKey Concept: Order of OperationsExample 1Example 2

6-3 Order of Operations



• 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 – 3 ÷6

10

Write the expression.

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

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

Parentheses first. (2 + 4) = 6

12 (4 2) 3 – ÷+

12 – 2



A. 16

B. 1

C. 8

D. 12

Find the value of 21 ÷ (3 + 4) + 5.


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

= 28 + 3

= 31

Replace x with 7 and y with 2.

Multiply and divide from left to right.Add.

Answer: 31



A. 19

B. 11

C. 21

D. 12

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



Five-Minute Check (over Lesson 6-3)Main IdeaCalifornia StandardsExample 1Example 2Example 3

6-4 Algebra: Solve Equations Mentally

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.




Step 1 Model the equation.

One Way: Use Models



Step 2 Find the value of c.

8 × c = 32

Answer: So, c = 4.

One Way: Use Models


8 × c = 32


You know that 8 × 4 = 32.8 × 4 = 32

Answer: So, c = 4.

Another Way: Mental Math



A. 6

B. 7

C. 8

D. 49

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.


16 ÷ s = 8

Answer: So, the value of s is 2.

Solve 16 ÷ s = 8.


16 ÷ 2 = 8

s = 2 You know that 16 ÷ 2 = 8.



A. 6

B. 7

C. 8

D. 9

Solve 36 ÷ p = 6.


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

Variable

Expression

6 friends bought 24 t-shirts

Let t = the number of t-shirts bought per person.

6 × t = 24


Solve the equation.

6 × t = 24

Answer: So each person bought 4 t-shirts.


6 × 4 = 24

t = 4



A. 7

B. 8

C. 9

D. 10

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?



Five-Minute Check (over Lesson 6-4)Main IdeaCalifornia StandardsExample 1: Problem-Solving Investigation

6-5 Problem-Solving Investigation: Choose a Strategy



• 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.


60

30+ 30

lesson 1lesson 2minutes per week


Solve

Find how many minutes Matt has lessons each week.


240

60× 4

minutes per weekweeks per monthminutes 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



• 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 = 104 × 5 = 206 × 5 = 30

Rule: Multiply by 5.

Equation: x × 5 = y




A. 8x = y

B. x + y = 8

C. 2x + 8 = y

D. x × 8 = 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.


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


x × 5 = y x × 5 = y

8 × 5 = $40

9 × 5 = $45 10 × 5 = $50

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

404550



A. $49, $64

B. $15, $16

C. $56, $64

D. $63, $72

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


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 = 112 ÷ 6 = 218 ÷ 6 = 3

Rule: Divide by 6.

Equation: c ÷ 6 = n



A. c ÷ 9 = n

B. c + 9 = n

C. c + n = 9

D. c – 9 = 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.


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

c ÷ 6 = n c ÷ 6 = n

24 ÷ 6 = 4

30 ÷ 6 = 5 36 ÷ 6 = 6

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

456



A. 4, 5

B. 5, 6

C. 7,8

D. 5, 7

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




6-7 Balanced Equations



• 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


6r ÷ 6 = 24 ÷ 6

r = 4

Write the equation.

Divide each side by 6.

So, r = 4.


Check


6r = 24

6 × 4 = 24

24 = 24



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


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

q ÷ 7 = 4

q ÷ 7 × 7 = 4 × 7

q = 28

Write the equation.

Multiply each side by 4.

So, q = 28.



Check


q ÷ 7 = 4

28 ÷ 7 = 4

4 = 4



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

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


Write the equation.

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 5 × 10 × 4 = 50 × .

5 × 10 × 4 = 50 ×5 × 10 × 4 = 50 ×




A. 8

B. 5

C. 3

D. 40

Find the missing number in 8 × 5 × 3 = 40 × .


Write the equation.

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 2 × 12 ÷ 4 = 24 × .

2 × 12 ÷ 4 = 24 ×2 × 12 ÷ 4 = 24 ×




A. 4

B. 11

C. 44

D. 2

Find the missing number in 4 × 11 ÷ 2 = 44 × .




Five-Minute Checks

Multiplication and Division Equations


Lesson 6-1 (over Chapter 5)Lesson 6-2 (over Lesson 6-1)Lesson 6-3 (over Lesson 6-2)Lesson 6-4 (over Lesson 6-3)Lesson 6-5 (over Lesson 6-4)Lesson 6-6 (over Lesson 6-5)Lesson 6-7 (over Lesson 6-6)


A. composite

B. prime

C. neither

(over Chapter 5)

Tell whether 13 is composite, prime, or neither.


A. composite

B. prime

C. neither

(over Chapter 5)



(over Chapter 5)

A. composite

B. prime

C. neither



(over Chapter 5)

A. composite

B. prime

C. neither



(over Chapter 5)

A. composite

B. prime

C. neither



A. 18

B. 14

C. 40

D. 80

(over Lesson 6-1)

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

m × 10


A. 1.5

B. 6

C. 12

D. 36

(over Lesson 6-1)


3 × (n ÷ m)


A. 6

B. 16

C. 24

D. 64

(over Lesson 6-1)


(12 ÷ m) × n


(over Lesson 6-1)

A. 6

B. 16

C. 24

D. 64


(n × m) ÷ 2


A. 7 bars

B. 5 bars

C. 3 bars

D. 1 bar

(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?


(over Lesson 6-3)

A. 6

B. 11

C. 13

D. 14

Find the value of each expression.

4 + (5 × 2) – 1


A. 12

B. 15

C. 24

D. 36

(over Lesson 6-3)


6 + 6 × 3


A. 6

B. 7

C. 8

D. 22

(over Lesson 6-3)


(17 – 3) – (2 × 4)


A. 9

B. 10

C. 21

D. 22

(over Lesson 6-3)


(21 ÷ 3) + 3


A. 4

B. 20

C. 5

D. 6

(over Lesson 6-4)

Solve each equation mentally.

5 × x = 25


A. 8

B. 48

C. 49

D. 7

(over Lesson 6-4)


56 ÷ m = 8


A. 21

B. 3

C. 24

D. 7

(over Lesson 6-4)


r ÷ 7 = 3


(over Lesson 6-4)

A. 3

B. 45

C. 4

D. 36


k × 9 = 36


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.

(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. 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

(over Lesson 6-6)

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

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Education

Math Gr4 Ch6