lesson 6 multiply and divide rational numbers...

Introduction

©Curriculum Associates, LLC Copying is not permitted.44

Use What You Know

Lesson 6 Multiply and Divide Rational Numbers

Lesson 6Multiply and Divide Rational Numbers

In Lesson 4, you learned the rules for multiplying and dividing integers. Take a look at this problem.

Vera is given three fractions. She wants to find the two fractions that have the greatest product. Which two fractions does Vera multiply together to find the greatest product?

252

45

352

Use the math you already know to solve the problem. Begin by solving a simpler problem.

a. Use what you know about multiplying and dividing integers to find the product of

2 10 ·· 5 3 20 ·· 5 . You can think of 2 10 ·· 5 3 20 ·· 5 as 22 3 4, which equals .

b. Find the product of 2 10 ·· 5 3 1 2 15 ·· 5 2 . Think of 2 10 ·· 5 3 1 2 15 ·· 5 2 as 3 which

equals .

c. Think about multiplying 2 2 ·· 5 3 1 2 3 ·· 5 2 . Will the product be positive or negative? Explain

your reasoning.

d. Explain how you can use the rules you know for multiplying positive and negative integers to multiply positive and negative fractions.

7.NS.A.2a

7.NS.A.2b

7.NS.A.2c

©Curriculum Associates, LLC Copying is not permitted. 45

Find Out More


In previous lessons you performed operations with integers. Now you will learn about performing operations with other types of rational numbers. A rational number is a number that can be written as the quotient of two integers. Rational numbers include positive and negative fractions and decimals, and also integers since an integer can be written as a fraction with 1 as the denominator: 26 5 2 6 ·· 1 .

When you multiply positive and negative fractions, you multiply the numerator by the numerator and then the denominator by the denominator, all the while using the rules for multiplying integers.

In the problem on the previous page, you multiply the fractions to find the greatest product.

2 3 ·· 5 3 4 ·· 5 5 2 12 ·· 25 4 ·· 5 3 1 2 2 ·· 5 2 5 2 8 ·· 25 2 3 ·· 5 3 1 2 2 ·· 5 2 5 6 ·· 25

So, 2 3 ·· 5 3 1 2 2 ·· 5 2 has the greatest product. It is the only product that is a positive number.

You also use the rules for multiplying positive and negative numbers when you divide fractions. Look at the examples below.

2 2 ·· 5 4 4 ·· 5 5 2 2 ·· 5 3 5 ·· 4 2 3 ·· 10 4 1 2 4 ·· 5 2 5 2 3 ·· 10 3 1 2 5 ·· 4 2 5 2 10 ·· 20 5 15 ·· 40

5 2 1 ·· 2 5 3 ·· 8

The rules for multiplying and dividing integers also apply to decimals.

23.5 3 (22.4) 5 8.4 3.5 3 (22.4) 5 28.4 23.5 3 2.4 5 28.4

24.25 4 (20.2) 5 21.25 4.25 4 (20.2) 5 221.25 24.25 4 0.2 5 221.25

Reflect1 How are the rules for dividing two negative fractions related to the rules for multiplying

two negative fractions? Will the answer be positive or negative? Explain.

Modeled and Guided Instruction

Learn About


Lesson 6


Multiplying Rational Numbers

Read the problem below. Then explore different ways to understand it.

Alicia, Emma, and Nick dive into an extremely deep pool. Alicia dives to a position of

2 9 ·· 4 yards relative to the surface of the water. Emma dives to a position twice as deep

as Alicia. Nick dives to a position 2 ··

3

as deep as Alicia. What are Emma’s and Nick’s

positions, relative to the surface of the water?

Picture It You can use models to understand the problem.

0

–1

–2

–3

–4

–5

Alicia’s Dive

94

–

0

–1

–2

–3

–4

–5

Emma’s Dive

184

−

94

–

0

–1

–2

–3

–4

–5

34

–

64

–

94

–

Nick’s Dive

Model It You can also use multiplication to understand the problem.

Alicia’s position: 2 9 ·· 4 yards

Emma’s position: 2 3 Alicia’s position 5 2 3 1 2 9 ·· 4 2 yards

Nick’s position: 2 ·· 3 3 Alicia’s position 5 2 ·· 3 3 1 2 9 ·· 4 2 yards

©Curriculum Associates, LLC Copying is not permitted. 47Lesson 6 Multiply and Divide Rational Numbers

Connect It Now you will solve the problem on the previous page.

2 Look at the number line representing Emma’s dive. What do the two arrows represent?

3 Look at Model It. Why do you multiply Alicia’s position by 2 to find Emma’s position?

4 Look at the number line representing Nick’s dive. Why are there three equal arrows ending at the position of Alicia’s dive? How will you use the arrows to find Nick’s position?

5 Look at Model It. Why do you multiply Alicia’s position by 2 ·· 3 to find Nick’s position?

6 Evaluate the products to find Emma’s position and Nick’s position.

2 3 1 2 9 ·· 4 2 5 2 ·· 3 3 1 2 9 ·· 4 2 5

7 Explain how you can use the rules for multiplying positive and negative numbers to check that your answers to problem 6 have the correct sign.

Try It Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

8 The temperature at noon is 08C. The temperature changes by 2 3

·· 2 8C every hour for

4 hours. Write an expression to represent this situation. What is the temperature

at 4:00 pm?

9 2 5

·· 6 3 1 2 3 ·· 10 2 5

Modeled and Guided Instruction

Learn About


Lesson 6


Dividing Rational Numbers

Read the problem below. Then explore different ways to understand it.

The volume of water in Pete’s pool is changing by 20.05 gallon per hour due to a leak. How many hours will it take for the volume of water in the pool to change by a total of 20.3 gallon?

Picture It You can use a bar model to understand the problem.

20.1 20.1 20.1

20.05 20.05 20.05 20.05 20.05 20.05

Model It You can use repeated subtraction to help understand the problem.

Total change in water volume 5

Sum of the changes in water volume each hour

20.3 5 0.05 1 (20.05) 1 (20.05) 1 (20.05) 1 (20.05) 1 (20.05)

Model It You can use division to help understand the problem.

Total change in water volume 4

Changes in water volume each hour 5

Number of hours

20.3 4 (20.05) 5 ?


Connect It Now you will solve the problem from the previous page using division.

10 Look at Picture It. What do the three bars labeled 20.1 represent? What do the bars

labeled 20.05 represent?

11 Why are the bars in Picture It labeled with negative numbers?

12 Look at Model It. How do you decide how many times to add 2 0.05?

13 How many hours will it take for the volume of water in the pool to change by a total of

20.3 gallon?

14 Explain how you can use the rules for dividing a negative number by a negative number to determine the sign of the quotient 20.3 4 (20.05).

Try It Use what you just learned to solve these problems. Show your work on a separate sheet of paper.

15 Rebecca’s bank statement shows a change of 2$1.50 in her account each week.

How many weeks will it take before the total change is 2$12?

16 The elevation of a small submarine changes by 2 3

·· 8 mile after each exploration. After

how many explorations will the submarine’s elevation have changed by 21 1 ·· 2 miles?

Guided Practice

Practice


Lesson 6


Multiplying and Dividing Rational Numbers

Study the example below. Then solve problems 17–19.

Example

What is the product of these three fractions: 2 3 ·· 8 , 2 2 ·· 3 , 2 5 ·· 2 ?

Look at how you could show your work using multiplication.

1 2 3 ·· 8 2 3 1 2 2 ·· 3 2 3 1 2 5 ·· 2 2 5 ?

1 1 6 ··· 24 2 3 1 2 5 ·· 2 2 5 ?

2 30 ··· 48 5 2 5 ·· 8

Solution

17 Lily records the change in her dog’s weight over 4 weeks.

My Dog’s Weight Change

Week 1 2 3 4

Weight Change (in pounds)

2 3 ·· 16 1 ·· 16 2 5 ·· 16 2 1 ·· 16

What is the dog’s average weekly weight change?

Show your work.

Solution

2 5 ·· 8

Pair/ShareUse multiplication rules to confirm that the answer should be negative.

Pair/ShareDiscuss what step you decided to take first in solving this problem.

To find the average of a set of numbers, do you multiply or divide?

The student multiplies the first two fractions, then multiplies by the third fraction.


18 Mitch is cutting wood to make a garden marker. Each garden marker

changes the length of the piece of wood by 20.8 foot. The length of

wood changes by a total of 24 4 ·· 5 feet after Mitch cuts all the garden

markers. How many pieces of wood did Mitch cut?

Show your work.

Solution

19 The amount of shore at a local beach changes an average of 2 3 ·· 4 foot

each year. What is the change in the amount of shore after 7 years?

A 5 1 ·· 4 feet

B 3 ·· 28 foot

C 2 3 ·· 28 foot

D 25 1 ·· 4 feet

Rashad chose C as the correct answer. How did he get that answer?

How can you write 20.8 as a fraction?

If the amount of shore is decreasing, will your answer be positive or negative?

Pair/ShareDiscuss how you can check that your answer is reasonable.

Pair/ShareDoes Rashad’s answer make sense? Why or why not?

Independent Practice

Practice


Lesson 6


Multiplying and Dividing Rational Numbers

Solve the problems.

1 The average of a group of numbers is 20.75. Which statement must be true?

A There must be 4 numbers in the group.

B All of the numbers in the group must be negative.

C The sum of the numbers in the group must be negative.

D The sum of the numbers in the group could be positive.

2 A clogged bathtub drains at a constant rate. The amount of water in the bathtub changes

by 23.75 gallons in one hour. What is the change in the amount of water in the bathtub

after 1 ·· 3 of an hour?

A 21.25 gallons

B 22.25 gallons

C 26.25 gallons

D 211.25 gallons

3 Mara’s family is driving to her grandmother’s house. The family travels 239.4 miles between the hours of 9:10 am to 1:40 pm. What is an equation that Mara can use to determine their average rate of travel, R, rounded to the nearest mile per hour? Use the given numbers and operations to complete an equation.

21 3 4

4.5 9.10 1.40239.4

5 R

Self Check


Go back and see what you can check off on the Self Check on page 1.

4 Victor claims that when 1 ·· 6 is divided by a fraction, the result will always be greater than 1 ·· 6 .

Write a number from 1 through 9 in the boxes below to both support and contradict

Victor’s claim.

Supports Victor’s Claim Contradicts Victor’s Claim

1 ·· 6 4 1 ·· 6 4

5 Frank, Leah, and Tom play a trivia game. Frank’s score is 22. Leah’s score is 3 ·· 4 of Frank’s score.

Tom’s score is 2 ·· 3 of Leah’s score. What is Tom’s score?

Show your work.

Answer

6 Write all the division problems that will have a negative quotient using two fractions from

this list: 2 1 ·· 2 , 4 ·· 5 , 2 3 ·· 8 . Then evaluate one of your problems. What number must be in all of your

problems? Why?

Show your work.

Answer

lesson 6 multiply and divide rational numbers...

Documents