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Practice Lesson 4 Multiply and D

ivide Positive and Negative Integers

Unit 1

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©Curriculum Associates, LLC Copying is not permitted. 29Lesson 4 Multiply and Divide Positive and Negative Integers

Name:

Lesson 4

Multiply and Divide Positive and Negative Integers

Prerequisite: Add and Subtract Positive and Negative Integers

Study the example showing how to add and subtract positive and negative integers. Then solve problems 1–6.

1 Geneva has 24 points Then her spin lands on blue What is her score now? Explain

2 Geneva and Juan start a new game Each player has 0 points

a. Geneva’s first two spins are yellow and red What is her score now? Explain

b. Juan’s first two spins are red and green What is his score now? Explain

c. Which player has a greater score now? Explain

Example

Geneva and Juan are playing a game that involves a spinner The table shows how the player’s score changes according to the color the spinner lands on

Color Red Yellow Blue Green

Number of Points 12 24 13 21

Juan has 32 points Then his spin lands on yellow What is his score now?

Yellow means Juan’s score changes by 24 points

32 2 4 5 32 1 (24) 5 28 Juan’s score is 28 points

26 27 28 29 30

24

31 32 33 34

2929

27 points; Landing on blue means she gains 3 points, and 24 1 3 5 27.

22; 0 2 4 5 24; 24 1 2 5 22

1; 0 1 2 5 2; 2 1 (21) 5 1

Juan; 1 > 22, so Juan’s score is greater.

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Solve.

3 Describe a situation that the expression 24 1 (21) could represent Find the sum and tell what it represents in terms of the situation

4 A water bird is fl ying 11 feet above the surface of a pond It dives 15 feet down and then rises 3 feet to catch a fi sh What is the new position of the bird relative to the surface of the water?

Show your work.

Solution:

5 Steve’s average long jump distance is 13 ft He uses a table to keep track of the distance he jumps in six long jumps during his next practice

Jump Number 1 2 3 4 5 6

Distance (ft) 14 11 15 12 13 10

Difference from Average of 13 (ft)

1 22

Complete the table Then use the results to explain whether Steve’s overall long jump performance on these six jumps is better or worse than average

6 Is the expression u23u 1 u4u equal to u23 1 4u? Justify your answer

30

Possible answer: The temperature in a freezer is 24°F. The temperature decreases by 1°F.

The sum, 25, represents the new temperature, 25°F.

11 2 15 5 11 1 (215) 5 24, or 4 feet below the surface 24 1 3 5 21, or 1 foot below the surface

Worse than average; Possible explanation: I added the differences together. The total

difference of 23 shows that Steve jumped 3 feet less than what he’d expect to jump from

his average.

No; Possible explanation: u23u 1 u4u 5 3 1 4 5 7; u23 1 4u 5 u1u 5 1; 7 1

The bird is now 1 foot below the surface of the pond.

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©Curriculum Associates, LLC Copying is not permitted. 31Lesson 4 Multiply and Divide Positive and Negative Integers

Name: Lesson 4

Multiplying Integers

Study the example problem showing how to multiply positive and negative integers. Then solve problems 1–8.

1 Rewrite the addition of groups as a multiplication of groups

(24) 1 (24) 1 (24) 1 (24) 1 (24) 5

2 Complete: Adding 5 groups of (24) is the same

as 5 groups of (24)

3 What is the total distance he dives in feet?

4 Use a model to multiply 5 ? (23)

Show your work.

Example

Aidan likes to dive on vacation He stops to take a picture every 4 feet he dives How has his elevation changed relative to the surface of the water after he takes 5 pictures?

You can think of the change in Aidan’s elevation between pictures as 24 Aidan takes 5 pictures, so there are 5 groups of 24

Use the model to show the total change in Aidan’s elevation as jumps on a number line Start at 0 and make jumps of 24

The number 220 represents the total change in Aidan’s elevation

(24) 1 (24) 1 (24) 1 (24) 1 (24) 5 220

222 220 218 216 214 212 210 28 26 24 22 0

24 24 24 24 24

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5 · (24)

multiplying

20 feet

Possible model: (23) 1 (23) 1 (23) 1 (23) 1 (23) 5 215. Another possible model is a

number line with 5 jumps of 23 from 0 to 215.

©Curriculum Associates, LLC Copying is not permitted.32 Lesson 4 Multiply and Divide Positive and Negative Integers

Solve.

5 A swimming pool has 500 gallons of water in it and is leaking The change in the amount of water in the pool is 23 gallons per hour How many gallons of water will the pool contain after 6 hours?

Show your work.

Solution:

6 Write the fi rst two products in the table below Then use the pattern in the products to write the last two products in the table

2 ? (22) 1 ? (22) 0 ? (22) (21) ? (22) (22) ? (22)

Product 0

7 Complete each statement about multiplying positive and negative integers using the word positive or negative. For each statement, provide an example

positive ? positive 5 positive

positive ? negative 5

negative ? negative 5

8 Mrs Krin has $300 deducted from her checking account every month for her car payment She also has $150 deducted every month for her insurance After 1 year, by how much do these payments change her checking account balance? Explain how you found your answer

32

23 gallons each hour for 6 hours 5 23 · (6) 5 218 gallons

500 gallons 1 (218 gallons) 5 482 gallons

2$5,400; The payments can be represented by negative numbers. The car payment changes

her balance by 2$300 per month, and her insurance changes the balance by 2$150 per

month. I added 2$300 and 2$150 to get 2$450 for each month and then multiplied by

12 months. 2$450 ? 12 5 2$5,400.

The pool will contain 482 gallons of water after 6 hours.

Example: 7 ? 6 5 42

24 22 2 4

negative; Example: 27 ? 6 5 242

positive; Example: 27 ? 26 5 42

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©Curriculum Associates, LLC Copying is not permitted.34 Lesson 4 Multiply and Divide Positive and Negative Integers

Solve.

4 Multiplication and division are related operations Look at these examples

2 ? 4 5 8 (22) ? (24) 5 8 22 ? 4 5 28 2 ? (24) 5 28

8 4 4 5 2 8 4 (24) 5 22 28 4 4 5 22 28 4 (24) 5 2

Think about the signs of the factors and the sign of the product in the multiplication problems How do the signs of the numbers you are dividing relate to their quotients?

5 Describe a situation that the expression 215 4 (25) could represent Find the quotient and tell what it represents in terms of the situation

6 The low temperatures in degrees Fahrenheit in Marsh City during a week in January were:

23, 25, 3, 7, 22, 22, 25

What was the average low temperature for that week?

Show your work.

Solution:

7 The quotient x 4 5 is a negative integer Name two integers that x could represent and fi nd the quotients

34

If both numbers have the same sign (both are negative or positive), the quotient is positive.

If the numbers have different signs (one is negative and the other is positive), the quotient

is negative.

Possible answer: During a football game, the Lions lost a total of 15 yards as a result of a

number of 5-yard penalties. 215 4 (25) 5 3. The quotient, 3, represents the number of

5-yard penalties.

23 1 (25) 1 3 1 7 1 (22) 1 (22) 1 (25) 5 27

27 4 7 5 21

The average low temperature was 21°F.

Possible answer: 25 and 210; 25 4 5 5 21 and 210 4 5 5 22

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©Curriculum Associates, LLC Copying is not permitted. 33Lesson 4 Multiply and Divide Positive and Negative Integers

Name: Lesson 4

Dividing Integers

Study the example problem showing how to divide positive and negative integers. Then solve problems 1–7.

1 Draw Chen’s thermometer Be sure to label each section

2 Chen labels another thermometer that uses 10 diff erent sections Complete the table for this thermometer

Total number of degrees (°F)

4Number

of sections5

Degrees ineach section (°F)

240 5

3 A scuba diver descends 48 feet in 4 minutes What is the diver’s average change in position per minute relative to where she started?

Example

Chen is labeling a thermometer that is used to measure very cold temperatures from 0°F to 240°F He divides it into 8 equal sections How many degrees does each section represent?

Think: I need to divide 240 by 8, so I can ask 8 ? ? 5 240

8 ? (25) 5 240

This means that 240 4 8 5 25

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A sample diagram is shown. The thermometer should

show a range from 0 to 240 with each section

representing 25°F.

248 feet 4 4 minutes 5 212 feet per minute

4 10 24

0

25

210

215

220

225

230

235

240

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©Curriculum Associates, LLC Copying is not permitted.36 Lesson 4 Multiply and Divide Positive and Negative Integers

Solve.

4 Kain made two number cubes to use in a game The faces on each cube contain the numbers 1, 22, 3, 24, 5, and 26 After Kain rolls the two cubes, he multiplies the two numbers

a. Give an example of two numbers that Kain could roll to get a positive product

b. Give an example of two numbers that Kain could roll to get a negative product

What is true about the signs of two factors if their product is positive?

5 Savannah solves each of the following problems as shown below

a. 26 ? 12 4 (24) 5 18

b. 8 ? (23) 4 6 5 24

c. 240 ? (22) 4 (210) 5 8

d. 27 ? 5 ? (22) 4 5 5 14

Are the answers correct? Explain any incorrect answers

Remember to pay careful attention to the signs of numbers as you find quotients and products.

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Possible answer: 22 and 3; accept any pair of numbers in

which one is negative and one is positive.

Possible explanation: Savannah is correct for

problems a, b, and d. The answer for problem c is

28, so Savannah’s answer has the wrong sign.

Possible answer: 22 and 24; accept either two

negative numbers or two positive numbers.

©Curriculum Associates, LLC Copying is not permitted. 35Lesson 4 Multiply and Divide Positive and Negative Integers

Name:

1 A whale dives at a speed of 3 feet per second What is the change in the position of the whale relative to where it started after 12 seconds?

A 236 feet C 4 feet

B 24 feet D 36 feet

Multiply and Divide Positive and Negative Integers

Solve the problems.

3 Myra withdraws the same amount of money from her checking account each week In 4 weeks, she withdraws a total of $200 Which equation represents the amount of money her account changes by each week?

A 2200 4 (24) 5 50

B 2200 4 4 5 250

C 2200 4 4 5 50

D 2200 4 (24) 5 250

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

2 Tell whether each equation is True or False

a. 27 ? 8 5 7 ? (28) u True u False

b. 27 ? (28) 5 7 ? 8 u True u False

c. 7 ? (28) 5 7 ? 8 u True u False

Will your answer be positive or negative?

How can the signs of the factors in each multiplication equation help you solve this problem?

What can the signs of the numbers in a division problem tell you about the quotient?

Lesson 4

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Possible answer: Sam forgot that the quotient of a negative number divided by

a positive number is a negative number.