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Practice Lesson 4 Scientific Notation

Unit 1

Practice and Prob

lem Solvin

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nit 1 Expressions an

d Equations (Exp

onents) an

d the N

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Key

B Basic M Medium C Challenge

©Curriculum Associates, LLC Copying is not permitted. 31Lesson 4 Scientific Notation

Name: Scientific Notation

Lesson 4

Prerequisite: Multiply and Divide by Powers of Ten

Study the example problem showing how to multiply a decimal number by a power of ten. Then solve problems 1–7.

1 Write the missing powers of 10

a. 3 0 005 5 5 3 0 005 5 0 5

b. 0 09 3 5 0 9 0 009 3 5 9

2 Jane says that multiplying a decimal by 100 is the same as multiplying the decimal by three factors of 10 Is Jane correct? Explain your answer

3 Complete the equations

a. 0 002 3 100 5

b. 0 05 3 1,000 5

Example

Find 1,000 3 0 006

Start by breaking 1,000 into the product of tens Then, multiply

1,000 3 0 006 5 10 3 10 3 10 3 0 006

5 10 3 10 3 0 06

5 10 3 0 6

5 6

This means that 1,000 3 0 006 5 6

Vocabularypower of ten a number

that can be written as a

product of tens

10 5 10

100 5 10 3 10

1,000 5 10 3 10 3 10

3131

10

100

1,000

1,000

No; Possible explanation: There are two factors

of 10 in 100, so multiplying by 100 is the same as

multiplying by two factors of 10.

0.2

50

B

B

B

©Curriculum Associates, LLC Copying is not permitted.32 Lesson 4 Scientific Notation

Solve.

4 Remember that division is the inverse of multiplication Complete the table below to show dividing 7 by powers of 10

Ones • Tenths Hundredths Thousandths

7 • 0 0 0

• 7 4 10

• 7 4 100

• 7 4 1,000

5 Daryl explains why 8 2 4 10,000 is 0 00082 Complete his explanation

There are factors of 10 in 10,000 Each time I divide a number by a factor of 10, the result is 10 times than the original number For example, if I divide 8 2 by 10, the result is 0 82 This means that 0 82 is 10 times than 8 2 So, 8 2 4 10,000 is 0 00082

6 Complete the equations

a. 0 004 3 100 5 0 004 3 102 5

b. 0 4 3 1,000 5 0 4 3 5

c. 600 4 100 5 600 4 102 5

d. 0 6 4 5 0 6 4 101 5

7 Yara multiplies and divides a certain number by the same power of 10 The product she gets is 40,000 and the quotient she gets is 0 000004 Find Yara’s number and the power of 10 she used Explain your reasoning

32

M

M

B

less

less

four

0.4

400103

6

0.0610

0.4 and 105; Possible explanation: You can start by noticing that 40,000 is a number in the

ten thousands and 0.000004 is a number in the millionths, so you can divide and multiply

them by a number that is between 10,000 and 1,000,000 to get Yara’s number. If you

divide 40,000 by 100,000, you get 0.4. If you multiply 0.000004 by 100,000, you get 0.4, so

Yara used 0.4 and 100,000, or 105.

C

0

0

0

7

0

0

0

7

0

0

0

7

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otation

©Curriculum Associates, LLC Copying is not permitted. 33Lesson 4 Scientific Notation

Name: Lesson 4

Write Numbers in Scientifi c Notation

Study the example problem showing how to write a number in scientific notation. Then solve problems 1–10.

1 Suppose that more than 96,000,000 passengers used the airport in the example Write 96,000,000 in scientifi cnotation

2 Write 64,500 in scientifi c notation

3 Describe how you would write 5 04 3 106 in standard form

4 Is 21 5 3 103 in scientifi c notation? Explain

Example

In one year, more than 90,000,000 passengers used the Hartsfield-Jackson Atlanta International Airport in Atlanta, Georgia Write 90,000,000 in scientific notation

A number written in scientific notation is the product of a number that is greater than or equal to 1 and less than 10 and a power of 10

You can think of 90,000,000 as 9 3 10,000,000 You can write 10,000,000 as a product of seven 10s, so 10,000,000 written with an exponent is 107

This means that 90,000,000 written in scientific notation is 9 3 107

Vocabularyscientific notation a

way to express a number

as the product of a

number greater than or

equal to 1 but less than

10 and a power of ten

5 4 3 106

1 08 3 1025

33

9.6 3 107

6.45 3 104

Possible answer: Because the power is 106, I multiply

5.04 by 1,000,000, so: 5.04 3 106 5 5.04 3 1,000,000

5 5,040,000.

No; The factor 21.5 is greater than 10.

B

B

M

B

©Curriculum Associates, LLC Copying is not permitted.34 Lesson 4 Scientific Notation

Solve.

5 The number 0 003 written in scientifi c notation is 3 3 1023 Why is the exponent negative?

6 Write 0 00052 in scientifi c notation

7 Is 4 23 3 1025 greater than 1 or less than 1? Explain how you know

8 Sylvester says that 0 000002015 written in scientifi c notation is 2 015 3 106 Do you agree with him? Explain why or why not

9 Explain how you could write the mixed number 45 3 ·· 8 in scientifi c notation

10 Jalil makes a generalization about writing numbers in scientifi c notation Complete his statement When you write a number greater than or equal to 1 in scientifi c notation, the exponent of the power of 10 will be

When you write a number between 0 and 1 in scientifi c notation, the exponent of the power of 10 will be

34

M

Possible answer: 0.003 is the same as 3 3 0.001 and 0.001 is written with an exponent as

1023, so 0.003 written in scientific notation is 3 3 1023.

B5.2 3 1024

positive

negative

M

M

C

M

less than 1; Possible explanation:

1025 5 1 ··· 105 5 1 ··· 100,000 5 0.00001. So 4.23 3 1025 = 4.23 3 0.00001 5 0.0000423, which is

less than 1.

No; 0.000002015 is the same as 2.015 3 0.000001 and 0.000001 is written with an

exponent as 1026, so 0.000002015 written in scientific notation is 2.015 3 1026.

Possible explanation: First write the mixed number 45 3 ·· 8 as a decimal: 45 3 ·· 8 5 45.375. Then

write 45.375 in scientific notation by using 101 as the power of 10: 45 3 ·· 8 5 45.375 5

4.5375 3 101.

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Unit 1

Practice Lesson 4 Scientific Notation

©Curriculum Associates, LLC Copying is not permitted. 35Lesson 4 Scientific Notation

Name: Lesson 4

Compare Numbers in Scientifi c Notation

Study the example problem showing how to compare two numbers written in scientific notation. Then solve problems 1–7.

1 Which of the two methods shown in the example problem do you prefer? Explain why

2 The total area of Ohio is about 4 5 × 104 square miles The total area of Ohio is about how many times the total area of Rhode Island?

3 Which is greater, 9 × 1022 or 3 × 1024? How many times greater is the number you chose than the other number? Explain your reasoning

Example

The total area of Rhode Island is about 1 5 3 103 square miles The total area of Georgia is about 6 3 104 square miles The total area of Georgia is about how many times the total area of Rhode Island?

Use Standard Form

1 5 3 103 5 1,500

6 3 104 5 60,000

Compare by dividing: 60,000 ······ 1,500 5 40

60,000 is 40 times 1,500

Use Scientific Notation

Compare the parts of 1 5 3 103 and 6 3 104

6 ·· 1 5 5 4 104 ··· 103 5 10423 5 101

Combine the results: 4 3 101 5 40

6 3 104 is 40 times 1 5 3 103

The total area of Georgia is about 40 times the total area of Rhode Island

35

Possible answer: I prefer the scientific notation method. The numbers you are comparing

are smaller.

The total area of Ohio is about 30 times the total area of Rhode Island.

9 3 1022; Possible explanation: 9 3 1022 is 300 times 3 3 1024 because 0.09 4 0.0003 5 300.

B

B

M

©Curriculum Associates, LLC Copying is not permitted.36 Lesson 4 Scientific Notation

Solve.

4 The total area of Kansas that is covered by water is about 5 3 102 square miles The total area of Alaska that is covered by water is about 9 5 3 104 square miles The total water area of Alaska is about how many times the total water area of Kansas?

Show your work.

Solution:

5 The average masses of several insects or animals are shown in the table The average mass of a hummingbird is about how many times the average mass of a housefl y?

6 Look at the table in problem 5

a. Write the masses of the ant, bee, and housefl y in order from greatest mass to least mass

b. The mass of the insect with the greatest mass is about how many times the mass of the insect with the least mass?

7 The population of the United States is about 3 2 3 108 The population of the United States is about 80 times the population of Oregon Write the population of Oregon in scientifi c notation

Insect or Animal Average Mass (g)

Ant 4 3 1023

Bee 1 3 1021

Housefly 1 2 3 1022

Ruby-throated hummingbird

3 6

36

The total water area of Alaska is about 190 times the total water area of Kansas.

about 300 times

1 3 1021, 1.2 3 1022, 4 3 1023

about 25 times

4 3 106

Possible work:

9.5 ··· 5 5 1.9

104 ··· 102 5 10422 5 102

1.9 3 102 5 190

M

M

C

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Unit 1 Exp

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nit 1Practice Lesson 4 Scientific N

otation

©Curriculum Associates, LLC Copying is not permitted. 37Lesson 4 Scientific Notation

Name: Lesson 4

1 Which of the following expressions is equivalent to 4,325,000,000?

A 4 325 3 1029 C 4 325 3 106

B 4 325 3 1026 D 4 325 3 109

Carson chose A as the correct answer How did he get that answer?

Scientifi c Notation

Solve the problems.

3 Last year a restaurant chain spent 3 3 3 106 dollars opening new restaurants This year the restaurant will spend 9 9 3 105 dollars Which statement is true?

A The restaurant spent $330,000 dollars last year

B This year the restaurant will spend $9,900,000

C The restaurant spent 6 6 3 106 dollars more this year than last year

D This year the restaurant will spend about 0 3 times the amount it did last year

2 The mass of Earth’s moon is about 7 3 1022 kilograms The mass of Jupiter is about 1 89 3 1027 kilograms The mass of Jupiter is about how many times the mass of Earth’s moon?

Show your work.

Solution:

Will the exponent be positive or negative?

How can the parts of each number help you to compare?

How can you write numbers in standard form?

37

M

Possible work: 1.89 ···· 7 5 0.27

1027 ···· 1022 5 1027222 5 105

0.27 3 105 5 27,000

C

M

Jupiter’s mass is about 27,000 times the moon’s mass.

He forgot that when you write a number greater than or equal to 1 in scientific

notation, the exponent in the power of 10 will be positive.

©Curriculum Associates, LLC Copying is not permitted.38 Lesson 4 Scientific Notation

6 The area of the Pacifi c Ocean is about 1 56 3 108 square kilometers The area of the East China Sea is about 1 2 3 106 square kilometers Tell whether each statement is True or False

a. The area of the Pacific Ocean is about 15,600,000 square kilometers u True u False

b. The area of the Pacific Ocean is about 130 times the area ofthe East China Sea u True u False

c. The area of the East China Sea is about 130 times the areaof the Pacific Ocean u True u False

d. The area of the East China Sea is about 1,200,000 square kilometers u True u False

Solve.

4 Which of the following numbers is NOT in scientifi c notation?

A 4 5 3 10212

B 3 025 3 1029

C 0 21 3 107

D 1 1 3 1010

5 Write 0 0000003105 in scientifi c notation Explain how you found your answer

Show your work.

Solution:

Is the number between 0 and 1 or is it greater than 1? What does that tell you about the number in scientific notation?

How do you compare numbers in scientific notation?

What does it mean for a number to be in scientific notation?

38

3.105 3 1027.

3

3

3

3

B

M

B

0.0000003105 5 3.105 3 0.0000001

5 3.105 3 1027

0.0000003105 is the same as 3.105 3 0.0000001 and 0.0000001 is written with an exponent as 1027, so 0.0000003105 written in scientific notation is 3.105 3 1027