Scientific Notation - Everyday Math - Login€¢ Translate numbers from scientific notation to standard and number-and-word notation. ... It is read as four times ten to the third

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eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

552 Unit 7 Exponents and Negative Numbers

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 94–98

Scientific NotationObjective To introduce scientific notation.

��

Key Concepts and Skills• Explore the place value of numbers written

as powers of 10.

[Number and Numeration Goal 1]

• Translate numbers from scientific notation

to standard and number-and-word notation.


• Use number patterns to solve problems

involving exponents.

[Patterns, Functions, and Algebra Goal 1]

Key ActivitiesStudents use powers of 10 to write

numbers in expanded notation. They solve

multiplication expressions containing

exponents and translate numbers written in

scientific notation into standard and number-

and-word notation. Students practice writing

and comparing numbers written in scientific

notation by playing Scientific Notation Toss.

Ongoing Assessment: Recognizing Student Achievement Use journal page 217. [Number and Numeration Goal 1]

Key Vocabularyexpanded notation � scientific notation

MaterialsMath Journal 2, pp. 214–217

Student Reference Book, p. 329

Study Link 7� 2

Class Data Pad � slate � per partnership:

2 six-sided dice

Math Boxes 7� 3Math Journal 2, p. 218

Students practice and maintain skills

through Math Box problems.

Study Link 7� 3Math Masters, p. 194

Students practice and maintain skills

through Study Link activities.

READINESS

Using Place Value to Rename NumbersMath Masters, p. 195

Students rename numbers using place

value and number-and-word notation.

EXTRA PRACTICE

Writing Numbers in Expanded NotationMath Masters, p. 196

Students write whole numbers in expanded

notation as addition and multiplication

expressions.

ELL SUPPORT

Comparing Notations for NumbersDifferentiation Handbook, p. 149

Students compare and contrast the terms

standard notation and exponential notation.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

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Date Time

Complete the following pattern.

1. 102 = 10 ∗ 10 = 100

2. 103 = 10 ∗ 10 ∗ 10 = 1,000

3. 104 = 10 ∗ 10 ∗ 10 ∗ 10 = 10,000

4. 105 = 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 = 100,000

5. 106 = 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 = 1,000,000

Use the answers to Problems 1–5 to help you complete the following.

6. 2 ∗ 102 = 2 ∗ 100 = 200

7. 3 ∗ 103 = 3 ∗ 1,000 = 3,000

8. 4 ∗ 104 = 4 ∗ 10,000 = 40,000

9. 6 ∗ 105 = 6 ∗ 100,000 = 600,000

10. 8 ∗ 106 = 8 ∗ 1,000,000 = 8,000,000

When you write a number as the product of a number and a power of 10, you are using

scientific notation. Scientific notation is a useful way to write large or small numbers.

Many calculators display numbers one billion or larger with scientific notation.

Example: In scientific notation, 4,000 is written as 4 ∗ 103.

It is read as four times ten to the third power.

Write each of the following in standard notation and number-and-word notation.

Standard Notation Number-and-Word Notation

11. 5 ∗ 103 = 5,000 5 thousand

12. 7 ∗ 102 = 700 7 hundred

13. 2 ∗ 104 = 20,000 20 thousand

14. 5 ∗ 106 = 5,000,000

5 million

Scientific NotationLESSON

7�3

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Math Journal 2, p. 214

Student Page

Lesson 7�3 553

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

(Math Journal 2, p. 214)

Ask students to share their solutions for Problems 1–10. Ask a volunteer to write 236 on the Class Data Pad using expanded notation. 236 = 200 + 30 + 6 Show the use of powers of 10 to write numbers in expanded notation. Write 236 = (2 ∗ 102) + (3 ∗ 101) + (6 ∗ 100) on the Class Data Pad. Ask another volunteer to evaluate the expressions in parentheses. 236 = (2 ∗ 10 ∗ 10) + (3 ∗ 10) + (6 ∗ 1) = (2 ∗ 100) + (3 ∗ 10) + (6 ∗ 1) = 200 + 30 + 6 Ask students what observations or connections they notice between the number sentences. Point out that the expanded notation expressions contain powers of 10 written in standard notation, as products of 10s, and in exponential notation.

As a class, read the introduction to scientific notation on the journal page. Problems 6–14 are given in scientific notation.

Do Problems 11–14 as a class. Ask volunteers to rename the power of 10 as a product of 10s. Then carry out the multiplication.

Example: 5 ∗ 103 = 5 ∗ (10 ∗ 10 ∗ 10) = 5 ∗ 1,000 = 5,000

▶ Translating Scientific Notation PARTNER ACTIVITY

(Math Journal 2, pp. 215 and 216)

Science Link Ask a student to select an event from journal page 215 and read, in scientific notation, how many years

ago the event took place. Demonstrate how to use the place-value chart on page 216 to write the number in standard notation.

Math MessageComplete Problems 1–10 on page 214 in your journal. Reminder: Calculations with exponents are done before other factors are multiplied.

Study Link 7�2 Follow-UpPartners share their solutions. Then have students answer the following questions on their slates:

• One thousand equals what power of 10? 103

• Which prefix means thousand? kilo-

• What is another name for 106? 1 million

• Which prefix means million? mega-

• What does the prefix tera- mean? trillion

• 1 trillion equals what power of 10? 1012

Mental Math and Reflexes Use your slate procedures for problems such as the following:

3.4 ∗ 10 = 34

3.4 ∗ 102 = 340

3.4 ∗ 103 = 3,400

2.41 ∗ 102 = 241

2.41 ∗ 103 = 2,410

0.241 ∗ 102 = 24.1

54 ∗ 103 = 54,000

5,400 ÷ 103 = 5.4

5,400 ÷ 104 = 0.54

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Date Time

Work with a partner to answer the questions. Write your answers in standard notation.

11. According to the estimates by scientists, about how many years passed

from the formation of Earth until the first signs of life?

12. About how many years passed between the appearance of the first fish

and the appearance of forests and swamps?

13. According to the geological record, about how long did dinosaurs roam on Earth?

Billion 100 M 10 M Million 100 Th 10 Th Thousand 100 10 One

109 108 107 106 105 104 103 102 101 100

1 2 3 4 5 6 7 8 9

10

History of Earth continuedLESSON

7�3

About 185,000,000 years



5 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 2 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 6 5 0 0 0 0 0 0 6 0 0 0 0 0 0 8 0 0 0 0 0 2 0 0 0 0

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Student Page

Date Time

Each digit in a number has a value depending on its place in the numeral.

Example: 2,784

4 ones or 4 * 1 or 4

8 tens or 8 * 10 or 80

7 hundreds or 7 * 100 or 700

2 thousands or 2 * 1,000 or 2,000

Numbers written in expanded notation are written as addition expressions

showing the value of the digits.

1. a. Write 2,784 in expanded notation as an addition expression.

b. Write 2,784 in expanded notation as the sum of multiplication expressions.

c. Write 2,784 in expanded notation as the sum of multiplication expressions using powers of 10.

2. Write 987 in expanded notation as an addition expression.

3. Write 8,945 in expanded notation as the sum of multiplication expressions.

4. Write 4,768 in expanded notation as the sum of multiplication expressions using powers of 10.

5. a. Write 6,125 in expanded notation as an addition expression.

b. Write 6,125 in expanded notation as the sum of multiplication expressions.

c. Write 6,125 in expanded notation as the sum of multiplication expressions using powers of 10.

Expanded NotationLESSON

7�3 �

2,784 = 2,000 + 700 + 80 + 4

2,784 = (2 * 1,000) + (7 * 100) + (8 * 10) + (4 * 1)

2,784 = (2 * 103) + (7 * 102) + (8 * 101) + (4 * 100)

987 = 900 + 80 + 7

8,945 = (8 * 1,000) + (9 * 100) + (4 * 10) + (5 * 1)

4,768 = (4 * 103) + (7 * 102) + (6 * 101) + (8 * 100)

6,125 = 6,000 + 100 + 20 + 5

6,125 = (6 * 1,000) + (1 * 100) + (2 * 10) + (5 * 1)

6,125 = (6 * 103) + (1 * 102) + (2 * 101) + (5 * 100)

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Student Page


Example: The first fish appeared about 4 ∗ 108 years ago. To express this number of years in standard notation, find 108 on the place-value chart and write 4 beneath it, followed by the appropriate number of zeros in the cells to the right. From the chart, 4 ∗ 108 can easily be read as four hundred million.

Ask partners to complete Problems 1–8 on the chart. Circulate and assist.

The scientific notations for Problems 5 and 7 contain decimals. Discuss the meanings of decimals in scientific notation. For example, to convert 6.5 ∗ 107 to standard notation, think of number-and-word notation. The 6 represents 6 ten millions, or 60 million. The 0.5 represents half of 1 ten million, or 5 million. So 6.5 ∗ 107 = 65 million. Write the 6 in the 107 column, the 5 in the 106 column, and complete the row with 0s. This shows that 6.5 ∗ 107 = 65,000,000.

List the following numbers on the Class Data Pad, and ask volunteers to rename the numbers using decimals.

� 15 hundred 1.5 thousand

� 35 thousand 3.5 ten thousand

� 230 million 2.3 hundred million

Ask students to explain how they would write these answers in scientific notation. Thousands is 103, so 1.5 thousand = 1.5 ∗ 103; ten thousands is 104, so 3.5 ten thousand = 3.5 ∗ 104; hundred millions is 108, so 2.3 hundred million = 2.3 ∗ 108. Add students’ responses to the Class Data Pad.

Ask partners to complete the journal page. Circulate and assist.

Ongoing Assessment: Journal

Page 217 �

Problems 1–5Recognizing Student Achievement

Use journal page 217, Problems 1–5 to assess students’ understanding of

place value and their ability to translate numbers written in standard notation

to expanded notation. Students are making adequate progress if they have

accurately converted the numbers to expanded notation. Some students may be

able to write in expanded notation using powers of 10.


▶ Reviewing Expanded Notation

INDEPENDENT ACTIVITY


Ask volunteers to identify the value of each digit in 2,784. Ask other volunteers to write the solutions for Problem 1 on the board. Discuss how the number sentences are related.

Expect students to reference ideas from the Math Message discussion. Then have students complete the page.

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Math Boxes LESSON

7�3

Date Time

4. Complete the “What’s My Rule?” table,

and state the rule.

2. Charlene has 2�5

8� yards of fabric. The

curtain she is making requires 3�3

4� yards.

How much more fabric does she need?

6. Lilia did 2�3

4� hours of homework on

Saturday and �3

4� hour of homework on

Sunday. What is the total time she spent

on homework over the weekend?

1. Circle the fractions that are equivalent to .

15�40

4�9

8�3

9�24

6�12

3�8

3. Find the missing numerator or denominator.

5. Solve.

a. � 32 / (16 / 2)

b. � (32 / 16) / 2

c. (6.5 � 8.3) / (3 � 1) �

d. (4 � 12) � 8 � 56

7.4

1

4

—5

2

—9

2

28 7

16 41 0.25

20 5

0 0

Rule

�47

8

a. �4�10

b. �42�66

c. �35�40

f. �6�27

—7

3d. �9

�21

—11

7

e. �5�20

1

4

1�18

� yards

3�12

� hours

66 67 71

231 23259

219 70


Student Page

STUDY LINK

7�3 Interpreting Scientific Notation

8

Name Date Time

Scientific notation is a short way to represent large and small numbers.

In scientific notation, a number is written as the product of two factors.

One factor is a whole number or a decimal. The other factor is a power of 10.

Scientific notation: 4 � 104

Meaning: Multiply 104 (10,000) by 4.

4 º 104 � 4 º 10,000 � 40,000

Number-and-word notation: 40 thousand

Scientific notation: 6 º 106

Meaning: Multiply 106 (1,000,000) by 6.

6 º 106 � 6 º 1,000,000 � 6,000,000

Number-and-word notation: 6 million

Complete the following statements.

1. The area of Alaska is about 6 º 105, or thousand, square miles.

The area of the lower 48 states is about 3 º 106, or million, square miles.

2. There are about 6 º 109, or billion, people in the world.

3. It is estimated that about 5 º 108, or , people speak English as

their first or second language.

4. In Bengal, India, and Bangladesh there are about 2.6 º 108, or ,

people who speak Bengali.

5. At least 1 person in each of 1 � 107 households, or ,

watches the most popular TV shows.

Source: The World Almanac and Book of Facts, 2000

10 million

260 million

500 million

6

3

600

Guides for Powers of 10

103 one thousand

106 one million

109 one billion

1012 one trillion

Practice

6. 5 � (32 � 42) � 7. 3 � (9 � 16) �

8. 2 � (9 � h) � 20 9. g � (72 � 22) g � 45h � 1

75125

Math Masters, p. 194

Study Link Master

Lesson 7�3 555

▶ Playing Scientific-Notation Toss PARTNER ACTIVITY

(Student Reference Book, p. 329)

Have students read the directions on page 329 in the Student Reference Book. Ask a volunteer to demonstrate how the game is played.

2 Ongoing Learning & Practice

▶ Math Boxes 7�3



Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-1. The skills in Problems 2 and 6 preview Unit 8 content.

▶ Study Link 7�3 INDEPENDENT

ACTIVITY (Math Masters, p. 194)

Home Connection Students practice reading and interpreting numbers written in scientific notation. Then they write the numbers in number-and-word notation.

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Using Place Value to 15–30 Min

Rename Numbers(Math Masters, p. 195)

To explore the use of place value and number-and-word notation to rename numbers, have partners complete name-collection boxes. Refer students to the place-value chart on Math Masters, page 195.

Pose the following questions:

● What is 1 _ 10 of 10? 1

● What is 1 _ 10 of 100? 10

● What is 1 _ 10 of 1,000? 100

● What is 1 _ 10 of 10,000? 1,000

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Example: 1.

2. 3.

py

gg

p

LESSON

7�3

Name Date Time

Using Place Value to Rename Numbers

Billions Millions Thousands Ones

100 10 1 100 10 1 100 10 1 100 10 1

1. 1 3 0 0 2. 1 8 0 0

3. 1 4 0 0 0 0 0

4. 1 6 0 0 0 0 0

Sample answers:

Write the numbers from the name-collection box tag in the place-value chart.

Then follow the pattern in Problem 1 to complete each name-collection box.

1,300

1,000 + 3001 thousand 3 hundred

13 hundred1 300 _

1,000 thousands

1 3 _ 10

thousands1.3 thousands

1,800

1,000 + 800

1 thousand 8 hundred

18 hundred

1 800 _

1,000 thousands

1.8 thousands

1 8 _ 10 thousands

1,600,000

16 hundred-thousands

1.6 millions

1,000,000 + 600,0001

6

_ 10 millions

1,400,000

1.4 million

1,000,000 + 400,000

14 hundred-thousands

1 400,000

_ 1,000,000 millions

1 400 _ 1,000

millions

1 4 _ 10

millions

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Teaching Master

LESSON

7�3

Name Date Time

Writing in Expanded Notation

A Standard Notation: 325

B Expanded Notation as an addition expression: 300 � 20 � 5

C Expanded Notation as the sum of multiplication expressions:

(3 º 100) � (2 º 10) � (5 º 1)

D Expanded Notation as the sum of multiplication expressions

using powers of 10: (3 º 102) � (2 º 101) � (5 º 100)

Write each number below in the other three possible ways, as shown above.

1. a. 5,314

b.

c.

d.

2. a.

b. 2,000 � 700 � 50 � 6

c.

d.

3. a.

b.

c. (9 º 100) � (8 º 10) � (3 º 1)

d.

4. a.

b.

c.

d. (7 º 103) � (4 º 102) � (5 º 101) � (2 º 100)

(7 º 1,000) � (4 º 100) � (5 º 10) � (2 º 1)

7,000 � 400 � 50 � 2

7,452

(9 º 102) � (8 º 101) � (3 º 100)

900 � 80 � 3

983

(2 º 103) � (7 º 102) � (5 º 101) � (6 º 100)

(2 º 1,000) � (7 º 100) � (5 º 10) � (6 º 1)

2,756

(5 º 103) � (3 º 102) � (1 º 101) � (4 º 100)

(5 º 1,000) � (3 º 100) � ( 1 º 10) � ( 4 º 1 )

5,000 � 300 � 10 � 4


Teaching Master

Adjusting the Activity


Explain that just as we think of the place-value of each column as 10 times that of the column to its right, we can also think of the place-value of each column as 1 _ 10 of the column to its left.

We can use these relationships to rename numbers. In the example on the Math Masters page, we can think how many hundreds in 1,300? and rename it as 13 hundred. If we think how many thousands in 1,300, we can rename it as 1.3 thousand. Since 100 is 1 _ 10 of 1,000, then 300 is 3 _ 10 of 1,000.

Ask students to first write the numbers from the name-collection box tags in the place-value chart and then follow the pattern in the example to complete the name-collection boxes for these numbers.

Have students share their answers. Consider making posters to display the completed name-collection boxes.

EXTRA PRACTICE


▶ Writing Numbers in 15–30 Min

Expanded Notation(Math Masters, p. 196)

Students practice writing whole numbers in expanded notation as addition expressions and multiplication expressions.

Have students write expanded notation as multiplication expressions for

decimals. Suggestions: 9.56, 87.125, 392.394

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

ELL SUPPORT

SMALL-GROUP ACTIVITY

▶ Comparing Notations 15–30 Min

for Numbers(Differentiation Handbook, p. 149)

To provide language support for number notations, ask students to compare and contrast the terms standard notation and exponential notation. Have students use the Venn diagram found on Differentiation Handbook, page 149. See the Differentiation Handbook for more information.

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Documents

Scientific Notation - Everyday Math - Login€¢ Translate numbers from scientific notation to standard and number-and-word notation. ... It is read as four times ten to the third