Scientific Notation
Objective To introduce scientific notation.
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Ongoing Learning & Practice
Key Concepts and Skills
Math Boxes 7 3
• Explore the place value of numbers written
as powers of 10. Math Journal 2, p. 218
Students practice and maintain skills
through Math Box problems.
[Number and Numeration Goal 1]
• Translate numbers from scientific notation
to standard and number-and-word notation. [Number and Numeration Goal 1]
• Use number patterns to solve problems
involving exponents. Study Link 7 3
Math Masters, p. 194
Students practice and maintain skills
through Study Link activities.
[Patterns, Functions, and Algebra Goal 1]
Key Activities
Students 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 numberand-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 Vocabulary
expanded notation scientific notation
Materials
Math Journal 2, pp. 214–217
Student Reference Book, p. 329
Study Link 72
Class Data Pad slate per partnership:
2 six-sided dice
Advance Preparation
Teacher’s Reference Manual, Grades 4–6 pp. 94–98
552
Unit 7
Exponents and Negative Numbers
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Using Place Value to Rename Numbers
Math Masters, p. 195
Students rename numbers using place
value and number-and-word notation.
EXTRA PRACTICE
Writing Numbers in Expanded Notation
Math Masters, p. 196
Students write whole numbers in expanded
notation as addition and multiplication
expressions.
ELL SUPPORT
Comparing Notations for Numbers
Differentiation Handbook, p. 149
Students compare and contrast the terms
standard notation and exponential notation.
Mathematical Practices
SMP1, SMP2, SMP4, SMP5, SMP6, SMP7, SMP8
Getting Started
Content Standards
5.NBT.2
Mental Math and Reflexes
Math Message
Use your slate procedures for problems such as
the following:
Complete Problems 1–10 on page 214 in your
journal. Reminder: Calculations with exponents are done before
other factors are multiplied.
3.4 ∗ 10 = 34
3.4 ∗ 102 = 340
3.4 ∗ 103 = 3,400
2.41 ∗ 10 = 241
2.41 ∗ 103 = 2,410
0.241 ∗ 102 = 24.1
2
54 ∗ 103 = 54,000
5,400 ÷ 103 = 5.4
5,400 ÷ 104 = 0.54
Study Link 7 2 Follow-Up
Partners share their solutions. Then have students
answer the following questions on their slates:
3
• One thousand equals what power of 10? 10
• Which prefix means thousand? kilo6
• What is another name for 10 ? 1 million
• Which prefix means million? mega• What does the prefix tera- mean? trillion
12
• 1 trillion equals what power of 10? 10
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.
Student Page
Date
Time
LESSON
73
Scientific Notation
Complete the following pattern.
1.
102 = 10 ∗ 10 = 100
1,000
10 ∗ 10 ∗ 10 ∗ 10
10,000
=
∗
∗
∗
∗
10
10
10
10
10
100,000
=
=
4. 10
∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 = 1,000,000
5. 10 = 10
2.
103 = 10 ∗ 10 ∗ 10 =
3.
104 =
5
6
Use the answers to Problems 1–5 to help you complete the following.
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)
6.
2 ∗ 102 = 2 ∗ 100 = 200
7.
3 ∗ 103 = 3 ∗
8.
4 ∗ 104 =
9.
6 ∗ 105 =
10.
8 ∗ 106 =
1,000 = 3,000
4
10,000 = 40,000
∗
6
∗ 100,000 = 600,000
8
∗ 1,000,000 = 8,000,000
Scientific notation is a useful way to write very large or very small numbers. A number is
in scientific notation when it is written as a product of two factors as follows: One factor is
greater than or equal to 1 and less than 10. The other factor is a power of 10.
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
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.
5,000
700
=
20,000
=
= 5,000,000
11.
5 ∗ 103 =
12.
7 ∗ 102
13.
2 ∗ 104
14.
5 ∗ 106
Number-and-Word Notation
5 thousand
7 hundred
20 thousand
5 million
Math Journal 2, p. 214
209-247_EMCS_S_MJ2_G5_U07_576434.indd 214
8/25/11 8:48 AM
Lesson 7 3
553
Student Page
Date
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.
Time
LESSON
History of Earth
73
continued
Billion 100 M 10 M Million 100 Th 10 Th Thousand 100 10 One
1
2
109
108
107
106
105
104
103
102
101
5
4
0
0
4
3
2
1
0
0
0
0
5
0
6
0
0
0
0
0
0
5
6
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
3
4
5
6
7
8
9
10
100
0
0
0
0
0
0
0
0
0
0
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.
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?
About 1,000,000,000 years
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?
About 100,000,000 years
About 185,000,000 years
List the following numbers on the Class Data Pad, and ask
volunteers to rename the numbers using decimals.
15 hundred 1.5 thousand
Math Journal 2, p. 216
209-247_EMCS_S_MJ2_G5_U07_576434.indd 216
8/25/11 8:48 AM
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.
▶ Reviewing Expanded Notation
(Math Journal 2, p. 217)
Student Page
Date
Time
LESSON
73
Expanded Notation
Each digit in a number has a value depending on its place in the numeral.
Example:
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.
2,784
4
8
7
2
INDEPENDENT
ACTIVITY
ones
or 4 ∗ 1
or 4
tens
or 8 ∗ 10
or 80
hundreds or 7 ∗ 100
or 700
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.
2,784 = 2,000 + 700 + 80 + 4
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.
3
2
1
0
2,784 = (2 ∗ 1,000) + (7 ∗ 100) + (8 ∗ 10) + (4 ∗ 1)
2,784 = (2 ∗ 10 ) + (7 ∗ 10 ) + (8 ∗ 10 ) + (4 ∗ 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.
3
2
1
0
987 = 900 + 80 + 7
8,945 = (8 ∗ 1,000) + (9 ∗ 100) + (4 ∗ 10) + (5 ∗ 1)
4,768 = (4 ∗ 10 ) + (7 ∗ 10 ) + (6 ∗ 10 ) + (8 ∗ 10 )
5. a.
6,125 = 6,000 + 100 + 20 + 5
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.
3
2
1
0
6,125 = (6 ∗ 1,000) + (1 ∗ 100) + (2 ∗ 10) + (5 ∗ 1)
6,125 = (6 ∗ 10 ) + (1 ∗ 10 ) + (2 ∗ 10 ) + (5 ∗ 10 )
Math Journal 2, p. 217
554
Unit 7
Journal
Page 217
Problems 1–5
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.
[Number and Numeration Goal 1]
Write 6,125 in expanded notation as an addition expression.
EM3cuG5MJ2_U07_209-247.indd 217
Ongoing Assessment:
Recognizing Student Achievement
1/19/11 7:42 AM
Exponents and Negative Numbers
Student Page
▶ Playing Scientific-Notation Toss
PARTNER
ACTIVITY
Date
Time
LESSON
Math Boxes
73
(Student Reference Book, p. 329)
1.
3
Circle the fractions that are equivalent to _
8.
9
_
6
_
Have students read the directions on page 329 in the Student
Reference Book. Ask a volunteer to demonstrate how the game
is played.
8
_
24
12
4
_
3
5
Charlene has 2_
8 yards of fabric. The
3
curtain she is making requires 3_
4 yards.
How much more fabric does she need?
2.
15
_
9
40
1_18 yards
66 67
3.
10
5
35
c. _
40
= 7
▶ Math Boxes 7 3
42
b. _
66
d.
8
5 = 1
_
e.
2
4
_
a.
2 Ongoing Learning & Practice
20
f.
4
=
7
9 =
_
21
6 =
_
27
Rule
3
a.
b.
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.
1
20
5
0
0
231 232
3
Lilia did 2_
4 hours of homework on
3
Saturday and _
4 hour of homework on
Sunday. What is the total time she spent
on homework over the weekend?
= 32 (16 2)
3_12 hours
= (32 16) 2
(6.5 + 8.3) (3 – 1) =
d.
(4 ∗ 12) + 8 =
7
4
0.25
9
6.
c.
16
2
Solve.
4
1
28
÷4
7
59
5.
Complete the “What’s My Rule?” table,
and state the rule.
4.
11
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 218)
71
Find the missing numerator or denominator.
7.4
56
219
▶ Study Link 7 3
70
Math Journal 2, p. 218
INDEPENDENT
ACTIVITY
EM3cuG5MJ2_U07_209-247.indd 218
1/19/11 7:42 AM
(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
▶ Using Place Value to
PARTNER
ACTIVITY
15–30 Min
Rename Numbers
(Math Masters, p. 195)
Study Link Master
Name
Date
STUDY LINK
73
Time
Interpreting Scientific Notation
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 number
greater than or equal to 1 and less than 10. The other factor is a power of 10.
Scientific notation: 4 ∗ 104
Meaning:
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.
Guides for Powers of 10
Multiply 104 (10,000) by 4.
4 ∗ 104 = 4 ∗ 10,000 = 40,000
Number-and-word notation:
40 thousand
Multiply 106 (1,000,000) by 6.
103
one thousand
106
one million
109
one billion
1012
one trillion
Scientific notation: 6 ∗ 106
Meaning:
8
6 ∗ 106 = 6 ∗ 1,000,000 = 6,000,000
Number-and-word notation: 6 million
Complete the following statements.
Pose the following questions:
1 of 10? 1
● What is _
10
●
1 of 100? 10
What is _
●
1 of 1,000? 100
What is _
●
10
What is
10
1
_
10
of 10,000? 1,000
1.
The area of Alaska is about 6 ∗ 105, or
600
thousand, square miles.
3
million, square miles.
6
billion, people in the world.
It is estimated that about 5 ∗ 10 , or 500 million , people speak English as
The area of the lower 48 states is about 3 ∗ 106, or
9
2. There are about 6 ∗ 10 , or
3.
8
their first or second language.
4.
In Bengal, India, and Bangladesh there are about 2.6 ∗ 108, or
people who speak Bengali.
5.
At least 1 person in each of 1 ∗ 107 households, or
watches the most popular TV shows.
260 million ,
10 million
,
Source: The World Almanac and Book of Facts, 2000
Practice
6.
5 ∗ (32 + 42) =
8. 2 ∗ (9 + h) = 20
125
h=1
7.
3 ∗ (9 + 16) =
9.
g = (7 - 2 )
2
2
75
g = 45
Math Masters, p. 194
187-220_EMCS_B_MM_G5_U07_576973.indd 194
8/16/11 3:45 PM
Lesson 7 3
555
Teaching Master
Name
Date
LESSON
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
1 of the column to its left.
place-value of each column as _
10
Time
Using Place Value to Rename Numbers
73
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.
Billions
100
10
Millions
1
100
10
Thousands
1
100
1.
2.
1
1
3.
4.
1.
Example:
4
6
10
100
10
1
1
1
0 0
0 0
3
8
0
0
0
0
0
0
0
0
0
0
Sample answers:
2.
1,300
1,800
1,000 + 300
1 thousand 3 hundred
1,000 + 800
1 thousand 8 hundred
18 hundred
800
1_
1,000 thousands
1.8 thousands
8
1_
10 thousands
13 hundred
300
1_
1,000 thousands
3
1_
10 thousands
1.3 thousands
3.
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
3 of 1,000.
1 of 1,000, then 300 is _
100 is _
10
10
Ones
1
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.
4.
1,600,000
1.4 million
1,000,000 + 400,000
14 hundred-thousands
400,000
millions
1_
1,000,000
400
millions
1_
1,000
4
millions
1_
10
16 hundred-thousands
1.6 millions
1,000,000 + 600,000
6
1_
10 millions
py g
g
p
1,400,000
INDEPENDENT
ACTIVITY
EXTRA PRACTICE
▶ Writing Numbers in
Math Masters, p. 195
187-220_EMCS_B_MM_G5_U07_576973.indd 195
15–30 Min
Expanded Notation
9/27/11 3:50 PM
(Math Masters, p. 196)
Students practice writing whole numbers in expanded notation as
addition expressions and multiplication expressions.
Adjusting the Activity
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
Teaching Master
Name
Date
LESSON
73
Standard Notation: 325
B
Expanded Notation as an addition expression: 300 + 20 + 5
C
Expanded Notation as the sum of multiplication expressions:
(3 ∗ 100) + (2 ∗ 10) + (5 ∗ 1)
D
Expanded Notation as the sum of multiplication expressions
using powers of 10: (3 ∗ 102) + (2 ∗ 101) + (5 ∗ 100)
▶ Comparing Notations
b.
c.
d.
2. a.
2,756
c.
(2 ∗ 1,000) + (7 ∗ 100) + (5 ∗ 10) + (6 ∗ 1)
d.
(2 ∗ 103) + (7 ∗ 102) + (5 ∗ 101) + (6 ∗ 100)
3. a.
983
b.
900 + 80 + 3
c.
(9 ∗ 100) + (8 ∗ 10) + (3 ∗ 1)
d.
(9 ∗ 102) + (8 ∗ 101) + (3 ∗ 100)
4. a.
7,452
b.
7,000 + 400 + 50 + 2
c.
(7 ∗ 1,000) + (4 ∗ 100) + (5 ∗ 10) + (2 º 1)
d.
(7 ∗ 103) + (4 ∗ 102) + (5 ∗ 101) + (2 ∗ 100)
Math Masters, p. 196
187-220_EMCS_B_MM_G5_U07_576973.indd 196
556
Unit 7
3/29/11 2:03 PM
Exponents and Negative Numbers
V I S U A L
SMALL-GROUP
ACTIVITY
15–30 Min
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.
5,000 + 300 + 10 + 4
(5 ∗ 1,000) + (3 ∗ 100) + (1 ∗ 10) + (4 ∗ 1)
(5 ∗ 103) + (3 ∗ 102) + (1 ∗ 101) + (4 ∗ 100)
2,000 + 700 + 50 + 6
T A C T I L E
(Differentiation Handbook, p. 149)
5,314
b.
for Numbers
Write each number below in the other three possible ways, as shown above.
1. a.
K I N E S T H E T I C
ELL SUPPORT
Time
Writing in Expanded Notation
A