Exponential Notation
for Powers of 10
Objective To introduce number-and-word notation for large
numbers and exponential notation for powers of 10.
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Practice
EM Facts
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Teaching the Lesson
Key Concepts and Skills
• Explore place value using powers of 10. [Number and Numeration Goal 1]
• Write and translate numbers in and
between standard and exponential notation. [Number and Numeration Goal 4]
• Compare exponential notation and standard
notation for positive powers of 10. [Number and Numeration Goal 4]
• Describe the number patterns inherent to
powers of ten. [Patterns, Functions, and Algebra Goal 1]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing First to 100
Student Reference Book, p. 308
Math Masters, pp. 456–458
per partnership: 2 six-sided dice,
calculator
Students practice solving open
number sentences.
Math Boxes 7 2
Math Journal 2, p. 213
Geometry Template
Students practice and maintain skills
through Math Box problems.
Key Activities
Study Link 7 2
Students use standard notation,
number-and-word notation, and exponential
notation to represent large numbers.
Math Masters, p. 191
Students practice and maintain skills
through Study Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Finding Patterns in Powers of 10
Math Masters, p. 192
Students complete a powers-of-10 table and
describe patterns they see in the table.
ENRICHMENT
Introducing Negative Exponents
and Powers of 0.1
Student Reference Book, p. 7
Math Masters, p. 193
Students explore patterns and notation
of negative exponents.
EXTRA PRACTICE
Multiplying Decimals by Powers of 10
Students solve problems involving the
multiplication of decimals by powers of 10.
Ongoing Assessment:
Recognizing Student Achievement
Use the Math Message. [Number and Numeration Goal 4]
Ongoing Assessment:
Informing Instruction See page 549.
Key Vocabulary
number-and-word notation powers of 10
Materials
Math Journal 2, p. 212
Student Reference Book, p. 5
Study Link 7 1
Math Masters, p. 433
transparency of Math Masters, p. 433 slate
Advance Preparation
For Part 1, copy the place-value chart from the top of journal page 212 on the board. If possible, use
semipermanent chalk, or make a transparency of Math Masters, page 433. Make copies of it available to
students. It will also be used in Lesson 7-3. For Part 3, extend the display to include negative powers of 10.
For the Math Message, draw 3 name-collection boxes on the board and label 100, 1,000, and 1,000,000. For
a mathematics and literacy connection, obtain a copy of Can You Count to a Googol? by Robert E. Wells.
Teacher’s Reference Manual, Grades 4–6 pp. 94–98
Lesson 7 2
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Getting Started
Mental Math and Reflexes
Math Message Use slates. Dictate numbers and have students identify
digits in given places.
On a half-sheet of paper, make name-collection
boxes for 100; 1,000; and 1,000,000. Write three different
names in each box. Use exponential notation at least once.
63 0,726. Circle the 10-thousands digit. Underline the
hundred-thousands digit.
26 3,014,613. Circle the 10-millions digit. Underline the
ten-thousands digit.
4 3,269,432.89. Circle the 10-millions digit. Underline the
tenths digit.
Study Link 7 1 Follow-Up
Have partners share answers and resolve any
differences.
Ongoing Assessment:
Recognizing Student Achievement
Math Message
Use the Math Message to assess students’ familiarity with writing exponential
notation for powers of 10 and their ability to write equivalent names for numbers.
[Number and Numeration Goal 4]
1 Teaching the Lesson
WHOLE-CLASS
DISCUSSION
▶ Math Message Follow-Up
Have students share their answers. Write the different names in
name-collection boxes on the board. Answers should include the
following:
1,000
100: 10 ∗ 10; _; 1 hundred; 102
10
10,000
1,000: 10 ∗ 10 ∗ 10; _
; 1 thousand; 103
10
5,000,000
1,000,000: 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10; _; 1 million; 106
5
Ask students to describe the kinds of notation that are included
on the board. Examples of powers of 10 written in exponential
notation are 102, 103, and 106. Examples of powers of 10 written
in number-and-word notation are 1 hundred, 1 thousand, and
1 million.
Explain that number-and-word notation is often used to express
large numbers using a few numerals and one or two words (for
example, 25 billion, 5 hundred thousand), because long strings of
zeros can be hard to read. Write number-and-word notation along
with the example 25 billion on the board or transparency. Write
standard notation along with the example 25,000,000,000. Ask
students to compare the two ways of expressing the same number.
Discuss how to translate from number-and-word notation to
standard notation.
One way: 25 billion = 25 ∗ 1,000,000,000 = 25,000,000,000
548
Unit 7
Exponents and Negative Numbers
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Student Page
Another way: Use a place-value chart to position the leading
digits. Then add zeros to complete the number.
Date
LESSON
72
Time
Guides for Powers of 10
Study the place-value chart below.
Have volunteers write number-and-word notations for the class
to write in standard notation.
Periods
Millions
Hundred
Ten
Billions millions millions
109
▶ Introducing Exponential
WHOLE-CLASS
DISCUSSION
Notation for Powers of 10
(Student Reference Book, p. 5)
Refer students to page 5 of the Student Reference Book. As a
class, discuss the presented definition of powers of 10 — whole
numbers that can be written using only 10s as factors.
For example, 1,000 = 10 ∗ 10 ∗ 10 = 103.
Ask students to look at the Powers of 10 Chart on the page and
share their ideas about what patterns might help them figure out
standard notation for powers of 10. Guide them to observe that
the number of zeros in a power of 10, written in standard notation,
is equal to the exponent of that number, written in exponential
notation. For example, 1,000,000 has 6 zeros, so the exponent of
the power of 10 is 6; 1,000,000 = 106.
108
107
Thousands
Millions
Hundred
thousands
106
105
Ones
Ten
thousands Thousands Hundreds Tens
104
103
102
Ones
101 100
In our place-value system, the powers of 10 are grouped into sets of three: ones,
thousands, millions, billions, and so on. These groupings, or periods, are helpful
for working with large numbers. When we write large
numbers in standard notation, we separate these groups
Prefixes
of three with commas.
teratrillion (1012)
gigabillion (109)
There are prefixes for the periods and for other important
megamillion
(106)
powers of 10. You know some of these prefixes from your
kilothousand (103)
work with the metric system. For example, the prefix kilo- in
hectohundred (102)
kilometer identifies a kilometer as 1,000 meters.
decaten (101)
Use the place-value chart for large numbers and the
unione (100)
decitenth (10–1)
prefixes chart to complete the following statements.
centihundredth (10–2)
Example:
millithousandth (10–3)
3
micromillionth (10–6)
1 kilogram equals 10
, or one thousand , grams.
nanobillionth (10–9)
thousand
1.
The distance from Chicago to New Orleans is about 103, or one
2.
A millionaire has at least 10
3.
A computer with 1 gigabyte of RAM memory can hold approximately 10
one
4.
trillion
, miles.
dollars.
9
, or
, bytes of information.
A computer with a 1 terabyte hard drive can store approximately 10
one
5.
billion
6
12 ,
or
, bytes of information.
According to some scientists, the hearts of most mammals will beat about 109, or
one
billion
, times in a lifetime.
Math Journal 2, p. 212
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The next three periods to the left of billions are trillions, then
quadrillions, then quintillions.
●
How many zeros are needed to write 1 trillion in standard
notation? 12 zeros
●
How many times will 10 appear in the repeated factor
expression? 12 times
●
How many periods are to the right of trillions? 4 periods
●
What is the relationship between the number of periods to the
right of trillions and the exponent when 1 trillion is written in
exponential notation? Each period has 3 digits, so 1 trillion
would have 3 digits ∗ 4 periods, or 12 zeros.
Record a few examples on the board, and ask students to write
these numbers in exponential or standard notation. Suggestions:
10,000 104; 100,000 105; 10 101; 10,000,000 107
103 1,000; 102 100; 105 100,000; 1010 10,000,000,000
▶ Using Guides for Powers of 10
PARTNER
ACTIVITY
(Math Journal 2, p. 212; Student Reference Book, p. 5;
Math Masters, p. 433)
Have students read the introductory paragraphs on journal
page 212. Use the example to discuss how to use the place-value
chart and the table of prefixes to work with powers of 10. Mention
that these guides are also found on the inside front cover of their
journals. Assign the problems on the rest of the page.
Ongoing Assessment:
Informing Instruction
Watch for students who have difficulty
identifying the exponents for Problems 3
and 4. Suggest that they use the placevalue chart on the journal page to first write
the number in standard notation and then
count the 0s to determine the exponent.
Alternatively, use a transparency of Math
Masters, page 433 and have students use
copies of the page to practice writing in
standard notation.
Lesson 7 2
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Student Page
Date
Time
LESSON
72
1.
2 Ongoing Learning & Practice
Math Boxes
Measure the length and width of each of the following objects to the nearest half inch.
Answers vary.
a.
in.
width
dictionary
in.
length
in.
d.
palm of your hand
in.
length
2.
b.
piece of paper
length
c.
183
width
width
▶ Playing First to 100
in.
(your choice)
in.
length
in.
width
in.
Amanda collects dobsonflies. Below are the lengths, in millimeters, for the flies in her collection.
PARTNER
ACTIVITY
(Student Reference Book, p. 308; Math Masters,
pp. 456–458)
95, 107, 119, 103, 102, 91, 115, 120, 111, 114, 115, 107, 110, 98, 112
117–119
a.
Circle the stem-and-leaf plot below that represents this data.
Stems
(100s and 10s)
Leaves
(1s)
9
158
9
15888
2377
10
237
10
23777
11
0124559
11
012459
11
0124555
12
0
12
0
12
0
158
Find the following landmarks for the data.
110
Median:
3.
Leaves
(1s)
Leaves
(1s)
10
9
b.
Stems
(100s and 10s)
Stems
(100s and 10s)
91
Minimum:
Range:
4.
Measure ∠P to the nearest degree.
29
∠P measures about
19° .
107, 115
Calculate the sale price.
Regular
Price
P
Mode(s):
Algebraic Thinking Students practice solving open number
sentences by playing First to 100. This game was introduced in
Lesson 4-7. For detailed instructions, see Student Reference Book,
page 308.
▶ Math Boxes 7 2
Discount
Sale
Price
$12.00
25%
$7.99
25%
$80.00
40%
$19.99
25%
$9.00
$5.99
$48.00
$14.99
204
(Math Journal 2, p. 213)
51
213
Math Journal 2, p. 213
EM3cuG5MJ2_U07_209-247.indd 213
INDEPENDENT
ACTIVITY
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lessons 7-4 and 7-6. The skill in
Problem 4 previews Unit 8 content.
1/19/11 7:42 AM
▶ Study Link 7 2
INDEPENDENT
ACTIVITY
(Math Masters, p. 191)
Home Connection Students are asked to memorize the
Guides for Powers of 10 and answer questions about
them.
3 Differentiation Options
Study Link Master
Name
Date
STUDY LINK
Time
READINESS
Guides for Powers of 10
72
There are prefixes that name powers of 10. You know some of them from the
metric system. For example, kilo- in kilometer (1,000 meters). It’s helpful to
memorize the prefixes for every third power of 10 through one trillion.
4–6
376
Powers of 10
Memorize the table below. Have a friend quiz you. Then cover the table, and try
to complete the statements below.
Standard
Notation
Number-and-Word
Notation
Exponential
Notation
1,000
1 thousand
103
kilo-
1 million
106
mega-
1,000,000,000
1 billion
109
giga-
1 trillion
1012
tera-
1,000,000,000,000
9
billion
1.
More than 10 , or one
2.
One thousand, or 10
3.
Astronomers estimate that there are more than 1012, or one
stars in the universe.
3
More than one million, or 10
5.
A kiloton equals one
To investigate patterns in powers of 10, have students
complete the table on the Math Masters page and describe
the patterns they identify in the table.
, people live in China.
6
trillion
,
ENRICHMENT
, copies of The New York Times are sold every day.
▶ Introducing Negative Exponents
thousand , or 10 3 , metric tons.
million , or 10 6 , metric tons.
A megaton equals one
24 ∗ 3
48 =
PARTNER
ACTIVITY
15–30 Min
and Powers of 0.1
Practice
Find the prime factorization of each number, and write it using exponents.
7.
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
L
LV
VING
VIN
V
IIN
NG
N
G
1
, feet is a little less than _5 of a mile.
4.
6.
15–30 Min
(Math Masters, p. 192)
Prefix
1,000,000
▶ Finding Patterns in
PARTNER
ACTIVITY
8.
60 =
(Student Reference Book, p. 7; Math Masters, p. 193)
22 ∗ 3 ∗ 5
Write each number in expanded notation.
9.
10.
3,264 =
675,511 =
3,000 + 200 + 60 + 4
600,000 + 70,000 + 5,000 + 500 + 10 + 1
Math Masters, p. 191
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550
To apply students’ understanding of exponents, have
them explore the patterns and notation of negative
exponents. Read and discuss Student Reference Book,
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Unit 7
Exponents and Negative Numbers
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Teaching Master
Name
Date
LESSON
1 ∗_
1 ∗_
1 = 0.1 ∗ 0.1 ∗ 0.1 = 0.13 = 0.001
10-3 = _
10
10
10
Sample answer: The number of zeros in the standard
notation matches the exponent in the power of 10.
5.
4.
Describe a pattern in the number of zeros used in the standard notation that you used to complete the table.
Sample answer: The value of the digit 1 becomes 10 times as
great as its value in the previous column.
Describe what happens to the value of the digit 1 when you move one column to the right.Sample answer:
1
The value of the digit 1 becomes _
10 of its value in the previous column.
Describe what happens to the value of the digit 1 when you move one column to the left.
Sample answer: Each time you move one column to the
right, the decimal point moves 1 place to the left.
3.
1.
Describe what happens to the decimal point in the standard notation as you move one column to the
right in the table.
103
104
105
106
2.
Describe at least one pattern you used to complete the table.
Sample answer:
Each time you move one column to the right, you divide by 10.
100
102
101
10 ∗ 10 10 ∗ 1
10 ∗ 10 ∗ 10 ∗
10 ∗ 10 ∗ 10
This equation also shows that negative powers of 10 are also
positive powers of 0.1.
ten
thousand
10 ∗ 10 ∗
10 ∗ 10
10
10,000
1 =_
1
1
10-3 = _
=_
3
10 ∗ 10 ∗ 10
1,000
100,000
Negative exponents can be used to express negative powers of 10.
one hundred
thousand
10 ∗ 10 ∗ 10 ∗
10 ∗ 10
1 =_
1
1
2-4 = _
=_
2∗2∗2∗2
16
24
1,000,000
1 =_
1
1
4-3 = _
=_
4∗4∗4
64
43
one million
1 =_
1 =_
1
5-2 = _
5∗5
25
52
1,000
one
thousand
Find the patterns and complete the table below. Do not use your Student Reference Book.
100
Suggestions:
one
hundred
10
ten
1
one
1
10 ∗ _
10
72
Use examples to discuss converting between exponential notation
with negative exponents and fractions.
Time
Powers of 10
10 ∗ 10 ∗ 10
page 7. Emphasize that negative exponents are another way to
represent numbers that are less than 1.
Math Masters, p. 192
Discuss the table on Math Masters, page 193. Students work with
their partners, using the table to answer the questions that follow.
Briefly go over the answers.
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Links to the Future
Negative exponents and powers of 0.1 will be investigated further in Sixth Grade
Everyday Mathematics. The Enrichment activity is provided for exposure only.
EXTRA PRACTICE
▶ Multiplying Decimals by
Powers of 10
WHOLE-CLASS
DISCUSSION
Teaching Master
5–15 Min
Name
Date
LESSON
72
To offer students more practice multiplying decimals by powers
of 10, pose problems like those below. For each problem, have
students write the original problem, rewrite the problem with
the power of 10 written in standard notation, and then solve the
problem.
●
●
●
Our base-ten place-value system works for decimals as well as for whole numbers.
Ones
.
Tenths
Hundredths
Thousandths
10s
1s
.
0.1s
0.01s
0.001s
1
1
1
1
_
∗_
∗
Example: 10-2 = _
=_
10 ∗ 10 = 10
10 = 0.1 0.1 = 0.01
102
Very small decimals can be hard to read in standard notation, so people often use
number-and-word notation, exponential notation, or prefixes instead.
2.3 ∗ 101 2.3 ∗ 10; 23
Guides for Small Numbers
Number-and-Word
Notation
3
35.1 ∗ 10 35.1 ∗ 1,000; 35,100
1 tenth
4
40.7 ∗ 10 40.7 ∗ 10,000; 407,000
5
0.52 ∗ 10 0.52 ∗ 100,000; 52,000
Standard
Notation
Exponential Notation
10
-1
1
=_
10
0.1
Prefix
deci-
1 hundredth
1
10-2 = _
10 ∗ 10
0.01
centi-
1 thousandth
1
10-3 = _
10 ∗ 10 ∗ 10
0.001
milli-
1 millionth
1
10-6 = __
10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10
0.000001
micro-
1 billionth
1
10-9 = ___
10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10
0.000000001
nano-
1 trillionth
1
10-12 = ____
10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 ∗ 10 0.000000000001 pico-
Use the table above to complete the following statements.
1.
g
Have students explain the relationship between multiplying by a
power of 10 and the placement of the decimal point in the product.
Tens
Negative powers of 10 can be used to name decimal places.
A fly can beat its wings once every 10-3 seconds, or once every one thousandth
of a second. This is one
p
●
Time
Negative Powers of 10
2.
py g
This is 10
3.
milli
second.
Earth travels around the sun at a speed of about one inch per microsecond.
6 second, or a millionth
-
of a second.
Electricity can travel one foot in a nanosecond, or one
This is 10
billionth
of a second.
9 second.
-
4.
In 10
12 second, or one picosecond, an air molecule can spin once.
This is one
trillionth of a second.
Math Masters, p. 193
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Lesson 7 2
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Describe what happens to the value of the digit 1 when you move one column to the left.
Describe what happens to the value of the digit 1 when you move one column to the right.
Describe a pattern in the number of zeros used in the standard notation that you used to complete the table.
4.
5.
100
3.
101
1
10 ∗ _
10
one
Date
Copyright © Wright Group/McGraw-Hill
187-220_EMCS_B_MM_G5_U07_576973.indd 192
Describe what happens to the decimal point in the standard notation as you move one column to the
right in the table.
10 ∗ 10 ∗ 10
one
hundred
1
2.
10,000
72
Describe at least one pattern you used to complete the table.
100,000
LESSON
1.
1,000,000
Find the patterns and complete the table below. Do not use your Student Reference Book.
Name
Time
Powers of 10
192
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