Big Numbers
Objective To provide practice reading, writing, and comparing
O
llarge numbers using patterns in the base-ten place-value system.
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Key Concepts and Skills
Analyzing a Data Table
• Read and write whole numbers to hundred
billions. [Number and Numeration Goal 1]
Math Journal 1, p. 128
Students analyze data on
Internet users.
• Identify digits and their values in whole
numbers to hundred billions. [Number and Numeration Goal 1]
• Use multiplication to solve a
multistep problem. [Operations and Computation Goals 3 and 4]
• Make reasonable estimates. [Operations and Computation Goal 6]
Key Activities
Students use a place-value chart to help
them read and write numbers up to the
billions place. Students use dot paper to
explore the relationships among a thousand,
a million, and a billion.
Common
Core State
Standards
Ongoing Assessment:
Informing Instruction See page 359.
Math Boxes 5 8
Math Journal 1, p. 129
Students practice and maintain skills
through Math Box problems.
Study Link 5 8
Math Masters, p. 163
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use a Math Log or Exit Slip (Math
Masters, page 388 or 389). Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing High-Number Toss
Student Reference Book, p. 252
Math Masters, p. 487 (optional)
calculator 1 six-sided die
Students practice place-value skills.
ENRICHMENT
Estimating the Number of Dots
and the Weight of Paper Needed
to Fill the Classroom
Math Masters, p. 164
calculator
Students apply their understanding of the
relationships among thousands, millions,
and billions.
ENRICHMENT
Exploring Big Numbers in How Much
Is a Million?
Math Masters, p. 165
Students explore big numbers.
[Operations and Computation Goal 3]
Key Vocabulary
million billion
Materials
Math Journal 1, pp. 126 and 127
Student Reference Book, p. 4
Study Link 57
Math Masters, p. 162; p. 388 or 389
(optional)
1 ream of copy paper 1 empty carton used
to pack 10 reams of paper calculator
Advance Preparation
For Part 1, draw a place-value chart on the board like the one at the bottom of journal page 126. Make the columns as
long as possible. You will use this chart in several lessons. If possible, use semipermanent chalk. For the second optional
Enrichment activity in Part 3, obtain a copy of How Much Is a Million? by David M. Schwartz (Mulberry Books, 1993).
Teacher’s Reference Manual, Grades 4–6 pp. 59, 60, 259, 260
Lesson 5 8
355
Mathematical Practices
SMP1, SMP2, SMP3, SMP4, SMP5, SMP6, SMP7
Content Standards
Getting Started
4.OA.2, 4.OA.3, 4.NBT.1, 4.NBT.2
Mental Math and
Reflexes
Math Message
Have students display a number on their
calculators for their partners to read. Have
them also take turns dictating numbers for
their partners to display on their calculators.
Study Link 5 7
Follow-Up
Read page 4 in your Student
Reference Book. Be prepared to
discuss how commas in large numbers
are helpful.
For Problem 7, have students discuss
which multiplication method they prefer
and explain why they chose it.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Student Reference Book, p. 4)
On the board, draw four sets of three dashed lines, separated by
commas. Write labels next to the commas as shown below. Remind
students that in large numbers, groups of three digits
are separated by commas.
,
nd
th
ou
m
sa
illi
n
llio
bi
,
on
,
Use this as a template to practice reading and writing numbers.
Write 14,413,236,610 from the Student Reference Book example
on the template. The strategy for reading this or any other large
number is simple.
Read each group of digits separated by commas as a 3-digit
number.
Student Page
Whole Numbers
Read the appropriate label associated with the comma to the
right for each group of three digits.
Study the place-value chart below. Look at the numbers that name the
places. As you move from right to left along the chart, each number is
10 times as large as the number to its right.
1 4,4 1 3,2 3 6,6 1 0
bi
nd
3
sa
1s
ones
0
ou
10s
tens
9
on
100s
hundreds
1
n
1,000s
thousands
8
llio
10,000s
ten thousands
illi
Any number, no matter how large or small, can be written using one or
more of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. A place-value chart
is used to show how much each digit in a number is worth. The place for
a digit is its position in the number. The value of a digit is how much
it is worth according to its place in the number.
m
Place Value for Whole Numbers
The
The
The
The
The
value
value
value
value
value
of
of
of
of
of
the
the
the
the
the
8
1
9
0
3
is
is
is
is
is
th
The number 81,903 is shown in the place-value chart
above. It is read “eighty-one thousand, nine hundred three.”
80,000 (8 ∗ 10,000).
1,000 (1 ∗ 1,000).
900 (9 ∗ 100).
0 (0 ∗ 10).
3 (3 ∗ 1).
In larger numbers, look for commas that separate groups of 3 digits. The
commas help you identify the thousands, millions, billions, and so on.
Last year, the U.S. Mint made 14,413,236,610 pennies.
billions
100
millions
thousands
ones
10
1
,
100
10
1
,
100
10
1
,
1
4
,
4
1
3
,
2
3
6
,
100 10
6
1
1
Erase the number and write other examples. Have volunteers read
the numbers aloud. Students indicate thumbs-up if they agree
with the answer. Suggestions:
●
5,000 5 thousand
●
900,000 900 thousand
●
123,450 123 thousand, 450
●
9,000,000 9 million
●
9,500,000 9 million,
500 thousand
The number is read as “14 billion, 413 million, 236 thousand, 610.”
Read each number to yourself. What is the value of the 5 in each number?
2. 82,500,000
3. 71,054
4. 3,657,000
Check your answers on page 340.
Student Reference Book, p. 4
356
23,000,000,000 23 billion
●
52,405,072 52 million,
405 thousand, 72
●
183,007,694,718
183 billion, 7 million,
694 thousand, 718
0
Read from left to right. Read “billion” at the first comma, “million”
at the second comma, and “thousand” at the last comma.
1. 35,104
●
Unit 5 Big Numbers, Estimation, and Computation
Student Page
Reverse the procedure; that is, name numbers and ask volunteers
to write the numerals on the template.
Date
Time
LESSON
Reading and Writing Big Numbers
5 8
䉬
Each row in the place-value chart shows a number. Use words to write the name
for each number below the chart.
1.
Tell students that in this lesson they will explore the relationships
among a thousand, a million, and a billion.
Billions
100B 10B
Millions
1B
7
a.
Reading and Writing
WHOLE-CLASS
ACTIVITY
1
d.
Big Numbers
a.
(Math Journal 1, p. 126)
c.
, 100Th 10Th 1Th
,
100
,
4
1
0
,
0
6
5
,
2
0
0
5
1
,
8
0
0
,
0
0
0
2
3
,
0
0
0
,
0
0
5
,
1
4
0
2
3
,
4
5
6
,
7
8
9
,
0
1
2
51 million, 800 thousand
23 billion, 5 thousand, 140
123 billion, 456 million, 789 thousand, 12
d.
Use digits to write these numbers in the place-value chart below.
2.
The place-value chart on journal page 126 separates places into
groups of three and labels these groups as ones, thousands,
millions, and billions. The chart also includes commas to
separate the groups of three digits.
a.
400 thousand, 500
b.
208 million, 350 thousand, 600
c.
16 billion, 210 million, 48 thousand, 715
d.
1 billion, 1 million, 1 thousand, 1
Billions
100B 10B
Millions
1B
, 100M 10M 1M
Thousands
b.
1
c.
d.
2
6 , 2
1 , 0
0
1
0
Ones
, 100Th 10Th 1Th
4
3
0
0
a.
The chart includes a label for each individual place at the top of
its column. For example, the labels for the thousands columns
are 100Th (100 thousands), 10Th (10 thousands), and 1Th
(1 thousands). Use the following example to show students that
the place-value name for each column indicates how much a digit
in that column is worth.
Thousands
10
0
7 billion, 400 million, 65 thousand, 200
b.
It is more convenient to use a place-value chart than to label
commas as thousand, million, and so on.
Ones
, 100M 10M 1M
b.
c.
Thousands
4
8 ,
0 ,
1 ,
0
5
4
0
0
0
8
1
,
100
10
1
,
,
,
,
5
6
7
0
0
0
1
0
0
0
5
1
Math Journal 1, p. 126
Ones
100Th
10Th
1Th
,
100
10
1
4
0
0
,
0
0
0
4
0
,
0
0
0
4
,
0
0
0
This 4 is worth
4 one thousands, or 4 thousand.
Student Page
Date
Time
LESSON
5 8
䉬
This 4 is worth
4 ten thousands, or 40 thousand.
How Much Are a Million and a Billion?
1.
How many dots are on the 50-by-40 array page?
2.
How many dots would be on
This 4 is worth
4 hundred thousands, or 400 thousand.
a.
5 pages?
b.
50 pages?
c.
500 pages?
10,000
100,000
1,000,000
2,000
dots
4
dots
dots
dots
CO
PY
PA
PE
R
Æ
Enter numbers in the place-value chart on the board and have
students read them. Then name several numbers, and ask
volunteers to write them in the chart.
夹
3.
Assign journal page 126 to the class.
4.
Links to the Future
Each package of paper, or ream, contains 500 sheets. How many dots
would be on the paper in
a.
1 ream? (Hint: Look at Problem 2.)
b.
10 reams? (1 carton)
c.
100 reams? (10 cartons)
d.
1,000 reams? (100 cartons)
dots
dots
dots
dots
Use digits to write these numbers in the place-value chart below.
a.
999 thousand
b.
1,000 thousand
Billions
100B 10B
In Lessons 2-3 and 2-4, students worked with numbers through the hundredmillions place. This is the first exposure to numbers in the billions. Reading,
writing, and identifying digits and their values in numbers beyond 1,000,000,000
is a Grade 5 Goal.
1,000,000
10,000,000
100,000,000
1,000,000,000
1B
c.
, 100M 10M 1M
b.
d.
9
1 , 0
9
0
1
9
0
1,000 million
d.
Thousands
, 100Th 10Th 1Th
a.
c.
999 million
Millions
,
,
,
9
0
0
0
9
0
0
0
9
0
0
0
Ones
,
100
10
1
,
,
,
,
0
0
0
0
0
0
0
0
0
0
0
0
Math Journal 1, p. 127
Lesson 5 8
357
Teaching Master
Name
Date
LESSON
Exploring the Relationships
Time
A 50-by-40 Array
58
among a Thousand, a Million,
and a Billion
PARTNER
ACTIVITY
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VING
VI
VIN
IN
ING
(Math Journal 1, p. 127; Math Masters, p. 162)
If you have a ream of copy paper and a packing carton, display
them. Tell the class that a ream contains 500 sheets and that a
full carton contains 10 reams. Working with a partner, students
complete journal page 127.
py g
Compare strategies and answers. Use the ream and empty carton
to support the discussion. Have students share how they found the
number of dots on Math Masters, page 162. Sample answers:
g
p
There are 20 squares with 100 dots each; that is 20 [100s],
or 2,000.
The dots form a 50-by-40 array; that is 50 [40s], or 50 ∗ 40,
or 2,000.
Math Masters, p. 162
EM3cuG4MM_U05_139-176.indd 162
12/28/10 1:39 PM
There are 1,000 dots on a half-sheet; that is 2,000 dots on a
full sheet.
Review strategies for determining that there are 1 million dots
in a ream:
There are 1,000 dots per half-sheet and 1,000 half-sheets per
ream of 500; that is 1,000 [1,000s], or 1 million.
There are 2,000 dots per sheet and 500 sheets; that is
500 [2,000s], or 1 million.
Use the answers to Problems 3a and 3d to illustrate that 1 billion
is equivalent to 1,000 million:
1 ream contains paper with a total of 1 million dots
(Problem 3a), so 1,000 reams must contain paper with 1,000
million dots. But 1,000 reams contain paper with 1 billion dots
(Problem 3d), so 1,000 million and 1 billion are the same
number.
Student Page
Date
Time
LESSON
Internet Users
58
The table below shows the estimated number of people who used the Internet
in several different countries in 2004.
Internet Users in 2004
Country
Users
France
25,470,000
Greece
2,710,000
Hungary
2,940,000
Italy
25,530,000
Poland
10,400,000
Spain
13,440,000
Source: The World Factbook
1.
Which of these countries had the most Internet users?
2.
Which of these countries had the fewest Internet users?
3.
Ongoing Assessment:
Recognizing Student Achievement
Italy
Greece
About how many more Internet users were there in France than in Spain? Sample
answer:
25,000,000 - 13,000,000 = 12,000,000
12,000,000
Number model:
Answer:
4.
5.
Write true or false.
a.
There were more Internet users in Greece,
Poland, and Hungary combined than in Spain.
b.
There were about 6 times as many Internet
users in France as there were in Hungary.
true
Why do you think the Internet usage data is rounded to the nearest 10,000
instead of an actual count?
The United States had about 62 times as many Internet users as Hungary. About how many
Internet users were there in the United States? Sample answers:
62 ∗ 2,940,000 = 182,280,000
182,280,000
Number model:
Answer:
Math Journal 1, p. 128
106-136_EMCS_S_MJ1_G4_U05_576361.indd 128
358
Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess
students’ ability to use extended multplication facts in a problem-solving situation.
Have students describe how they determined on journal page 127, Problem 3a
that a ream of paper would have 1 million dots. Students are making adequate
progress if they can explain in words, with pictures, or with number models how
2,000 dots on one page can be used to determine the number of dots on 500
pages. Some students may be able to explain how they used this information
to solve problems 3b–3d.
[Operations and Computation Goal 3]
false
Sample answer: The number of users changes often, so it is
hard to get an accurate count.
6.
Math Log
or Exit Slip
8/25/11 8:31 AM
Unit 5 Big Numbers, Estimation, and Computation
Student Page
2 Ongoing Learning & Practice
Analyzing a Data Table
Date
Time
LESSON
Math Boxes
5 8
䉬
1. a.
1
4
Measure the line segment to the nearest ᎏᎏ inch.
R
S
6
About
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 128)
2.
Social Studies Link Students answer questions about the
number of Internet users in several countries in Region 2.
inches
b.
Draw a line segment that is half as long as the one above.
c.
How long is the line segment you drew?
Estimate the product. Write a number
model to show how you estimated.
a.
3
About
4,508
Sample answers:
40 ⴱ 80 ⫽ 3,200
9
4
0
2
4
+
4
4 5 0
75 ⴱ 32
80 ⴱ 30 ⫽ 2,400
Ongoing Assessment: Informing Instruction
184
4.
Write each number using digits.
5.
a.
seven hundred six thousandths
b.
three and four hundredths
Math Boxes 5 8
3.04
A.
9
B.
8
C.
7
D.
6
175
Math Journal 1, p. 129
INDEPENDENT
ACTIVITY
Study Link Master
Name
Mixed Practice Math Boxes in this lesson are linked
with Math Boxes in Lessons 5-6 and 5-10. The skill in
Problem 5 previews Unit 6 content.
Date
STUDY LINK
58
䉬
Writing/Reasoning Have students write a response to the
following: Explain how you determined the number of pies
for Problem 5. Sample answer: Because 75 / 8 = 9 R3 or
9_38 , there are enough apples for only 9 pies.
INDEPENDENT
ACTIVITY
(Math Masters, p. 163)
Home Connection Students solve a place-value puzzle
involving the four basic operations.
Time
Place-Value Puzzle
4
Use the clues below to fill in the place-value chart.
Billions
100B 10B
Study Link 5 8
18
27 28
(Math Journal 1, p. 129)
8
6
0
0
0
8
8
You have 75 apples. You need 8 apples
to make a pie. How many pies can you
bake?
0.706
When students have completed the journal page, ask them to
look at Problems 3 and 6. Ask: How is the way you compared the
number of Internet users in France and in Spain different from
how you compared the number of Internet users in the United
States and in Hungary? Sample answer: For France and Spain,
I used subtraction to compare. But for the United States and
Hungary, I used multiplication. Point out that the difference
between the numbers of Internet users in the United States and
in Hungary is so large that it is almost equal to the number for
the United States. When differences are this big, it is usually more
useful to use multiplication to compare the numbers, because
multiplication gives a better sense of how much bigger one
quantity is than another.
⫽ 46 ⴱ 98
º
3 6
3
5
Number model:
Watch for students who use estimation strategies rather than calculate exact
answers to solve the problems. Have them explain why estimation is appropriate.
128
41 ⴱ 83
Number model:
b.
inches
Multiply. Use the partial-products method.
3.
1B
Millions
,
100M 10M
9 2
Thousands
1M
,
1 0 6
100Th 10Th 1Th
Ones
,
9 5 4
100
10
1
8 7 3
1.
1
Find ᎏ2ᎏ of 24. Subtract 4. Write the result in the hundreds place.
2.
1
Find ᎏ2ᎏ of 30. Divide the result by 3. Write the answer in the ten-thousands place.
3.
Find 30 ⫼ 10. Double the result. Write it in the one-millions place.
4.
Divide 12 by 4. Write the answer in the ones place.
5.
Find 9 º 8. Reverse the digits in the result. Divide by 3.
Write the answer in the hundred-thousands place.
6.
Double 8. Divide the result by 4. Write the answer in the one-thousands place.
7.
In the one-billions place, write the even number greater than 0
that has not been used yet.
8.
Write the answer to 5 ⫼ 5 in the hundred-millions place.
9.
In the tens place, write the odd number that has not been used yet.
10.
Find the sum of all the digits in the chart so far.
Divide the result by 5, and write it in the ten-billions place.
11.
Write 0 in the empty column whose place value is less than billions.
12.
Write the number in words. For example, 17,450,206 could be written as
“17 million, 450 thousand, 206.”
92 billion, 106 million, 954 thousand, 873
Practice
13.
15.
74 º 5 ⫽
1,656
370
⫽ 92 º 18
14.
16.
3,168 ⫽ 396 º 8
2,632
56 º 47 ⫽
Math Masters, p. 163
Lesson 5 8
359
Teaching Master
Name
LESSON
58
䉬
Date
Time
Suppose you filled your classroom from floor to ceiling with dot paper
(2,000 dots per sheet).
1.
3 Differentiation Options
A Roomful of Dots
180–184
About how many dots do you think there would be on all the paper
needed to fill your classroom? Make a check mark next to your guess.
Answers vary.
less than a million
between a million and a half billion
between half a billion and a billion
more than a billion
2.
One ream of paper weighs about 5 pounds and has 500 sheets of paper.
About how many pounds would the paper needed to fill your classroom weigh?
Make a check mark next to your guess.
READINESS
Playing High-Number Toss
15–30 Min
(Student Reference Book, p. 252; Math Masters,
p. 487)
Answers vary.
less than 100,000 pounds
PARTNER
ACTIVITY
between 100,000 pounds and 500,000 pounds
between 500,000 pounds and a million pounds
To provide experience with place-value skills, have students play
High-Number Toss. See Lesson 2-7 for additional information.
more than a million pounds
3.
Now, work with your group to make more accurate estimates for Problems 1 and 2.
Explain what you did.
Sample answers:
56,700,000,000 dots
My group’s estimates:
Number of dots:
Weight of the paper:
ENRICHMENT
283,500 lb
Sample answer:
A ream of paper is about 2 in. high, so 6 reams will be
about 1 ft tall. It will take about 60 reams to reach the ceiling.
The floor is about 27 reams by 35 reams, so 945 reams will
cover it. To fill the room, it will take about 56,700 reams.
Estimating the Number of Dots
SMALL-GROUP
ACTIVITY
15–30 Min
and the Weight of Paper Needed
to Fill the Classroom
(Math Masters, p. 164)
Math Masters, p. 164
To apply students’ understanding of the relationships
among thousands, millions, and billions, have them devise
and carry out a strategy to see how many dots (on dot
paper) would fill a classroom.
Points of reference:
Approximately 56,700 reams of paper would be needed to
fill a 25-by-25-by-10 ft classroom (a 625 sq ft classroom with
a 10 ft ceiling).
About 283,500 pounds, or 140 tons, of paper would fill the
same classroom. 56,700 reams ∗ 5 pounds = 283,500 pounds.
Name
LESSON
58
䉬
Date
Time
How Much Is a Million?
David M. Schwartz, the author of How Much Is a Million?, used 7 pages of his book to show
approximately 100,000 tiny stars.
Exploring Big Numbers in
He wrote, “If this book had a million tiny stars, they would fill seventy pages.... If this book
had a billion tiny stars, its pages spread side by side would stretch almost ten miles.... If you
put a trillion of our stars onto a gigantic roll of paper, it would stretch all the way from New
York to New Zealand.” Sample answers:
1.
How Much Is a Million?
PARTNER
ACTIVITY
5–15 Min
(Math Masters, p. 165)
About how many pages would be needed to show a trillion tiny stars? Explain.
About 70,000,000 pages show 1 trillion stars. 1 million
stars fill 70 pages. 1 billion stars fill 70 º 1,000 ⫽ 70,000 pages.
1 trillion stars fill 70,000 º 1,000 ⫽ 70,000,000 pages.
2.
ENRICHMENT
Describe a strategy you could use, other than counting each star, to find the
number of tiny stars on one page of How Much Is a Million?
Divide 100,000 by 7 pages. (100,000 / 7 ⫽ 14,286) Or, there are
133 rows and 108 columns per page. (133 º 108 ⫽ 14,364)
Math Masters, page 165
360
Unit 5 Big Numbers, Estimation, and Computation
Literature Link To further explore students’ understanding
of the relationships among thousands, millions, and billions
to millions, billions, and trillions, have students read and answer
questions about How Much Is a Million? by David M. Schwartz
(Mulberry Books, 1993).