Comparing Millions,
Billions, and Trillions
Objectives To provide experience with comparing the relative
sizes
of 1 million, 1 billion, and 1 trillion and using a sample to
s
make an estimate.
www.everydaymathonline.com
ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Read and write large numbers. [Number and Numeration Goal 1]
• Compare order of magnitude for large
numbers. [Number and Numeration Goal 6]
• Make reasonable estimates for whole
number multiplication problems. [Operations and Computation Goal 6]
Key Activities
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing High-Number Toss
Student Reference Book, p. 320
Math Masters, p. 487
per partnership: 1 die 1 sheet of
paper
Students practice concepts of place
value and standard notation by writing
and comparing large numbers.
Ongoing Assessment:
Recognizing Student Achievement
Ongoing Assessment:
Informing Instruction See page 134.
Key Vocabulary
sample
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Playing Number Top-It (7-Digit Numbers)
Student Reference Book, p. 326
Math Masters, pp. 491 and 492
per partnership: 4 each of number cards 0–9
(from the Everything Math Deck, if available)
Students apply place-value concepts to form,
read, and compare large numbers.
EXTRA PRACTICE
[Numbers and Numeration Goals 1 and 6]
Comparing Powers of 10 Using
Place Value
Math Boxes 2 10
Math Masters, p. 66B
Students apply place value for powers of 10.
Use Math Masters, page 487. Students review time conversion factors.
They count the number of times they can
tap their desks in 10 seconds and estimate
how long it would take to tap 1 million times.
Students then estimate how long it would
take to tap 1 billion and 1 trillion times.
Curriculum
Focal Points
Math Journal 1, p. 58
Students practice and maintain skills
through Math Box problems.
Study Link 2 10
Math Masters, p. 61
Students practice and maintain skills
through Study Link activities.
ENRICHMENT
Applying Estimation Strategies
Math Masters, p. 62
Students make time estimates and identify
the number models used for their estimation
strategies.
Materials
Math Journal 1, p. 57
Study Link 29
Class Data Pad blank paper or construction
paper markers or crayons watch or timer
with second hand calculator
Advance Preparation
For Part 1, use the Class Data Pad to record and display the Mental Math and Reflexes problems and responses.
For the optional Readiness activity in Part 3, make one game mat for each partnership by copying, cutting, and
taping together Math Masters, pages 491 and 492.
For a mathematics and literary connection, obtain a copy of How Much Is a Million? by David M. Schwartz
(HarperCollins Publishers, 1985).
Teacher’s Reference Manual, Grades 4–6 pp. 256–264
132
Unit 2
Estimation and Computation
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Getting Started
Mental Math and Reflexes
Math Message
Have students practice conversions between units of time. Use the Class
Data Pad to record the correct responses. Keep this display up for students
to refer to during the lesson.
Explain the strategy you
would use to find the number of
minutes in one year.
How many seconds are in...
How many hours are in...
1 minute? 60
3 minutes? 180
100 minutes? 6,000
How many minutes are in...
Study Link 2 9
Follow-Up
1 day? 24
2 days? 48
200 days? 4,800
Have partners share answers
and resolve any differences.
How many days are in...
1 hour? 60
5 hours? 300
50 hours? 3,000
1 year? 365, except 366 in leap years
10 years? about 3,650
100 years? about 36,500
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Have students in small groups discuss their individual strategies.
The group then decides which steps they would use to find the
number of minutes in a year. Each group should make a poster,
using construction paper and markers or crayons, to list their
steps and place the poster on display. Allow time for students to
read the displayed group posters. Ask volunteers to identify the
similarities and differences in the strategies. Guide the class
discussion to focus on summarizing the strategies. Sample
answers: Convert a year into a unit of time that can be converted
to minutes; convert a year into days; convert these days into
hours and the hours into minutes.
Student Page
Date
Time
LESSON
2 10
䉬
Millions, Billions, and Trillions
Useful Information
NOTE Include a walkabout in the follow-up to this Math Message: Display the
1 billion is 1,000 times 1 million.
1 million ⴱ 1 thousand ⫽ 1 billion
1,000,000 ⴱ 1,000 ⫽ 1,000,000,000
group posters in separate areas of your classroom, and allow students time to
browse until they have read all the posters.
1 trillion is 1,000 times 1 billion.
1 billion ⴱ 1 thousand ⫽ 1 trillion
1,000,000,000 ⴱ 1,000 ⫽ 1,000,000,000,000
1 minute ⫽ 60 seconds 1 hour ⫽ 60 minutes
1 year ⫽ 365 days (366 days in a leap year)
1 day ⫽ 24 hours
Make a guess: How long do you think it would take you to tap your desk 1 million times,
without any interruptions?
Answers vary.
▶ Solving a Tapping Problem
with Sampling Strategies
WHOLE-CLASS
DISCUSSION
PROBLEM
PRO
P
RO
R
OB
BLE
BL
L
LE
LEM
EM
SOLVING
SO
S
OL
O
LV
VING
VIN
ING
(Math Journal 1, p. 57)
Check your guess by doing the following experiment.
Sample answers:
1.
Take a sample count.
Record your count of taps made in 10 seconds.
2.
Calculate from the sample count.
At the rate of my sample count, I expect to tap my desk:
a.
times in 1 minute.
(Hint: How many 10-second intervals are there in 1 minute?)
40 taps
240
14,400 times in 1 hour.
345,600 times in 1 day (24 hours).
3
d. At this rate it would take me about
b.
Ask students to refer to the Useful Information chart on journal
page 57. Pose several questions from the first row of information
to highlight these large number relationships for students. What
is 1,000 times 1 million? 1 billion One billion has what
relationship to 1 trillion? One trillion is 1,000 times 1 billion.
Ask a volunteer to read the question labeled Make a guess. Ask
students to guess how long it would take them to tap their
desks 1 million times without any interruptions. Have students
c.
full 24-hour days to tap my
desk 1 million times.
3.
Suppose that you work 24 hours per day tapping your desk. Estimate how long it
would take you to tap 1 billion times and 1 trillion times.
a.
It would take me about
b.
It would take me about
8 years to tap my desk 1 billion times.
(unit)
8,000 years to tap my desk 1 trillion times.
(unit)
Math Journal 1, p. 57
Lesson 2 10
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Student Page
record their responses on the journal page. Conduct a quick
survey of the class for their responses.
Games
High-Number Toss
Materials 䊐 1 six-sided die
䊐 1 sheet of paper for each player
Players
2
Skill
Place value, exponential notation
Object of the game To make the largest numbers possible.
Directions
1. Each player draws 4 blank lines on a
sheet of paper to record the numbers that
come up on the rolls of the die.
Player 1: ______
______
______
|
______
Player 2: ______
______
______
|
______
2. Player 1 rolls the die and writes the number on any of his or
her 4 blank lines. It does not have to be the first blank—it can
be any of them. Keep in mind that the larger number wins!
3. Player 2 rolls the die and writes the number on one of his or
her blank lines.
4. Players take turns rolling the die and writing the number
3 more times each.
5. Each player then uses the 4 numbers on his or her blanks to
build a number.
Note
♦ The numbers on the first 3 blanks are the first 3 digits of
If you don’t have a die,
you can use a deck of
number cards. Use all
cards with the numbers
1 through 6. Instead of
rolling the die, draw the
top card from the
facedown deck.
the number the player builds.
♦ The number on the last blank tells the number of zeros
that come after the first 3 digits.
6. Each player reads his or her number. (See the place-value
chart below.) The player with the larger number wins the
round. The first player to win 4 rounds wins the game.
Hundred
Millions
Ten
Millions
,
Millions
Hundred
Thousands
First three digits
Ten
Thousands
Thousands
,
Hundreds
Tens
3
2 | 6 ⫽ 132,000,000 (132 million)
Player 2: 3
5
6 | 4 ⫽
Encourage students to think about what information they would
need to make a more educated guess, and then ask volunteers
to explain strategies that could be used to gather additional
information.
Ones
Number
of zeros
Player 1: 1
Discuss the difference between a guess and an estimate. Use the
discussion to clarify for students that a guess is an opinion that
you might state without the support of other information. An
estimate is based on some knowledge about the subject and is
often called an educated guess. Students can only guess the
amount of time it would take to tap 1 million times until they
collect additional information with which to make an estimate.
Ongoing Assessment: Informing Instruction
3,560,000 (3 million, 560 thousand)
Watch for students who are having difficulty developing a strategy. Explain that
this is another situation for which obtaining the exact answer is impossible, such
as the Estimation Challenge from Lesson 2-1. Recall the strategies students
used in that lesson and have them use similar approaches here.
Player 1 wins.
Student Reference Book, p. 320
Students might suggest strategies such as the following:
Count how many times you can tap your desk in a set amount
of time, such as 10 seconds.
Time how long it takes you to tap a certain number of times,
such as 100 times.
Pick a reasonable number of taps for a set amount of time,
and make an estimate based on that rate, such as 3 taps per
second.
When the class found and used the median step length, they
were using a sample. Ask whether students know of any other
situations where samples are used.
Game Master
Name
Date
Time
1 2
4 3
High-Number Toss Record Sheet
Hundred
Millions
Ten
Millions
Millions
Round
Sample
,
Hundred
Thousands
Ten
Thousands
>, <, =
Player 1
1 3 2
|
6
132, 000, 000
>
,
Hundreds
Tens
Ones
Player 2
3 5 6
|
4
3, 560,000
1
|
|
2
|
|
3
|
|
4
|
|
5
|
|
NOTE A sample of anything is a small piece or part that is intended to give
information about the whole thing. Consumers use product samples to decide
whether products suit their needs. Pollsters use population samples to estimate
information for the whole population. The finger-tapping samples here are time
samples: The count of taps in a 10-second period is a sample used to determine
a tapping rate, and the rate is then used to estimate how long it would take to
make large numbers of taps.
Each student will take a 10-second sample count of their own
finger tapping. Practice taking sample counts by timing students
as they tap and count for 10 seconds.
py g
g
p
Thousands
320
Math Masters, p. 487
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Unit 2
Estimation and Computation
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▶ Using Sampling to Make
PARTNER
ACTIVITY
an Estimate
(Math Journal 1, p. 57)
Have partners complete the journal page. They begin by finding
their individual sample counts. Partners take turns. While one
partner taps and counts the taps for 10 seconds, the second partner
keeps time for 10 seconds, signaling when to start and stop.
Partners then use their 10-second sample counts to estimate the
number of taps they could make in 1 minute; in 1 hour; and in
1 day. Encourage students to use their calculators, as needed.
Next students use their estimates to calculate the approximate
number of days it would take to tap 1 million times. Encourage
students to devise their own solution strategies. One possible
approach is to divide 1 million by the number of taps per day.
▶ Making Time Estimates for
PARTNER
ACTIVITY
1 Billion and 1 Trillion Taps
(Math Journal 1, p. 57)
In Problem 3 on journal page 57, students estimate the time it
would take to tap 1 billion and 1 trillion times. Remind students
that they can use the relationships between 1 million, 1 billion,
and 1 trillion found in the Useful Information chart to help them
estimate. They will also need to decide whether to report their
estimates for 1 billion and 1 trillion taps as days or years.
NOTE Expect that the tapping rate for most students will be about 40 times in 10
seconds. At this rate they will tap about 250 times in 1 minute (6 ∗ 40, rounded up);
15,000 times in 1 hour (60 ∗ 250); 350,000 times in 1 day (24 ∗ 15,000; rounded
down), and about 3 days, without interruptions, to tap 1 million times.
Student Page
Date
▶ Sharing and Discussing
SMALL-GROUP
DISCUSSION
the Results
Time
LESSON
䉬
1.
Find the missing numbers and landmarks for the set of numbers below.
48, 50, 51, 51, 57, 59, 60, 63, 69,
(Math Journal 1, p. 57)
a.
Range:
b.
Mode: 76
2. a.
When most students have completed the problems, have partners
form small groups to discuss their strategies. Then have the
groups report on the similarities and differences of the strategies
used as well as any notable experiences they encountered. Use the
following questions as a guide:
●
●
●
How does your estimate of the time for 1 million taps compare
with your initial guess?
Did you use your estimate for the number of taps in 1 day to
estimate how long it would take to tap 1 million times?
Did you use the time for 1 million taps to estimate the time for
1 billion and 1 trillion taps?
Math Boxes
2 10
76
28
, 76,
48
Minimum:
d.
Maximum: 76
119
Make up a set of at least twelve numbers
that have the following landmarks.
Maximum: 8
Mode: 6
Range: 6
Median: 5
5
4
3
2
1
0
Sample answer: 2, 2, 2,
3, 4, 5, 5, 6, 6, 6, 6, 8, 8
b.
76
c.
Make a bar graph of the data.
Sample answer:
1 2 3 4 5 6 7 8
119 122
3.
Use the map on page 354 of your Student Reference Book to answer the questions.
Choose the best answer.
a.
What is the shortest distance between San Francisco and San Antonio? About:
b.
What is the shortest distance between New York City and Chicago? About:
500 miles
1,000 miles
1,500 miles
2,000 miles
211
700 miles
4. a.
2:00
b.
800 miles
900 miles
1,000 miles
Circle the times for which the hands on a clock form an acute angle.
6:40
1:30
12:50
Circle the times for which the hands on a clock form
an obtuse angle.
8:00
1:20
5:15
10:30
139
Math Journal 1, p. 58
Lesson 2 10
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Study Link Master
Place-Value Puzzles
STUDY LINK
2 10
䉬
Millions
28
Thousands
Ones
HundredTenMillions HundredTenThousands Hundreds Tens Ones
millions millions
thousands thousands
Use the clues to solve the puzzles.
Puzzle 1
䉬 The value of the digit in the thousandths place is equal to the sum of the measures of
the angles in a triangle (180°) divided by 30.
To conclude Part 1 ask students: If one person could tap 24 hours
per day without stopping, would it be possible to tap 1 trillion
times? No Aim follow-up questions at getting students to support
their responses. They would still need many more years than are
in a normal lifetime. An exit question might be: Do you feel that
your informed estimate was more reasonable than your guess?
䉬 If you multiply the digit in the tens place by 1,000; the answer will be 9,000.
䉬 Double 35. Divide the result by 10. Write the answer in the tenths place.
1
䉬 The hundreds-place digit is ᎏ2ᎏ the value of the digit in the thousandths place.
䉬 When you multiply the digit in the ones place by itself, the answer is 0.
䉬 Write a digit in the hundredths place so that the sum of all six digits in this number is 30.
3
What is the number?
9
0
.
7
5
6
2 Ongoing Learning & Practice
Puzzle 2
䉬 Double 12. Divide the result by 8. Write the answer in the thousands place.
䉬 If you multiply the digit in the hundredths place by 10, your answer will be 40.
Copyright © Wright Group/McGraw-Hill
䉬 The tens-place digit is a prime number. If you multiply it by itself, the answer
is 49.
䉬 Multiply 7 and 3. Subtract 12. Write the answer in the thousandths place.
䉬 Multiply the digit in the hundredths place by the digit in the thousands place. Subtract 7
from the result. Write the digit in the tenths place.
䉬 The digit in the ones place is an odd digit that has not been used yet.
䉬 The value of the digit in the hundreds place is the same as the number of
sides of a quadrilateral.
3
What is the number?
,
4
7
1
5
.
4
9
Check: The sum of the answers to both puzzles is 3,862.305.
Practice
9,340
44,604
244
19 R2
3.
7,772 ⫹ 1,568 ⫽
4.
472 ⫺ 228 ⫽
5.
826 º 54 ⫽
6.
59 / 3 ∑
61
Math Masters, p. 61
▶ Playing High-Number Toss
PARTNER
ACTIVITY
(Student Reference Book, p. 320; Math Masters, p. 487)
High-Number Toss provides students with the opportunity to
apply their knowledge of place value and standard notation to
create, write, read, and compare large numbers. Provide students
with a reminder box on the board noting that < means less than
and > means greater than.
Ongoing Assessment:
Recognizing Student Achievement
Math Masters
Page 487
Use the Record Sheet for High-Number Toss (Math Masters, page 487) to
assess students’ knowledge of place value and comparing numbers. Students
are making adequate progress if they correctly insert the relational symbols
between the two numbers in five rounds of the game.
[Numbers and Numeration Goals 1 and 6]
▶ Math Boxes 2 10
(Math Journal 1, p. 58)
Student Page
Games
Number Top-It
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 2-8. The skill in Problem 4
previews Unit 3 content.
(7-Digit Numbers)
Materials 䊐 number cards 0—9 (4 of each)
䊐 one Place-Value Mat
(Math Masters, pp. 491 and 492)
Players
2 to 5
Skill
Place value for whole numbers
INDEPENDENT
ACTIVITY
Object of the game To make the largest 7-digit numbers.
▶ Study Link 2 10
Directions
1. Shuffle the cards and place the deck number-side down on
the table.
2. Each player uses one row of boxes on the place-value mat.
INDEPENDENT
ACTIVITY
(Math Masters, p. 61)
3. In each round, players take turns turning over the top card
from the deck and placing it on any one of their empty
boxes. Each player takes a total of 7 turns, and places 7
cards on his or her row of the game mat.
Home Connection Students use their knowledge of place
value and number relationships to solve number puzzles.
4. At the end of each round, players read their numbers aloud
and compare them to the other players numbers. The player
with the largest number for the round scores 1 point. The
player with the next-largest number scores 2 points, and so on.
5. Players play 5 rounds for a game. Shuffle the deck between
each round. The player with the smallest total number of
points at the end of 5 rounds wins the game.
Example
Andy and Barb played 7-digit Number Top-It. Here is the result for one
complete round of play.
Place-Value Mat
Ones
1
4
2
5
5
4
5
2
3
Tens
0
2
2
3
6
7
7
7
9
9
4
5
0
Barb
4
6
7
4
Andy
Hundred
Ten
Thousands Thousands Thousands Hundreds
1
Millions
4
Andy s number is larger than Barb s number. So Andy scores 1 point for this
round, and Barb scores 2 points.
Student Reference Book, p. 326
136
Unit 2
Estimation and Computation
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Teaching Master
Name
3 Differentiation Options
Date
Time
Number Stories and Estimation
LESSON
2 10
䉬
䉬 Read each number story carefully.
䉬 Write an open number sentence to use in estimating.
䉬 Answer the question.
READINESS
▶ Playing Number Top-It
PARTNER
ACTIVITY
Example:
15–30 Min
Open number sentence:
(7-Digit Numbers)
It is said that the Aztec king, Montezuma, drank about 50 cups of chocolate per day. Did he
drink more or less than 10 gallons of chocolate in a week? (Hint: 16 cups 1 gallon)
Answer:
Certain varieties of seahorses can move 10.5 inches per minute. At this rate,
could these seahorses be able to travel 6 yards in 1 hour?
1.
a.
(Student Reference Book, p.326; Math Masters,
pp. 491 and 492)
b.
To review place-value concepts, have students play Number Top-It
(7-Digit Numbers).
a.
b.
▶ Comparing Powers of 10
Sample answer:
120 º 5 feet traveled in 1 minute
No
Answer:
Open number sentence:
In 1960, the Triton became the first submarine to circumnavigate the world. It
covered 36,014 miles in 76 days. Is that more or less than 100 miles per day?
3.
SMALL-GROUP
ACTIVITY
a.
b.
5–15 Min
Sample answer:
10.5 º 60 36 yards traveled in 1 hour
Yes
Answer:
Open number sentence:
Orville Wright completed the first airplane flight on December 17, 1903. He
traveled 120 feet in 12 seconds. If he had been able to stay in the air for a full
minute, would he have traveled 1 mile? (Hint: 1 mile 5,280 feet)
2.
EXTRA PRACTICE
10 º 16 Number of cups in 10 gallons
more
Sample answer:
100 º 76 total miles for 76 days at 100 miles per day
more
Answer:
Open number sentence:
Source: The Kids’ World Almanac of Records and Facts
Using Place Value
(Math Masters, p. 66B)
Math Masters, p. 62
To provide additional practice with place value and understanding
the relationships between powers of 10, have students complete
Math Masters, page 66B.
ENRICHMENT
▶ Applying Estimation Strategies
INDEPENDENT
ACTIVITY
5–15 Min
(Math Masters, p. 62)
To apply students’ ability to use estimation strategies, have them
solve problems that involve converting situational information into
open number sentences. Direct students to focus on making
informed estimates.
Teaching Master
Name
Date
LESSON
Time
Using Place Value to Compare Powers of 10
2 10
1 meter
10 decimeters
100 centimeters
1,000 millimeters
1 centimeter
0.01 meter
0.1 decimeter
10 millimeters
Use the information in the conversion table to respond to each statement
below. Complete each statement with one of the following phrases:
1
1
10 times, 100 times, _
of, _
of
10
100
10 times the size of a decimeter.
1
___
100 of
1 centimeter is
the size of a meter.
10 times the size of a millimeter.
1 centimeter is
1
__
10 of
1 decimeter is
the size of a meter.
1
___
100 of
1 millimeter is
the size of a decimeter.
1.
1 meter is
2.
3.
4.
5.
Write two of your own statements using the information in the table.
Answers vary.
6.
7.
Complete the table below by making the appropriate conversions.
millimeters
centimeters
decimeters
meters
8.
9,743
3,000
1,750
97.43
30
17.5
9.743
9.
974.3
300
10.
11.
175
3
1.75
In Problem 10, explain what happens to the value of the digit 5 when you go
from millimeters to centimeters, and then from decimeters to meters.
Sample answer: You divide by 10 with each
conversion; so each time, you move the decimal
point one position to the left. So with each move, the
1
digit 5 is worth __
10 as much as before.
Math Master, p. 66B
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Lesson 2 10
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Name
Date
LESSON
Time
Using Place Value to Compare Powers of 10
2 10
1 meter
10 decimeters
100 centimeters
1,000 millimeters
1 centimeter
0.01 meter
0.1 decimeter
10 millimeters
Use the information in the conversion table to respond to each statement
below. Complete each statement with one of the following phrases:
1
1
10 times, 100 times, _
of, _
of
10
100
1.
1 meter is
2.
1 centimeter is
the size of a meter.
3.
1 centimeter is
the size of a millimeter.
4.
1 decimeter is
the size of a meter.
5.
1 millimeter is
the size of a decimeter.
the size of a decimeter.
Write two of your own statements using the information in the table.
6.
7.
Complete the table below by making the appropriate conversions.
Copyright © Wright Group/McGraw-Hill
millimeters
8.
centimeters
meters
9,743
9.
3
10.
11.
decimeters
175
In Problem 10, explain what happens to the value of the digit 5 when you go
from millimeters to centimeters, and then from decimeters to meters.
66B
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