Objective
To discuss sensible ways of reporting a count when a
large number of items has been counted.
1
materials
Teaching the Lesson
Key Activities
Students discuss the reliability of large population counts, such as numbers of marathon
runners and attendance figures for sports events. They use these figures to review and
practice rounding to a given place.
Math Journal 1, p. 132
Study Link 5 9
Key Concepts and Skills
• Read and write whole numbers; identify digits and their values.
[Number and Numeration Goal 1]
• Describe differences between estimates and exact counts.
[Operations and Computation Goal 6]
• Round large numbers to a given place.
[Operations and Computation Goal 6]
• Use data presented in a table.
[Data and Chance Goal 2]
Key Vocabulary
rounding (to a certain place)
2
materials
Ongoing Learning & Practice
Students take a 50-facts test. They use a line graph to record individual and class
scores. Then students find the median and calculate the mean of class scores.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Recognizing Student Achievement Use the 50-facts test.
Math Journal 1, p. 133
Study Link Master (Math Masters, p. 169)
Teaching Aid Masters (Math Masters,
pp. 410, 414, and 416)
pen or colored pencil
[Operations and Computation Goal 3]
3
materials
Differentiation Options
READINESS
Students find the halfway
point on a number line
and practice plotting a
number greater than or
less than half.
ENRICHMENT
EXTRA PRACTICE
Students visually round
numbers represented on
a bar graph.
Students practice rounding
whole numbers.
Teaching Masters (Math Masters,
pp. 170 and 171)
5-Minute Math, pp. 15, 91, and 92.
Technology
Assessment Management System
50-facts test
See the iTLG.
Lesson 5 10
367
Getting Started
Mental Math and Reflexes
Math Message
In preparation for rounding, pose questions such as:
A newspaper reported that
20,344 participants ran in the
2004 Boston Marathon. Do you
think that number is exactly
correct? Be ready to explain
your answer.
Is 86 closer to 80 or 90? 90
Is 538 closer to 500 or 600? 500
Is 541 closer to 540 or 550? 540
Is 786 closer to 780 or 790? 790
Is 2,670 closer to 2,000 or 3,000? 3,000
Is 2,670 closer to 2,600 or 2,700? 2,700
Study Link 5 9
Follow-Up
Include several questions in which the given number is exactly halfway between
the lower and higher numbers. Suggestions:
• Is 75 closer to 70 or 80?
• Is 250 closer to 200 or 300?
Acknowledge students who recognize that these numbers are exactly halfway
between the other two numbers. Remind them that halfway numbers are usually
rounded to the higher number.
Have partnerships discuss
what names were not used
in the name-collection boxes.
Ten, 10, and 101, which are
all names for 10.
1 Teaching the Lesson
Math Message Follow-Up
NOTE These are the official counts of the
number of entrants in each race, not the
number of actual starters or finishers. A count
of finishers is likely to be more accurate than a
count of entrants or starters, because
officials check off the runners’ numbers as
they cross the finish line.
WHOLE-CLASS
DISCUSSION
On the board, write the reported numbers of runners for the 1994
and 2004 Boston Marathons:
1994: 9,059
2004: 20,344
Have students discuss the reliability of these numbers.
●
Do you think exactly that many people ran in each race? Why
or why not? Probably not. It is difficult to keep track of every
person who competes. A person may register to run but then
not run for many possible reasons.
●
How do you think the counts of runners were obtained? When
runners register for a race, each person is given a number to
wear, and that person’s name and number are recorded on
a list.
In counting large numbers of items (especially things that change
constantly), it is probably impossible to come up with a number
that is exactly right, down to the last single item. Large counts
that are reported without rounding appear to be more accurate
than they really are. In these cases, a rounded count is a more
honest report about the number of things counted.
Tell students that in this lesson they will practice rounding whole
numbers to a certain place.
368
Unit 5 Big Numbers, Estimation, and Computation
Reviewing Rounding
WHOLE-CLASS
DISCUSSION
Use the marathon counts to review rounding a number to a
certain place. Remind students about the steps in rounding a
number. For example, to round 20,344 to the nearest hundred
follow these steps:
1. Mark the digit in the place
you are rounding to.
20,344
2. Replace all digits to the right
of this digit with zeros.
This is the lower number.
20,300
3. Add 1 to the digit in the place
you are rounding to.
This is the higher number.
20,400
4. Ask yourself: “Is the number I
am rounding closer to the lower
number or the higher number?”
lower
5. Round to the number that is
closer. If the number you
are rounding is halfway between
the lower and higher numbers,
round to the higher number.
20,300
Work together as a class to round both marathon counts to the
nearest thousand, to the nearest hundred, and to the nearest ten.
Lower
number
Higher
number
Rounded
number
Round 9 ,059 to the
nearest thousand.
9 ,000
10 ,000
9 ,000
Round 9,059 to the
nearest hundred.
9,000
Round 9,059 to the
nearest ten.
9,050
9,060
9,060
Round 20,344 to
the nearest thousand.
20,000
21,000
20,000
Round 20,344 to
the nearest hundred.
20,300
20,400
20,300
Round 20,344 to
the nearest ten.
20,340
ELL
Adjusting
the Activity
Use number lines to help students visualize
this rounding method. For example:
9,100
9,100
9,059
9,000
10,000
9,059
9,000
AUDITORY
20,350
9,500
9,050
KINESTHETIC
9,100
TACTILE
VISUAL
20,340
Summarize by asking students to pretend they are writing a
newspaper report of the 1994 (or 2004) marathon. Ask: Which
version of the marathon count would you report—9,059, 9,060,
9,100, or 9,000? Explain your answer. Sample answer: The
number of runners who actually started the race might have
differed from the official registrant list by at least 10 to 20,
and maybe even by 100. This suggests rounding to the nearest
hundred, or 9,100, because it is a more realistic count.
Lesson 5 10
369
Student Page
Date
Rounding Baseball Team
Time
LESSON
Evaluating Large Numbers
5 10
Attendance Figures
1. Round the attendance figures in the table to the nearest hundred-thousand.
182 183
2004 Major League Baseball Home Game Attendance
for 10 Teams
Total Home Game
Attendance
Attendance Rounded
to the Nearest 100,000
Atlanta Braves
2,603,484
Baltimore Orioles
2,682,917
2,600,000
2,700,000
2,700,000
2,600,000
2,700,000
3,100,000
3,500,000
3,500,000
3,000,000
2,400,000
Team
Boston Red Sox
2,650,063
Cleveland Indians
2,616,940
Colorado Rockies
2,737,838
Los Angeles Dodgers
3,131,255
New York Yankees
3,465,807
Seattle Mariners
3,540,658
St. Louis Cardinals
3,011,216
Texas Rangers
2,352,397
PARTNER
ACTIVITY
(Math Journal 1, p. 132)
Students complete journal page 132. To support English language
learners, discuss the meaning of attendance figures. Clarify the
difference between the meaning of figure in this context compared
to a geometric figure.
Source: Information Please—Baseball Attendance
2. How do you think attendance figures for major league baseball games are obtained?
Adjusting the Activity
Sample answer: By the turnstile that counts how many
people go through each gate, or by the number of tickets sold
or turned in at the stadium
Pose the following question: The teams shown in the table on journal
page 132 have between 25,000 and 50,000 attendees per game. Inaccurate
counting could well lead to a 2 percent error in the number of attendees or an error
of about 1,000 per game. Each team plays about 81 home games. Ask: What
might be the total error in attendees for an entire season? As large as 50,000 to
100,000 per season This explains why rounding to the nearest 100,000 is a very
realistic thing to do.
3. Do you think exactly 2,737,838 people were at the home games played by the
Sample answer:
No. Some people might not go through the turnstile. Some
people who bought tickets might not go. With so many people,
mistakes are easily made.
Colorado Rockies? Explain your answer.
4. You rounded the figures in the table above to the nearest hundred-thousand. Which teams
have the same attendance figures based on these rounded numbers?
The Braves and Indians; the Orioles, Red Sox, and Rockies;
and the Yankees and Mariners
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
132
Math Journal 1, p. 132
2 Ongoing Learning & Practice
Taking a 50-Facts Test
WHOLE-CLASS
ACTIVITY
(Math Masters, pp. 410, 414, and 416)
See Lesson 3-4 for details regarding the administration of the
50-facts test and the recording and graphing of individual and
class results.
Student Page
Date
Time
LESSON
5 10
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
1
1. a. Measure the line segment to the nearest inch.
4
T
4 2
Use the 50-facts test to assess students’ automaticity with multiplication facts
through 10 10. Students are making adequate progress if they demonstrate
automaticity with the 0, 1, 2, 5, 10, and square facts and use strategies
to compute the remaining facts up to 10 10. Some students may demonstrate
automaticity with multiplication facts through 10 10.
G
1
About
inches
b. Draw a line segment that is half as long as the one above.
1
c. How long is the line segment you drew?
About
2 4
inches
128
[Operations and Computation Goal 3]
3. Multiply. Use the partial-products method.
2. Estimate the product. Write a number
model to show how you estimated.
3,827
50-Facts Test
43 89
a. 37 91
8
4
0
6
4
+
2
3 8 2
Sample answers:
40 90 3,600
Number model:
º
3 2
3
2
b. 53 17
Number model:
50 20 1,000
9
3
0
0
0
7
7
184
Math Boxes 5 10
(Math Journal 1, p. 133)
18
5. There are 60 trading cards. Each student
4. Write fourteen and three-tenths using
gets 5 cards. How many students get
trading cards?
digits. Fill in the circle next to the best
answer.
12
A. 14.03
students
B. 14.003
C. 14.310
D. 14.3
27
28
175
133
Math Journal 1, p. 133
370
INDEPENDENT
ACTIVITY
Unit 5 Big Numbers, Estimation, and Computation
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lessons 5-6 and 5-8. The skill in
Problem 5 previews Unit 6 content.
Study Link Master
Study Link 5 10
INDEPENDENT
ACTIVITY
(Math Masters, p. 169)
Name
Date
STUDY LINK
5 10
Time
Rounding
182 183
1. Round the seating capacities in the table below to the nearest thousand.
Women’s National Basketball Association
Seating Capacity of Home Courts
Home Connection Students round basketball stadium
seating capacities to the nearest thousand. They round
population census data to the nearest million.
3 Differentiation Options
Team
Seating Capacity
Charlotte Sting
24,042
Cleveland Rockers
20,562
Detroit Shock
22,076
New York Liberty
19,763
Phoenix Mercury
19,023
Sacramento Monarchs
17,317
San Antonio Stars
18,500
Seattle Storm
17,072
Rounded to the
Nearest 1,000
24,000
21,000
22,000
20,000
19,000
17,000
19,000
17,000
2. Look at your rounded numbers. Which stadiums have about the same capacity?
Mercury and Stars; Monarchs and Storm
3. Round the population figures in the table below to the nearest million.
U.S. Population by Official Census from 1940 to 2000
INDEPENDENT
ACTIVITY
READINESS
Using Number Lines to Find
5–15 Min
the Halfway Point
Year
Population
1940
132,164,569
1960
179,323,175
1980
226,542,203
2000
281,421,906
Rounded to the
Nearest Million
132,000,000
179,000,000
227,000,000
281,000,000
Source for both tables: The World Almanac and Book of Facts 2004
(Math Masters, p. 170)
Practice
4.
To explore locating large numbers on a number line, have students
find halfway points on segments of number lines. Then have them
plot a number less than or greater than the halfway number.
4,152
692 º 6
5.
798
38 º 21
6. 44 º 73 3,212
169
Math Masters, p. 169
Teaching Master
Name
Date
LESSON
5 10
1.
Time
Number Lines
182 183
For each number line, record the number that is halfway between the lower
and higher number. Then plot a number that is less than the halfway number.
Sample answers:
32
a.
35
30
lower number
b.
880
lower number
2.
40
higher number
halfway number
884
885
890
higher number
halfway number
For each number line, record the number that is halfway between the lower
and higher number. Then plot a number that is greater than the halfway number.
3,499
a.
3,400
lower number
3,450
71,750
b.
71,000
lower number
3.
3,500
higher number
halfway number
Make up a problem of your own.
lower number
71,500
halfway number
72,000
higher number
Answers vary.
halfway number
higher number
Math Masters, p. 170
Lesson 5 10
371
Teaching Master
Name
LESSON
5 10
Date
Time
ENRICHMENT
Rounding Bar Graph Data
Each bar represents the 2003 population of a country.
76
Population 2003
70
Rounding Bar Graph Data
INDEPENDENT
ACTIVITY
5–15 Min
(Math Masters, p. 171)
Number of People in Millions
60
To apply students’ understanding of rounding, have them use
population data represented on a bar graph to practice visually
rounding numbers. Direct students’ attention to the break near
the bottom of the vertical axis of the graph. Students should note
that the vertical axis begins at 30 million, not at 0.
50
40
In Problem 1, students organize the information in a table.
Sample answer:
30
France
Italy
Poland
Spain
Countries
United
Kingdom
Source: The World Factbook
1.
Estimate the population of each country to the nearest 10 million,
to the nearest 5 million, and to the nearest 1 million.
Organize your information in a table on the back of this sheet.
2.
Describe the strategy you used to round to the nearest million.
Population in 2003
Sample answer: I looked at each bar to see
which million mark it was closer to and then
rounded to that number. If it was exactly
halfway between, then I rounded up to the
higher million.
Math Masters, p. 171
Country
Nearest 10 million
Nearest 5 million
Nearest 1 million
France
60 million
60 million
61 million
Italy
60 million
60 million
58 million
Poland
40 million
40 million
39 million
Spain
40 million
40 million
41 million
U.K.
60 million
60 million
60 million
Have students discuss the advantages and disadvantages of
rounding to each place specified in the problem. Sample answer: It
is easier to round to the nearest 10 million because the 10 millions
are marked on the graph. If the bars are almost the same height,
it is easier to compare them if you round to the nearest million.
EXTRA PRACTICE
5-Minute Math
SMALL-GROUP
ACTIVITY
5–15 Min
To offer students more experience with rounding numbers, see
5-Minute Math, pages 15, 91, and 92.
372
Unit 5 Big Numbers, Estimation, and Computation