Estimating Sums
Objectives To provide practice deciding whether estimation
O
is
i appropriate in a given situation; and to provide practice
estimating sums.
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Teaching the Lesson
Key Concepts and Skills
• Estimate sums. [Operations and Computation Goal 6]
• Compare appropriate situations for the use
of exact answers and estimates. [Operations and Computation Goal 6]
• Use a travel map to find driving distance
and driving time. [Data and Chance Goal 2]
Key Activities
Students discuss an example of a problem
that can be solved by estimation. They use a
travel map to determine approximate
distances and times between cities. Then
they estimate the total distance and the total
driving time for a trip.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
1 2
4 3
Playing Product Pile-Up
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Student Reference Book, p. 259
per partnership: 8 each of number
cards 1–10 (from the Everything
Math Deck, if available)
Students maintain automaticity with
multiplication facts.
Finding “Closer-To” Numbers
with Base-10 Blocks
Math Boxes 5 3
Solving a Traveling Salesperson Problem
Math Journal 1, p. 111
Students practice and maintain skills
through Math Box problems.
Study Link 5 3
Math Masters, p. 144
Students practice and maintain skills
through Study Link activities.
Ongoing Assessment:
Informing Instruction See page 328.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 113. [Operations and Computation Goal 6]
Key Vocabulary
estimation round
Math Masters, p. 145
base-10 blocks
Students explore rounding numbers.
ENRICHMENT
Math Journal 1, p. 112
Math Masters, p. 146
Students use estimation skills to find the
shortest route between four cities.
EXTRA PRACTICE
Solving Elapsed-Time Problems
Math Journal 1, p. 112
demonstration clock (optional) calculator (optional)
Students solve elapsed-time problems.
ELL SUPPORT
Building a Math Word Bank
Differentiation Handbook, p. 140
Students add the terms estimation and round
to their Math Word Banks.
Materials
Math Journal 1, pp. 112 and 113
Study Link 52
Math Masters, p. 143
calculator demonstration clock (optional)
Advance Preparation
Make one copy of Math Masters, page 143 for every two students. Cut them in half and place them near
the Math Message.
Teacher’s Reference Manual, Grades 4–6 pp. 260–264
Lesson 5 3
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Getting Started
Mental Math and Reflexes
Write multidigit addition problems on the board. Students estimate the sums. They indicate “thumbs-up” if
problems and greater than 1,000 for the
and
problems.
the answer is greater than 100 for the
Have students explain their strategies. Suggestions: Sample answers are given.
52 + 49 50 + 50 = 100
22 + 49 20 + 50 = 70
37 + 54 35 + 55 = 90
21 + 47 + 68 20 + 50 + 70 = 140
786 + 293 800 + 300 = 1,100
496 + 257 500 + 250 = 750
865 + 439 900 + 400 = 1,300
572 + 314 600 + 300 = 900
316 + 145 + 459
125 + 239 + 353
673 + 314 + 249
588 + 467 + 218
300 + 150 + 450 = 900
100 + 250 + 350 = 700
700 + 300 + 250 = 1,250
600 + 500 + 200 = 1,300
Math Message
Study Link 5 2 Follow-Up
Take an answer sheet (Math Masters, page 143)
and complete it.
Have students discuss Problem 5. Point out that these
are comparison problems involving multiplication.
Have students discuss the patterns they found in their
answers.
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Masters, p. 143)
Ask students what methods they used to solve the problem.
Ask questions like the following:
5,980
2,420
2,420
1,380
1,040
One method is to add the thousands and
then the hundreds.
●
Did anyone figure out the exact number of miles traveled
using paper and pencil?
●
Using a calculator?
●
Did anyone use estimation?
On the board, list the distances traveled and ask someone who
estimated the total distance to describe his or her estimation
method.
Name
LESSON
53
䉬
Date
Time
Flight Coupons
The airline you are using on the World Tour will give you a $200 discount coupon for every
15,000 miles you fly. Suppose you have flown the distances shown in the table below.
Washington, D.C. ∑ Cairo
Cairo ∑ Accra
Accra ∑ Cairo
Cairo ∑ Budapest
Budapest ∑ London
1.
Have you flown enough miles to get a discount coupon?
2.
Describe the strategy you used to solve the problem.
5,980
2,420
2,420
1,380
1,040
mi
mi
mi
mi
mi
no
Sample answer: I added the thousands and got 11,000.
Then I added the hundreds and got 2,000. 11,000 ⫹ 2,000 ⫽
13,000. 13,000 is less than 15,000.
Math Masters, page 143
If no one mentions it, demonstrate the following method on the
board: Add the thousands first (11,000 miles). Next add the
hundreds (2,000 miles). (See margin.) The estimated total
distance, using thousands and hundreds, is 11,000 + 2,000, or
13,000 miles —far short of the 15,000 miles required to get a
discount coupon.
One idea that should emerge from this discussion is that it is
not necessary to find the exact total number of miles to answer
the question. This point needs to be emphasized. Students will
be reluctant to use estimation unless they are convinced of its
usefulness in certain situations.
Ask those students who used estimation to check their estimates
by using a calculator to find the exact number of miles they have
traveled so far.
326
Unit 5 Big Numbers, Estimation, and Computation
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Note that the method of estimation described previously is
consistent with the left-to-right partial-sums algorithm, in which
the sums of the thousands, the hundreds, the tens, and the ones
are recorded separately and then added. (See margin.)
Partial-Sums Algorithm
5,980
2,420
In this lesson students use their estimation skills to figure out
how many days are needed to reach a destination during a driving
trip that includes four U.S. cities. Students will estimate mileage
and time to find the solution.
Examining a Travel Map
2,420
1,380
+ 1,040
WHOLE-CLASS
DISCUSSION
Add thousands:
11,000
Add hundreds:
2,000
Add tens:
(Math Journal 1, p. 112)
240
Add ones:
Social Studies Link Ask students to turn to the Estimated
U.S. Distances and Driving Times map on journal page 112.
Briefly discuss the information the map provides.
0
+
Add partial sums:
13,240
The map displays distances and driving times between major U.S.
cities. These are based on travel from the center or downtown of
one city to the center or downtown of another city.
The number above the line between two cities is the distance in
miles between the cities based on the fastest or most commonly
traveled roads.
The notation below the line is an estimate of the time it would
take to drive the distance under normal conditions at the
posted speed limits.
Example: The map indicates that the distance between Chicago
and St. Louis is 302 miles. The notation “5:40” means that the
trip would take about 5 hours and 40 minutes.
Chicago
IN
483
9:05
St. Louis
5 3
䉬
30
5:4 2
0
425
8:0
0
Student Page
Date
LESSON
4
29 5
5:3
MO
366
6:55
288
5
5:3
IL
275
5:10
Time
Planning a Driving Trip
Use the map on journal page 112. Start at your hometown. Plan a driving trip that
takes you to 4 other cities on the map. If your hometown is not on the map, find the
nearest city on the map to your hometown. Start your driving trip from this city.
Example: Start in Chicago. Drive to St. Louis, Louisville,
Birmingham, and then New Orleans.
KY
1.
Record your routes, driving distances, and driving times in the table.
From…To
Louisville
NOTE Point out that although the cities on the map are connected by straight
Driving Distance
(miles)
Adjusting the Activity
ELL
This lesson provides multiple opportunities to introduce words that may
be new to English language learners, such as route, driving distance, driving
time, sensible, and destination. As you encounter each word during the lesson,
be sure to write it on the board and discuss its meaning.
K I N E S T H E T I C
T A C T I L E
Sample answers:
Driving Time
Rounded Time
(hours:minutes)
(hours)
Chicago to St. Louis
3
0
2
5 : 40
6
St. Louis to Louisville
2
7
5
5 : 10
5
Louisville to Birmingham
3
6
7
6 : 55
7
Birmingham to New Orleans
3
4
2
6 : 25
6
lines, the roads themselves do not necessarily run in a straight line.
A U D I T O R Y
180–183
2.
Estimate how many miles you will drive in all.
About
3.
Estimate how many hours you will drive in all.
About
4.
Tell how many days it will take to complete the trip
if you plan to drive about 8 hours each day and
then stop somewhere for the night.
3
1,300 miles
24 hours
days
夹
5.
Explain how you solved Problems 2–4.
Sample answer: I rounded each distance to the
nearest hundred, and then added them. I added
the rounded times to get the estimated hours. I
divided the estimated 24 hours by 8 to get 3 days.
V I S U A L
Math Journal 1, p. 113
Lesson 5 3
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Discuss distance and time measurements.
Adjusting the Activity
When rounding times to the nearest
hour, remind students that 30 minutes is
one-half of an hour; it is halfway between two
consecutive hours. For example, 2:30 is
halfway between 2:00 and 3:00. A time that
is less than 30 minutes after the hour is
rounded down. A time that is 30 or more
minutes after the hour is rounded up. Have
students use clocks to work through several
problems. For example:
lower
time
higher
time
rounded
time
Round 5:20 to
nearest hour.
5: 00
6: 00
5: 00
Round 3:45 to
nearest hour.
3: 00
4: 00
4: 00
Round 6:30 to
nearest hour.
6: 00
7: 00
7: 00
AUDITORY
KINESTHETIC
TACTILE
●
How do you think the distances and travel times between cities
were obtained? Up-to-date maps were used to find routes and
distances between cities. A typical speed, such as 55 miles per
hour, was assumed for all drivers and all routes.
●
If the distances and times were measured again, would it be
reasonable to expect the same results? Expect the same, or very
close, results. There may be two or more equally good routes,
with slightly different distances and times.
●
What is a sensible way of reporting distances and travel times?
Would you see exact numbers? Would you use estimates within
a certain range or numbers rounded to a certain place? If so,
what is a sensible range or place to round to? Although the
reported distances are fairly accurate, it is probably not sensible to report long distances between cities to the nearest mile.
It might be better to round distances to the nearest 10 miles.
Since driving times are even less reliable due to weather,
traffic, road construction, and so on, it is probably sensible to
think that driving times may be off by up to _12 hour for each
5 hours reported on the travel map. Therefore, round times to
the nearest hour.
Planning a Trip
VISUAL
PARTNER
ACTIVITY
(Math Journal 1, pp. 112 and 113)
PROBLEM
PRO
P
RO
R
OBL
BLE
B
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
VING
VIN
IIN
NG
Ask students to read the instructions for “Planning a Driving
Trip” on journal page 113. Make sure they understand the
purpose of the activity—to determine how many days it will take
them to reach their destination. Do they need to find the exact total
driving time, or will an estimate do?
You will probably want to model a sample trip with the class. For
each city-to-city part of the trip, record the distance and driving
time given on the travel map on journal page 112.
To estimate the total distance, find the sum of the distances
by adding only the hundreds and tens. Ignore digits in the
ones place.
To estimate the total driving time, first round each time to the
nearest hour. Then add the rounded times.
Have students complete Problem 5 independently.
Ongoing Assessment:
Informing Instruction
As students complete Problem 4, watch for
those who think they must reach one of the
cities named by driving exactly 8 hours each
day. It is assumed that overnight stops are
made anywhere along the route when about
8 hours have been driven. Remind students
that the 8-hour average is used to calculate
the days it takes to reach the destination, not
the absolute driving time each day.
328
Ongoing Assessment:
Recognizing Student Achievement
Journal
page 113
Problem 5
Use journal page 113, Problem 5 to assess students’ ability to explain how
they used estimation to solve addition problems. Students are making adequate
progress if their explanation involves “close-but-easier” numbers. Some students
may be able to describe more than one way to solve the problems.
[Operations and Computation Goal 6]
Unit 5 Big Numbers, Estimation, and Computation
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Student Page
Date
2 Ongoing Learning & Practice
Playing Product Pile-Up
Time
LESSON
Math Boxes
53
1.
2
7
6
4
5
1
PARTNER
ACTIVITY
2.
A number has
in
in
in
in
in
in
the
the
the
the
the
the
hundreds place,
tenths place,
hundredths place,
ones place,
tens place, and
thousandths place.
Students play Product Pile-Up to maintain automaticity with
multiplication facts. See Lesson 4-3 for additional information.
b.
72
450
160
+ 1,000
1,682
Math Boxes 5 3
(Math Masters, p. 144)
1, 2, 5, 10, 25, 50
b.
Which of these factors are prime?
2 and 5
6,205
A.
850 miles
B.
900 miles
C.
450 miles
D.
800 miles
78
6. a.
If 1 centimeter on a map represents
200 miles, what do 4.5 centimeters
represent? Fill in the circle next to the
best answer.
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 5-1. The skill in Problem 6
previews Unit 6 content.
Study Link 5 3
List all the factors of 50.
10 11
5.
Writing/Reasoning Have students write a response to the
following: Explain how you solved Problem 5. Sample answer:
I multiplied 4 ∗ 200 to get 800 miles. I knew that 1 centimeter
represents 200 miles, so 0.5 centimeters represents 100 miles.
800 + 100 = 900
17
4. a.
15
240
350
+ 5,600
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 111)
e.
31
Solve mentally or with a paper-and-pencil
algorithm.
a.
90 ∗ 70 =
d.
2 5 4 . 7 6 1
3.
120
6,300
60 = 3,000
50 ∗
500 ∗ 8 = 4,000
700 = 56,000
80 ∗
3 ∗ 40 =
b.
c.
Write the number.
(Student Reference Book, p. 259)
Solve mentally.
a.
Sara collected 30 leaves. On the way
to school, she lost 2 of them. At school
she and her 6 friends shared them
equally. How many leaves did each
person get?
4
b.
leaves
Ava and her 3 sisters shared 24 mints
equally. How many mints did each
sister get?
6
43
mints
20
111
Math Journal 1, p. 111
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INDEPENDENT
ACTIVITY
ELL
Home Connection Students estimate sums. If the
estimate is greater than or equal to 1,500, students find
the exact sum. If the estimate is less than 1,500, students
do not need to solve the problem. Emphasize to English language
learners that they do not have to solve all the problems.
Study Link Master
Name
STUDY LINK
53
3 Differentiation Options
Date
Time
Estimating Sums
For all problems, write a number model to estimate the sum.
181
If the estimate is greater than or equal to 1,500, find the exact sum.
If the estimate is less than 1,500, do not solve the problem.
READINESS
Finding “Closer-To” Numbers
SMALL-GROUP
ACTIVITY
5–15 Min
Sample answers are given for number models.
1,601
1,824
1. 867 + 734 =
2. 374 + 962 + 488 =
3.
Number model:
850 + 750 = 1,600
400 + 1,000 + 500 = 1,900
382 + 744 =
4.
Number model:
with Base -10 Blocks
(Math Masters, p. 145)
Number model:
318 + 295 + 493 =
600 + 650 + 350 = 1,600
6.
Number model:
To provide experience with estimation skills by finding the “closerto” number when rounding, have students use base-10 blocks
to build
the nearest multiple of 10 or 100 greater than the number.
800 + 700 = 1,500
8.
Number model:
756 + 381 + 201 =
132 + 692 + 803 =
1,627
Number model:
700 + 200 + 400 = 1,300
9.
the number to be rounded,
the nearest multiple of 10 or 100 less than the number, and
694 + 210 + 386 =
1,547
845 + 702 =
Number model:
300 + 300 + 500 = 1,100
7.
1,595
Number model:
400 + 750 = 1,150
5.
581 + 648 + 366 =
100 + 700 + 800 = 1,600
10.
575 + 832 =
Number model:
Number model:
750 + 400 + 200 = 1,350
600 + 800 = 1,400
Practice
11.
60 ∗ 80 =
4,800
12.
30 ∗ 70 =
2,100
13.
50 ∗ 900 =
45,000
14.
40 ∗ 800 =
32,000
Math Masters, p. 144
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Lesson 5 3
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Teaching Master
Name
Date
LESSON
Time
53
Solving a Traveling
You can use base-10 blocks to help you round numbers.
Example: Round 64 to the nearest ten.
Think: What multiples of 10 are nearest to 64?
If I take the ones (cubes) away, I would have 60.
If I add more ones to make the next ten, I would have 70.
(Math Journal 1, p. 112; Math Masters, p. 146)
Build models for 60 and 70.
64
To apply students’ understanding of estimating sums,
have them use journal page 112 to plan a route
connecting four cities, beginning in Washington State and
ending in Maine. The route should have the shortest total driving
distance. To extend this problem, students may also find the route
with the shortest total driving time.
70
Think: Is 64 closer to 60 or 70? 64 is closer to 60. So, 64 rounded to the nearest ten is 60.
Build models to help you choose the closer number.
1.
Round 87 to the nearest ten.
List the three numbers you will build models for:
87 is closer to
2.
90
80
,
87 , 90
90 .
. So, 87 rounded to the nearest ten is
Round 43 to the nearest ten.
43 , 50
So, 43 rounded to the nearest ten is 40 .
List the three numbers you will build models for:
43 is closer to
3.
40
.
40
,
Round 138 to the nearest ten.
EXTRA PRACTICE
130 , 138 , 140
140 . So, 138 rounded to the nearest ten is 140 .
List the three numbers you will build models for:
138 is closer to
4.
Solving Elapsed-Time Problems
Round 138 to the nearest hundred.
100 , 138 , 200
100 . So, 138 rounded to the nearest hundred is 100 .
2/26/11 12:06 PM
NOTE To teach a standard
procedure for rounding
whole numbers to the nearest
ten and hundred, see
www.everydaymathonline.com.
Date
LESSON
Encourage students to share their strategies. Suggestions:
A salesperson plans to visit several cities. To save time and money, the trip should be as
short as possible. If the salesperson were visiting only a few cities, it would be possible to
figure the shortest route in a reasonable time. But what if the trip includes 10 cities? There
would be 3,628,800 possible routes! Computer scientists are trying to find ways to solve this
problem on the computer without having to do an impossible number of calculations.
Brynn is taking a trip from New York City, NY to Bangor, ME.
If she leaves at 7:00 A.M. and travels through Boston, MA, what
time will she arrive in Bangor? 3:05 P.M.
Think like a computer. Imagine that you begin a trip in Seattle and have to visit Denver,
Birmingham, and Bangor for business.
1.
Estimate to find the shortest route that would include each city. Use the map on journal
page 112.
2.
Describe your route between each of the four cities.
Esteban is delivering produce from Seattle, WA to San
Francisco, CA by way of Salt Lake City, UT. How long will the
trip take him? 29 hr, 20 min
Sample answer: Seattle to Boise to Salt Lake City to Denver
(totals 1,361 miles); then Denver to Dallas to Birmingham
(totals 1,431 miles); then Birmingham to Washington, D.C.,
to New York City to Boston to Bangor (totals 1,408 miles).
The total mileage is 4,200 miles.
ELL SUPPORT
Try This
3.
2:40
3:05
Time
A Traveling Salesperson Problem
53
25 min
10:40
Teaching Master
4 hr
40 min
10:00
7:00
3 hr
Name
15–30 Min
To provide extra practice with elapsed-time problems, you may
wish to use the context of journal page 112 to pose word problems.
Students may use a demonstration clock, calculators, or anything
else that may help. Some students may find it helpful to use an
open number line to illustrate the strategy of counting up in hours
and minutes. In the first suggestion given below, students might
draw a diagram like the one shown here. Students mark off the
starting time and count up in hours and then in minutes, while
recording the actual and elapsed time.
Math Masters, p. 145
139-176_EMCS_B_MM_G4_U05_576965.indd 145
SMALL-GROUP
ACTIVITY
(Math Journal 1, p. 112)
List the three numbers you will build models for:
138 is closer to
15–30 Min
Salesperson Problem
Build a model for 64 with base-10 blocks.
60
INDEPENDENT
ACTIVITY
ENRICHMENT
“Closer To” with Base-10 Blocks
Describe a route that includes each city that would take the shortest amount
of time.
py g
Sample answer: Looking at the map, the route described
above should be the shortest because the shortest distances
are also shown as the shortest times.
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
5–15 Min
(Differentiation Handbook, p. 140)
g
p
Math Masters, p. 146
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To provide language support for estimation, have students use
the Word Bank Template found on Differentiation Handbook,
page 140. Ask students to write the terms estimation and round,
draw pictures representing the terms, and write other words
that describe them. See the Differentiation Handbook for
more information.
Unit 5 Big Numbers, Estimation, and Computation
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