Objectives
To explain why decimals are useful; and to guide
estimation of sums and differences of decimals.
1
materials
Teaching the Lesson
Key Activities
Students list examples of decimals used in everyday life and sort them into categories.
They estimate sums and differences of decimals to answer questions about a bicycle trip.
Key Concepts and Skills
ⵧ Math Journal 1, p. 85
ⵧ Student Reference Book, p. 26
ⵧ Study Link 4 3
ⵧ slate
䉬
• Read and interpret decimals through tenths. [Number and Numeration Goal 1]
• Compare whole numbers and decimals. [Number and Numeration Goal 6]
• Estimate sums and differences of decimals; explain the strategies used.
[Operations and Computation Goal 6]
• Use a table of data to answer questions. [Data and Chance Goal 2]
Key Vocabulary trip meter • speedometer
Ongoing Assessment: Informing Instruction See page 257.
Ongoing Assessment: Recognizing Student Achievement Use journal page 85.
[Operations and Computation Goal 6]
2
materials
Ongoing Learning & Practice
Students play Number Top-It (Decimals) to practice comparing and ordering decimals.
Students practice and maintain skills through Math Boxes and Study Link activities.
ⵧ Math Journal 1, p. 86
ⵧ Student Reference Book, p. 256
ⵧ Study Link Master (Math Masters, p. 113)
ⵧ Game Masters (Math Masters, p. 490;
pp. 491 and 506, optional)
ⵧ number cards 0–9 (4 of each per
partnership)
3
materials
Differentiation Options
READINESS
Students estimate decimal
sums in a money context.
ENRICHMENT
Students use estimation to
solve problems involving
mileage.
ENRICHMENT
Students use estimation to
solve a decimal magic
square puzzle.
ⵧ Teaching Masters (Math Masters,
pp. 114–117)
ⵧ Teaching Aid Master (Math Masters,
p. 428)
ⵧ quarters, dimes
Technology
Assessment Management System
Journal page 85, Problems 4 and 5
See the iTLG.
Lesson 4 4
䉬
255
Getting Started
Mental Math and Reflexes
Math Message
Have students give a whole-number estimate for decimal
addition and subtraction problems. Suggestions: Sample answers:
Describe two examples in which decimals
are used in real-life situations.
3.8 9.9
14
3.6 4.5
8
25.5 11. 9
38
2.7 8.1
11
8.8 8.9
18
48.7 20.3
69
Study Link 4 3 Follow-Up
10.5 6.6
4
16.8 9.9
7
26.2 10.8
15
12.6 9.8
3
23.7 8.8
15
62.6 50.8
12
Have students compare answers and
decide whether partners have found
appropriate numbers for Problems 3–8.
䉬
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Discuss students’ answers. Sort the uses they suggest into three
categories: measurements, money, and other. The majority of
responses are likely to be measurements.
Tell students that in this lesson they will explore some uses
of decimals.
䉴 Discussing Why Decimals
WHOLE-CLASS
DISCUSSION
Are Useful
(Student Reference Book, p. 26)
Student Page
Decimals and Percents
Decimals
Mathematics in everyday life involves more than just whole
numbers. We also use decimals and fractions to name
numbers that are between whole numbers.
Both decimals and fractions are used to name a part of a whole
thing or a part of a collection. Decimals and fractions are also
used to make more precise measurements than can be made
using only whole numbers.
Fractional parts of a dollar are almost always written as
decimals. The receipt at the right shows that lunch cost
between 25 dollars and 26 dollars. The “64” in the cost
names a part of a dollar.
Have students read page 26 in the Student Reference Book. Guide
students as they read by asking them to share one thing they
learned after each paragraph.
Most students have seen a trip meter on the speedometer of a
car. Discuss the purposes of this instrument. The numbers that
represent whole numbers of miles traveled are usually in one
color, and the numbers that represent tenths of miles are in
another color.
You probably see many other uses of decimals every day.
Weather reports give rainfall amounts in decimals. The average
annual rainfall in New Orleans, Louisiana, is 66.28 inches.
Digital scales in supermarkets show the weight of fruits,
vegetables, and meat with decimals.
Discuss what information you would get if a trip meter showed
only whole numbers of miles traveled.
Winners of many Olympic events are often decided by times
measured to hundredths, and sometimes even thousandths,
of a second. Florence Griffith-Joyner’s winning time for the
100-meter run in 1988 was 10.54 seconds.
0
Many sports statistics use decimals. In 1993, basketball player
Michael Jordan averaged 32.6 points per game. In 1901,
baseball player Napoleon Lajoie had a batting average of .426.
Cars have instruments called odometers that measure
distance. The word odometer comes from the Greek
words odos, which means road, and metron, which
60 70
means measure. The odometer at the right shows
80
50
90
12,963, which means the car has traveled
40
90 110 130
at least 12,963 miles. The trip meter above
30
100
70
150
50
it is more precise and shows tenths of a mile
6
110
20
170
30
traveled. The trip meter at the right shows
190 120
10
the car has traveled at least 45.6 miles
kph
10
mph
since it was last reset to 0.
Decimals use the same base-ten place-value system that whole
numbers use. The way you compute with decimals is very
similar to the way you compute with whole numbers.
Student Reference Book, p. 26
256
Unit 4 Decimals and Their Uses
●
0
4
5
If the trip meter showed 45 miles, what would this tell you
about the distance you have traveled so far? It would show that
you have traveled at least 45 miles but less than 46 miles.
Decimals make it possible to make more precise measurements.
For example, on a trip meter that shows tenths of a mile, a
reading of 45.6 miles tells you that the car has traveled at least
45.6 miles but less than 45.7 miles.
0
0
4
5
6
Student Page
Date
LESSON
4 4
䉬
Time
A Bicycle Trip
Diego and Alex often take all-day bicycle
trips together. During the summer, they took a
3-day bicycle tour. They carried camping gear
in their saddlebags for the two nights they would
be away from home.
Travel Log
Distance Traveled
Alex had a trip meter that showed miles traveled
in tenths of miles. He kept a log of the distances
they traveled each day before and after lunch.
Timetable
Before lunch
After lunch
Day 1
27.0 mi
31.3 mi
Day 2
36.6 mi
20.9 mi
Day 3
25.8 mi
27.0 mi
Use estimation to answer the following questions.
Do not work the problems out on paper or with a
calculator.
Day 1
Day 3
3. During the whole trip, did they travel more miles before or after lunch? Before lunch
1. On which day did they travel the most miles?
Ongoing Assessment: Informing Instruction
2. On which day did they travel the fewest miles?
夹
夹
4. Estimate the total distance they traveled. Choose the best answer.
Watch for students who do not think of decimals as numbers with a value
between two consecutive whole numbers. You can reinforce this concept by
asking such questions as “20.964 is between which two consecutive whole
numbers?” and by asking students to fill in missing numbers on a number line.
Write the word consecutive on the board with an example.
less than 150 miles
between 150 and 180 miles
between 180 and 200 miles
more than 200 miles
5. Explain how you solved Problem 4.
Sample answer: I rounded the distances. They traveled about
60 miles on Day 1, about 60 miles on Day 2, and about 50 miles
on Day 3. 60 + 60 + 50 = 170.
6. On Day 1, about how many more miles did they travel after lunch than before lunch?
About 4 more miles
䉴 Estimating Decimal Sums
7. Diego said that they traveled 1.2 more miles before lunch on Day 1 than on Day 3. Alex
PARTNER
ACTIVITY
(Math Journal 1, p. 85)
disagreed. He said they traveled 2.2 more miles. Who is right? Explain your answer.
Diego; I counted up from 25.8 to 26.8 (1.0 more).
Then I counted 0.2 more to 27. 1.0 0.2 1.2
85
Read the first part of “A Bicycle Trip” on journal page 85 aloud,
and discuss what is meant by estimation. Have partnerships finish
the page. Emphasize that they are to answer the questions by
using estimation, not by computing exact answers.
Math Journal 1, p. 85
Adjusting the Activity
Direct students to the table on journal page 85. For each day, have
students draw a circle around the before lunch and after lunch distances to
emphasize that students need to consider both numbers when determining the
total distance traveled each day.
Travel Log
Distance Traveled
Timetable
Day 1
A U D I T O R Y
䉬
Before lunch
After lunch
27.0 mi
31.3 mi
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
Allow time for students to share solution strategies.
䉯 One approach to Problem 3 is to estimate the sum of the
distances traveled before and after lunch and then compare
the totals. Another possibility: Diego and Alex traveled about
4 more miles after lunch on Day 1 and about 1 more mile on
Day 3. However, they traveled so many more miles before
lunch on Day 2 that this more than offsets the results for
Days 1 and 3.
䉯 For Problem 7, a good strategy is to count up from 25.8 to 26.0
(0.2 more) and then from 26.0 to 27.0 (1.0 more) for a total of
1.2 miles.
Lesson 4 4
䉬
257
Student Page
Date
Time
LESSON
Math Boxes
4 4
䉬
1. a. What is the maximum number of movies
Number of Movies
Viewed in a Month
a student viewed in a month?
Number of Students
5
0
b. What is the minimum number of movies?
3
2
c. What is the mode?
d. What is the median?
7
6
5
4
3
2
1
0
1
0
2
3
4
5
Number of Movies
73
2. Solve mentally or with a paper-and-pencil
3. If 2 centimeters on a map represent
algorithm.
a.
50 kilometers, then
814
123
b.
937
25 km.
75 km.
4 cm represent 100 km.
0.5 cm represent 12.5 km.
8.5 cm represent 212.5 km.
a. 1 cm represents
754
396
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 85, Problems 4 and 5 to assess students’ ability to estimate
sums of decimals and explain their estimation strategy. Students are making
adequate progress if they are able to describe a strategy that involves “close-buteasier ” numbers. Some students may estimate the total distance traveled on
each of the three days (about 58, 57, and 53 miles) and then estimate the sum
of these totals. Others may round each distance to the nearest ten and find the
sum (30 30 40 20 30 30 180).
b. 3 cm represent
358
c.
d.
e.
10–15
夹
Journal
page 85
Problems 4 and 5
[Operations and Computation Goal 6]
145
5. Solve mentally.
4. Write 40 quarters and 3 dimes in
$4.03
b. 6 ⴱ 90 54
540
$4.30
c.
5ⴱ8
$10.30
d.
$40.30
e. 4 ⴱ 4 dollars-and-cents notation. Choose
the best answer.
a. 6 ⴱ 9 f.
40
400
4 ⴱ 40 2 Ongoing Learning & Practice
50 ⴱ 8
16
160
17
86
䉴 Playing Number Top-It
PARTNER
ACTIVITY
(Decimals)
Math Journal 1, p. 86
(Student Reference Book, p. 256; Math Masters, pp. 490 and 506)
Students play Number Top-It (Decimals) to practice comparing
2-place decimals. The version for more than 2 players provides
students with practice ordering sets of decimals. Consider having
students record a few rounds of play on Math Masters, page 506.
Adjusting the Activity
Have students use Math Masters, page 491 to play a 3-place-decimal
version of the game.
A U D I T O R Y
䉬
K I N E S T H E T I C
䉬
T A C T I L E
䉬
V I S U A L
Study Link Master
Name
Date
STUDY LINK
Time
4 4
The table below shows the five longest railroad tunnels in the world.
Tunnel
Location
䉴 Math Boxes 4 4
䉬
Railroad Tunnel Lengths
䉬
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 86)
Year Completed
Length in Miles
Seikan
Japan
1988
33.46
Channel
France/England
1994
31.35
Moscow Metro
Russia
1979
19.07
London Underground
United Kingdom
1939
17.30
Dai-Shimizu
Japan
1982
13.98
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 4-2. The skill in Problem 5
previews Unit 5 content.
Use estimation to answer the following questions.
1.
Which two tunnels have a combined length of about 60 miles?
2.
Which of the following is closest to the combined length of all five tunnels?
Choose the best answer.
3.
4.
Seikan Tunnel
and
Channel Tunnel
Less than 90 miles
Between 90 and 130 miles
Between 130 and 160 miles
More than 160 miles
Sample answer:
I rounded the tunnel lengths to “close-but-easier ” numbers and
added 35 30 20 15 15 115 to find the total length.
About how many miles longer is the Channel Tunnel than the Moscow Metro Tunnel?
12
miles
Try This
The Cascade Tunnel in Washington State is the longest railroad tunnel in the United
1
States. It is about 4 the length of the Seikan. About how long is the Cascade Tunnel?
8
About
miles
Practice
6.
䉬
INDEPENDENT
ACTIVITY
(Math Masters, p. 113)
Explain how you solved Problem 2.
About
5.
䉴 Study Link 4 4
190 b 200
b
10
7.
g 500 225
g
725
Math Masters, p. 113
258
Unit 4 Decimals and Their Uses
Home Connection Students estimate sums and
differences of lengths of the world’s longest railroad
tunnels. To support English language learners, discuss
the meaning of the word tunnel.
Teaching Master
Name
3 Differentiation Options
READINESS
䉴 Estimating Cost of Purchase
Date
LESSON
4 4
䉬
Time
Will I Run Out of Gas?
You are driving with your family from Denver, Colorado, to Des Moines, Iowa.
You know the following:
PARTNER
ACTIVITY
181
䉬
Your car’s gasoline tank holds about 12.1 gallons.
䉬
Your car uses about 1 gallon of gasoline for every 30 miles on the highway.
䉬
You start your trip with a full tank.
Here is a map of the route you follow.
Missouri
River
Numbers indicate
miles between cities.
5–15 Min
(Math Masters, pp. 114, 115, and 428)
Des Moines
Adair
Avoca
60
Omaha
40
50 North
50
Lincoln
Platte
50
Ogallala
Iowa
Sterling
100 Kearney110
80
130
Colorado
To explore estimation using decimals, have students estimate the
total cost of items. Students cut apart the item slips on Math
Masters, page 114. Ask them to place the slips facedown. At each
turn, a student flips over three slips, estimates the total cost of the
items, and records a number model for their estimate on Math
Masters, page 115. Provide bills (Math Masters, page 428),
quarters, and dimes so that students can model amounts that are
close to the cost of the items and combine these to find the total.
NOTE The item slips from Math Masters, page 114 are used again in Lesson 4-6
in an optional Readiness activity. Store them for use in that lesson.
Missouri
Nebraska
Kansas
Denver
1.
About how many gallons of gasoline would
your car use traveling from Denver to Sterling?
2.
When you get to Ogallala, you would expect your gas tank to be
a.
3.
almost empty.
b.
1
about 4 full.
About
1
about 2 full.
c.
4
d.
gallons
3
about 4 full.
Is it OK to wait until you get to Kearney to buy more gas? Explain.
No. Sample answer: You would be taking a chance waiting.
The distance from Denver to Kearney is about 360 miles, and
you may or may not be able to drive that distance on one tank.
4.
You stop at North Platte to buy more gasoline. If you buy
7.6 gallons, about how many gallons are there in your tank now?
5.
Could you get to Des Moines from North Platte without running
out of gas if you filled your gasoline tank just one more time?
About
11
gallons
yes
Sample answers: Avoca,
Omaha, Lincoln, or Kearney
If so, where would you stop?
Math Masters, p. 116
ENRICHMENT
䉴 Solving Gasoline Mileage
INDEPENDENT
ACTIVITY
5–15 Min
Problems
(Math Masters, p. 116)
To apply students’ understanding of estimating with decimals,
have them use estimation and mental arithmetic strategies to
solve mileage problems. Emphasize that the data on Math
Masters, page 116 are approximations:
䉯 The numbers on the map are not exact distances: they have
been rounded to the nearest 10 miles.
Name
䉯 The number of miles a car travels on 1 gallon of gasoline
(called gas mileage) varies, depending on driving conditions.
ENRICHMENT
䉴 Solving a Decimal Magic
INDEPENDENT
ACTIVITY
15–30 Min
Square Puzzle
(Math Masters, p. 117)
To apply students’ understanding of place value and addition of
decimals, have students complete a magic square. Students use
estimation to place decimals so that the sum of the numbers in each
row, column, and diagonal of the magic square is equal to 6.5. Have
students describe how they decided to place the decimal points; for
example, “I knew that on the diagonal, the 80 had to be 0.80
because the sum is 6.5, and 8 and 80 are each greater than 6.5.”
LESSON
4 4
䉬
Date
Time
Decimal Magic Square
Insert decimal points so that the sum of the numbers in each row, column,
and diagonal is equal to 6.5.
3 0
16
9 0
2 2 15 0
2 0
8 0
21
14
7 0
2 5 13 0 10 19 0
2 4
12
5 0
11
4 0
17 10 0 2 3
18
34 –37
2 0
6 0
Math Masters, page 117
Lesson 4 4
䉬
259