Objective
To extend methods for whole-number addition and
subtraction to decimals.
1
materials
Teaching the Lesson
Key Activities
Students discuss different methods in which to add and subtract decimals, including
modeling with base-10 blocks and using algorithms.
Key Concepts and Skills
• Model decimals through hundredths with base-10 blocks.
[Number and Numeration Goal 1]
• Express the values of digits in decimals.
[Number and Numeration Goal 1]
• Add and subtract decimals to the hundredths place.
[Operations and Computation Goal 2]
• Judge the reasonableness of solutions to decimal addition and subtraction problems.
[Operations and Computation Goal 6]
ⵧ Math Journal 1, p. 87
ⵧ Study Link 4 4
䉬
ⵧ Teaching Master (Math Masters,
p. 118)
ⵧ Teaching Aid Masters (Math
Masters, pp. 427 and 428)
ⵧ base-10 blocks
ⵧ quarters, nickels, dimes,
pennies (optional)
ⵧ slate
See Advance Preparation
Ongoing Assessment: Recognizing Student Achievement Use Math Masters,
page 118. [Number and Numeration Goal 1]
Ongoing Assessment: Informing Instruction See page 263.
2
materials
Ongoing Learning & Practice
Students analyze circle graphs.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Differentiation Options
READINESS
Students use a decimal version of the
number grid to model decimal addition
and subtraction.
ENRICHMENT
Students compute various distances on a
hiking trail.
Additional Information
Advance Preparation For Part 1, copy and cut apart Math Masters, page 118 so that each
student has one answer sheet for the Math Message. Place these sheets near the Math
Message.
260
Unit 4 Decimals and Their Uses
ⵧ Math Journal 1, pp. 88 and 89
ⵧ Study Link Master (Math Masters,
p. 119)
ⵧ Teaching Masters (Math Masters,
pp. 120 and 121)
ⵧ Teaching Aid Master (Math
Masters, p. 427)
ⵧ Number Grid Poster
Technology
Assessment Management System
Math Masters, page 118
See the iTLG.
Getting Started
Mental Math and Reflexes
Pose decimal addition and subtraction problems within a money context. Suggestions:
$0.50 ⫹ $0.75 ⫽ $1.25
$0.30 ⫹ $0.60 ⫽ $0.90
$1.00 ⫺ $0.70 ⫽ $0.30
$0.80 ⫺ $0.40 ⫽ $0.40
Math Message
$1.20 ⫹ $0.25 ⫽ $1.45
$1.18 ⫹ $0.10 ⫽ $1.28
$1.75 ⫺ $1.25 ⫽ $0.50
$1.41 ⫺ $0.30 ⫽ $1.11
夹
$1.39 ⫹ $0.46 ⫽ $1.85
$2.40 ⫹ $0.63 ⫽ $3.03
$0.64 ⫺ $0.33 ⫽ $0.31
$0.45 ⫺ $0.28 ⫽ $0.17
Study Link 4 4 Follow-Up
䉬
Take an answer sheet (Math
Masters, page 118) and
complete it.
Ask students to share the strategies they used to solve the problems.
For Problem 5, students may be interested in knowing that the Cascade
Tunnel was completed in 1929 and is 7.79 miles long.
You may want to pose subtraction problems involving the years in which the tunnels
were completed. For example: How many years before the completion of the Channel
Tunnel was the London Underground completed? 55 years
1 Teaching the Lesson
䉴 Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
(Math Masters, p. 118)
Have students discuss why the answer to the problem is incorrect.
There are many ways to explain the mistake. Mention the
following, if no one brings them up:
䉯 Model the problem with base-10 blocks (or pictures of
base-10 blocks).
Name
LESSON
4 5
䉬
Math Message
What’s wrong with this problem?
What is the correct answer?
+
0.76
Date
+ 0.2
This gives a total of 9 longs and 6 cubes, or 0.96.
0.76
⫹ 0.2
0.78
夹
Time
Sample answer: The digits are not in
the correct columns. Six hundredths
plus 2 tenths is not 8 hundredths.
The correct answer is 0.96.
Math Masters, p. 118
䉯 Write the problem in dollars-and-cents notation.
0.76 ⫽ $0.76 and 0.2 ⫽ $0.20. Think of the 7 in $0.76 as
7 dimes and the 6 as 6 pennies. Think of the 2 in $0.20 as
2 dimes and the 0 as no pennies. This gives a total of 9 dimes
and 6 pennies, or $0.96.
䉯 Think in terms of place value.
0.76 ⫽ 7 tenths and 6 hundredths, and 0.2 ⫽ 2 tenths.
This gives a total of 9 tenths and 6 hundredths, or 0.96.
䉯 Rename 0.2 as 0.20 so that both addends name hundredths.
Then use an addition algorithm.
.76
⫹ 0.2
→
→
0.76
⫹ 0.20
0.96
(0.2 ⫽ 0.20)
Lesson 4 5
䉬
261
Ongoing Assessment:
Recognizing Student Achievement
Math Message
夹
Use the Math Message to assess students’ understanding of the
values of decimal digits. Students are making adequate progress if
their responses indicate that the digit 6 stands for or represents 6 hundredths
and the digit 2 stands for or represents 2 tenths. Some students may be
able to describe how a ballpark estimate can be used to check the answer
to the problem.
[Number and Numeration Goal 1]
䉴 Adding and Subtracting
WHOLE-CLASS
ACTIVITY
Decimals Using an Algorithm
Ask: Is it possible to use the same methods for adding and
subtracting decimals that you use for whole numbers? yes
As with whole numbers, all digits of a given place value must
be lined up correctly.
One way to make sure the digits align correctly is to rename the
numbers so that each has the same number of digits after the
decimal point. For example, if adding or subtracting decimals in
tenths and hundredths, rename the tenths as hundredths by
adding a zero to the end of the numbers. When the digits are
aligned correctly, the decimal points will also align.
Pose several decimal addition and subtraction problems. Ask
students to model their answers with base-10 blocks (or symbols).
Suggestions:
2.63 3.5 ?
17 5.1 ?
8.1 4.72 ?
9 0.09 ?
The zeros in boldface have been appended so both numbers have
the same number of digits after the decimal point.
2.63
3.50
6.13
17.0
05.1
22.1
8.10
4.72
3.38
9.00
0.09
8.91
Links to the Future
Do not be concerned if students use manipulatives such as base-10 blocks or
bills and coins to add and subtract decimals. Students will be expected to do so
without the use of manipulatives in Grade 5.
262
Unit 4 Decimals and Their Uses
Student Page
䉴 Practicing Decimal Addition
INDEPENDENT
ACTIVITY
and Subtraction
Date
LESSON
4 5
䉬
Time
Decimal Addition and Subtraction
Add or subtract mentally or with a paper-and-pencil algorithm.
Pay attention to the and symbols.
3.88
3. 2.4 3.01 0.26 5.67
5. 19 1.9 20.9
1. 2.05 1.83 (Math Journal 1, p. 87)
Students solve decimal addition and subtraction problems.
2.
4.
6.
34–37
5.84
2.31 1.88 0.43
1 0.67 0.33
3.04 2.8 ELL
Adjusting the Activity
Have base-10 blocks, coins and bills (Math Masters, page 428),
and a decimal number grid (Math Masters, page 427) available. Encourage
students to think in terms of the partial-sums algorithm.
2.05
1.83
Add the 1s:
Add the 0.1s:
Add the 0.01s:
Find the total:
A U D I T O R Y
21
0.0 0.8
0.05 0.03
3 0.8 0.08
䉬
→
→
→
→
3.00
0.80
0.08
7. Choose one of the problems from above. Explain the method you used
to solve the problem.
3.88
K I N E S T H E T I C
䉬
T A C T I L E
䉬
Sample answer: Problem 6; I rewrote the problem as
$1.00 $0.67. Then I mentally thought how I would
make change. $0.03 $0.05 $0.25 $0.33.
V I S U A L
87
Math Journal 1, p. 87
Ongoing Assessment: Informing Instruction
Watch for students who do not correctly align the digits when adding and
subtracting. All digits of a given place value must be written in the same column.
Encourage students to use computation grid paper and record the place-value
heading above each column.
2 Ongoing Learning & Practice
Students compare population data presented in circle graphs. To
support English language learners, discuss the terms population,
urban, and rural.
LESSON
4 5
䉬
Time
Circle Graphs
Percent urban is the number of people out of 100 who live in towns or cities. Percent
rural is the number of people out of 100 who live in the countryside. Each circle graph
below represents the percent of the urban and rural population of an African country.
Burundi
Cameroon
Central African
Republic
Congo
rural
rural
Lesotho
urban
Namibia
rural
rural
rural
urban
urban
Rwanda
urban
Links to the Future
Gabon
rural
rural
urban
South Africa
urban
Uganda
an
(Math Journal 1, p. 88)
Student Page
Date
urban
rural
urb
INDEPENDENT
ACTIVITY
urb
an
䉴 Analyzing Circle Graphs
rural
urban
rural
Source: The United Nations
Creating and interpreting circle graphs are Grade 5 and Grade 6 Goals.
1. For each pair, circle the country with the larger urban population.
a. Congo
Uganda
b. Rwanda
Gabon
c. Burundi
South Africa
d. Namibia
Lesotho
2. Which country has the greatest percentage of people living in urban areas?
3. Which two countries have the greatest percentage
Burundi, Uganda
of people living in rural areas?
1
2
Gabon
4. Which two countries have about of their people living
1
in urban areas and of their people living in rural areas?
2
Congo, Cameroon
Try This
5. Write a question that can be answered from the information in the graphs. Then answer the
question.
Which country has about two-thirds
of its population living in rural areas?
Answer: Namibia
Question:
88
Math Journal 1, p. 88
Lesson 4 5
䉬
263
Student Page
Date
䉴 Math Boxes 4 5
Time
LESSON
䉬
Math Boxes
4 5
䉬
2. a. Measure the length of this line segment
1
to the nearest centimeter.
2
1. Insert , , or .
a.
b.
c.
d.
e.
0.96 0.4
0.50 0.500
1.3 1.09
0.85 0.86
0.700 0.007
5.5
About
3 centimeters long.
32 33
49 , 56 , 63
a. 28, 35, 42,
7
Rule:
32 , 24 , 16
8
81, 72 , 63, 54 , 45, 36
Rule: 9
b. 56, 48, 40,
Rule:
c.
Writing/Reasoning Have students write a response to the
following: Explain how you found the value of S in Problem
4e. Sample answer: Since I knew the whole (2.43) and
one of the parts (1.06), I subtracted 1.06 from 2.43 to find the
value of S.
128
4. Solve each open sentence.
a. 5.9 T 5
T
b. 9.4 K 3
K
c. 0.81 M 0.43
M
d. F 2.1 6.8
F
e. 2.43 S 1.06
S
R 12.2 4.65
R
f.
0.9
6.4
0.38
8.9
1.37
16.85
148
160 161
䉴 Study Link 4 5
䉬
6. Add mentally or with a paper-and-pencil
5. Add 9 tens, 8 hundredths, and 3 tenths
algorithm.
to 34.53.
124.91
What is the result?
a.
6
40
150
1,000
b.
1,196
(Math Journal 1, p. 89)
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 4-7. The skill in Problem 6
previews Unit 5 content.
cm
b. Draw a line segment
3. Fill in the missing numbers.
INDEPENDENT
ACTIVITY
54
180
240
800
(Math Masters, p. 119)
1,274
36
INDEPENDENT
ACTIVITY
10 11
89
Math Journal 1, p. 89
Home Connection Students add and subtract decimals.
They also write , , or symbols to make true number
sentences.
Encourage students to continue bringing examples of decimals to
display in the Decimals All Around Museum.
3 Differentiation Options
READINESS
䉴 Investigating a Decimal
SMALL-GROUP
ACTIVITY
5–15 Min
Version of the Number Grid
Study Link Master
Name
Date
STUDY LINK
(Math Masters, p. 427)
Time
Addition and Subtraction of Decimals
4 5
䉬
Add or subtract. Show your work.
1.
96.45 23.96 3.
9.87 4.69 120.41
5.18
2.
1.06 0.4 4.
0.4 0.37 1.46
0.03
34 –37
To explore the use of a visual organizer for understanding the
base-ten place-value system for decimals, have students use a
decimal version of the number grid.
Have students compare the number grid poster with the decimal
version. Ask: What are some similarities and differences? Possible
answers: Patterns in the digits are similar in that the hundredths
digit stays the same as you move down a column, and the tenths
digit stays the same as you move across a row. The numbers
increase by 0.01 as you move a step to the right; the numbers
increase by 0.1 as you move a step down.
Write , , or to make each statement true.
1.04 0.03
8.3 4.7
Sample answers:
2.33 4.21 6.54
Name two 3-digit numbers whose sum is 6.54.
6.83 5.31 1.52
Name two 3-digit numbers whose difference is 1.52.
5.
2.78 9.1
7.
13.62 4.9
9.
10.
Practice
11.
13 7 s
s
13.
36 / p 6
p
3.36 8.49
6.
0.08 0.97
9.4 1.33
8.
9.4 5.6
6
6
12.
8 º g 24
g 14.
m/98
m
3
72
Math Masters, p. 119
264
Unit 4 Decimals and Their Uses
Teaching Aid Master
Ask students to solve addition or subtraction problems by counting
on the grid.
Name
Date
Time
Number Grid (Decimal Version)
0
Examples:
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
䉯 Write 0.02 0.07 on the board.
0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20
Have students put their fingers on 0.02 and count by
hundredths as they move their fingers 7 steps to the
right—one step for each hundredth. 0.09
0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40
0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50
䉯 Write 0.14 0.10 on the board.
0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60
0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70
Have students put their fingers on 0.14 and count by
hundredths as they move their fingers 10 steps to the
right—one for each hundredth. Or, move down one row
for each tenth. 0.24
0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80
0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90
0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
PARTNER
ACTIVITY
ENRICHMENT
䉴 Solving Hiking Trail Problems
5–15 Min
(Math Masters, pp. 120 and 121)
Math Masters, p. 427
To apply students’ understanding of computation with decimals
to the hundredths place, have them find distances on a hiking map.
Teaching Master
Name
LESSON
4 5
䉬
Teaching Master
Date
Time
Name
A Hiking Trail
Date
LESSON
4 5
䉬
The Batona Trail is a hiking trail in
southern New Jersey. The Batona
Hiking Club measured the trail very
carefully and found that it is about
47.60 kilometers long.
Map of Batona Trail
A Hiking Trail
Time
continued
The following table shows distances from several points of interest from
the north to the south end of the trail. Fill in the missing distances.
34 –37
Carpenter Spring
Batona Trail
Deep Hollow
Pond
The trail crosses several roads, so it
can be reached by car at a number
of places.
70
Point of Interest
Lebanon
Headquarters
& Fire Tower
N
72
L
AI
TR
Go to Math Masters, page 121.
563
Carpenter Spring is at the north end
of the trail. Washington Road, near
Batsto, is at the trail’s south end.
Pakim
Pond
y
Ha
a
Ro
d
Batsto River
Carranza
Memorial
Scale of Kilometers
FOREST
Quakerbridge
New
Jersey
d
oa
BATSTO
2
1.91
45.69
Route 70
3.37
Lebanon Headquarters
4.66
g
hin
as
R
ton
W
Batsto Historical Area
54
Area of
this map
9.91
Route 72
12.10
Route 563
14.04
Route 532
19.53
Carranza Memorial
Hay Road
STATE
Batsto
Lake
47.60
Deep Hollow Pond
Apple Pie Hill Fire Tower
0 1 2 3 4
WHARTON
Distance from
Washington Road (km)
0
Pakim Pond
532
NA
CHATSWORTH
BATO
Apple Pie Hill
Fire Tower
Distance from
Carpenter Spring (km)
Carpenter Spring
21.31
27.80
33.05
44.23
42.94
37.69
35.50
33.56
28.07
26.29
19.80
14.55
Quakerbridge
37.92
9.68
Washington Road
47.60
0
How can you check your answers?
Sample answer: Finding the sum of the two
entries on each line should give you the
distance of the whole trail: 47.60 km.
Source: Batona Hiking Club of Philadelphia
Math Masters, p. 120
Math Masters, p. 121
Lesson 4 5
䉬
265