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260 Unit 4 Decimals and Their Uses
Advance PreparationFor 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.
Teacher’s Reference Manual, Grades 4–6 pp. 119 –126
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]
Key ActivitiesStudents discuss different methods in which
to add and subtract decimals, including
modeling with base-10 blocks and using
algorithms.
Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 118. [Number and Numeration Goal 1]
Ongoing Assessment: Informing Instruction See page 263.
MaterialsMath Journal 1, p. 87
Student Reference Book, pp. 178–178B
Study Link 4�4
Math Masters, p. 118; pp. 427 and 428
(optional)
base-10 blocks � quarters, nickels, dimes,
pennies (optional) � slate
Analyzing Circle GraphsMath Journal 1, p. 88
Students compare population data
presented in circle graphs.
Math Boxes 4�5Math Journal 1, p. 89
Students practice and maintain skills
through Math Box problems.
Study Link 4�5Math Masters, p. 119
Students practice and maintain skills
through Study Link activities.
READINESS
Investigating a Decimal Version of the Number GridMath Masters, p. 427
Number-Grid Poster
Students use a decimal version of the
number grid to model decimal addition and
subtraction.
ENRICHMENTSolving Hiking Trail ProblemsMath Masters, pp. 120 and 121
Students compute various distances on a
hiking trail.
Teaching the Lesson Ongoing Learning & Practice Differentiation Options
Decimal Addition andSubtraction
Objective To extend methods for whole-number addition and
subtraction to decimals.s
�������
eToolkitePresentations Interactive Teacher’s
Lesson Guide
Algorithms Practice
EM FactsWorkshop Game™
AssessmentManagement
Family Letters
CurriculumFocal Points
Common Core State Standards
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Lesson 4�5 261
Mental Math and Reflexes Pose decimal addition and subtraction problems within a money context. Suggestions:
LESSON
4 �5
Name Date Time
Math Message
What’s wrong with this problem?
What is the correct answer?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.
�0.76
� 0.2
0.78
Math Masters, p. 118
1 Teaching the Lesson
� 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:
� Model the problem with base-10 blocks or pictures of base-10 blocks. (See margin.) This gives a total of 9 longs and 6 cubes, or 0.96.
� 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.
� 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.
� Rename 0.2 as 0.20 so that both addends name hundredths. Then use an addition algorithm.
0.76 → 0.76+ 0.2 → + 0.20 (0.2 = 0.20)
0.96
PROBLEMBBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEEEEMMMMLEBLELBLEBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBBBBBLBLBBLBLBLLBLLLLPROPROPROPROPROPROPROPROPROPROPROPPRPROPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROOROOPPPPPPP MMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELELELEEEEEEEEEELLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRPROBLEMSOLVING
BBBBBBBBBBBBBBBBBBBB ELEELELEMMMMMMMMMOOOOOOOOOBBBLBLBLBBBLBBLOOROROROORORORORORORORORO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRGGGGGLLLLLLLLLLLLLVINVINVINVINNNNVINVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOLOLOLOO VVINVINLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLVVVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVLLLLLLLLVVVVVVVVVLLLVVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING
Getting Started
Study Link 4�4 Follow-Up Draw students’ attention to Problems 4 and 5. Problem 4 describes what should be added to the length of one tunnel to get the length of another. This is an example of a comparison situation involving addition. Problem 5 describes what one tunnel length should be multiplied by to get another tunnel length. This is an example of a comparison situation involving multiplication.
Descriptions of these problem types are on Student Reference Book, pages 178–178B. Refer to these pages as you lead a discussion about the difference between these two types of comparisons. You might suggest that students sketch a situation diagram for each problem.
$0.50 + $0.75 = $1.25
$0.30 + $0.60 = $0.90
$1.00 - $0.70 = $0.30
$0.80 - $0.40 = $0.40
$1.20 + $0.25 = $1.45
$1.18 + $0.10 = $1.28
$1.75 - $1.25 = $0.50
$1.41 - $0.30 = $1.11
$1.39 + $0.46 = $1.85
$2.40 + $0.63 = $3.03
$0.64 - $0.33 = $0.31
$0.45 - $0.28 = $0.17
Math Message Take an answer sheet (Math Masters, page 118 ) and complete it.
�
++0.76 0.2
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262 Unit 4 Decimals and Their Uses
Links to the Future
Algorithm Project In this lesson,
students use various methods to add
and subtract decimals. To teach U.S.
traditional addition and subtraction of
decimals, see Algorithm Projects 2 and 4
on pages A5 and A15.
Ongoing Assessment: Math Message �
Recognizing Student Achievement
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]
� Adding and Subtracting WHOLE-CLASS ACTIVITY
Decimals Using an AlgorithmAsk: 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 17.0 8.10 9.0 0 + 3.50 + 05.1 - 4.72 - 0.09 6.13 22.1 3.38 8.91
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.
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Links to the Future
Adjusting the Activity
Decimal Addition and SubtractionLESSON
4 �5
Date Time
Add or subtract mentally or with a paper-and-pencil algorithm.
Pay attention to the � and � symbols.
1. 2.05 � 1.83 � 2. 3.04 � 2.8 �
3. 2.4 � 3.01 � 0.26 � 4. 2.31 � 1.88 �
5. 19 � 1.9 � 6. 1 � 0.67 � 0.3320.9
0.435.67
5.843.88
7. Choose one of the problems from above. Explain the method you used
to solve the problem.
make change. $0.03 � $0.05 � $0.25 � $0.33.
$1.00 � $0.67. Then I mentally thought how I would
Sample answer: Problem 6; I rewrote the problem as
34–37
Math Journal 1, p. 87
Student Page
Circle GraphsLESSON
4 �5
Date Time
Percent urban is the number of people out of 100 who live in towns or cities. Percentrural 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.
Source: The United Nations
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
of people living in rural areas?
4. Which two countries have about �1
2� of their people living
in urban areas and �1
2� of their people living in rural areas?
5. Write a question that can be answered from the information in the graphs. Then answer the
question.
Question:
Answer: Namibia
of its population living in rural areas?
Which country has about two-thirds
Congo, Cameroon
Burundi, Uganda
Gabon
urb
an
rural
Burundi
urbanrural
Cameroon
urbanrural
Central AfricanRepublic
urbanrural
Congo
urban
rural
Gabon
urban
rural
Lesotho
urbanrural
Namibia
urban
rural
Rwanda
urbanrural
South Africa
urb
an
rural
Uganda
Try This
Math Journal 1, p. 88
Student Page
Lesson 4�5 263
� Practicing Decimal Addition INDEPENDENTACTIVITY
and Subtraction(Math Journal 1, p. 87)
Students solve decimal addition and subtraction problems.
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: 2 + 1 → 3.00
Add the 0.1s: 0.0 + 0.8 → 0.80
Add the 0.01s: 0.05 + 0.03 → + 0.08
Find the total: 3 + 0.8 + 0.08 → 3.88
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
Ongoing Assessment: Informing InstructionWatch 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
� Analyzing Circle Graphs INDEPENDENTACTIVITY
(Math Journal 1, p. 88)
Students compare population data presented in circle graphs. To support English language learners, discuss the terms population, urban, and rural.
Creating and interpreting circle graphs are Grade 5 and Grade 6 Goals.
ELL
ELL
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264 Unit 4 Decimals and Their Uses
STUDY LINK
4�5 Addition and Subtraction of Decimals
34–37
Name Date Time
Add or subtract. Show your work.
1. 96.45 � 23.96 � 2. 1.06 � 0.4 �
3. 9.87 � 4.69 � 4. 0.4 � 0.37 � 0.035.18
1.46120.41
Write �, �, or � to make each statement true.
5. 2.78 � 9.1 3.36 � 8.49 6. 0.08 � 0.97 1.04 � 0.03
7. 13.62 � 4.9 9.4 � 1.33 8. 9.4 � 5.6 8.3 � 4.7
9. Name two 3-digit numbers whose sum is 6.54. � � 6.54
10. Name two 3-digit numbers whose difference is 1.52. � � 1.52
11. 13 � 7 � s s � 12. 8 º g � 24 g �
13. 36 / p � 6 p � 14. m / 9 � 8 m � 726
36
5.316.83
4.212.33��
��
Sample answers:
Practice
Math Masters, p. 119
Study Link Master
89
Math Boxes LESSON
4 �5
Date Time
1. Insert �, �, or �.
a. 0.96 0.4
b. 0.50 0.500
c. 1.3 1.09
d. 0.85 0.86
e. 0.700 0.007�
�
�
�
�2. a. Measure the length of this line segment
to the nearest �1
2� centimeter.
About cm
b. Draw a line segment
3 centimeters long.
5.5
3. Fill in the missing numbers.
a. 28, 35, 42, , ,
Rule:
b. 56, 48, 40, , ,
Rule:
c. 81, , 63, , 45,
Rule:
365472
162432
6356494. 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 �
f. R � 12.2 � 4.65 R � 16.85
1.37
8.9
0.38
6.4
0.9
5. Add 9 tens, 8 hundredths, and 3 tenths
to 34.53.
What is the result? 124.91
6. Add mentally or with a paper-and-pencil
algorithm.
a. 6 b. 54
40 180
150 240
� 1,000 � 800
32 33 128
160 161
36
148
10 11
1,196 1,274
�7
�8
�9
Math Journal 1, p. 89
Student Page
� Math Boxes 4�5 INDEPENDENTACTIVITY
(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.
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.
� Study Link 4�5 INDEPENDENTACTIVITY
(Math Masters, p. 119)
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 SMALL-GROUP ACTIVITY
� Investigating a Decimal 5–15 Min
Version of the Number Grid(Math Masters, p. 427)
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.
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Name Date Time
Number Grid (Decimal Version)
0
0.100.090.080.070.060.050.040.030.020.01
0.200.190.180.170.160.150.140.130.120.11
0.300.290.280.270.260.250.240.230.220.21
0.400.390.380.370.360.350.340.330.320.31
0.500.490.480.470.460.450.440.430.420.41
0.600.590.580.570.560.550.540.530.520.51
0.800.790.780.770.760.750.740.730.720.71
0.700.690.680.670.660.650.640.630.620.61
0.900.890.880.870.860.850.840.830.820.81
1.000.990.980.970.960.950.940.930.920.91
Math Masters, p. 427
Teaching Aid Master
Lesson 4�5 265
LESSON
4�5
Name Date Time
A Hiking Trail continued
34–37
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.
How can you check your answers?
Sample answer: Finding the sum of the twoentries on each line should give you the distance of the whole trail: 47.60 km.
Batona Trail
Point of Interest Distance from Distance fromCarpenter Spring (km) Washington Road (km)
Carpenter Spring 0 47.60
Deep Hollow Pond 1.91 45.69
Route 70 3.37 44.23Lebanon Headquarters 4.66 42.94Pakim Pond 9.91 37.69Route 72 12.10 35.50Route 563 14.04 33.56
Route 532 19.53 28.07Apple Pie Hill Fire Tower 21.31 26.29Carranza Memorial 27.80 19.80
Hay Road 33.05 14.55Quakerbridge 37.92 9.68
Washington Road 47.60 0
Math Masters, p. 121
Teaching MasterLESSON
4 �5
Name Date Time
A Hiking Trail
542
532
72
70
Carpenter Spring
Map of Batona Trail
Lebanon
Headquarters
& Fire TowerN
CHATSWORTH
FOREST
BATSTO Batsto Historical Area
STATE
Apple Pie Hill
Fire Tower
Scale of Kilometers
0 1 2 3 4
Quakerbridge
Area of
this map
Source: Batona Hiking Club of Philadelphia
NewJersey
Wash
ington R
oad
BATONA
TRAIL
Deep HollowPond
PakimPond
BatstoLake
Bats
to R
iver
563
Carranza
Memorial
Hay R
oad
WHARTON
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.
The trail crosses several roads, so it
can be reached by car at a number
of places.
Carpenter Spring is at the north end
of the trail. Washington Road, near
Batsto, is at the trail’s south end.
Go to Math Masters, page 121.
Math Masters, p. 120
Teaching Master
Ask students to solve addition or subtraction problems by counting on the grid.
Examples:
� Write 0.02 + 0.07 on the board.
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
� Write 0.14 + 0.10 on the board.
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
ENRICHMENT PARTNER ACTIVITY
� Solving Hiking Trail 5–15 Min
Problems(Math Masters, pp. 120 and 121)
To apply students’ understanding of computation with decimals to the hundredths place, have them find distances on a hiking map.
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