Decimal Addition and Subtraction



Objective To extend methods for whole-number addition and subtraction to decimals. s

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ePresentations

eToolkit

Algorithms Practice

EM Facts Workshop Game™

Teaching the Lesson

Family Letters

Assessment Management

Common Core State Standards

Ongoing Learning & Practice

Key Concepts and Skills

Analyzing Circle Graphs

• Model decimals through hundredths with base-10 blocks.  

Math Journal 1, p. 88 Students compare population data presented in circle graphs.

[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.  

Math Boxes 4 5 

Math Journal 1, p. 89 Students practice and maintain skills through Math Box problems.

[Operations and Computation Goal 2]

• Judge the reasonableness of solutions to decimal addition and subtraction problems.  [Operations and Computation Goal 6]

Study Link 4 5 

Math Masters, p. 119 Students practice and maintain skills through Study Link activities.

Key Activities

Curriculum Focal Points

Interactive Teacher’s Lesson Guide

Differentiation Options READINESS

Investigating a Decimal Version of the Number Grid Math Masters, p. 427 Number-Grid Poster Students use a decimal version of the number grid to model decimal addition and subtraction. ENRICHMENT

Solving Hiking Trail Problems Math Masters, pp. 120 and 121 Students compute various distances on a hiking trail.

Students 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. Materials Math Journal 1, p. 87 Student Reference Book, pp. 178–178B Study Link 44 Math Masters, p. 118; pp. 427 and 428 (optional) base-10 blocks  quarters, nickels, dimes, pennies (optional)  slate

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.

Teacher’s Reference Manual, Grades 4–6 pp. 119 –126

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Unit 4

Decimals and Their Uses

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Getting Started Mental Math and Reflexes Pose decimal addition and subtraction problems within a money context. Suggestions: $0.50 + $0.75 = $1.25 $1.20 + $0.25 = $1.45 $0.30 + $0.60 = $0.90 $1.18 + $0.10 = $1.28 $1.00 - $0.70 = $0.30 $1.75 - $1.25 = $0.50 $0.80 - $0.40 = $0.40 $1.41 - $0.30 = $1.11

Math Message 

$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 

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.

Take an answer sheet (Math Masters, page 118 ) and complete it.

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.

1 Teaching the Lesson

 Math Message Follow-Up (Math Masters, p. 118)

WHOLE-CLASS ACTIVITY PROBLEM PRO P RO R OB BLE BL L LE LEM EM SOLVING SO S OL O LV VIN IIN NG

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.

+ 0.76

+ 0.2

 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.

Name LESSON

4 5 䉬

Time

Math Message

What’s wrong with this problem? What is the correct answer? 0.76 ⫹ 0.2 0.78

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.

夹

Date

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

 Rename 0.2 as 0.20 so that both addends name hundredths. Then use an addition algorithm. 0.76 + 0.2

→ →

0.76 + 0.20 0.96

(0.2 = 0.20)

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Ongoing Assessment: Recognizing Student Achievement

Math Message



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]

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.

 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.

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Unit 4 Decimals and Their Uses

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Student Page

 Practicing Decimal Addition

INDEPENDENT ACTIVITY

and Subtraction

Date

Time

LESSON

4 5 䉬

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.

(Math Journal 1, p. 87)

2.05 ⫹ 1.83 ⫽

2. 4. 6.

34–37

5.84 2.31 ⫺ 1.88 ⫽ 0.43 1 ⫺ 0.67 ⫽ 0.33 3.04 ⫹ 2.8 ⫽

Students solve decimal addition and subtraction problems.

Adjusting the Activity

ELL

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

2+1 0.0 + 0.8 0.05 + 0.03 3 + 0.8 + 0.08 

→ → → →

3.00 0.80 + 0.08

7.

3.88

K I N E S T H E T I C



T A C T I L E



Choose one of the problems from above. Explain the method you used to solve the problem.

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

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 Date

Time

LESSON

4 5 䉬

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

Lesotho

urban

Namibia

rural

Rwanda

urban

Links to the Future

rural

urban

rural

urban

South Africa

urban

Uganda an

rural

Gabon rural

urban rural

urban

rural

urb

Students compare population data presented in circle graphs. To support English language learners, discuss the terms population, urban, and rural.

Student Page

an

(Math Journal 1, p. 88)

INDEPENDENT ACTIVITY ELL

urb

 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 of people living in rural areas?

4.

Which two countries have about ᎏᎏ of their people living 2 1 in urban areas and ᎏᎏ of their people living in rural areas?

1

2

Gabon

Burundi, Uganda 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:

Math Journal 1, p. 88

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Student Page Date



Math Boxes

4 5 䉬

1.

 Math Boxes 4 5

Time

LESSON

Insert , , or .

2. a.

 0.4 0.50  0.500 1.3  1.09 0.85  0.86 0.700  0.007

Measure the length of this line segment 1 to the nearest  centimeter.

b. c. d. e.

5.5

About b.

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

Draw a line segment 3 centimeters long.

32 33

3.

128

Fill in the missing numbers.

49 , 56 , 63

b.

9.4  K  3

K

c.

0.81  M  0.43

M

Rule:

d.

F  2.1  6.8

F

81,

e.

2.43  S  1.06

S

f.

R  12.2  4.65

R

7

56, 48, 40,

32 , 24 , 16

8 72 , 63, 54 , 45, 36 9 Rule:

c.

Solve each open sentence. 5.9  T  5

Rule: b.

4.

a.

28, 35, 42,

a.

T

Add 9 tens, 8 hundredths, and 3 tenths to 34.53.

148

6.

124.91

What is the result?

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.

0.9 6.4 0.38 8.9 1.37 16.85

160 161

5.

(Math Journal 1, p. 89)

2

0.96

a.

INDEPENDENT ACTIVITY

 Study Link 4 5

Add mentally or with a paper-and-pencil algorithm. a.

6 40 150  1,000

b.



1,196



54 180 240 800

INDEPENDENT ACTIVITY

(Math Masters, p. 119)

1,274

36

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

 Investigating a Decimal 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 ⫽

5–15 Min

Version of the Number Grid

Study Link Master Name

SMALL-GROUP ACTIVITY

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.

⬎ ⬎

3.36 ⫹ 8.49

6.

0.08 ⫹ 0.97

9.4 ⫺ 1.33

8.

9.4 ⫺ 5.6

Practice 11.

13 ⫽ 7 ⫹ s

s⫽

13.

36 / p ⫽ 6

p⫽

6 6

12.

8 º g ⫽ 24

g ⫽

14.

m/9⫽8

m⫽

3 72

Math Masters, p. 119

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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

 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

 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

 Solving Hiking Trail

5–15 Min

Problems

Math Masters, p. 427

(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.

Teaching Master Name LESSON

4 5 䉬

Teaching Master

Date

Time

Name

A Hiking Trail

Date

LESSON

4 5 䉬

Map of Batona Trail

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.

A Hiking Trail

Point of Interest

Lebanon Headquarters & Fire Tower

N

Pakim Pond 72

IL

A 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.

Distance from Carpenter Spring (km) 0

47.60

Deep Hollow Pond

1.91

45.69

Route 70

3.37

Lebanon Headquarters

4.66

44.23 42.94 37.69 35.50

Pakim Pond

d oa

yR

Ha Batsto River

Carranza Memorial

9.91

FOREST Quakerbridge New Jersey

ad

BATSTO

12.10

Route 563

14.04

33.56

Route 532

19.53

28.07 26.29

Carranza Memorial Hay Road

STATE

Batsto Lake

Route 72

Apple Pie Hill Fire Tower 0 1 2 3 4 Scale of Kilometers

n gto

Ro

hin

as

W

Batsto Historical Area

2

54

Area of this map

Distance from Washington Road (km)

Carpenter Spring

532

NA CHATSWORTH BATO Apple Pie Hill Fire Tower

WHARTON

34 –37

Batona Trail

Deep Hollow Pond

70

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.

Carpenter Spring

The trail crosses several roads, so it can be reached by car at a number of places.

Time

21.31

27.80

19.80

33.05

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

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