Addition of Whole Numbers and Decimals... Lesson 2 2 85 Addition of Whole Numbers and Decimals Objectives To review place-value concepts and the use of the partial-sums and column-addition

www.everydaymathonline.com

Lesson 2�2 85

Addition of Whole Numbers and Decimals

Objectives To review place-value concepts and the use of the

partial-sums and column-addition methods.

Advance PreparationPlan to spend two days on this lesson. Distribute copies of the computation grid on Math Masters, page 415

for students to use as they do addition problems. Make and display a poster showing expanded notation for a

whole number and a decimal.

Teacher’s Reference Manual, Grades 4–6 pp. 119–122

Key Concepts and Skills• Write numbers in expanded notation.

[Number and Numeration Goal 1]

• Use paper-and-pencil algorithms for

multidigit addition problems.

[Operations and Computation Goal 1]

• Make magnitude estimates for addition.


Key ActivitiesStudents review place-value concepts and

write numbers in expanded notation. They

review addition of whole numbers and

decimals with the partial-sums and column-

addition methods.

Ongoing Assessment: Recognizing Student Achievement Use journal page 33. [Operations and Computation Goal 1]

Key Vocabularyplace � value � digit � algorithm � partial-

sums method � place value � expanded

notation � column-addition method

MaterialsMath Journal 1, pp. 32 and 33

Student Reference Book, pp. 13, 14, 28–30,

and 35

Study Link 2�1

Math Masters, p. 415

slate

Playing Addition Top-It(Decimal Version)Student Reference Book, p. 333

per partnership: 4 each of the number

cards 1–10 (from the Everything Math

Deck, if available); 2 counters

Students practice place-value

concepts, use addition methods, and

compare numbers.

Math Boxes 2�2Math Journal 1, p. 34

Students practice and maintain skills

through Math Box problems.

Study Link 2�2Math Masters, p. 36

Students practice and maintain skills

through Study Link activities.

READINESS

Building Numbers with Base-10 BlocksMath Masters, p. 37

base -10 blocks

Students use base -10 blocks to explore the

partial-sums method of addition.

ENRICHMENTUsing Place Value to Solve Addition ProblemsMath Masters, p. 38

Students apply place-value and addition

concepts to solve problems.

ELL SUPPORT

Building a Math Word Bank Differentiation Handbook, p. 142

Students define and illustrate the term

expanded notation.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

085_EMCS_T_TLG1_U02_L02_576825.indd 85085_EMCS_T_TLG1_U02_L02_576825.indd 85 2/2/11 5:24 PM2/2/11 5:24 PM

Check your answers on page 434.

1. What is the value ofthe digit 2 in each ofthese numbers?a. 20,006.8b. 0.02c. 34.502

2. Using the digits 9, 3, and 5, what isa. the smallest decimal that you can write?b. the largest decimal less than 1 that

you can write?c. the decimal closest to 0.5 that you

can write?

Decimals and Percents

You use facts about the place-value chart each time you maketrades using base-10 blocks.

Left to Right in the Place-Value ChartStudy the place-value chart below. Look at the numbers thatname the places. As you move from left to right along the chart,each number is �1

10� as large as the number to its left.

one 100 � �110� of 1,000

one 10 � �110� of 100

one 1 � �110� of 10

one �110� � �1

10� of 1

one �1100� � �1

10� of �1

10�

one �1,0100� � �1

10� of �1

100�

Suppose that a flat is worth 1. Then a long is worth �1

10�, or 0.1; and a cube is worth �1

100�, or 0.01.

You can trade one long for ten cubes because one �1

10� equals ten �1

100�s.

You can trade ten longs for one flat because ten �1

10�s equals one 1.

You can trade ten cubes for one long because ten �1

100�s equals one �1

10�.

For this example:

A flat is worth 1.

A long is worth �110� or 0.1.

A cube is worth �1100�,

or 0.01.

Student Reference Book, p. 30

Student Page

35 30 + 5

0.35 3 ∗ ( 1 _ 10 ) + 5 ∗ ( 1

_ 100 )

52 50 + 2

0.52 5 ∗ ( 1 _ 10 ) + 2 ∗ ( 1

_ 100 )

241 200 + 40 + 1

0.241 2 ∗ ( 1 _ 10 ) + 4 ∗ ( 1

_ 100 ) +

1 ∗ ( 1

_ 1,000 )

162 100 + 60 + 2

0.467 4 ∗ ( 1 _ 10 ) + 6 ∗ ( 1

_ 100 ) +

7 ∗ ( 1

_ 1,000 )

0.109 1 ∗ ( 1 _ 10 ) +

9 ∗ ( 1

_ 1,000 )

0.708 7 ∗ ( 1 _ 10 ) +

8 ∗ ( 1

_ 1,000 )

0.084 8 ∗ ( 1

_ 100 ) + 4 ∗ ( 1

_ 1,000 )

7,904 7,000 + 900 + 4

Adjusting the Activity

86 Unit 2 Estimation and Computation

Getting Started

● On Day 1 of this lesson, students should

complete the Mental Math and Reflexes and

the Math Message. They should review and

discuss the partial-sums addition method.

● On Day 2 of this lesson, do the Study

Link Follow-Up. Then review and discuss

the column-addition method. Finally, have

students complete the Part 2 activities.

Mental Math and Reflexes Read the numbers orally and have students write them in expanded notation on their slates. Remind students that expanded notation expresses a number as the sum of the values of each digit. For example, 906 is equivalent to 9 hundreds + 0 tens + 6 ones, and 0.796 is equivalent to 7 tenths + 9 hundredths + 6 thousandths. In expanded notation, 906 is written as 900 + 6, and 0.796 may be written as 0.7 + 0.09 + 0.006 or as 7 ∗ ( 1

_ 10 ) + 9 ∗ ( 1

_ 100 ) + 6 ∗ ( 1

_ 1,000 ). Encourage students to write the decimal numbers in fraction notation. Sample answers are given.

Math MessageUse the information on Student Reference Book, pages 28–30 to solve the Check Your Understanding Problems on the bottom of page 30.

Study Link 2�1 Follow-Up Have partners discuss their strategies and identify one thing that they did the same and one thing that they did differently. Have volunteers share their findings.

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

(Student Reference Book, pp. 28–30)

Ask students to use the information they read in the Student Reference Book to think of one true statement they could make about the base-ten number system.

Refer students to the place-value chart on page 30 of the Student

Reference Book. Ask them to look over the headings on the chart and describe

any patterns they see. The numbers decrease in size from left to right; the

columns on the right side of the chart have a decimal point and the left side does

not; the 0s increase by one for each column as you move outward from the

center in either direction.

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

Survey the class and use their responses to discuss the following:

� Each place has a value that is 10 times the value of the place to its right. For example, 1,000 is 10 times as much as 100; 100 is 10 times as much as 10; 10 is 10 times as much as 1; 1 is 10 times as much as 0.1; and 0.1 is 10 times as much as 0.01.

� Each place has a value that is one-tenth the value of the place to its left. For example, 100 is 1 _ 10 of 1,000; 10 is 1 _ 10 of 100; 1 is 1 _ 10 of 10; 0.1 is 1 _ 10 of 1; and 0.01 is 1 _ 10 of 0.1.

Ask students how these relationships guide them in writing the decimals in Problem 2 on Student Reference Book, page 30. Sample answers: Place the largest digits rightmost when forming the smallest decimal; place the largest digits leftmost when forming the largest decimal; place the 5 in the tenths place and the other

EM3cuG5TLG1_086-090_U02L02.indd 86EM3cuG5TLG1_086-090_U02L02.indd 86 1/3/11 2:53 PM1/3/11 2:53 PM

Lesson 2�2 87

two digits so the larger is to the right to form the decimal that is closest to 0.5. Explain that knowing these relationships also helps with comparing and ordering numbers by their relative sizes.

Ask students to listen closely as you read the numbers from Problem 1 on Student Reference Book, page 30. Tell them that you will include some mistakes. Read the numbers as 200,068; 0.2; and, 34.052. For each number, ask students to tell a partner what mistake was made. Then ask volunteers to describe the mistake and to read the number correctly. 200,068—no decimal point; 0.2—decimal point in the wrong position; 34.052 reverses the tenths and hundredths.

Tell students that clocks operate on a base-60 number system for minutes, and base-12 or base-24 (with military clocks) for hours. Have volunteers compare these systems and the base-10 systems.

▶ Reviewing Algorithms: WHOLE-CLASSDISCUSSION

Partial-Sums Method(Math Journal 1, p. 32; Student Reference Book,

pp. 13, 29, and 35; Math Masters, p. 415)

Most fifth-grade students have mastered an algorithm of their choice for addition. If they are comfortable with that algorithm, there is no reason for them to change it. However, all students are expected to know the partial-sums method for addition. This method helps students develop their understanding of place value and addition. In the partial-sums method, addition is performed from left to right, column by column. The sum for each column is recorded on a separate line. The partial sums are added either at each step or at the end.

Ask students to read Student Reference Book, page 29 and then write the numbers 348 and 177 in expanded notation. 300 + 40 + 8 = 348 and 100 + 70 + 7 = 177 Provide additional examples for students to write in expanded notation if needed. Then refer to page 13 in the Student Reference Book and demonstrate adding 348 + 177 using the partial-sums method. Ask students to describe any relationships they see between the expanded notation and the partial-sums method. Sample answer: Both methods use the value of the digits.

Ask students to write the numbers 4.56 and 7.9 in expanded notation. 4 + 5 ∗ 0.1 + 6 ∗ 0.01 = 4.56 and 7 + 9 ∗ 0.1 = 7.9 Provide additional decimal examples for students to write in expanded notation if needed. Then refer to page 35 in the Student Reference Book and demonstrate adding 4.56 + 7.9 using the partial-sums method. Ask: What are the similarities and differences between expanded notation and the partial-sums method with whole numbers and with decimals? Sample answers: Both methods for whole numbers and decimals use the value of the digits. With decimals, you have to line up the places correctly, either by affixing 0s to the end of the numbers, or by aligning the digits in the ones place.

Have students independently solve the problems on journal page 32 and then check each other’s answers.

NOTE Display a poster showing the

expanded notation of a whole number and

of a decimal to provide students with a readily

accessible example.

Expanded Notation

Number Expanded Form

34.15 3 ∗ 10 + 4 ∗ 1 + 1 ∗ 0.1 + 5 ∗ 0.01

27.94 2 ∗ 10 + 7 ∗ 1 + 9 ∗ 0.1 + 4 ∗ 0.01

18.795 1 ∗ 10 + 8 ∗ 1 + 7 ∗ 0.1 +

9 ∗ 0.01 + 5 ∗ 0.001

72.089 7 ∗ 10 + 2 ∗ 1 + 0 ∗ 0.1 +

8 ∗ 0.01 + 9 ∗ 0.001

Expanded form may also be written with fractions

instead of decimals. For example: 34.15 = 3 ∗ 10 +

4 ∗ 1 + 1 ∗ ( 1 _

10 ) + 5 ∗ ( 1

_ 100 ).

Language Arts Link The word

algorithm is used to name a step-by-

step procedure for solving a

mathematical problem. The word is derived

from the name of a ninth-century Muslim

mathematician, Al-Khowarizimi. Encourage

students to research the etymology of other

mathematical terms.

Algorithm Project The focus of this

lesson is the partial-sums and column-addition

methods for adding whole numbers and

decimals. To teach U.S. traditional addition

with whole numbers and with decimals, see

Algorithm Project 1 on page A1 and Algorithm

Project 2 on page A6.

086-090_EMCS_T_TLG1_U02_L02_576825.indd 87086-090_EMCS_T_TLG1_U02_L02_576825.indd 87 2/2/11 5:24 PM2/2/11 5:24 PM

Adding with Partial SumsLESSON

2 �2

Date Time

Write the following numbers in expanded notation.

1. 432:

2. 56.23

Write an estimate for each problem. Then use the partial-sums method to find the

exact answer.

Example:

Estimate: 400

400 � 30 � 2

325.022�134.527�400.00050.0009.0000.5000.0400.009

459.549

3. Estimate: 4. Estimate: 100700

5. Estimate: 6. Estimate: 7060

214

� 475.2

600.080.09.00.2

689.2

10.31

32.04

� 59.61

90.0011.000.900.06

101.96

28.765

� 31.036

50.0009.0000.7000.0900.011

59.801

47.84

� 21.023

60.0008.0000.8000.0600.003

68.863

50 � 6 � 0.2 � 0.03

Math Journal 1, p. 32

Student Page

Methods for AdditionLESSON

2 �2

Date Time

Solve Problems 1–5 using the partial-sums method. Solve the rest of the problems

using any method you choose. Show your work in the space on the right. Compare

answers with your partner. If there are differences, work together to find the correct

solution.

1. 714 � 465 �

2. 253 � 187 �

3. � 5,312 � 3,687

4. 3,416 � 2,795 �

5. 475 � 139 � 115 �

6. � 217 � 192 � 309 � 536

7. 38.47 � 9.58 �

8. � 32.06 � 65.1

9. 43.46 � 7.1 � 2.65 �

10. Alana is in charge of the class pets. She spent

$ 43.65 on hamster food,

$ 37.89 on rabbit food,

$ 2.01 on turtle food, and

$ 7.51 on snake food.

How much did she spend on pet food?

Estimate:

Solution: $91.06

About $90

53.21

97.16

48.05

1,254

729

6,211

8,999

440

1,179

�


Student Page


▶ Reviewing Algorithms:

SMALL-GROUP ACTIVITY

Column-Addition Method(Student Reference Book, pp. 13 and 35;

Math Masters, p. 415)

The column-addition method is similar to the traditional algorithm most adults know. It can become an alternate method for students who are still struggling with addition.

Demonstrate the method using examples like those on pages 13 and 35 of the Student Reference Book. In this method, each column of numbers is added separately, and in any order.

� If adding results in a single digit in each column, the sum has been found.

� If the sum in any column is a 2-digit number, it is renamed and part of it is added to the sum in the column on its left.

This adjustment serves the same purpose as “carrying” in the traditional algorithm.

Ask students to compare the examples of column addition on pages 13 and 35 of the Student Reference Book. Assign each small group one of the following problems to solve using the column addition method. Encourage students to use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Have students write about their method and explain the reasoning they used to solve the assigned problem. Volunteers share the group’s solution using the board or a transparency.

39 + 23 62 607 + 46 + 239 892 7,069 + 3,481 10,550

0.7 + 0.29 0.99 1.56 + 8.72 10.28 48.26 + 7.94 56.2

▶ Adding Whole Numbers PARTNER ACTIVITY

and Decimals(Math Journal 1, p. 33; Student Reference Book,

pp. 13, 14, and 35; Math Masters, p. 415)

Encourage students to estimate and solve the problems independently and then check each other’s work. Using the computation grid helps students line up digits and/or decimal points. Encourage students to try the methods described on pages 13, 14, and 35 of the Student Reference Book.

Ongoing Assessment: Journal

Page 33 �Problems 1 and 2Recognizing Student Achievement

Use journal page 33 to assess students’ ability to solve multidigit addition

problems. Students are making adequate progress if they correctly use the

partial-sums method to solve Problems 1 and 2.


NOTE Remind students always to say aloud,

or to themselves, the numbers that they are

adding when they use the partial-sums

method. For example: 500 + 200, not 5 + 2;

70 + 60, not 7 + 6.


STUDY LINK

2�2 Number Hunt

Name Date Time

Reminder: A means Do not use a calculator.

Use the numbers in the following table to answer thequestions below. You may not use a number more than once.

1. Circle two numbers whose sum is 832.

2. Make an X in the boxes containing three numbers whose sum is 57.

3. Make a check mark in the boxes containing two prime numbers whose sum is 42.

4. Make a star in the boxes containing two numbers whose sum is 658.

5. Make a triangle in the boxes containing two numbers whose sum is 105.7.Explain how you found the answer.

Solve Problems 6–9 using any method you want. Show your work in the space below.

6. 3,804 � 768 � 7. 2.83 � 1.57 �

8. 33 � 148 � 65 � 9. 1.055 � 0.863 � 1.9182464.44,572

10. 73 � 26 � 11. 727 � 519 �

12. 27 � 9 � 13. 4 �3�4� ∑ 8 R2320847

★

★X ✓ X

X

Sample answers:19 85.2 533 571

88.2 525 20 17.5

400 261 20.5 125

7 23 901 30

✓

Sample answer: Since the sum has a 7 inthe tenths place, look for numbers withtenths that add to 7: 85.2 � 20.5 � 105.7;and 88.2 � 17.5 � 105.7.

Practice

13–17

Math Masters, p. 36

Study Link Master

Math Boxes LESSON

2�2

Date Time

1. Round to the nearest tenth.

a. 45.52 �

b. 60.18 �

c. 123.45 �

d. 38.27 �

e. 56.199 � 56.2

38.3

123.5

60.2

45.5

6. Match.

2. Multiply.

a. 7 � 10 �

b. 7 � 60 �

c. 70 � 60 �

d. 8 � 10 �

e. 8 � 70 �

f. 80 � 70 � 5,600

560

80

4,200

420

70

4. Complete.

a. 354 � 300 � 50 �

b. 867 � 800 � � 7

c. 975 � � 70 � 5

d. 1,256 � 1,000 � � � 6

e. 6,704 � 6,000 � � 4 700

50200

900

60

43. The temperature at midnight was 25�F.

The wind chill temperature was 14�F.

How much warmer was the actual

temperature than the wind chill

temperature?

11�F

5. Complete.

a. A person 74 in. tall is ft in.

b. A person who runs a mile runs

ft.

c. A person who runs 1,760 yd runs

ft.

d. A person who grew �12

� ft over

the summer grew in. 6

5,280

5,280

26

46

15–17203

184 139

13

18

a. Straight angle

b. Obtuse angle

c. Right angle

d. Acute angle

<90°

>90°

90°

180°


Student Page

Lesson 2�2 89

▶ Sharing Results WHOLE-CLASSDISCUSSION

(Math Journal 1, pp. 32 and 33)

Bring the class together to share solutions. Some possible discussion questions include the following:

● What are some of the advantages or disadvantages of different methods for addition?

● When might a particular method be useful? When might it not be useful?

● How did students use their estimates on journal page 32? Did they make estimates for any subsequent problems?

Ask students to explain the reasoning they used to solve some of the problems on Math Journal 1, page 33.

2 Ongoing Learning & Practice

▶ Playing Addition Top-It (Decimal Version)

PARTNER ACTIVITY

(Student Reference Book, p. 333; Math Masters, p. 493)

Addition Top-It (Decimal Version) provides practice adding decimals, comparing numbers, and understanding place-value concepts. Direct students to play a decimal version of Addition Top-It, Student Reference Book, page 333. Use this variation:

� Each player draws 4 cards and forms 2 numbers that each has a whole-number portion and a decimal portion. Players should consider how to form their numbers to make the largest sum possible. Use counters or pennies to represent the decimal point.

� Each player finds the sum of the 2 numbers and then writes the sum in expanded form.

� Each player records his or her sum on Math Masters, page 493 to form a number sentence using >, <, or =.

� The player with the largest sum takes all of the cards.

▶ Math Boxes 2�2

INDEPENDENT ACTIVITY

(Math Journal 1, p. 34)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-4. The skill in Problem 6 previews Unit 3 content.

Writing/Reasoning Have students write a response for the following: Leroy rounded 56.199 to 60. Rosina said that he was incorrect. Do you agree or disagree with Rosina? Sample

answer: It depends on whether Leroy intended to round to the nearest 10 or the nearest whole number; 56.199 rounded to the nearest 10 is 60; 56.199 rounded to the nearest whole number is 56.

NOTE Remind students of the benefits of

making estimates prior to solving problems.

Estimation as an ongoing practice helps

students to become flexible with mental

computation and to check their answers for

reasonableness.


LESSON

2�2

Name Date Time

Modeling with Base-10 Blocks

Copyright

© W

right

Gro

up/M

cG

raw

-Hill

Example:

Work with a partner. Choose a problem below. Use the base-10 blocks to model the

problem. Have your partner solve the problem and record the answer using the partial-sums

method. Compare your model with your partner's solution. Reverse roles and continue until

all problems are solved.

1. 456

� 53

Add 100s 400Add 10s 100Add 1s 9

3. 271

� 653

Add 100s 800Add 10s 120Add 1s 4

100s

+

10s 1s

500 70 6+ +

100s 10s 1s

2 3 1

� 3 4 5

Add 100s 5 0 0

Add 10s 7 0

Add 1s 6

5 7 6

509

924

2. 764

� 208

Add 100s 900Add 10s 60Add 1s 12

4. 521

� 455

Add 100s 900Add 10s 70Add 1s 6

972

976

Math Masters, p. 37

Teaching Master

LESSON

2�2

Name Date Time

Place-Value Strategies

Use your favorite addition algorithm to solve the first problem in each column. Then use the

answer to the first problem in each column to help you solve the remaining problems.

10,0605,240

d. 7,401

� 2,699

d. 3,058

� 2,181

c. 7,401

� 2,689

c. 3,058

� 2,582

b. 7,401

� 2,669

b. 3,058

� 2,082

a. 7,401

� 2,679

a. 3,058

� 2,282

2. 7,401

� 2,659

1. 3,058

� 2,182

3. Explain the strategy you used to solve the problem sets above.

I identified which digit in the second numberchanged. Then I adjusted the sum of the original problem by that amount.

5,340 10,080

5,140 10,070

5,640 10,090

5,239 10,100

Math Masters, p. 38

Teaching Master


▶ Study Link 2�2

INDEPENDENT ACTIVITY

(Math Masters, p. 36)

Home Connection Students practice finding sums. Students can solve problems using any method they choose.

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Building Numbers with 15–30 Min

Base-10 Blocks(Math Masters, p. 37)

To explore place value and expanded notation using concrete models, have students use base-10 blocks to model numbers. After students complete Math Masters, page 37, discuss the relationship between expanded notation and base-10 block representations. Have students also share how they used the blocks to add the numbers.

ENRICHMENT PARTNER ACTIVITY

▶ Using Place Value to Solve 15–30 Min

Addition Problems(Math Masters, p. 38)

To apply students’ understanding of place value and addition algorithms, have them use the sum of one problem to help them find the solution of other problems. After students complete Problems 1 and 2 on Math Masters, page 38, discuss their strategies. Ask students to explain how these strategies might be useful when solving addition problems. Sample answer: I could find related addends that are easier to work with than those in the original problem and use what I know about place value to help me solve the problem.

ELL SUPPORT

SMALL-GROUPACTIVITY

▶ Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 142)

To provide language support for number notation, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the term expanded notation, draw pictures relating to the term, and write other related words. See the Differentiation Handbook for more information.


Documents

Addition of Whole Numbers and Decimals... Lesson 2 2 85 Addition of Whole Numbers and Decimals Objectives To review place-value concepts and the use of the partial-sums and column-addition