Paper-and-Pencil Addition Strategies€¦ · of systematic paper-and-pencil procedures— algorithms—that can be applied to any addition problem, including those with 3- and 4-digit

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288 Unit 4 Addition and Subtraction

Advance PreparationPlan to spend a total of two days on this lesson and three days on Lesson 4�9.

Teacher’s Reference Manual, Grades 1–3 pp. 102–105, 190

Paper-and-PencilAddition Strategies

Objectives To guide children as they develop paper-and-pencil

strategies for adding 2- and 3 -digit numbers; and to demonstrate

using estimation to check if answers are reasonable.

s

Key Concepts and Skills• Share solution strategies for finding the

sum of multidigit numbers mentally.

[Operations and Computation Goal 2]

• Use base-10 blocks to add two multidigit

numbers.


• Practice addition fact extensions.


• Estimate sums by changing the addends to

“close but easier” numbers.


Key ActivitiesChildren solve multidigit addition problems,

record their work with paper and pencil,

share solution strategies, and use ballpark

estimates to check whether their answers

are reasonable. Children practice adding

2-, 3 -, and 4-digit numbers.

Ongoing Assessment: Recognizing Student Achievement Use journal page 105. [Operations and Computation Goal 3]

Key Vocabularyballpark estimate

MaterialsMath Journal 1, p. 105

My Reference Book, pp. 92–94

Home Link 4�7

base-10 blocks: 18 cubes, 18 longs, and

6 flats (optional) � paper � Class Data Pad

(optional)

Playing the Fact Extension GameMy Reference Book, pp. 134 and 135

per partnership: 4 each of number

cards 0–9 (from the Everything Math

Deck, if available), 1 die, calculator �

1 sheet of paper per player

Children practice fact extensions.

Math Boxes 4�8Math Journal 1, p. 106

Children practice and maintain skills

through Math Box problems.

Home Link 4�8Math Masters, pp. 111 and 112

Children practice and maintain skills

through Home Link activities.

READINESS

Modeling Addition and Subtraction Properties of 10snumber grid � 1 penny or small object

per child

Children practice adding and subtracting

10s using a visual model.

ENRICHMENTFinding New Methods to Solve Multidigit Addition Problemsnumber line, number grid, and base-10 blocks

(optional) � large paper or poster board

Children find different methods to solve 2- or

3-digit addition problems.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 142

Children add the term ballpark estimate to

their Math Word Banks.

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

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Adjusting the Activity

Lesson 4�8 289

57 ∑ 6022 ∑ +20 80

For the problem 57 + 22, suggest changing

the first addend to 60 and the second addend

to 20. Because 60 + 20 = 80, any answers

that are not close to 80 are likely to be

incorrect.

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

Have children share and explain their answers. Explain to the class that they will use estimating strategies to find ballpark estimates—an answer that may not be exact but that is close enough. To support English language learners, discuss the meaning of ballpark.

� Discussing the Use of Ballpark WHOLE-CLASSDISCUSSION

Estimates to Check Answers(My Reference Book, pp. 92–94)

Remind the class that one way to decide whether an answer is reasonable is to make a ballpark estimate. One way to estimate a sum is to change the addends to close-but-easier numbers and then to add those numbers. To support English language learners, clarify the meaning of close-but-easier. You may want to demonstrate on a number line how some numbers are close to others but easier to work with than the others. To review estimation concepts, you may wish to read My Reference Book, pages 92–94 with your class.

Draw a “ballpark” on the board (a square oriented like a baseball field).

Figure out the answer and write it where home plate would be. As children

suggest possible ballpark estimates, the class should decide if the number is

near enough to be in the ballpark, in which case, the number would go inside the

diamond. If the number would not be in the ballpark, it would be written

outside the ballpark and appropriately far away.

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

ELL

ELL

ELL

Getting Started

Math Message Make an estimate for the answer to each problem. Be prepared to tell how you found your answer. 75 + 8; 57 + 22

Home Link 4�7 Follow-Up Review answers as necessary. Ask children to explain how they determined the nearest inch and their strategies for finding the total length of the path.

Mental Math and Reflexes Pose pairs of problems similar to the following:

30 + 40 = ? 70

? = 50 + 20 70

? = 60 + 30 90

30 + 46 = ? 76

? = 58 + 20 78

? = 60 + 37 97

124 + 30 = ? 154

268 + 50 = ? 318

1,210 + 40 = ? 1,250

● On Day 1 of this lesson, children should

complete the Math Message Follow-Up,

discuss the use of ballpark estimates to

check answers, and solve addition problems.

● On Day 2 of this lesson, children should

solve addition problems with two

multidigit numbers. Then have children

complete Part 2 activities.

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Adjusting the Activity

NOTE Although problems written in a

horizontal format probably encourage more

varied responses than problems written in a

vertical format, the vertical format is often

more efficient when children need to perform

more complex computations.

NOTE In Lesson 4-6, children discussed

mental arithmetic strategies for adding multidigit

numbers. In this lesson, children continue to

develop and share their solution strategies for

solving addition problems and keep a paper-

and-pencil record of their solutions.

The goal is to develop, over time, a number

of systematic paper-and-pencil procedures—

algorithms—that can be applied to any

addition problem, including those with 3- and

4-digit addends.

Unit

� Solving Addition Problems; SMALL-GROUP ACTIVITY

Keeping a Paper-and-Pencil RecordWrite problems like the following on the board, some in a horizontal format and some in a vertical format. Ask someone to suggest a unit and record it in a unit box. Be sure that the class number grid or individual number grids are available for the children to use.

29 + 37 66 76 + 23 99

52 + 29 81 163 + 56 219

26 + 74 100 219 + 352 571

Solve the first problem as a class. Have children share solution strategies. Write the numbers 29, 37, and 66 on the board. Help children discover that number families, just like fact families, have four related number sentences. For example, the following four number sentences relate 29, 37, and 66 by addition and subtraction:

29 + 37 = 66 37 + 29 = 66 66 - 37 = 29 66 - 29 = 37

Ask: What addition number sentences describe these numbers? 29 + 37 = 66; 37 + 29 = 66 What subtraction number sentences can you write using the three numbers? 66 – 37 = 29; 66 – 29 = 37

Have children work on the remaining problems together in small groups, recording their work with paper and pencil and checking whether each answer is reasonable by making a ballpark estimate. Children may also use the related subtraction number sentences to check their answers.

After groups have done two or three problems, bring the class together to share solution strategies. Record successful strategies. When most children seem to understand, let groups complete the rest of the problems. Then bring the class together for discussion.

Have children act out the problems with base-10 blocks, using longs

and cubes to represent each addend separately. Then have children combine

the longs and cubes into a single pile. If 10 or more cubes are in the pile,

children replace 10 cubes with one long (regrouping). Then they count the blocks

to find the sum.

If children are curious, show them how to reduce problems with 3-digit addends

to 2-digit addition. For example, with 219 + 352, add the hundreds first:

200 + 300 = 500. Record or remember this. That leaves 19 + 52. Find this

sum. 71 Add 500 to get the final answer. 571

It is often convenient to think of 3- and 4-digit addition problems as

dollars-and-cents problems. Example:

To add: 2,354 Think: $23.54

+ 4,667 + 46.67

Add the dollars: 23 Add the cents: 54

+ 46 + 67

60 110

9 11

69 121

Combine dollars and cents: $69 + $1.21 = $70.21.

So 2,354 + 4,667 = 7,021.

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

ELL

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Try This

327� 146

52� 28

Addition PracticeLESSON

4 � 8

Date Time

Write a number model to show the ballpark estimate. Solve the problem. Show your work in the workspaces.

Ballpark estimate:40 � 30 � 70





54� 79133 473

Add. In each problem, use the first sum to help you find the other two sums.

7. 17 � 8 �

17 � 8 � 25 �

17 � 8 � 25 � 12 �

9. 19 � 6 �

19 � 6 � 5 �

19 � 6 � 5 � 70 � 10030

25

6250

25 8. 15 15 15� 9 9 9

� 6 6� 223052

10. 24 24 24� 4 4 4

� 7 7� 35

2835

70

39� 2665

18� 4563 80

1. 2. 3.

4. 5.

6. Ballpark estimate:

��

24

115� 32147

Unit

Sample estimates

330 � 150 � 480

Math Journal 1, p. 105

Student Page

Lesson 4�8 291

Ongoing Assessment:Recognizing Student Achievement

Use journal page 105, Problems 1–3 to

assess children’s ability to make

reasonable estimates. Children are making

adequate progress if they are near the actual

answer in Problems 1–3. Some children may

find closer estimates.


Journal

Page 105 �Problems 1–3

Children usually use three major types of strategies:

Counting Up

My problem: 47 + 33 = ?

Start at 47. Count up 30 more: 47 57 67 77

Add on 3 more: + 3

The answer is 80: 80

Combining Groups (1s, 10s ...) Separately

My problem: 29 + 37 = ?

Add the tens: 20 + 30 = 50

Add the ones: 9 + 7 = 16

Put these together. 50 + 16. The answer is 66: 66

Adjusting and Compensating

My problem: 52 + 29 = ?

30 is close to 29, just 1 more: 30

52 plus 30 is 82: 52 + 30 = 82

Take away 1, because I added 30 instead of 29: -1

The answer is 81: 81

� Finding the Sum of Two PARTNER ACTIVITY

Multidigit Numbers(Math Journal 1, p. 105)

Ask partners to solve problems like the following:

63 + 27 90 26 + 31 57 54 + 82 136 123 + 38 161

Share solution strategies as a class. Model counting up, combining groups, and adjusting and compensating, if no one mentions them.

Have partners work together to solve the addition problems on journal page 105. Tell children to show their computations in the workspaces on the journal page.

Children should record their answers and write a number model for their ballpark estimate to check their work. Remember that ballpark estimates can vary depending on which close, but easier to add, numbers are used. In Problems 7–10, children can use the first sum to find the second sum, the second sum to find the third sum, and the third sum to find the fourth sum.

Circulate and listen to the interactions as children work on the problems, but let them figure out the answers with as little help from you as possible. Suggest that children try one of the procedures that you recorded on the board earlier.

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Math BoxesLESSON

4 � 8

Date Time

1. How much?

‰ÍÍÍÍÂÂÎÎÎ$ 25.78

§‡‡

3. 25 books. Bought 15 more. How many now? books

Fill in the diagram and write anumber model.

40

6. Draw a rectangle. Two sidesare 3 cm long and two sidesare 5 cm long.

2. The temperature was 73�F. Itgot 13�F colder. What is the temperature now? �F Fill in the diagram and write anumber model.

60

4. What time is it?

:

What time will it be in a half hour?

: 4510

1510

5. Draw a line segment 6 cm long. Underneath it, draw a line segment that is 2 cm longer.

Start EndChange

73 �13 ?73 � 13 � 60

Start EndChange

25 �15 ?25 � 15 � 40

12 12

3

4567

8

9

1011

116 –118

116–118

54

Math Journal 1, p. 106

Student Page

Adjusting the Activity For the Fact Extension Game, suggest that children may add more than

one zero.

AUDITORY � KINESTHETIC � TACTILE � VISUAL

Fact Extension GameMaterials ❑ number cards 0–9 (4 of each)

❑ 1 six-sided die❑ 1 calculator❑ 1 sheet of paper for each player

Players 2

Skill Finding sums of 2-digit numbers andmultiples of 10

Object of the game To have the higher total.

Directions

1. Shuffle the cards. Place the deck number-sidedown on the table.

2. Each player draws 2 cards from the deck andmakes the larger 2-digit number.

3. Players take turns rolling the die and makinganother 2-digit number by using the number on the die in the tens place and a zero in theones place.

4. Each player adds his or her 2 numbers and records the sum on a sheet of paper.

Games

My Reference Book, p. 134

Student Page

2 Ongoing Learning & Practice

� Playing the Fact PARTNER ACTIVITY

Extension Game(My Reference Book, pp. 134 and 135)

Children practice fact extension by playing the Fact Extension Game. See My Reference Book pages 134 and 135 for the rules of the game.

� Math Boxes 4�8 INDEPENDENTACTIVITY

(Math Journal 1, p. 106)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-6. The skills in Problems 5 and 6 preview Unit 5 content.

Writing/Reasoning Have children draw, write, or verbalize their answers to the following: Explain how you know that your answer for Problem 2 is correct. Sample

answers: I know when the temperature gets colder the number goes lower. I used my number grid and put my finger on 73. I went back to 63 and jumped back 3 more. I stopped at 60; I counted back on the thermometer.

� Home Link 4�8 INDEPENDENTACTIVITY

(Math Masters, pp. 111 and 112)

Home Connection Children add two multidigit numbers and record their strategies.

The Family Note on the Home Link explains paper-and-pencil strategies other than the traditional right-to-left method.

3 Differentiation Options

READINESS SMALL-GROUP ACTIVITY

� Modeling Addition and 15–30 Min

Subtraction Properties of 10sTo provide experience with adding and subtracting 10s using a visual model, have children use the number grid to model the pattern. Begin by posing problems with numbers that are multiples of 5 or 10.

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Name Date Time

Add.1. 40 � 36 � 2. 20 � 80 � 3. � 53 � 30

4. 60 � 60 � 5. � 50 � 48 6. � 70 � 20

Write a number model to show your ballpark estimate.

Add. Show your work in the workspaces.

Check your work. Sample estimates:

90981208310076

HOME LINK

4 �8 Addition Strategies continued



46 � 25 �


123 � 46 �

120 � 50 � 170

50 � 30 � 80

34� 59

30 � 60 � 908. Ballpark estimate:

17 � 68 �


56 � 27 �


318� 226

300 � 200 � 500

60 � 30 � 90

20 � 70 � 90

93

85

83

544

71

169

Practice Unit

Math Masters, p. 112

Home Link Master

Lesson 4�8 293

Name Date Time

Everyday Mathematics encourages children to use a variety of strategies to solve computationproblems. By doing so, children are developing a sense for numbers and operations ratherthan simply memorizing a series of steps.

We suggest that you give your child an opportunity to explore and choose additionstrategies that he or she feels comfortable using. At some point, you may want to share themethod that you know from your own school experience; please allow your child some timeto use his or her own methods before doing so.

Below are three examples of methods that your child might use to solve 2-digit additionproblems.

FamilyNote

HOME LINK

4 �8 Addition Strategies

Encourage your child to use a ballpark estimate as a way to check whether an answer to acomputation problem makes sense. For example, in 34 � 59, 34 is close to 30 and 59 isclose to 60. 30 � 60 � 90 is your ballpark estimate. “90 is close to my answer 93, so 93 is a reasonable answer.”

Please return the second page of this Home Link to school tomorrow.

Counting On

47 � 33 � ? “My problem”

47 57 67 77 “Start at 47. Count up 30 more.”

� 3 “Add on 3 more.”

80 “The answer is 80.”

Combining Groups (1s, 10s, ...) Separately

29 � 37 � ? “My problem”

20 � 30 � 50 “Add the tens.”

9 � 7 � 16 “Add the ones.”

66 “Put these together. The answer is 66.”

Adjusting and Compensating

52 � 29 � ? “My problem”

30 “30 is close to 29, just 1 more.”

52 � 30 � 82 “52 plus 30 is 82.”

� 1 “Take away 1, because I added 30 instead of 29.”

81 “The answer is 81.”

Math Masters, p. 111

Home Link MasterExample: 30 + 10 = 40

Using the number grid, children place a penny on 30. Children use the penny to help them add 10 or jump 10 spaces forward. Ask: What number are you on now? 40 Have children move a finger down the column from 30 to 40, pointing out that when adding ten, you go down one space on the number grid.

Example: 25 – 10 = 15

Again using the number grid, children lay a penny on 25. Children use the penny to help them subtract 10 or jump 10 spaces back. Ask: What number are you on now? 15

Have children move a finger up from 25 to 15 on the number grid. Point out that when subtracting 10 you go up one space on the number grid. Demonstrate that the digit in the tens place changes by 1 while the digit in the ones place stays the same when adding and subtracting 10.

Continue posing problems, working up to adding or subtracting 10 to or from any 2-digit number. For example, 17 + 10 = . 27 Then, add or subtract 100 or 1,000 to or from any number. For example, 23 + 100 = . 123

ENRICHMENT SMALL-GROUP ACTIVITY

� Finding New Methods 30+ Min

to Solve Multidigit Addition ProblemsTo further explore solving 2-digit addition problems, have children find different methods to solve 2- or 3-digit addition problems.

Small groups of children work together to pose a 2- or 3-digit addition problem. They find as many methods as they can to solve the problem. Possible manipulatives include the number grid, a number line, or base-10 blocks. They may also discover paper-and-pencil methods, such as counting up, combining groups, or adjusting and compensating. Invite children to create a poster of their methods to share with the class.

ELL SUPPORT SMALL-GROUPACTIVITY

� Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 142)

To provide language support for estimation, use the Word Bank Template found on Differentiation Handbook, page 142. Ask children to write the term ballpark estimate. Children describe the terms with words, pictures, and real-life examples. See the Differentiation Handbook for more information.

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Paper-and-Pencil Addition Strategies€¦ · of systematic paper-and-pencil procedures— algorithms—that can be applied to any addition problem, including those with 3- and 4-digit

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