Paper-and-Pencil
Addition Strategies
Objectives To guide children as they develop paper-and-pencil
strategies
for adding 2- and 3-digit numbers; and to demonstrate
s
using estimation to check if answers are reasonable.
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Teaching the Lesson
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. [Operations and Computation Goal 2]
• Practice addition fact extensions. [Operations and Computation Goal 2]
• Estimate sums by changing the addends to
“close but easier” numbers. [Operations and Computation Goal 3]
Key Activities
Children 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.
Family
Letters
Assessment
Management
Ongoing Learning & Practice
1 2
4 3
Playing the Fact Extension Game
My 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 8
Math Journal 1, p. 106
Children practice and maintain skills
through Math Box problems.
Home Link 4 8
Math Masters, pp. 111 and 112
Children practice and maintain skills
through Home Link activities.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 105. [Operations and Computation Goal 3]
Key Vocabulary
ballpark estimate
Materials
Math Journal 1, p. 105
My Reference Book, pp. 92–94
Home Link 47
base-10 blocks: 18 cubes, 18 longs, and
6 flats (optional) paper Class Data Pad
(optional)
Advance Preparation
Plan to spend a total of two days on this lesson and three days on Lesson 49.
Teacher’s Reference Manual, Grades 1–3 pp. 102–105, 190
288
Unit 4
Addition and Subtraction
Common
Core State
Standards
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Modeling Addition and Subtraction
Properties of 10s
number grid 1 penny or small object
per child
Children practice adding and subtracting
10s using a visual model.
ENRICHMENT
Finding New Methods to Solve Multidigit
Addition Problems
number 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 Bank
Differentiation Handbook, p. 142
Children add the term ballpark estimate to
their Math Word Banks.
Mathematical Practices
SMP1, SMP3, SMP5, SMP6, SMP7, SMP8
Content Standards
Getting Started
2.NBT.5, 2.NBT.6, 2.NBT.7, 2.NBT.8, 2.NBT.9
Mental Math and Reflexes
Math Message
Pose problems similar to the following:
Make an estimate for the answer to each problem.
Be prepared to tell how you found your answer.
75 + 8; 57 + 22
30 + 40 = ? 70
? = 50 + 20 70
? = 60 + 30 90
30 + 46 = ? 76
? = 58 + 20 78
? = 60 + 37 97
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.
124 + 30 = ? 154
268 + 50 = ? 318
1,210 + 40 = ? 1,250
1 Teaching the Lesson
Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
ELL
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
Estimates to Check Answers
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.
● On
WHOLE-CLASS
DISCUSSION
ELL
(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.
Adjusting the Activity
ELL
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
57 ∑ 60
22 ∑ +20
80
T A C T I L E
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.
V I S U A L
Lesson 4 8
289
Solving Addition Problems;
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.
SMALL-GROUP
ACTIVITY
Keeping a Paper-and-Pencil Record
Write 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
Unit
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
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 paperand-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.
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.
Adjusting the Activity
ELL
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:
+ 46
60
9
69
Combine dollars and cents: $69 + $1.21 = $70.21.
So 2,354 + 4,667 = 7,021.
A U D I T O R Y
290
Unit 4 Addition and Subtraction
K I N E S T H E T I C
54
+ 67
110
11
121
T A C T I L E
V I S U A L
Student Page
Date
Children usually use three major types of strategies:
Time
LESSON
Addition Practice
48
Counting Up
Write a number model to show the ballpark estimate.
Solve the problem. Show your work in the workspaces.
47 + 33 = ?
My problem:
Sample estimates
The answer is 80:
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:
7. 17 + 8 =
25
8.
50
17 + 8 + 25 + 12 =
9. 19 + 6 =
10.
30
19 + 6 + 5 + 70 =
100
15
+9
24
62
25
19 + 6 + 5 =
30
24
+4
28
15
9
+6
30
24
4
+7
15
9
6
+ 22
52
24
4
7
+ 35
35
70
Math Journal 1, p. 105
52 + 30 = 82
Take away 1, because I added 30 instead of 29:
330 + 150 = 480
473
147
17 + 8 + 25 =
52 + 29 = ?
52 plus 30 is 82:
6. Ballpark estimate:
Add. In each problem, use the first sum to help you find the other two sums.
66
30 is close to 29, just 1 more:
Try This
327
+ 146
115
+ 32
133
Adjusting and Compensating
My problem:
120 + 30 = 150
54
+ 79
50 + 30 = 80
80
5. Ballpark estimate:
50 + 80 = 130
3. Ballpark estimate:
52
+ 28
63
4. Ballpark estimate:
Combining Groups (1s, 10s ...) Separately
20 + 50 = 70
18
+ 45
65
80
2. Ballpark estimate:
40 + 30 = 70
39
+ 26
+3
Add on 3 more:
1. Ballpark estimate:
Start at 47. Count up 30 more: 47 57 67 77
Unit
EM3cuG2MJ1_U04_80_111.indd 105
12/22/10 12:01 PM
-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 and the second sum to find the
third sum.
Ongoing Assessment:
Recognizing
Journal
Student
Page 105 Achievement Problems 1–3
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
answers in Problems 1–3. Some children
may find closer estimates.
[Operations and Computation Goal 3]
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.
Lesson 4 8
291
Student Page
Games
2 Ongoing Learning & Practice
Fact Extension Game
Materials ❑
❑
❑
❑
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 and
multiples of 10
Playing the Fact
Extension Game
PARTNER
ACTIVITY
(My Reference Book, pp. 134 and 135)
Object of the game To have the higher total.
Children practice fact extension by playing the Fact Extension
Game. See My Reference Book pages 134 and 135 for the rules of
the game.
Directions
1. Shuffle the cards. Place the deck number-side
down on the table.
2. Each player draws 2 cards from the deck and
makes the larger 2-digit number.
Math Boxes 4 8
3. Players take turns rolling the die and making
another 2-digit number by using the number
on the die in the tens place and a zero in the
ones place.
(Math Journal 1, p. 106)
4. Each player adds his or her 2 numbers
and records the sum on a sheet of paper.
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.
My Reference Book, p. 134
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.
Adjusting the Activity
For the Fact Extension Game,
suggest that children may add more than
one zero.
AUDITORY
KINESTHETIC
TACTILE
INDEPENDENT
ACTIVITY
VISUAL
Home Link 4 8
INDEPENDENT
ACTIVITY
(Math Masters, pp. 111 and 112)
Home Connection Children add two multidigit numbers
and record their strategies.
Student Page
Date
Time
LESSON
4 8
䉬
Math Boxes
2. The temperature was 73⬚F. It
1. How much?
got 13⬚F colder. What is the
temperature now? 60 ⬚F
Fill in the diagram and write a
number model.
Change
Start
End
‡ ‡ §
‰ÍÍÍÍ
ÂÂÎÎÎ
$ 25.78
73 ⫺13
?
73 ⫺ 13 ⫽ 60
3. 25 books. Bought 15 more.
How many now?
40
4. What time is it?
books
10
Fill in the diagram and write a
number model.
Change
Start
End
25 ⫹15
?
25 ⫹ 15 ⫽ 40
5. Draw a line segment
116 –118
:
15
What time
will it be in
a half hour?
10
:
11
10
12
2
3
8
4
7
6
5
116–118
6. Draw a rectangle. Two sides
6 cm long. Underneath it,
draw a line segment that is
2 cm longer.
are 3 cm long and two sides
are 5 cm long.
54
Math Journal 1, p. 106
292
Unit 4 Addition and Subtraction
3 Differentiation Options
1
9
45
The Family Note on the Home Link explains paper-andpencil strategies other than the traditional right-to-left
method.
READINESS
Modeling Addition and
SMALL-GROUP
ACTIVITY
15–30 Min
Subtraction Properties of 10s
To 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.
Home Link Master
Example: 30 + 10 =
Name
40
Date
4 8
䉬
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 =
Everyday Mathematics encourages children to use a variety of strategies to solve computation
problems. By doing so, children are developing a sense for numbers and operations rather
than simply memorizing a series of steps.
Family
Note
We suggest that you give your child an opportunity to explore and choose addition
strategies that he or she feels comfortable using. At some point, you may want to share the
method that you know from your own school experience; please allow your child some time
to use his or her own methods before doing so.
Below are three examples of methods that your child might use to solve 2-digit addition
problems.
Counting On
15
47 ⫹ 33 ⫽ ?
“My problem”
47 57 67 77
“Start at 47. Count up 30 more.”
⫹3
“Add on 3 more.”
80
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
“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
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
Finding New Methods
Time
Addition Strategies
HOME LINK
“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.”
Encourage your child to use a ballpark estimate as a way to check whether an answer to a
computation problem makes sense. For example, in 34 ⫹ 59, 34 is close to 30 and 59 is
close 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.
Math Masters, p. 111
SMALL-GROUP
ACTIVITY
30+ Min
to Solve Multidigit
Addition Problems
To 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-andpencil 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.
Home Link Master
Name
Date
HOME LINK
4 8
䉬
Addition Strategies
Building a Math Word Bank
SMALL-GROUP
ACTIVITY
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.
continued
Unit
Practice
Add.
1. 40 ⫹ 36 ⫽
4. 60 ⫹ 60 ⫽
ELL SUPPORT
Time
76
120
100
3.
⫽ 50 ⫹ 48
6.
2. 20 ⫹ 80 ⫽
5.
98
83
90
⫽ 53 ⫹ 30
⫽ 70 ⫹ 20
Write a number model to show your ballpark estimate.
Add. Show your work in the workspaces.
Check your work.
Sample estimates:
7. Ballpark estimate:
30 ⫹ 60 ⫽ 90
8. Ballpark estimate:
20 ⫹ 70 ⫽ 90
17 ⫹ 68 ⫽
34
⫹ 59
85
93
9. Ballpark estimate:
50 ⫹ 30 ⫽ 80
46 ⫹ 25 ⫽
71
11. Ballpark estimate:
120 ⫹ 50 ⫽ 170
123 ⫹ 46 ⫽
169
10. Ballpark estimate:
60 ⫹ 30 ⫽ 90
56 ⫹ 27 ⫽
83
12. Ballpark estimate:
300 ⫹ 200 ⫽ 500
318
⫹ 226
544
Math Masters, p. 112
Lesson 4 8
293