Addition of Whole Numbers and Decimals



Objectives To review place-value concepts and the use of the partial-sums and column-addition methods.

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ePresentations

eToolkit

Algorithms Practice

EM Facts Workshop Game™

Teaching the Lesson 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.   [Operations and Computation Goal 6]

Family Letters

Assessment Management

Common Core State Standards

Ongoing Learning & Practice 1 2 4 3

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.

Key Activities

Math Boxes 2 2

Students review place-value concepts and write numbers in expanded notation. They review addition of whole numbers and decimals with the partial-sums and columnaddition methods.

Math Journal 1, p. 34 Students practice and maintain skills through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use journal page 33. 



Study Link 2 2 

Math Masters, p. 36 Students practice and maintain skills through Study Link activities.

[Operations and Computation Goal 1]

Curriculum Focal Points

Interactive Teacher’s Lesson Guide

Differentiation Options READINESS

Building Numbers with Base-10 Blocks Math Masters, p. 37 base -10 blocks Students use base -10 blocks to explore the partial-sums method of addition. ENRICHMENT

Using Place Value to Solve Addition Problems Math 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.

Key Vocabulary place  value  digit  algorithm  partialsums method  place value  expanded notation  column-addition method

Materials Math Journal 1, pp. 32 and 33 Student Reference Book, pp. 13, 14, 28–30, and 35 Study Link 21 Math Masters, p. 415 slate

Advance Preparation Plan 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

Lesson 2 2 

85

Mathematical Practices

SMP1, SMP2, SMP3, SMP6, SMP7

Content Standards

Getting Started

5.NBT.1, 5.NBT.3a, 5.NBT.3b, 5.NBT.7

Mental Math and Reflexes

Math Message

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 1 + 1 + 1 9 ∗ (_ 6 ∗ (_ as 7 ∗ (_ 10 ) 100 ) 1,000 ). Encourage students to write the decimal numbers in fraction notation. Sample answers are given. 1 35 30 + 5 241 200 + 40 + 1 0.109 1 ∗ (_ 10 ) + 1 _ 1 1 1 1 _ _ _ _ ∗ 9 ( 1,000 ) 0.35 3 ∗ ( 10 ) + 5 ∗ ( 100 ) 0.241 2 ∗ ( 10 ) + 4 ∗ ( 100 ) + 1 1 _ ∗ 1 ( ) 0.708 7 ∗ (_ 52 50 + 2 1,000 10 ) + 1 _ ∗ 8 ( 1,000 ) 1 1 _ 162 100 + 60 + 2 0.52 5 ∗ (_ 10 ) + 2 ∗ ( 100 ) 1 1 1 _ _ 0.084 8 ∗ (_ 0.467 4 ∗ ( 10 ) + 6 ∗ ( 100 ) + 100 ) + 1 _ 1 _ ∗ 4 ( 1,000 ) 7 ∗ ( 1,000 ) 7,904 7,000 + 900 + 4

Use 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 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

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

(Student Reference Book, pp. 28–30)

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.

● On

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.

Adjusting the Activity Student Page

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.

Decimals and Percents You use facts about the place-value chart each time you make trades using base-10 blocks. Suppose that a flat is worth 1. Then a long is 1 1 , or 0.1; and a cube is worth _ , or 0.01 worth _ 10 100

For this example: A flat

You can trade one long for ten cubes because 1 1 one _ equals ten _ s. 10 100

is worth 1.

A U D I T O R Y

1 A long is worth _ or 0.1. 10

You can trade ten longs for one flat because 1 ten _ s equals one 1. 10

A cube is worth or 0.01.

You can trade ten cubes for one long because 1 1 ten _ s equals one _ . 100 10

1 one _ 10

1 =_ of 1 10

1 of 100 =_ 10

1 one _ 100

1 1 =_ of _ 10 10

one 1

1 of 10 =_ 10

1 1 1 =_ of _ one _ 1,000 10 100

a. 20,006.8 b. 0.02 c. 34.502



V I S U A L

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

2. Using the digits 9, 3, and 5, what is a. the smallest decimal that you can write?

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

b. the largest decimal less than 1 that you can write? c. the decimal closest to 0.5 that you can write?

Student Reference Book, p. 30 025_054_EMCS_S_SRB_G5_DEC_576515.indd 30

Unit 2

T A C T I L E

 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.

Check your answers on page 434.

86



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

one 10

1. What is the value of the digit 2 in each of these numbers?

K I N E S T H E T I C

100

Left to Right in the Place-Value Chart Study the place-value chart below. Look at the numbers that name the places. As you move from left to right along the chart, 1 each number is _ as large as the number to its left. 10

1 one 100 = _ of 1,000 10



1 _ ,

3/8/11 4:57 PM

Estimation and Computation

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. After students have checked their answers to Problem 2, have them write the decimals using number names. 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, read the number correctly, and write it using number names. 200,068—no decimal point; 0.2—decimal point in the wrong position; 34.052 reverses the tenths and hundredths.

▶ Reviewing Algorithms:

WHOLE-CLASS DISCUSSION

Partial-Sums Method

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.

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

(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.

Expanded form may also be written with fractions instead of decimals. For example: 34.15 = 3 ∗ 10 + 1 1 _ 4 ∗ 1 + 1 ∗ (_ 10 ) + 5 ∗ ( 100 ).

NOTE Display a poster showing the number in standard form, expanded form, and using number names to provide students with readily accessible examples.

Language Arts Link The word algorithm is used to name a step-bystep 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.

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.

Lesson 2 2 

87

Student Page Date

▶ Reviewing Algorithms:

Time

LESSON

2 2 䉬

Adding with Partial Sums

Column-Addition Method

Write the following numbers in expanded notation. 1.

432:

2.

56.23

400  30  2 50  6  0.2  0.03

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

400

Estimate:

Estimate:

700

4.

Estimate:

Estimate:

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.

100

10.31 32.04  59.61

214  475.2

600.0 80.0 9.0 0.2 689.2 5.

90.00 11.00 0.90 0.06 101.96

60

6.

Estimate:

28.765  31.036

47.84  21.023

50.000 9.000 0.700 0.090 0.011 59.801

60.000 8.000 0.800 0.060 0.003 68.863

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

70

 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.

Math Journal 1, p. 32

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.

Student Page Date

Time

LESSON

2 2 䉬

Methods for Addition

夹

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.

8,999

1,179 440  5,312  3,687

6,211 729 5. 475  139  115  1,254  217  192  309  536 6. 48.05 7. 38.47  9.58  97.16  32.06  65.1 8. 53.21 9. 43.46  7.1  2.65  4.

10.

3,416  2,795 

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:

(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.

325.022 134.527 400.000 50.000 9.000 0.500 0.040 0.009 459.549 3.

SMALL-GROUP ACTIVITY

About $90 $91.06

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: Recognizing Student Achievement



Journal Page 33 Problems 1 and 2

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. [Operations and Computation Goal 1]

Math Journal 1, p. 33

88

Unit 2

Estimation and Computation

Student Page

▶ Sharing Results

WHOLE-CLASS DISCUSSION

(Math Journal 1, pp. 32 and 33)

Date

Time

LESSON

Math Boxes

22 䉬

1.

Round to the nearest tenth. a.

45.52 

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

b.

60.18 

c.

123.45 

d.

38.27 

●

e.

56.199 

●

●

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

2.

45.5 60.2 123.5 38.3 56.2 46

3.

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

The temperature at midnight was 25F. The wind chill temperature was 14F. How much warmer was the actual temperature than the wind chill temperature?

11F

b.

7 ⴱ 60 

c.

70 ⴱ 60 

d.

8 ⴱ 10 

e.

8 ⴱ 70 

f.

80 ⴱ 70 

5.

Complete.

a.

354  300  50 

b.

867  800 

c.

975 

60

4 7

900  70  5 1,256  1,000  200  50  6 6,704  6,000  700  4

A person 74 in. tall is

b.

A person who runs a mile runs

5,280

ft

2

in.

Match. a.

Straight angle

90°

c.

Right angle

90°

d.

Acute angle

180°

ft.

A person who runs 1,760 yd runs ft.

A person who grew the summer grew

1  2

ft over

6

in.

184

139

Math Journal 1, p. 34

PARTNER ACTIVITY

(Decimal Version)

NOTE Remind students of the benefits of

(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:

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.

 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.

Study Link Master Name

 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 >,