244 Unit 4 Decimals and Their Uses

Teaching the Lesson materials

Key ActivitiesStudents review basic concepts and notation for decimals for tenths and hundredths.

Key Concepts and Skills• Read and write decimals through hundredths. [Number and Numeration Goal 1]• Model decimals through hundredths with base-10 blocks. [Number and Numeration Goal 1]• Name the fractional part of the ONE represented by a base-10 block.

[Number and Numeration Goal 2]• Rename fractions with 10 and 100 in the denominator as decimals.

[Number and Numeration Goal 5]

Key Vocabulary ONE • whole • unit • tenth • hundredth Ongoing Assessment: Informing Instruction See page 246.

Ongoing Assessment: Recognizing Student Achievement Use a Math Log orExit Slip. [Number and Numeration Goal 1]

Ongoing Learning & Practice materialsStudents play Baseball Multiplication to practice multiplication facts.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students play Base-10 Exchange toexplore the relationship among hundredths, tenths, and ones.

Students explore the concept of the whole, or the ONE.

� Teaching Master (Math Masters, p. 111)

� Game Master (Math Masters, p. 458)� per group: base-10 blocks (1 flat,

20 longs, 20 cubes); two 6-sided dice

ENRICHMENTREADINESS

3

� Math Journal 1, p. 81� Student Reference Book, pp. 231 and 232� Study Link Master (Math Masters, p. 110)� Game Master (Math Masters, p. 460) � per partnership: two 6-sided dice and

4 pennies; calculator or º, / Facts Table(optional)

2

� Math Journal 1, p. 80� Study Link 4 �1� Teaching Aid Masters (Math Masters,

p. 388 or 389; p. 426)� Transparency (Math Masters,

p. 426; optional)� base-10 blocks � slate

See Advance Preparation

1

Objective To review basic concepts and notation for decimals

through hundredths.

Technology Assessment Management System

Math Log or Exit SlipSee the iTLG.

Additional InformationAdvance Preparation Place a flat, a long, and a cube near the Math Message. Write flat,long, and cube on slips of paper and use them to label the base-10 blocks.

� Math Message Follow-Up(Math Masters, p. 426)

Have students share their answers. Review relationships amongthe base-10 blocks.

� To demonstrate that 1 f lat � 10 longs, show a copy or anoverhead transparency of Math Masters, page 426, and coverthe 10-by-10 square with longs.

� To demonstrate that 1 long � 10 cubes, cover one of the 10-strips with cubes.

� Explain that because 1 long � 10 cubes, 1 f lat must have 10 times as many cubes, or 100 cubes.

In this lesson students revisit the connection between decimalsand fractions, and the relationship of both to the ONE.

� UnderstandingFraction ConceptsRemind students that a fraction is always a fraction of something,such as �

12� of an apple, �

15� of the crayons in a box, or �

34� of a yard.

We refer to this “something” as the ONE, or the whole; for measures, “something” refers to the unit. To support Englishlanguage learners, write the word whole on the board and explainits meaning in this context. Clarify the difference between wholeand hole.

WHOLE-CLASS

ACTIVITY

WHOLE-CLASS

ACTIVITY

1 Teaching the Lesson

Name Date Time

Base-10 Grids

Math Masters, p. 426

Teaching Aid Master

Lesson 4�2 245

Getting Started

Mental Math and Reflexes Practice counting decimal amounts. Use base-10 blocks on the overhead to model the following:

Use longs to count by tenths, starting from zero: “One-tenth, two-tenths, three-tenths, ...” Stop every so often to askhow to write the decimal. Stop after nine-tenths and ask what comes next. ten-tenths, or oneCount by hundredths from zero. Stop at nine-hundredths and ask what comes next. ten-hundredths, or one-tenthUse cubes to count by tenths from other numbers (0.8, 1.5, and so on), again stopping occasionally to write the numbers or ask what comes next. Focus on counting through landmarks like 1.0 or 2.0.Count by hundredths from other numbers.Count backward by tenths and hundredths, focusing on counting through landmarks.

Math Message Write the answers.

1 flat � longs 1 flat � cubes

1 long � cubes

Study Link 4�1 Follow-Up Have partners compare answers. Note that foreach problem, some answers depend on previousanswers. On the back of the page have students usewords to write the numbers they found in Problems 1–3.

10010

10

246 Unit 4 Decimals and Their Uses

Links to the Future

Illustrate the idea of the ONE with the following examples:

� Stuart drank three-fourths of a carton of juice. In this situation,the carton of juice is the ONE, or whole.

� Mary had 5 dollars. She spent half of it on baseball cards. Inthis situation, Mary’s 5 dollars is the ONE, or whole.

Ask the following questions:● If we think of a flat as the ONE, what fraction of a flat is a

long? �110�, because 10 longs � 1 flat (Write 0.1 on the board.

Remind students that this is another way of writing �110�; 0.1 is

read as “one-tenth.” Next to 0.1, write “one-tenth” and “�110�.”)

● If we think of a flat as the ONE, what fraction of a flat is a cube? �1

100�, because 100 cubes � 1 flat (Write 0.01 on the board.

Remind students that this is another way of writing �1100�; 0.01 is

read as “one-hundredth.” Next to 0.01, write “one-hundredth” and “�1

100�.”)

In Unit 7 of Fourth Grade Everyday Mathematics students will continue toexplore the whole as they use pattern blocks and counters to find the ONE forgiven fractions of sets and regions.

� Modeling Decimals with Base-10 Blocks(Math Masters, p. 426)

Use the following steps to review decimal notation for tenths andhundredths.

Modeling fractions with base-10 blocksStudents use base-10 blocks to model numbers containing tenthsand hundredths on copies of Math Masters, page 426 while you doso on the transparency.

Ask: If the flat is ONE, then what are

● 3 longs? �130� ● 4 longs and 2 cubes? �1

4020�

● 5 longs? �150�, or �

12� ● 2 flats, 3 longs, and 4 cubes? 2�1

3040�

● 13 cubes? �11030� ● 1 flat and 6 cubes? 1�1

600�

Ongoing Assessment: Informing InstructionWhen base-10 blocks were used in Unit 2 to model whole numbers, the cubewas the ONE, the long was 10, the flat was 100, and the big cube was 1,000.Watch for students who may not have made the transition to using base-10blocks to model fractions with the flat as the new ONE. Draw a sketch of a flaton the board and label it ONE. Throughout the lesson, draw and label sketchesto support students’ transition to the new ONE.

WHOLE-CLASS

ACTIVITY

0.1 one-tenth �110�

0.01 one-hundredth �1100�

Adjusting the Activity

Adjusting the Activity

Representing numbers with base-10 blocks Have students use as few base-10 blocks as possible to shownumbers. If base-10 blocks are not available, students can drawthe blocks to record their answers. See their symbols on journalpage 80. Suggestions:

� �120� 2 longs � 2�1

10� 2 flats and 1 long

� �170� 7 longs � 3�1

2040� 3 flats, 2 longs, and 4 cubes

� �1300� 3 cubes � 5�1

30� 5 flats and 3 longs

� �13050� 3 longs and 5 cubes

Reviewing decimal notation Have students write numbers in decimal notation. Suggestions:

� �130� 0.3 � �1

400� 0.04 � �1

60� 0.6 � 2�1

80� 2.8

� �12070� 0.27 � 7�1

5090� 7.59 � �1

40� 0.4 � 3�1

200� 3.02

Compare the decimals for �140� and �1

400�. Emphasize the need for

the second 0 in 0.04 to name �1400�. Without this zero, the value of

the decimal would be �140�.

Have students think of tenths and hundredths in terms of dollars-and-cents notation when asked to compare decimals such as 0.4 and 0.04. Indollars-and-cents notation, the first digit to the right of the decimal point standsfor dimes, or tenths of a dollar, and the second digit stands for pennies, orhundredths of a dollar.

• How many dimes are in $1.00? 10 What fraction of a dollar is 1 dime? �110�

How would you write 4 dimes in dollars-and-cents notation? $0.40

• How many pennies are in $1.00? 100 What fraction of a dollar is 1 penny? �1100�

How would you write 4 pennies in dollars-and-cents notation? $0.04

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

Reviewing reading decimals for tenths and hundredths Remind students that decimals are read the same way as their fraction equivalents: 0.3 is read as 3 tenths, 0.27 as27 hundredths, and 0.04 as 4 hundredths.

Students should read numbers like 4.78 as four and seventy-eighthundredths rather than four point seven eight. While the secondreading is correct and is commonly used, especially for longer decimals, students can benefit from the more formal reading atthis stage in their learning.

ELL

Have students adjust their voice to be louder or lower to emphasize the -th in the decimal names.

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

Lesson 4�2 247

248 Unit 4 Decimals and Their Uses

81

Math Boxes LESSON

4 � 2

Date Time

2. Solve mentally or with a paper-and-pencilalgorithm.

a. 391 b. 983� 467 � 494

0012345

1 2 3 4 5 6 7 8 9 10 11 12Number of Blocks from School

How Many Blocks from School?

Num

ber o

f Stu

dent

s

3. If 1 inch on a map represents 40 miles,then

a. 2 in. represent mi.

b. 4 in. represent mi.

c. 5 in. represent mi.

d. 2 �12� in. represent mi.

e. 1�34� in. represent mi.70

10020016080

1. a. What is the maximum number of blocksany student lives from school?

b. What is the minimum number of blocks?

c. What is the mode?

d. What is the median number of blocks?

3

31

12

73

10–15 145

17

858 489

4. Write as dollars and cents.

a. 20 dimes � $ .

b. 20 nickels � $ .

c. 20 quarters � $ .

d. 10 quarters and 7 dimes �

$ . 203

005001002

5. Solve mentally.

a. 5 � 3 �

b. 5 � 30 �

c. � 5 � 9

d. � 50 � 9

e. 6 � 7 �

f. 60 � 70 � 4,20042

45045

15015

Math Journal 1, p. 81

Student Page

5. Write 53 thousandths in decimal notation. 0.053Try This

80

Tenths and HundredthsLESSON

4 � 2

Date Time

27 28Base-10 Block Symbol Value

Flat 1

Long �110�, or 0.1

Cube �1100�, or 0.01

1. Complete the table.

�190�

1.36

Base-10 Blocks Fraction Notation Decimal Notation

0.2�120�

�1900� 0.09

0.91�1

3060�

2. Write each number in decimal notation.

Example: �130� �

a. �140� � b. �1

7010� � c. 32�1

600� �

3. Draw base-10 blocks to show each number. Draw as few blocks as possible.

Example: 0.3

a. 0.43 b. 2.16 c. 0.07

4. Write each of the following in decimal notation.

a. 8 tenths b. 82 hundredths c. 38 and 5 tenths 38.50.820.8

32.060.710.40.3

�

Math Journal 1, p. 80

Student Page

� Practicing with Tenths and Hundredths(Math Journal 1, p. 80)

Students solve problems involving the representation of numberswith base-10 blocks, fractions, and decimals.

Ongoing Assessment:Recognizing Student AchievementUse a Math Log or an Exit Slip (Math Masters, page 388 or 389) toassess students’ understanding of the values of decimal digits. Ask

students to explain the difference between 0.9 and 0.09 on journal page 80,Problem 1. Students are making adequate progress if their responses make reference to the value of the 9 in 0.9 9 tenths and the value of the 9 in 0.09 9 hundredths. Some students may support their explanations by referring to thepictures on journal page 80 or by stating the decimals in terms of money amounts.

[Number and Numeration Goal 1]

� Playing Baseball Multiplication(Student Reference Book, pp. 231 and 232; Math Masters, p. 460)

Students play Baseball Multiplication to develop automaticity withmultiplication facts. See Lesson 3-3 for additional information.

� Math Boxes 4�2(Math Journal 1, p. 81)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-4. The skill in Problem 5previews Unit 5 content.

Writing/Reasoning Have students write a response to thefollowing: Explain the strategy you used in Problem 4d toconvert 10 quarters and 7 dimes into dollars and cents.

Sample answer: 10 � 25 cents � 250 cents, and 7 � 10 cents � 70cents. Total: 250 cents � 70 cents � 320 cents, or 3 dollars and20 cents.

� Study Link 4�2(Math Masters, p. 110)

Home Connection Students collect examples of decimals.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

Math Log or Exit Slip

INDEPENDENT

ACTIVITY

� Playing Base-10 Exchange(Math Masters, p. 458)

To explore the relationship among hundredths, tenths, and onesusing a concrete model, have students play Base-10 Exchange. Theobject of the game is to be the first player to make an exchange fora f lat (1).

1. The bank starts with 1 f lat, 20 longs, and 20 cubes.

2. Players take turns. A player rolls two dice, announces thetotal, takes that number of cubes (hundredths) from the bank,and places them on the mat.

3. Whenever possible, a player exchanges 10 cubes (hundredths)for a long (tenth) from the bank.

4. The first player to exchange 10 longs (tenths) for 1 f lat (1) wins.

Have students describe the exchanges they make. For example, “I have 12 cubes, so I exchange 10 of them for 1 long. Now I have1 long and 2 cubes.”

� Exploring the ONE(Math Masters, p. 111)

To apply students’ understanding of the whole or ONE using aconcrete model, have them identify the value of base-10 blockswhen different combinations are designated as ONE. They are also asked to find the whole, or ONE, when a fraction is given.

Have students review their answers to Problem 1. Ask: Whatwould �1

100� look like? Sample answer: It would be very small. You

would need to cut a cube into 10 equal pieces. One piece is �1100�.

Ask: What would �1,0100� look like? Sample answer: Cut a cube into

100 equal parts. Each part is �1,0100�.

5–15 Min

INDEPENDENT

ACTIVITYENRICHMENT

5–15 Min

PARTNER

ACTIVITYREADINESS

3 Differentiation Options STUDY LINK

4�2 Decimals All Around

26

Name Date Time

Find examples of decimals in newspapers, in magazines, in books, or on food packages. Ask people in your family for examples.

Write your numbers below or, if an adult says you may, cut them out and tape them on thispage. Be sure to write what the numbers mean. For example, “The body temperature of ahibernating dormouse may go down to 35.6°F.”

Answers vary.

Practice

Write true or false.

1. 286 � 286 � 462 2. 907 � 709 � 200

3. 641 � 359 � 359 � 641 4. 2,345 � 198 � 2,969 � 822 truetruefalsefalse

Math Masters, p. 110

Study Link Master

LESSON

4�2

Name Date Time

The ONE

Use base-10 blocks to help you solve the following problems.

1. If is ONE, then what is ? What is ?

2. If is ONE, then what is ? What is ?

3. If is ONE, then what is ? What is ?

4. If is �1100�, then what is the ONE?

5. If is �110�, then what is the ONE?

6. If is �110�, then what is the ONE?

What is �1100�?

7. Explain how you solved Problem 6.

is ONE, then of 150 cubes is 1.5 cubes.is the ONE, and if fifteen longs or 150 cubesthen 10 times this (150 cubes or 15 longs) Sample answer: If 1 long and 5 cubes is ,

1.5 or 1�12� cubes

15 longs

30 longs or 3 flats

500 cubes or 5 flats

2

10

�210� �2

100�

�510�

�110�

�110�

�1100�

Math Masters, p. 111

Teaching Master

Lesson 4�2 249