Reading and Writing Small Numbers

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108 Unit 2 Operations with Whole Numbers and Decimals

Reading and Writing Small Numbers

Objective

To read and write small numbers in standard and expanded notations.

e

Advance Preparation

Allow three days for Lessons 21 and 22. Make at least one copy of Math Masters, page 410 per student.

Teacher’s Reference Manual, Grades 4–6 pp. 94–98

Key Concepts and Skills

• Apply place-value concepts to read, write, and interpret numbers less than 1.

[Number and Numeration Goal 1]

• Convert between standard and expanded notations.

• Apply extended facts and order of operations to express the value of digits in a number.

[Operations and Computation Goal 2]

Key Activities

Students read and write numbers to thousandths in standard notation and expanded notation. They also convert between these notations.

Key Vocabulary

standard notation expanded notation Materials

Math Journal 1, pp. 48–50 Study Link 21

Math Masters, p. 410

transparency of Math Masters, p. 410

Playing High-Number Toss (Decimal Version)

Student Reference Book, p. 324 Math Masters, p. 455

per partnership: 4 each of number cards 0–9 (from the Everything Math Deck, if available), calculator (optional) Students practice reading, writing, and comparing numbers through thousandths.

Ongoing Assessment:

Recognizing Student Achievement Use Math Masters, p. 455.

Math Boxes 22 Math Journal 1, p. 51

Students practice and maintain skills through Math Box problems.

Study Link 22

Math Masters, pp. 44 and 45 Students practice and maintain skills through Study Link activities.

READINESS

Modeling and Comparing Decimals Math Masters, pp. 46 and 411–413 scissors

Students use base-10 grids to model and compare decimals through thousandths.

ENRICHMENT

Decimals between Decimals Students explore the infinite number of decimals between any two given decimal numbers.

Teaching the Lesson Ongoing Learning & Practice

1 3 2 4

Differentiation Options

eToolkit

ePresentations Interactive

Teacher’s Lesson Guide Algorithms

Practice

EM Facts Workshop

Game™

Assessment Management Family

Letters

Curriculum Focal Points Common

Core State Standards

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Reading and Writing Numbers between 0 and 1

LESSON

22

Date Time

Math Message

A grain of salt is about 0.1016 millimeter long.

Write the number 0.1016 in the place-value chart below.

0 1 0 1 6

100 10 1 0 0 0 0 0 0

onesand tens

hundreds tenthshundredthsthousandthsten-thousandthshundred-thousandthsmillionths

.

.1 .01 .001 .0001 .00001 .000001

Write each of the following numbers in standard form.

1. four tenths 0.4

2. twenty-three hundredths 0.23

3. seventy-five thousandths 0.075

4. one hundred nine ten-thousandths 0.0109

5. eight hundredths 0.08

6. one and fifty-four hundredths 1.54

7. twenty-four and fifty-six thousandths 24.056

8. Write the word name for the following decimal numbers.

a. 0.00016 Sixteen hundred-thousandths

b. 0.000001 One millionth

28

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Math Journal 1, p. 48

Student Page

Getting Started

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS DISCUSSION

(Math Journal 1, pp. 48 and 49)

In the previous lesson, students used their knowledge of place-value concepts to read and write whole numbers. In this lesson, they will apply similar place-value concepts to read and write fractional quantities.

The number 0.1016 that students recorded in the chart is written in standard notation. Standard notation is a base-ten place- value numeration.

Ask students to look at 0.1016 and identify the digits in the following places:

tenths 1 hundredths 0 thousandths 1 ten-thousandths 6

Ask students to generate a few sentences using the different meanings of tenth. For example, Mary finished in tenth place;

A dime is a tenth of a dollar.

Have students work in pairs to look for patterns in the place-value chart. Then ask them to share the patterns they found. Patterns include:

Each place is _ ¹

10 the value of the place to its left and 10 times the value of the place to its right.

The value of each place in the place-value chart is a power of 10. The value of each place to the left of the decimal point is a product of 10s, for example, 100 = 10 ∗ 10. The value of each place to the right of the decimal point is a product of _ ¹

10 s (or 0.1s), for example, 0.01 = _ ¹

10 ∗ _ ¹

10 = 0.1 ∗ 0.1.

NOTE Multiplying a number by _ ₁₀¹ is the same as dividing the number by 10.

SOLVING

Math Message

Complete the Math

Message on journal page 48.

Study Link 2

1 Follow-Up

Briefly review answers.

Mental Math and Reflexes

Students divide numbers by 10 and record their answers on slates or dry-erase boards.

$10.00 / 10 $1.00 $1.00 / 10 $0.10

$0.30 / 10 $0.03 0.01 / 10 0.001

0.001 / 10 0.0001 0.0001 / 10 0.00001 Refer to the above problems during discussions of place-value chart patterns.

Adjusting the Activity

Have students highlight each th ending in the place-value chart on journal page 48.

Discuss the difference between tenth as an ordinal number and tenth as a fractional part.

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

Mathematical Practices SMP1, SMP3, SMP4, SMP6, SMP7 Content Standards

6.NS.6c, 6.NS.7a

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Reading and Writing Small Numbers

LESSON

22

Date Time

Complete the following sentences.

Example: A grain of salt is about 0.004, or four thousandths, of an inch long.

1. A penny weighs about 0.1, or

one tenth , of an ounce.

2. A dollar bill weighs about 0.035, or

thirty-five thousandths , of an ounce.

3. On average, fingernails grow at a rate of about 0.0028, or

twenty-eight ten-thousandths, of a centimeter per day.

4. Toenails, on average, grow at a rate of about 0.0007, or seven ten-thousandths , of a centimeter per day.

5. It takes about 0.005, or five thousandths , of a second for a smell to transfer from the nose to the brain.

6. A baseball thrown by a major-league pitcher takes about 0.01, or one hundredth , of a second to cross home plate.

7. A flea weighs about 0.00017, or seventeen hundred-thousandths, of an ounce.

8. A snowflake weighs about 0.00000004, or four hundred-millionths, of an ounce.

9. About how many times heavier is a penny than a dollar bill?

About 3 times as heavy

10. About how many times faster do fingernails grow than toenails?

About 4 times as fast

Try This

28

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Student Page

50

Expanded Notation for Small Numbers

LESSON

22

Date Time

Write each of the following numbers in expanded notation.

Example: 5.96 ( 5 ∗ 1 ) + ( 9 ∗ 0 . 1 ) + ( 6 ∗ 0 .01 )

1. 0.847 (8 ∗ 0.1) + (4 ∗ 0.01) + (7 ∗ 0.001)

2. 3.093 (3 ∗ 1) + (9 ∗ 0.01) + (3 ∗ 0.001)

3. 25.3 (2 ∗ 10) + (5 ∗ 1) + (3 ∗ 0.1)

Give the value of the underlined digit in each number below.

Example: 2.3504 0 .05 , or 5 hundredths

4. 196.9665 5. 15.9994 6. 23.62173

7. 387.29046 0.2, or 2 tenths

Write each number as a fraction or mixed number.

Example 1: 0.735 Example 2: 3.41

8. 17.03 9. 235.075 10. 0.0543

11. Use extended facts to complete the following.

a. 1 tenth = 1 ÷ 10_b. 1 hundredth = 0.1 ÷ 10_c. 1 thousandth = 0.01 ÷ 10 0.00003, or 3 hundred-thousandths

0.09, or 9 hundredths 0.0005, or 5ten-thousandths

100 10 1 0 0 0 0 0 0

hundreds tenthshundredthsthousandthshundred-thousandthmillionths s

ten-thousandths

. .1 .01 .001 .0001 .00001 .000001

17 ^_100 ³ 235 _ 1,000⁷⁵ _ ⁵⁴³ 10,000 ⁷³⁵

_ _1,000 3^_100 ⁴¹

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Student Page

Adjusting the Activity

To read a small number such as 0.1016, underline the final digit and identify its place value. 0.1016, ten-thousandths Then read the number (ignoring the decimal point and any leading zeros) followed by the place value. One thousand sixteen ten-thousandths Writing the number as a fraction or a mixed number can also help students read it. For example, 6.738 as a mixed number is 6 _ _1,000⁷³⁸ and is read six and seven hundred thirty-eight thousandths.

To write a small number, such as three hundred fifty-nine ten-thousandths, in standard notation, the last digit of the decimal number should be in the place value named.

Ten-thousandths Draw a decimal point and mark spaces up to and including the place identified. 0. Write the number in the spaces so the final digit is written in the last space at the right. 0. 3 5 9 Fill any blank spaces with zeros. 0.0 3 5 9

NOTE In Everyday Mathematics, a zero appears to the left of the decimal point in any number greater than 0 and less than 1. This makes it easier to order decimal numbers, to draw attention to the decimal point, and to correspond with the display on most calculators.

Have students complete journal pages 48 and 49.

Have students use the place-value template (Math Masters, p. 410) as they work on the journal pages, Math Boxes, and Study Link for this lesson.

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

▶ Interpreting Expanded

WHOLE-CLASS ACTIVITY

Notation for Small Numbers

(Math Journal 1, p. 50; Math Masters, p. 410; Transparency of Math Masters, p. 410)

100 10 1 0 0 0 0 0 0

hundreds tenthshundredthsthousandthsten-thousandthshundred-thousandthsmillionths

0 4 9 5 .

.1 .01 .001 .0001 .00001 .000001

Display a transparency of Math Masters, page 410 and distribute one copy of the same page to each student. Ask students to write the number 0.495 in the place-value chart. Students will apply skills they used in the previous lesson to write this decimal number in expanded notation.

(4 ∗ 0.10) + (9 ∗ 0.01) + (5 ∗ 0.001)

Have students work independently to complete journal page 50.

0 0 3 5 .

ones tenths

hundredthsthousandths

9

ten-thousandths

Three hundred fifty-nine ten thousandths

6 7 3 8 .

ones ¹

10

1 1,000 1 100

Six and seven hundred thirty-eight thousandths

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51

Math Boxes

LESSON

22

Date Time

1. Write each of the following numbers using digits.

a. five and fifty-five hundredths

5.55

b. one hundred eight thousandths

0.108

c. two hundred six and nineteen ten-thousandths

206.0019

3. This line graph shows the average monthly rainfall in Jacksonville, Florida.

26 27

2. Write each number in expanded form.

a. 53.078 (5 ∗ 10) + (3 ∗ 1) +

(7 ∗ 0.01) + (8 ∗ 0.001)

b. 9.0402 (9 ∗ 1) + (4 ∗ 0.01)

+ (2 ∗ 0.0001)

0 1 2 3 4 5 6 7 8

Rainfall (in inches)

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Average Monthly Rainfall

in Jacksonville, Florida

4. Janessa is 3 years older than her brother Lamont.

a. If Janessa is 18 years old, how old is Lamont?

15 years old

b. How old is Janessa when she is twice as old as Lamont?

6 years old

212 5. Find the perimeter

of the square if s = 4.3 cm. Use the formula P = 4 ∗ s, where s represents the length of one side.

P = 17.2 cm s Which conclusion can you draw from the graph?

Fill in the circle next to the best answer.

A At least 10 months of the year, the average rainfall is less than 3.5 inches.

B The average rainfall increases from June through December.

C The average rainfall for May and November is about the same.

D Jacksonville gets more rain on average than Tampa.

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Student Page

STUDY LINK

2^䉬2 Writing Decimals

26–28

Name Date Time

1.Build a numeral. Write: 2.Build a numeral. Write:

9 in the thousandths place, 3 in the tenths place, 4 in the tenths place, 6 in the ten-thousandths place, 8 in the ones place, 4 in the hundredths place, 3 in the tens place, and 0 in the thousandths place, and 6 in the hundredths place. 1 in the ones place.

Answer: Answer:

. .

Write the following numbers in words.

3.0.8

4.0.95

5.0.05

6.0.067

7.4.0802

Write a decimal place value in each blank space.

8.Bamboo grows at a rate of about 0.00004, or four ,

kilometer per hour.

9.The average speed that a certain brand of catsup pours from the mouth of the bottle is

about 0.003, or three , mile per hour.

10.A three-toed sloth moves at a speed of about 0.068 to 0.098, or sixty-eight

to ninety-eight , mile per hour.

3 8 4 6 9 1 340 6

eight-tenths ninety-five hundredths

five-hundredths sixty-seven thousandths

hundred-thousandths

thousandths

thousandths thousandths

four and eight hundred two ten-thousandths

Math Masters, p. 44

Study Link Master

2 Ongoing Learning & Practice

▶ Playing High-Number Toss

^PARTNER_ACTIVITY

(Decimal Version)

(Student Reference Book, p. 324; Math Masters, p. 455)

Divide the class into pairs and distribute four each of number cards 0–9 to each pair, as well as a game record sheet (Math Masters, p. 455). Students may need to play a practice game.

At the end of each round and when finding the total score, allow students to use calculators, if needed.

Ongoing Assessment:

Math Masters Page 455

Recognizing Student Achievement

Use Math Masters, page 455 to assess students’ ability to compare decimals through thousandths. Students are making adequate progress if they are able to identify the larger number. Some students may not need a calculator to find the difference between scores.

▶ Math Boxes 2

2

INDEPENDENT ACTIVITY

(Math Journal 1, p. 51)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-4. The skills in Problems 4 and 5 preview Unit 3 content.

Writing/Reasoning Have students write a response to the following: Explain why each of the other three answers for Problem 3 is not the best choice. Sample answer: A is not correct because only 5 months have an average rainfall less than 3.5 inches. B is incorrect because the average rainfall is very low from October to December. D is not correct because there is no way to know what the average rainfall is for Tampa.

▶ Study Link 2

2

INDEPENDENT ACTIVITY

(Math Masters, pp. 44 and 45)

Home Connection Students practice place-value skills and write small numbers in standard and expanded notations.

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STUDY LINK

2^䉬2 Writing Decimals continued

Name Date Time

Write each of the following numbers in expanded notation.

Example: 2.756 ⫽ (2 º 1) ⫹ (7 º 0.1) ⫹ (5 º 0.01) ⫹ (6 º 0.001) 11.0.013

12.109.3527

13.Using the digits 0, 3, 6, and 8, write the greatest decimal number possible.

.

14.Using the digits 0, 3, 6, and 8, write the least decimal number possible.

.

100 10 1 0 0 0 0 0 0

ones tens

hundreds tenthshundredthsthousandthsten-thousandthshundred-thousandthsmillionths

. ^.1 ^.01 ^.001 ^.0001 ^.00001 ^.000001

and

(1º 0.01) ⫹ (3 º 0.001) (1 º 100) ⫹ (9 º 1) ⫹ (3 º 0.1) ⫹ (5 º 0.01)⫹ (2 º 0.001) ⫹ (7 º 0.0001) 8 6 3 0

0 3 6 8

Try This

Name the point on the number line that represents each of the following numbers.

15.0.66 16.0.6299 17.0.6 18.0.695

19.Refer to the number line above. Round 0.6299 to the nearest hundredth.

0.6 0.65 0.7

C A D B

D

0.63

A C B

20.0.01 ⫹ 0.006 ⫹ 0.0008 ⫽ ^21.0.7 ⫹ 0.04 ⫹ 0.0002 ⫽

22. ⫽ 40 ⫹ 5 ⫹ 0.009 23. ⫽ 0.50 ⫹ 0.080 ⫹ 0.00010

0.0168 0.7402

Practice

45.009 0.5801

Study Link Master

Teaching Master

LESSON

2^r2

Name Date Time

Modeling and Comparing Decimals

46

One way to compare decimals is to model them with base-10 grids.

The flat is the

whole, or 1.0. The long is

worth 0.1. The cube is

worth 0.01. The fractional part of the cube

is worth 0.001.

Another way to compare decimals is to draw pictures.

1. Use decimal models to complete the following.

1.0 = 0.10 ∗ 0.10 = 0.01 ∗ 0.01 = 0.001 ∗ Model the decimal numbers in each pair. Draw a picture to record each model.

Then compare the decimal numbers using <, >, or =.

2. 3.

0.3 0.14 1.56 1.562

4. 5. Model and record a decimal number that is between 0.41 and 0.42. Sample answer:

0.2 0.025

0.41 < < 0.42

The flat is the

whole, or 1.0. The long is

worth 0.1. The cube is

worth 0.01. The fractional part of the cube

is worth 0.001.

>

10 10 10

> <

0.418

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3 Differentiation Options

READINESS ^PARTNER_ACTIVITY

▶ Modeling and

^{5–15 Min}

Comparing Decimals

(Math Masters, pp. 46, 411–413)

To provide experience comparing decimals, have students use base-10 grids. Provide each pair with one copy of Math Masters, page 46, two copies of page 411, one copy of pages 412 and 413, and scissors. Review the worth of the flat, long, unit, and fractional parts of the unit, as shown on Math Masters, page 46. After

students have prepared their base-10 grids, ask them to record their models and compare the given decimal numbers using <, >, or =.

ENRICHMENT ^PARTNER

ACTIVITY

▶ Decimals between Decimals

^{5–15 Min}

Between any two decimal numbers, there is always another decimal number. Help students explore this concept by asking them to list 20 or more decimal numbers between a given pair of decimals. Suggested decimal pairs include:

0.1 and 0.2; 0.33 and 0.34; 2.561 and 2.562.

Have students describe any patterns or strategies they used to generate their lists.

Continue the discussion by asking students to comment about the relative positions of their answers on the number line. Ask questions such as the following:

● If a number is between 0.33 and 0.34, is it greater than or less than 0.33? Greater than 0.33 Should the number be to the left of 0.33 or to the right of 0.33 on the number line? To the right of 0.33

● If a number is between 0.33 and 0.34, is it greater than or less than 0.34? Less than 0.34 Should the number be to the left of 0.34 or to the right of 0.34 on the number line? To the left of 0.34

● Choose two of your decimal numbers between 2.561 and 2.562.

Which number would be farther to the right on a number line?

Answers vary.

Encourage students to sketch number lines and plot the decimal numbers to verify their relative positions. Have students make a general statement about the relative positions of two numbers on a number line. Sample answer: A greater number will be farther to the right on a number line.

NOTE Collect and store students’ base-10 grids for future use.

Reading and Writing Small Numbers