Teaching the Lesson materials

Key ActivitiesStudents fill in a place-value chart that shows place-value headings expressed as powers of 10. They use exponential notation to represent powers of 10.

Key Concepts and Skills• Read and write large numbers; identify the digits and their values.

[Number and Numeration Goal 1]• Use exponential notation to represent powers of 10. [Number and Numeration Goal 4]• Use expanded notation to represent powers of 10. [Number and Numeration Goal 4]• Identify and describe patterns in a place-value table. [Patterns, Functions, and Algebra Goal 1]

Key Vocabularyscientific notation • trillion • quadrillion • quintillion • sextillion • powers of 10 • exponent

Ongoing Assessment: Recognizing Student Achievement Use a Math Log or Exit Slip.[Patterns, Functions, and Algebra Goal 1]

Ongoing Learning & Practice materials

Students play Polygon Pair-Up to practice naming properties of polygons.

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

Differentiation Options materials

Students investigate powersof 10 with a calculator.

Students use exponentialnotation.

Students find various waysof using the word power.

� Teaching Master (Math Masters,p. 168)

� 5-Minute Math, pp. 2, 3, and 7 � calculator

ELL SUPPORTEXTRA PRACTICEENRICHMENT

3

� Math Journal 1, p. 131� Student Reference Book, p. 258� Study Link Master (Math Masters,

p. 167) � Polygon Pair-Up Property Cards and

Polygon Cards (Math Masters,pp. 496 and 497)

2

� Math Journal 1, p. 130� Student Reference Book, p. 271� Study Link 5�8� Teaching Aid Master (Math

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

p. 166; optional)� calculator

See Advance Preparation

1

Objective To introduce exponential notation for powers of 10

as a way of naming the values of places in our base-ten system.

Lesson 5�9 361

Technology Assessment Management System

Math Log or Exit SlipSee the iTLG.

Additional InformationAdvance Preparation For Part 1, draw a place-value chart on the board like the one on journalpage 130, or use a transparency of Math Masters, page 166.

362 Unit 5 Big Numbers, Estimation, and Computation

� Math Message Follow-Up(Student Reference Book, p. 271)

Divide students into small groups so they can compare answers.You might give them a few additional minutes to look for an evenlarger number. Then have the groups report their answers andwrite them in the place-value chart on the board.

Discuss the answers. The largest numbers in the World Toursection of the Student Reference Book are found on page 271.

The total population of the world is given in the table near thebottom of the page. Although this is not the largest number in theWorld Tour section, it is probably the largest number that moststudents will recognize at this time.

The largest number in the World Tour section is found in theother table on page 271. The approximate weight (mass) of Earthis listed as 6.6 � 1021 tons. This number is read as “6 point 6times 10 to the 21st power.”

Numbers written in this form are said to be in scientific notation. This is how the number looks, written out:

Write this 22-digit number on the board, including the labels forthousands, millions, billions, and so on. Point out the place-valuegroup names that are used for numbers larger than billions,namely trillions, quadrillions, quintillions, and sextillions.

Ask students to read the number by using the name labels writtenbeside the commas. 6 sextillion, 600 quintillion

6,600,000,000,000,000,000,000se

xtillio

n

quint

illion

quad

rillion

trillio

nbil

lion

million

thous

and

SMALL-GROUP

ACTIVITY

1 Teaching the Lesson

Getting Started

Math MessageFind the largest number you can in the World Toursection of your Student Reference Book.

Study Link 5�8 Follow-UpThe number in the place-value chart should be92,106,954,873. Accept either of the following ways to write the number:

� 92 billion, 106 million, 954 thousand, 873� ninety-two billion, one hundred six million, nine hundred

fifty-four thousand, eight hundred seventy-three

Mental Math and Reflexes Write large numbers through the billions on the board,and have volunteers read them aloud. Suggestions:

396,467 3,654,987 5,123,467,890283,950 17,834,567 8,312,945,607712,945 527,000,348 3,980,246,571

Ask questions such as:

• What digit is in the millions place?• What is the value of the digit x?• How many billions are there?

Facts About the World

Continents are large land masses. There areseven continents on the Earth, although Europeand Asia are sometimes thought of as onecontinent. Most continents contain manycountries, but there are no countries at allin Antarctica.

A country is a territory and the people who livethere under one government. The number ofcountries in the world often changes as countriessplit apart or join with other countries. At thistime, there are about 200 countries in the world.

Population is the number of people who live in a certain region. Population growth is the change in the population every year after all births and deaths are accounted for. The population growth rate is the increase(or decrease) in population per year, written asa percent.

The world’s population is now increasing by about 200,000 people per day, or about 75 million people per year. Over the last 40 years, the world’s population has about doubled. It reached the 6 billion mark in 1999. World population is expected to reachabout 9 billion people by the year 2050.

World Tour

North America 509,000,000 8.0% 8,300,000 14.8%South America 367,000,000 5.8 6,800,000 12.1Europe 799,000,000 12.5 4,100,000 7.3Asia 3,797,000,000 59.5 16,700,000 29.8Africa 874,000,000 13.7 11,500,000 20.5Australia 32,000,000 0.5 3,300,000 5.9Antarctica 0 0.0 5,400,000 9.6

World Totals 6,378,000,000 100.0% 56,100,000 100.0%(about 6.4 billion)

The Continents

Continent Population*

Percent of WorldPopulation

Area (sq miles)

Percent ofLand Area

*Data are for the year 2004. World population growth rate for the year 2004: about 1.2% per year

Dimensions of the EarthEquatorial circumference*:about 24,900 miles (40,000 kilometers)Equatorial diameter**:about 7,930 miles (12,760 kilometers)Volume: 2.6 x 1011 cubic miles (1.1 x 1012 cubic kilometers)Weight (mass): 6.6 x 1021 tons (6.0 x 1021 metric tons)Total world water area:about 139,433,000 square miles (361,129,000 square kilometers)*Circumference is the distance around a circle or sphere.

**Diameter is the distance measured by a straight line passing from one side of a circle or sphere, through the center, to the other side.

Student Reference Book, p. 271

Student Page

� Introducing Exponential Notation for Powers of 10(Math Journal 1, p. 130)

In preparation for later work with scientific notation, it is useful to show how place-value headings can be expressed as powers of10 using exponential notation.

1. Review the place-value headings that students are familiarwith on journal page 130—ones, tens, hundreds, and so on.Ask them to write the standard numerals in Row 1 of theplace-value chart.

2. Review the relationship between the value of each place and the place to its right. The value of each place is 10 timesthe value of the place to its right. Students complete the 10-times-as-much pattern in Row 2.

3. Tell students that numbers like 100, 1,000, and 10,000 arecalled powers of 10. They are numbers that can be expressedas products whose factors are 10. Have students fill in Row 3of the chart on their journal page.

4. Instead of repeating the factor 10, we can use the shorthandintroduced in the Math Message Follow-Up. For example, 10 � 10 can be written as 102 (read “10 to the second power”or “10 squared”). 10 � 10 � 10 can be written as 103 (read “10 to the third power” or “10 cubed”). The raised digit, calledthe exponent, tells how many times 10 is used as a factor.Ask students to fill in Row 4 of the chart on their journal page.

WHOLE-CLASS

ACTIVITY

Lesson 5�9 363

Links to the FutureThe discussions of exponential notation andscientific notation are exposures to thetopics. They will be revisited more formally inlater grades. The use of exponential notationis a Grade 5 Goal, and the use of scientificnotation is a Grade 6 Goal.

130

Pla

ce V

alu

e a

nd

Po

wers o

f10

LESSON

5�9

Date

Time

5

�

Fill in this place-value chart as follows:

1. Write standard numbers in Row 1.

2. In Row 2, write the value of each place to show that it is 10 times the value of the place to its right.

3. In Row 3, write the place values as products of 10s.

4. In Row 4, show the values as powers of 10. Use exponents. The exponent shows how many times 10 is used as a factor. It also shows how many zeros are in the standard number.

Hundred TenMillions

Thousands ThousandsThousands Hundreds Tens Ones

1,000,000 100,000 10,000 1,000 100 10 1

10 [100,000s] 10 [10,000s] 10 [1,000s] 10 [100s] 10 [10s] 10 [1s] 10 [tenths]

10 � 10 � 10 � 10 � 10 �10 � 10 � 10 � 10 10 � 10 � 10 10 � 10 1010 � 10 � 10 10 � 10 � 10

106 105 104 103 102 101 100

Math Journal 1, p. 130

Student Page

Math Masters, page 166 is identical to Math Journal 1, page 130.

Adjusting the Activity

364 Unit 5 Big Numbers, Estimation, and Computation

5. Have students explore the calculator key sequences used to raise 10 to a certain power.

TI-15:

Key Sequence: 10 3

Display: 1000

Casio fx-55:

Key Sequence: 3

Display: 1000

or

Key Sequence: 10 3

Display: 1000

6. Ask students to look for patterns in their completed charts.The patterns should include the following:

� The exponent tells how many zeros are in the number inRow 1. For example, look at the thousands place: the exponent is 3, and the number 1,000 has 3 zeros.

� The exponent tells how many columns are to the right. For example, look at the hundreds place: the exponent is 2.There are 2 columns to the right of the hundreds place—the tens place and the ones place.

� In moving left one place, the exponent increases by 1. Forexample, start in the thousands place. The exponent is 3.Move left one place to the ten-thousands place. The exponenthere is 4.

� In moving right one place, the exponent decreases by 1.For example, start in the hundreds place. The exponent is 2. Move right one place to the tens place. The exponenthere is 1.

Ask students to find another name for 10�1. Suggest that they extend the patterns in Rows 1 and 4. Each number in Row 1 is �1

10� of the number

to its left. So if a column is added to the right of the ones column, it would have�110� in Row 1 and would be labeled Tenths. Because the exponents decrease by 1

in moving one place to the right, the tenths place would have 10�1 in Row 4. 10�1 is another name for �1

10�.

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

NOTE The pattern in the table shows 100 isanother name for 1. This illustrates a moregeneral mathematical definition: Any non-zeronumber to the zero power is defined as 1. Forexample, 3470 � 1.

131

Math Boxes LESSON

5�9

Date Time

5. Multiply. Use a paper-and-pencil algorithm.

9 � 258 � 2,3226. Which of the angles below has a measure

of about 90 degrees? Circle it.

1. Estimate the sum. Write a number model to show how you estimated.a. 1,254 � 8,902 � 2,877

Number model:

b. 12,645 � 7,302 � 15,297

Number model:

35,00013,000�7,000�15,000�

1,000 � 9,000 � 3,000 � 13,000

3. Draw lines to match each word to thecorrect pair or pairs of line segments.

4. Complete.

Rule: Add 0.07

5 6

94 95 162–166

181

18 19 92 93

2. Which number sentence is true? Fill in thecircle next to the best answer.

A. 5,800,000 � 58 million

B. 62 million � 3,100,000,000

C. 103 � 1,000

D. 100,000 � 102

perpendicular

parallel

intersecting

Sample answers:

in out

1.29 1.36

6.47

5.17

12.66

7.93 8.0012.73

6.545.10

Math Journal 1, p. 131

Student Page

STUDY LINK

5�9 Many Names for Powers of 10

5

Name Date Time

1,000,000 10,000 1,000

100 10 10 [100,000s]

10 [10,000s] 106 10 [1,000s]

103 10 º 10 º 10 º 10 one thousand

105 10 º 10 º 10 º 10 º 10 10 [10s]

10 º 10 ten 101

10 [tenths] 100 1

7. 63 º 7 � 8. � 495 º 6 9. � 97 º 535,1412,970441Practice

Below are different names for powers of 10. Write the names in the appropriatename-collection boxes. Circle the names that do not fit in any of the boxes.

100,000

105

10 [10,000s]10 º 10 º 10 º 10 º 10

102

10010 [10s]10 º 10

one

1100

10 [tenths]

1 million

106

1,000,00010 [100,000s]

10 � 10 � 10

103

1,000one thousand

104

10,00010 [1,000s]

10 º 10 º 10 º 10

1. 2.

3. 4.

5. 6.

Math Masters, p. 167

Study Link Master

Lesson 5�9 365

Ongoing Assessment:Recognizing Student Achievement

Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess students’ ability to describe numeric patterns. Have students describe the pattern in one of the rows in the table on journal page 130. Students are makingadequate progress if they note that the value of each place is 10 times the value of the place to its right and �1

10� the value of the place to its left. Some students

may be able to extend this pattern past millions and ones.

[Patterns, Functions, and Algebra Goal 1]

� Playing Polygon Pair-Up(Student Reference Book, p. 258; Math Masters, pp. 496 and 497)

Students play Polygon Pair-Up to practice identifying properties ofpolygons. See Lesson 1-6 for additional information.

� Math Boxes 5�9(Math Journal 1, p. 131)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 5-11. The skill in Problem 6previews Unit 6 content.

Writing/Reasoning Have students write a response tothe following: For Problem 3, can intersecting lines be perpendicular lines? Explain. Sample answer: Yes.

Perpendicular lines intersect at right angles and intersecting linesintersect at any angle.

� Study Link 5�9(Math Masters, p. 167)

Home Connection Students identify and sort names forpowers of 10.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

Math Log or

Exit Slip

366 Unit 5 Big Numbers, Estimation, and Computation

LESSON

5�9

Name Date Time

Powers of 10 on a Calculator

Experiment to see what happens when your calculator can no longer display all of the digitsin a power of 10.

Clear your calculator’s memory, then program it to multiply over and over by 10 as follows:

Calculator Key Sequence

Calculator A and together

10 10 ...

Calculator B

10 ...

Op1Op1Op1Op1ÆOp1

ClearOn/Off

1. What is the largest power of 10 that your calculator can display before it switches fromdecimal notation to exponential notation? 1 billion (Calculator A); 10 million (Calculator B)

2. Write what the calculator displays after it switches from decimal notation to exponential notation. 9 1 10^10 (Calculator A); 1. 08 (Calculator B)

3. If there are different kinds of calculators in your classroom, is the largest power of 10 thatthey can display the same or different from your calculator? If it is different, tell how. Write your answer on the back of this page. Answers vary.

Try This

4. What is the smallest power of 10 that your calculator can display before it switches fromdecimal notation to exponential notation?0.0000000001 (Calculator A); 0.0000001 (Calculator B)

Explain what you did to find out.Sample answer: I followed the steps above but pressed thedivision key instead of the multiplication key.

5 213 215

Math Masters, p. 168

Teaching Master

� Investigating Powers of 10 on a Calculator(Math Masters, p. 168)

To further explore exponential notation, have students investigatehow calculators display powers of 10 and the patterns they show. If more than one kind of calculator is available, have studentscompare the ways the different calculators perform these functions.

� 5-Minute MathTo offer students more experience with exponential notation, see5-Minute Math, pages 2, 3, and 7.

� Building Background for Mathematics WordsTo provide language support for exponential notation, havestudents think of as many ways to use the word power as theycan. For example:

� The President of the United States has the power to veto billssent from Congress.

� The United States armed forces are a mighty power in the world.

� A battery powers this CD player.

� He powered up (or powered down) the computer.

� Her new car has power windows.

� 100 and 0.1 are powers of 10.

5–15 Min

SMALL-GROUP

ACTIVITYELL SUPPORT

5–15 Min

SMALL-GROUP

ACTIVITYEXTRA PRACTICE

5–15 Min

INDEPENDENT

ACTIVITYENRICHMENT

3 Differentiation Options