Lesson 5.9 Powers of 10

Objective To introduce exponential notation for powers of 10 as a way of naming the values of places in our base-ten system.

1 Teaching the Lesson materials Key Activities Math Journal 1, p. 130 Students fill in a place-value chart that shows place-value headings expressed as powers Student Reference Book, p. 271 of 10. They use exponential notation to represent powers of 10. Study Link 58 Key Concepts and Skills Teaching Aid Master (Math • Read and write large numbers; identify the digits and their values. Masters, p. 388 or 389) [Number and Numeration Goal 1] Transparency (Math Masters, p. 166; optional) • 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] calculator • Identify and describe patterns in a place-value table. [Patterns, Functions, and Algebra Goal 1] See Advance Preparation Key Vocabulary scientific 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] 2 Ongoing Learning & Practice materials Students play Polygon Pair-Up to practice naming properties of polygons. Math Journal 1, p. 131 Students practice and maintain skills through Math Boxes and Study Link activities. 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)

3 Differentiation Options materials ENRICHMENT EXTRA PRACTICE ELL SUPPORT Teaching Master (Math Masters, p. 168) Students investigate powers Students use exponential Students find various ways of 10 with a calculator. notation. of using the word power. 5-Minute Math, pp. 2, 3, and 7 calculator

Additional Information Advance Preparation For Part 1, draw a place-value chart on the board like the one on journal Technology page 130, or use a transparency of Math Masters, page 166. Assessment Management System Math Log or Exit Slip See the iTLG.

Lesson 59 361 Getting Started

Mental Math and Reflexes Math Message Write large numbers through the billions on the board, Find the largest number you can in the World Tour and have volunteers read them aloud. Suggestions: section of your Student Reference Book. 396,467 3,654,987 5,123,467,890 283,950 17,834,567 8,312,945,607 Study Link 58 Follow-Up 712,945 527,000,348 3,980,246,571 The number in the place-value chart should be Ask questions such as: 92,106,954,873. Accept either of the following • What digit is in the millions place? ways to write the number: • What is the value of the digit x? 92 billion, 106 million, 954 thousand, 873 • How many billions are there? ninety-two billion, one hundred six million, nine hundred fifty-four thousand, eight hundred seventy-three

1 Teaching the Lesson

SMALL-GROUP Math Message Follow-Up ACTIVITY (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 even larger number. Then have the groups report their answers and write them in the place-value chart on the board. Discuss the answers. The largest numbers in the World Tour section of the Student Reference Book are found on page 271. The total population of the world is given in the table near the Student Page bottom of the page. Although this is not the largest number in the World Tour section, it is probably the largest number that most World Tour students will recognize at this time. Facts About the World Dimensions of the Earth Continents are large land masses. There are Equatorial circumference*: seven continents on the Earth, although Europe about 24,900 miles The largest number in the World Tour section is found in the and Asia are sometimes thought of as one (40,000 kilometers) Equatorial diameter**: continent. Most continents contain many about 7,930 miles other table on page 271. The approximate weight (mass) of Earth countries, but there are no countries at all (12,760 kilometers) 11 21 in Antarctica. Volume: 2.6 x 10 cubic miles (1.1 x 1012 cubic kilometers) is listed as 6.6 10 tons. This number is read as “6 point 6 A country is a territory and the people who live Weight (mass): 6.6 x 1021 tons 21 there under one government. The number of (6.0 x 10 metric tons) times 10 to the 21st power.” countries in the world often changes as countries Total world water area: about 139,433,000 square miles split apart or join with other countries. At this (361,129,000 square kilometers) time, there are about 200 countries in the world. *Circumference is the distance around a circle or sphere. Numbers written in this form are said to be in scientific Population is the number of people who **Diameter is the distance live in a certain region. Population growth is measured by a straight line passing from one side of a notation. the change in the population every year after circle or sphere, through the This is how the number looks, written out: all births and deaths are accounted for. center, to the other side. The population growth rate is the increase (or decrease) in population per year, written as a percent. 6 600 000 000 000 000 000 000 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 ion ion ion ion ion ion ill ill ill ill ill ill the 6 billion mark in 1999. World population is expected to reach tr b about 9 billion people by the year 2050. int m sext qu thousand The Continents quadr Percent of World Area Percent of Continent Population* Population (sq miles) Land Area Write this 22-digit number on the board, including the labels for North America 509,000,000 8.0% 8,300,000 14.8% South America 367,000,000 5.8 6,800,000 12.1 Europe 799,000,000 12.5 4,100,000 7.3 thousands, millions, billions, and so on. Point out the place-value Asia 3,797,000,000 59.5 16,700,000 29.8 Africa 874,000,000 13.7 11,500,000 20.5 group names that are used for numbers larger than billions, Australia 32,000,000 0.5 3,300,000 5.9 Antarctica 0 0.0 5,400,000 9.6 namely trillions, quadrillions, quintillions, and sextillions. World Totals 6,378,000,000 100.0% 56,100,000 100.0% (about 6.4 billion) *Data are for the year 2004. World population growth rate for the year 2004: about 1.2% per year Ask students to read the number by using the name labels written Student Reference Book, p. 271 beside the commas. 6 sextillion, 600 quintillion

362 Unit 5 Big Numbers, Estimation, and Computation WHOLE-CLASS Introducing Exponential ACTIVITY Notation for Powers of 10 Links to the Future (Math Journal 1, p. 130) The discussions of exponential notation and scientific notation are exposures to the topics. They will be revisited more formally in In preparation for later work with scientific notation, it is useful later grades. The use of exponential notation to show how place-value headings can be expressed as powers of is a Grade 5 Goal, and the use of scientific 10 using exponential notation. notation is a Grade 6 Goal. 1. Review the place-value headings that students are familiar with on journal page 130—ones, tens, hundreds, and so on. Ask them to write the standard numerals in Row 1 of the place-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 times the 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 are called powers of 10. They are numbers that can be expressed as products whose factors are 10. Have students fill in Row 3 of the chart on their journal page. 4. Instead of repeating the factor 10, we can use the shorthand introduced 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, called the 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.

Student Page 130 Date Time LESSON 5 9

Hundred Ten 10 Place Value of and Powers Millions Thousands Hundreds Tens Ones Thousands Thousands

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 10 10 10 10 10 10 10

6 4 1 10 105 10 103 102 10 100

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. 5

Math Journal 1, p. 130

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

Lesson 59 363 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. NOTE The pattern in the table shows 100 is another name for 1. This illustrates a more The patterns should include the following: general mathematical definition: Any non-zero The exponent tells how many zeros are in the number in number to the zero power is defined as 1. For example, 3470 1. Row 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. For example, start in the thousands place. The exponent is 3. Move left one place to the ten-thousands place. The exponent here 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 exponent here is 1.

Adjusting the Activity

Ask students to find another name for 101. Suggest that they extend the patterns in Rows 1 and 4. Each number in Row 1 is 1 of the number 10 to its left. So if a column is added to the right of the ones column, it would have 1 in Row 1 and would be labeled Tenths. Because the exponents decrease by 1 10 in moving one place to the right, the tenths place would have 101 in Row 4. 1 1 10 is another name for 10. AUDITORY KINESTHETIC TACTILE VISUAL

364 Unit 5 Big Numbers, Estimation, and Computation Student Page

Date Time

LESSON Math Boxes Ongoing Assessment: Math Log or 5 9 Exit Slip 1. Estimate the sum. Write a number model 2. Which number sentence is true? Fill in the to show how you estimated. circle next to the best answer. Recognizing Student Achievement a. 1,254 8,902 2,877 A. 5,800,000 58 million Number model: Sample answers: Use a Math Log or an Exit Slip (Math Masters, page 388 or 389) to assess 1,000 9,000 3,000 13,000 B. 62 million 3,100,000,000

b. 12,645 7,302 15,297 students’ ability to describe numeric patterns. Have students describe the C. 103 1,000 Number model: pattern in one of the rows in the table on journal page 130. Students are making 13,000 7,000 15,000 D. 100,000 102 adequate progress if they note that the value of each place is 10 times the value 35,000 181 5 6 of the place to its right and 1 the value of the place to its left. Some students 10 3. Draw lines to match each word to the 4. Complete. may be able to extend this pattern past millions and ones. correct pair or pairs of line segments. Rule: Add 0.07 in out [Patterns, Functions, and Algebra Goal 1] perpendicular 1.29 1.36 parallel 6.47 6.54 5.10 5.17 intersecting 12.66 12.73 7.93 8.00 94 95 162–166

5. Multiply. Use a paper-and-pencil algorithm. 6. Which of the angles below has a measure of about 90 degrees? Circle it. 9 258 2,322 2 Ongoing Learning & Practice

PARTNER 18 19 92 93 Playing Polygon Pair-Up ACTIVITY 131 (Student Reference Book, p. 258; Math Masters, pp. 496 and 497) Math Journal 1, p. 131

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

INDEPENDENT Math Boxes 5 9 ACTIVITY (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 6 previews Unit 6 content. Writing/Reasoning Have students write a response to the following: For Problem 3, can intersecting lines be perpendicular lines? Explain. Sample answer: Yes. Study Link Master Name Date Time

Perpendicular lines intersect at right angles and intersecting lines STUDY LINK Many Names for Powers of 10 intersect at any angle. 5 9 Below are different names for powers of 10. Write the names in the appropriate name-collection boxes. Circle the names that do not fit in any of the boxes. 5

1,000,000 10,000 1,000 INDEPENDENT 100 10 10 [100,000s] 6 Study Link 5 9 ACTIVITY 10 [10,000s] 10 10 [1,000s] 103 10 º 10 º 10 º 10 one thousand (Math Masters, p. 167) 105 10 º 10 º 10 º 10 º 10 10 [10s]

10 º 10 ten 101 Home Connection Students identify and sort names for 10 [tenths] 100 1 powers of 10. 1.100,000 2. 102 105 100 10 [10,000s] 10 [10s] 10º10º10º10º10 10 º 10

3.1 million 4. one 106 1 1,000,000 100 10 [100,000s] 10 [tenths]

5.10 10 10 6. 104 103 10,000 1,000 10 [1,000s] one thousand 10 º 10 º 10 º 10

Practice

7. 63 º 7 441 8. 2,970 495 º 6 9. 5,141 97 º 53

Math Masters, p. 167

Lesson 59 365 Teaching Master

Name Date Time

LESSON Powers of 10 on a Calculator 5 213 5 9 215 Differentiation Options Experiment to see what happens when your calculator can no longer display all of the digits 3 in a power of 10.

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

Calculator Key Sequence INDEPENDENT Calculator AOn/Off andClear together ENRICHMENT ACTIVITY Op1 Æ 10Op1 10Op1 Op1 Op1 ... Calculator B Investigating Powers of 10 10 ... 5–15 Min

1. What is the largest power of 10 that your calculator can display before it switches from decimal notation to exponential notation? on a Calculator 1 billion (Calculator A); 10 million (Calculator B) (Math Masters, p. 168) 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) To further explore exponential notation, have students investigate 3. If there are different kinds of calculators in your classroom, is the largest power of 10 that they 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. how calculators display powers of 10 and the patterns they show.

Try This If more than one kind of calculator is available, have students

4. What is the smallest power of 10 that your calculator can display before it switches from decimal notation to exponential notation? compare the ways the different calculators perform these functions. 0.0000000001 (Calculator A); 0.0000001 (Calculator B)

Explain what you did to find out. Sample answer: I followed the steps above but pressed the EXTRA PRACTICE SMALL-GROUP division key instead of the multiplication key. ACTIVITY

5-Minute Math 5–15 Min

Math Masters, p. 168 To offer students more experience with exponential notation, see 5-Minute Math, pages 2, 3, and 7.

ELL SUPPORT SMALL-GROUP ACTIVITY Building Background for 5–15 Min Mathematics Words

To provide language support for exponential notation, have students think of as many ways to use the word power as they can. For example: The President of the United States has the power to veto bills sent 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.

366 Unit 5 Big Numbers, Estimation, and Computation