Exponential Notation for Powers of 10

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1 Expntial Notation for Objective To introduce number-and-word notation for large numbers and expntial notation for powers of 10. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Explore place value using powers of 10. [Number and Numeration Goal 1] Write and translate numbers in and between standard and expntial notation. Compare expntial notation and standard notation for positive powers of 10. Describe the number patterns inherent to powers of ten. [Patterns, Functions, and Algebra Goal 1] Key Activities Students use standard notation, number-and-word notation, and expntial notation to represent large numbers. Recognizing Student Achievement Use the Math Message Playing First to 100 Student Reference Book, p. 08 Math Masters, pp per partnership: 2 six-sided dice, calculator Students practice solving open number sentences. Math Boxes Math Journal 2, p. 21 Geometry Template Students practice and maintain skills through Math Box problems. Study Link Math Masters, p. 191 Students practice and maintain skills through Study Link activities. READINESS Finding Patterns in Math Masters, p. 192 Students complete a powers-of-10 table and describe patterns they see in the table. ENRICHMENT Introducing Negative Expnts and Powers of 0.1 Student Reference Book, p. 7 Math Masters, p. 19 Students explore patterns and notation of negative expnts. EXTRA PRACTICE Multiplying Decimals by Students solve problems involving the multiplication of decimals by powers of 10. Informing Instruction See page 549. Key Vocabulary number-and-word notation powers of 10 Materials Math Journal 2, p. 212 Student Reference Book, p. 5 Study Link 7 1 Math Masters, p. 4 transparency of Math Masters, p. 4 slate Advance Preparation For Part 1, copy the place-value chart from the top of journal page 212 on the board. If possible, use semipermanent chalk, or make a transparency of Math Masters, page 4. Make copies of it available to students. It will also be used in Lesson 7-. For Part, extend the display to include negative powers of 10. For the Math Message, draw name-collection boxes on the board and label 100, 1,000, and 1,000,000. For a mathematics and literacy connection, obtain a copy of Can You Count to a Googol? by Robert E. Wells. Teacher s Reference Manual, Grades 4 6 pp Lesson 547

2 Getting Started Mental Math and Reflexes Use slates. Dictate numbers and have students identify digits in given places. 6 0,726. Circle the 10-s digit. Underline the hundred-s digit. 26,014,61. Circle the 10-millions digit. Underline the ten-s digit. 4,269, Circle the 10-millions digit. Underline the tenths digit. Math Message On a half-sheet of paper, make name-collection boxes for 100; 1,000; and 1,000,000. Write three different names in each box. Use expntial notation at least once. Study Link 7 1 Follow-Up Have partners share answers and resolve any differences. Recognizing Student Achievement Math Message Use the Math Message to assess students familiarity with writing expntial notation for powers of 10 and their ability to write equivalent names for numbers. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Have students share their answers. Write the different names in name-collection boxes on the board. Answers should include the following: 100: 10 10; _ 1,000 ; 1 hundred; ,000: ; _ 10,000 ; 1 ; ,000,000: ; _ 5,000,000 ; 1 million; 10 6 Ask students to describe the kinds of notation that are included on the board. Examples of powers of 10 written in expntial notation are 10 2, 10, and Examples of powers of 10 written in number-and-word notation are 1 hundred, 1, and 1 million. Explain that number-and-word notation is often used to express large numbers using a few numerals and or two words (for example, 25 billion, 5 hundred ), because long strings of zeros can be hard to read. Write number-and-word notation along with the example 25 billion on the board or transparency. Write standard notation along with the example 25,000,000,000. Ask students to compare the two ways of expressing the same number. Discuss how to translate from number-and-word notation to standard notation. One way: 25 billion = 25 1,000,000,000 = 25,000,000, Unit 7 Expnts and Negative Numbers

3 Another way: Use a place-value chart to position the leading digits. Then add zeros to complete the number. Have volunteers write number-and-word notations for the class to write in standard notation. Introducing Expntial Notation for (Student Reference Book, p. 5) WHOLE-CLASS DISCUSSION Refer students to page 5 of the Student Reference Book. As a class, discuss the presented definition of powers of 10 whole numbers that can be written using only 10s as factors. For example, 1,000 = = 10. Ask students to look at the Chart on the page and share their ideas about what patterns might help them figure out standard notation for powers of 10. Guide them to observe that the number of zeros in a power of 10, written in standard notation, is equal to the expnt of that number, written in expntial notation. For example, 1,000,000 has 6 zeros, so the expnt of the power of 10 is 6; 1,000,000 = The next three periods to the left of billions are trillions, then quadrillions, then quintillions. How many zeros are needed to write 1 trillion in standard notation? 12 zeros How many times will 10 appear in the repeated factor expression? 12 times How many periods are to the right of trillions? 4 periods Date Study the place-value chart below. Time Guides for Periods Millions Thousands Ones Hundred Ten Hundred Ten millions millions s s Billions Millions Thousands Hundreds Tens Ones In our place-value system, the powers of 10 are grouped into sets of three: s, s, millions, billions, and so on. These groupings, or periods, are helpful for working with large numbers. When we write large numbers in standard notation, we separate these groups of three with commas. There are prefixes for the periods and for other important powers of 10. You know some of these prefixes from your work with the metric system. For example, the prefix kilo- in kilometer identifies a kilometer as 1,000 meters. Use the place-value chart for large numbers and the prefixes chart to complete the following statements. Example: 1 kilogram equals 10, or, grams. 1. The distance from Chicago to New Orleans is about 10, or, miles. 2. A millionaire has at least 10 dollars. Student Page 6. A computer with 1 gigabyte of RAM memory can hold approximately 10, or billion, bytes of information. 4. A computer with a 1 terabyte hard drive can store approximately 10, or trillion, bytes of information. Prefixes tera- trillion (10 12 ) giga- billion (10 9 ) mega- million (10 6 ) kilo- (10 ) hecto- hundred (10 2 ) deca- ten (10 1 ) uni- (10 0 ) deci- tenth (10 1 ) centi- hundredth (10 2 ) milli- th (10 ) micro- millionth (10 6 ) nano- billionth (10 9 ) 5. According to some scientists, the hearts of most mammals will beat about 10 9, or billion, times in a lifetime. Math Journal 2, p What is the relationship between the number of periods to the right of trillions and the expnt when 1 trillion is written in expntial notation? Each period has digits, so 1 trillion would have digits 4 periods, or 12 zeros. Record a few examples on the board, and ask students to write these numbers in expntial or standard notation. Suggestions: 10, ; 100, ; ; 10,000, ,000; ; ,000; ,000,000,000 Using Guides for (Math Journal 2, p. 212; Student Reference Book, p. 5; Math Masters, p. 4) Have students read the introductory paragraphs on journal page 212. Use the example to discuss how to use the place-value chart and the table of prefixes to work with powers of 10. Mention that these guides are also found on the inside front cover of their journals. Assign the problems on the rest of the page. Informing Instruction Watch for students who have difficulty identifying the expnts for Problems and 4. Suggest that they use the placevalue chart on the journal page to first write the number in standard notation and then count the 0s to determine the expnt. Alternatively, use a transparency of Math Masters, page 4 and have students use copies of the page to practice writing in standard notation. Lesson 549

4 Date. Measure P to the nearest degree. P Math Boxes P measures about 19. Student Page Time 1. Measure the length and width of each of the following objects to the nearest half inch. Answers vary. a. piece of paper b. dictionary length in. width in. length in. width in. c. palm of your hand d. (your choice) length in. width in. length in. width in. 2. Amanda collects dobsonflies. Below are the lengths, in millimeters, for the flies in her collection. 95, 107, 119, 10, 102, 91, 115, 120, 111, 114, 115, 107, 110, 98, a. Circle the stem-and-leaf plot below that represents this data. Stems Leaves Stems Leaves Stems Leaves (100s and 10s) (1s) (100s and 10s) (1s) (100s and 10s) (1s) b. Find the following landmarks for the data. Median: 110 Minimum: 91 Range: 29 Mode(s): 107, 115 Math Journal 2, p Calculate the sale price. Regular Discount Price Price Sale $ % $ % $ % $ % 18 $9.00 $5.99 $48.00 $ Ongoing Learning & Practice Playing First to 100 (Student Reference Book, p. 08; Math Masters, pp ) Algebraic Thinking Students practice solving open number sentences by playing First to 100. This game was introduced in Lesson 4-7. For detailed instructions, see Student Reference Book, page 08. Math Boxes (Math Journal 2, p. 21) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 7-4 and 7-6. The skill in Problem 4 previews Unit 8 content. Study Link (Math Masters, p. 191) INDEPENDENT Home Connection Students are asked to memorize the Guides for and answer questions about them. Study Link Master Name Date Time STUDY LINK Guides for There are prefixes that name powers of 10. You know some of them from the metric system. For example, kilo- in kilometer (1,000 meters). It s helpful to memorize the prefixes for every third power of 10 through trillion. Memorize the table below. Have a friend quiz you. Then cover the table, and try to complete the statements below. Standard Number-and-Word Expntial Notation Notation Notation Prefix Differentiation Options READINESS Finding Patterns in (Math Masters, p. 192) 15 0 Min PROBLEM SOLVING 1, kilo- 1,000,000 1 million 10 6 mega- 1,000,000,000 1 billion 10 9 giga- 1,000,000,000,000 1 trillion tera- 1. More than 10 9, or billion, people live in China. 2. One, or 10, feet is a little less than 1_ 5 of a mile.. Astronomers estimate that there are more than 10 12, or trillion, stars in the universe. 4. More than million, or 10 6, copies of The New York Times are sold every day. 5. A kiloton equals, or 10, metric tons. 6. A megaton equals million, or 10 6, metric tons. Practice Find the prime factorization of each number, and write it using expnts = = Write each number in expanded notation. 9.,264 =, ,511 = 600, , , Math Masters, p. 191 ENRICHMENT To investigate patterns in powers of 10, have students complete the table on the Math Masters page and describe the patterns they identify in the table. Introducing Negative Expnts and Powers of 0.1 (Student Reference Book, p. 7; Math Masters, p. 19) 15 0 Min To apply students understanding of expnts, have them explore the patterns and notation of negative expnts. Read and discuss Student Reference Book, 550 Unit 7 Expnts and Negative Numbers

5 page 7. Emphasize that negative expnts are another way to represent numbers that are less than 1. Use examples to discuss converting between expntial notation with negative expnts and fractions. Suggestions: 5-2 = 1_ = 1_ = 1_ = 1_ = 1_ = 1_ = 1_ = 1_ = 1_ 16 Negative expnts can be used to express negative powers of = 1_ = 1_ = 1_ 1,000 This equation also shows that negative powers of 10 are also positive powers of = 1_ 10 1_ 10 1_ 10 = = 0.1 = Discuss the table on Math Masters, page 19. Students work with their partners, using the table to answer the questions that follow. Briefly go over the answers. Name Date Time Find the patterns and complete the table below. Do not use your Student Reference Book. 1,000, ,000 10,000 1, ten hundred ten hundred million _ Teaching Master Describe at least pattern you used to complete the table. Sample answer: Each time you move column to the right, you divide by 10. Math Masters, p Describe what happens to the decimal point in the standard notation as you move column to the right in the table. Sample answer: Each time you move column to the right, the decimal point moves 1 place to the left.. Describe what happens to the value of the digit 1 when you move column to the left. Sample answer: The value of the digit 1 becomes 10 times as great as its value in the previous column. Sample answer: 4. Describe what happens to the value of the digit 1 when you move column to the right. The value of the digit 1 becomes 1_ 10 of its value in the previous column. 5. Describe a pattern in the number of zeros used in the standard notation that you used to complete the table. Sample answer: The number of zeros in the standard notation matches the expnt in the power of 10. Links to the Future Negative expnts and powers of 0.1 will be investigated further in Sixth Grade Everyday Mathematics. The Enrichment activity is provided for exposure only. EXTRA PRACTICE WHOLE-CLASS DISCUSSION Multiplying Decimals by 5 15 Min Teaching Master Name Date Time To offer students more practice multiplying decimals by powers of 10, pose problems like those below. For each problem, have students write the original problem, rewrite the problem with the power of 10 written in standard notation, and then solve the problem ; ,000; 5, ,000; 407, ,000; 52,000 Have students explain the relationship between multiplying by a power of 10 and the placement of the decimal point in the product. py g g p Negative Our base-ten place-value system works for decimals as well as for whole numbers. Tens Ones. Tenths Hundredths Thousandths 10s 1s. 0.1s 0.01s 0.001s Negative powers of 10 can be used to name decimal places. Example: 10-2 = 1_ 10 = 1_ = 1_ 10 1_ 10 = = 0.01 Very small decimals can be hard to read in standard notation, so people often use number-and-word notation, expntial notation, or prefixes instead. Guides for Small Numbers Number-and-Word Standard Expntial Notation Notation Notation Prefix 1 tenth 10-1 = 1_ deci- 1 hundredth 10-2 = 1_ centi- 1 th 10 - = 1_ milli- 1 millionth = micro- 1 billionth = nano- 1 trillionth = pico- Use the table above to complete the following statements. 1. A fly can beat its wings once every 10 - seconds, or once every th of a second. This is milli second. 2. Earth travels around the sun at a speed of about inch per microsecond. This is 10-6 second, or a millionth of a second. billionth. Electricity can travel foot in a nanosecond, or of a second. This is 10-9 second In 10 second, or picosecond, an air molecule can spin once. This is trillionth of a second. Math Masters, p. 19 Lesson 551

6 Name Date Time Find the patterns and complete the table below. Do not use your Student Reference Book. 1,000, ,000 10,000 1 hundred _ Describe at least pattern you used to complete the table. 2. Describe what happens to the decimal point in the standard notation as you move column to the right in the table.. Describe what happens to the value of the digit 1 when you move column to the left. 4. Describe what happens to the value of the digit 1 when you move column to the right. 5. Describe a pattern in the number of zeros used in the standard notation that you used to complete the table. Copyright Wright Group/McGraw-Hill 192

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