Comparing Millions, Billions, and Trillions

Transcription

1 Comparing Millions, Billions, and Trillions Objectives To provide experience with comparing the relative sizes of million, billion, and trillion and using a sample to make an estimate. 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 Read and write large numbers. [Number and Numeration Goal ] Compare order of magnitude for large numbers. [Number and Numeration Goal 6] Make reasonable estimates for whole number multiplication problems. [Operations and Computation Goal 6] Key Activities Students review time conversion factors. They count the number of times they can tap their desks in 0 seconds and estimate how long it would take to tap million times. Students then estimate how long it would take to tap billion and trillion times. Ongoing Assessment: Informing Instruction See page. Key Vocabulary sample Playing High-Number Toss Student Reference Book, p. 0 Math Masters, p. 8 per partnership: die sheet of paper Students practice concepts of place value and standard notation by writing and comparing large numbers. Ongoing Assessment: Recognizing Student Achievement Use Math Masters, page 8. [Numbers and Numeration Goals and 6] Math Boxes 0 Math Journal, p. 8 Students practice and maintain skills through Math Box problems. Study Link 0 Math Masters, p. 6 Students practice and maintain skills through Study Link activities. READINESS Playing Number Top-It (-Digit Numbers) Student Reference Book, p. 6 Math Masters, pp. 9 and 9 per partnership: each of number cards 0 9 (from the Everything Math Deck, if available) Students apply place-value concepts to form, read, and compare large numbers. EXTRA PRACTICE Comparing Powers of 0 Using Place Value Math Masters, p. 66B Students apply place value for powers of 0. ENRICHMENT Applying Estimation Strategies Math Masters, p. 6 Students make time estimates and identify the number models used for their estimation strategies. Materials Math Journal, p. Study Link 9 Class Data Pad blank paper or construction paper markers or crayons watch or timer with second hand calculator Advance Preparation For Part, use the Class Data Pad to record and display the Mental Math and Reflexes problems and responses. For the optional Readiness activity in Part, make one game mat for each partnership by copying, cutting, and taping together Math Masters, pages 9 and 9. For a mathematics and literary connection, obtain a copy of How Much Is a Million? by David M. Schwartz (HarperCollins Publishers, 98). Teacher s Reference Manual, Grades 6 pp. 6 6 Unit Estimation and Computation

2 Getting Started Mathematical Practices SMP, SMP, SMP, SMP, SMP6 Content Standards.NBT.,.MD. Mental Math and Reflexes Have students practice conversions between units of time. Use the Class Data Pad to record the correct responses. Keep this display up for students to refer to during the lesson. How many seconds are in... minute? 60 minutes? minutes? 6,000 How many minutes are in... hour? 60 hours? 00 0 hours?,000 How many hours are in... day? days? 8 00 days?,800 How many days are in... year? 6, except 66 in leap years 0 years? about,60 00 years? about 6,00 Math Message Explain the strategy you would use to find the number of minutes in one year. Study Link 9 Follow-Up Have partners share answers and resolve any differences. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Have students in small groups discuss their individual strategies. The group then decides which steps they would use to find the number of minutes in a year. Each group should make a poster, using construction paper and markers or crayons, to list their steps and place the poster on display. Allow time for students to read the displayed group posters. Ask volunteers to identify the similarities and differences in the strategies. Guide the class discussion to focus on summarizing the strategies. Sample answers: Convert a year into a unit of time that can be converted to minutes; convert a year into days; convert these days into hours and the hours into minutes. Date 0 Time Millions, Billions, and Trillions NOTE Include a walkabout in the follow-up to this Math Message: Display the group posters in separate areas of your classroom, and allow students time to browse until they have read all the posters. Solving a Tapping Problem with Sampling Strategies (Math Journal, p. ) PROBLEM SOLVING WHOLE-CLASS DISCUSSION Ask students to refer to the Useful Information chart on journal page. Pose several questions from the first row of information to highlight these large number relationships for students. What is,000 times million? billion One billion has what relationship to trillion? One trillion is,000 times billion. Ask a volunteer to read the question labeled Make a guess. Ask students to guess how long it would take them to tap their desks million times without any interruptions. Have students Make a guess: How long do you think it would take you to tap your desk million times, without any interruptions? Answers vary. Check your guess by doing the following experiment. Sample answers:. Take a sample count. Record your count of taps made in 0 seconds. 0 taps. Calculate from the sample count. At the rate of my sample count, I expect to tap my desk: a. 0 times in minute. (Hint: How many 0-second intervals are there in minute?) b.,00 times in hour. c.,600 times in day ( hours). Useful Information billion is,000 times million. trillion is,000 times billion. million thousand = billion billion thousand = trillion,000,000,000 =,000,000,000,000,000,000,000 =,000,000,000,000 minute = 60 seconds hour = 60 minutes day = hours year = 6 days (66 days in a leap year) d. At this rate it would take me about full -hour days to tap my desk million times.. Suppose that you work hours per day tapping your desk. Estimate how long it would take you to tap billion times and trillion times. a. It would take me about 8 years to tap my desk billion times. (unit) (unit) b. It would take me about 8,000 years to tap my desk trillion times. Math Journal, p. Lesson 0

3 Games High-Number Toss Materials six-sided die sheet of paper for each player Players Skill Place value, exponential notation Object of the game To make the largest numbers possible. Directions. Each player draws blank lines on a sheet of paper to record the numbers that come up on the rolls of the die. Player : Player :. Player rolls the die and writes the number on any of his or her blank lines. It does not have to be the first blank it can be any of them. Keep in mind that the larger number wins!. Player rolls the die and writes the number on one of his or her blank lines.. Players take turns rolling the die and writing the number more times each.. Each player then uses the numbers on his or her blanks to build a number. The numbers on the first blanks are the first digits of the number the player builds. Note If you don t have a die, you can use a deck of number cards. Use all The number on the last blank tells the number of zeros that come after the first digits. cards with the numbers through 6. Instead of rolling the die, draw the 6. Each player reads his or her number. (See the place-value chart below.) The player with the larger number wins the top card from the facedown deck. round. The first player to win rounds wins the game. record their responses on the journal page. Conduct a quick survey of the class for their responses. Discuss the difference between a guess and an estimate. Use the discussion to clarify for students that a guess is an opinion that you might state without the support of other information. An estimate is based on some knowledge about the subject and is often called an educated guess. Students can only guess the amount of time it would take to tap million times until they collect additional information with which to make an estimate. Encourage students to think about what information they would need to make a more educated guess, and then ask volunteers to explain strategies that could be used to gather additional information. Hundred Ten Hundred Ten Millions Millions Millions, Thousands Thousands Thousands, Hundreds Tens Ones Number First three digits of zeros Player : 6,000,000 ( million) Player : 6,60,000 ( million, 60 thousand) Player wins. Student Reference Book, p. 0 Ongoing Assessment: Informing Instruction Watch for students who are having difficulty developing a strategy. Explain that this is another situation for which obtaining the exact answer is impossible, such as the Estimation Challenge from Lesson -. Recall the strategies students used in that lesson and have them use similar approaches here. Name Date Time High-Number Toss Record Sheet Hundred Millions Ten Millions Game Master Hundred Ten, Thousands Millions, Thousands Thousands Hundreds Tens Ones Round Player >, <, = Player Sample 6 > 6,000,000,60,000 Students might suggest strategies such as the following: Count how many times you can tap your desk in a set amount of time, such as 0 seconds. Time how long it takes you to tap a certain number of times, such as 00 times. Pick a reasonable number of taps for a set amount of time, and make an estimate based on that rate, such as taps per second. When the class found and used the median step length, they were using a sample. Ask whether students know of any other situations where samples are used. NOTE A sample of anything is a small piece or part that is intended to give information about the whole thing. Consumers use product samples to decide whether products suit their needs. Pollsters use population samples to estimate information for the whole population. The finger-tapping samples here are time samples: The count of taps in a 0-second period is a sample used to determine a tapping rate, and the rate is then used to estimate how long it would take to make large numbers of taps. Each student will take a 0-second sample count of their own finger tapping. Practice taking sample counts by timing students as they tap and count for 0 seconds. py g g p 0 Math Masters, p. 8 Unit Estimation and Computation

4 Using Sampling to Make an Estimate (Math Journal, p. ) Have partners complete the journal page. They begin by finding their individual sample counts. Partners take turns. While one partner taps and counts the taps for 0 seconds, the second partner keeps time for 0 seconds, signaling when to start and stop. Partners then use their 0-second sample counts to estimate the number of taps they could make in minute; in hour; and in day. Encourage students to use their calculators, as needed. Next students use their estimates to calculate the approximate number of days it would take to tap million times. Encourage students to devise their own solution strategies. One possible approach is to divide million by the number of taps per day. Making Time Estimates for Billion and Trillion Taps (Math Journal, p. ) In Problem on journal page, students estimate the time it would take to tap billion and trillion times. Remind students that they can use the relationships between million, billion, and trillion found in the Useful Information chart to help them estimate. They will also need to decide whether to report their estimates for billion and trillion taps as days or years. NOTE Expect that the tapping rate for most students will be about 0 times in 0 seconds. At this rate they will tap about 0 times in minute (6 0, rounded up);,000 times in hour (60 0); 0,000 times in day (,000; rounded down), and about days, without interruptions, to tap million times. Sharing and Discussing the Results (Math Journal, p. ) SMALL-GROUP DISCUSSION When most students have completed the problems, have partners form small groups to discuss their strategies. Then have the groups report on the similarities and differences of the strategies used as well as any notable experiences they encountered. Use the following questions as a guide: How does your estimate of the time for million taps compare with your initial guess? Did you use your estimate for the number of taps in day to estimate how long it would take to tap million times? Did you use the time for million taps to estimate the time for billion and trillion taps? Date 0 Math Boxes Time. Find the missing numbers and landmarks for the set of numbers below. 8, 0,,,, 9, 60, 6, 69, 6, 6, 6 8 a. Range: c. Minimum: b. Mode: 6 d. Maximum: 6. a. Make up a set of at least twelve numbers that have the following landmarks. Maximum: 8 Range: 6 Mode: 6 Median: Sample answer:,,,,,,, 6, 6, 6, 6, 8, 8 b. Make a bar graph of the data. Sample answer: Use the map on page of your Student Reference Book to answer the questions. Choose the best answer. a. What is the shortest distance between San Francisco and San Antonio? About: 00 miles,000 miles,00 miles,000 miles b. What is the shortest distance between New York City and Chicago? About: 00 miles 800 miles 900 miles,000 miles. a. Circle the times for which the hands on a clock form an acute angle. :00 6:0 :0 :0 b. Circle the times for which the hands on a clock form an obtuse angle. 8:00 :0 : 0: Math Journal, p. 8 Lesson 0

5 Copyright Wright Group/McGraw-Hill STUDY LINK 0 Place-Value Puzzles 8 Millions Thousands Ones Hundred- Ten- Millions Hundred- Ten- Thousands Hundreds Tens Ones millions millions thousands thousands Use the clues to solve the puzzles. Puzzle The value of the digit in the thousandths place is equal to the sum of the measures of the angles in a triangle (80 ) divided by 0. If you multiply the digit in the tens place by,000; the answer will be 9,000. Double. Divide the result by 0. Write the answer in the tenths place. The hundreds-place digit is the value of the digit in the thousandths place. When you multiply the digit in the ones place by itself, the answer is 0. Write a digit in the hundredths place so that the sum of all six digits in this number is 0. What is the number? Puzzle Double. Divide the result by 8. Write the answer in the thousands place. If you multiply the digit in the hundredths place by 0, your answer will be 0. The tens-place digit is a prime number. If you multiply it by itself, the answer is 9. Multiply and. Subtract. Write the answer in the thousandths place. Multiply the digit in the hundredths place by the digit in the thousands place. Subtract from the result. Write the digit in the tenths place. The digit in the ones place is an odd digit that has not been used yet. The value of the digit in the hundreds place is the same as the number of sides of a quadrilateral. What is the number?,. 9 Check: The sum of the answers to both puzzles is,86.0. Practice Study Link Master.,,68 9, º, / 9 R Math Masters, p. 6 6 To conclude Part ask students: If one person could tap hours per day without stopping, would it be possible to tap trillion times? No Aim follow-up questions at getting students to support their responses. They would still need many more years than are in a normal lifetime. An exit question might be: Do you feel that your informed estimate was more reasonable than your guess? Ongoing Learning & Practice Playing High-Number Toss (Student Reference Book, p. 0; Math Masters, p. 8) High-Number Toss provides students with the opportunity to apply their knowledge of place value and standard notation to create, write, read, and compare large numbers. Provide students with a reminder box on the board noting that < means less than and > means greater than. Ongoing Assessment: Recognizing Student Achievement Math Masters Page 8 Use the Record Sheet for High-Number Toss (Math Masters, page 8) to assess students knowledge of place value and comparing numbers. Students are making adequate progress if they correctly insert the relational symbols between the two numbers in five rounds of the game. [Numbers and Numeration Goals and 6] Games Number Top-It (-Digit Numbers) Materials number cards 0 9 ( of each) one Place-Value Mat (Math Masters, pp. 9 and 9) Players to Skill Place value for whole numbers Object of the game To make the largest -digit numbers. Directions. Shuffle the cards and place the deck number-side down on the table.. Each player uses one row of boxes on the place-value mat.. In each round, players take turns turning over the top card from the deck and placing it on any one of their empty boxes. Each player takes a total of turns, and places cards on his or her row of the game mat.. At the end of each round, players read their numbers aloud and compare them to the other players numbers. The player with the largest number for the round scores point. The player with the next-largest number scores points, and so on.. Players play rounds for a game. Shuffle the deck between each round. The player with the smallest total number of points at the end of rounds wins the game. Example Andy and Barb played -digit Number Top-It. Here is the result for one complete round of play. Place-Value Mat Millions Hundred Ten Thousands Thousands Thousands Hundreds Tens Ones Math Boxes 0 (Math Journal, p. 8) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -8. The skill in Problem previews Unit content. Study Link 0 (Math Masters, p. 6) INDEPENDENT Home Connection Students use their knowledge of place value and number relationships to solve number puzzles. Andy Barb Andy s number is larger than Barb s number. So Andy scores point for this round, and Barb scores points. Student Reference Book, p. 6 6 Unit Estimation and Computation

6 Differentiation Options READINESS Playing Number Top-It (-Digit Numbers) (Student Reference Book, p.6; Math Masters, pp. 9 and 9) 0 Min To review place-value concepts, have students play Number Top-It (-Digit Numbers). EXTRA PRACTICE SMALL-GROUP Comparing Powers of 0 Using Place Value (Math Masters, p. 66B) Min Name Date Time 0 Number Stories and Estimation Read each number story carefully. Write an open number sentence to use in estimating. Answer the question. Example: It is said that the Aztec king, Montezuma, drank about 0 cups of chocolate per day. Did he drink more or less than 0 gallons of chocolate in a week? (Hint: 6 cups gallon) Open number sentence: 0 º 6 Number of cups in 0 gallons Answer: more. Certain varieties of seahorses can move 0. inches per minute. At this rate, could these seahorses be able to travel 6 yards in hour? a. Open number sentence: Sample answer: 0. º 60 6 yards traveled in hour Yes b. Answer:. Orville Wright completed the first airplane flight on December, 90. He traveled 0 feet in seconds. If he had been able to stay in the air for a full minute, would he have traveled mile? (Hint: mile,80 feet) a. Open number sentence: Sample answer: 0 º feet traveled in minute No b. Answer:. In 960, the Triton became the first submarine to circumnavigate the world. It covered 6,0 miles in 6 days. Is that more or less than 00 miles per day? a. Open number sentence: Sample answer: 00 º 6 total miles for 6 days at 00 miles per day more b. Answer: Source: The Kids World Almanac of Records and Facts Math Masters, p. 6 Teaching Master To provide additional practice with place value and understanding the relationships between powers of 0, have students complete Math Masters, page 66B. ENRICHMENT INDEPENDENT Applying Estimation Strategies (Math Masters, p. 6) Min To apply students ability to use estimation strategies, have them solve problems that involve converting situational information into open number sentences. Direct students to focus on making informed estimates. Teaching Master Name Date Time 0 Using Place Value to Compare Powers of 0 meter 0 decimeters 00 centimeters,000 millimeters centimeter 0.0 meter 0. decimeter 0 millimeters Use the information in the conversion table to respond to each statement below. Complete each statement with one of the following phrases: 0 times, 00 times, _ 0 of, _ 00 of. meter is the size of a decimeter.. centimeter is 00 of the size of a meter.. centimeter is the size of a millimeter.. decimeter is 0 of the size of a meter.. millimeter is 00 of the size of a decimeter. Write two of your own statements using the information in the table times Answers vary. 0 times. Complete the table below by making the appropriate conversions. millimeters centimeters decimeters meters 9, , ,0. In Problem 0, explain what happens to the value of the digit when you go from millimeters to centimeters, and then from decimeters to meters. Sample answer: You divide by 0 with each conversion; so each time, you move the decimal point one position to the left. So with each move, the digit is worth as much as before. 0 Math Master, p. 66B 0.. Lesson 0

7 Name Date Time 0 Using Place Value to Compare Powers of 0 meter 0 decimeters 00 centimeters,000 millimeters centimeter 0.0 meter 0. decimeter 0 millimeters Use the information in the conversion table to respond to each statement below. Complete each statement with one of the following phrases: 0 times, 00 times, _ 0 of, _ 00 of. meter is the size of a decimeter.. centimeter is the size of a meter.. centimeter is the size of a millimeter.. decimeter is the size of a meter.. millimeter is the size of a decimeter. Write two of your own statements using the information in the table. 6.. Complete the table below by making the appropriate conversions. Copyright Wright Group/McGraw-Hill millimeters centimeters decimeters meters 8. 9, In Problem 0, explain what happens to the value of the digit when you go from millimeters to centimeters, and then from decimeters to meters. 66B