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Revisiting Numbers Number CuuDuongThanCong.com https://fb.com/tailieudientucntt Mathematics in Context is a comprehensive curriculum for the middle grades It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928 This unit is a new unit prepared as a part of the revision of the curriculum carried out in 2003 through 2005, with the support of the National Science Foundation Grant No ESI 0137414 National Science Foundation Opinions expressed are those of the authors and not necessarily those of the Foundation Abels, M., Wijers, M., and Pligge, M (2006) Revisiting numbers In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context Chicago: Encyclopỉdia Britannica, Inc Copyright © 2006 Encyclopædia Britannica, Inc All rights reserved Printed in the United States of America This work is protected under current U.S copyright laws, and the performance, display, and other applicable uses of it are governed by those laws Any uses not in conformity with the U.S copyright statute are prohibited without our express written permission, including but not limited to duplication, adaptation, and transmission by television or other devices or processes For more information regarding a license, write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610 ISBN 0-03-038568-7 073 09 08 07 06 05 CuuDuongThanCong.com https://fb.com/tailieudientucntt The Mathematics in Context Development Team Development 2003–2005 Revisiting Numbers was developed by Mieke Abels and Monica Wijers It was adapted for use in American Schools by Margaret A Pligge Wisconsin Center for Education Freudenthal Institute Staff Research Staff Thomas A Romberg David C Webb Jan de Lange Truus Dekker Director Coordinator Director Coordinator Gail Burrill Margaret A Pligge Mieke Abels Monica Wijers Editorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator Margaret R Meyer Anne Park Bryna Rappaport Kathleen A Steele Ana C Stephens Candace Ulmer Jill Vettrus Arthur Bakker Peter Boon Els Feijs Dédé de Haan Martin Kindt Nathalie Kuijpers Huub Nilwik Sonia Palha Nanda Querelle Martin van Reeuwijk Project Staff Sarah Ailts Beth R Cole Erin Hazlett Teri Hedges Karen Hoiberg Carrie Johnson Jean Krusi Elaine McGrath CuuDuongThanCong.com https://fb.com/tailieudientucntt (c) 2006 Encyclopædia Britannica, Inc Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopædia Britannica, Inc Cover photo credits: (left to right) © William Whitehurst/Corbis; © Getty Images; © Comstock Images Illustrations 1, 2, 4, Christine McCabe/© Encyclopỉdia Britannica, Inc.; 11 (top, bottom) Jerry Kraus/© Encyclopỉdia Britannica, Inc.; (middle) Michael Nutter/© Encyclopỉdia Britannica, Inc.; 37, 54 Rich Stergulz Photographs © Tony Arruza/Corbis; Victoria Smith/HRW; (top) © Corbis; (middle) © Tim Davis/ Corbis; (bottom) R Clarke/Diomedea Images; Sam Dudgeon/HRW; (top) © Robert Galbraith/Reuters/Corbis; (bottom) © David Madison/NewSport/Corbis; 10, 14 PhotoDisc/Getty Images; 18 © Image 100; 20 Janice Carr/CDC; 23 Visuals Unlimited; 25, 36 Victoria Smith/HRW; 40 John Langfore/HRW CuuDuongThanCong.com https://fb.com/tailieudientucntt Contents Letter to the Student Section A Speed The Wave Make Some Waves Rates and Units Speed Records Reaction Time In a Jiffy Speed of Light Distance in Space Summary Check Your Work Section B ounce 25 27 29 32 33 Stir it up ounces 10 ounces Operations Funny Zero Negative Numbers Number Properties Summary Check Your Work Section E 16 17 20 22 23 Investigating Algorithms Multiplication Division Fraction Operations Summary Check Your Work Section D 2 8 10 12 13 Notation Base Ten Dilution Small Numbers Summary Check Your Work Section C vi 36 37 40 42 43 Reflections on Numbers Addition and Subtraction Multiplication and Division Random Number Activity Powers and Roots Summary Check Your Work 45 46 47 50 52 53 Additional Practice 54 Answers to Check Your Work 60 Contents v CuuDuongThanCong.com https://fb.com/tailieudientucntt Dear Student, Have you ever been at a competitive event where the race was too close to call? Electronic timers today are very precise and can split a second into a million parts Precision is extremely important to scientists as they map out unknown territories in outer space and inside the human body In the unit, Revisiting Numbers, you use will learn to use numbers more precisely You will further investigate ways to represent very large and very small numbers You will reflect on all number operations You will improve your precision working with number operations, by looking at related operations We hope you enjoy this unit Sincerely, The Mathematics in Context Development Team Neptune Venus vi Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt A Speed W ES T RA NO MP RT HE AS T R A P N R TH M O The Wave NW BROWNS SW 695 f t EAST SCOREBOAR D BROWNS WEST SCOREBOARD NE The Cleveland Browns Stadium can seat over 73,200 people Cleveland fans often create a wave—people stand up lifting their arms and then quickly sit down When one group sits down, an adjacent group takes over The wave moves around the entire stadium SE SO UT HW ES T R AM T SOU P HEA ST RA M P 933 ft a Estimate how many feet the wave travels one time around the stadium Describe how you made your estimate The stadium dimensions are 933 feet by 695 feet b How much time will it take the wave to go once around the stadium? Describe how you made your estimate c Use your estimates to find the average distance the wave travels in one second You may want to use a ratio table for your calculations Distance (in ft) Time (in sec) If a stadium wave travels 60 feet (ft) in seconds (sec), it travels with an average speed of 30 feet per second, or 30 ft/sec Section A: Speed CuuDuongThanCong.com https://fb.com/tailieudientucntt Make Some Waves For this activity, you will need: • • stop watch tape measure Discuss with a classmate how to find the average speed of a wave Share your idea with the class Decide on a plan to find the average speed of a wave in your classroom Record the time (in seconds) and distance (in feet) for a wave to travel through your classroom a Use the data from the activity to calculate the average speed of the wave in your class b Compare your class’s wave with the Cleveland Stadium wave Describe your findings Rates and Units A rate is the ratio of two different measuring units For example, you can express the rate of speed in miles per hour (mi/h), or in feet per second (ft/s) Using metric units, speed is usually expressed in kilometers per hour (km/h), or meters per second (m/s) Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Speed A What other rates you know? Copy this table and complete it Units Example Heart Rate Heartbeats/minute (bmp) My heart beats at a rate of 65 beats per minute Data Transfer Kilobytes/second (kB/s) The speed of the download stream is 1,024 kilobytes per second Population Density You may remember from the unit Ratio and Rates how you used ratio tables to express rates as a single number Stuart and Lexa are researchers who analyze average speeds of large crowds In a soccer stadium, they timed one wave taking 22 sec to travel 440 seats Each seat had a total width of ft a Calculate the average speed of the wave in seats per second b Compare the average speed from this research to the speeds you found in problem 2b How they compare? Rita found 30 ft/s as the average speed of the wave in her class She wants to know how fast this is in miles per hour Here is how she started to solve the problem ؋ 60 Distance (in ft) 30 Time (in sec) ؋ 60 Rita: “First, I multiplied by 60 to get the number of feet per 60 seconds, or per one minute.” a Explain Rita’s second step ؋ 60 ؋ 60 b Copy Rita’s ratio table and calculate the missing numbers Did you know? 5,280 ft are in mi? c Use this information to find the average speed of the wave into miles per hour (mi/h) Section A: Speed CuuDuongThanCong.com https://fb.com/tailieudientucntt A Speed You can use a similar technique to convert meters per second (m/s) into kilometers per hour (km/h) Kenny ran 60 m in 15 sec a Copy this ratio table and use the arrows to show the steps you have to make to find Kenny’s distance in one hour Distance (in m) 60 Time (in sec) 15 b What is Kenny’s average speed in meters per hour? What is Kenny’s average speed in km/h? c Will Kenny really be able to run that distance in one hour? Explain d Henri ran 50 m in 12 sec Is Henri’s average speed higher or lower than Kenny’s? Show your work Maddie: “I did the 5K run in 25 minutes I wonder how fast I ran in kilometers per hour.” Calculate Maddie’s average speed for the 5K run in km/h To compare speeds, you may have to change the units Changing kilometers per hour into miles per hour is easy if you have an speedometer like this one Explain why this speedometer is actually a double number line Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Additional Practice Section A Speed Speed of Sound Lena saw a flash of lightening and heard the thunder three seconds later She knows a rule to estimate the storm’s distance—one mile for every five seconds What estimate will Lena get for the storm’s distance? To understand why this works, you have to know that there is a big difference between the speed of light and the speed of sound Sound travels through air with a speed of about 760 mi/h Light travels almost 106 times faster than sound a What is the speed of light in mi/h? Write your answer as a numeral b Use the speed of sound to calculate the distance in miles that sound can travel in three seconds c Is Lena’s rule reasonable? Explain Peter is more familiar with metric units He uses this rule—one kilometer for every three seconds Which rule is more accurate, Lena’s rule or Peter’s rule? Explain Calculate the speed of sound in feet per second When you stand in front of a rock wall and you clap your hands, you may hear an echo The sound waves move from your hand to the wall, then reflect off the wall and travel the same distance back to you Kim heard an echo a half a second after she clapped her hands How far was Kim standing from the wall? You may want to use the speed of sound you calculated in problem 54 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt You can hear an echo when the time the sound travels from you to the wall and back again is more than 0.05 seconds What is the minimum distance you need to stand from the wall to hear an echo? Section B Notations Sound is actually a form of energy The intensity of sound is measured in Watts per square meter The softest sound that a human ear can detect has an intensity of ؋ 10-12 Watts per square meter (W/m2) Here are some sounds and an estimation of their intensity level Intensity (in W/m2) Times Greater Than Threshold of Hearing (TOH) TOH ؋ 10–12 Rustling leaves ؋ 10–11 10 Whisper ؋ 10–10 Library conversation ؋ 10–9 Normal conversation ؋ 10–6 Sound Source a Consider the relationship between the threshold of hearing (TOH) and the sound of rustling leaves Explain that the sound intensity of rustling leaves is ten times greater than the TOH b How many times greater is the intensity of a library conversation than the TOH? c How many times greater is the intensity of a normal conversation than the intensity of a library conversation? Additional Practice 55 CuuDuongThanCong.com https://fb.com/tailieudientucntt Additional Practice A new scale, the decibel scale, makes it easier to work with these small numbers A sound intensity of decibel (dB) represents the threshold of hearing Sound Source Intensity (in W/m2) Times Greater Than TOH Decibel Level Threshold of Hearing (TOH) ؋ 10–12 dB Rustling leaves ؋ 10–11 10 10 dB Whisper ؋ 10–10 20 dB Library conversation ؋ 10–9 30 dB Normal conversation ؋ 10–6 You can use the table on Student Activity Sheet to solve the following problems a How many decibels is the sound intensity of a normal conversation? Music using an MP3 player has an intensity of ؋ 10–2 W/m2 b How many decibels does a MP3 player produce? The threshold of pain is 1014 ؋ TOH This level causes pain and permanent hearing damage c What is the decibel level for threshold of pain? 56 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Additional Practice Section C Investigating Algorithms Throughout history, people have used different algorithms to the same calculations The gelosia or lattice method is an algorithm for multiplication used in Italy around 1500 A.D Today, some people use a shortened version of this algorithm Here is an example of the problem Harvey worked (24 ؋ 49) using the lattice method You might recall the product is 1,176 1 8 6 a Explain how the lattice algorithm works b Check your explanation by using the lattice method to multiply 52 ؋ 63 and 254 ؋ 647 Use a calculator to check your results a Compare the lattice algorithm with the algorithm Hattie used on page 26 What are the advantages of using the lattice method? What are the disadvantages? b Compare the lattice method with the method used by Clarence shown on page 27 Michael creates the following story to solve the division problem: ، ᎑᎑ ᎑᎑ miles I know that a city block is “I walked ᎑᎑ ᎑᎑ of a mile Each mile 1 ؋ .” has eight blocks, so instead of ᎑᎑ ، ᎑᎑ , I can calculate ᎑᎑ a Complete Michael’s story to rewrite the division problem ، ᎑᎑ ᎑᎑ as a multiplication problem ، b Solve the division problem: ᎑᎑ ᎑᎑ Additional Practice 57 CuuDuongThanCong.com https://fb.com/tailieudientucntt Additional Practice feet long He wants to cut the tape into Harvey has a piece of tape ᎑᎑ foot long pieces that are ᎑᎑ 4 a Write a division problem that matches this story b Solve the problem using Michael’s multiplication ، c How can you calculate ᎑᎑ ᎑᎑ 5? Section D Operations a Use the area model to illustrate the distributive property for finding the product of ؋ 24 b Use the area model to calculate ؋ 12.5 Show how you can use the distributive property to make some calculations easier Don’t forget to find each product a ؋ 27 c ؋ 99 b ؋ 49 d ؋ 28 Here are two multiplication problems illustrating the area model Copy and complete each model Include the multiplication problem associated with each model along with the correct answer 20 600 15 Make special number sentences for all whole numbers from through 10 Here are the rules for making the special number sentence • You must use the number exactly five times • You may choose any operation and use parentheses Here is an example for the whole number 2 = ؊ ᎑᎑ ؊ ᎑᎑ 58 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Additional Practice Section E Ref lections on Numbers Use Student Activity Sheet to assign a value to each grid-line intersection by multiplying the first coordinate by the second For example, if point A had the coordinates (3, 2), the value of point A would be because ؋ ؍6 Quadrant II Quadrant I y a Color the grid points that have a perfect square number Describe the location of all the perfect square numbers b Describe the location of all the negative numbers A –7 –6 –5 –4 –3 –2 –1 –1 –2 x c Color the grid points that have a prime number Describe the location of all the prime numbers –3 –4 –5 –6 –7 Quadrant III Quadrant IV How would you classify numbers that have a repeating decimal? Are they rational or irrational numbers? Can every repeating decimal be written as a fraction? Here is a sophisticated way to find the fraction that is associated with ᎑᎑ The notation 0.7 ᎑᎑ means that repeats indefinitely 0.7 (Subtracting) 10 ؋ 0.77777… ؍7.77777… ؊ ؋ 0.77777… ؍0.77777… ؋ 0.77777… ؍7 a Use ؋ 0.77777… ؍7 and what you know about the relationship between multiplication and division to complete ، ؍0.7777… b What fraction equals 0.7777…? ᎑᎑ Use a calculator c Apply this technique to find the fraction for 1.5 to check your result Additional Practice 59 CuuDuongThanCong.com https://fb.com/tailieudientucntt Section A Speed Helen’s average speed was km/h You may have used a ratio table to find your answer ؋2 Distance (in km) 18 Time (in hr) 42 ،9 36 1 ؋2 ،9 a The six means 6,000,000,000, which is six billion b 6.4 ؋ 109 The speed of the earth at the equator is about 1,042 mi/h Note that it doesn’t make sense to write this answer with decimals You may have used the following strategy The earth completes one revolution in one day, or 24 hours At the equator, the distance around the earth is about 2.5 ؋ 104 miles, which is 25,000 miles To find the speed in mi/h you can use a ratio table ، 24 Distance (in mi) 25,000 1,042 24 Time (in hr) ، 24 The average speed of the earth around the sun is 6.6 ؋ 104 mi/h, or 66,000 mi/h You may have used the following strategy The earth travels 5.8 ؋ 108 miles in 365 days One day ؍24 hours, so 365 days ؍8,760 hours ، 1,000 Distance (in mi) Time (in hr) ، 8.76 5.8 ؋ 108 5.8 ؋ 105 0.66 ؋ 105 or 66,000 8,760 ، 1,000 8.76 ، 8.76 60 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work Mercury’s average orbital speed is faster than Earth’s average orbital speed Your strategy may be different from this strategy: Mercury travels 48 km/s One hour ؍3,600 seconds, so that is my target ؋ 60 ؋ 60 Distance (in km) 48 2,880 172,800 Time (in sec) 60 3,600 ؋ 60 ؋ 60 Mercury’s average speed is about 17 ؋ 104 km/h Earth’s average speed is about 6.6 ؋ 104 mi/h Comparing 6.6 miles and 17 km, 17 km is more than 6.6 miles Mercury travels faster around the sun Here are the answers written in scientific notation along with one sample solution strategy a ؋ 106 (3 ؋ 104) ؋ (2 ؋ 102) b ؋ 104 (2 ؋ 1010) ، 106 ؍3 ؋ 104 ؋ ؋ 102 ( ؍3 ؋ 2) ؋ (104 ؋ 102) ؋ 106 ؍ ؍2 ؋ (1010 ، 106) ؍2؋ 104 c 2.45 ؋ 103 (2 ؋ 103) ؉ (4.5 ؋ 102) ؍2,000 ؉ 450 ؍2,450 Writing 2,450 in scientific notation is 2.45 ؋ 103 d 1.4 ؋ 103 (2 ؋ 103) ؊ (6 ؋ 102) ؍2,000 ؊ 600 ؍1,400 Writing 1,400 in scientific notation is 1.4 ؋ 103 Answers to Check Your Work 61 CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work a Your opinion might vary from these • I not think it is fair because it is unlikely that all of Larson’s timekeepers had a faster reaction time than all of Devitt’s timekeepers • It could be fair since the race was timed manually, and the timekeepers would have had different reaction times • I think it is fair because the swimmers were too close to distinguish actual finishing times More than likely, the judges decided that John Devitt finished first, so they had to award him the race in spite of the recorded times b Larson was about 18 cm behind, which is more than the length of a hand This distance is visible, but it occurs in 0.1 second Here is one strategy When Devitt finished, Larson had to swim 0.1 second Larson swam 100 meters in 55.2 seconds Using 100 m ؍10,000 cm to set up this ratio table: ، 55.2 Distance (in cm) Time (in sec) 10,000 ، 10 181.2 18 0.1 55.2 ، 55.2 ، 10 Larson was about 18 cm behind, which is more than the length of a hand In less than 0.1 seconds, I can understand why the manual timers had difficulty reacting to this visible distance Section B Notation a 1,000x means the hair is 1,000 times larger than its actual size b The thickness of the hair in the picture is about 3.8 cm c The actual thickness is about 0.0038 cm and written 3.8 ؋ 10-3 cm 62 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work a 1010 ، 105 ؍105 b 103 ، 105 ؍10-2 c 100 ، 100,000 ؍102 ، 105 or 10-3 d 10 ، 1,000,000,000 ؍101 ، 109 or 10-8 e 10-4 ، 10 ؍10-4 ، 101 or 10-5 f 10-6 ، 100 ؍10-6 ، 102 or 10-4 g 10-7 ، 103 ؍10-10 a 34,200 b 0.0342 a 1.6 ؋ 10-8 b (3.5 ؋ 103) ؋ (1.2 ؋ 10-2) ؍3.5 ؋ 1.2 ؋ 103 ؋ 10-2 ( ؍3.5 ؋ 1.2) ؋ (103 ؋ 10-2) ؍ 4.2 ؍ ؋ 101 42 a 0.00267 ؍2.67 ؋ 10-3 b 0.00000678 ؍6.78 ؋ 10-6 c 15 ، 20,000,000,000 ؍7.5 ؋ 10-10 Section C Investigating Algorithms The answer is 1,464, but you must show two different strategies Here are three sample strategies to calculate 24 ؋ 61 • Using Clarence’s algorithm: 20 ؋ 60 20 ؋ ؋ 60 4؋ ؍1,200 ؍ 20 ؍240 ؍ • Using a ratio table: 61 122 20 24 244 1,220 1,464 24 ؋ 61 ؍1,464 Answers to Check Your Work 63 CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work • Using the area model: 60 20 1,200 20 240 Adding up all the parts: 1,200 ؉ 20 ؉ 240 ؉ ؍1,464,so 24 ؋ 61 ؍1,464 a Everybody in your class probably made up a different context Compare and discuss your answer with another student Here are some contexts that fit with 3,000 ، 28 • Three thousand marbles shared equally among 28 students How many marbles will each student get? • There are three thousand cubes to pack in boxes Each box holds 28 cubes How many boxes you need to pack all the cubes? b Your strategy might match one of these • Using a ratio table: 28 100 2,800 196 107 2,996 3,000 ، 28 ؍107 with left over • Using mental computation: 100 ؋ 28 ؍2,800 ؋ 30 ؍210, so ؋ 28 ؍210 ؊ 14 ؍196 So 3,000 divided by 28 is 107 c Answers will vary, depending on your context in part a Sample responses based on sample contexts in part a: • • In the first story, there are four marbles left over (In reality, four people will probably receive an extra marble each, but that cannot work for this problem because then the marbles wouldn’t be shared equally!) In the second story, you need 108 boxes since you have to pack all of the cubes into boxes The last box holds only four cubes, so the remainder is important 64 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work a 50 30 1,500 60 350 14 b Here are two possible multiplication problems (50 ؉ 2) ؋ (30 ؉ 7) ؍1,924 or 52 ؋ 37 ؍1,924 Here are the answers along with some explanations a ، 1᎑᎑5 ؍30 Changing both to fifths: ، 1᎑᎑5 ؍30 ᎑᎑᎑ ، ᎑᎑ 30 , so the answer is 30 ᎑᎑ fits 30 times in ᎑᎑᎑ ؍22 b ᎑᎑ ، ᎑᎑ Changing both to eighths: ؍22 ᎑᎑ ᎑᎑᎑ ، ᎑᎑ 8 ، ᎑᎑ ؍22 ، ؍22 ؍3 c ، ᎑᎑ Changing both to thirds: ؍15 5 ، ᎑᎑ ᎑᎑᎑ 3 ، ᎑᎑ fits into 15 If you want to know how many times ᎑᎑ ᎑᎑᎑ 3 , you can calculate 15 ، ؍3 a Different stories are possible, for example: • I have 10 1᎑᎑ kilograms of peanuts, and I want to divide them kilogram in portions of ᎑᎑ , which is 42 ᎑᎑᎑᎑ b 10 1᎑᎑ is the same as 10 ᎑᎑ 4 = 42 c 10 1᎑᎑ ᎑᎑᎑᎑ ، ᎑᎑ 4 ، ᎑᎑ ؍42 ، ؍14 Answers to Check Your Work 65 CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work km/h The average speed is ᎑᎑ ؋3 Distance (in km) Time (in hr) 22 72 34 ؋3 Section D ،2 ،2 Angles Many explanations are possible Here are two If neither explanation is like yours, discuss your explanation with another student • Because the calculator gives me an error message when I enter ، • Because there is no one answer that works all the time Some people might think that ، = because usually when you divide a number by itself, you get an answer of Another person might think ، ؍0 because you have nothing to share and no one to share with A third person might think ، ؍7 because rewriting ، ؍7 works as a multiplication problem (7 ؋ ؍0) You cannot have multiple answers for the same division problem Maybe you started to use your calculator to find the answer But if you take a closer look at the last parenthesis, you see ؊ 9, which is And multiplying by always gives as an answer –12 –4 ، ، –2 ، – 32 66 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work –2 – 12 ؋ ، –10 –10 ؊ a ؋ 18 ؍108 ؋ 18 ؍6 ؋ (10 ؉ 8) ؍6 ؋ 10 ؉ ؋ ؍ 108 b ؋ 21᎑᎑ ؍7 ᎑᎑ 1) ؋ ᎑᎑ ؍3 ؋ (2 ؉ ᎑᎑ ؍3 ؋ ؉ ؋ 1᎑᎑ ؍ ؉ 11᎑᎑ or ᎑᎑ c ؋ 12 1᎑᎑ ؍87 ᎑᎑ 1) ؋ 12 1᎑᎑ ؍7 ؋ (10 ؉ ؉ ᎑᎑ ؍7 ؋ 10 ؉ ؋ ؉ ؋ 1᎑᎑ ؍ 70 ؉ 14 ؉ 1᎑᎑ ؍87 1᎑᎑ Answers to Check Your Work 67 CuuDuongThanCong.com https://fb.com/tailieudientucntt Answers to Check Your Work Section E Ref lections on Numbers Yes, it is possible to make numbers smaller by multiplying Here are two examples: • Multiplying a positive number by a number between and results in a smaller number Consider that 16 ؋ 0.25 ؍4, and is smaller than 16 • Multiplying a positive number by a negative number results in a smaller number Consider that 14 ؋ –5 – ؍70, and –70 is smaller than 14 a rational b rational c rational d yes a No Depending on Pablo’s calculator setting, he might get 78.53981634 cm2, and Josie’s answer will be 78.5 cm2 b Neither of them will get an exact answer since π is an irrational number and only has a decimal approximation The only exact area is 25π cm2 However, Pablo’s answer uses more precision because his calculator’s value for π uses more significant digits a irrational b rational (In the real number Yes An example would be – Ί or – Ί 52 system, the number under the radical sign cannot be negative For all negative radical numbers, the negative sign must be placed in front of the radical symbol; this holds true for both and – Ί ) rational and irrational numbers like – Ί 68 Revisiting Numbers CuuDuongThanCong.com https://fb.com/tailieudientucntt ... write Encyclopædia Britannica, Inc., 331 North LaSalle Street, Chicago, Illinois 60610 ISBN 0-0 3-0 3856 8-7 073 09 08 07 06 05 CuuDuongThanCong.com https://fb.com/tailieudientucntt The Mathematics... used hand-held stopwatches, it was very difficult to rank evenly matched competitors At the 1960 Olympic games in Rome, Australia’s John Devitt and America’s Lance Larson finished neck-and-neck... lived in the same area as the Sumerians, used a base-60 number system It is not sure why they chose 10 11 60 One reason might be because 12 base-60 makes divisional operations Day Night easy since
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