Facts and Factors Number 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., de Lange, J., and Pligge, M.,A (2006) Facts and Factors In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in Context Chicago: Encyclopædia 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 0-03-038564-4 073 09 08 07 06 05 The Mathematics in Context Development Team Development 2003–2005 Facts and Factors was developed by Meike Abels and Jan de Lange 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 (c) 2006 Encyclopædia Britannica, Inc Mathematics in Context and the Mathematics in Context Logo are registered trademarks of Encyclopỉdia Britannica, Inc Cover photo credits: (all) © Getty Images Illustrations (top) Michael Nutter/© Encyclopỉdia Britannica, Inc.; (bottom) Holly Cooper-Olds; 2, 3, 4, 13 Christine McCabe/© Encyclopỉdia Britannica, Inc.; 18, 24 (left), 25, 27, 34 (left), 36 Holly Cooper-Olds; 38 Christine McCabe/© Encyclopỉdia Britannica, Inc.; 45, 50 (top) Holly Cooper-Olds; 51, 56 Christine McCabe/© Encyclopỉdia Britannica, Inc Photographs Sam Dudgeon/HRW Photo; © Richard T Nowitz/Corbis; 8, (top) Victoria Smith/HRW; (bottom) R Stockli, A Nelson, F Hasler, NASA/GSFC/NOAA/USGS; 12 Victoria Smith/HRW; 13 (top) Sam Dudgeon/HRW Photo; (bottom) PhotoDisc/Getty Images; 14 (top left) PhotoDisc/ Getty Images; (top right) G K & Vikki Hart/ PhotoDisc/Getty Images; 15 © ImageState; 30 © Corbis; 37 Sam Dudgeon/HRW Photo; 38, 39 Victoria Smith/HRW; 40 Stephanie Friedman/HRW; 41 © PhotoDisc/Getty Images; 44 Don Couch/ HRW Photo; 49 Sam Dudgeon/HRW Photo; 55 Archives Académie des Sciences, photo Suzanne Nagy; 56 Lisa Woods/HRW Contents Letter to the Student Section A Base Ten Hieroglyphics Times Ten Large Numbers Exponential Notation Scientific Notation Summary Check Your Work Section B 24 12 24 27 29 30 32 33 Square and Unsquare Square Unsquare Cornering a Square Not So Square Summary Check Your Work Section E 13 17 17 21 22 23 Prime Numbers Upside-Down Trees Primes Prime Factors Cubes and Boxes Summary Check Your Work Section D 10 11 Factors Pixels Facts Factors Changing Positions Summary Check Your Work Section C vi 35 37 37 40 42 43 More Powers The Legend of the Chess Board Powers of Two Powers of Three Different Bases Back to the Egyptians Summary Check Your Work 44 46 48 48 50 52 53 Additional Practice 54 Answers to Check Your Work 60 Contents v Dear Student, The numbers we use today are widely used by people all over the world This might surprise you since there are about 190 independent countries in the world, speaking over 5,000 different languages! This was not always the case In the unit Facts and Factors, you will investigate how ancient civilizations wrote numbers and performed number computations Looking into the past will help you make more sense of the way you write and compute with numbers You will look into other numbering systems in use today You will investigate some properties of digital photographs By doing so, you will learn more about the properties of numbers How many different pairs of numbers can you multiply to find a product of 36? How about for a product of 51 or 53? You will expand your understanding of all the real numbers We hope you enjoy this unit Sincerely, The Mathematics in Context Development Team vi Facts and Factors A Base Ten Hieroglyphics MEDITERRANEAN SEA Alexandria N Rosetta Giza Memphis S SINAI AR AB LOWER EGYPT IA N Tell El-Amarna Thebes Karnak Abydos Valley of UPPER the Kings Luxor EGYPT Edfu Aswan 1st Cataract Philae D SE Abu Simbel SE 2nd Cataract Nil e RT R iver This hieroglyph A RT Tropic of Cancer DE 0 SE DE N Here is his latest work The hieroglyphs on the stone represent the number 1,333,331 RE r YA He carved little pictures called hieroglyphs to record information LIB At this time, Horus was the best stone carver of his village E Heliopolis Cairo N i l e R iv e Step back in time to a world without computers, calculators, and television; to Egypt around 3000 B.C DE QAT P R TA ES RA SI ON W 100 NUBIA 200 mi 100 200 300 km is an astonished man Perhaps he is astonished because he represents a very large number What number does the astonished man represent? Section A: Base Ten A Base Ten Here is the number 3,544 written in hieroglyphics How would Horus write your age? And 1,234? Today, we use the Arabic system and the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and to represent any number Complete the table on Student Activity Sheet to compare the Egyptian hieroglyphs with the Arabic numerals we use today Egyptian Hieroglyph Egyptian Description Arabic Numeral English Word one vertical stroke a heel bone a coil or rope lotus flower pointing finger tadpole an astonished man What number is represented in this drawing? How would Horus write 420? And 402? How many Egyptian hieroglyphs you need to draw the number 999? Facts and Factors Base Ten A You found these three pieces of a stone containing Egyptian hieroglyphs What number they represent when placed altogether? Today, Peter found these three tiles lying on the ground by an abandoned house Can you figure out the address of this house? Why or why not? What are the differences between our Arabic system of writing and using numbers and the Egyptian system? Times Ten 10 a Draw the Egyptian number that is ten times as large as this one b Describe what the ancient Egyptians would to multiply a number by ten In our Arabic number system, numerals in a number are called digits Digits have a particular value in a number For example, in the number 379: The digit has a value of hundreds The digit has a value of tens The digit has a value of ones Section A: Base Ten A Base Ten You can expand the number 379 with words as hundreds and tens and ones or as ؋ 100 ؉ ؋ 10 ؉ ؋ 11 Expand the following numbers in the same way a 628 b 2,306 c 256 d 2,560 12 Compare your answer to 11c and d What you notice? The pictures here compare multiplying a number by 10 for both number systems Ancient Egyptian Hieroglyphics vs Arabic Number System 537 ؋ 10 ؋ 10 5370 Sasha looks at the hieroglyphics and notices, “When you multiply a number by 10, you only have to change each hieroglyph into a hieroglyph of one value higher.” 13 a Explain what Sasha means Use an example in your explanation b What is the value of in 537? And what is the value of in 5,370? c What is the value of in 537? And in 5,370? d Explain what happens to the value of the digits when you multiply by ten e Calculate 26 ؋10 and 2.6 ؋10 f Does your explanation from d hold for problem e? If not, revise your explanation Facts and Factors E More Powers Products of Powers You can completely factor any number into a product of prime factors Sometimes, when the factors repeat, you can write this number as a product of powers For example: 5,625 ؍3 ؋ ؋ ؋ ؋ ؋ ؍32 ؋ 54 You can combine a product of powers with the same base into one base and power For example: ؋ 32 33 ؍35 ؋ ؋ ؋ ؋ ؍35 2؉3 If you want to calculate a product of powers with different bases, then you have to calculate the powers first and then multiply There are not any shortcuts because the bases are different For example: 53 ؋ 102 125 ؋ 100, which is 12,500 The Binary System The binary system is based on powers of two There are only two digits in the binary system, and To write a number in the binary system, you only need to write the number as a sum of powers of For example: 24 23 22 21 20 5؍ ؉ 20 ؍ ؉ ؍ ؉0 ؉ 20 ؍1 ؋ 22 + ؋ 21 + ؋ 20 In the binary system, you can write as 1012 (read as “one, zero, one, base 2.) 52 Facts and Factors Write 10,000 as a product of powers Write the prime factorization of each number If possible, use exponential notation to write each factorization as a product of powers a 288 b 900 c 1764 Calculate 23 ؋ 52 Use the table with powers of three to calculate: Powers of Three 31 32 b 2187 ؋ 33 27 c ؋ ؋ ؋ 27 ؋ 81 34 81 d 94 35 243 36 729 37 2,187 38 6,561 39 19,683 310 59,049 a 27 ؋ 81 Compare our decimal numbers to binary numbers Why you think we use base ten rather than base two? Be specific Section E: More Powers 53 Additional Practice Section A Base Ten a About how old is a person who is a million seconds old? b About how old is a person who is a billion seconds old? Explain your strategy for calculating the answer c What happened about a million days ago? Explain how you found your answer On a transatlantic flight, the speed of an airplane is about 1,000 km per hour If it were possible, a plane traveling at this speed would need 16 days to fly to the moon Use this information to calculate the distance from the earth to the moon The distance from the earth to the sun is about 400 times the distance between the earth and the moon a How many days would that same plane need to fly from the earth to the sun? b What is the distance between the earth and the sun (in km)? Write your answer in scientific notation and as a single number Which one of the following is the largest number? Explain you reasoning 0.4 ؋ 1011 Section B 40 ؋ 108 400 ؋ 106 Factors What is the smallest natural number that has exactly five factors? Explain how you found it 54 Facts and Factors You cannot view the last digit of this 11-digit number 84 355 216 015 What digit can you place in the open position to have a number: a divisible by 5? b divisible by and 5? c divisible by but not by 2? d divisible by and 3? You can write the number 10,000 as a product of two numbers in many different ways Here are two different ways: 10,000 ؍1,000 ؋ 10 and 10,000 ؍400 ؋ 25 Write 10,000 as a product of three numbers, so that none of the numbers is divisible by ten Find two different possibilities In 1845, Bertrand conjectured: “For every whole number greater than three, there is at least one prime between that number and its double.” 1822–1900 Joseph Bertrand was a French mathematician interested in prime numbers, geometry, and probability In 1855, he translated Gauss’ s work on error analysis into French In 1856, he was appointed as a professor at the École Polytechnique Later he also became a professor at the Collège de France From 1874 until the end of his life, he was a distinguished member of the Paris Academy of Sciences Verify Bertrand’s conjecture by checking all the appropriate numbers less than 21 Organize your work so that someone else can understand Bertrand’s conjecture Additional Practice 55 Additional Practice Section C Prime Numbers In 1742, the Russian mathematician Christian Goldbach conjectured: “Every even integer larger than two can be written as the sum of two prime numbers.” For example, = + Goldbach’s conjecture has been tested for all values up to 1014, but no one has been able to prove it yet! a You can write the even number as ؍3 ؉ or ؍1 ؉ One of these sums doesn’t verify Goldbach’s conjecture Which one? Why? b You can verify Goldbach’s conjecture for 28 with the sum 28 ؍23 ؉ Is this the only possibility? Investigate other possibilities c Verify Goldbach’s conjecture by checking all the even numbers less than 21 Organize your work so someone else can understand Goldbach’s conjecture Place the eight numbers, 0, 1, 2, 3, 4, 5, 6, and on the eight vertices of the 3-D shape so that the sum of any two adjacent vertices is a prime number Adjacent vertices are physically connected A prime day is when both the month and the day are prime numbers For example, May 23 is a prime day because both (5 and 23) are prime numbers a What is the first prime day of the year? And the last one? b How many prime days are there in a year? Completely factor each of the following numbers into a product of primes a 900 56 Facts and Factors b 2,079 c 12,121 Additional Practice Square and Unsquare Section D a Explain how you can use the area model in this drawing to calculate (7 –12– )2 b Use graph paper to copy this drawing and fill in all the missing information ? ? c Calculate (7 –12– )2 Choose a strategy to calculate: a (12 –1– )2 b ? (21 –– )2 ? 64 cm2 24 cm2 Here are two squares and two rectangles The number on each shape is the area of that shape You can use all four shapes to form one large square a What is the side length of the large square? Show your work and make a sketch of the large square cm2 24 cm2 b Suppose you could reshape the two identical rectangles to form a large square What is the area of this square? What is the length of one side? In this drawing, the dark yellow shapes are squares The area of each square is indicated 121 cm2 Explain how the total area of all of the shapes is a square number 16 cm2 cm2 Additional Practice 57 Additional Practice Section E More Powers Three of these statements are true and three are false a 43 ؋ 82 ؍4 ؋ ؋ ؋ ؋ c (11 –1– )2 ؋ (11 –1– )3 ؍ (11 –1– )5 e 54 ؋ 24 ؍108 b (6 –1– )2 ؍36 –1– d 26 ؋ 25 ؍ 230 f 34 ؋ ؍36 Try to decide which are true and which are false without calculations Explain your reasoning Completely factor the following numbers into a product of prime numbers: a 10 b 26 c 77 d 50 During math class, Mr Shawn asked Peter, “How many different rectangles can you make that have an area of 26 square inches?” Peter quickly answered, ”If the sides are the counting numbers, then there are four possibilities.” a Which possibilities did Peter think of? Can you explain how Peter was able to answer so quickly? b How many possible rectangles can you make with areas of 10 in2 and 77 in2, respectively? Explain how you can quickly find all possibilities c Consider all the possible rectangles with an area of 50 in2 Did you find all the possible rectangles quickly? Explain why or why not In communications, electronics, and physics, a kilo stands for 103 For example, kilometer = 103 meters or 1,000 meters In Information Technology (IT) and data storage, a kilo stands for 210 For example, kilobyte = 210 bytes 58 Facts and Factors Additional Practice This table explains the prefixes kilo, mega, and giga IT Terminology one kilobyte kB ؍210 bytes one megabyte mB ؍220 bytes one gigabyte gB ؍ bytes a Calculate how many bytes are in one kilobyte Estimate your answer using a power of ten b How many bytes are in one megabyte? Write your answer in scientific notation Estimate your answer using a power of ten (You may want to use a calculator for this.) c How many kilobytes are in one megabyte? How you know? The relationship between kilobytes and megabytes holds true for megabytes and gigabytes One gigabyte is more than 1,000 times one megabyte d How many bytes are in one gigabyte? Write your answer as a power of two In problem 21 of Section E, you learned how to read a binary clock Sketch a binary clock and color the lights so the time displayed is 3:12 P.M Additional Practice 59 Section A Base Ten a 1,000 ؋ 10 ؋ 10 ؍100,000 b 1,000 ؋ 1,000 ؍1,000,000 c 63.7 ؋ 10 ؍637 d 63.7 ؋ 100 ؍6,370 e 0.58 ؋ 1,000 ؍580 a Here are five sample products of one billion 1,000 ؋ 1,000,000 1,000 ؋ 1,000 ؋ 1,000 10 ؋ 10 ؋ 10 ؋ 10 ؋ 10 ؋ 10 ؋ 10 ؋ 10 ؋ 10 10 ؋ 100,000,000 100 ؋ 10,000,000 You might have others; check with a classmate to make sure the product is 1,000,000,000 b Here are five sample products for 2,270,000 2,270 ؋ 1,000 22.7 ؋ 100,000 227 ؋ 10,000 227 ؋ 100 ؋ 100 227 ؋ 10 ؋ 1,000 You might have others; check with a classmate to make sure the product is 2,270,000 a 10 or 10,000,000, or ten million 10 b 10 or 10,000,000,000 or ten billion c 10 or 1,000,000 or one million (10 ؋ 100 ؋ 1,000 ؍101 ؋ 102 ؋ 103) d 10 or 1,000,000,000 or one billion (1,000,000 ؋ 10,000 ؍106 ؋ 104) (1,000 ؋ 1,000,000 ؍103 ؋ 106) a 2.25؋104 ➞ 22.5؋103 ➞ 225؋102 ➞ 2,250؋10 ➞ 22,500 b Check the work of the classmate who solved your problem 60 Facts and Factors Answers to Check Your Work a Both calculators display 5.1 and 06; 5.1 is the first factor between and 10; 06 is the exponent of 10 The difference is the second display uses an E to designate the exponent of ten; the first one displays the exponent of ten as a small number in the upper right corner b 5.1 ؋ 10 or 5.1 million or 5,100,000 Section B Factors Yes, groups of three work because the sum of the digits of 945 is 18: ؉ ؉ ؍18, and 18 is divisible by No, groups of six will not work “Divisible by 6” means that the number 945 has to be divisible by and by Because 945 is not an even number, it is not divisible by 2, so it is not divisible by a 1, 3, 5, and 15 b 1, 2, 4, 8, 16, and 32 c and 53 d and 17 a The number you wrote can be even or odd but must not be a perfect square number One sample number that has an even number of factors is 20; the factors of 20 are 1, 2, 4, 5, 10, and 20 b The number you wrote must be any perfect square number Sample numbers with an odd number of factors are 25 or 100 c A perfect square number will always have an odd number of factors There are 10 perfect square numbers from through 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 Answers to Check Your Work 61 Answers to Check Your Work Section C Prime Numbers There are many ways to calculate with an arithmetic tree In all cases, your final answer is 1,400 Here are two different ways: ؋ 35 ؋ 20 ؋ 10 ؋ 20 ؋ 700 ؋ 1,400 ؋ 200 ؋ 1,400 The numbers 12, 39, and 51 are all composite numbers Sample reasoning: Prime numbers have exactly two factors, and itself Composite numbers are numbers larger than one that are not prime One way to find out whether or not a number is a composite number is to use the rules for divisibility 12 is an even number, so it is divisible by and has more factors than and 12 19 ؍1 ؋ 19; 19 has no other factors than and 19, so 19 is prime 39 ؍3 ؋ 13, so 39 has more factors than and 39 51 is divisible by because the sum of the digits is 6, and is divisible by 3, so 51 has more factors than and 51 62 Facts and Factors Answers to Check Your Work a 99 ؍3 ؋ ؋ 11 99 ؍ b 750 ؍2 ؋ ؋ ؋ ؋ 750 ؋ 11 ( ؍3 ؋ 3) ؋ 11 ؋ 10 ؍3 ؋ ؋ 11 ؋ 75 ؋ 25 ؋ c 264 ؍2 ؋ ؋ ؋ ؋ 11 264 132 66 33 11 11 A strategy to solve these problems is to find all factors of the number of centimeter cubes first a The factors of are: 1, 2, 4, and Three possible dimensions are: cm by cm by cm, cm by cm by cm, and cm by cm by cm b The factors of 50 are: 1, 2, 5, 10, 25, and 50 Three possible dimensions are: cm by cm by 25 cm, cm by cm by 10 cm, and cm by cm by cm Answers to Check Your Work 63 Answers to Check Your Work Section D Square and Unsquare a The base is b The exponent is c 25 ؍2 ؋ ؋ ؋ ؋ ؍32 සස ≈ 34.64 in a The side length is ͱසස 1200 සස ≈ 10.95 in b The side length is ͱසස 120 c The side length is ͱසස 12 ≈ 3.46 in d The side length is ͱසස 1.2 ≈ 1.095 in., or 1.10 in සස ≈ 0.364 in., or 0.35 in e The side length is ͱසස 0.12 f If the area is 100 times as small, then the side length is ten times as small Compare, for example, a and c or c and e one city block square mile a The area of one city block is –– 64 Sample reasoning: One city block is –1– mile by –1– mile square mile mile In one square mile (see drawing), you can fit eight rows of eight city blocks This makes rows ؋ blocks or 64 blocks If 64 city blocks fit in one square mile, then of a square –– the area of one city block is 64 mile mile 36 , or –– b –3– ؋ –1– ––– ؍ 64 16 Here is a way to calculate –3– ؋ –1– using city blocks 12 mile –3– mile is city blocks 8 mile –1– miles are 12 city blocks (8 ؉ 4) –– ؋ –1– is the same as blocks ؋ 12 blocks, or 36 blocks square mile, 36 city Since city block is –– 64 blocks is 64 Facts and Factors 36 ––– 64 square mile Answers to Check Your Work –1– ؋ –1– ؍8 –1– 3 1 1 6 One sample strategy using the area model: The four parts total –1– , (6 ؉ 1؉ 1؉ –1– ) Section E More Powers Many answers are possible Here are three samples: 10,000 ؍24 ؋ 54 10,000 ؍102 ؋ 102 10,000 ؍22 ؋ 52 ؋ 102 Make sure you use a product of powers; 10,000 ؍104 is one power and not a product of powers a 288 ؍2؋ ؋ ؋ ؋ ؋ ؋ ؍25 ؋ 32 b 900 ؍2 ؋ ؋ ؋ ؋ ؋ ؍22 ؋ 32 ؋ 52 c 1764 ؍2 ؋ ؋ ؋ ؋ ؋ ؍22 ؋ 32 ؋ 72 23 ؋ 52 ؍2 ؋ ؋ ؋ ؋ ؍200 a 27 ؋ 81 ؍2,187 Explanation: You can use the table to find 27 ؍33 and 81 ؍34 27 ؋ 81 ؍33 ؋ 34 ؍37 In the table, 37 ؍2,187 Answers to Check Your Work 65 Answers to Check Your Work b 2,187 ؋ ؍6,561 Explanation: You can use the table to find 2,187 ؍37 and ؍31 2,187 ؋ ؍37 ؋ 31 ؍38 In the table, 38 ؍6,561 c ؋ ؋ ؋ 27 ؋ 81 ؍59,049 Explanation: You can use the table to find ؍30; ؍31; ؍32; 27 ؍33 and 81 ؍34 ؋ ؋ ؋ 27 ؋ 81 ؍30 ؋ 31 ؋ 32 ؋ 33 ؋ 34 ؍310 In the table, 310 ؍59,049 d 38 ؍6,561 Explanation: 94 ؍9 ؋ ؋ ؋ ؍32 ؋ 32 ؋ 32 ؋ 32 ؍38 In the table, 38 ؍6,561 66 Facts and Factors ... tenths hundredths hundreds For example: 79 .54 ؋ 10 9 ؋ 10 79 .54 ؋ 10 ؍ 79 5.4 79 .54 79 5.4 10 Facts and Factors 1 ᎑᎑᎑᎑ ؍ 7 ؋ 10 ؉ ؋ ؉ ؋ ᎑᎑᎑ 10 ؉ ؋ 100 ؍ 7 ؋ 100 ؉ ؋ 10 ؉ ؋ ؉ ؋ ᎑᎑᎑ 10 ؋ 10 Exponential... number 2,638, 577 as “two million, six hundred thirty-eight thousand, five hundred seventy-seven.” 17 How you read 4, 370 ,000? And 1,500,000,000? There are different ways to read and write large... of in 5 37? And what is the value of in 5, 370 ? c What is the value of in 5 37? And in 5, 370 ? d Explain what happens to the value of the digits when you multiply by ten e Calculate 26 ؋10 and 2.6