Facts and Factors

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Facts and

FactorsNumber


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Mathematics in Contextis a comprehensive curriculum for the middle grades.It was developed in 1991 through 1997 in collaboration with the Wisconsin Centerfor Education Research, School of Education, University of Wisconsin-Madison andthe Freudenthal Institute at the University of Utrecht, The Netherlands, with thesupport of the National Science Foundation Grant No. 9054928.

This unit is a new unit prepared as a part of the revision of the curriculum carriedout in 2003 through 2005, with the support of the National Science FoundationGrant No. ESI 0137414.

National Science FoundationOpinions 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: Encyclopdia Britannica, Inc.

Copyright 2006 Encyclopdia 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 notin conformity with the U.S. copyright statute are prohibited without our express

written permission, including but not limited to duplication, adaptation, andtransmission by television or other devices or processes. For more informationregarding a license, write Encyclopdia Britannica, Inc., 331 North LaSalle Street,Chicago, Illinois 60610.

ISBN 0-03-038564-4

1 2 3 4 5 6 073 09 08 07 06 05


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The Mathematics in ContextDevelopment TeamDevelopment 20032005

Facts and Factorswas 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 DekkerDirector Coordinator Director Coordinator

Gail Burrill Margaret A. Pligge Mieke Abels Monica WijersEditorial Coordinator Editorial Coordinator Content Coordinator Content Coordinator

Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie Kuijpers

Beth R. Cole Anne Park Peter Boon Huub NilwikErin Hazlett Bryna Rappaport Els Feijs Sonia PalhaTeri Hedges Kathleen A. Steele Dd de Haan Nanda QuerelleKaren Hoiberg Ana C. Stephens Martin Kindt Martin van Reeuwijk Carrie Johnson Candace UlmerJean Krusi Jill VettrusElaine McGrath


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(c) 2006 Encyclopdia Britannica, Inc. Mathematics in Context

and the Mathematics in ContextLogo are registered trademarksof Encyclopdia Britannica, Inc.

Cover photo credits: (all) Getty Images

Illustrations1 (top) Michael Nutter/ Encyclopdia Britannica, Inc.; (bottom)Holly Cooper-Olds; 2,3,4,13 Christine McCabe/ EncyclopdiaBritannica, Inc.; 18, 24 (left), 25, 27, 34 (left), 36 Holly Cooper-Olds;38 Christine McCabe/ Encyclopdia Britannica, Inc.; 45, 50 (top)Holly Cooper-Olds; 51, 56 Christine McCabe/ EncyclopdiaBritannica, Inc.

Photographs3 Sam Dudgeon/HRW Photo; 6 Richard T. Nowitz/Corbis; 8, 9 (top)

Victoria Smith/HRW; (bottom) R. Stockli, A. Nelson, F. Hasler,NASA/GSFC/NOAA/USGS; 12Victoria 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 SamDudgeon/HRW Photo; 38, 39Victoria Smith/HRW; 40 StephanieFriedman/HRW; 41 PhotoDisc/Getty Images; 44 Don Couch/HRW Photo; 49 Sam Dudgeon/HRW Photo; 55Archives Acadmiedes Sciences, photo Suzanne Nagy; 56 Lisa Woods/HRW


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Contents

Contents v

Letter to the Student vi

Section A Base Ten

Hieroglyphics 1

Times Ten 3Large Numbers 6Exponential Notation 7Scientific Notation 8Summary 10Check Your Work 11

Section B Factors

Pixels 13Facts 17Factors 17Changing Positions 21Summary 22Check Your Work 23

Section C Prime Numbers

Upside-Down Trees 24Primes 27Prime Factors 29Cubes and Boxes 30Summary 32Check Your Work 33

Section D Square and UnsquareSquare 35Unsquare 37Cornering a Square 37Not So Square 40Summary 42Check Your Work 43

Section E More Powers

The Legend of the Chess Board 44Powers of Two 46Powers of Three 48Different Bases 48Back to the Egyptians 50Summary 52Check Your Work 53

Additional Practice 54

Answers to Check Your Work 60

6

12

24

32

2

2


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vi Facts and Factors

Dear Student,

The numbers we use today are widelyused by people all over the world.This might surprise you since thereare about 190 independent countries in the world, speaking over5,000 different languages! This was not always the case. In theunit Facts and Factors, you will investigate how ancient civilizationswrote numbers and performed number computations. Looking intothe past will help you make moresense of the way you write andcompute with numbers. You will

look into other numberingsystems in use today.

You will investigate some properties of digital photographs. By doingso, you will learn more about the properties of numbers. How manydifferent pairs of numbers can you multiply to find a product of 36?How about for a product of 51 or 53? You will expand yourunderstanding of all the real numbers.

We hope you enjoy this unit.

Sincerely,

The Mathematics in Context Development Team


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Section A: Base Ten 1

ABase Ten

Hieroglyphics

Tropic of Cancer

RED

SEA

M E D I T E R R A N E A N S E A

QATTAR

A

DEPR

ESSI

ON

Alexandria

GizaMemphis

Abydos

Edfu1st Cataract

2nd Cataract

Valley ofthe Kings

Abu Simbel

Rosetta

HeliopolisCairo

TellEl-Amarna

KarnakThebes

Luxor

Aswan

Philae

SINAI

LIBYAN

DESERT

ARABIA

N

DESERT

LOWER

EGYPT

UPPER

EGYPT

NUBIA

Nile

River

Nile

Riv

er

N

S

W E

0 100 200 300 km

0 100 200 mi

This hieroglyph is an astonished man. Perhaps he is astonished

because he represents a very large number.

1. What number does the astonished man represent?

Here is his latest work. The hieroglyphs onthe stone represent the number 1,333,331.

Step back in time to a world withoutcomputers, calculators, and television;to Egypt around 3000 B.C.

At this time, Horus was the best stonecarver of his village.

He carved little pictures calledhieroglyphs to record information.


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Egyptian Egyptian Arabic English

Hieroglyph Description Numeral Word

vertical stroke 1 one

a heel bone

a coil or rope

lotus flower

pointing finger

tadpole

an astonishedman

2 Facts and Factors

Base TenA

Here is the number 3,544 written in hieroglyphics.

2. 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 9 to represent any number.

3. Complete the table on Student Activity Sheet 1 to compare theEgyptian hieroglyphs with the Arabic numerals we use today.

4. What number is represented in this drawing?

5. How would Horus write 420? And 402?

6. How many Egyptian hieroglyphs do you need to draw thenumber 999?


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A

You found these three pieces of a stone containing Egyptianhieroglyphs.

7. What number do they represent when placed altogether?


Base Ten

Times Ten

Today, Peter found these three tiles lyingon the ground by an abandoned house.

8. Can you figure out the address of thishouse? Why or why not?

9. What are the differences between ourArabic system of writing and usingnumbers and the Egyptian system?

10. a. Draw the Egyptian number that isten times as large as this one.

b. Describe what the ancientEgyptians would do to multiplya 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 3 has a value of 3 hundreds.The digit 7 has a value of 7 tens.The digit 9 has a value of 9 ones.


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You can expand the number 379 with words as 3 hundreds and 7 tensand 9 ones or as 3 100 7 10 9 1.

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 do you notice?

The pictures here compare multiplying a number by 10 for bothnumber systems.

Ancient Egyptian Hieroglyphics vs. Arabic Number System

Sasha looks at the hieroglyphics and notices, When you multiply anumber by 10, you only have to change each hieroglyph into ahieroglyph of one value higher.

13. a. Explain what Sasha means. Use an example in yourexplanation.

b. What is the value of 7 in 537? And what is the value of7 in 5,370?

c. What is the value of 3 in 537? And in 5,370?

d. Explain what happens to the value of the digits when youmultiply by ten.

e. Calculate 2610 and 2.6 10.

f. Does your explanation from d hold for problem e?If not, revise your explanation.

4 Facts and Factors

Base TenA

537

5370

10 10


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A

The Egyptian number system was not well suited for decimal orfraction notation. The decimal notation we use today was developedalmost 4,000 years later. A Dutch mathematician, Simon Stevin,invented the decimal point.

14. a. Explain the value of each digit in the number 12.574.

b. Write 7 100 6 1 4 110 51

1000 as a single number.

If you multiply a decimal number by 10, the value of each digit ismultiplied by 10.

Consider the product of 57.38 10.

57.38 10 573.8

57.38 5 10 7 1 3 110 81

100

573.8 5 100 7 10 3 1 8 110

15. Calculate each product without using a calculator.

a. 4.8 10

b. 4.8 10 10

c. 6.37 10 10

d. 9.8 10 10 10

e. 1.25 1,000

f. 0.57892 1,000


Base Ten

10

hundre

ds

tens

ones

tenths

hundre

dths

5 7 3 8

5 7 3 8

10


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In 2004, the population of the United States was about 292 million,and the world population was about 6 billion.

16. Write these populations using only numerals.

Notice that commas separate each group of three digits. Thismakes the numbers easy to read. You read the number 2,638,577as two million, six hundred thirty-eight thousand, five hundredseventy-seven.

17. How do you read 4,370,000? And 1,500,000,000?

There are different ways to read and write large numbers. Forexample, you can read 3,200,000 as: three million, two hundred

thousand or simply as 3.2 million.18. Write at least two different ways you can read each number.

a. 6,500,000

b. 500 million

c. 1.2 thousand

d. 750,000

6 Facts and Factors

Base TenA

Large Numbers

Numerals Words

1 one10 ten

100 one hundred1,000 one thousand

10,000 ten thousand100,000 one hundred thousand

1,000,000 one million10,000,000 ten million

100,000,000 one hundred million1,000,000,000 one billion

10,000,000,000 ten billion100,000,000,000 one hundred billion

1,000,000,000,000 one trillion


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A

19. Find each product and write your answers using only words.

a. One million times ten

b. One hundred times one hundred

c. One thousand times one thousand

20. a. How many thousands are in one million?

b. How many thousands are in one billion?

c. How many millions are in one billion?

d. Use numbers such as 10, 100, 1,000, and so on, to write fivedifferent multiplication problems for which the answer is1,000,000.

21. Suppose you counted from one to one million and every countwould last one second. How long would this take?

To save time writing zeroes and counting zeroes, scientists invented aspecial notation, called exponential notation.

The number 1,000 written in exponential notation is 103 (read asten raised to the third power or ten to the third).

1,000103 because 1,000101010

In 103, the 10 is the base, and the 3 is the exponent.

22. Write each number in exponential notation.

a. 100 b. 1,000,000,000 c. 10,000,000,000

23. Write each number in numerals and words.

a. 104 b. 101 c. 106


ABase Ten

Exponential Notation

103exponent

base


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6.4 09 6.4 E 09

How does your calculator display large numbers? To find out, answerthe following:

24. a. On a calculator, enter 9s until all places on the display areoccupied. Record the number displayed in your notebook.

b. Without using the calculator, what happens when you add 1to this number? Calculate the answer in your notebook. Writeyour answer in exponential notation. Identify the base andthe exponent.

c. Now, use your calculator to add 1 to the large numberdisplayed (the one with all 9s). Record the new numberdisplayed.

d. Explain what each part of the number displayed means.

e. In your notebook, calculate the product of 2,000,000,0003,000,000,000. Verify your calculation using your calculator.If needed, revise your answer for part d.

8 Facts and Factors

Base TenA

Scientific Notation

For very large numbers, most calculators switch to scientific notation(Sci) mode. The display shows a number between 1 and 10 and apower of ten.

Calculators display scientific notation in a variety of ways. Here are

two different calculator displays for the 2004 world populationof 6,400,000,000 people.

The number that is displayed is the product: 6.4 109.


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4 06 3.8 04

A

25. a. Write 6.4 109 in numerals and words.

b. What numbers are displayed here?

The distance from the earth to the moon is approximately240 thousand miles.

26. How would your calculator display this distance in scientificnotation?


Base Ten


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10 Facts and Factors

Base Ten

The Arabic Number System you use todayis a positional system using the numerals0 through 9. The position of each digit in anumber determines its value. You can readthe number 79.54 as seventy-nine andfifty-four hundredths.

You can expand the number 79.54 as:

7 10 9 1 5 110 41100

or 70 9 510

4

100.

A

Multiplying by Ten

If you multiply a decimal number by 10, the value of each digit ismultiplied by 10.

For example: 79.54 10

79.54 10 795.4

79.54

7

10

9

1

5

1

10

4

1

100

795.4 7 100 9 10 5 1 4 110

10

hundre

ds

tens

ones

tenths

hundre

dths

7 9 5 4

7 9 5 4

10

tens

ones

tenths

hundredths

7 9 . 5 4

4 hundredths5 tenths

9 ones7 tens


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Scientific Notation

Exponential notation is a shorter way to write repeated multiplication.

For example: 10 10 10 10 10 10 10 107.

You can read 107 as ten to the seventh power or ten to the seventh.

In 107, the 10 is the base, andthe 7 is the exponent.

Calculators display very large numbers using scientific notation.The number is displayed as a product of a number between 1 and 10and a power of ten.

A calculator displaying represents

the product 4.5 107 4.5 10,000,000

45,000,000

1. Calculate the following without using a calculator.

a. 1,000 10 10 d. 63.7 100

b. 1,000 1,000 e. 0.58 1,000

c. 63.7

102. a. Use numbers such as 10, 100, 1,000, and so on, to write five

different multiplication problems for which the answer is onebillion.

b. Write five more multiplication problems similar to those in parta, but for which the answer is 2,270,000.

Exponential Notation

107exponent

base

4.5 07


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Base Ten

3. Calculate the following and write your answers three different

ways: in exponential notation, as a single number, and in words.a. 104 103

b. 1,000,00010,000

c. ten one hundred one thousand

d. one thousand one million

4. a. Fill in the missing exponents and then write the answer as asingle number.

2.25 104

22.5 10?

225 10?

b. Make up a problem similar to the one in a. Ask a classmate tosolve your problem.

Here are two different calculatordisplays of the same number.

5. a. Explain what is displayed.

b. Write this number as asingle number.

Write a short paragraph for a school newsletter describing the

benefits of using scientific notation for very large numbers.

A

5.1 06 5.1 E 06


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Jacqui and Nikki are friends. They used to beneighbors, but Nikki moved to Cleveland. Nowthey maintain their friendship by using theInternet. They send e-mail to each other andchat online at least once a day.

Today after school, Jacqui checks her e-mail. Afterabout three minutes, she realizes Nikkis messageis taking longer than usual to download. Afterwaiting impatiently for ten minutes, Jacqui asksher brother, Dave, what can I do? Look at that baron the computer screen!

Section B: Factors 13

BFactors

Pixels

Nikkis e-mail included a picture with hernew puppy.

Dave remarks, Its a cute picture, but thesize of the file is too large. Send her an

e-mail and tell her that she has to makethe files smaller before she sends them.

Jacqui says, Dave, how can she do that?I dont even know how to do that.

Dave shares what he knows about digitalpictures.

Here is a screen shot of the bar on Jacquis computer after 12 minutes.

1. Estimate how many more minutes Jacqui will have to wait todownload this message completely. Show how you foundyour answer.


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A digital picture is made up of many little colored squares. These littlesquares are PICture ELements, or pixels.


FactorsB

The number of pixels determines the file size:

the more pixels, the larger the file size.

Here is a smaller file of Nikki and her dog.The number of pixels has decreaseddramatically: you can now see the pixels.

You will now investigate the effect of changingthe number of pixels per inch (ppi).

Pictures 1, 2, and 3 are the same picture.

Picture 1 has side lengths of two inches.

2. a. How many pixels do you count along one inch?

b. What is the total number of pixels in Picture 1?

Picture 2 shows the same pixel pattern but uses morepixels per inch (ppi).

3. a. How many pixels per inch are in Picture 2?

b. Without counting, what can you tell about thenumber of pixels per inch in Picture 3?

Compare Pictures 1, 2, and 3.

4. Describe how the pictures are the same and howthey are different.

You probably didnt find the total number of pixelsby counting all the small squares. For counting thepixels in Picture 1, you may have multiplied 12 12.Whenever you multiply a number by itself, you aresquaring the number.

5. Why do you think the expression squaring anumber is used?

Picture 1

Picture 2

Picture 3


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The picture Nikki included with her e-mail had 400 ppi and dimen-sions of 3 in. by 4 in.

8. How many total pixels were in the picture Nikki e-mailed? Show

your calculations.

In the unit Expressions and Formulas, you used arithmetic trees tohelp organize your calculations.

9. Explain how each arithmetic tree relates to problem 8.

10. a. Without changing the size of her picture (3 in. by 4 in.), Nikkireduced the number of pixels to 200 ppi. How many totalpixels make up Nikkis new picture?

b. Use the information from Nikkis picture to copy and completethis table.

c. In the table, the number of ppi is cut in half. What happens tothe total number of pixels?

Jacqui waited about 48 minutes for Nikkis original picture to download.

11. a. What would have been the download time if the picture had200 ppi instead of 400 ppi?

b. And if the picture had 100 ppi?


FactorsB

4003

______?

______?

______?

4004

43

______?

______?

______?

400400

ppi

400

200

100

Total Number of Pixels


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Jacqui prints 24 square pictures. She wants to use all 24 pictures tomake a rectangular display in her room.

She begins to investigate all possible arrangements so she can

choose the one she wants. First, she sketches one rectangulararrangement.

Then she decides to make a list of all possible arrangements.

Jacquis 24 pictures:

One possible rectangular arrangement of 24 pictures: 6 acrossand 4 vertical:

List of all possible rectangular arrangements:

1 by 24 6 by 42 by 12 8 by 33 by 8 12 by 24 by 6 24 by 1

She asks Dave if she has them all. Dave sees the list and says, I think1 by 24 is the same as 24 by 1.

14. Do you agree with Dave? Why or why not?


FactorsB


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Jacqui decides to draw one ofher picture arrangements ongraph paper.

Here is a graph showing all ofthe rectangular arrangementsin Jacquis list.

15. a. Explain what the graphshows.

b. How would you labelthe axes?

c. Describe what each pairof coordinates has incommon.

Since 3 8 24, 3 and 8 arefactors of 24.

16. List all of the possiblefactors of 24.

How can you be sure youhave them all?


BFactors

2

2

4

6

8

10

12

14

16

18

20

22

24

0 4 6 8 10 12 14 16 18 20 22 24

2

2

4

6

8

10

12

14

16

18

20

22

24

0 4 6 8 10 12 14 16 18 20 22 24

(1, 24)

(2, 12)

(4, 6)

(8, 3)

(3, 8)

(6, 4)

(12, 2)

(24, 1)


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17.a. Create a graph showing all the points that represent factors of25. How many points are on this graph?

b. Create a graph showing all the points that are factors of 23.

How many points are on this graph?c. Describe a relationship between the number of points on the

graph and the number of factors.

18. a. Which numbers will always have an odd number of factors?

b. Which numbers will always have an even number of factors?

c. For what number does the graph of factors have exactly onepoint?

19. a. Find at least five numbers with exactly two factors.

b. What do you notice about the factors of the five numbers youfound in part a?

The numbers you found in problem 19a are called prime numbers.They have exactly two factors: the number one and the number itself.You will further investigate prime numbers in the next section.

You may have discovered an easy way to list all of the factors of anumber.

Rosa, Lloyd, and Rachel, are finding all of the factors of 36.

Here is their work.

20. a. If you continue Rosas list, how will you know when to stop?

b. Finish Rachels work to find all of the factors of 36.

c. Use one of these strategies to find all of the factors of 96.


FactorsB

Lloyd:

All factors of 36 are

1, 36, 2, 18, 3, 12,

Rosa:

1 and 36

2 and 18

3 and 12 . . .

Rachel

1 2 3 12 18 36


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Factors

Squaring

Multiplying a number by itself is squaringa number.

Two ways to indicate the squaring of a number, such as 3, are 32 and3^2. Both represent 3 3, which gives an answer of 9.

The numbers that result from squaring a number are called squarenumbersor perfect square numbers.

Factors

5 is a factor of 30 because 30 divided by 5 is a whole number.30 5 6 and 5 6 30, so 6 is another factor of 30. All thefactors of 30 are:

Divisibility

To see if a number is divisible by a certain number, you can followsome rules of divisibility.

A number is divisible:

by 2 if the last digit is even,by 3 if the sum of the digits is divisible by three,

by 5 if the last digit is a zero or a five,by 9 if the sum of its digits is divisible by nine.

Prime Number

A number is a prime number if it has exactly two factorsthe numberitself and the number one.

B

1 2 3 5 6 10 15 30


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1. At Green Middle School, there are 945 students. Is it possible tosplit up all of the students into groups of three? Into groups ofsix?

2. Find all of the factors of:

a. 15 c. 53

b. 32 d. 17

3. a. Give an example of a number that has an even number offactors.

b. Give an example of a number that has an odd number offactors.

c. What name do you give the numbers having an odd numberof factors?

4. List all numbers from 1 to 100 that are perfect square numbers.

Consider these statements.

All even numbers have 2 as a factor. Therefore, there are noeven primes.

An even number divided by an even number is even.

Tell whether each statement is true or false. Justify your reasoning.


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In Section B, you used an arithmetic tree to organize your calcula-tions.

1. Use an arithmetic tree to calculate 2 5 7 7.

Here are two different arithmetic trees to calculate 5 5 2 6 3.

2. a. Will they both give the same result? Why or why not?

b. Which arithmetic tree would you prefer to use? Why?

CPrime Numbers

Upside-Down Trees

6 32

______?

______?

______?

______?

55

36

______?

______?

______?

255

______?

As I was going to St. Ives,I met a man with seven wives.Every wife had seven sacks,Every sack had seven cats,Every cat had seven kits.

Kits, cats, sacks, and wives,

How many were going to St. Ives?

77

______?

______?

______?

77


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You can write 150 as a product of two factors.

150 3 50

Both numbers, 3 and 50, are factors of 150.3. a. Explain why 10 is a factor of 150.

b. What is a factor? Use your own words todescribe factor.

Section C: Prime Numbers 25

CPrime Numbers

6

12

24

32

2

2

These special arithmetic trees are called factor trees. In these factortrees, you will only see multiplication signs. Here is the beginning ofa factor tree for the number 1,560.

5. a. Copy and complete the factor tree for the number 1,560.Take the branches out as far as possible.

b. How will you know when you are completely finished withthe tree?

c. Use the end numbers to write 1,560 as a product of factors.

d. Would you use the number 1 as an end number? Why or whynot?

An upside-down arithmetic tree can help you towrite a number as a product of factors.

4. a. What information does the upside-downarithmetic tree give you?

b. Use the end numbers (the numbers atthe end of the tree) to write 24 as aproduct of factors.

2 5

10 156

1,560

2355150

a product offour factors


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When you have taken a factor tree out as far as possible, you havecompletely factored the original number. The number 1 is a factor ofevery number, but it is not necessary to include 1s in a factor tree.

6. Completely factor each number. Use a factor tree to write eachnumber as a product of the end numbers.

a. 56 c. 420

b. 285 d. 3,432

Hakan and Alberta each begin a factor tree to completely factor 1,092.

Hakan realizes that 1,092 is Alberta realizes that 1,092 iseven, so he starts his tree divisible by both 2 and 3, solike this. it is divisible by 6. She starts

her tree like this.

7. a. In your notebook, finish Hakans and Albertas factor trees.

b. Do you get the same factors at the ends of the branches ofboth trees?

8. a. Refer back to all of the trees you have made so far and compilea list of all the end numbers.

b. You learned another name for these end numbers in Section B.What is it?

c. Find at least three other possible end numbers that are notalready on your list for part a.


Prime NumbersC

1,092

2 546

1,092

6 182


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The end numbers of all factor trees are prime numbers. In Section B,you discovered that prime numbers have exactly two factors, the

number one and the number itself.

Numbers that are not prime numbers are called composite numbers.The number 1 is neither a prime number, nor a composite number.

The ancient Greeks used prime numbers. Eratosthenes discovered amethod to extract all of the prime numbers from 1 to 100. Beginningwith a list of 100 numbers, he sifted out the prime numbers by cross-ing off multiples of numbers.


CPrime Numbers

Primes

The multiples of 2 are 2, 4, 6, 8, 10, and so on.

9. a. What is the next multiple of 2?b. List the first five multiples of 3.

c. Are there any numbers common to both lists? Explain.


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Use Student Activity Sheet 2 and problems 1015 to recreateEratosthenes method for extracting the prime numbers.

10. a. Circle the number 2 and put an X through all of the othermultiples of 2.

b. The numbers with an X through them are not prime.Why not?

11. a. Circle 3 and put an X through all other multiples of 3.

b. Explain why you do not need to put an X through all of themultiples of 4.

c. Do you need to cross out multiples of 6? Explain why.

d. Pablo went through these steps and said, I cannot findany number that is divisible by 12 that has not been

crossed out. Is Pablo correct? Explain your answer.e. Marisa argues that even if you extended the table to the

number 1,000, all numbers in the table that are divisibleby 24 would already have been crossed out. Do you agree?Explain.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100


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12. a. Circle 5 and put an X through all other multiples of 5 thathave not been crossed out.

b. What is the first number you put an X through?

c. Circle 7. Without looking at the table, name the firstmultiple of 7 that you will have to put an X through. Howwere you able to determine this number? Now cross outthe other multiples of 7.

d. Why is it unnecessary to cross out all of the multiples of 8, 9,and 10?

13. a. Circle 11. What multiple of 11 will you put an X throughfirst?

b. Circle all numbers that have not been crossed out.

c. What numbers did you circle?

d. In what columns do these circled numbers appear?

14. a. Explain why you crossed out only multiples of primenumbers.

b. Explain why you needed to cross out multiples of primesonly up to the number 11.

The number 8 can be completely factored into a product of primenumbers: 8 2 2 2.

15. a. Write each composite number between 2 and 10 as a productof prime numbers.

b. Do you think it is possible to write all numbers by using onlyprime numbers and multiplication?

By using factor trees, you can find all of the prime factors of anumber.

16. a. Use the factor tree method to find the prime factorsof 156.

b. Write 156 as a product of prime factors.


Prime Factors


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Here is another method you can use to find all of the primefactors of a number.

156

278 239 313 131

17. a. Compare this method with the tree method.

b. Use this method to find all prime factors of 72.


Prime NumbersC

Cubes and Boxes

Helena manages the shipping departmentfor Learning Is Fun, Inc., a company thatmakes centimeter cubes for use in schools.

18. a. One type of box holds 24 cubes.What are the possible dimensionsof this box?

b. Another type of box holds 45 cubes.Can this box have the same heightas a box that holds only 24 cubes?Explain why or why not.

In order to be able to stack the boxes easily, Learning Is Fun wouldlike the boxes to have the same length and width. Every box shippedis completely filled with centimeter cubes.

19. Is it possible for the two types of boxes in problem 18 to have thesame length and width? Explain and give the length, width, andheight of both types of boxes.


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Learning Is Fun also packages cubes in two different large-sizedboxes. One box holds 210 cubes, and the other holds 315 cubes. Thelarger boxes have to be completely filled with centimeter cubes.

20. a. Is it possible for these two boxes to have the same height?Explain your answer.

b. Helena wants the boxes to have the same dimensions for thebottom so the boxes stack easier. Is this possible? If so, whatare the possible dimensions for the bottom?

c. What information do you have to know about the numbers210 and 315 in order to help you answer parts a and b above?

Learning Is Fun now wants to make an extra large box to hold 525cubes.

21. a. What are possible dimensions for this box? Name at leastthree possibilities.

b. Is it possible to make boxes for 210, 315, and 525 cubes withthe same dimensions for the bottom? Explain your answer.

c. How can prime factorization help you to solve this problem?


CPrime Numbers


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Prime Numbers

In this section, you used factor trees and other methods to completelyfactor composite numbers into a product of prime factors. The endnumbers of the trees are prime numbers.

Prime Numbers

Prime numbers have exactly two factors, the number one and thenumber itself.

Composite Numbers

Numbers that are not prime are called composite numbers.

The number 1 is neither a prime number nor a composite number.

A product of factors

You can write 150 as a product of four factors:

150 2 3 5 5

The numbers 2, 3, and 5 are factors of 150.

You can also write 150 as a product of two factors.

150 3 50

Another factor of 150 is 50.

All of the factors of 150 are 1, 2, 3, 5, 6,10,15, 25, 30, 50, 75, and 150.

The prime factors of 150 are 2, 3, and 5.

C

a product of four factors

2 3 5 5


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Finding Prime Factors

You learned two methods to find all of the prime factorsof a number.

By using factor trees, you can find all prime factors of a number.

Here is another method to find all the prime factors of a number.

140 270 235 57 71

You can use all the end numbers to completely factor 140 as aproduct of primes.

140 2 2 5 7The prime factors of 140 are 2, 5, and 7.

1. Use an arithmetic tree to calculate 5 7 4 5 2.

2. Which of these numbers are composite numbers? Explain youranswer.

12 19 39 51

75

2 35

702

140


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Think back to Section B, where you squared numbers. In this section,you will continue squaring numbers using the context of area.

1. a. Draw a square with the dimensions 3 cm by 3 cm.

b. How many squares (1 cm by 1 cm) completely cover thesquare you just drew?

c. Explain how squaring is related to the area of the square you

drew in a.

2. a. Copy and complete this table filling in the area of the squarewith side lengths going from 1 cm through 10 cm.

b. Is this table a ratio table? Explain why or why not.

c. Use the grid on Student Activity Sheet 3 to graph the

information from your table. Connect all points witha smooth curve.

d. Describe the curve of your graph. Explain what this curvetells you. Keep this graph. You will use it again in problem 7.

For problems 37, use centimeter graph paper.

3. a. Draw a square with the dimensions 1 cm by 1 cm.

b. What is the area of this square?

c. Draw a square with the dimensions 12

cm by 12

cm.

d. Use your two drawings to explain that 12 1

2 1

4.

Now you will look at larger squares.

4. a. Draw a square with the dimensions 1 12

cm by 112

cm.

b. Use this drawing to calculate the area of the square.(Remember the unit Reallotment.)

Section D: Square and Unsquare 35

DSquare and Unsquare

Square

10987654321Length of Side (in cm)

Area of Square (in cm2)


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The number 1 12

is called a mixed number. It is a whole number andfraction combined.

5. a. Use a drawing to calculate the area of a square with side

lengths of 212 cm.

b. Use a drawing to calculate 312 3 1

2.

c. What does (4 12

)2 mean? Calculate (4 12

)2.

d. Calculate (512

)2.

6. Use your results of problems 4 and 5 to add five more pointsto your graph of problem 2c.


Square and UnsquareD

Nicole uses the pattern in her

answers to problem 5 to say,There is a pattern to squaringthese halves! Look, if I want tocalculate 61

2 6 1

2, I just calculate

6 7 and then add 14

.

7. a. Show how you can use your graph to see whether or notNicoles idea makes sense.

b. Use a drawing of a square with side lengths of 6 12 cm to showthat Nicole is right. Will Nicoles idea always work? How do you

know?

c. Use Nicoles idea to calculate 9 12 9 1

2.

d. Use your graph from problem 2 to check whether or not youranswer to c is reasonable.

e. Use that same graph to estimate the area of a square with sidelengths of 3.8 cm.


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9. a. At the beginning of the conversation, Kay and Juanita disagree.Who do you think is right? Explain.

b. How did Rick find the number 5.6568542 with his calculator?



Kay, Juanita, and Rick are having a conversation about theside length of the square with an area of 32 cm2.

Kay: Juanita, I dont think 5.6 cm or 5.7 cm is precise

enough. If we take the number out to more decimalplaces, we will get the exact length. Lets try 5.65because it is exactly halfway between 5.6 and 5.7.

Juanita: I dont think that will help, Kay. A number withdecimals multiplied by itself will never give a wholenumber as an answer.

Rick: I figured it out on my calculator and got 5.6568542.That has to be the exact answer.

Kay: Great job, Rick! Lets check it out.

5.6568542

Juanita forgot her calculator and writes 5.6568542 on a piece of paper.She starts figuring whether or not Ricks number is the exact length of

the side of the square. Rick uses his calculator to check the number hefound.

10. a. How could Juanita check the number without a calculator?Based on Juanitas computation, is Ricks number the exactlength of the side of the square?

b. How could Rick check the number with his calculator? Dothe same with your calculator. Write down your keystrokesand result.


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14. a. For which numbers listed here would it be easy to find thesquare root? Write down the square roots of the numbers youchoose.

24 49 121 120 81 72

1 64 2.5 0.25 225 525

b. Consider the numbers that you did not choose in part a.Use your calculator to approximate the square roots ofthese numbers.

15. How can you tell whether or not you can give an exact numberfor a square root?

16. a. How can you find what whole number is the closest to24?Explain this without the use of a calculator.

b. Draw a number line from 6 to 6 and place the followingnumbers on the number line.

36 5 5 5 5 6 17 half of 50


DSquare and Unsquare

52

You will now use a calculator to find the side length of asquare with an area of 52 cm2. The length of this side (or theside length of any square) can be found by taking the squareroot of the area.

11. What does the square root key do?

12. Use Student Activity Sheet 5 to investigate the squareroot of 52. Write a paragraph describing your findingsand what you think about the exact value of52.

13. Draw a square that has an area of 20 cm2. Explain thestrategy you used.


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Not So Square

The floor of Nathans room is 2 12

m by 4 12

m. His room will be

redecorated, and the floor will be redone. In order to estimate the

cost of the new floor covering, Nathan estimates the area of the

floor to be about 814 m

2

.17. a. How did Nathan arrive at his answer?

b. Show that this answer cannot be right.

c. On graph paper, make a scale drawing of the floor of Nathansroom. Use the scale drawing to calculate the area of the floor.


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Square and Unsquare

SquareTo find the area of a square,you can square the side lengths.For example, if the side length of a square is 5 cm, then thearea is 5 cm 5 cm, or 52 cm2, or 25 cm2.

Unsquare

If you know the area of a square, you can find the side length of thesquare by unsquaring the area. Unsquaring the area is alsofinding the square root of the area.

For example:

If the area is 52 m2, then the length of theside is 52 m. Using the button on yourcalculator, you find that 52 7.211.

You can write the side length 7.2 m.Think about how to round your answer!

Mixed Numbers

A mixed number is a number that is a sum of a whole numberand a fraction.

For example, 3 12 is a mixed number because 312 3 12 .

You can use the area model to multiply mixed numbers.

3 12 8 1

2

The four parts total 29 34

(24 4 1 12

14

).

So 3 12 8 1

2 29 3

4.

D

Area 52 m2

Side Length 52 m2

8

1

4

3

12

12

1224

14


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1. a. What is the base in 25? b. What is the exponent in 25?

c. Calculate 25.

2. What are the side lengths of squares with each area given?

a. 1200 in2 b. 120 in2 c. 12 in2

d. 1.2 in2 e. 0.12 in2

f. Compare your answers for ae. What do you notice?

3. In Springville, streets and avenues form the city

blocks. The blocks are very regular and looka lot like a grid. Each city block in Springvilleis usually 1

8of a mile long.

a. What is the area of one city block?

b. Calculate 38 1 1

2.

4. Calculate 3 12 2 1

3.

18

mile

In this pattern, you can see foursquares. Without measuringor making calculations, whatrelationship do you noticebetween the red squares?

Find the area andthe side lengths ofeach of the four

squares. Compare thearea and the side lengths.What other relationshipsdo you notice?


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Sissa invented the game of chess in India during the 6th century.

Sissas ruler was so pleased with the new game that he offered him areward in gold.

Sissa asked for a reward in rice and suggested that he collect rice for64 days (the number of squares on the chess board). Sissa lovedpatterns and asked for:

one grain of rice on the first day,two grains of rice on the second day,four grains on the third day,eight grains on the fourth day,and so on, doubling the number of grains each time.

The ruler was pleased that Sissa requested such a small reward!

1. a. Estimate how many grains of rice Sissa will receive on the64th day.

b. What would you have to do to calculate the total amount ofrice Sissa plans to collect?

c. What is your opinion of this reward?

EMore Powers

The Legend of the Chess Board


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In the beginning of this unit, you worked with powers of ten.You wrote repeated factors of ten in exponential notation:10 10 10 10 is written as 104. You can write numbers withbases other than 10 using exponential notation. For example,

125 is 5 5 5, so 125 53.

2. Completely factor these numbers into a product of prime numbers.Write the prime factorization using exponential notation.

a. 8 b. 81 c. 1,024

3. a. Calculate 35.

b. Which number is larger, 32 or 23? Explain why.

c. Which number is larger, 42 or 24? Explain why.

Section E: More Powers 45

EMore Powers


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You can solve the rice problem using powers of two. This table is setup for the first ten days of the 64 days Sissa will receive rice.

4. a. Describe a pattern in the first two columns.

b. You can write the number of grains of rice for the first day as

a power of two. How? Explain your reasoning.5. Explain how you can find the answer to the rice problem in 1a if

you use exponential notation.

Use the patterns of the last three columns to help you answer thesequestions.

6. a. What is the total number of grains after five days? And aftersix days?

b. How many grains will Sissa receive on Day 11?

c. How many grains will Sissa have in total on Day 11? Explainyour work.

d. What is the relationship between the number of grains per dayand the total number of grains?


More PowersE

Powers of Two

Number of Grains of Rice

Day Each Day Running Sum Running Total

1 1 1

2 21 2 1 2 3

3 22 4 3 4 7

4 23 8 7 8 15

5 24 16

6 25 32

7 26 64

8 27 128

9 28 256

10 29 512


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EMore Powers

On Day 19, Sissa will get 262,144 grains.

7. a. Write this number as a power of two.

b. On Day 19, how many grains will he have in total? Explain.

c. Write your answer to b using powers of two.

Five students used powers of two to write the total number of grainsafter 64 days.

Here is their work.

8. For each student, explain whether the work is right or wrong.

263 1 + 263

Bea

264

Cici

2263 1

Deron

264 1

Eva

20+ 21+ 22+ 23+ 24+ ...... + 263

Ali


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More PowersE

11. a. Write a rule for multiplying with powers ofthree.

b. Does your rule apply to powers of ten?Illustrate this with an example.

c. Calculate 52 103. Does your rule work in thiscase? Why or why not?

This table can be used to find products of powers of 3.

9. Explain how you can use this table to verify thatthe product of 243 729 is 177,147.

10. Use the table to calculate:

a. 9 243b. 6,561 6,561

c. 3 19,683

Powers of Three

Different Bases

243 729 177,147

33333 333333

324

324

162

...

2

2

In Section C, you used two different ways to factor anumber completely into a product of prime factors.

12. a. Write the prime factorization of 324.b. What are the two different prime factors of 324?

c. Kathie finds the prime factors of 288 and writes288 25 32. Explain what she did.

d. Write your answer to a as a product of powers,using two different prime factors.

Powers of Three

31

332 9

33 27

34 81

35 243

36 729

37 2,187

38 6,561

39

19,683310 59,049

311 177,147

312 531,441

313 1,594,323

314 4,782,969

315 14,348,907

316 43,046,721

317 129,140,163318

319

320


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EMore Powers

The prime factorization of 400 is 2 5 2 5 2 2.You can write the prime factorization of 400 as a product of powers.Using the prime factors 2 and 5, the prime factorization of 400 is 24 52.

13. Write the prime factorization of each number. If possible, useexponential notation to write each factorization as a product ofpowers.

a. 216 b. 6,125 c. 1,000

Joshua, Brenda, Veronique, and Pete calculated 10 23. Here istheir work.

14. For each student, explain whether the work is right or wrong.

Kian has ten cubes, each with the dimensions 2 cm by 2 cm by 2 cm.

15. a. What is the total volume of all ten of Kians cubes?

b. How does problem 15a relate to problem 14?

16. a. Calculate 5 43.

b. Calculate 32 103.

c. Write a rule for multiplying with powers with different bases.

10 x 23=

203 = 8,000

Joshua

10 x 23=

10 x 6 = 60

Brenda

10 X23 =10 X8 = 80

Veronique

101 x 23 =

204= 160,000

Pete


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EMore Powers

Both the Egyptian method and the ratio table method show how youcan write the number 13 as a sum of different powers of two. Since13 1 4 8, you can write 13 as a sum of powers of two:13 20 22 23.

19. a. Use your answer to problem 18c to write the number 11 as asum of powers of two.

b. Use your answer to problem 18d to write the number 18 as asum of powers of two.

You can write any whole number as a sum of powers of two. To findthese powers of two, you can use a table. Any power of two is notused more than once.The target number is located in the upperleft-hand position of the table.

20. a. Explain what information these two schemas display.

b. Use Student Activity Sheet 6 to write all the whole numbersfrom 1 through 15 as a sum of powers of two.

You just rewrote fifteen numbers of the decimal system (base 10) intoa binary system (base 2).

The number 5 written in the binary system is 101, read as one, zero,one. The number 12 written in the binary system is 1100. The binarysystem uses only two digits: 0 and 1. The prefix bi in binary meanstwo.

524 23 22 21 20

1 0 1

1224 23 22 21 20

1 1 0 0

Here is a special binary clock with little blue lightsshining to display the current time. Each light is

either on (1) or off (0).21. a. The third column shows the number 4.

How would you explain this to someone?

b. What time does the clock show now?Show your work.


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More Powers

Products of PowersYou can completely factor any number into a product of primefactors. Sometimes, when the factors repeat, you can write thisnumber as a product of powers. For example:

5,625 3 3 5 5 5 5

32 54

You can combine a product of powers with the same baseintoone base and power. For example:

If you want to calculate a product of powers with different bases, thenyou have to calculate the powers first and then multiply. There are notany shortcuts because the bases are different. For example:

53 102

125 100, which is 12,500

The Binary SystemThe binary system is based on powers of two. There are only twodigits in the binary system, 0 and 1. To write a number in the binarysystem, you only need to write the number as a sum of powers of 2.For example:

5 4 1 22 20 22 0 20

1 22 + 0 21 + 1 20

In the binary system, you can write 5 as 1012

(read as one, zero, one,base 2.)

E

524 23 22 21 20

1 0 1

3

2

3

3

3

5

333 3533 23


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1. Write 10,000 as a product of powers.

2. Write the prime factorization of each number. If possible, useexponential notation to write each factorization as a product ofpowers.

a. 288 b. 900 c. 1764

3. Calculate 23 52.

4. Use the table with powers of three

to calculate:a. 27 81

b. 2187 3

c. 1 3 9 27 81

d. 94

Compare our decimal numbers to binary numbers. Why do you thinkwe use base ten rather than base two? Be specific.

Powers of Three

31 3

32 9

33 27

34 81

35 243

36 729

37 2,187

38 6,561

39 19,683

310 59,049


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1. 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 youfound your answer.

On a transatlantic flight, the speed of an airplane is about 1,000 kmper hour. If it were possible, a plane traveling at this speed wouldneed 16 days to fly to the moon.

2. Use this information to calculate the distance from the earth tothe moon.

The distance from the earth to the sun is about 400 times the distancebetween the earth and the moon.

3. a. How many days would that same plane need to fly from theearth 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.

4. Which one of the following is the largest number? Explain youreasoning.

1. What is the smallest natural number that has exactly five factors?Explain how you found it.

400 10640 1080.4 1011


Additional Practice

Section Base TenA

Section FactorsB


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You cannot view the last digit of this 11-digit number.

2. What digit can you place in the open position to have a number:

a. divisible by 5?

b. divisible by 2 and 5?

c. divisible by 9 but not by 2?

d. divisible by 2 and 3?

You can write the number 10,000 as a product of two numbers in manydifferent ways. Here are two different ways: 10,000 1,000 10 and10,000 400 25.

3. Write 10,000 as a product of three numbers, so that none of thenumbers is divisible by ten. Find two different possibilities.

In 1845, Bertrand conjectured:

For every whole number greater than three, there is at least oneprime between that number and its double.


84 355 216 015

18221900

4. Verify Bertrands conjecture bychecking all the appropriatenumbers less than 21. Organizeyour work so that someoneelse can understand Bertrandsconjecture.

Joseph Bertrand was a Frenchmathematician interested in primenumbers, geometry, and probability.In 1855, he translated Gauss s workon error analysis into French. In 1856,he was appointed as a professor atthe cole Polytechnique. Later he alsobecame a professor at the Collge deFrance. From 1874 until the end of hislife, he was a distinguished member

of the Paris Academy of Sciences.


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In 1742, the Russian mathematician Christian Goldbach conjectured:

Every even integer larger than two can be written as the sumof two prime numbers.

For example, 8 = 5 + 3.

Goldbachs conjecture has been tested for all values up to 1014, but noone has been able to prove it yet!

1. a. You can write the even number 6 as 6 3 3 or 6 1 5.One of these sums doesnt verify Goldbachs conjecture.Which one? Why?

b. You can verify Goldbachs conjecture for 28 with the sum28 23 5. Is this the only possibility? Investigate otherpossibilities.

c. Verify Goldbachs conjecture by checking all the evennumbers less than 21. Organize your work so someone elsecan understand Goldbachs conjecture.


Additional Practice

Section Prime NumbersC

2. Place the eight numbers, 0, 1, 2, 3, 4, 5, 6, and 7 onthe eight vertices of the 3-D shape so that the sumof any two adjacent vertices is a prime number.Adjacent vertices are physically connected.

A prime dayis when both the month andthe day are prime numbers. For example,May 23 is a prime day because both

(5 and 23) are prime numbers.3. a. What is the first prime day of the

year? And the last one?

b. How many prime days are there in a year?

4. Completely factor each of the following numbers into a product ofprimes.

a. 900 b. 2,079 c. 12,121


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1. a. Explain how you can use the area model inthis drawing to calculate (7 12

)2.

b. Use graph paper to copy this drawing andfill in all the missing information.

c. Calculate (7 12

)2.

2. Choose a strategy to calculate:

a. (12 12

)2

b. (21 12

)2

Here are two squares and two rectangles. Thenumber on each shape is the area of that shape.You can use all four shapes to form one largesquare.

3. a. What is the side length of the large square?Show your work and make a sketch of the

large square.b. Suppose you could reshape the two identical

rectangles to form a large square. What isthe area of this square? What is the lengthof one side?


Additional Practice

Section Square and UnsquareD

7

1

2

7? ?

? ?12

In this drawing, the dark yellow shapesare squares.

The area of each square is indicated.4. Explain how the total area of all

of the shapes is a square number.

9 cm2

24 cm2

24 cm2

4 cm2

121 cm2

16 cm2

9cm2


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Three of these statements are true and three are false.

a. 43 82 4 4 4 8 8 b. (6 12 )2 36 12c. (11 1

2)2 (11 1

2)3 (11 1

2)5 d. 26 25 230

e. 54 24 108 f. 34 9 36

1. Try to decide which are true and which are false withoutcalculations. Explain your reasoning.

2. Completely factor the following numbers into a product of primenumbers:

a. 10 b. 26 c. 77 d. 50

During math class, Mr. Shawn asked Peter, How many differentrectangles can you make that have an area of 26 square inches?

Peter quickly answered, If the sides are the counting numbers, thenthere are four possibilities.

3. a. Which possibilities did Peter think of? Can you explain howPeter 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 quicklyfind all possibilities.

c. Consider all the possible rectangles with an area of 50 in2.Did you find all the possible rectangles quickly? Explain whyor why not.

In communications, electronics, and physics, a kilo stands for 103.For example, 1 kilometer = 103 meters or 1,000 meters.

In Information Technology (IT) and data storage, a kilo stands for 210.

For example, 1 kilobyte = 2

10

bytes.


Additional Practice

Section More PowersE


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This table explains the prefixes kilo, mega, and giga.

4. a. Calculate how many bytes are in one kilobyte. Estimate youranswer using a power of ten.

b. How many bytes are in one megabyte? Write your answer inscientific notation. Estimate your answer using a power of ten.(You may want to use a calculator for this.)

c. How many kilobytesare in one megabyte? How do you know?

The relationship between kilobytes and megabytes holds true formegabytes and gigabytes. One gigabyteis more than 1,000 timesone megabyte.

d. How many bytes are in one gigabyte? Write your answer as apower of two.

5. In problem 21 of Section E, you learned how to read a binaryclock. Sketch a binary clock and color the lights so the timedisplayed is 3:12 P.M.


Additional Practice

IT Terminology

one kilobyte 1 kB 210 bytesone megabyte 1 mB 220 bytes

one gigabyte 1 gB ..... bytes


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5. a. Both calculators display 5.1 and 06; 5.1 is the first factorbetween 1 and 10; 06 is the exponent of 10. The difference isthe 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. 5.1 106 or 5.1 million or 5,100,000

1. Yes, groups of three work because the sum of the digits of 945is 18: 9 4 5 18, and 18 is divisible by 3.

No, groups of six will not work. Divisible by 6 means that thenumber 945 has to be divisible by 3 and by 2. Because 945 is notan even number, it is not divisible by 2, so it is not divisible by 6.

2. a. 1, 3, 5, and 15

b. 1, 2, 4, 8, 16, and 32

c. 1 and 53

d. 1 and 17

3. a. The number you wrote can be even or odd but must not be

a perfect square number. One sample number that has aneven 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 offactors.

4. There are 10 perfect square numbers from 1 through 100: 1, 4, 9,16, 25, 36, 49, 64, 81, and 100.


Answers to Check Your Work

Section FactorsB


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3. a. 99 3 3 11 b. 750 2 3 5 5 5

c. 264 2 2 2 3 11

4. A strategy to solve these problems is to find all factors of thenumber of centimeter cubes first.

a. The factors of 8 are: 1, 2, 4, and 8.Three possible dimensions are:1 cm by 1 cm by 8 cm,1 cm by 2 cm by 4 cm, and2 cm by 2 cm by 2 cm.

b. The factors of 50 are: 1, 2, 5, 10, 25, and 50.

Three possible dimensions are:1 cm by 2 cm by 25 cm,1 cm by 5 cm by 10 cm, and

2 cm by 5 cm by 5 cm.



2 5

10

55

3 25

75

75099 9 11

(3 3) 11

3 3 11

264

13266

33

11

1

2

2

2

3

11


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1. a. The base is 2.

b. The exponent is 5.

c. 25 2 2 2 2 2 32

2. a. The side length is 1200 34.64 in.

b. The side length is 120 10.95 in.

c. The side length is 12 3.46 in.

d. The side length is 1.2 1.095 in., or 1.10 in.

e. The side length is 0.12 0.364 in., or 0.35 in.

f. If the area is 100 times as small, then the side length is tentimes as small. Compare, for example, a and c or c and e.



Section Square and UnsquareD

1 mile

1 square mile 1 mile

one city block

1 mile12

mile38

3. a. The area of one city block is 164

square mile.

Sample reasoning:

One city block is 18

mile by 18

mile.

In one square mile (see drawing), you can fit

eight rows of eight city blocks. This makes8 rows 8 blocks or 64 blocks.

If 64 city blocks fit in one square mile, thenthe area of one city block is 1

64of a square

mile.

b. 38 1 1

2

3664

, or 916

Here is a way to calculate 38 1 1

2using city

blocks.38 mile is 3 city blocks.1 1

2miles are 12 city blocks (8 4).

38 1 1

2is the same as

3 blocks 12 blocks, or 36 blocks.

Since 1 city block is 164

square mile, 36 city

blocks is 3664

square mile.


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4. 3 12 2 1

3 8 1

6

One sample strategy using the area model:

The four parts total 8 16

, (6 1 1 16

).

1. 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 onepower and not a product of powers.

2. a. 288 2 2 2 2 2 3 3 25 32

b. 900 2 2 3 3 5 5 22 32 52

c. 1764 2 2 3 3 7 7 22 32 72

3. 23 52 2 2 2 5 5 200

4. 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.



Section More PowersE

213

3

12

6

116

1


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Facts and Factors

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