Facts and Factors

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Facts and Factors


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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.

© 2010 Encyclopædia Britannica, Inc. Britannica, Encyclopædia Britannica, the thistle logo, Mathematics in Context, and the Mathematics in Context logo are registered trademarks of Encyclopædia Britannica, Inc.

All rights reserved.

No part of this work may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage or retrieval system, without permission in writing from the publisher.

International Standard Book Number 978-1-59339-939-9 Printed in the United States of America

Abels, M., de Lange, J., & Pligge, M. A. (2010). Facts and factors.

In Wisconsin Center for Education Research & Freudenthal Institute (Eds.), Mathematics in context. Chicago: Encyclopædia Britannica, Inc.

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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 Project Staff

Sarah Ailts Margaret R. Meyer Arthur Bakker Nathalie Kuijpers

Beth R. Cole Anne Park Peter Boon Huub Nilwik

Erin Hazlett Bryna Rappaport Els Feijs Sonia Palha

Teri Hedges Kathleen A. Steele Dédé de Haan Nanda Querelle

Karen Hoiberg Ana C. Stephens Martin Kindt Martin van Reeuwijk

Carrie Johnson Candace Ulmer

Jean Krusi Jill Vettrus

Elaine McGrath

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Cover photo credits: (all) © Getty Images Illustrations

1 (top) Michael Nutter/© Encyclopædia Britannica, Inc.; (bottom) Holly Cooper-Olds; 2, 3, 4, 13 Christine McCabe/© Encyclopædia Britannica, Inc.; 18, 24 (left), 25, 27, 34 (left), 36 Holly Cooper-Olds;

38 Christine McCabe/© Encyclopædia Britannica, Inc.; 45, 50 (top) Holly Cooper-Olds; 51, 56 Christine McCabe/© Encyclopædia Britannica, Inc.


3 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; 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/

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Contents v

Letter to the Student vi

Section A Base Ten

Hieroglyphics 1

Times Ten 3

Large Numbers 6

Exponential Notation 7 Scientific Notation 8

Summary 10

Check Your Work 11

Section B Factors

Pixels 13

Facts 17

Factors 17

Changing Positions 21

Summary 22

Check Your Work 23

Section C Prime Numbers

Upside-Down Trees 24

Primes 27

Prime Factors 29

Cubes and Boxes 30

Summary 32

Check Your Work 33

Section D Square and Unsquare

Square 35

Unsquare 37

Cornering a Square 37

Not So Square 40

Summary 42

Check Your Work 43

Section E More Powers

The Legend of the Chess Board 44

Powers of Two 46

Powers of Three 48

Different Bases 48

Back to the Egyptians 50

Summary 52

Check Your Work 53

Additional Practice 54 Answers to Check Your Work 60

6 12 24

3 2 2 2

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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, T

Thhee MMaatthheemmaattiiccss iinn CCoonntteexxtt DDeevveellooppmmeenntt TTeeaamm

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Section A: Base Ten 1

A Base Ten


Tropic of Cancer RE





Giza Memphis


Edfu 1st Cataract

2nd Cataract Valley of the Kings

Abu Simbel Rosetta

Heliopolis Cairo

Tell El-Amarna

Karnak Thebes

Luxor Aswan Philae SINAI












Riv er


R iver




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 on the stone represent the number 1,333,331.

Step back in time to a world without computers, calculators, and television;

to Egypt around 3000 B.C.

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

He carved little pictures called hieroglyphs 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 astonished man

Base Ten


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 numerals0, 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 the Egyptian 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 the number 999?

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You found these three pieces of a stone containing Egyptian hieroglyphs.

7. What number do they represent when placed altogether?

Section A: Base Ten 3

Base Ten

Times Ten

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

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

9. What are the differences between our Arabic system of writing and using numbers and the Egyptian system?

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

b. Describe what the ancient Egyptians would do 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 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 tens and 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 both number systems.

Ancient Egyptian Hieroglyphics vs. Arabic Number System

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 7 in 537? And what is the value of 7 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 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.

Base Ten




 10  10

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The Egyptian number system was not well suited for decimal or fraction notation. The decimal notation we use today was developed almost 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  101  5  10001 as a single number.

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

Consider the product of 57.38 10.

57.38  10  573.8

57.38  5  10  7  1  3  101  8 1001 573.8  5  100  7  10  3  1  8  101

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

Section A: Base Ten 5

Base Ten


hundreds tens ones tenths hundredths

5 7 3 8

5 7 3 8


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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. This makes the numbers easy to read. You read the number 2,638,577 as “two million, six hundred thirty-eight thousand, five hundred seventy-seven.”

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

There are different ways to read and write large numbers. For example, 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

Base Ten


Large Numbers

Numerals Words 1 one 10 ten

100 one hundred 1,000 one thousand 10,000 ten thousand

100,000 one hundred thousand 1,000,000 one million

10,000,000 ten million

100,000,000 one hundred million 1,000,000,000 one billion

10,000,000,000 ten billion

100,000,000,000 one hundred billion 1,000,000,000,000 one trillion

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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 five different multiplication problems for which the answer is 1,000,000.

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

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

The number 1,000 written in exponential notation is 103(read as

“ten raised to the third power” or “ten to the third”).

1,000 103because 1,000 10 10 10

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

Section A: Base Ten 7

Base Ten A

Exponential Notation

10 3



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

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

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

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

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

e. In your notebook, calculate the product of 2,000,000,000 ⴛ 3,000,000,000. Verify your calculation using your calculator.

If needed, revise your answer for part d.

Base Ten


Scientific Notation

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

Calculators display scientific notation in a variety of ways. Here are two different calculator displays for the 2004 world population of 6,400,000,000 people.

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

6.4 E 09 6.4


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

b. What numbers are displayed here?

The distance from the earth to the moon is approximately 240 thousand miles.

26. How would your calculator display this distance in scientific notation?

Section A: Base Ten 9

Base Ten





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Base Ten

The Arabic Number System you use today is a positional system using the numerals 0 through 9. The position of each digit in a number determines its value. You can read the number 79.54 as “seventy-nine and fifty-four hundredths.”

You can expand the number 79.54 as:

7  10  9  1  5  101  4 1001 or 70  9  105  1004 .


Multiplying by Ten

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

For example: 79.54  10

79.54  10  795.4

79.54  7  10  9  1  5  101  4 1001

795.4  7  100  9  10  5  1  4  101  10

hundreds tens ones tenths hundredths

7 9 5 4

7 9 5 4


tens ones tenths hundredths

7 9 . 5 4

4 hundredths 5 tenths

9 ones 7 tens

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Section A: Base Ten 11

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 107as “ten to the seventh power” or “ten to the seventh.”

In 107, the 10 is the base, and the 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 10 and a power of ten.

A calculator displaying represents the product 4.5  107 4.5  10,000,000


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  10

2. a. Use numbers such as 10, 100, 1,000, and so on, to write five different multiplication problems for which the answer is one billion.

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

Exponential Notation

10 7



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. 1,000,000 10,000

c. ten  one hundred  one thousand d. one thousand  one million

4. a. Fill in the missing exponents and then write the answer as a single number without an exponent.

2.25  104 22.5  10? 225  10?

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

Here are two different calculator displays of the same number.

5. a. Explain what is displayed.

b. Write this number as a single number.

Write a short paragraph for a school newsletter describing the benefits of using scientific notation for very large numbers.


5.1 06 5.1 E 06

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Jacqui and Nikki are friends. They used to be neighbors, but Nikki moved to Cleveland. Now they maintain their friendship by using the Internet. They send e-mail to each other and chat online at least once a day.

Today after school, Jacqui checks her e-mail. After about three minutes, she realizes Nikki’s message is taking longer than usual to download. After waiting impatiently for ten minutes, Jacqui asks her brother, “Dave, what can I do? Look at that bar on the computer screen!”

Section B: Factors 13

B Factors


Nikki’s e-mail included a picture with her new puppy.

Dave remarks, “It’s a cute picture, but the size of the file is too large. Send her an e-mail and tell her that she has to make the files smaller before she sends them.”

Jacqui says, “Dave, how can she do that?

I don’t even know how to do that.”

Dave shares what he knows about digital pictures.

Here is a screen shot of the bar on Jacqui’s computer after 12 minutes.

1. Estimate how many more minutes Jacqui will have to wait to download this message completely. Show how you found your answer.

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



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 decreased dramatically: you can now see the pixels.

You will now investigate the effect of changing the 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 more pixels per inch (ppi).

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

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

Compare Pictures 1, 2, and 3.

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

You probably didn’t find the total number of pixels by counting all the small squares. For counting the pixels in Picture 1, you may have multiplied 12  12.

Whenever you multiply a number by itself, you are squaringthe number.

5. Why do you think the expression “squaring a number” is used?

Picture 1

Picture 2

Picture 3

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Two ways to indicate squaring the number 12 are 122or 12^2. Both represent 12 ⴛ 12, which gives an answer of 144.

Picture 1 has 12 pixels along each side, for a total of 122, or 144 total pixels.

Numbers like 144, which result from squaring a number, are called square numbersorperfect square numbers.

6. Find at least five different perfect square numbers. Share your list with a classmate. See if each of you can guess the number before it was squared.

Earlier, you compared the same pixel pattern for three different sized pictures. The pictures became smaller, but the total number of pixels did not change.

If you want to reduce the size of a picture file, then you must reduce the total number of pixels. You will now investigate ways to reduce the number of pixels by changing the number of pixels per inch (ppi).

This square picture of a pink rose has sides of 1 inch.

7. a. What is the total number of pixels if there are 200 ppi?

b. What is the total number of pixels if there are 100 ppi?

Note that the sides of the picture stay 1 in.

And 50 ppi? And 25 ppi?

c. Copy this table and record your answers from a and b in column 2. Describe how the pixels per inch (ppi) in column 1 change from row to row.

d. How does the total number of pixels decrease as the number of pixels per inch is cut in half?

e. The download time decreases as the total number of pixels decrease.The download time for a 200 ppi picture is16 seconds.

Use this information to fill in the last column of your table.

Section B: Factors 15

Factors B

ppi Total Number of Pixels Download Time 200

100 50 25

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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 to help 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.), Nikki reduced the number of pixels to 200 ppi. How many total pixels make up Nikki’s new picture?

b. Use the information from Nikki’s picture to copy and complete this table.

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

Jacqui waited about 48 minutes for Nikki’s original picture to download.

11. a. What would have been the download time if the picture had 200 ppi instead of 400 ppi?

b. And if the picture had 100 ppi?



400 3


______? ______?

400 4

4 3


______? ______?

400 400

ppi 400 200 100

Total Number of Pixels

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Here is Nikki’s picture reduced too much — it has just five pixels per inch.

Images appear nicely on a computer screen if there are at least 72 pixels per inch.

Section B: Factors 17

Factors B



p s S

In the previous problem, you might have calculated the total number of pixels by using division: 480,000  4 ___ , or multiplication: 4  ___  480,000.

Division and multiplication operations relate to each other in this way.

Using either operation, you found that the total number of pixels decreased from 480,000 to 120,000. Two number sentences for this context are 480,000  4  120,000, and 4  120,000  480,000.

The whole numbers 4 and 120,000 are called factorsof 480,000.

12. a. Find four different factors of 48.

b. Can you find a factor of 45 without making a calculation?


c. How do you know that 2 is not a factor of 45?

You may remember some divisibilityrules. Divisibility rules involve division of whole numbers without any remainders. Here are three divisibility rules:

• A number is divisible by 3 if the sum of the digits is divisible by 3.

• A number is divisible by 4 if the last two digits form a number that is divisible by 4.

• A number is divisible by 9 if the sum of its digits is divisible by 9.

13. a. Is 2,520 divisible by 3? By 4? By 9?

b. Is 2,520 divisible by 5? Write a rule for divisibility by 5.

c. How can you check whether or not 2,520 is divisible by 6?

d. What other rules for divisibility do you know?

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Jacqui prints 24 square pictures. She wants to use all 24 pictures to make 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 rectangular arrangement.

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

Jacqui’s 24 pictures:

One possible rectangular arrangement of 24 pictures: 6 across and 4 vertical:

List of all possible rectangular arrangements:

1 by 24 6 by 4 2 by 12 8 by 3 3 by 8 12 by 2 4 by 6 24 by 1

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

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



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Jacqui decides to draw one of her picture arrangements on graph paper.

Here is a graphshowing all of the rectangular arrangements in Jacqui’s list.

15. a. Explain what the graph shows.

b. How would you label the axes?

c. Describe what each pair of coordinates has in common.

Since 3  8  24, 3 and 8 are factorsof 24.

16. List all of the possible factors of 24.

How can you be sure you have them all?

Section B: Factors 19

Factors B

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)

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17.a. Create a graph showing all the points that represent factors of 25. 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 one point?

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

b. What do you notice about the factors of the five numbers you found 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 a number.

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

Here is their work.

20. a. If you continue Rosa’s list, how will you know when to stop?

b. Finish Rachel’s work to find all of the factors of 36.

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




All factors of 36 are 1, 36, 2, 18, 3, 12, …  


1 and 36 2 and 18 3 and 12 . . .


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In this activity, each of the students standing will be holding a card with a number from a special set.

You will need cards numbered from 1 to the total number of students in class.

Follow these steps:

Step i. Each student receives a card and stands up.

Step ii. Does the number 2 go into the number on your card?

If the answer is YES, then that student must sit down;

otherwise, the student remains standing.

Step iii. Does the number 3 go into the number on your card?

If the answer is YES, then change your position.

If standing, sit down; if sitting, stand up.

Check that you are playing the game correctly by discussing these questions:

After Step iii, is the student with the number 5 standing or sitting?

Is the student with the number 12 standing or sitting?

If everyone agrees, continue asking Does the number ___ go into the number on your card? Don’t forget to change your position whenever you answer YES.

Step iv. Does the number 4 go into the number on your card?

If you answer YES, then change your position.

If standing, sit down; if sitting, stand up.

Continue these steps asking whether the number on the card is divisible by 5, then 6, then 7, and so on, until you reach the total number of students in the class.

21. a. What numbers belong to the students who are standing at the end? What is common to these numbers?

b. If you did this activity with 100 students, what numbers would the students who are standing at the end be holding?

Section B: Factors 21

Changing Positions

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Multiplying a number by itself is squaring a number.

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

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


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 the factors of 30 are:


To see if a number is divisible by a certain number, you can follow some 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 factors—the number itself and the number one.


1 2 3 5 6 10 15 30

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Section B: Factors 23 1. At Green Middle School, there are 945 students. Is it possible to

split up all of the students into groups of three? Into groups of six?

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 of factors.

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

c. What name do you give the numbers having an odd number of 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 no even 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?

C Prime Numbers

Upside-Down Trees

6 3




 ______? ______?

5 5

3 6


______? ______?

2 5



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?

7 7


______? ______?

7 7

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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 to describe “factor.”

Section C: Prime Numbers 25

Prime Numbers C

6 12 24

3 2

2 2

These special arithmetic trees are called factor trees. In these factor trees, you will only see multiplication signs. Here is the beginning of a 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 with the 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 why not?

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

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

b. Use the “end numbers” (the numbers at the end of the tree) to write 24 as a product of factors.

2  5

10  156

1,560 2  3  5  5  150

a product of four factors

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

6. Completely factor each number. Use a factor tree to write each number 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 is even, so he starts his tree divisible by both 2 and 3, so like this. it is divisible by 6. She starts

her tree like this.

7. a. In your notebook, finish Hakan’s and Alberta’s factor trees.

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

8. a. Refer back to all of the trees you have made so far and compile a 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 not already on your list for part a.

Prime Numbers



2 546


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 a method to extract all of the prime numbers from 1 to 100. Beginning with a list of 100 numbers, he sifted out the prime numbers by cross- ing off multiples of numbers.

Section C: Prime Numbers 27

Prime Numbers C


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 10–15 to recreate Eratosthenes’ method for extracting the prime numbers.

10. a. Circle the number 2 and put an X through all of the other multiples 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 the multiples of 4.

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

d. Pablo went through these steps and said, “I cannot find any 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 divisible by 24 would already have been crossed out. Do you agree?


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 that have 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 first multiple of 7 that you will have to put an X through. How were you able to determine this number? Now cross out the 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 through first?

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 prime numbers.

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

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

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

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

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

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

b. Write 156 as a product of prime factors.

Section C: Prime Numbers 29

Prime Factors

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

—— 2156

—— 278

—— 339

—— 1313 1

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

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

Prime Numbers


Cubes and Boxes

Helena manages the shipping department for Learning Is Fun, Inc., a company that makes centimeter cubes for use in schools.

18. a. One type of box holds 24 cubes.

What are the possible dimensions of this box?

b. Another type of box holds 45 cubes.

Can this box have the same height as 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 would like the boxes to have the same length and width. Every box shipped is completely filled with centimeter cubes.

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

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Learning Is Fun also packages cubes in two different large-sized boxes. One box holds 210 cubes, and the other holds 315 cubes. The larger 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 the bottom so the boxes stack easier. Is this possible? If so, what are the possible dimensions for the bottom?

c. What information do you have to know about the numbers 210 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 525 cubes.

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

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

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

Section C: Prime Numbers 31

Prime Numbers C

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Prime Numbers

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

Prime Numbers

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


a product of four factors

2  3  5  5

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Section C: Prime Numbers 33

Finding Prime Factors

You learned two methods to find all of the prime factors of 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.

—— 2140

—— 270

—— 535

—— 77 1

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

140  2  2  5  7

The 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 your answer.

12 19 39 51



2  35




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Prime Numbers

3. Use any method you like to completely factor each number into a product of primes.

a. 99 b. 750 c. 264

4. a. What are the possible dimensions of a box that can completely be filled with eight centimeter cubes? Name at least three possibilities.

b. What are the possible dimensions of a box for 50 centimeter cubes? Name at least three possibilities.

Write a five-sentence verse similar to the one above that begins:

“Five students had five friends. Each friend had….” At the conclusion, find the total number of the things you mention in the verse.


Seven houses contain seven cats.

Each cat kills seven mice.

Each mouse had eaten seven ears of grain.

Each ear of grain would have produced seven hekats of wheat.

What is the total of all of these?

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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 the square 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 square with 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 with a smooth curve.

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

For problems 3–7, 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 1––

2 cm by 1––

2 cm.

d. Use your two drawings to explain that 1––

2 1––2 1––4. Now you will look at larger squares.

4. a. Draw a square with the dimensions 11––

2 cm by 11––

2 cm.

b. Use this drawing to calculate the area of the square.

(Remember the unit Reallotment.)

Section D: Square and Unsquare 35

D Square and Unsquare


10 9 8 7 6 5 4 3 2 Length of Side (in cm) 1

Area of Square (in cm2)

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The number 11––

2 is called a mixed number. It is a whole number and fraction combined.

5. a. Use a drawing to calculate the area of a square with side lengths of 21––

2 cm.

b. Use a drawing to calculate 31––

2  31––2. c. What does (4––1

2)2mean? Calculate (41––

2)2. d. Calculate (5––1


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

Square and Unsquare


Nicole uses the pattern in her answers to problem 5 to say,

“There is a pattern to squaring these halves! Look, if I want to calculate 6––1

2  6––12, I just calculate 6  7 and then add ––14.”

7. a. Show how you can use your graph to see whether or not Nicole’s idea makes sense.

b. Use a drawing of a square with side lengths of 61––

2 cm to show that Nicole is right. Will Nicole’s idea always work? How do you know?

c. Use Nicole’s idea to calculate 9––1

2  9––12.

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

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

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Square and Unsquare D

Use Student Activity Sheet 4. Cut out the 8 cm by 8 cm grid. What is the area of this shape?

Fold all four corners so that they meet in the center.

What is the shape of this folded paper? What is its area? Measure the length of each side of the shape with a ruler. (Hint: You might want to look at the back of the shape.)

Fold all four of the new corners so that they meet in the middle. Repeat this process until you have looked at a total of five shapes. Each time you fold the four corners, write down the name of the shape, the area of the shape, and the length of one of its sides.

How does the area change each time you fold to make a new shape?

In the activity, you measured a side length of a square with an area of 32 square centimeters (cm2). Mina did the same activity and mea- sured the length as 5.6 cm. When Justin did the activity, he measured the length as 5.7 cm.

8. a. How do your measurements compare with Mina’s and Justin’s measurements?

b. When Vance looked at Mina’s and Justin’s answers, he commented that they were close to the correct answer, but not exact. How could he tell?

Section D: Square and Unsquare 37


Cornering a Square

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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?

Square and Unsquare


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

Kay: “Juanita, I don’t think 5.6 cm or 5.7 cm is precise enough. If we take the number out to more decimal places, we will get the exact length. Let’s try 5.65 because it is exactly halfway between 5.6 and 5.7.”

Juanita: “I don’t think that will help, Kay. A number with decimals multiplied by itself will never give a whole number 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! Let’s check it out.”


Juanita forgot her calculator and writes 5.6568542 on a piece of paper.

She starts figuring whether or not Rick’s number is the exact length of the side of the square. Rick uses his calculator to check the number he found.

10. a. How could Juanita check the number without a calculator?

Based on Juanita’s computation, is Rick’s number the exact length of the side of the square?

b. How could Rick check the number with his calculator? Do the same with your calculator. Write down your keystrokes and result.

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

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 of these numbers.

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

16. a. How can you find what whole number is the closest to 24?

Explain this without the use of a calculator.

b. Draw a number line from –6 to 6 and place the following numbers on the number line.

36 5 5 5 5 6 17 half of 50

Section D: Square and Unsquare 39

Square and Unsquare D


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

11. What does the square root key do?

12. Use Student Activity Sheet 5 to investigate the square root of 52. Write a paragraph describing your findings and what you think about the exact value of 52 . 13. Draw a square that has an area of 20 cm2. Explain the

strategy you used.

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Square and Unsquare


Not So Square

The floor of Nathan’s room is 2 ––1

2 m by 41––

2 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 8 1––

4 m2.

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 Nathan’s room. Use the scale drawing to calculate the area of the floor.

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Section D: Square and Unsquare 41

Square and Unsquare D

During the fall, Nathan earns extra money working at the apple orchard. In one hour, he fills 3 ––1

2 bushels of apples. How many bushels will he fill after working 6 ––1

2 hours?

A solution to this problem involves calculating 3 ––1

2  6 ––12. Although bushels of apples and hours are involved, you can use the area model to make a calculation. In this case, the area is 3 ––1

2  6 ––12, and the rectangle is 3 ––1

2 by 6 ––1 2.


1 2

18 3

1 2

18. a. Copy the area model above and use it to find the number of bushels of apples Nathan will fill after working for 6 ––1

2 hours.

b. Use the area model to calculate 3 ––1

2  4 ––12 . c. Use the area model to calculate 5 ––1

2  11 ––12.

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