Facts and factors grade 7

Letter to the Student vi Section A Base Ten Hieroglyphics 1 Large Numbers 6 Exponential Notation 7 Scientific Notation 8 Check Your Work 23 Section C Prime Numbers Upside-Down Trees 24 P

Facts and Factors

Number

Mathematics in Context is a comprehensive curriculum for the middle grades

It was developed in 1991 through 1997 in collaboration with the Wisconsin Center for Education Research, School of Education, University of Wisconsin-Madison and the Freudenthal Institute at the University of Utrecht, The Netherlands, with the support of the National Science Foundation Grant No 9054928.

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

National Science Foundation

Opinions expressed are those of the authors and not necessarily those of the Foundation.

Abels, M., de Lange, J., and Pligge, M.,A (2006) Facts and Factors.

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

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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

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Letter to the Student vi

Section A Base Ten

Hieroglyphics 1

Large Numbers 6 Exponential Notation 7 Scientific Notation 8

Check Your Work 23

Section C Prime Numbers

Upside-Down Trees 24

Prime Factors 29 Cubes and Boxes 30

Check Your Work 33

Section D Square and Unsquare

Cornering a Square 37 Not So Square 40

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

3 2 2 2

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 over5,000 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 intothe 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 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 your

understanding of all the real numbers

We hope you enjoy this unit

Sincerely,

T

Th hee M Ma atth heem ma attiiccss iin n C Co on ntteex xtt D Deevveello op pm meen ntt T Teea am m

Giza Memphis

Abu Simbel

Rosetta

Heliopolis Cairo

Tell El-Amarna Karnak Thebes

Luxor

Aswan Philae

SINAI

L

IBAN

DE S R

A A

I A N

D S R

LOWER EGYPT

UPPER EGYPT

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 withoutcomputers, calculators, and television;

Egyptian Egyptian Arabic English Hieroglyph Description Numeral Word

a heel bone

a coil or rope lotus flower pointing finger tadpole

an astonished man

Base Ten

A

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

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 usingnumbers 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

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.

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 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

The Egyptian number system was not well suited for decimal or fraction 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  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

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 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 hundredthousand” or simply as “3.2 million.”

18 Write at least two different ways you can read each number.

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

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 is1,000,000

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 aspecial 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.

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 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 Writeyour 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 3,000,000,000 Verify your calculation using your calculator

If needed, revise your answer for part d.

Base Ten

A

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 aretwo different calculator displays for the 2004 world population

of 6,400,000,000 people

The number that is displayed is the product: 6.4  109

4 06 3.8 04

A

25 a Write 6.4  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?

Base Ten

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 anumber determines its value You can readthe number 79.54 as “seventy-nine and fifty-four hundredths.”

You can expand the number 79.54 as:

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

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

base

4.5 07

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

2.25  10422.5  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

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 and chat online at least once a day.

Today after school, Jacqui checks her e-mail Afterabout three minutes, she realizes Nikki’s message

is taking longer than usual to download Afterwaiting impatiently for ten minutes, Jacqui asksher brother, “Dave, what can I do? Look at that bar

on the computer screen!”

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

I don’t even know how to do that.”

Dave shares what he knows about digitalpictures

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

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

Factors

B

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 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 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

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

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

Numbers like 144, which result from squaring a number, are called

square numbersor perfect 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 sizedpictures The pictures became smaller, but the total number of pixelsdid 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 b in column 2.

Describe how the pixels per inch (ppi) in column 1 changefrom 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

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

dimen-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 totalpixels 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?

Factors

B

400 3



 ?

? ?

400 4



4 3



 ?

? ?

400 400



ppi 400 200 100

Total Number of Pixels

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

Division and multiplication operations relate to each other in this way.Using either operation, you found that the total number of pixelsdecreased from 480,000 to 120,000 Two number sentences for thiscontext 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?

Explain

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

You may remember some divisibilityrules Divisibility rules involvedivision of whole numbers without any remainders Here are threedivisibility 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 numberthat 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?

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 canchoose the one she wants First, she sketches one rectangulararrangement

Then she decides to make a list of all possible arrangements

Jacqui decides to draw one ofher picture arrangements ongraph paper.

Here is a graphshowing all ofthe rectangular arrangements

Since 3  8  24, 3 and 8 are

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 anumber

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.

In this activity, each of the students standing will be holding a cardwith 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 thesequestions:

● 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 totalnumber 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?

Changing Positions

Squaring

Multiplying a number by itself is squaring a number.

Two ways to indicate the squaring of a number, such as 3, are 32and3^2 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.

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 the factors 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

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

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

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

calcula-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?

?

 ?

?

5 5



3 6



 ?

? ?

2 5

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



You can write 150 as a product of two factors.

150  3  50Both 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.”

C

Prime Numbers

6 12 24

3 2

2 2

These special arithmetic trees are called factor trees In these factortrees, 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 towrite 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 aproduct of factors

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 each

number as a product of the end numbers

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.

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

C

Prime 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.

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?Explain

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 primenumbers: 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 anumber

16 a Use the factor tree method to find the prime factors

of 156

b Write 156 as a product of prime factors.

Prime Factors

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 Numbers

C

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 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 wouldlike 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, andheight of both types of boxes