fractions - weeblyclass4f.weebly.com/uploads/2/3/8/4/23841562/fractions.pdffractions as we do today....

$: Fractions - Weeblyclass4f.weebly.com/uploads/2/3/8/4/23841562/fractions.pdffractions as we do today. Uses of Fractions The numbers 1 2, 2 3, 5 4, 7 1, and 1 2 0 5 0 are all fractions$
Fractions

Arab mathematicians

began to use the horizontal

fraction bar around the

year 1200.

They were the first to write

fractions as we do today.

Uses of Fractions

The numbers �12

�, �23

�, �54

�, �71

�, and �12050

� are all fractions. A fraction is

written with two whole numbers that are separated by a fraction

bar. The top number is called the numerator. The bottom

number is called the denominator. The numerator of a fraction

can be any whole number. The denominator can be any whole

number except 0.

When naming fractions, name the numerator first, then name

the denominator.

Fractions were invented thousands of years ago to name

numbers between whole numbers. People needed these

in-between numbers for making careful measurements.

Here are some examples of measurements that use fractions:

�23

� cup, �34

� hour, �190� km, and 13 �

12

� lb.

Fractions are also used to name parts of wholes. The whole

might be one single thing, like a pizza. Or, the whole might be

a collection of things, like all the students in a classroom. The

whole is sometimes called the ONE.

42 forty-two

Fractions

�34

� yard

�13

� mile

�58

� of a pizza

Mrs. Blake’s Classroom�12

� of the students are girls.

Name the whole, or ONE, for each statement.

“Derek ate �58

� of the pizza.”

The whole is the entire pizza.

The fraction, �58

�, names the part of the pizza that Derek ate.

“In Mrs. Blake’s classroom, �12

� of the students are girls.”

The whole is the collection of all students in Mrs. Blake’s classroom.

The fraction, �12

�, names the part of that collection that are girls.

In Everyday Mathematics, fractions are used in other ways that

may be new to you. Fractions are used in the following ways:

♦ to show rates (such as cost per ounce)

♦ to compare (such as comparing the weights of two animals)

♦ to name percents (�12

� is 50%)

♦ to show divisions (15 � 3 can be written �135�)

♦ to show the scale of a map or a picture

♦ to show probabilities

�12

� teaspoon

Number Sense 1.5; Number Sense 1.7

Here are some other examples of uses of fractions:

♦ Study the recipe shown at the right. Many of the amounts

listed in the recipe include fractions.

♦ This spinner has �13

� of the circle colored red, �14

� colored blue,

and �152� colored green.

If we spin the spinner many times, it will land on red about�13

� of the time. It will land on blue about �14

� of the time.

And it will land on green about �152� of the time.

The probability that the spinner will land on a color that

is not green is �172�.

♦ If a map includes a scale, you can use the scale to

estimate real-world distances. The scale on the map

shown here is given as 1:10,000. This means that every

distance on the map is �10,

1000� of the real-world distance.

A 1 centimeter distance on the map stands for a real-

world distance of 10,000 centimeters (100 meters).

forty-three 43

Fractions

Size Chart for Women’s Shoes

Heel-to-toe length (in.) Size

8�11

56� to 9�

116� 6

9�126� to 9�

136� 6 �

12

�

9�146� to 9�

166� 7

9�176� to 9 �

196� 7 �

12

�

9�11

06� to 9�

11

16� 8

9�11

26� to 9�

11

46� 8�

12

�

9�1156� to 10�

116� 9

♦ Fractions are often used to describe

clothing sizes. For example, women’s

shoes come in sizes 3, 3 �12

�, 4, 4 �12

�, and

so on, up to 14.

Part of a size chart for women’s

shoes is shown at the right. It gives

the recommended shoe size for

women whose feet are between 9

and 10 inches long.

♦ A movie critic gave the film Finding Cosmo a rating of

3�12

� stars (on a scale of 0 to 4 stars).


Look at the collection of counters.

What fraction of the counters is red?

There are 12 counters in all.

Five of the counters are red.

Five out of 12 counters are red.

This fraction shows what part of the collection is red.

Sally ate half a pizza. Is that a lot?

The answer depends on how big the pizza was. If the

pizza was small, then �12

� is not a lot. If the pizza was

large, then �12

� is a lot.

Name the fraction of counters that are each shape in the collection above.

1. Circles 2. Triangles 3. Squares

Check your answers on page 341.

Understanding the many ways people use fractions will help

you solve problems more easily.

Fractions for Parts of a Whole

Fractions are used to name a part of a whole thing that is

divided into equal parts. For example, the circle at the right has

been divided into 8 equal parts. Each part is �18

� of the circle.

Three of the parts are blue, so �38

� (three-eighths) of the circle

is blue.

In Everyday Mathematics, the whole thing that is divided into

equal parts is called the ONE. To understand a fraction used to

name part of a whole, you need to know what the ONE is.

Fractions for Parts of a Collection

A fraction may be used to name part of a collection of things.

44 forty-four

Fractions


When you pick a ball out of this jar without looking,

the chance of getting a red ball is �58

�. The chance of getting a

blue ball is �38

�.

probability of picking a red ball �

probability of picking a blue ball �

number of red balls

total number of balls�

number of blue balls

total number of balls�

�58

�

�38

�

To understand a fraction that is used to name part of a

collection, you need to know how big the whole collection is.

forty-five 45

Fractions

Only half of Sam’s cousins can come to his party.

Is that many people?

It depends on how many cousins Sam has. If Sam has only

4 cousins, then 2 cousins are coming; that’s not many

people. But if Sam has 24 cousins, then 12 cousins are

coming. That’s many people.

Fractions in Measuring

Fractions are used to make more careful measurements.

Think about the inch scale on a ruler. Suppose the spaces

between the whole-inch marks are left unmarked. With a

ruler like this, you can measure only to the nearest inch.

Now suppose the 1-inch spaces are divided into quarters by �12

�-inch and �14

�-inch marks. With this ruler, you can measure

to the nearest �12

� inch or to the nearest �14

� inch.

To understand a fraction used in a measurement, you need

to know what the unit is. To say, “Susan lives �12

� from here”

makes no sense. Susan might live half a block away or

half a mile. The unit in measurement is like the ONE when

fractions are used to name a part of a whole.

Fractions in Probability

A fraction may tell the chance that an event will happen.

This chance, or probability, is always a number from 0 to 1.

An impossible event has a probability of 0; it has no chance

of happening. An event with a probability of 1 is sure to happen.

An event with a probability of �12

� has an equal chance of

happening or not happening.

Ruler has inch marks only. You can measure to the nearest inch.

Ruler has �12

�- and �14

�-inch marks.

You can measure to the nearest

�12

� inch or to the nearest �14

� inch.


The word fraction is

derived from the Latin

word frangere, which

means “to break.”

Fractions are sometimes

called “broken numbers.”

Fractions and Division

Division problems can be written using a slash / instead of the

division symbol �. For example, 21 � 3 can be written 21 / 3.

Division problems can also be written as fractions. One of the

many uses of fractions is to show divisions. The example below

shows that 21 � 3 can be written as the fraction �231�.

46 forty-six

Fractions

21 � 3 � 7 Show that �231� � 7 also.

This is the whole, or ONE. This is the whole after dividing

it into 3 equal parts. Each part

is �13

� of the whole.

The picture below shows that 21 thirds make 7 wholes. So, �231� � 7.

You can rename any fraction by dividing on your calculator.

To rename �231�, think of it as a division problem and divide:

Press 21 3 . The answer in the display will show 7,

which is another name for �231�.

21 � 3 � 21 / 3

and

21 � 3 � �231�

Fractions less than 1 can also be thought of as divisions.

Show that 3 � 4 � �34

�.

Think of 3 � 4 as an equal-sharing problem.

Suppose 4 friends want to share 3 oranges.

They could cut or divide each orange into 4 equal parts.

Each person gets �34

� of an orange. So, 3 � 4 � �34

�.


A length of 6 centimeters on the original will be 3 centimeters on the copy.

� �

12

�

copy size

original size

Fractions in Rates and Ratios

Fractions are often used to name rates

and ratios.

A rate compares two numbers with

different units. For example, 30 miles per

hour is a rate that compares distance with

time. It can be written as �310

hmoiulers

�.

A ratio is like a rate, but it compares two

quantities that have the same unit.

Other Uses of Fractions

Fractions are used to compare distances on maps to distances

in the real world, and to describe size changes.

forty-seven 47

Fractions

Rate Example

speed (jogging)

price

conversion of units

Ratio Example

won / lost record

rainy days comparedto total days

Find the real-world distance from Clay St. to S. Lake St.

Measure this distance on the map. It is 6 cm.

Each distance on the map is �20,1000� of the real-world

distance. So, the real-world distance equals

20,000 times the map distance. The real-world

distance � 20,000 * 6 cm � 120,000 cm.

100 cm � 1 m. So, 1,200 cm � 12 m and

120,000 cm � 1,200 m.

The distance from Clay St. to S. Lake St.

is 1,200 m.

distance

time�

7 blocks

4 minutes

cost

quantity�

99¢

3 erasers

distance in yards

distance in feet�

1 yard

3 feet

games (won)

games (lost)�

6

8

(rainy) days

(total) days�

11

30


Note

Even though they are

called improper, there is

nothing wrong about

improper fractions.

Do not avoid them.

Mixed Numbers

Numbers like 1 �12

�, 2 �35

�, and 4 �38

� are called mixed numbers.

A mixed number has a whole-number part and a fraction part.

In the mixed number 2 �35

�, the whole-number part is 2 and the

fraction part is �35

�. A mixed number is equal to the sum of the

whole-number part and the fraction part: 2 �35

� � 2 � �35

�.

Mixed numbers are used in many of the same ways that

fractions are used.

Mixed numbers can be renamed as fractions. For example, if

a circle is the ONE, then 2 �35

� names 2 whole circles and �35

� of

another circle.

If you divide the 2 whole circles into fifths, then you can see

that 2 �35

� � �153�.

To rename a mixed number as a fraction, first rename 1 as a

fraction with the same denominator as the fraction part. Then

add all of the fractions.

For example, to rename 4 �38

� as a fraction, first rename 1 as �88

�.

Then 4 �38

� � �88

� � �88

� � �88

� � �88

� � �38

� � �385�.

Fractions like �153� and �

88

� are called improper fractions.

An improper fraction is a fraction that is greater than or equal

to 1. In an improper fraction, the numerator is greater than or

equal to the denominator.

A proper fraction is a fraction that is less than 1. In a proper

fraction, the numerator is less than the denominator.

48 forty-eight

Fractions


Write a mixed number for each picture.

1. 2.

Write an improper fraction for each mixed number.

3. 1 �34

� 4. 2 �13

� 5. 3 �58

�


On Ms. Klein’s bus route, she picks up 24 students, 18 boys and 6 girls.

Equivalent Fractions

Two or more fractions that name the same number are called

equivalent fractions.

forty-nine 49

Fractions

The four circles below are the same size, but they are divided into different

numbers of parts. The green areas are the same in each circle. These circles show

different fractions that are equivalent to �12

�.

The fractions �12

�, �24

�, �36

� and �48

� are all equivalent. They are just different names for the

part of the circle that is green.

You can write: �12

� � �24

� �12

� � �48

� �24

� � �48

�

�12

� � �36

� �24

� � �36

� �36

� � �48

�

The fractions �264� , �

132� , and �

14

� are all equivalent.

You can write �264� � �

132� � �

14

�.

2 equal parts

1 part green

�12

� of the circle

is green.

24 equal groups

Each group is �214�

of the total.

6 groups of girls

�264� of the students

are girls.

12 equal groups

Each group is �112�

of the total.

3 groups of girls

�132� of the students

are girls.

4 equal groups

Each group is �14

�

of the total.

1 group of girls

�14

� of the students

are girls.

4 equal parts

2 parts green

�24

� of the circle

is green.

6 equal parts

3 parts green

�36

� of the circle

is green.

8 equal parts

4 parts green

�48

� of the circle

is green.


Find fractions that are equivalent to �12

84�.

� �192� � �

68

� � �34

�

1. a. What fraction of this rectangle is shaded?

b. Give two other fractions for the shaded part.

2. Name 3 fractions that are equivalent to �12

�.

3. Name 3 fractions that are equivalent to �24400

�.


Rules for Finding Equivalent Fractions

Here are two shortcuts for finding equivalent fractions.

Using Multiplication

If the numerator and the denominator of a fraction are both

multiplied by the same number (not 0), the result is a fraction

that is equivalent to the original fraction.

Fractions

Change �25

� to an equivalent fraction.

Multiply the numerator and the denominator of �25

� by 3.

In symbols, you can write � �165� .

So, �25

� is equivalent to �165�.

�25

� is red.

�165� is red.

2 * 35 * 3

6 � 315 � 3

18 � 224 � 2

18 � 324 � 3

18 � 624 � 6

Using Division

If the numerator and the denominator of a fraction are both

divided by the same number (not 0), the result is a fraction that

is equivalent to the original fraction.

To understand why division works, use the example shown above.

But start with �165� this time and divide both numbers in the

fraction by 3: � �25

�

The division by 3 “undoes” the multiplication by 3 that we did

before. Dividing both numbers in �165� by 3 gives an equivalent

fraction, �25

�.

50 fifty Number Sense 1.5; Number Sense 1.7

1. True or false?

a. �12

� � �198� b. �

58

� � �23

02� c. �

25

� � �14

20� d. �

02

� � �1

020�

2. a. Use the table to find 3 other fractions that are equivalent to �15

�.

b. Add 2 more equivalent fractions that are not in the table.


Table of Equivalent Fractions

This table lists equivalent fractions. All the fractions in a row

name the same number. For example, all the fractions in the

last row are names for the number �78

�.

Every fraction in the first column is in simplest form. A fraction

is in simplest form if there is no equivalent fraction with a

smaller numerator and smaller denominator.

fifty-one 51

Fractions

Note

Every fraction is either

in simplest form or is

equivalent to a fraction

in simplest form.

Lowest terms

means the same

as simplest form.

SimplestName Equivalent Fraction Names

Under normal conditions,�15

� of the length of a

telephone pole should

be in the ground.

0 (zero) �01

� �02

� �03

� �04

� �05

� �06

� �07

� �08

� �09

�

1 (one) �11

� �22

� �33

� �44

� �55

� �66

� �77

� �88

� �99

�

�12

� �24

� �36

� �48

� �150� �

162� �

174� �

186� �

198� �

12

00�

�13

� �26

� �39

� �142� �

155� �

168� �

271� �

284� �

297� �

13

00�

�23

� �46

� �69

� �182� �

11

05� �

11

28� �

1241� �

12

64� �

12

87� �

23

00�

�14

� �28

� �132� �

146� �

250� �

264� �

278� �

382� �

396� �

14

00�

�34

� �68

� �192� �

1126� �

12

50� �

12

84� �

2218� �

23

42� �

23

76� �

34

00�

�15

� �120� �

135� �

240� �

255� �

360� �

375� �

480� �

495� �

15

00�

�25

� �140� �

165� �

280� �

12

05� �

13

20� �

1345� �

14

60� �

14

85� �

25

00�

�35

� �160� �

195� �

1220� �

12

55� �

13

80� �

2315� �

24

40� �

24

75� �

35

00�

�45

� �180� �

11

25� �

1260� �

22

05� �

23

40� �

2385� �

34

20� �

34

65� �

45

00�

�16

� �122� �

138� �

244� �

350� �

366� �

472� �

488� �

594� �

16

00�

�56

� �11

02� �

11

58� �

2204� �

23

50� �

33

06� �

3452� �

44

08� �

45

54� �

56

00�

�18

� �126� �

234� �

342� �

450� �

468� �

576� �

684� �

792� �

18

00�

�38

� �166� �

294� �

1322� �

14

50� �

14

88� �

2516� �

26

44� �

27

72� �

38

00�

�58

� �11

06� �

12

54� �

2302� �

24

50� �

34

08� �

3556� �

46

04� �

47

52� �

58

00�

�78

� �11

46� �

22

14� �

2382� �

34

50� �

44

28� �

4596� �

56

64� �

67

32� �

78

00�


1. Name a fraction or mixed number for each mark labeled A, B, and C on the

ruler above.

2. What is the length of this nail?

a. in quarter inches b. in eighths of an inch

c. in sixteenths of an inch


Equivalent Fractions on a Ruler

Rulers marked in inches usually have tick marks of different lengths. The

longest tick marks on the ruler below show the whole inches. The marks used

to show half inches, quarter inches, and eighths of an inch become shorter

and shorter. The shortest marks show the sixteenths of an inch.

Every tick mark on this ruler can be named by a number of sixteenths. Some

tick marks can also be named by eighths, fourths, halves, and ones. The

picture below shows the pattern of fraction names for a part of the ruler.

This pattern continues past 1 inch, with mixed numbers naming the tick marks.

52 fifty-two

Fractions


� is less than

� is greater than

� is equal to

�35

� � �38

�

�45

� � �35

� because 4 � 3. �29

� � �79

� because 2 � 7.

�12

� � �13

� because halves are bigger than thirds.

�38

� � �34

� because eighths are smaller than fourths.

Comparing Fractions

When you compare fractions, you have to pay attention to both

the numerator and the denominator.

Like Denominators

Fractions are easy to compare when they have the same

denominator. For example, to decide which is larger, �78

� or �58

�,

think of them as 7 eighths and 5 eighths. Just as 7 bananas

is more than 5 bananas, and 7 dollars is more than 5 dollars,

7 eighths is more than 5 eighths.

To compare fractions that have the same denominators, just

look at the numerators. The fraction with the larger numerator

is larger.

Like Numerators

If the numerators of two fractions are the same, then the

fraction with the smaller denominator is larger. Remember, a

smaller denominator means the ONE has fewer parts and each

part is bigger. For example, �35

� � �38

� because fifths are bigger

than eighths, so 3 fifths is more than 3 eighths.

fifty-three 53

Fractions

�58

� � �78

� or �78

� � �58

�

Note

Fractions with like

denominators have the

same denominator.

�14

� and �34

� have like

denominators.

Fractions with like

numerators have the

same numerator.

�23

� and �25

� have

like numerators.

Use of the symbol � for

“equal to” dates back to

1571. Use of the symbols

� and � for “greater

than” and “less than”

dates back to 1631.

Number Sense 1.7.2

Compare. Write �, �, or � in each box.

1. �35

� �37

� 2. �23

� �49

� 3. �38

� �58

� 4. �26

� �25

�


Unlike Numerators and Unlike Denominators

Several strategies can help you compare fractions when

both the numerators and the denominators are different.

Comparing to �12

� Compare �37

� and �58

�.

Notice that �58

� is more than �12

�

and �37

� is less than �12

�.

So, �37

� � �58

�.

Comparing to Comparing fractions to 0 or 1 can also be helpful.

0 or 1 For example, �78

� � �34

� because �78

� is closer to 1.

( �78

� is �18

� away from 1 but �34

� is �14

�

away from 1. Since eighths are

smaller than fourths, �78

� is closer to 1.)

Using Equivalent One way to compare fractions that

Fractions always works is to find equivalent fractions that

have the same denominator. For example, to

compare �58

� and �35

�, look at the table of equivalent

fractions on page 51. The table shows that both

fifths and eighths can be written as 40ths:

�58

� � �24

50� and �

35

� � �24

40�. Since �

24

50� � �

24

40�, you

know that �58

� � �35

�.

Using Decimal Using decimal equivalents is another

Equivalents way to compare fractions that always

works. For example, to compare �25

� and �38

�,

use a calculator to change both fractions

to decimals:

�25

�: Key in: 2 5 Answer: 0.4

�38

�: Key in: 3 8 Answer: 0.375

Since 0.4 � 0.375, you know that �25

� � �38

�.

54 fifty-four

Fractions

Note

Remember that fractions

can be used to show

division problems.

� a � bab

Number Sense 1.9

�38

� � �18

� � �48

� � � �12

�4 � 48 � 4

�170� � �

130� � �

140� � � �

25

�4 � 2

10 � 2

Adding and Subtracting Fractions

Like Denominators

Adding or subtracting fractions that have the same

denominator is easy: Just add or subtract the numerators,

and keep the same denominator.

You can use division to put the answer in simplest form.

Unlike Denominators

When you are adding and subtracting fractions that have unlike

denominators, you must be especially careful. One way is to

model the problem with pattern blocks. Remember that

different denominators mean the ONE is divided into different

numbers (and different sizes) of parts.

fifty-five 55

Fractions

�13

� � �16

� � ?

If the hexagon is ONE, then the rhombus is �13

� and the triangle is �16

�.

When you put one rhombus and one triangle together, you

will find that they make a trapezoid. If the hexagon is ONE,

then the trapezoid is �12

�.

So, �13

� � �16

� � �12

�.

�56

� � �23

� � ?

If the hexagon is ONE, then �56

� is 5 triangles and �23

� is 2 rhombuses.

To take away �23

� (2 rhombuses) from �56

� (5 triangles), you would

need to take away 4 triangles.

Then there would be 1 triangle or �16

� left.

So, �56

� � �23

� � �16

�.

�13

� � �16

� � �12

�

�56

� � �23

� � �16

�

�23

��56

�

Number Sense 1.7

Clock Fractions

A clock face can be used to model fractions with 2, 3, 4, 5, 6, 10,

12, 15, 20, 30, or 60 in the denominator.

Fractions

�15

� hour �

12 minutes

�23

� hour �

40 minutes

�610� hour �

1 minute

�152� hour �

25 minutes

Note

Thousands of years ago,

the ancient Babylonians

divided the day into

24 hours, the hour into

60 minutes, and the

minute into 60 seconds.

This system for keeping

time is a good model

for working with

many fractions.

A clock face can help in solving simple fraction addition and

subtraction problems.

�13

� � �16

� � ?

�13

� hour � 20 minutes

�16


�13

� � �16

� � �12

�

�34

� � �13

� � ?

�34


�13


�34

� � �13

� � �152�

Using a Calculator

Some calculators can add and subtract fractions.

�38

� � �14

� � ?

Key in: 3 8 1 4 ; or 3 8 1 4 Answer: �58

�

Solve. Use pattern blocks or clock faces to help you.

1. �56

� � �12

� 2. �23

� � �14

� 3. �172� � �

14

� 4. �12

� � �23

�


56 fifty-six Number Sense 1.7

A furlong is a unit of

distance, equal to �18

� mile.

It is often used to

measure distances in

horse and dog races.

To add a distance in

furlongs (eighths of a

mile) and a distance

given in tenths of a

mile, you could rename

the fractions using

8 * 10 � 80 as a

like denominator.

�14

� � �23

� � ?

A quick way to find a like denominator for these fractions

is to multiply the denominators: 4 * 3 � 12.

Rename �14

� and �23

� as 12ths:

�14

� � � �132�

�23

� � � �182�

So, �14

� � �23

� � �132� � �

182� � �

11

12�.

�34

� � �25

� � ?

A like denominator for these fractions is 4 * 5 � 20.

Rename �34

� and �25

� as 20ths:

�34

� � � �12

50�

�25

� � � �280�

So, �34

� � �25

� � �12

50� � �

280� � �

270�.

Sometimes tools like pattern blocks or clock faces are not

helpful for solving a fraction addition or subtraction problem.

Here is a method that always works.

Using a Like Denominator

To add or subtract fractions that have different denominators,

first rename them as fractions with a like denominator. A quick

like denominator to use is the product of the denominators.

If two fractions are renamed so that they have the same

denominator, that denominator is called a common

denominator.

fifty-seven 57

Fractions

Add or subtract.

1. �24

� � �15

� 2. �58

� � �12

� 3. �172� � �

14

� 4. �12

� � �13

� � �14

�


1 * 34 * 32 * 43 * 4

3 * 54 * 5

2 * 45 * 4

Number Sense 1.5

Use any method to solve these problems.

1. 6 * �23

� 2. 3 * �45

� 3. �34

� * 6 4. 4 * �34

� 5. �45

� * 5


Multiplying Fractions and Whole Numbers

There are several ways to think about multiplying a whole

number and a fraction.

Using a Number Line

One way to multiply a whole number and a fraction is to think

about “hops” on a number line. The whole number tells how

many hops to make, and the fraction tells how long each hop

should be. For example, to solve 4 * �23

�, imagine taking 4 hops on

a number line, each �23

� unit long.

Using Addition

You can use addition to multiply a fraction and

a whole number. For example, to find 4 * �23

�, draw

4 models of �23

�. Then add up all of the fractions.

58 fifty-eight

Fractions

Using Fraction of an Area

You can think of multiplying with a fraction as finding the

fraction of an area. For example, to solve 4 * �23

� (which is the

same as �23

� * 4), find �23

� of an area that is 4 square units.

The rectangle on the left has an area of 4 square units.

The shaded area of the rectangle on the right has an area of �83

� square units (8 small rectangles, each with an area of �13

�.)

So, �23

� of the rectangle area � the shaded area � �83

�.

The word “of” in problems likethese means multiplication.

4 * �23

� � 2 �23

�

4 * �23

� = �23

� + �23

� + �23

� + �23

� � �83

�

4 squares �23

� * 4 � �83

�

�35

� of 20 means �35

� * 20.

�16


� * 18.

�34


� * 24.

4 * �23

� � �23

� * 4 � �83

�


Finding a Fraction of a Set

You can think of multiplication with fractions as finding a

fraction of a set. For example, think of the problem �25

� * 30 as

“What is �25

� of 30¢?” One way to solve this problem is first to

find �15

� of 30, and then use that answer to find �25

� of 30.

fifty-nine 59

Fractions

�25


� * 30.

5 equal groups, with 6 in each group

�25

� * 30 � ? Think of the problem as “What is �25

� of 30?”

Step 1: Find �15

� of 30.

To do this, divide the 30 pennies into 5 equal groups.

Then count the number of pennies in one group.

30 � 5 � 6, so �15

� of 30 is 6.

Step 2: Next find �25

� of 30.

Since �15

� of 30 is 6, �25

� of 30 is 2 * 6 � 12.

�25

� * 30 � �25

� of 30 � 12

3 equal groups, with 5 in each group

�23

� * 15 � ? Think of the problem as “What is �23

� of 15?”

Step 1: Find �13

� of 15.

Divide 15 pennies into 3 equal groups.

15 � 3 � 5, so �13

� of 15 is 5.

Step 2: Next find �23

� of 15.

Since �13

� of 15 is 5, �23

� of 15 is 2 * 5 � 10.

�23

� * 15 � �23

� of 15 � 10

Find each answer.

1. �14

� * 28 � ? 2. �35

� of 20 � ? 3. 16 * �58

� � ?

4. Rita and Hunter earned $12 raking lawns. Since Rita did most of the work,

she got �23

� of the money. How much did each person get?


Number Sense 1.5

Rename as fractions: 0, 12, 15.3, 3.75, and 25%.

0 � �

01

� 12 � �

112� 15.3 � �

11503

� 3.75 � �

31

70

50

� 25% � �

12050

�

Negative Numbers and Rational Numbers

People have used counting numbers (1, 2, 3, and so on) for

thousands of years. Long ago people found that the counting

numbers did not meet all of their needs. They needed numbers

for in-between measures such as 2�

12

� inches and 6�

56

� hours.

Fractions were invented to meet these needs. Fractions can

also be renamed as decimals and percents. Most of the numbers

you have seen are fractions or can be renamed as fractions.

However, even fractions did not meet every need. For example,

problems such as 5 � 7 and 2�

34

� � 5�

14

� have answers that are less

than 0 and cannot be named as fractions. (Fractions, by the way

they are defined, can never be less than 0.) This led to the

invention of negative numbers. Negative numbers are numbers

that are less than 0. The numbers ��

12

�, �2.75, and �100 are

negative numbers. The number �2 is read “negative 2.”

Negative numbers serve several purposes:

♦ To express locations such as temperatures below zero on

a thermometer and depths below sea level

♦ To show changes such as yards lost in a football game

♦ To extend the number line to the left of zero

♦ To calculate answers to many subtraction problems

The opposite of every positive number is a negative number,

and the opposite of every negative number is a positive

number. The number 0 is neither positive nor negative;

0 is also its own opposite.

The diagram at the right shows this relationship.

The rational numbers are all the

numbers that can be written or renamed

as fractions or as negative fractions.

60 sixty

Fractions

Note

Every whole number

(0, 1, 2, and so on) can

be renamed as a fraction.

For example, 0 can be

written as �

01

�. And 8 can

be written as �

81

�.

Numbers like �2.75 and

�100 may not look like

negative fractions, but

they can be renamed as

negative fractions.

�2.75 � � �

141�, and

�100 � ��

101

0�

Note

Number Sense 1.8

This method will work

for most of the decimal

numbers you see. But it

will not work for every

decimal number. For

example, 0.4444…

cannot be written as

a fraction with a

denominator of 10, 100,

1,000, or any other

power of 10.

Note

�12

� � � �

15000

� � 0.50

�

45

� � � �

180� � 0.8

�

34

�: Key in: 3 4 Answer: 0.75 �

58

�: Key in: 5 8 Answer: 0.625

�

49

�: Key in: 4 9 Answer: 0.4444… �

131�: Key in: 3 11 Answer: 0.2727…

Fractions, Decimals, and Percents

Fractions, decimals, and percents are different ways to write

numbers. Sometimes it is easier to work with a fraction instead

of a decimal or a percent. Other times it is easier to work with

a decimal or a percent.

Renaming a Fraction as a Decimal

You can rename a fraction as a decimal if you can find an

equivalent fraction with a denominator of 10, 100, or 1,000.

This only works for certain fractions.

Another way to rename a fraction as a decimal is to divide

the numerator by the denominator. You can use a calculator

for this division.

Renaming a Decimal as a Fraction

To change a decimal to a fraction, write the decimal as a

fraction with a denominator of 10, 100, or 1,000. Then you can

rename the fraction in simplest form.

sixty-one 61

Fractions

Remember that

� a � b is true for

any fraction .

Note

The U.S. Constitution did

not take effect until 9 of

the 13 original states had

approved it.

�193� � 0.69 and

�

193� � 69%

(decimal and percent

rounded to 2 digits)

Write each decimal as a fraction.

For 0.5, the rightmost digit is 5, which is in the 10ths place.

So, 0.5 � �

150�, or �

12

�.

For 0.307, the rightmost digit is 7, which is in the 1,000ths

place. So, 0.307 � �

13,00070

�.

For 4.75, the rightmost digit is 5, which is in the

100ths place. So, 4.75 � �

41

70

50

� (a fraction) or 4 �

17050

� or 4 �

34

�

(mixed numbers).

1 * 502 * 50

4 * 25 * 2

a

ba

b

Number Sense 1.6

0

Renaming a Decimal as a Percent

To rename a decimal as a percent, try to write the decimal as a

fraction with a denominator of 100. Then use the meaning of

percent (number of hundredths) to rename the fraction as a percent.

Fractions

Rename each decimal as a percent.

0.5 � 0.50 � �15000

� � 50% 0.01 � �1

100� � 1% 1.2 � 1.20 � �

11

20

00

� � 120%

Rename each percent as a fraction in simplest form.

50% � �15000

� � �12

� 75% � �17050

� � �34

� 1% � �1

100� 200% � �

21

00

00

� �

Rename each percent as a decimal.

45% � �14050

� � 0.45 120% � �11

20

00

� � 1.20, or 1.2 1% � �1

100� � 0.01

Rename each fraction as a percent.

�12

� � 0.50 � �15000

� � 50% �35

� � 0.60 � �16000

� � 60% �38

� � 0.375 � �3170.05

� � 37.5%

Renaming a Percent as a Decimal

To rename a percent as a decimal, try to rename it as a fraction

with a denominator of 100. Then rename the fraction as a decimal.

Renaming a Percent as a Fraction

To rename a percent as a fraction, try to write it as a fraction

with a denominator of 100.

Renaming a Fraction as a Percent

To rename a fraction as a percent, try to rename it as a fraction

with a denominator of 100. Then rename the fraction as a percent.

Write each number as a fraction, a decimal, and a percent.

1. �12

� 2. 0.75 3. 10% 4. �45

�


62 sixty-two Number Sense 1.6

��21

Musicians make patterns of sound to create music. Mathematics

can help us understand how both sound and music are created.

Sound, Music, and Mathematics

Sound

Every sound you hear begins with a vibration—a back and forth

motion. For musical instruments to produce sound, something must

be set in motion.

The rate at which a string, a drum head,

or a column of air vibrates is called the

frequency. Higher frequency vibrations

produce higher-pitched notes. Frequency

is measured in Hertz (Hz), or “vibrations

per second.” The human ear can hear

vibrations from about 15 Hz to 20,000 Hz.

The sound of a guitar starts whena person plucks or strums the strings.Each vibrating string moves backand forth at the same rate until itstops moving. When the stringsstop vibrating, the sound stops.

➤

The sound of a drumstarts when a personbeats the drum head.When the drum headstops vibrating, thesound stops.

The sound of a flute starts when a person blowsacross the mouthpiece. A column of air movesback and forth inside the flute. When the playerstops blowing, the columnof air stops vibratingand the sound ofthe flute stops.

sixty-three 63

➤

➤

Measurement and Geometry 3.2; Mathematical Reasoning 1.1

64 sixty-four

Instrument Length and Pitch

Many instruments rely on a vibrating column of air to make

sound. A longer column of air vibrates at a lower frequency and

makes a deeper- or lower-pitched note. Shorter vibrating air

columns make higher-pitched notes.

Here are some instruments you may haveheard, along with the frequency of thelowest note that can be played on theinstrument. What happens to the frequencyas the instruments get shorter?

➤

bassoon, 58 Hz clarinet, 139 Hz oboe, 233 Hz piccolo, 587 Hz


sixty-five 65

A recorder can play a range ofpitches. By covering all of the fingerholes on a recorder, the musiciancreates the longest possible columnof air, and the lowest-pitched note.With all holes uncovered, a highnote is produced.

The piccolo has a very shortcolumn of air within it, so itproduces high-pitched notes.Piccolos produce notes in therange of about 600 to 4,000 Hz,which humans can hear easily.

This pan flute, from Peru, is playedby blowing across the edges ofhollow tubes of different lengths.Short tubes produce high-pitchednotes, and long tubes producelow-pitched notes. The playerslides the instrument from sideto side to change notes.

➤

Because the alto saxophone is muchlonger than the piccolo, its sound islower-pitched. Saxophones use areed, which is a carefully-shaped pieceof cane. The musician blows into themouthpiece, which causes the reed tovibrate. This starts the vibration ofthe column of air.

➤

➤

➤


66 sixty-six

Percussion Instruments

Drums are percussion instruments. The size of the instrument

affects the pitch it can play. The size and tightness of the drum

head and the materials that the drum head is made from also

affect the pitch.

A drummer holds a West Africantalking drum, or donno, between theupper arm and the body. Squeezingthe strings with the upper armtightens the drumhead and raises the pitch of the drum. Releasing thestrings loosens the drumhead andlowers the pitch of the drum.

➤

➤

In a trap set, the largest drum—thebase or “kick” drum—produces thelowest-pitched notes. Each drum canbe tuned up or down by tighteningor loosening the heads.

The steel drum, from the Caribbeanisland of Trinidad, is made by cutting offthe top of a steel oil barrel. Each smallrounded section of the drum head isshaped to play a different pitched note.The pitch of the instrument can be veryhigh because the small metal sectionsvibrate rapidly.

➤


sixty-seven 67

Stringed Instruments

The pitch of the notes that a stringed instrument can play is

related to the length, diameter, and tension of the strings.

This man is tuning his stringedinstrument. Tightening a stringraises the pitch. Loosening astring lowers the pitch.

Compared to the violin, the cellohas longer strings of greaterdiameters. It is designed to playlow-pitched notes.

When a musician winds a stringtighter around its tuning peg,the string is tightened and thepitch becomes higher.

The violin, the smallest member ofthe string family, has short stringswith small diameters. It is designedto play high-pitched notes. Whena player presses down on a string,the vibrating part is shortenedand the pitch becomes higher.

➤

➤

➤

➤


The Piano

Looking closely at the way a piano works can help you see some

of the mathematical relationships in music.

68 sixty-eight

An octave begins and ends on a note with the samename. For example, the keys between “Middle C”and the C to the right of it represent 1 octave. Thereare 8 octaves on most pianos. The names of thewhite and black keys in an octave repeat eight times.

➤

A piano’s sound begins when a player presses a key.This causes a felt-covered wooden hammer to hit thestrings for that key. The strings then vibrate to producesound. Each key produces a note with a different pitch.

CMiddle

Cone octave

What patterns can you find in music? How

have you seen mathematics used in music?

➤

As you move to the right on thepiano keyboard, the frequenciesget higher. What patterns do yousee in the frequencies?

➤ This tuning fork vibrates 440 times per second. A piano tuner tightens or loosens the A4 stringuntil its pitch exactly matches the pitch of thevibrating tuning fork. Then all other strings aretightened or loosened based on that note.

➤


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