Math 5 English 2009 Book 1

7/31/2019 Math 5 English 2009 Book 1

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Mathematics

Fifth Form Primary

First Term

Author

Gamal Fathy Abdel-sattar

For


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Dar El Shorouk 2009

8 Sebaweh el Masry St.

Nasr City, Cairo, Egypt

Tel.: (202) 24023399

Fax: (202) 24037567

E-mail: [email protected]

www.shorouk.com

ISBN: 978 - 977 - 09 - 2664 - 6

Deposit No.: 15345 / 2009

Illustrations: Mahmoud Hafez/Khalid Abdel Aziz

Art Direction and Cover Design: Hany Saleh

Designers: Ahmed Yassin/Ahmed Hekal

Project Manager: Ahmed Bedeir


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Introduction

It gives us pleasure to introduce this book to our students of the fth form primary, hoping

that it will fulll what we aimed for in regards of simplicity of the information included and clarity.We hope it helps train our generations to be able to think scientically and be innovative.

The aspirations of the human have exceeded the limits of Earth and reached out into Outer

Space. Every day and night, satellites and information networks report on current events from

all over the world.

Due to technological progress, learning sources have become plentiful and various, and

learning medias have also become numerous and more various than before. This has also

caused teaching aids to become more complex, valuable and of greater impact.

While composing this book, the ollowing was taken into consideration:

Since studying number has not been enough for solving various life problems, so we must

start studying mathematics that uses symbols instead of numbers to solve such problems.

The use of images, shapes and colors to clarify mathematical concepts and properties of

shapes.

Integrating and linking between mathematics and other subjects.

Designing educational situations that facilitate the use of active learning strategies and

problem - solving skills.

Display lessons in a way that allows students to deduce and construe information on their

own.

The book includes real-life issues, educational activities and situations related to problems

environment, health, population issues in addition to the development of values such as

human rights, equality, justice and developing concepts of Patriotism.

Giving a variety of evaluation exercises at the end of each lesson, a test at the end of each

unit and examinations at the end of the book.

Include portfolio models to implement the Overall (Comprehensive) Educational Assessment

Employ technological methods.

This book has included our units:

Unit 1: Numbers - It aims at presenting the approximation, multiplication and division of

decimals and fractions.

Unit 2: Sets - it presents the meaning of the set and operations on sets.

Unit 3: Geometry - it focuses on constructing the circle, the triangle and its altitudes.

Unit 4: Probability - it aims at investigating experiments and outcomes.

While explaining the topics included in this book, it was taken into consideration

that it must be as simple as possible with a wide variety o exercises to provide the

students with the opportunity to think and create.

The Author


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Revision

The numbers that are combined in

addition are called addends and

together they form a new number

called a sum.

The number being subtracted is

called a subtrahend. the number

being subtracted from is called aminuend. the new number left after

subtracting is called a remainder or

difference.

Division is the process of finding

out how many times one number, the

divisor, will fit into another number,

the dividend. The division sentence

results in a quotient.

1 metre = 100 centimetres

1 decimetre = 10 centimetres

1 centimetre = 10 millimetres

1 kilometre = 1000 metres.

1 kilogram = 1000 grams.

1 ton = 1000 kilograms.

1 litre = 1000 millitres

1 day = 24 hours

1 hour = 60 minutes.

1 minute =60 seconds

prime numbers are counting

numbers that can be divided

by only two numbers: 1 and

themselves.

Perimeter of a square

= side length 4

Perimeter of a rectangle

= (length + width) 2

Area of a square

= side length itself

Area of a rectangle

= length width

addends

sum

2

2

4

+

3

1

4

+

minuend

subtrahend

difference

4

2

2

4

1

3

=

divisor

10

quotient

4

dividend

40


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Revision

Basic skills

First: Choose the correct answer:

1 The value of 3 in the number 347 is ......... [700 , 400 , 300]

2 The place value of 9 in 3972 is ......... [units, tens, hundreds]

3 667 ......... < 498 + 152 [7 , 17 , 27]

4 The height of the classroom door in metres is ......... [2 , 4 , 6]

5 The height of the greatest pyramid in metres is ......... [50 , 180 , 400]

6 50 > 5 ......... [11 , 10 , 9]

7 When it is seven o'clock, the angle between the hands of the clock

is ......... [acute, right, obtuse]

8 35

= ......... [1

5+ 3

5, 16

20, 1 2

5]

9 The perimeter of the school playground is ......... [100 km, 1 km, 100 m]

10 The probability of appearance of 2 on the upper face of a die when it

is thrown once is ......... [ 12

, 13

, 16

]

11 The H.C.F. of 12 and 18 is ......... [3 , 6 , 9 , 72]

Your correct answer of the following questions is a pass to begin

studying this book. If your answers are incorrect you have to follow

a special training.


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12 The L.C.M. of 12 and 20 is ......... [4 , 30 , 36 , 60]

13 The perimeter of the square whose side length 6 cm equals ......... cm.

[12 , 18 , 24 , 36]

14 7 m2 = ......... [70 dm2 , 700 cm2 , 700 dm2 , 49 dm2]

15 12.585e......... to the nearest tenth [13 , 12.6 , 12.59 , 12.5]

16 7 251 309 + 748 691 = ......... [8 milliard, 8 million, 8 thousand]

17 XYZ is a triangle in which, m ( dX) = 40, m (dY) = 30, then it is

......... triangle. [a right - angled, an obtuse - angled, an acute - angled]

18 The value of 7 in 123.579 is ......... [7 , 70 , 0.07 , 700]

19 256.104 = 256 + 0.1 + ......... [0.04 , 0.4 , 0.004]

20 Number of axes of symmetry of a square is ......... [0 , 2 , 3 , 4]

Second: Complete:

1 47.85 + ......... = 100

2 33.3 ......... = 12.008

3 93608.2 + 18905 = ..............e.............. to the nearest hundred.

4 453.64 72.317 = ..............e.............. to the nearest tenth.

5 1, 5, 9, 13, ......... , .........

6 The smallest number from the following numbers 1.3 , 3.2 , 10.04 ,

3.12 , 3.215 , and 1.12 is .........


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Revision

7 2.8 > .........

8 The name of the gure is .........

9 14

of a day = ......... hours = ......... minutes.

10 The prime factors of 350 are ........., ........., and .........

Third: Answer the ollowing questions:

1 ( a ) On the lattice, draw the triangle ABC, where A (1, 5), B (1, 8),

C (5, 5). What is the type of the triangle according to the measures

of its angles?

( b ) Omar has 45 pounds, He bought a ball for LE 9.75 and a book for

PT 840. How much money was left with him?

2 ( a ) Draw the triangle ABC, right angled at B where BC = 8 cm and

AB = 6 cm. Determine the mid - point M of AC.

( b ) A rectangular - shaped piece of land with dimensions 3 km and

2 km, it is needed to be surrounded by a wire fence. the cost of

one metre of wire fence equals 8 pounds. what is the total cost

of the fence?

3 The following table shows the number of pupils in each grade.

Represent these data by a histogram.

Grade First Second Third Four

Numberof pupils

80 60 100 70


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Unit 1: Numbers and operations

Lesson 1 : Approximating to the nearest hundredth 2

Lesson 2 : Approximating to the nearest thousandth 4

Lesson 3 : Multiplying decimals 6

Lesson 4 : Multiplying decimals by 10 , 100 and 1000 12

Lesson 5 : Dividing by - 3 digit number 14

Lesson 6 : Lesson 6 Innite division 16

Lesson 7 : Dividing decimals by 10 , 100 and 1000 18

Lesson 8 : Dividing by a decimal 20Lesson 9 : Comparing and ordering fractions 22

Lesson 10 : Multiplying fractions 24

Lesson 11 : Dividing fractions 30

Unit 2: Sets

Lesson 1 : Introduction to sets 38

Lesson 2 : Set notation 40

Lesson 3 : Types of sets 44

Lesson 4 : Representing sets by venn diagram 46

Lesson 5 : Subsets 48

Lesson 6 : Operations on sets 52

Unit 3: Geometry

Lesson 1 : Geometric patterns 70

Lesson 2 : Constructing a circle 74

Lesson 3 : Constructing a triangle 78

Lesson 4 : Constructing the heights of the triangle 80

Unit 4: Probability

Lesson 1 : Investigating experiments and outcomes 92

Lesson 2 : Certain and impossible events 98

First Term Examinations.

Contents


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Numbers and operationsunit

1Ancient Egyptians wrote all their fractions

so that they had a numerator of 1.

Unit Objectives

Lessons o the unit

Ater studying this unit the student should be able to:deduce the value of approximating a numeral decimal to the nearest

hundredth or thousandth. multiply a number or a decimal by a decimal or a numeral decimal. multiply decimal and a numeral decimal by 10.100, and 1000. perform ending division operation by a 3 - digit number.

nd the quotient of an innite division approximated to the nearest tenth and hundredth. divide decimals and numeral decimals by 10 ,100 and 1000 divide decimals and numeral decimals by a decimal. use common denominators to compare between the fractions. multiply fractions and mixed numbers. divide fraction and mixed number.

Lesson 1 Approximating to the nearest hundredth

Lesson 2 Approximating to the nearest thousandth

Lesson 3 Multiplying decimals

Lesson 4 Multiplying decimals by 10 , 100 and 1000

Lesson 5 Dividing by - 3 digit number

Lesson 6 Innite division

Lesson 7 Dividing decimals by 10 , 100 and 1000

Lesson 8 Dividing by a decimal

Lesson 9 Comparing and ordering fractions

Lesson 10 Multiplying fractions

Lesson 11 Dividing fractions

3

4= 1

4+ 1

2

5

6= 1

3+ 1

2

7

12 =1

3 +1

4 18 + 14

+


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2

Approximating to the

nearest hundredth

Approximate each of the following numbers to the nearest hundredth

(1) 6.972 (2) 74.158 (3) 138.835

1 The number 6.972 lies between 6.97 and 6.98 and is nearer to 6.97

than 6.98

2 The number 74.158 lies between 74.15 and 74.16 and is nearer to

74.16 than 74.15

3 The number 138.835 lies in the middle between 138.83 and 138.84

then the number 6.972e 6.97 to the nearest hundredth.

then the number 74.158e 77.16 to then nearest hundredth.

then the number 138.835e 138.84

6.972 6.975

6.97 middle 6.98

Deduce a rule to show approximation to the nearest hundredth, then

complete.

Look at the digit to the right o that place.

If it is 5 or more, cancel the decimal part after the hundredths place and

add .............. to the .......................... digit.

If it is less than 5, cancel the decimal part after the .......................... place.

74.158

middle74.1674.15

74.155

middle138.83 138.84

138.835

lesson

1


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3

unit 1

1 Notice the position o each o the ollowing numbers on the

number line, then complete.

2 Determine the position o each o the ollowing numbers on

the number line, then complete.

3 Approximate each o the ollowing to the nearest hundredth.

( a ) 4.908 ( d ) 12.723 ( g ) 39 31000

( b ) 13.575 ( e ) 104.086 ( h ) 94 7500

( c ) 147.041 ( f ) 23.3729 ( i ) 31 9250

4 Discover directly the error in each approximated result to the

nearest hundredth give reason.

( a ) 73.625 e 73.62 ( c ) 2.222 + 5.555e 8

( b ) 200.081e 200.07 ( d ) 762.3 267.212e 495.089

Exercise (1 1)

( a )

( a ) 143.297

( b ) 50.052

( b )

to the nearest

hundredth.

to the nearest

hundredth.

6.732

e............

19.146

e............

143.297e............ to the nearest hundredth.

50.052e............ to the nearest hundredth.

6.732 6.735

6.73middle

6.74

19.146

19.14middle

19.15

143.29 143.30

50.05 50.06


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4

Approximate each of the following numbers to the nearest thousandth

(1) 5.1873 (2) 53.2307 (3) 831.2345

Complete.

1 The number 5.1873 lies between 5.187 and 5.188 and is nearer to

5.187 than 5.188

2

3

then the number 5.1873e 5.187 to the nearest thousandth.

The number 53.2307e.......................... to the nearest thousandth.

The number 831.2345e.......................... to the nearest thousandth..

5.1873

5.187 middle

Deduce a rule to show approximation to the nearest thousandth, then

complete.

Look at the digit to the right o that place.

If it is 5 or more, cancel the decimal part after the .......................... place and

add .......................... to the .......................... digit.

If it is less than 5, cancel the decimal part after the .......................... place.

middle53.2300 53.2310

53.2305 53.2307

middle831.2340

831.2345

5.1875

5.188

831.2350

Approximating to the

nearest thousandth

lesson

2


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5

unit 1

1 Approximate each o the ollowing numbers to the nearest

thousandth.

( a ) 12.6245 ( d ) 144.1014 ( g ) 17 2310000

( b ) 1.0409 ( e ) 21.3495 ( h ) 94 12910000

( c ) 0.0673 ( f ) 19.9996 ( i ) 8 95000

2 Find The result o each o the ollowing then approximate it to

the nearest thousandth.

( a ) 35.241 + 6.0344 ( c ) 42.5667 25.36

( b ) 17 34

+ 71.0075 ( d ) 8 2.5116

3 Complete:

( a ) The number 83.7695e 83.7700 to the nearest ................

( b ) The number 1.2939 e 1.294 to the nearest ................

( c ) The number 521.291 e 521.3 to the nearest ................

( d ) The number 152.23 e 150 to the nearest ................

4 Complete with suitable digits.

( a ) 6.7321 + 9.8661e 16.59 to the nearest thousandth.

( b ) 1.2376 + 1.6689e 2. 0 to the nearest thousandth.

( c ) 9.866 7.214e 2.6 to the nearest hundredth.

( d ) 13.001 7.123e .8 to the nearest hundredth.

( e ) 7 0.6 + 56 . e 49.8 to the nearest

tenth.

Exercise (1 2)


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6

Estimating products

Mental Math

As part of the preparation for his

space flight, karim studies the space

shuttle operators Manual. It statesthat 1.8 kilograms of oxygen are used

per day for each crew member.

How much oxygen per day would be needed for 7 crew members?

Number of people kilograms per person = Total

7 1.8 = ?

Estimate to place the decimal point in the product.

1 Multiply as with whole

numbers.

2 Estimate to place the decimal point in the

product.

2 7 = 14

So, 7 1.8 = 12.6 12.6

1 . 8

7

126

1 4 . 72 5 . 8

8 5 376

Complete:

Round 14.72 to 15 Round off 5.8 to ............... 15 .............. = ..............

So, 14.72 5.8 = 85.376 85.376 is closer to ..............

Multiplying decimalslesson

3

is closer to

14 than 1.26

Round off 1.8 to 2


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7

unit 1

Practice

Choose a , b, or c.

3 The fuel cells in the space shuttle produce about 0.84 of a gallon

of water each hour. How much water would be produced in

93.5 hours?

( a ) 7854 ( b ) 785.4 ( c ) 78.54

4 Each crew member of the space shuttle uses 3.08 kilograms of

Nitrogen each day. How much would be used by 5 crew members?

( a ) 154 ( b ) 15.4 ( c ) 1.54

2 Estimate to place the decimal point in the underlined actor.

( a ) 7.5 23 = 17.25 ( d ) 4.25 33 = 14.025

( b ) 10.2 24 = 24.48 ( e ) 15.6 204 = 31.824

( c ) 88 6.3 = 55.44 ( f ) 122 34 = 41.48

1 Estimate to place the decimal point in the product.

( a ) ( c ) ( e )

( b ) ( d ) ( f )

6 . 9

3

207

4 . 8

1 . 3

624

8 . 3

2

166

1 5 . 85

4 . 3

6 8 155

9 . 04

7 . 9

7 1 416

5 1 . 2

3 . 04

1 5 5 648

Problem solving: Applications


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8

Multiplying decimals

The inside distance between the rails on some model railroads is about

1.6 cm. If each 1 cm on the model is about 0.9m on a real railroad,

about how far apart are the rails on a real railroad?

The rails on a real railroad are about 1.44m apart.

2decimal places

1decimal place

3decimal places

2decimal places

2decimal places

4decimal places3decimal places

0decimal place

3decimal places

Since each centimeter on the

model stands for same actual

distance, we multiply.

1 Multiply as with whole

numbers.

2 Write the product so it has as manydecimal places as the sum of the

decimal places in the factors.

1decimal place

1decimal place

2decimal places

1 . 6

0 . 9

144

1 . 6

0 . 9

1 . 44

( a ) ( c )

( b )

9 . 43 0 . 6

..............

1 . 32 0 . 87

9241056

................

0 . 276

3

.................

Complete:

More examples


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9

unit 1

Practice

Multiply

( a ) ( d ) ( g ) 6.8 3.2

( b )

( c )

( e )

( f )

( h ) 9.7 0.56

( i ) 4.75 0.9

4 . 27

0 . 7

..............

2 . 41

0 . 68

..............

1 . 374

6

.................

46 . 7 5

8 . 68

.................

9 . 4

6 . 8

..........

6 . 461

28

..................

Place the decimal point in each product.

( a ) 4.3 86 = 3698 ( c ) 69.5 0.47 = 32665

( b ) 2.3 6.4 = 1472 ( d ) 3.57 59.4 = 212058

Mental Math

1 A snail travels about 0.05 kilometers per hour. A spider travels 62.4

times as fast the snail. How fast does the spider travel?

2 Some needed data is missing from the problem below. Make up the

needed data and solve the problem.

A six - car model train is 73.2 cm long. How long is the actual train?



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10

Zeros in the products

A person who walks slowly might travel 6 km per hour. A fast snail might

travel 0.008 times as fast.

How fast does the snail travel?

Since the snail travel 0.008

times as fast, we multiply.

sometimes you need to write more

zeros in the product to have the correct

number of decimal places.

3decimal places

0decimal place

3decimal places

0 . 0 08

6

0 . 048

The snail travel 0.048km per hour.

0 . 09

0 . 6

0 . 0 54

37

0 . 0 02

..................

0 . 0 03

2

..................

435

0 . 0002......................

0 . 2

0 . 04...............

1 . 5

0 . 04

...............

2decimal places

1decimal place

3decimal places

1decimal place

2decimal places3decimal places

3decimal places

0decimal place

3decimal places

( a ) ( d )

( b ) ( e )

( c ) ( f )

Complete:

More examples


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11

unit 1

Practice

1 Place the decimal point in the answers. You may have towrite zeros in the product.

2 Multiply.

( a )

( a )

( c )

( c )

( f )

( e )

( g ) 4.3 0.007

( i ) 3.04 0.016

( b )

( b )

( d )

( e )

( d )

( f )

( h ) 5.7 0.18

0 . 1

0 . 7

7

1 . 5

0 . 4

60

0 . 09

0 . 3

27

6 . 2

0 . 01

62

0 . 0 6

0 . 3

..............

0 . 28

0 . 5

140

2 . 05

0 . 02

..............

8 . 1

0 . 0 6

..............

0 . 0 08

7

56

2 . 3

0 . 0 04..................

590

0 . 0001

......................

57

0 . 0 03..................

1 The smallest known insect is a beetle 0.02 centimeters long. Suppose

that 12 of these beetles were lined up in a row. What would be the

total length?

2 The height of a common flea is 1.5 millimeters. It can jump 130

times its own height. How high can it jump?



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12

The owner of a Jewelry store sells

a very popular digital watch for

LE 29.95. What will the stores total

amount of sales be for 10 watches?

10

Watches

100

Watches

1000

Watches

100 watches? 1000 watches? Are

these calculators answers reasonable?

The calculator answer seems reasonable.

What do you notice?

299.5 2995 29950

Complete:

29.95 10

30 10 = 300

29.95 100

30 100 = .............

29.95 1000

30 1000 = .............

To multiply by move the decimal point

10

100

1000

1

.........

.........

Place to the right

Multiplying decimals

by 10, 100, and 1000

lesson

4

LE 29.95


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13

unit 1

Practice

1 Multiply.( a ) 3.54 10 ( e ) 2.74 100 ( i ) 4.376 1000

( b ) 4.8 10 ( f ) 0.68 100 ( j ) 0.762 1000

( c ) 0.65 10 ( g ) 54.8 100 ( k ) 1000 0.81

( d ) 10 0.8 ( h ) 100 0.9 ( l ) 1000 6.7

2 Multiply then match.

4 Join the equal results.

3 What must you do to the

irst number to get the

second number?

4.635 100000

4.463 10000

4.7 100

4.703 1000

4.635 10

7.4 1000

0.074 1000

0.74 10000

0.0074 1000

7.4 10

0.74 10

(1)

(2)

(3)

(4)

(5)

( a )

( b )

( c )

( d )

( e )

lies between 400 and 500





First number Second number

4.3 430.24

6.08

240

608


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14

The country depends on tourism or much o

its income, so we must treat tourists well.

A tourist group travelled from Cairo to Aswan to visit its ancient monuments.

there were 337 tourists, and the total cost of the trip for the whole group

was 42125 pounds. Find the cost of the trip for each tourist.

The total cost number o tourists = the cost or each tourist

42125 337 = ?

42125 = 421 hundreds + 25 units

421 hundreds 337 = one

hundred and the remainder is 84,

8400 + 25 = 8425

= 842 tens + 5 units.

842 337 =

2 tens and the remainder is 168,

1680 + 5 = 1685

1685 337 = 5

The cost for each tourist = 125 pounds

Check: 337 125 = .........

42125

33700

8425

6740

1685

1685

0

125337 42100 + 25

100 337

20 337

5 337

Dividing by 3 - digit numberlesson

5


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15

unit 1

Practice

1 2 25625

...............

625

............

............

2125 25600 + 25

200 125

62 tens + 5 units

Check: 125 ......... = 25625

20192

...............

............

............

............

30631 20190 + 2

30 631

.......... 631

Check: 631 ......... = 20192

Note: the division operation is carried out without a remainder.

In this case we say that the division operation is nite.

3 Divide( a ) 6188 221 ( d ) 50478 141

( b ) 6266 241 ( e ) 89614 518

( c ) 16796 323 ( f ) 15660 435

Complete:

4 A truck can carry 265 watermelons. Find the number of trip neededto transport 54060 watermelons.

5 A factory produces 235 pieces of cloth monthly. In How many months

does it produce 26555 pieces of cloth?

6 A shopkeeper saves LE 337 each month which he deposits in his bank

account. After how many years will he have saved LE 16176?



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16

Adel and Soad are members

of schools agricultural society.

They divided a piece of ground

as shown in the figure.

They planted 3 squares with

yellow flowers, and a square with

red flowers.

The number of squares containing

yellow owers represents the fraction

3

4and is written as decimal as follows

The square containing red owers

represents the fraction 14

and is

written as decimal as follows

3

4= 3 25

4 25= 75

100= 0.75

or

14

= 1 254 25

=......

...... = ...........

or

3.0

2.8

0.20

0.20

0

0.754

1.0

...........

0.20

...........

0

0.2 ...4

Convert 37

to a decimal fraction approximating

the result to two decimal places, then to onedecimal place.

Solution3

7= 0.43 to the nearest hundredth

3

7= 0.4 to the nearest tenth

3.02.8

0.20

0.14

0.060

0.056

0.004

0.428

7

Infnite divisionlesson

6


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17

unit 1

Practice

1 Complete.

( a ) ( b )

8

17= ........to the nearest tenth

8

17= ........to the nearest hundredth

3092

412= ........to the nearest tenth

3092

412= ........to the nearest hundredth

8.0

6.8

...............

...............

...............

0.4....17

Note: The division operation is carried out with a remainder.

In this case we say that the division operation is innite.

3092..............

208.0..............

2.000

1.648

7.504412

7 tens 412

5 tenths 412

..............

2 Divide each o the ollowing, approximating the quotient to

two decimal places, then to one decimal place.

( a ) 2 3 ( d ) 11 125 ( g ) 19912 152

( b ) 5 11 ( e ) 13 123 ( h ) 36128 612

( c ) 9 35 ( f ) 12929 517 ( i ) 77649 143

3 If the calender year is 365 days, how many calender years are there

in 8775 days?

4 Hanys father bought a at for LE 96 888. He paid LE 10 000 in cash,

and paid the rest in 125 equal installments. Find to the nearest LE

the value of each instalment.



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18

A pilot whale weighed 734.83 kg.

This is 10 times an average mans

weight, 100 times a small dogs

weight, and 1000 times a rabbits

weight. To find these weights,

we divide. Are these calculatoranswers reasonable?

Mans weight =

whales weight 10

Dogs weight =

whales weight ......

Rabbits weight =

whales weight ......

The calculator answer seems reasonable.

73.483 7.3483 0.73483

Complete:

734.83 10

700 10 = 70

734.83 100

700 100 = .........

734.83 1000

700 1000 = .........

To divide by move the decimal point

10

100

1000

1

.........

.........

Place to the left

Place to the left

Place to the left

Dividing decimals by

10,100, and 1000

What must you do to the irst number to get the second number?

( a ) 73 , 0.73 ( b ) 600 , 0.6 ( c ) 5.6 , 0.56

lesson

7


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19

unit 1

Practice

2 Put the suitable sign (< , = , >).

( a ) 27.65 10 2.765 10

( b ) 4034 1000 403.4 10

( c ) 608.3 100 608.7 10

( d ) 4.162 100 4162 100

3 Join the equal results.

4 Complete:

96.7 100

96.7 10 96.7 10009.67 10

967 100 967 10 000

0.2654

..................

..................

..................

..................

..................

..................

..................

..................

54.071

..................

..................

..................

..................

7253.4

760

10 10 10

1000 100 10

1 Divide.( a ) 9.6 10 ( e ) 8.7 100 ( i ) 86.3 1000

( b ) 27.54 10 ( f ) 536.5 100 ( j ) 68.3 100

( c ) 0.7 10 ( g ) 496.4 1000 ( k ) 29.74 10

( d ) 34.2 100 ( h ) 387.25 1000 ( l ) 456.8 1000


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20

Study these division examples - look for the pattern.

When multiplying the dividend and the divisor by the same number,

The quotient does not change.

7

5687

560

10 8 10 56

80

10 0.7 10 5.6

5.60.7

Divide: 5.6 0.7

or 5.60.7

= 5.60.7

1010

= 567

= 8

8

567

( c ) 30.24 3.6 = 30.24 ..........

3.6 ..........= ............ = ............

( d ) 76.5 7.65 = 76.5 ..........

7.65 ..........= ............ = ............

( e ) 2.16 7.2 = 2.16 ..........

7.2 ..........= ............ = ............

Complete:

( a ) 34 . 4 0 . 4 = 344 4 = ............344

3224

24

0

834

Dividing by a decimallesson

8

Example

so

lution

100 8 100 56

56008007

( b ) 3 . 175 0 . 25 = ............ ............ = ............


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21

unit 1

Practice


1 Find the quotient o each o the ollowing.

( a ) 98.4 8.2 ( d ) 4.2 0.06

( b ) 4.794 1.7 ( e ) 0.7684 0.34

( c ) 18.45 4.5 ( f ) 114.45 1.09

2 Find the result o each o the ollowing.

( a ) (42.566 25.36) 0.7 ( d ) (25.42 3.1) + 0.7

( b ) 5.78 + (228.92 9.7) ( e ) (85.132 50.72) 1.4

( c ) (67.495 + 23.45) 0.05 ( f ) (50.84 6.2) + 18.2

3 A cyclist covered 38.7 km in 4.5 hours.

How many kilometers can he cover in

one hour?

5 The length of an orbit on one ight of the

space shuttle was 25905.24 miles.

The shuttle traveled at a speed of 285.3

miles per minute. How long did it take

the space shuttle to make one orbit?

4 If LE 382.5 is distributed among some poor people and each of

them takes LE 4.5 Find the number of poor people.


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22

Which is greater, 23

or 34

?

When the denominators are

different, write equivalent

fractions with the same

denominator.

2

3

8

12

9

12

3

4

Write the fractions in order from the smallest to the greatest.5

6 ,7

8 ,2

3

Complete: 56

= 10....

= ....18

= 2024

7

8= 14

....= 21

24

2

3= 4

...= ...

9= 8

...= 16

24

Since the ascending order of the numerators is 16, ...... , ......

So, 16

24< 20

24< 21

24

Then the ascending order of the fractions is 23

, ........

, ........

Put the suitable sign (< , = , >) or each :

Since 9 > 8 , 912

> ..............

So, 34

> 23

Complete:2

3= 4

...= ...

9= ...

12

3

4= 6

...= ....

12same

denominator

Comparing and ordering ractionslesson

9

( a ) 12

= 48

3

8= 3

8

( b ) 68

= 1824

9

12= 18

24

1

2 3

8

6

8 9

12


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23

unit 1

1 Put the suitable sign (< , = , >) or each :

( a ) 45

34

( e ) 2 14

2 13

( b ) 58

23

( f ) 1 38

1 25

( c ) 56

78

( g ) 4 712

4 23

( d ) 35

23

( h ) 7 6 69

2 Write in order rom the smallest to the greatest.

( a ) 25

, 34

, 310

( c ) 1 29

, 56

, 1 13

( b ) 56

, 34

, 78

( d ) 4 58

, 4 35

, 4 34

3 Arrange each o the ollowing in a descending order.

( a ) 79

, 56

, 23

( c ) 5 38

, 5 34

, 6 12

( b ) 12

, 34

, 23

( d ) 2 25

, 2 13

, 279

4 One day, Ramy walked 1 78

kilometers and Hoda walked 1 916

kilometers. Which distance was greater?

5 On three different days Sameh swam 516

kilometer, 78

kilometer and

3

4kilometer. Arrange the distances in an ascending order.

Write a problem comparing two mixed numbers. Ask the others to solve it.

Exercise (1 3)


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24

Practice

Finding parts

Samir has colored 12

of

the circle, then he cut out

1

3of the colored part.

16 of the circle has been

cut out.

13

of 12

= ........

Multiplying Fractions

1 Use the drawing to complete each sentence.

2 Draw a picture, then complete the sentence.

( a ) 13

of 23

( c ) 15

of 56

( e ) 23

of 35

( b ) 14

of 45

( d ) 25

of 27

( f ) 34

of 58

1

3 of

3

4 is ........1

4 of

2

5 is ........

( a ) ( b )

lesson

10

1

2

1

3 of1

2


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25

unit 1

Practice

3 Make a model: Is 12

of 13

the same as 13

of 12

?

Multiplying ractions

This drawing shows that

3

5of 3

4= 9

20

If you want to find 35

of 34

, Multiply 35

34

= ?

To multiply by fractions, multiply the numerators, then multiply the

denominators.

3

5 3

4= 3 3

...........= 9

...., 3

5 3

4= 9

5 4= ....

....

3

4

3

4

3

5

1

3of 12

1

2

1

2of 13

1

3

1 Multiply then write the answer in the simplest orm.

( a ) 1

8

2

3

= ........ ( c ) 1

2

4

5

= ........ ( e ) 2

5

1

4

= ........

( b ) 47

38

= ........ ( d ) 23

12

= ........ ( f ) 910

34

= ........

2 Find the missing actors.

( a ) 35

........ = 615

( c ) 27

........ = 1049

( e )........ 38

= 524

( b ) 910

........ = 12

( d )........ 59

= 736

( f ) 13

........ = 215


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26

Practice

Salwa is learning how to be a pastry

chef. She practices making roses

with a pastry tube for 34

of an hour

each day. she practices for 6 days

every week. How many hours

does she practice each week?

Multiplying ractions and whole numbers

Multiply 6 34

= ?

Write the whole number as a fraction: 6 = 61

multiply the numerators. 61

34

= .... ....

= ....

multiply the denominators. 61

34

= 18 .... ....

= ........

Write a mixed number for the answer 61

34

= 184

= ... ...2

Salwa practices 4 12

hours a week.

Multiply then write the answers in the simplest orm.

( a ) 4 34

( d ) 25

7 ( g ) 3 45

( b ) 48

7 ( e ) 8 56

( h ) 9 56

( c ) 6 28

( f ) 13

5 ( i ) 8 23


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27

unit 1

Practice

Saber owns a bakery. He works

7 12

hours each day. He bakes

bread 56

of this time. He spends

the rest of his day serving

customers. How many hours

a day does saber bake?

Multiplying ractions and mixed numbers

Multiply5

6 7 1

2= ?

Write the mixed number as a fraction: 7 12

= 152


152

= .... ....

= ....


152

= 75 .... ....

= ........

Write a mixed number for the answer 56 152 = 7512 = ......4

Saber bakes 6 14

hours a day.

1 Multiply . Write the answers in the simplest orm.

( a ) 25

5 12

( d ) 3 23

56

( g ) 2 16

34

( b ) 34

4 14

( e ) 5 13

37

( h ) 9 13

26

( c ) 78

7 14

( f ) 4 14

23

( i ) 34

8 23

2 Find the missing whole number in each problem.

( a ) 3 12

....... = 7 ( b ) 4 13

....... = 13 ( c ) 10 14

....... = 41


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28

Practice

Multiplying Mixed numbers

Ahmed and Dalia attend Cooking

class. Today they are learning

how to make a pie. The recipe

calls for 2 12

cups of flour. They

need to make 1 12

times the

recipe. How much flour should

they use?

Multiply 11

2 2 1

2= ?

Write the mixed numbers as fractions: 1 12

= 32

, 2 12

= 52


52

= .... ....

= ....


52

= 15 .... ....

= ........

Write a mixed number for the answer 32

52

= 154

= ... ...4

They should use 3 34

cups of flour.

Multiply then write the answers in the simplest orm.

( a ) 2 34

1 23

( d ) 3 25

4 12

( g ) 26 45

23

( b ) 4 12

1 78

( e ) 2 12

1 110

( h ) 21 78

3 13

( c ) 3 12

1 26

( f ) 3 12

1 26

( i ) 31 35

4 35


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29

unit 1

1 Reham is installing ceramic tiles on 23

of a bath room wall. one -

half of the ceramic tiles are yellow. How much of the bath room

wall will have yellow tiles?

2 Dina is installing linoleum tiles on 34

of the family room floor. She

has completed 23

of the job. What part of the floor is now

covered?

3 Hekal takes an inventory of the clothes at the shop. He finds that

suits make up 12

of the total stock. Womens suits make up 35

of all the suits. What part of the stores inventory is made up of

womens suits?

4 Peter practices decorating cakes for 34

of an hour each day. How

many hours does he practice in 7 days?

5 A recipe calls for 34

of a cup of flour. Laila makes 3 12

times the

recipe. How much flour does she need.

6 Eman works in the Teen Trends shop. All cotton fashions make

up 58

of the stock she sells. Cotton shirts make up 23

of this stock.

What part of the total stock is made up of cotton shirts.

7 Of 40 students in a cooking class, 58

are preparing to be chefs.

How many students is this?

8 Faiza is making spaghetti sauce. The recipe calls for 1 34

cups of

water, she wants to make 4 12

times the recipe. How much water

should she use?

Problem Solving. Applicatoins.

Exercise (1 4)


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30

Farida is a chef at the seashore

restaurant. she is making fruit salad.

The recipe says to cut all the fruit into

quarters. She has 3 slices of oranges.

How many fourths are there in

3 slices oranges?

You can count to nd how many

Divide the 2 12

squares into 2 equal

parts and shade one of them. The

fraction for the shaded parts is 54

which is 1 14

.

Illustrate 2 12

2

reciprocals

3 14

= 12

2 12

2 = 52

........

= ........

= .... ...4

3 41

= ........ there are 12quarters in

3 slices

reciprocals

Dividing Fractionslesson

11


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31

unit 1

Practice

How many 18

s are there in 34

?

How many 34

s are there in 4 12

apples?

Since we want to know how many

eighths are there in 34

, we divide

Count to nd how many?

There are 6 eighths in 34

3

4 1

8= 3

4 ....

....= ........

4 12

34

= 92

........

= ........

1 Divide. write the answers in lowest terms.

( a ) 23

16

( d ) 18

43

( g ) 19

1 12

( b ) 34

58

( e ) 712

16

( h ) 2 45

1 34

( c ) 45

13

( f ) 4 12

12

( i ) 4 27

1 514

2 The perimeter of a square is 611

m. Find the length of each side of

the square.

3 Alaa divided 79

of a cake equally between his son and his daughter.

What fraction of this cake did each of them take?

4 How many persons can share 4 pizzas if each person gets 12

of

a pizza?



40/119

32

1 Arrange the products o the ollowing rom the smallest to the

greatest. Use the sign


41/119

33

unit 1

4 Guess and check

can you find a decimal

for and a decimal

for so that their sum is 0.9 and their product is 0.18?

5 Discovering a pattern.

Do you see a pattern in these statements?

0.1089 9 = .........

0.10989 9 = .........

0.109989 9 = .........

Give the next two statements.

6 ( a ) Choose two decimals between 0 and 1 . Find their product.

( b ) Choose two decimal numbers greater than 1 . Find their product.

Is the product greater than or less than

the two factors? Try other examples.

Do you get the same results?

Is the product always greater than or

less than the two factors?

Is the product greater than 1 or less than the

two factors? Try other examples. Do youget the same results?

Is the product always greater than or

less than the two factors.

Is the product greater than or less than one?

+ = 0.9

= 0.18

0.2 0.7

1.2 1.7


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34

7 Find a number in the box or each pair o clues.

8 ( a ) Find the maximum product using the numbers 6, 8, 7, and 4

. = ..........

( b ) Find the minimum product using the numbers 6, 3, 1, and 9

. = ..........

( c ) Who am I?

(1) If you divide me by 8 then you divide the result by 2, you willget 6.4

(2) I am less than half the product of 4.25 and 4.4 by 5.63

( d ) Complete the table.

( a ) The sum of two numbers is 0.4 and their product is 0.04

( b ) The sum of two numbers is 2.4 and their product is 1.44.

( c ) The sum of two numbers is 0.02and their product is 0.0001

( d ) The sum of two numbers is 0.22 and their product is 0.0121

0.01 0.11 0.2 1.2

10.5 31.2 5.42

2.5

18.72

7.046

29.4


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35

unit 1

Unit test

Answer the ollowing questions:

1 Choose the correct answer.

( a ) 2.5 100 = .......... [ 250 , 25 , 0.25 , 0.025 ]

( b ) 1.8 0.09 = .......... [ 0.162 , 0.972 , 1.89 , 162 ]

( c ) 34.8 100 = .......... [ 3.480 , 348 , 3.48 , 0.348 ]

( d ) 9.64 4 = .......... [ 241 , 2.41 , 1.94 , 38.56 ]

( e ) A rope of length 10.5 m is cut into 7 pieces of equal length. Howlong is each piece? [ 15 m , 7 m , 1.5 m , 73.5 ]

( f ) If 478 = 23 20 + 18, then 478 20 equals

[ 20.39 , 20.9 , 23.18 , 23.9 ]

( g ) 2 18

e.......... approximated to the nearest hundredth

[ 2.1 , 2.13 , 2.12 , 2 ]

( h ) (3.69 3) 2 = .......... [ 2.64 , 2.46 , 0.246 , 1.23 ]

( i ) (0.325 + 91

4 ) 100 = .......... [ 0.9575 , 0.09575 ,322

300 , 0.95 ]

( j ) 14

23

25

= .......... [ 15

, 110

, 115

, 515

]

2 Complete.

( a ) 21 (7.02 1.8) = ..........

( b ) 3.6 , 5 15

, 6.8, .......... , ..........

( c ) 3.75 1000 = 37.5 ..........

( d ) 16

.......... = 14

( e ) ...2

45

= 65

( f ) 23

is the reciprocal of ..........

( g ) 76.52 .......... = 7.652

( h ) .......... 1000 = 5.619


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36

3 ( a ) From the table opposite,

choose ve numbers whose

product is 1

( b ) Put the suitable sign ().

(1) 45

23

(5) 3.2 kg 3200 gm

(2) 3 67

2 67

(6) 5.142 100 5142 100

(3) 7 13

2 13

(7) 806.7 100 806.7 10

(4) 34

23

57

(8) 2.4 10 0.24 1000

4 ( a ) Mariam went to the market. She bought 4.5kilograms of sh each

for LE 12, and 6 kilograms of apples each for LE 5.5. How many

pounds did she pay?

( b ) Ahmed turned on the water tap and forget to turn it off. If 1.45 litres

of water are wasted each hour, calculate the amount of water

wasted in 4 hours. How would you advise Ahmed?

5 ( a ) Arrange in an ascending order: 1

2

, 5

7

, 4

5

.

( b ) A big barrel has 113 34

kg of oil, and we want to distribute the oil

in bottles, the capacity of each one is 1 14

kg of oil. How many

bottles are needed for that?

0

5

7 2

0

5

7 2

3

4

6

7

1

5

1

2

5

8

1

10

8

9

2

3


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Venn diagrams are named after the English

mathematician venn (1834 - 1923) Who

first showed how useful they could be in

work on sets.

Setsunit

2

Lessons o the unit

Ater studying this unit the student should be able to:

Recognize Mathematical concept of set.

Recognize the concept of element in the set.

Express a set by listing and common property methods.Recognize the types of sets: empty - nite - innite.

Represent sets by Venn diagram.

Recognize the concept of two equal sets, subsets and containment relation.

Include completion of numerical patterns by deducing the relation between the

components of the pattern.

Solve relative problems.

Lesson 1 Introduction to sets

Lesson 2 Set notation

Lesson 3 Types of sets

Lesson 4 Representing sets by venn diagram

Lesson 5 Subsets

Lesson 6 Operations on sets

Gone Venn

Unit Objectives


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38

There are many words which we use to show a collection of things.

For example, we talk of

The letters in the word tomato represents a set because it is defined

well, its elements are t, o, m, a. (Note that t and o appear only once

when listing the elements of a set, none of them are repeated).

Introduction to sets

Example

lesson

1

a shoal of fish

a flock of geese

a head of cattle

a crowd of people

We are familiar with collections of objects such that "a set of pupils in

the class", " a set of teachers in the school" , a set of tools and so on.

In mathematics when we use the word set, we mean a well-defined

collection of objects. Each object of a set is called a member or an

element of the set.

"Foods which taste nice" does not represent a set since, some peoplemay like bananas and others may not.


47/119

39

unit 2

1Mention the elements o each o the ollowing sets.

2 State, giving reasons, which o the ollowing is a set and whichis not a set, mention the elements o those that are sets.

( a ) Tall men living in Cairo.

( b ) even numbers between 11 and 20.

( c ) Fruits you have eaten in the last 12 hours.

( d ) The ngers on your left hand.

( e ) Intelligent pupils in the class.

( f ) The letters in the English alphabet.( g ) Things in your bag.

( h ) Days of the week.

( i ) The letters in the word Mathematics.

( j ) Clever people living in Egypt.

( k ) Short pupils in your class.

( l ) Good manners.

( a ) ( c )

( b ) ( d )

Exercise (2 1)

set of birds

set of animals

set of children

set of flowers

AhmedSamir Soha


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40

A pair of braces { } is used to designate a set with the elements listed

or written inside the braces. The braces mean the set of or the set

whose elements are.

Capital letters are used to designate sets:

The symbol q indicates that an object is not an element of the set.

The symbol p is used to denote that an object is an element of the

set.

Small letters may name elements of sets such as:

The expression {1, 3, 5, 7, 9} is read The set whose

elements are one, three, five, seven, nine and may

be described as the set of one - digit odd numbers

or the set of odd digits.

B = {1, 2, 3, 4, 5, 6, 7, 8, 9} reads B is the set

whose elements are one, two, three, four, five, six ,

seven, eight, nine.

5 p B means Five is an element of set B.

R = {m , a , t , h}

12 q B means Twelve is not an element of set B.

Set notationlesson

2


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41

unit 2

Sets which contain exactly the same elements are called equal sets.

Sets which contain the same number of elements are called equivalent

sets.

{4, 2, 3} and {3, 4, 2} are equal sets. The elements

may be listed or written in any order. It is not allowed

to repeat an element when listing them.

{1, 2, 3, 4} and {1, 3, 5, 7} are equivalent sets.

1 Express each o the ollowing sets by listing its elements:

( a ) Set of digits in the number 3501.

A = {......., ......., ......., .......}

( b ) Set of letters in the word address.B = {......., ......., ......., ......., .......}

( c ) Set of digits in the number 9.

C = {.........}

( d ) Set of the original four directions.

.................................................................

2 Express each o the ollowing sets in words:( a )X = {2, 4, 6, 8}

The set whose elements are

......., ......., ......., .......

( b ) Y = {5, 10, 15}

.................................................................

Practice


50/119

42

( c ) A = {Nageeb, Nasser, Sadat, Mobarak}

.................................................................

( d ) B = {e, t, w}

.................................................................

3 List the elements o each o the ollowing sets:

4 Put the suitable symbol ( p or q ) :

A = { , , , }

B = { , , }

C = {1, 3, 5, 7, 9}

D = {0, 2, 4, 6, 8}

( a ) 3 > a

A ={......., ......., .......}

( b ) 7 + X < 11

X ={......., ......., ......., .......}

( c ) 17 - y > 12

Y ={......., ......., ......., ......., .......}

( d ) b < 1

B = {.......}

( a ) ....... A ( f ) 9 ....... D

( b ) ....... A ( g ) ....... B

( c ) ....... A ( h ) ....... A

( d ) 1 ....... C ( i ) 7 ....... D

( e ) 8 ....... D ( j ) 13 ....... C


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43

unit 2

1 List all the elements in each o the ollowing sets.

( a ) A = {months of the year beginning with j }.

( b ) B = {letters in the word Zaghlool}.

( c ) C = {arabic countries in Africa}.

( d ) D = {First ve letters of the English alphabet}.

2 Which o these sets are equal to T i T = {b, c, a}?

A = {First three letters of the English alphabet}.

B = {letters in the word cab}.

C = {First three letters in the word back words}.

D = {a, b, c, d}. E = {c, b, a}

3 Read, or write in words, each o the ollowing.

( a ) D = {Kennedy, johnson, Nixon}.

( b ) X = {z, i, a, e, b, n}.

( c ) B = {1, 3, 5, 7}.

( d ) E = {2, 3, 5, 7}.

( a ) 7 p C

( b ) 51 p C

( c ) 24 q C

( d ) 97 q C

( e ) 23 p C

( f ) 31 q C

4 I C = {all prime numbers}, which o the ollowing statements

are true?

5 ( a ) Are {2, 7, 9} and {9, 3, 7} equal sets? Equivalent sets?

( b ) Are {5, 1, 6, 8, 3} and {8, 6, 1, 3, 5} equal sets? Equivalent sets?( c ) Are {4, 8, 12, 16, 20} and {16, 20, 8, 4} equal sets? Equivalent sets?

6 ( a ) Using the listing method, Find the two possible ways of writing

the set of digits in the number 87787.

( b ) Using the listing method nd the ve other sets that are equal

to {m, t, s}

Exercise (2 2)


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44

Try to list the elements of each of the following sets.

A = {Prime numbers less than 3}

B = {Whole numbers between 11 and 16}

C = {Whole numbers divisible by 3}

D = {Whole numbers between 19 and 20}

Sets may contain one element, a definite number of elements, an

unlimited number of elements or no elements.

A set containing no elements is called the null set or empty set and is

denoted by the symbol "" or { }.

A set that contains a countable number of elements is called a inite

set. we can easily count the number of its elements.

A set that contains an uncountable number of elements is called an

ininite set. we can not actually count its elements.

{Cats that can fly} = { } =

{0} is not an empty set.

{Letters in the word "Good"} = {G, o, d}

{ Whole numbers } = {1, 2, 3, ...}

Note: a row of dots... is used to show that more numbers follow, but they

have not all been listed.

Types o setslesson

3


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45

unit 2

1 State whether each set is inite, ininite or empty.

( a ) { Letters used in writing this book}.

( b ) {People living in Egypt}.

( c ) {Cats with three heads}.

( d ) { even numbers between 11 and 12}.

( e ) {Whole numbers greater than 1000000}.

( f ) {Prime numbers that are even}.

( g ) {even numbers}.

( h ) {Egyptian pound notes}.

2 Give the irst our elements o each o the ollowing sets.

( a ) { Whole numbers greater than 3}.

( b ) {odd numbers greater than 100}.

( c ) {numbers that can be divided by ten without a remainder}.

( d ) {Prime numbers}.

3 Write, using braces, the set o common elements. I the set is

empty, Write { } or

( a ) {1, 3, 5, 7, 9, 11} , {1, 2, 3, 4, 5, 6, 7, 8}.

( b ) {2, 3, 4, 5, 6} , {even numbers less than 10}.

( c ) {1, 4, 9, 17} , {Prime numbers less than 12}.

( d ) {stick, mango, knife, , } , {Fruits}.

( e ) {People more than two metres tall}, {Pupils in your class}.

Exercise (2 3)


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46

Practice

The elements of any finite set can be represented by a set of points

over which we write the elements of the set on a white paper, then circle

them by a suitable geometric shape as a circle, square, triangle or a

loop such as the one in the example. A = {1, 3, 5, 7, 9}

A 1 3

5

9

7

A1

3 5

9

7

A 1

3

5

9

7

A 1

3

59

7

Representing sets by venn diagram

1 List the elements in each of the following sets:

A = {........., ........., ........., .........}

Is p A? ......

Is p A? ......

A = {......, ......, ......, ......, ......}

Is 2 p B? ......

Is 9 p B? ......

( a ) ( b )A B0

4

6

8

2

lesson

4


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47

unit 2

2 If X = {7, 9, 15, 3, 5} , Y = {3, 5, 11, 13, 19}

Then the following figure represents the two setsX

and Y , completethe venn diagram.

3 There may be more than two loops in a venn diagram., They may

overlap or intersect in many different ways.

Two possible ways are shown.

Y3

5

X

1

2 3

7

8

9

5

4

B

A

C

a

b

dg

f

e

c

X

Y

Z

ab

d

g

f

e

c


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48

We could make a number of

sets from the elements of u

= {cat, dog, elephant, monkey,

horse, lion} Such that:

{ Horse, Lion}, {elephant} or

{monkey, cat, lion}. These sets

are said to be subsets of u.

They can be written as:

( a ) {horse, lion} u

( b ) {elephant}u

( c ) {monkey, cat, lion}u

( d ) {tiger, goose} u

( e ) {tiger, monkey} u

Note that: monkey pu but tiger qu.

is a subset of

is not a subset of

The universal set

The universal set containing all the elements that can be used in a

question is called the universal set. It is written as u.

Subsetslesson

5


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unit 2

Practice

1 In the venn diagram:

( a ) List the elements of the three

sets X, Y and Z.

(1) X = {......., .......}

(2) Y = {......., ......., .......}

(3) Z = {......., ......., ......., .......}

( b ) Put the suitable sign ( or ).

(1) X....... Y (3) Y.......X

(2) X.......Z (4) Y ....... Z

2 A = {Letters in the English alphabet}.

B = {a, b, c, d, e, h, i, k, o, s, t, u, x}

( a ) State whether the following are true or false. Give reasons(1) A B (3) {a, b, k}A

(2) B A (4) {a} B

( b ) Represent the sets A and B in the venn diagram

Remark: Since the empty set does not contain any element, thus it is considered a subset of any

other set, {0}, {a, b, c}, {1, 2, 3, ...}.

X

ZY

7

3 1

5

( c ) List any three subsets of B that have four elements.


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50

1 X

= {a, b, c, d} , Y = {a, b, c, e}Write the elements o X and Y in

the venn diagram.

( a ) Is X Y?

( b ) Is Y X?

2 Write the elements in the venn

diagram given that:

u = {a, b, c, d, e, h, x, y}

X = {a, b, c, d, e}

Y = {a, b}

3 List

( a ) The elements of u

( b ) The elements of A

( c ) The elements of B

( d ) The elements of A that are in B

( e ) Is u?

4 Put the suitable sign ( or ).

( a ) {1}....... {1, 3}

( b ) {4, 5}....... {54}

( c ) {0, 1}....... {10, 15}

( d )....... {1, 2, 3}

X Y

XY

AB

1

6

4

2

5

113

7

10

12

9

8

Exercise (2 4)


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unit 2

5 Put the suitable sign ( p, q, , ).

(a) b ....... {b, c}

(b) {b}....... {b, c}

(c) {a, b}....... {b, a}

(d) 1 ....... {0, 10}

(e) ....... {0}

( f ) {38}....... {6, 3, 8}

6 Find the number "X" so that these statements are all correct.

(a) {9, 4} {X, 5, 9}

(b) {7, 9} {5, 7, X}

(c) {1, 3, 7} {1, 3, 4, X}

(d) {10, 13, 12} {X, 11, 12, 13}

7 X = {Letters in the word "cover" } , Y = { Letters in the word

"recover"}.

(a) Is X Y?

(b) Does X = Y?

Give reasons for your answers.


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Notice and complete.

1 A = {factors of 15},

B = { factors of 21}.

A } B = {Common factors of

15 and 21}

A } B = {...... , ......}

Operations on setslesson

6

Set A intersects set B

A } B

Addition, subtraction, multiplication, and division are said to be operations

on numbers. We are now going to meet two operations on sets: intersection

and union.

} means intersection of sets.

Set A Set ASet B Set B

Venn diagramA B

intersection: A } B

A B

Intersection o sets


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53

unit 2

We notice that:

There are four cases showing the combination of any two sets: The two

sets are intersecting, one of the two sets contains the other one, The

two sets are equal or the two sets are disjoint.

2 X = {digits of the number 12304}

= {... , ... , ... , ... , ...},

Y = {digits of the number 102}

= {... , ... , ...},

X} Y = {... , ... , ...}.

3 D = {Letters of "cat"},

E = {Letters of "act"}

D} E = {... , ... , ...}.

4 C = { , , },

F = { , }

C} F = ...

Containment: Y X

Equality: D = E

Disjoint: C} F =

X

XY

Y

D

FC

Intersection of two sets is the set which contains all

the common elements belonging to the two sets.

A} B = {x : xp A and xp B}

AB

E


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54

Properties o Intersection

1 X = {pupils with full marks in math}

Y= {pupils with full marks in Science}.

Represent on the venn diagrams.X} Y = {Pupils with full marks in both Math and science}.

X} Y = {.......... , ..........}

Y}X = {Pupils with full marks in both science and math}.

Y}X = {.......... , ..........}

What do you notice?

YEhab

HusseinYousef

Hatem

Adel

Ahmed Yousef

Hatem

Said

X

X Y

Y X

Commutative property of intersection

........ = ........


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55

unit 2

2 A = {readers of stories}, B = {players Gymnastics}

C = {Players of ping - pong}.

(A} B) } C = {.......... , ..........}} {.......... , .......... , .......... , ..........}

= {..........}

A } (B } C) = {.......... , .......... , .......... , ..........}} {.......... , ..........}

= {..........}

What do you notice?

Ahmed

Said

Aly

Ayman

AAhmed

Said

Samy

Hassan

B

Said

Samy

Baker

Aly

C

Represent on the venn diagram

A

C

B

A

C

B

Associative property of intersection

........ = ........


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1 The venn diagram below shows sets X, Y, and Z

2 The venn diagram opposite shows sets

A, B, and C. List the elements o:

( a ) A} B ( c ) C} A

( b ) B} C ( d ) A} B} C

( a )

( b )

( c )

( d )

3 Using the symbol "}", Write down what the shaded part in

each o the ollowing igures represents:

List the elements o:

( a ) X} Y ( c ) Y} Z( b ) X} Z ( d ) X} Y} Z

Exercise (2 5)

X Y Z

12

17

13

2 1

8

4

5

9

a

c

d

e b

h

f

g

A B

C

X

YA B

C

ED

Y

Z

X


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57

unit 2

4 The venn diagram opposite shows sets X and Y.

Put the suitable sign (p, q, , )

( a ) 3 ....... (X} Y)

( b ) {1, 2, 5}....... (X} Y)

( c ) {3}....... (X} Y)

( d ) {3, 4}....... (X} Y)

2

5

1

3

4

Y

X

5 Mark or the correct statement and or the incorrect one.

If A = {1, 2, 3, 4} , B= {3 , 4}, and C = {1 , 4}, then

( a ) 2 p A} B

( b ) 3 p A} B

( c ) 1 q A} B

( d ) A} B = B

( e ) B} C A

( f ) A} B} C =

6 u = {cow, horse, camel, dove, duck, cat, dog}

X = {animals that feed on grass}

Y = {birds}

Z = {animals whose names begin with the letter c}

( a ) List each of the sets X, Y, and Z.

( b ) List each of: X} Y, Y} Z , X} Z

( c ) Draw a venn diagram for the sets X, Y, and Z.

AB C

132 4


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Set A joins set B

A{ B

{ Means union of sets

Set A Set B

Notice and complete.

1 A = { , , },

B = {...... , ...... , ......}

A{ B = {... , ... , ... , ... , ... , ...}

2 C = {5, 6, 7, 8}

D = {... , ... , ... , ... , ...}

C} D = {... , ...}

C{ D = {... , ... , ... , ... , ... , ... , ...}

Venn diagram

Union: A{ B

A B

C ... D

CD

5

8

2

1

3

7

6

Union o sets


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59

unit 2

3 X = {digits of the number 9705},

Y = {digits of the number 95}

X} Y = {... , ...}

X{ Y = {... , ... , ... , ...}

4 D = {Letters of sing}

E = {Letters of sign}

D} E = {... , ... , ... , ...}

D{ E = {... , ... , ... , ...}

X

XY

Y

We notice that:In all cases; the union of two sets consists of the elements of one of

the sets, together with the elements from the second set that are not

included in the first set. Elements are not repeated if they are in both

sets.

Union of two sets A and B is that set which contains all elements

belonging A or B.

A{ B = {x : xp A or xp B}

BA BA BBAA

A{ B is coloured in each diagram

D = E

D

E


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60

Properties o Union

1 A = {Players of football},

B = {Players of Handball}

Represent on the venn diagrams.

A{ B = {Players of football or handball}.

A{ B = {...... , ...... , ...... , ...... , ...... , ...... , ......}

B{ A = {Players of handball or football}.

B{ A = {...... , ...... , ...... , ...... , ...... , ...... , ......}

What do you notice?

Aly

Ayman

Hamed

Alaa

A

Hany

Samy

Alaa

B

A B

B A

Commutative property of union

........ = ........


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61

unit 2

2 If X {9, 4, 5, 2} , Y = {4, 1, 5, 3}, Z = {4, 5, 7, 8}

Then complete:

( a ) X{ Y = {... , ... , 4, 5, ... , ...} , (X{ Y){ Z= {... , ... , 4, 5, ... , ... , ... , ...}

( b ) Y{ Z = {... , ... , 4, 5, ... , ...} , X{ (Y{ Z)

= {... , ... , 4, 5, ... , ... , ... , ...}

What do you notice?

( c ) X{ Y = {... , ... , 4, 5, ... , ...} , (X{ Y)} Z = {... , ...}

(Y} Z) = {... , ...} , X{ (Y} Z) = {.... , ... , ... , ...}

Is (X{ Y)} Z the same as X{ (Y} Z) ? ... why?

This can easily be seen

from a venn diagram.

5

4

X YX Y

Z

5

4

Associative property of union

........ = ........

X Y

Z

X Y

Z

................................


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62

The complement o a set

Consider the set of pupils in your

class as the universal set u.

Let B be the set of boys and G

the set of girls.

The complement of set B is those

pupils in u that are not in B which

is written as B, then B = G.

Similarly G = B , since G is the set of those pupils in u that are not

girls, hence they are boys.

1 u = {1, 2, 3, 4, 5, 6, 7} , A = {1, 2, 3}, and B = {4, 5, 6}

then A = { 4, 5, 6, 7} ,

B = {......., ......., ......., .......}

2 In the venn diagram, u is the universal set,

then A = {......., .......},

B = {......., ......., .......},

(A{ B) = {.......},

(A} B) = {......., ......., ......., .......}

Practice

9

8

A B1

37

A

B

1

6

4

2

5

3

7


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63

unit 2

Dierence o two sets

The difference of two sets A and B is

the set of elements that are in A but

not in B. It is written as A B

difference

A B = { , }

B A = { , }

1 If A = {1, 2, 3, 4, 5, 6} , B = {4, 5, 6, 7, 9}

then A B = {......., ......., .......}

B A = {......., .......}

2 Use the following venn diagrams to list:

A B = {......., ......., .......}

B A = {......., .......}

A B = ..........

B A = ..........

A B = ..........

B A = ..........

Practice

A B

What do you notice?

..............................

9

4 5

6

A B

1

3

2

7

AB

5

8

2

1

3

7

6

BA

2

1

3

B

2

10

A3

96


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64

1 X = {6 , 7} , Y = {6, 7, 9} , Z = {7, 8, 9, 10}

List each o the ollowing sets:

( a ) X} Y , X} Z , X} Y} Z

( b ) X{ Y , Y{ Z , X{ Y{ Z

( c ) Y X , Z Y , X Y

3 Using the two symbols} ,{ write down what the coloured part in

each of the following gures represents.

2 The gure opposite is a venn diagram

for the sets X, Y and Z.

List each o these sets:

( a ) X{ Y , X} Y , X} Y} Z

( b ) X{ Z , Y{ Z , X{ Y{ Z

( c ) X , Y , Z

( a ) ( d )

( b ) ( e )

( c ) ( f )

Exercise (2 6)

BA

A B

A B

A B

C

BA

C

X Y

Z

XY

Z

1

2 5

34

6

7

8


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65

unit 2

4 Write each of the following sets using the symbols:

} ,{ and the letters X, Y, and Z.

( a ) {2, 3, 5}

( b ) {2, 5, 7}

( c ) {2 , 5}

( d ) {2, 3, 5, 7}

( e ) {1, 2, 3, 4, 5, 7, 8, 9}

XYZ

1

3

2

5

4

9

8

7

5 The gure opposite is a venn diagram for the sets X, Y, and Z.

Mark or the correct statement and or the incorrect one.

( a )X

}

Y = Y( b ) Z X

( c ) Y Z

( d ) (Z{

Y)

X

( e ) X

( f ) (Z} Y) X

6 u = { Hayam, Eman, Fouad, Hoda, Hamed, Gehad, Cairo}

X = { Words including the letters H or h}

Y = { Words including the letter "d" }

( a ) List the elements in X and list the elements in Y.

( b ) Use a venn diagram to show the words in u, X, and Y.

YZ2

4

3

X

5


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66

1 I u = { 1, 2, 3, ..., 19}, A = {3, 9, 11, 13} , D = {1, 5, 13, 15},

N = {2, 6, 10, 14}, R = {3, 7, 9, 11} , S = {5, 11, 15, 19}.

and T = , Find each o the ollowing:

( a ) A{ N ( d ) D{ A ( g ) S} N

( b ) R{ S ( e ) D} S ( h ) R{ A

( c ) S{ T ( f ) N} R ( i ) R} D

2 Use B = { 1, 2, 3, 4, 8}, G = {3, 4, 5, 7} , and

H = {2, 4, 8} to show that:

( a ) B} G = G} B ( b ) G{ H = H{ G

3 Use R = {1, 5, 6, 8, 9, 12}. S = {2, 4, 6, 8, 10, 12}, and

T = {1, 4, 6, 8, 9} to show that:

( a ) (R} S)} T = R} (S} T)

( b ) (S} T)} R = S} (T} R)

4 Use A = {0, 3, 4, 7, 8, 9}. E = {1, 3, 5, 7, 9}, and

R = {0, 2, 4, 7, 8} to show that:

( a ) A{ (E} R) = (A{ E)} (A{ R)

( b ) A} (E{ R) = (A} E){ (A} R)

A} B is read "the intersection of A and B" or "A cap B".

A{ B is read "the union of A and B" or "A cup B".

Activity


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67

unit 2

2 ( a ) Put the suitable symbol ( p, q, , or ).

(1) 9 ....... { 3, 6, 9, 12}

(2) { , }....... { , , , }

(3) ....... {0}

(4) {b, k}....... {Letters of the word "Book"}

( b ) I {X, 3, 4, 7} = {7, y, 6, 3} then:

X y = ....... X + y = ....... X y = .......X

y= .......

Unit test

1 Complete

B A

A{ B

p ......

q ......

...... A

...... B

... A{ B

... A{ B


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68

3 ( a )

( b ) I u = {0, 1, 2, 3, ..., 9} , A = {2, 4, 6, 7},

B = {1, 3, 7} , and E = {3, 4, 7, 9} , use a venn diagram to

illustrate each o the ollowing:

1 A } E 4 E{ B

2 E{ A 5 B} A

3 B} E 6 A { B

What elements belong to set u ?

to set M? to set N? write the resulting

set, Listing the elements or:

1 M{ N 2 N} M 3 M N 4 M

u

M N2

3 47

5

9

8

6

4 ( a ) I A = {1, 2, 3}, B = {2, 0, 3, 1}, and C = {digits o the number

123}. What is the relation between:

(1) A and B? (2) B and C? (3) A and C?

( b ) State the subsets o the set {5, 7, 9}

5 ( a ) Complete:

(1) If 4 p {2 , X , 5} , then X = .......

(2) If b q {7 , 9} , then b =.......

(3) If 3 q {1 , y , 4} , then y = .......

( b ) Represent the sets X = {1, 5} , Y = {1, 3, 5}, and Z = {1, 3, 5, 7}

by a venn diagram.


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Lessons o the unit

Ater studying this unit the student should be able to:

Use the compasses for drawing circle

Recognize the diameter as the longest chord in the circle

Draw a triangle given the lengths of its three sides

Draw the altitudes of a triangle

Signify the use of some computer programs in drawing some geometric shapes

Recognize geometric patterns, complete their elements, and form new geometric

patterns on his own

Lesson 1 Geometric patterns

Lesson 2 Constructing a circle

Lesson 3 Constructing a triangle

Lesson 4 Constructing the altitudes of the triangle

The compasses is used to draw

a circle, it is composed of two arms:

one of them ends in a sharp point the

other ends in a pencil. The two arms

are joined together at the top.

Geometryunit

3

Unit Objectives


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70

Geometric patternslesson

1Sometimes you must find a pattern to solve a problem. you have seen

patterns that are made up a numbers. Other patterns are made up of

geometric figures.

What are the next two figures in this pattern?

What are the next two figures in this pattern?

What is the order of the figures?

Two triangles and then two circles.

The next two figures are circles.

What is the size and position of the figures?

A large triangle and then a smaller triangle that is upside down.

These are the next two figures.

Think

Think


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71

unit 3

Problems

1 Draw the next our igures or each pattern.

( a )

( b )

( c )

( d )

( e )

( f )

( g )

( h )

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........

........UU UU


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72

2 Draw the next igure in each pattern.

3 Compare the igures to see what change took place in the

igure, then draw the next igure in each pattern.

( a )

( b )

( a )

( b )

( e )

( c )

( d )

........

........

E EE


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73

unit 3

4 Choose the next igure in the pattern.

Portolio

Form new geometric patterns on your own.

( a )

( b )

( c )

( d )

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4


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74

Constructing a circlelesson

2A ferris wheel suggests a circle. All

the points on a circle are the same

distance from the centre.

How many more could you draw?

How many more could you draw?

Use your compasses. Put the

metal tip on a point. Swing the

pencil around.

Draw a line segment that joins

the centre and a point on the

circle. You have drawn a radius.

Your have constructed a circle

point A is the centre. This is

circle A.

Draw a line segment through

the centre that joins two

points on the circle. You have

drawn a diameter.

To draw any circular object, you can use

a compasses.

Step 1 Step 3

Step 2 Step 4

A

A radius B

AB

C

D

dia

mete

r


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75

unit 3

Practice

1 Name the radius and the diameter o each circle where "M" is

the centre.

2 Use a compass and centimeter ruler to draw a circle with:

( a ) radius 3 cm ( c ) diameter 4 cm

( b ) radius 3.5 cm ( d ) diameter 8 cm

3 In the igure opposite, complete:

( a ) A B is a ................ in the circle.

( b ) B C is a ................ in the circle.

( c ) The point ......... is a the centre of the circle.

( d ) A D is a ................ in the circle.

( e ) The line segments ......... , ......... , and ......... are radii in the circle.

N LM

E

Z

M

X

Y

D

M

B

C

Any line segment that intersects the circle at two points and

does not pass through the centre

is called a chord.

C D is a chord in the circleD

C

M

B A

D

C


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76

1 Complete the table.

2 In the fgure opposite,

( a ) What is the name of the circle?

( b ) How long is radius A N ?

( c ) How long is radius A M ?

( d ) If you drew another radius for circle A, how long would it be?

( e ) How long is diameter L N ?

( f ) How is the length of diameter L N related to the length of

radius L A ?

3 In the fgure opposite,

( a ) Name the segments that

are chords.

( b ) Name the longest chord.

4 Draw a line segment with the length given. use it as a radius

to construct a circle.

( a ) 2.5 cm ( b ) 5 cm ( c ) 4.5 cm

Exercise (3 1)

Radius 3 cm 5 cm ..... ..... 18 cm ..... 1.8 cm .....

Diameter ..... ..... 16 cm 22 cm ..... 6.8 cm ..... 9.4 cm

M

A

L

N

R

C

T

V

u

S

W


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77

unit 3

5 Try to draw a fgure similar to the ollowing fgures:

( a ) ( b )

7 Mark or the correct sentence and or the incorrect one.

8 Use a compass. Design a logo or your ith grade class.

( a ) The length of N Z is greater than the

length of L M.

( b ) L M is a diameter in the circle with the

centre N.

( c ) L N and N Z are equal in length.

( d ) The radius of the circle is the longest line

segment can be drawn in the circle.

6 If each side of the square is

10 cm, what is the length of

a radius of circle w?

W

A

D

B

C

M LN

Z


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78

Constructing a trianglelesson

3Drawing a triangle given its side lengths

Draw a triangle A B C is which AB = 6 cm,

BC = 5 cm, and CA = 3.5 cm. you can use

your ruler and compass to help you draw

the triangle

Use your ruler to draw A B

with Length 6 cm

Set your compass to 5 cm

and with B as a centre, draw

an arc.

Reset the compass to 3.5 cm

and with A as a centre, draw

another arc to intersect the

first arc at C.

Draw A C and B C, then ABC

is the required triangle.

Step 1

Step 2

Step 3

Step 4

6 cm BA

6 cmA B 6 cm

5cm3

.5cm

A B

C

6 cmA B

C

C

A6 cm

3.5

cm 5cm

B


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79

unit 3

1 Draw and label a triangle KLM, in

which KM = 5 cm,

KL = 7 cm, and ML = 6 cm

4 Draw the triangle A B C in which AB = 8 cm, BC = 5 cm and

CA = 6 cm. What type of A B C according to its angles?

5 Draw the triangle A B C in which AB = 10 cm, BC = CA = 7 cm.

What type of A B C according to its sides?

6 Draw the triangle X Y Z in which XY = YZ = ZX = 6 cm.

What do you notice?

I make sure that each interior angle of the triangle is ...........

2 Draw and label an equilateral

triangle A B C of side 7 cm.How can you check if the

triangle that you have drawn

is accurate?

3 Draw the triangle A B C sin which AB = 4 cm, BC = 3 cm and

AC = 5 cm. What type of this triangle, according to its angles?

Exercise (3 2)

Try to draw triangle A B C, shown

opposite, on a piece of paper. Are you

able to draw it?, What do you notice?

Discuss your findings with your class.7 cm

6060

6 cm6 cm

C

BA

5 cm

6 cm7 cm

M

L

K

7 cm

7 cm7 cm

CA

B


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80

Constructing the altitudes

o the triangle

lesson

4The altitudes o an acute - angled triangle

Draw the acute - angled triangle

ABC, put the edge of the ruler on

a side of the triangle, say B C.

Put the edge of one side of a set

square on B C. Move it to slide

along the edge of the ruler untilthe point A coincides with the

edge of the set square. Draw

A D, then A DB C.

The length of A D is called the

height of the triangle.

In the same way draw from B

and C two other line segments

to represent the two other line

segments to represent the two

other altitudes of the triangle.

Step 1 Step 2

Step 3

Note that

The three altitudes interset at

a point M inside the triangle.

For each altitude there is

a corresponding base.

10 2 3 4 5 6 7

A

B C

10 2 3 4 5 6 7

A

B D C

A

F E

B D

M

C


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unit 3

Note that

The three altitudes intersect at

M outside the triangle.

Note that

The three altitudes intersect at B.

The altitudes o the obtuse - angled triangle

The altitudes o the right - angled triangle

In A B C, A B B E, to draw the

third altitude we draw B DA C

To construct the three altitudes

of the obtuse - angled triangle

A B C we follow the same steps

as shown before.

Complete:

1 B C is the corresponding base to the altitude ........

2 A B is the corresponding base to the altitude ........

3 ........ is the corresponding base to the altitude B E

Complete:

1 A B is the corresponding base to B C.

2 B C is the corresponding base to ........

3 ........ is the corresponding base to B D.

A

B

D

C

A

M

D

E

F

CB


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82

1 Draw and label each of the following triangles, use a ruler and

a set square to draw their altitudes, then measure the length of

each altitude.

3 Draw the triangle A B C in which AB = 5 cm, BC = 6 cm and

AC = 4 cm. Draw the altitudes of A B C. Then measure their lengths.

4 Draw the triangle A B C in which AB = BC = 7.5 cm and AC = 4 cm.

Draw the altitudes of A B C, then measure their lengths.

2 In the gure opposite, A B C D

is a rectangle. Draw the third

altitude in the two triangles

A B E and D C E

5 Draw the triangle A B C in which AB = 5 cm, BC = 6 cm and the

measure of B = 120 , draw the three altitudes, then determinethe corresponding base to each altitude.

6 Draw the line segment B C where BC = 5 cm. D is the mid point of

B C , draw D A perpendicular to B C where DA = 6 cm. Measure

Documents

Math 5 English 2009 Book 1