Grade5- Math by Ahmed Kamal

Mathematics For

Fifth Form Primary First Term

Author Gamal Fathy Abdel-sattar

ÂŠ Dar El Shorouk 2009 8 Sebaweh el Masry St. Nasr City, Cairo, Egypt Tel.: (202) 24023399 Fax: (202) 24037567 E-mail: dar@shorouk.com 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

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

that it will fulfill 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 scientifically 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 following 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 four 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 of exercises to provide the students with the opportunity to think and create.

The Author

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 a minuend. 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

addends

+ 2

+ 1 sum

4 4 – 2 2

3 4

minuend 4 subtrahend – 1 difference

40 ÷ 10 = dividend

quotient divisor

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 ■

Revision Basic skills

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.

First: Choose the correct answer:

1 The value of 3 in the number 347 is ......... 2 The place value of 9 in 3972 is .........

[700 , 400 , 300] [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]

3 5

[

= .........

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 .........

11 The H.C.F. of 12 and 18 is .........

[

1 2

1 3

1 6

]

[3 , 6 , 9 , 72]

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 12.585 e ......... to the nearest tenth 16 7 251 309 + 748 691 = .........

[13 , 12.6 , 12.59 , 12.5] [8 milliard, 8 million, 8 thousand]

17 XYZ is a triangle in which, m ( d X) = 40°, m (d Y) = 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 .........

Revision

7 2.8 > ......... 8 The name of the figure 9

1 4

is .........

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

10 The prime factors of 350 are ........., ........., and ......... Third: Answer the following 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. Grade

First

Second

Third

Four

Number of pupils

100

Represent these data by a histogram.

Contents Unit 1: Numbers and operations 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 : Lesson 6 Infinite 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

Unit 2: Sets 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

Unit 3: Geometry Lesson 1 : Geometric patterns

Lesson 2 : Constructing a circle

Lesson 3 : Constructing a triangle

Lesson 4 : Constructing the heights of the triangle

Unit 4: Probability Lesson 1 : Investigating experiments and outcomes

Lesson 2 : Certain and impossible events

First Term Examinations.

unit

Numbers and operations

Ancient Egyptians wrote all their fractions so that they had a numerator of 1. 3 = 4 5 = 6 7 = 12

1 4 1 3 1 3

+ + +

1 2 1 2 1 4

+ 1 8

1 4

Unit Objectives After studying this unit the student should be able to: 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. ● find the quotient of an infinite 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. ● deduce

Lessons of the unit 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 Infinite 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

lesson

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

6.972 6.97

6.975 6.98

middle

then the number 6.972 e 6.97 to the nearest hundredth.

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

74.155 74.15

74.158 74.16

middle

then the number 74.158 e 77.16 to then nearest hundredth.

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

middle

138.84

then the number 138.835 e 138.84 Deduce a rule to show approximation to the nearest hundredth, then complete. Look at the digit to the right of 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. 2

unit 1 Exercise (1 – 1)

1 Notice the position of each of the following numbers on the number line, then complete. ( a )

6.732 6.73

6.735 6.74

middle

( b )

19.146 19.14

19.15

middle

6.732 e ............

to the nearest hundredth.

19.146 e ............

to the nearest hundredth.

2 Determine the position of each of the following numbers on the number line, then complete. ( a ) 143.297

143.29

143.30

143.297 e ............ to the nearest hundredth. ( b ) 50.052

50.05

50.06

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

3 Approximate each of the following to the nearest hundredth. ( a ) 4.908

( d ) 12.723

( g ) 39

( b ) 13.575

( e ) 104.086

( h ) 94

( c ) 147.041

( f ) 23.3729

( i ) 31

3 1000 7 500

9 250

4 Discover directly the error in each approximated result to the nearest hundredth give reason. ( a ) 73.625 e 73.62

( b ) 200.081 e 200.07

( c ) 2.222 + 5.555 e 8

( d ) 762.3 – 267.212 e 495.089 3

Approximating to the nearest thousandth

lesson

Approximate each of the following numbers to the nearest thousandth (1) 5.1873 (2) 53.2307 (3) 831.2345

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

5.1873 5.187

5.1875 5.188

middle

then the number 5.1873 e 5.187 to the nearest thousandth. â&#x2013; Complete.

53.2305 53.2300

middle

53.2307 53.2310

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

831.2345 831.2340

middle

831.2350

The number 831.2345 e.......................... to the nearest thousandth.. Deduce a rule to show approximation to the nearest thousandth, then complete. Look at the digit to the right of 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. 4

unit 1 Exercise (1 – 2)

1 Approximate each of the following numbers to the nearest thousandth. 23 10000 94 129 10000 8 9 5000

( a ) 12.6245

( d ) 144.1014

( g ) 17

( b ) 1.0409

( e ) 21.3495

( h )

( c ) 0.0673

( f ) 19.9996

(i)

2 Find The result of each of the following then approximate it to the nearest thousandth. ( a ) 35.241 + 6.0344 ( b ) 17

3 4

( c ) 42.5667 – 25.36

+ 71.0075

( d ) 8 – 2.5116

3 Complete: ( a ) The number 83.7695 e 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. to the nearest thousandth.

( a ) 6.7321 + 9.8661 e 16.59 0

( b ) 1.2376 + 1.6689 e 2.

to the nearest hundredth.

( c ) 9.866 – 7.214 e 2.6 ( d ) 13.001 – 7.123 e ( e ) 7

0.6

tenth.

+ 56

to the nearest thousandth.

.8 .

to the nearest hundredth. e

49.8 to the nearest

lesson

Multiplying decimals

Estimating products Mental Math As part of the preparation for his

space flight, karim studies the space

shuttle operator’s Manual. It states that 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 ×

Multiply as with whole numbers.

1.8

Estimate to place the decimal point in the product.

2 × 7 = 14

1.8 × 7 126

Round off 1.8 to 2

So, 7 × 1.8 = 12.6

Estimate to place the decimal point in the product.

12.6

is closer to 14 than 1.26

1 4 . 72 × 5.8 8 5 376

Complete: Round 14.72 to 15

Round off 5.8 to ...............

So, 14.72 × 5.8 = 85.376 6

15 × .............. = ..............

85.376 is closer to ..............

unit 1 Practice

1 Estimate to place the decimal point in the product. ( a )

6.9 × 3 207

( c )

4.8 ×1 . 3 624

( e )

9 . 04 ×7 . 9 7 1 416

( b )

8.3 × 2 166

( d )

1 5 . 85 × 4.3 6 8 155

(f)

51 .2 × 3 . 04 1 5 5 648

2 Estimate to place the decimal point in the underlined factor. ( 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

Problem solving: Applications 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 7

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? Since each centimeter on the model stands for same actual distance, we multiply.

Multiply as with whole numbers.

1.6 ×0 . 9 144

Write the product so it has as many decimal places as the sum of the decimal places in the factors.

1.6 ×0 . 9 1 . 44

1 decimal place 1 decimal place 2 decimal places

The rails on a real railroad are about 1.44m apart. More examples Complete: ( a ) 9 . 4 3 ×0 . 6

2 decimal places

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

3 decimal places

0 . 276 × 3

3 decimal places

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

3 decimal places

( b )

1 decimal place

0 decimal place

( c )

1 . 32 × 0 . 87 924 1056

2 decimal places

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

4 decimal places

2 decimal places

unit 1 Practice Multiply ( a )

4 . 27 ×0 . 7

( d )

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

( b )

1 . 374 × 6

2 . 41 × 0 . 68

( g ) 6.8 × 3.2

..........

( e )

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

( c )

9.4 ×6 . 8 46 . 75 × 8 . 68

( h ) 9.7 × 0.56

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

(f)

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

6 . 461 × 28

( i ) 4.75 × 0.9

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

Mental Math 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

Problem solving: Applications

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? 9

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. 0 . 008 × 6 0 . 048

3 decimal places 0 decimal place 3 decimal places

sometimes you need to write more zeros in the product to have the correct number of decimal places.

The snail travel 0.048km per hour. More examples Complete: ( a )

( b )

( c )

0 . 09 ×0 . 6 0 . 054

2 decimal places

0.2 × 0 . 04

1 decimal place

( d )

1 decimal place

37 × 0 . 002 ..................

3 decimal places

( e )

2 decimal places

435 × 0 . 0002

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

3 decimal places

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

0 . 003 × 2

3 decimal places

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

3 decimal places

0 decimal place

(f)

1.5 × 0 . 04 ...............

unit 1 Practice

1 Place the decimal point in the answers. You may have to write zeros in the product. ( a )

0 . 09 ×0 . 3 27

( c )

0.1 ×0 . 7 7

( e )

6.2 × 0 . 01 62

( b )

0 . 28 ×0 . 5 140

( d )

1.5 ×0 . 4 60

(f)

0 . 008 × 7 56

( d )

2 . 05 × 0 . 02

( g ) 4.3 × 0.007

2 Multiply. ( a )

0 . 06 ×0 . 3 ..............

( b )

57 × 0 . 003

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

( e )

2.3 × 0 . 004

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

( c )

590 × 0 . 0001

( h ) 5.7 × 0.18

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

(f)

8.1 × 0 . 06

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

( i ) 3.04 × 0.016

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

Problem solving: Applications

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?

Multiplying decimals by 10, 100, and 1000

lesson

The owner of a Jewelry store sells a very popular digital watch for LE 29.95. What will the store’s total amount of sales be for 10 watches?

LE 29.95

100 watches? 1000 watches? Are

these calculators answers reasonable?

299.5 10 Watches

2995 100 Watches

29950 1000 Watches

■ Complete: 29.95 × 10

29.95 × 100

30 × 10 = 300

29.95 × 1000

30 × 100 = .............

30 × 1000 = .............

The calculator answer seems reasonable.

●

What do you notice?

To multiply by

100 move the decimal point 1000

1 ......... .........

Place to the right

unit 1 Practice

1 Multiply.

( a ) 3.54 × 10 ( e ) 2.74 × 100 ( b ) 4.8 × 10 ( f ) 0.68 × 100 ( c ) 0.65 × 10 ( g ) 54.8 × 100 ( d ) 10 × 0.8 ( h ) 100 × 0.9

( i ) 4.376 × 1000 ( j ) 0.762 × 1000 ( k ) 1000 × 0.81 ( l ) 1000 × 6.7

2 Multiply then match. (1) 4.635 × 100000

( a ) lies between 400 and 500

(3) 4.7 × 100

( c ) lies between 400000 and 500000

(2) 4.463 × 10000 (4) 4.703 × 1000 (5) 4.635 × 10

( b ) lies between 40 and 50

( d ) lies between 40000 and 50000 ( e ) lies between 4000 and 5000

3 What must you do to the first number to get the second number?

First number

Second number

4.3

0.24

240

6.08

608

4 Join the equal results. 7.4 × 1000

7.4 × 10 0.74 × 10

0.074 × 1000

0.0074 × 1000 0.74 × 10000 13

lesson

Dividing by 3 - digit number

The country depends on tourism for much of 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 of tourists = the cost for 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 = ......... 14

125 337 42125 – 33700

8425

– 6740 1685

– 1685 0

42100 + 25 100 × 337 20 × 337 5 × 337

unit 1 Practice Complete:

2 125 25625 –

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

–

625

25600 + 25

30 631 20192

200 × 125

–

62 tens + 5 units

30 × 631

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

–

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

..........

× 631

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

Check: 125 × ......... = 25625

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

20190 + 2

Check: 631 × ......... = 20192

Note: the division operation is carried out without a remainder. In this case we say that the division operation is finite.

3 Divide ( a ) 6188 ÷ 221

( d ) 50478 ÷ 141

( c ) 16796 ÷ 323

( f ) 15660 ÷ 435

( b ) 6266 ÷ 241

( e ) 89614 ÷ 518

Problem solving: Applications

4 A truck can carry 265 watermelons. Find the number of trip needed to 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?

lesson

Infinite division

Adel and Soad are members of school’s 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 flowers represents the fraction 3 4

and is written as decimal as follows

3 4

3 × 25 4 × 25

75 100

● The square containing red flowers

represents the fraction

written as decimal as follows

= 0.75

0.75 4 3.0 – 2.8

0.20 – 0.20

■ Convert

3 7

to a decimal fraction approximating the result to two decimal places, then to one decimal place. Solution 3 7 3 7

= 0.43 to the nearest hundredth = 0.4 to the nearest tenth

1 4

1 × 25 4 × 25

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

0.2 ... 4 1.0 – ........... –

0.20

...........

0.428 7 3.0 – 2.8 0.20 – 0.14 0.060 – 0.056 0.004

1 4

and is

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

unit 1 Practice

1 Complete. ( a )

0.4.... 17 8.0 – 6.8 –

( b )

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

7.504 412 3092 – .............. –

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

8 17 8 17

= ........ to the nearest tenth = ........ to the nearest hundredth

3092 412 3092 412

208.0

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

2.000 – 1.648

7 tens × 412 5 tenths × 412

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

= ........ to the nearest tenth = ........ to the nearest hundredth

Note: The division operation is carried out with a remainder. In this case we say that the division operation is infinite.

2 Divide each of the following, approximating the quotient to two decimal places, then to one decimal place. ( a ) 2 ÷ 3 ( d ) 11 ÷ 125

( g ) 19912 ÷ 152

( c ) 9 ÷ 35 ( f ) 12929 ÷ 517

( i ) 77649 ÷ 143

( b ) 5 ÷ 11 ( e ) 13 ÷ 123

( h ) 36128 ÷ 612

Problem solving: Applications

3 If the calender year is 365 days, how many calender years are there in 8775 days?

4 Hany’s father bought a flat 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.

Dividing decimals by 10,100, and 1000

lesson

A pilot whale weighed 734.83 kg. This is 10 times an average man’s weight, 100 times a small dog’s weight, and 1000 times a rabbit’s weight. To find these weights, we divide. Are these calculator answers reasonable? ■ Complete: Man’s weight = whale’s weight ÷ 10

Dog’s weight = whale’s weight ÷ ......

73.483

Rabbit’s weight = whale’s weight ÷ ......

7.3483

734.83 ÷ 10

0.73483

734.83 ÷ 100

700 ÷ 10 = 70

734.83 ÷ 1000

700 ÷ 100 = .........

700 ÷ 1000 = .........

The calculator answer seems reasonable. To divide by

100

1000

move the decimal point

1 ......... .........

Place to the left Place to the left Place to the left

● What must you do to the first number to get the second number? ( a ) 73 , 0.73 18

( b ) 600 , 0.6

( c ) 5.6 , 0.56

unit 1 Practice

1 Divide. ( a ) 9.6 ÷ 10 ( e ) 8.7 ÷ 100 ( b ) 27.54 ÷ 10 ( f ) 536.5 ÷ 100 ( c ) 0.7 ÷ 10 ( g ) 496.4 ÷ 1000 ( d ) 34.2 ÷ 100 ( h ) 387.25 ÷ 1000

( i ) 86.3 ÷ 1000 ( j ) 68.3 ÷ 100 ( k ) 29.74 ÷ 10 ( l ) 456.8 ÷ 1000

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. 96.7 ÷ 100

967 ÷ 100

967 ÷ 10 000

96.7 ÷ 10

9.67 ÷ 10

96.7 ÷ 1000

4 Complete:

× 10

0.2654

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

54.071

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

÷ 1000

÷ 100

7253.4 760

÷ 10

lesson

Dividing by a decimal

Study these division examples - look for the pattern. 7 80 560

7 8 56

10 × 8

7 800 5600 100 × 8 100 × 56

10 × 56

When multiplying the dividend and the divisor by the same number, The quotient does not change. Example Divide: 5.6 ÷ 0.7 solution

0.7

5.6

8 7 56

10 × 0.7 10 × 5.6 or

5.6 0.7

10 10

56 7

■ Complete: ( a ) 34 . 4 ÷ 0 . 4 = 344 ÷ 4 = ............ ( b ) 3 . 175 ÷ 0 . 25 = ............ ÷ ............ = ............ ( 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 × ..........

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

83 4 344 – 32

24 – 24

unit 1 Practice

1 Find the quotient of each of the following. ( a ) 98.4 ÷ 8.2

( b ) 4.794 ÷ 1.7

( c ) 18.45 ÷ 4.5

( d ) 4.2

÷ 0.06

( e ) 0.7684 ÷ 0.34

( f ) 114.45 ÷ 1.09

2 Find the result of each of the following. ( a ) (42.566 – 25.36) ÷ 0.7

( d ) (25.42 ÷ 3.1) + 0.7

( c ) (67.495 + 23.45) ÷ 0.05

( f ) (50.84 ÷ 6.2) + 18.2

( b ) 5.78 + (228.92 ÷ 9.7)

( e ) (85.132 – 50.72) ÷ 1.4

Problem solving: Applications

3 A cyclist covered 38.7 km in 4.5 hours.

How many kilometers can he cover in one hour?

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.

5 The length of an orbit on one flight 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?

lesson

Comparing and ordering fractions

2 3

...

....

● Which is greater,

3 4

...

When the denominators are different, write equivalent fractions with the same denominator. ■ Complete:

2 3

...

3 4

...

Since 9 > 8 ,

2 3

3 4

8 12

9 12

same denominator

12 9 12

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

So,

3 4

2 3

● Write the fractions in order from the smallest to the greatest. ■ Complete:

5 6 5 6 7 8 2 3

7 8

2 3 10

= = =

....

...

....

...

18 21 24 9

20 24

...

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

2 3

● Put the suitable sign (< , = , >) for each ( a )

1 2

3 8

= =

4 8

3 8

1 2

3 8

( b )

6 8

9 12

.... ....

= =

18 24

6 8

9 12

unit 1 Exercise (1 – 3) 1 Put the suitable sign (< , = , >) for each

( a )

4 5

3 4

( e ) 2

1 4

1 3

( b )

5 8

2 3

(f)1

3 8

2 5

( c )

5 6

7 8

( g ) 4

7 12

2 3

( d )

3 5

2 3

( h ) 7

6 9

2 Write in order from the smallest to the greatest. ( a )

2 5

3 4

3 10

( c ) 1

2 9

5 6

( b )

5 6

3 4

7 8

( d ) 4

5 8

,1 3 5

1 3

3 4

3 Arrange each of the following in a descending order. ( a )

7 9

5 6

2 3

( c ) 5

3 8

3 4

( b )

1 2

3 4

2 3

( d ) 2

2 5

1 3

4 One day, Ramy walked 1

7 8

1 2

27 9

kilometers and Hoda walked 1

kilometers. Which distance was greater?

5 On three different days Sameh swam 3 4

5 16

kilometer,

7 8

9 16

kilometer and

kilometer. Arrange the distances in an ascending order.

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

lesson

Multiplying Fractions

Finding parts Samir has colored

1 2

the circle, then he cut out 1 3 1 6

of the colored part.

1 3

1 2

of the circle has been

1 3

cut out. ■

1 2

.... ....

Practice

1 Use the drawing to complete each sentence. ( a )

( b )

1 3

3 4

1 4

is ........

2 5

is ........

2 Draw a picture, then complete the sentence. ( a )

1 3

2 ( c ) 1 3 5

5 6

( e )

2 3

3 5

( b )

1 4

4 ( d ) 2 5 5

2 7

(f)

3 4

5 8

1 2

unit 1 Multiplying fractions

3 5

This drawing shows that 3 5

3 4

9 20

3 4

3 5

If you want to find

3 4

3 , 4

Multiply

3 5

3 4

To multiply by fractions, multiply the numerators, then multiply the denominators. 3 5

3 4

3×3

...........

....

3 5

3 4

9 5×4

.... ....

Practice

1 Multiply then write the answer in the simplest form. ( a )

1 8

2 3

= ........

( c )

1 2

4 5

= ........

( e )

2 5

1 4

= ........

( b )

4 7

3 8

= ........

( d )

2 3

1 2

= ........

(f)

9 10

3 4

= ........

2 7

× ........ =

2 Find the missing factors. ( a )

3 5

( b )

9 10

× ........ = × ........ =

6 15

( c )

1 2

( d ) ........ ×

3 Make a model: Is

1 2

1 3

5 9

10 49 7 36

the same as

1 3

1 2

5 24

× ........ =

2 15

( e ) ........ ×

(f)

1 3

1 2

1 3

3 8

1 2

1 3

1 2

Multiplying fractions and whole numbers Salwa is learning how to be a pastry chef. She practices making roses with a pastry tube for

3 4

of an hour

each day. she practices for 6 days every week. How many hours does she practice each week?

■ Multiply 6 ×

3 4

Write the whole number as a fraction: 6 = 6 1

3 4

multiply the denominators. 6 ×

3 4

multiply the numerators.

Write a mixed number for the answer Salwa practices 4

1 2

6 1

.... × ....

....

.... ....

.... × ....

6 1

3 4

18 4

...

hours a week.

Practice Multiply then write the answers in the simplest form. ( a ) 4 × ( b )

4 8

× 7

( c ) 6 × 26

3 4

2 8

( d )

2 5

( g ) 3 ×

4 5

( h ) 9 ×

5 6

× 5

(i)8×

2 3

× 7

( e ) 8 × (f)

1 3

5 6

...

unit 1 Multiplying fractions and mixed numbers Saber owns a bakery. He works 7

1 2

hours each day. He bakes

bread

5 6

of this time. He spends

the rest of his day serving customers. How many hours a day does saber bake? ■ Multiply

5 6

×7

1 2

Write the mixed number as a fraction: 7 ×

15 2

multiply the denominators. 5 ×

15 2

multiply the numerators.

5 6 6

Write a mixed number for the answer Saber bakes 6

1 4

1 2

.... × ....

....

.... ....

.... × ....

5 6

15 2

75 12

...

hours a day.

Practice

1 Multiply . Write the answers in the simplest form. ( a )

2 5

×5

1 2

( d ) 3

2 3

5 6

( g ) 2

1 6

3 4

( b )

3 4

×4

1 4

( e ) 5

1 3

3 7

( h ) 9

1 3

2 6

( c )

7 8

×7

1 4

(f)4

1 4

2 3

×8

2 3

(i)

3 4

2 Find the missing whole number in each problem. ( a ) 3

1 2

× ....... = 7

( b ) 4

1 3

× ....... = 13

( c ) 10

1 4

× ....... = 41 27

Multiplying Mixed numbers Ahmed and Dalia attend Cooking class. Today they are learning how to make a pie. The recipe calls for 2

1 2

cups of flour. They 1 2

need to make 1

times the

recipe. How much flour should they use? ■ Multiply 1

1 2

× 2

1 2

Write the mixed numbers as fractions: 1 multiply the numerators.

3 2

5 2

multiply the denominators.

3 2

5 2

Write a mixed number for the answer They should use 3

3 4

1 2

3 2

, 2

.... × ....

....

.... ....

.... × ....

3 2

5 2

15 4

1 2

5 2

...

cups of flour.

Practice Multiply then write the answers in the simplest form. ( a ) 2

3 4

×1

2 3

( d ) 3

2 5

×4

1 2

( b ) 4

1 2

×1

7 8

( e ) 2

1 2

×1

( c ) 3

1 2

×1

2 6

(f)3

1 2

×1

( g ) 26

4 5

1 10

( h ) 21

7 8

×3

1 3

2 6

( i ) 31

3 5

×4

3 5

2 3

unit 1 Exercise (1 – 4) Problem Solving. Applicatoins. 1 Reham is installing ceramic tiles on

2 3

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 2 3

has completed

covered?

3 4

of the family room floor. She

of the job. What part of the floor is now

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

1 2

of the total stock. Women’s suits make up

3 5

of all the suits. What part of the store’s inventory is made up of women’s suits?

4 Peter practices decorating cakes for

3 4

of an hour each day. How

many hours does he practice in 7 days?

5 A recipe calls for

3 4

of a cup of flour. Laila makes 3

recipe. How much flour does she need.

1 2

times the

6 Eman works in the Teen Trends shop. All cotton fashions make up 5 of the stock she sells. Cotton shirts make up 2 of this stock. 8 3 What part of the total stock is made up of cotton shirts. 7 Of 40 students in a cooking class, How many students is this?

5 8

are preparing to be chefs.

8 Faiza is making spaghetti sauce. The recipe calls for 1 water, she wants to make 4

should she use?

1 2

3 4

cups of

times the recipe. How much water

lesson

Dividing Fractions

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 find how many 3÷

1 4

= 12

3×

4 1

= ........

there are 12 quarters in 3 slices

reciprocals ● Illustrate 2

1 2

÷2

■ Divide the 2 1 squares into 2 equal 2 parts and shade one of them. The fraction for the shaded parts is 2

1 2

5 4

÷ 2 =

which is 1 5 2

.... ....

reciprocals 30

1 4

. .... ....

....

...

unit 1 1 8

● How many

’s are there in

3 4

■ Since we want to know how many eighths are there in 3 4

1 8

● How many

3 4

.... ....

, we divide

= ........ There are 6 eighths in

‘s are there in 4

1 2

3 4

apples?

■ Count to find how many? 4

1 2

3 4

9 2

.... ....

= ........

Practice

1 Divide. write the answers in lowest terms. ( a )

2 3

1 6

( d )

1 8

4 3

( g )

1 9

÷1

1 2

( b )

3 4

5 8

( e )

7 12

1 6

( h ) 2

4 5

÷1

3 4

( c )

4 5

1 3

(f)4

1 2

(i)4

2 7

÷1

5 14

Problem solving: Applications

2 The perimeter of a square is the square.

3 Alaa divided

6 11

m. Find the length of each side of

7 9

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 a pizza?

1 2

Activities

1 Arrange the products of the following from the smallest to the greatest. Use the sign <: ( a )

2 5

of 80

( b ) 70 ×

4 9

( c )

6 11

× 77

( d ) 63 ×

The order is .......... < .......... < .......... < ..........

3 7

2 Fill in the circles Multiply

3 8

Start

by 2

Multiply

1 2

1 5

act

Subtr

Multiply by 1 1 by 1 3

2 16

Add 1 3

6 16

Multiply

by 1

Multiply

5 6

by 1

End

1 11

3 Use a Calculator to multiply a whole number by a fraction. Example: 488 ×

3 4

Rewrite the problem

488 × 3 ÷ 4

4 8 8 × 3 ÷ 4 =

366

Practice: Find the products. ( a ) 366 × 32

5 6

( b ) 648 ×

7 8

means 3 ÷ 4

3 4

( c ) 936 ×

5 6

( d ) 852 ×

7 12

unit 1

4 Guess and check can you find a decimal

= 0.9

for

and a decimal

= 0.18

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. Is the product greater than 1 or less than the two factors? Try other examples. Do you get the same results?

0.2 × 0.7

Is the product always greater than or less than the two factors.

Is the product greater than or less than one? ( 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.

1.2 × 1.7

Do you get the same results?

Is the product always greater than or less than the two factors?

7 Find a number in the box for each pair of clues. 0.01 0.11

0.2 1.2

( 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

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 will get 6.4 (2) I am less than half the product of 4.25 and 4.4 by 5.63 ( d ) Complete the table. ×

10.5

31.2

5.42

2.5 18.72 7.046 29.4 34

unit 1

Unit test Answer the following questions:

1 Choose the correct answer. ( a ) 2.5 × 100 = ..........

( b ) 1.8 × 0.09 = .......... ( c ) 34.8 ÷ 100 = .......... ( d ) 9.64 ÷ 4

= ..........

[ 250 , 25 , 0.25 , 0.025 ]

[ 0.162 , 0.972 , 1.89 , 162 ]

[ 3.480 , 348 , 3.48 , 0.348 ]

[ 241 , 2.41 , 1.94 , 38.56 ]

( e ) A rope of length 10.5 m is cut into 7 pieces of equal length. How long is each piece?

[ 15 m , 7 m , 1.5 m , 73.5 ]

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

( g ) 2

1 8

[ 20.39 , 20.9 , 23.18 , 23.9 ]

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

( h ) (3.69 ÷ 3) × 2 = .......... ( i ) (0.325 + 9 (j)

1 4

2 3

2 5

1 ) 4

÷ 100 = ..........

= ..........

[ 2.1 , 2.13 , 2.12 , 2 ]

[ 2.64 , 2.46 , 0.246 , 1.23 ] 322 , 0.95 300 1 , 1 , 5 10 15 15

[ 0.9575 , 0.09575 , [ 1 , 5

2 Complete. ( a ) 21 – (7.02 × 1.8) = .......... ( b ) 3.6 , 5

1 5

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

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

1 6

÷ .......... =

( e )

...

(f)

2 3

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

4 5

6 5

1 4

( g ) 76.52 ÷ .......... = 7.652 ( h ) .......... ÷ 1000 = 5.619

] ]

3 ( a ) From the table opposite,

1 2 5 8 1 10

choose five numbers whose product is 1 ( b ) Put the suitable sign (<, =, >). (1)

4 5

(2) 3

6 7

(3) 7 × (4)

3 4

–2 1 3

2 3

3 4

5 5 1 5

8 9 2 3

7 7

0 0 6 7

2 2

2 3

6 7

(6) 5.142 × 100

5142 ÷ 100

1 3

(7) 806.7 ÷ 100

806.7 × 10

5 7

(8) 2.4 × 10

0.24 × 1000

(5) 3.2 kg

3200 gm

4 ( a ) Mariam went to the market. She bought 4.5kilograms of fish 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: ( b ) A big barrel has 113

3 4

1 2

5 7

4 . 5

kg of oil, and we want to distribute the oil

in bottles, the capacity of each one is 1 bottles are needed for that?

1 4

kg of oil. How many

unit

Sets

Venn diagrams are named after the English

mathematician venn (1834 - 1923) Who first showed how useful they could be in work on sets.

Gone Venn

Unit Objectives After 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 - finite - infinite. ● 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.

Lessons of the unit 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

lesson

Introduction to sets

There are many words which we use to show a collection of things. For example, we talk of

a head of cattle

a shoal of fish

a crowd of people

a flock of geese

●●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 people may like bananas and others may not. Example he letters in the word “tomato” represents a set because it is defined T 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). 38

unit 2 Exercise (2 – 1)

1 Mention the elements of each of the following sets. ( a )

( c )

set of birds ( b )

Samir

Soha

Ahmed

set of children ( d )

set of animals

set of flowers

2 State, giving reasons, which of the following is a set and which is not a set, mention the elements of 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 fingers 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. 39

lesson

Set notation

●●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”. 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.

●●Capital

letters are used to designate sets: 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”.

●●The

set.

symbol “p” is used to denote that an object is an element of the 5 p B means “Five is an element of set B.”

●●The

symbol “q” indicates that an object is not an element of the set. 12 q B means “Twelve is not an element of set B.

●●Small

letters may name elements of sets such as: R = {m , a , t , h}

unit 2 ●●Sets

which contain exactly the same elements are called equal 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.

●●Sets

sets.

which contain the same number of elements are called equivalent {1, 2, 3, 4} and {1, 3, 5, 7} are equivalent sets.

Practice

1 Express each of the following 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 of the following sets in words: ( a ) X = {2, 4, 6, 8}

The set whose elements are ......., ......., ......., .......

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

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

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

3 List the elements of each of the following sets: ( c ) 17 - y > 12

( a ) 3 > a A = {......., ......., .......}

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

( b ) 7 + X < 11

( d ) b < 1

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

B = {.......}

4 Put the suitable symbol ( p or q ) : A={ B={

, ,

} }

C = {1, 3, 5, 7, 9} D = {0, 2, 4, 6, 8}

( a )

....... A

(f) 9

( b )

....... A

( g )

( c )

....... A

( h )

.......

....... B

.......

( d ) 1 ....... C

(i) 7

( e ) 8 ....... D

( j ) 13 ....... C

unit 2 Exercise (2 – 2)

1 List all the elements in each of the following 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 five letters of the English alphabet}.

2 Which of these sets are equal to T if 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 of the following. ( 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}.

4 If C = {all prime numbers}, which of the following statements are true? ( a ) 7 p C

( b ) 51 p C

( c ) 24 q C

( d ) 97 q C

( e ) 23 p C ( f ) 31 q C

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 find the five other sets that are equal to {m, t, s}

lesson

●●Try

Types of sets

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 { }.

{Cats that can fly} = { } = ∅ {0} is not an empty set. ●●A

set that contains a countable number of elements is called a finite

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

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

set that contains an uncountable number of elements is called an

infinite set. we can not actually count its elements. { 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. 44

unit 2 Exercise (2 – 3)

1 State whether each set is finite, infinite 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 first four elements of each of the following 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 of common elements. If 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}. 45

lesson

Representing sets by venn diagram

●●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×

9×

3×

5×

9×

7×

A 1

3×

1× 7×

5×

3×

9× 7× 5×

1×

9×

3×

7×

Practice

1 List the elements in each of the following sets: ( a )

( b ) B

0× 8×

6×

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

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

Is 2 p B? ......

Is 46

2×

4×

p A? ...... p A? ......

Is 9 p B? ......

5×

unit 2

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

Then the following figure represents the two sets X and Y , complete the venn diagram.

3×

5×

Y ×

here may be more than two loops in a venn diagram., They may 3 T overlap or intersect in many different ways. Two possible ways are shown. A

1×

B 2× 4×

8×

5×

3× 7×

9×

X a×

b×

c× g ×

e×

d×

Z C ( a ) What number is in both A and B, but not in C? ( b ) What numbers are in C but not in A or B? ( c ) What letter is not in X but is in Y and Z?

4 Think. Logical Reasoning. Give the letter or letters that are: ( a ) in

, but not in

( b ) in

and

, but not in

( c ) in

and

, and in

( d ) in

and

, but not in

a×

c×

b× f

d× e×

g ×

lesson

Subsets

The universal set ●●The

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

●●We

could make a number of

question is called the universal set. It is written as u. sets from the elements of

= {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} ⊂

is a subset of ( b ) {elephant} ⊂ u ( c ) {monkey, cat, lion} ⊂ u ( d ) {tiger, goose} ⊄

is not a subset of ( e ) {tiger, monkey} ⊄

Note that: monkey p u but tiger q u. 48

unit 2 Practice

1 In the venn diagram: ( a ) List the elements of the three sets X, Y and Z.

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

Z 7×

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

Y 3×

X 1×

5×

(3) Z = {......., ......., ......., .......} ( b ) Put the suitable sign ( ⊂ or ⊄ ). (1) X ....... Y

(2) X ....... Z

(3) Y ....... X (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 ×

× ×

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

Exercise (2 – 4)

1 X = {a, b, c, d} , Y = {a, b, c, e} Write the elements of 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

1×

( c ) The elements of B

2×

( e ) Is ∅ ⊂ u?

6×

( d ) The elements of A that are in B

4 Put the suitable sign ( ⊂ or ⊄ ). ( a ) {1} ....... {1, 3}

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

( c ) {0, 1} ....... {10, 15} ( d ) ∅ ....... {1, 2, 3}

4× 5×

3×

10 × 12 ×

9× 7× 8×

11 ×

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.

lesson

Operations on sets

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. Intersection of sets } means intersection of sets. Set A

Set B

Set A

Set B

Set A intersects set B A } B

Notice and complete.

1 A = {factors of 15},

B = { factors of 21}. A } B = {Common factors of 15 and 21}

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

Venn diagram ×

× ×

B ×

× ×

intersection: A } B

unit 2 X

2 X = {digits of the number 12304} = {... , ... , ... , ... , ...},

× ×

Y = {digits of the number 102}

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

× ×

X } Y = {... , ... , ...}. Containment: Y ⊃ X

3 D = {Letters of "cat"}, E = {Letters of "act"}

F={

, ,

, }

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

4 C={

Equality: D = E

F × ×

C } F = ...

Disjoint: C } F = ∅

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.

Intersection of two sets is the set which contains all the common elements belonging to the two sets.

A } B = {x : x p A and x p B} 53

Properties of Intersection

1 X = {pupils with full marks in math} Y= {pupils with full marks in Science}. X

Adel ×

Ahmed ×

Yousef × Said ×

Hatem ×

Ehab × Yousef ×

Hatem × Hussein ×

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

Y ×

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

Y ×

× ×

Y } X = {.......... , ..........} What do you notice? Commutative property of intersection ........

= ........

unit 2

2 A = {readers of stories}, B = {players Gymnastics} C = {Players of ping - pong}. A

Ayman × Aly ×

Ahmed ×

Said ×

Aly ×

Hassan × Samy ×

Samy × Baker ×

Represent on the venn diagram A

B ×

× ×

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

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

What do you notice?

Associative property of intersection ........

= ........

Exercise (2 – 5)

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

Y 2×

4×

Z 9×

5×

12 ×

13 ×

1×

17 ×

8×

List the elements of: ( a ) X } Y

( b ) X } Z

( c ) Y } Z

( d ) X } Y } Z

2 The venn diagram opposite shows sets A A, B, and C. List the elements of: ( a ) A } B

( b ) B } C

a×

( c ) C } A

d× c×

( d ) A } B } C

e×

h×

b× g ×

C 3 Using the symbol "}", Write down what the shaded part in each of the following figures represents: ( a )

( b )

Y Z

( c )

( d )

C E

unit 2

4 The venn diagram opposite shows sets X and Y. X

Put the suitable sign (p, q, ⊂,⊄ ) ( a ) 3 ....... (X } Y)

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

3×

2×

4×

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

1×

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

5×

5 Mark ✓ for the correct statement and ✗ for 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

A 2×

3×

4×

1×

( 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. 57

Union of sets { Means union of sets Set A

Set B

Set A joins set B A{B

Venn diagram

Notice and complete.

1 A={

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

Union: A { B

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

7×

8× 5× 6×

2×

1× 3×

C ... D

unit 2 X

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

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

A { B is coloured in each diagram

A { B = {x : x p A or x p B} 59

Properties of Union

1 A = {Players of football}, B = {Players of Handball}

Ayman ×

Aly ×

Hany ×

Alaa ×

Hamed ×

Samy ×

Represent on the venn diagrams. A { B = {Players of football or handball}. A

B ×

× × × ×

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

A ×

× ×

B { A = {...... , ...... , ...... , ...... , ...... , ...... , ......} What do you notice? Commutative property of union ........

= ........

Alaa ×

unit 2

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

4×

5×

4× 5× ×

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

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

( b ) Y { Z = {... , ... , 4, 5, ... , ...} , X { (Y { Z) What do you notice? Associative property of union ........ = ........

( 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. X

Z ................

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

The complement of 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 girls, hence they are boys.

u that are not

Practice

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

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

then A´ = { 4, 5, 6, 7} , B´ = {......., ......., ......., .......}

1×

3×

2× 4×

7×

6×

5×

2 In the venn diagram, u is the universal set, then A´ = {......., .......}, B´ = {......., ......., .......}, (A } B)´ = {.......}, (A { B)´ = {......., ......., ......., .......} 62

A 7×

1×

B 8×

3×

9×

unit 2 Difference of 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={

} ,

What do you notice?

}

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

≠ ...............

Practice

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

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

2×

4×

3×

6×

1×

B – A = {......., .......}

B 5×

9×

7×

2 Use the following venn diagrams to list: A

3×

6×

2×

9×

10 ×

7×

8×

5× 6×

2×

1× 3×

2×

3× 1×

A – B = {......., ......., .......}

A – B = ..........

B – A = {......., .......}

B – A = ..........

Exercise (2 – 6)

1 X = {6 , 7} , Y = {6, 7, 9} , Z = {7, 8, 9, 10} List each of the following sets: ( a ) X } Y , X } Z , X } Y } Z

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

( c ) Y – X , Z – Y , X – Y

2 The figure opposite is a venn diagram

1×

for the sets X, Y and Z.

2×

5×

4× 3× 6×

List each of these sets: ( a ) X { Y , X } Y , X } Y } Z

8×

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

7×

( c ) X´ , Y´ , Z´

3 Using the two symbols } , { write down what the coloured part in each of the following figures represents. ( a )

( b )

( c )

( d )

( e )

(f)

C A

Z 64

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}

1×

( c ) {2 , 5}

3×

2× 5× 4×

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

7×

8×

9×

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

5 The figure opposite is a venn diagram for the sets X, Y, and Z. Mark ✓ for the correct statement and ✗ for the incorrect one. ( a ) X } Y = Y ( b ) Z ⊃ X

2×

4×

3×

( c ) Y ⊃ Z 6

( d ) (Z { Y) ⊃ X

( e ) ∅ ⊂ X 5×

( f ) (Z } Y) ⊄ X

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.

Activity 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".

1 If 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 of the following: ( 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) 66

unit 2

Unit test 1 Complete

A{B

q ......

...... B

... A { B

...... A

... A { B

p ......

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 ) If {X, 3, 4, 7} = {7, y, 6, 3} then: X – y = ....... X + y = .......

× y = .......

= ....... 67

3 ( a )

2× M 3× 5×

9×

M { N

4× 7×

6×

What elements belong to set u ?

to set M? to set N? write the resulting 8×

N } M

set, Listing the elements for: 3

M – N

M´

( b ) If 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 of the following: 1

A } E

E { B

E { A

B } A

B } E

A { B

4 ( a ) If A = {1, 2, 3}, B = {2, 0, 3, 1}, and C = {digits of the number 123}. What is the relation between: (1) A and B?

(2) B and C?

(3) A and C?

( b ) State the subsets of 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. 68

unit

Geometry

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.

Unit Objectives After 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 ●

Lessons of the unit Lesson 1 Geometric patterns

Lesson 2 Constructing a circle

Lesson 3 Constructing a triangle

Lesson 4 Constructing the altitudes of the triangle

lesson

Geometric patterns

Sometimes 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?

Think What is the order of the figures?

Two triangles and then two circles. The next two figures are circles.

What are the next two figures in this pattern?

Think 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. 70

unit 3 Problems

1 Draw the next four figures for each pattern. ( a )

........

( b )

........

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

( c )

( d )

( e )

(f)

( g )

........

( h )

........

2 Draw the next figure in each pattern. ( a )

........

( b )

........

3 Compare the figures to see what change took place in the figure, then draw the next figure in each pattern. ( a )

( d )

( e )

( c )

( b )

unit 3

4 Choose the next figure in the pattern. ( a )

( b )

( c )

( d )

Portfolio Form new geometric patterns on your own. 73

lesson

Constructing a circle

A ferris wheel suggests a circle. All

the points on a circle are the same distance from the centre.

To draw any circular object, you can use a compasses. Step 1

Step 3

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. A radius B

How many more could you draw? Step 2

Step 4

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. D

ete r

dia m

â&#x2013;

How many more could you draw? 74

unit 3 Practice

1 Name the radius and the diameter of each circle where "M" is the centre. X

C N

M Z

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

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 circle

M C

3 In the figure 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. 75

Exercise (3 – 1)

1 Complete the table. Radius

3 cm

5 cm

Diameter

.....

16 cm 22 cm

18 cm

.....

1.8 cm

.....

6.8 cm

.....

9.4 cm

2 In the figure opposite,

L M

( a ) What is the name of the circle? A

( 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 figure opposite,

( a ) Name the segments that C

are chords. ( b ) Name the longest chord.

V W

4 Draw a line segment with the length given. use it as a radius to construct a circle. ( a ) 2.5 cm 76

( b ) 5 cm

( c ) 4.5 cm

unit 3

5 Try to draw a figure similar to the following figures: ( a )

( b )

6 If each side of the square is

10 cm, what is the length of

a radius of circle w?

7 Mark ✓ for the correct sentence and ✗ for the incorrect one. ( 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.

Z M

( d ) The radius of the circle is the longest line segment can be drawn in the circle.

8 Use a compass. Design a logo for your fifth grade class. 77

lesson

Constructing a triangle

Drawing a triangle given its side lengths cm

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

3.5 A

the triangle

BC = 5 cm, and CA = 3.5 cm. you can use your ruler and compass to help you draw

■

6 cm

Step 1

Step 3

Use your ruler to draw A B with Length 6 cm

Reset the compass to 3.5 cm and with A as a centre, draw another arc to intersect the first arc at C. C

6 cm

Step 2

Step 4

Set your compass to 5 cm and with B as a centre, draw an arc.

Draw A C and B C , then ABC is the required triangle. C cm

3.5

6 cm

unit 3 Exercise (3 – 2) L

1 Draw and label a triangle KLM, in which KM = 5 cm,

7 cm

KL = 7 cm, and ML = 6 cm

6 cm

5 cm

2 Draw and label an equilateral triangle A B C of side 7 cm.

How can you check if the

7 cm

is accurate?

7 cm

triangle that you have drawn

7 cm

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

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?

4 Draw the triangle A B C in which AB = 8 cm, BC = 5 cm and CA = 6 cm. What type of Δ 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 Δ 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?

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.

6 cm

60°

6 cm

7 cm

60°

Constructing the altitudes of the triangle

lesson

The altitudes of an acute - angled triangle Step 1

Step 2

Draw the acute - angled triangle

Put the edge of one side of a set

ABC, put the edge of the ruler on

square on B C . Move it to slide along the edge of the ruler until

a side of the triangle, say B C .

the point A coincides with the

edge of the set square. Draw A D , then A D ⊥ B C . A

In the same way draw from B

and C two other line segments

segments to represent the two other altitudes of the triangle. A

●

The three altitudes interset at a point M inside the triangle.

●

Note that

The length of A D is called the height of the triangle.

to represent the two other line

Step 3

For each altitude there is a corresponding base.

unit 3 The altitudes of the obtuse - angled triangle To construct the three altitudes

of the obtuse - angled triangle A B C we follow the same steps

as shown before.

Note that ●

B F

The three altitudes intersect at

M outside the triangle.

■

Complete: 1

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

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

3 ........

is the corresponding base to the altitude B E A

The altitudes of the right - angled triangle D

In Δ A B C, A B ⊥ B E , to draw the third altitude we draw B D ⊥ A C Note that ●

■

The three altitudes intersect at B.

Complete: 1

A B is the corresponding base to B C .

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

3 ........

is the corresponding base to B D . 81

Exercise (3 – 3)

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. ( a )

( b )

( c )

5 cm

6 cm 9 cm

2 In the figure opposite, A B C D

6 cm 30°

6 cm

12 cm

30° A

is a rectangle. Draw the third altitude in the two triangles A B E and D C E

3 Draw the triangle A B C in which AB = 5 cm, BC = 6 cm and AC = 4 cm. Draw the altitudes of Δ 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 Δ A B C, then measure their lengths.

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 determine the 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 the length of each of A B and A C . What do you notice? 82

unit 3 Activities

â?&#x2039; Use the computer program or a sheet of

card - board to make a set of congruent isosceles right - angled triangles. â?&#x2039; Use the needed number of triangles to form each of the following

shapes, and write in each case the number of triangles you used.

GeoGebra http://www.geogebra.org/cms/index.php?option=com_frontpage&Itemid=1

2 A quadrilateral has for sides:

A B , B C , C D , and A D and Four angles: ∠ A, ∠ B, ∠ C and ∠ D.

Four types of quadrilateral have been made below by putting together all seven pieces of the famous Tangram puzzle. Square

All sides have the same length All angles are right

Rectangle

Two pairs of sides have the same length All angles are right

Parallelogram

Two pairs of sides have the same length Two pairs of sides are parallel

Trapezium

One pair of sides are parallel

Rhombus Another quadrilateral called rhombus, can not be made with all the pieces. Study the properties of each quadrilateral. All sides have the same length

unit 3 Write square, rectangle, parallelogram, rhombus or Trapezium to describe each quadrilateral below. ( a )

( d )

( g )

( b )

( e )

( h )

( c )

(f)

(i)

3 Which of these strips could you use to form these quadrilaterals?

( a ) a rectangle

( d ) a rhombus

( b ) a square

( e ) a trapezium

( c ) a parallelogram Which quadrilateral has: ( f ) All sides the same length and four right angles? ( g ) Only one pair of parallel sides? ( h ) Four right angles, but not all sides the same length? 86

unit 3

4 Use the computer program or tracing paper to make 4 copies of this quadrilateral.

Put the four pieces together to make: 1

a square

a rhombus

a square with a small square hole in it.

Designs â&#x2014;?

These designs were made using only a compass. Try to draw them.

â&#x2014;?

Make other designs of your own. colour them in an interesting way. 88

unit 3

Unit test 1 Complete: ( a ) In the figure opposite, 1

The length of radius B O is ........ cm.

The length of diameter B A is ........ cm.

The diameter of the circle is ........ times as long as the radius.

( b ) Draw the next figure in each pattern

2 ( a ) Draw a circle with a diameter of 6 cm. ( b ) Draw the triangle A B C, where AB = AC = 5 cm and BC = 4 cm. Draw A D ⊥ B C to meet it at D. Measure the length of A D

3 ( a ) Draw the missing figure.

.................. 89

(â&#x20AC;&#x2030;bâ&#x20AC;&#x2030;) The 7 shapes of the tangram puzzle fit into this square, there

are 5 triangles, 1 square and 1 parallelogram 1

Arrange triangles "a" and

"b" to make a square

Arrange triangles "c" and "d" to make triangle "e"

Arrange shapes "c" , "f", and "d" to make a rectangle

Arrange shapes "c", "f", and "d" to make a parallelogram

4 Some quadrilaterals have special names and properties, write the name of each quadrilateral: Quadrilateral

Properties

.....

Exactly one pair of opposite sides parallel.

.....

Two pairs of opposite sides parallel and congruent.

.....

Parallelogram with all sides congruent.

.....

Parallelogram with four right angles.

.....

Rectangle with all sides congruent.

Example

unit

Probability

Probability means the chance or likelihood that something will happen. In Math, probability is a number that is used to describe that chance. The probability that heads will come up is one chance in two.

Unit Objectives After studying this unit the student should be able to: â&#x2014;? Carry

out simple probability experiments

â&#x2014;? Predict â&#x2014;? Test

the outcomes of simple probability experiments

the truth of simple probability experiments

Lessons of the unit Lesson 1 Investigating experiments and outcomes Lesson 2 Certain and impossible events

Investigating Experiments and Outcomes

lesson

Ramy is tossing a number cube. He wonders which number will be on the top most often. Recording the frequency of numbers tossed is an example of an experiment. Each possible result in an experiment is an outcome. Working together Materials: number cube with faces labeled 1 - 6

A Toss the cube. What number is on the top? Toss the cube again. What is the outcome?

B Complete this frequency table. outcome

Tally

1 2 3 4 5 6

frequency

C Toss the cube. Mark a tally in the table to show the outcome. Toss the cube and record the outcome until 20 tallies have been marked.

D Count the tallies for each number. Record the total

for each in the frequency column.

Sharing results

1 Combine the results of each groupâ&#x20AC;&#x2122;s experiment into a class table. 2 Which outcome occurred most often? Least often? 3 Are the chances of tossing each number the same ? explain. 92

unit 5 Investigating making predictions Mariam is tossing a coin. What are the possible outcomes? What if Mariam tossed the coin 20 times? How many times do you think the coin would land heads up? Working together Materials: a coin

A How many times do you think outcomes a coin will land heads up if it is tossed 20 times? Record your prediction.

Tally

frequency

Heads Tails

B Toss a coin 20 times. Record

your results in a frequency table like the one at the right.

Sharing your results

1 Combine the results of each groupâ&#x20AC;&#x2122;s experiment and make a class table.

2 How many times did the coin land heads up? tail up?

3 Compare your experimental results with your prediction. How close was your prediction to the actual results?

Use the results of your experiment, the class results, and your prediction to write each of the following: number of heads number of tosses

number of tails Number of tosses

The spinner game The spinner is a circular disc divided into circular sectors. A Pointer is pinned at the middle of the spinner. The pointer rotates till it comes to rest at one of the sectors. If the pointer rests on one of the dividers, (Lines between sectors), the experiment should be repeated again.

Finding the probabilities

1 What is the probability of getting an odd number with one spin of the pointer?

2 3

There are 5 equally

There are 3 chances

The probability

likely outcomes:

in 5 of getting an odd

of getting an odd

1, 2, 3, 4, 5

number

number is

There are 2 chances of

The probability of getting

getting an even number.

an even number is

There are 5 chances in 5

The probability of getting

of getting a number less

a number less than 6 is

than 6

3 5

....

.... ....

........

There are 0 chances in

The probability of getting

5 of getting a number

a number greater than 5

greater than 5

....

........

unit 5

5 Each outcome is equally likely in the following experiments. Give the missing information in each row Experiment

Draw a card

Outcomes

1 2 3 4 3

without looking

Toss a cube with sides

numberd 1 - 6

Chances

There

chance in

Probability

1 The probability ......

of getting a 3

of getting a 3 is ......

1 2 3

There are

4 5 6

getting an odd add number is number.

......

A B C

There are 2

The probability

of getting

a vowel

......

The probability

chances in 6 of of getting an

Toss a cube

that has one

of the letters A, B, C, D, E, F on each face

D E F

chances in ...... of getting

a vowel (A or E). is .......

C B

6 Suppose you draw one of these cards without looking. ( a ) What are the possible outcomes? ( b ) Are the outcomes equally likely?

( c ) What is the probability of getting an odd number? ( d ) What is the probability of getting a red card?

10 8 3

7 Suppose you draw a marble without looking. ( a ) What are the possible outcomes? ( b ) Do you have a better chance of getting a blue marble or a yellow marble? ( c ) What is the probability of getting a blue marble?

8 Suppose you spin the pointer. ( a ) Which letter do you think you would

get most often in 12 spins? ( b ) Which color do you think you would get most often in 12 spins?

( c ) What is the probability that you will get an A? ( d ) What is the probability that the pointer will stop on a red space? 96

unit 5 Check your understanding

9 What if these cards are in a box and you choose one of them?

11 9 7 3 5 1

( a ) What are the possible outcomes?

( b ) What is the probability of choosing the card with the number 9? ( c ) What is the probability of choosing a red card?

( d ) What is the probability of choosing a card with a number less than 6?

10 Five checkers are placed in a bag. Two checkers are red and three are black. Karim thinks he will pick a red one. What are his chances?

( a ) Complete: There are 5 checkers so there are

..........

Possible

outcomes. Each checker has the same chance of being picked. Two checkers are red so there are .......... Favorable outcomes.

The probability of choosing a red checkers is .......... The probability of choosing a black checkers is .......... ( b ) (1) How many checkers would you put in a bag to make 10 possible outcomes?

(2) What if the probability of choosing a red checker was

4 10

How many red checkers would be in the bag? How many black checkers?

Show what you know What if there are 9 checkers in a bag? the probability of picking a black is

1 3

. How many black checking are in the bag?

lesson

Certain and Impossible Events

What if you picked one marble

Without looking? Is it likely that you will pick the blue marble? Why?

Numbers can be used instead of

words to describe whether an

outcome is possible, impossible,

or certain.

Outcomes

Chance

Blue marble

Possible

Red marble

Possible

White marble

impossible

Any marble

certain

Probability ....

10 ....

= ......

....

= ......

10 10

1 The probability of an event is always 0, or ........, or a number between ........

and ........

2 An impossible event has a probability of ........ 3 A certain event has a probability of ........ 4 What is the probability of March having 32 days? 5 What is the probability of April having 30 days? 98

unit 5 Check your understanding

1 Tell whether each of the following means that the probability is 0, close to 0, close to 1 or 1 ( a ) rarely

( b ) never

( c ) always

( d ) probably

( e ) The school bus will start in the morning. ( f ) The sun will set in the evening. ( g ) Summer vacation will last 4 months. ( h ) We will all have perfect attendance in our school next year.

2 Choose the best probability for each. ( a ) impossible

[0,

1 3

2 3

,1]

( b ) Sure

[0,

1 4

3 4

,1]

( c ) doubtful

[0,

1 5

4 5

,1]

( d ) expected

[0,

1 10

9 10

,1]

3 Describe the numbers on a cube for which the probability of rolling a 3 is: 1 2

( a ) 1

( c )

( b ) Close to 0

( d ) Close to 1

Show what you know Use examples from every day life to explain why possible and probable do not always mean the same thing. 99

Unit test 1 Choose the correct answer: ( a ) If a coin is tossed 600 times, then the nearest expected number for the tail to appear is ......... 1

220

255

298

356

( b ) A basket contains cards numbered from 1 to 20., If a card was drawn at random, what is the probability that the number written on the card is divisible by 6? 1

3 2 20

4 20

5 20

6 20

( c ) If there are 3 blue, 6 violet, 7 orange and 8 red sectors on a spinner, what is the probability that the pointer stopping on the red color? 1

1 2 8

1 24

1 2

1 3

( d ) When rolling a die once, the probability of a number greater than 4 appearing on the top side is ......... 1

1 2 6

1 3

1 2

( e ) Gamal is in a grade 5 class of 36 students. 16 of them are girls, if a student is selected at random from the class, What is the probability that the student is a boy? 1

4 2 9

1 2

5 9

1 36

( f ) A letter of the word "Ahmed" is selected randomly. what is the probability of selecting the letter "d"? 1

100

1 2 5

1 4

1 2

unit 5

2 ( a ) Pick a card without looking., Find each probability.

Triangle

hexagon

quadrilateral

Polygon

( b ) Use the following cards.

1 1 2

What is the probability of getting a 3 on one draw?

What is the probability of getting an even number?

3 Use the spinner opposite, ( a ) What if you spin the spinner 20 times?

Predict the number of times blue will be chosen.

1 4

( b ) What is the probability of an odd number? ( c ) What is the probability of red?

4 A die is tossed once, what is the probability of the appearance of each of the following on the top face of the die? ( a ) an odd number greater than 2? ( b ) A number between 0 and 9? ( c ) A prime number? ( d ) Zero ?

5 3 6 101

First Term Examinations Test One Answer the following questions

1 Choose the correct answer. ( a ) Round off 289316 to the nearest thousand.

[290000 , 289300 , 289310 , 289000]

( b ) Find the value of 10.57 ÷ 9 and round it off to 2 decimal places.

[1.20 , 1.18 , 1.17 , 1.16]

( c ) What fraction of 3 hours is 24 minutes?

[

2 5

( d ) Which of the following fractions is greater than

1 2

[

3 5

4 15

6 13

( e ) A rectangle consists of 4 parts where A is half

1 5

2 15

]

7 15

9 22

]

B CD

of the rectangle, B is half of A and C is half of B. D has the same size as C. Which 2 parts add up to rectangle?

5 8

of the

[B and C , B and D , A and B , A and D]

( f ) Which statement or statements are false?

102

Every trapezoid is a quadrilateral

Every rectangle is a square

Every parallelogram is a rhombus [

only ,

and

, and

]

Examinations 2 Complete: ( a ) If 0.3 × 20 = 0.3 × 2 × 10 = 0.6 × 10 = 6 then 0.44 × 30 = 0.44 × 3 × ......... = ......... × 10 = ......... ( b ) If 28 ÷ 200 = 28 ÷ 2 ÷ 100 = 14 ÷ 100 = 0.14 then 36 ÷ 4000 = 36 ÷ ......... ÷ 1000 = ......... ÷ 1000 = ......... ( c ) 3 ( d )

4 9

7 20

metres = ......... centimetres

÷ 6 = .........

( e ) If X = {1, 2, 3} and X ≠ D, then D = .........

3 ( a ) A tourist group of 2795 tourists reached Cairo Airport to visit Luxor and Aswan. They all got into a train from cairo station. If each train

carriage holds 215 passengers, Find the number of carriages they got into.

( b ) A worker earns LE 1 LE 9

3 4

1 2

per hour. When he finished his work he got

. How many hours did he work?

4 ( a ) Draw the triangle XYZ such that XY = YZ = ZX = 7 cm. where do the altitudes meet?

( b ) Draw a circle with radius 3.5 cm, draw the diameter A B , draw another circle with radius B C , C ∈ A C equals the diameter of the first circle. what is the length of A C ?

( c ) These sets are defined by their common property; list the elements: 1

Numbers greater than 6 but less than 10

Letters in the word "element".

5 ( a ) If 2 ∉ { x } , find x. ( b ) A box contains 18 balls,

1 6

of them are red,

1 3

of them are blue and

the rest are green. A ball is drawn at random from the box. What colour has the greatest chance to be drawn?

103

Test Two Answer the following questions:

1 Choose the correct answer. ( a ) Which of the following fractions is equivalent to 12 sevenths?

1 7

2 7

, 12

1 7

( b ) Which of the following expressions gives the largest value?

5 7

]

[(24 + 6) ÷ (2 × 3) , (24 + 6) ÷ 2 × 3 , 24 + 6 ÷ (2 × 3) , 24 + 6 ÷ 2 × 3]

( c ) The digit 6 in the number 35.867 is in the ......... place.

[Tens, Tenths, hundreds, hundredths]

( d ) The chord of the circle M is ......... [ A C , A M , A B , M B]

( e ) 2

1 4

×1

2 3

= .........

1 4

3 4

( f ) If u = {1 , 2 , 3 , 4} and A = {1 , 2 , 3} , then A´ = .........

2 12

]

[4,{4},{3,4},{1,4}]

2 Complete: ( a ) If 0.8 × 200 = 0.8 × 2 × 100 = 1.6 × 100 = 160 then 0.7 × 400 = 0.7 × ......... × 100 = ......... × 100 = ......... ( b ) If 0.23 = 2.3 ÷ 10 = 23 ÷ 100 = 230 ÷ 1000 then 0.68 = 6.8 ÷ ......... = 68 ÷ ......... = 680 ÷ ......... 104

7 12

Examinations ( c )

The figure which has only 1

pair of parallel lines is .........

( e )

3 10

The name of each of the figures above are ........., ........., .........

( d ) 4

1 3

minutes = ......... seconds.

÷ 3 = .........

3 ( a ) Rania made some juice. she gave

1 4

of it away to his neighbour

and poured the rest equally into 9 bottles, what fraction of the juice did each bottle contain? ( b ) A box contains 4 white balls, 3 blue balls and 5 red balls, all of which are of equal size. When one ball is drawn randomly from the box find the probability of: 1

blue ball

red ball

4 ( a ) Draw a circle with radius 4.5 cm, draw the chord A B of length 6 cm, draw m (∠ BAC) = 90º to meet the circle at C. measure the length of A C . ( b ) State whether these sets are equal or not: 1

{3, 8, 2} and {2, 3, 8}

{letter in the word "cat"} and {a, c, t}

5 ( a ) Two boxes contain cards .In the first, the cards are numbered from 1 to 25 and in the second, the cards are numbered from 1 to 125, Which of the boxes gives a greater chance for the card numbered 14, explain your answer. ( b ) Draw the triangle ABC in which AB = 8 cm, BC = 6 cm and AC = 10 cm. measure the altitudes of the triangle. 105

Test Three Answer the following questions:

1 Complete: ( a ) If 0.3 × 20 = 0.3 × 2 × 10 = 0.6 × 10 = 6 then 0.4 × 30 = 0.4 × ......... × 10 =......... × 10 = ......... ( b ) If 0.6 ÷ 20 = 0.6 ÷ 2 ÷ 10 = 0.3 ÷ 10 = 0.03 then 0.06 ÷ 20 = 0.06 ÷ ......... ÷ 10 = ......... ÷ 10 = ......... ( c ) ( b ) If {1 , x} ⊂ {1 , 2 , 5 , 9] , then x = ......... ( d ) 4 ( e ) (f)

5 8

1 4

litres = ......... ml

÷ 10 = ......... The figures which have

2 pairs of parallel lines

and equal opposite sides are ......... 2

The figure which has 2 pairs of parallel lines and 4 equal sides is .........

2 ( a ) Eman ate

1 6

of a cake. she gave

2 5

of the remainder to her sister.

What fraction of the cake did Eman's sister receive? ( b ) Complete the pattern:

1 10000

1 1000

then deduce the seventh term.

1 100

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

3 Choose the correct answer: ( a ) 6 × 2

1 3

= .........

1 3

, 14 , 42 , not given]

( b ) Which of these are proper subsets of {g, h, f} ? 106

[{f} , {f, g, h} , {

} , {gh}]

Examinations ( c ) If 1260 = 43 × 29 + 13, then 1260 ÷ 43 = .........

[43.13 , 29.3 , 29.13 , 29]

( d ) The following fractions

= .........

and

are in their simplest form then [28 , 25 , 24 , 23]

( e ) A rectangle is 15.95 m by 8.25 m, then the best estimate for its area is about .........

[48 m , 48 m2 , 128 m , 128 m2]

( f ) A box contains cards numbered from 1 to 20, if a card is drawn

randomly, what is the probability that the card numbered is divisible by 6?

[

3 20

4 20

5 20

6 20

]

4 ( a ) Draw a circle of centre M with radius 4 cm, draw the two radii M Y and M X with an angle 60°, draw X Y Measure the length of X Y .

( b ) Draw ∆ A B C, in which AB = 9.8 cm, BC = 7 cm, AC = 5 cm, Draw the altitudes of this triangles.

( c ) State which of the following are sets and which are NOT sets: 1 2

Triangle, 3

, y

Letters in the word committee

even numbers

Clever people living in Cairo

5 ( a ) If A = {1, 3, 5, 7} , B = {3, 5}, C = {5} , D = {5, 7, 9} show which of the following statements are true or false? 1

B ⊂ A

A is an infinite set

∅ ⊂ D

CqB

D⊂A

( b ) In a class of 45 students, there are 5 more girls than boys. If one

of the students is chosen randomly, Find the probability of this student being a girl.

107

Test Four Answer the following questions:

1 Complete: ( a ) If 82 ÷ 200 = 82 ÷ 2 ÷ 100 = 41 ÷ 100 = 0.41 then 69 ÷ 3000 = 69 ÷ ......... ÷ 1000 = ......... ÷ 1000 = ......... ( b ) If 168.9 = 16.89 × 10 = 1.689 × 100 then 35.6 = 3.56 × ......... = 0.356 × ......... ( c ) 0.97 × 0.05 = ......... ( d ) To find out about how much she makes in one week, Dina divided her monthly salary of LE 980 by 4 About how much does she earn in one week? ( e ) 2

5 6

days = ......... hours

( f ) If 5 q { x , 1 , 4 } , then x = .........

2 Choose the correct answer. ( a ) 1

2 3

×1

1 5

= .........

( b ) Which polygon is a rhombus? T

3 8

11 15

,2,1

2 15

]

[T,R,Z,M]

( c ) Estimate the product: 0.976 × 6.9 [10 , 7 , 6 , 0] ( d ) If 5304 ÷ 136 = 39, then ........ ÷ 1.36 = 390 [530400 , 53.040 , 530.4 , 53040] ( e ) The decimal which is included between 0.6 and 0.7 is ........ [0.71 , 0.59 , 0.61 , 0.72] ( f ) The place value of the digit 4 in the number 3.0042 is ........ [ tens, tenths, thousands, thousandths] 108

Examinations 3 ( a ) Marwa brought LE 60 to the market. She spent on meat and

1 4

1 3

of her money

of her money on vegetables. How much did she

spend altogether? ( b ) A teacher bought a piece of cloth 10.5 metres long to be distributed equally among excellent girls. she gave a piece of length 1.5 metres to each girl to put on her chest. How many excellent girls are there?

4 ( a ) The cards shown were put in a box, A card was drawn randomly what is the probability of this

8 13 4

1 6 3 5

card being a prime number? ( b ) Represent the following sets X = {3 , 4 , 5 , 6}, Y = {7 , 5 , 6 , 8 , 9} and Z = {10 , 9 , 8} on a Venn diagram then list: X – Y , Z – Y. If u = {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9} then find (X } Y)´

5 ( a ) Put p , q , ⊂ , ⊄ to make each statement true. 1

5 ......... {even numbers}

{birds} ......... {animals}

{a , b} ......... {a , c , b}

buffalo ......... {animals}

( b ) Draw the triangle ABC where AB = 7.5 cm, BC = 10 cm and CA = 8 cm, draw the altitude from A to B C and measure its length. ( c ) Draw a circle M with diameter AB = 10 cm and the chord BC = 5 cm. what is the type of triangle ABC and triangle MBC?

109

Answers Test One 1 ( a ) 289000

( d )

( b ) 1.17 ( c )

2 15

Test Two 3 5

1 ( a ) 1

(f)

and

( d ) A B

( b ) (24 + 6) ÷ 2 × 3 ( e ) 3

( e ) A and D

5 7

( c ) hundredths

(f){4}

2 ( a ) 4 , 2.8 , 0.028

2 ( a ) 10 , 1.32 , 13.2 ( b ) 4 , 9 , 0.009

( b ) 10 , 100 , 1000

( c ) 335

( c )

( d )

2 27

( e ) {1} , answers may vary

3 ( a ) Number of carriages = 2795 ÷ 215 = 13 ( b ) Number of hours of work =9

3 4

÷1

1 2

4 ( a ) construction, the altitudes meet inside the triangle. ( b ) construction, AC = 14 cm

( c )

{7, 8, 9}

{e, l, m, n, t}

5 ( a ) Answer may very; 3 ( b ) green. 110

3 4

Parallelogram, rhombus,

trapezium ( d ) 260

3 ( a ) She poured = 1 – each bottle = ( b )

3 12

1 10 1 = 3 4 4 9= 1 12

( e )

1 4

3 4

5 12

yes

4 ( a ) construction, AC e 6.7 cm ( b )

yes

5 ( a ) the first box contains cards less than the second, then

the first box gives a chance greater. (

1 25

( b ) construction,

1 ) 125

6 , 8 and 4.8 cm

Answers

Test Three

Test Four

1 ( a ) 3 , 1.2 , 12

1 ( a ) 3 , 23 , 0.023

( b ) 2 , 0.03 , 0.003

( b ) 10 , 100

( c ) 0.0485

( c ) answer may vary; 2

( d ) LE 245

( e ) 68

( f ) answer may vary; 6

( d ) 4250 ( e ) (f)

1 16

2 ( a ) 2

A and B

2 ( a ) The remainder = 1 – she received = ( b )

1 10

,1;

2 5 1 3

1 6

= 5 6

5 6

3 ( a ) 14

( d ) 23

( b ) { f }

( e ) 128 m2

( c ) 29.3

3 20

(f)

4 ( a ) XY= 4 cm

( b ) M

( e ) 0.61

( c ) 7

( f ) thousandths

3 ( a ) She spent = she spent =

the seventh term is 100

5 ( a )

( b )

set

not set

true

false

true

false

false =

5 9

= 35 altogether

= 10.5 ÷ 1.5 = 7

4 ( a )

5 9

( b ) Venn diagram, X – Y = {3 , 4} Z – Y = {10} (X } Y)´ = {1 , 2 , 10}

25 45

1 + 1 = 7 3 4 12 7 × 60 12

( b ) Number of girls

( b ) construction. ( c )

( d ) 530.4

5 ( a )

⊂

( b ) construction, the height e 5.9 cm ( c ) construction, right angled triangle, equilateral triangle. 111