1A_Ch0(1). 0.3Multiples and Factors A Multiples B Factors C Prime Numbers and Prime Factors Index 1A_Ch0(2) D Index Notation

1A_Ch0(1)

0.3 Multiples and Factors

A Multiples

B Factors

C Prime Numbers and

Prime Factors

Index

1A_Ch0(2)

D Index Notation

0.4 Fractions

A Introduction

B Proper and Improper

Fractions

C Equal Fractions

Index

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D Comparing Fractions

E Arithmetic Operations with

Fractions

0.5 Choosing Appropriate Measuring Tools and Units

A Choosing Appropriate

Measuring Tool

B Choosing Appropriate Unit for

Measurement

Index

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Numbers

0.1 Numbers

Index

Example

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1. Natural numbers are the numbers we used in counting

and they are 1, 2, 3, 4, 5, 6, ... .

2. The first five whole numbers are 0, 1, 2, 3 and 4.

3. The whole numbers 0, 2, 4, 6 and 8 are examples of

even numbers. They are divisible by the number 2.

4. The whole numbers 1, 3, 5, 7, 9 are examples of odd

numbers. When these numbers are divided by 2, there

is always a remainder 1.

From numbers 0 to 8, list all the

(a) natural numbers, (b) whole

numbers,

(c) even numbers, (d) odd numbers.

Key Concept 0.1.1

Index

1A_Ch0(6)0.1 Numbers

(a) Natural numbers : 1, 2, 3, 4, 5, 6, 7, 8

(b) Whole numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8

(c) Even numbers : 2, 4, 6, 8

(d) Odd numbers : 1, 3, 5, 7

The Four Fundamental Arithmetic Operations (+, –, ×, ÷)

0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)

Index

Example

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1. In 4 + 7 = 11, the number 11 is called the sum of 4 and 7.

2. In 11 – 7 = 4, the number 4 is called the difference

between 11 and 7.

3. In 4 × 7 = 28, the number 28 is called the product of 4

and 7.

4. In 28 ÷ 7 = 4, the number 4 is called the quotient and 7

is called the divisor.

Find (a) 13 + 8 (b) 92 – 9

(c) 36 × 4 (d) 56 ÷ 8

Index

1A_Ch0(8)

(a) 13 + 8 = 21

(b) 92 – 9 = 83

(c) 36 × 4 = 144

(d) 56 ÷ 8 = 7

0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)

Find 5 + 4 × 2.

5 + 4 × 2

Index

1A_Ch0(9)

= 5 + 8

= 13

Fulfill Exercise Objective

+, –, × and ÷

0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)

Find 250 × 5 ÷

25.

250 × 5 ÷ 25

Index

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= 1 250 ÷ 25

= 50


+, –, × and ÷


Key Concept 0.2.1

Brackets


Index

Example

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1. Brackets are used to indicate the priority of operations.

2. We should always do the operations within the brackets

first.

3. If more than one pair of brackets are used, the general

rule is to use ( ) for the first arithmetic operation, then

[ ] and then { }.

Find (a) (20 – 8) ÷ (5 – 3) (b) 3 × (8 + 3) – (4 – 2)

Index

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(b) 3 × (8 + 3) – (4 – 2) = 3 × 11 – 2

= 33 – 2

= 31

(a) (20 – 8) ÷ (5 – 3) = 12 ÷ 2

= 6


Find 6 × {100 ÷ [(6 – 2) × 5] – 5}.

6 × {100 ÷ [(6 – 2) × 5] – 5}

Index

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= 6 × {100 ÷ [4 × 5] – 5}

= 6 × {100 ÷ 20 – 5}

= 6 × {5 – 5}

= 6 × 0

= 0



Expressions involving brackets.

Key Concept 0.2.2

Several verbs used to describe the arithmetic operations


Index

Example

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Divide 20 by 4,or 20 is divided by 4

Multiply 7 by 3,or 7 times 3

Subtract 6 from 10,or 10 minus 6

Add 5 to 2,or 2 plus 5

Arithmetic operationsTerms and descriptions

2 + 5

10 – 6

7 × 3

20 ÷ 4

Index

1A_Ch0(15)


Write down the result of each of the following.

(a) Find the difference when 7 is subtracted from 18.

(b) Find the product of 6 and 12.

(c) When 100 is divided by 5, find the quotient.

(a) 18 – 7 = 11

(b) 6 × 12 = 72

(c) 100 ÷ 5 = 20 Key Concept 0.2.3

Multiples


Index

Example

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‧ When we multiply a number by the natural numbers

1, 2, 3, 4 and so on, we get multiples of that number.

E.g. The first 4 multiples of 6 are : 6, 12, 18 and 24.

A)

Index 0.3

List the first four multiples of 5.

Index

0.3 Multiples and Factors 1A_Ch0(17)

5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20

∴

∴ The first four multiples of 5 are 5, 10, 15 and 20.

5 10 1515 20

5 × 1 5 × 2 5 × 3 5 × 4

Index

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Write down the first 5 multiples of each of the following numbers.


15

12

9

6

Multiples

6 × 1 , 6 × 2 , 6 × 3 , 6 × 4 , 6 × 5

9 × 1 , 9 × 2 , 9 × 3 , 9 × 4 , 9 × 5

12 × 1 , 12 × 2 , 12 × 3 , 12 × 4 , 12 × 5

15 × 1 , 15 × 2 , 15 × 3 , 15 × 4 , 15 × 5

6 , 12 , 18 , 24 , 30

9 , 18 , 27 , 36 , 45

12 , 24 , 36 , 48 , 60

15 , 30 , 45 , 60 , 75

Key Concept 0.3.1

Factors


Index

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1. When a given number is expressed as a product of

two or more natural numbers, then each of these

natural numbers is a factor of the given number.

2. In general, the method of division can be used to test

whether a number is a factor of another number.

E.g. Since 48 is divisible by 4, 4 is a factor of 48.

B)

Note :


Index

Example

1A_Ch0(20)

B)

i. All numbers are divisible by 1, therefore 1 is a

factor of any number.

ii. All even numbers are divisible by 2, therefore 2 is

a factor of any even number.

iii. Any number (except 0) is divisible by itself,

therefore any number is a factor of itself.

Index 0.3

Determine whether 5 is a factor of 30.

Since 30 ÷ 5 = 6, we say 30 is divisible by 5.

Therefore 5 is a factor of 30.

Index

0.3 Multiples and Factors 1A_Ch0(21)

Index

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Write down all the factors of each of the following numbers.

32

22

12

4

Factors

1 × 4 , 2 × 2 , 4 × 1

1 × 12 , 2 × 6 , 3 × 4 , 4 × 3 , 12 × 1

1 × 22 , 2 × 11 , 11 × 2 , 22 × 1

1 × 32 , 2 × 16 , 4 × 8 , 8 × 4 , 16 × 2 , 32 × 1

1 , 2 , 4

1 , 2 , 3 , 4 , 6 , 12

1 , 2 , 11 , 22

1 , 2 , 4 , 8 , 16 , 32


Key Concept 0.3.2

Prime Numbers and Prime Factors


Index

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1. A prime number is a natural number (other than 1)

which is not divisible by any natural number except

1 and itself.

2. Consider the factors of 24, 2 and 3 are prime

numbers and they are therefore called prime factors

of 24.

C)

Example

Index 0.3

Index

1A_Ch0(24)

List all the prime numbers

(a) from 80 to 150, (b) from 200 to

250.

(a) The prime numbers from 80 to 150 :

83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149

(b) The prime numbers from 200 to 250 :

211, 223, 227, 229, 233, 239, 241


Express 120 as a product of prime factors.

120

Index

1A_Ch0(25)


Prime factors.

2 120

2 60

2 30

3 15

5


= 2 × 60

= 2 × 2 ×

30= 2 × 2 × 2 × 15

= 2 × 2 × 2 × 3 × 5

Key Concept 0.3.3

Index Notation


Index

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1. When a number is multiplied by itself several times,

we can express the product using the index notation.

2. Consider the index notation 72, 73, 74 and 75. The

number 7 is called the base and the numbers 2, 3, 4

and 5 are each called the index.

D)

Example

Index 0.3

Index

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Using index notation, express each of the following numbers as a

product of prime factors.


(a) 50 (b) 132

(c) 180 (d) 225

(a) 50 (b) 132= 2 × 52 = 22 × 3 × 11

(c) 180 (d) 225= 22 × 32 × 5 = 32 × 52

Key Concept 0.3.4

Introduction

0.4 Fractions

Index

Example

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A)

‧ The number or is called a fraction in which 4

is called the denominator and 1 or 3 is called the

numerator of the fraction.

41

43

Index 0.4

Index

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8 triangles are shaded.

0.4 Fractions

Shade of the triangles below. How many triangles have you

shaded?

41

Key Concept 0.4.1

Proper and Improper Fractions

Index

1A_Ch0(30)

B)

0.4 Fractions

1. Examples of proper fractions are : , and , w

here the numerator of the fraction is smaller than the

denominator.

2

1

4

1

5

2

2. Examples of improper fractions are : , and ,

where the numerator of the fraction is greater than or

equal to the denominator.

3

3

3

4

11

12

3. Examples of mixed numbers are : , and ,

where the fraction is written as a sum of a whole nu

mber and a proper fraction.

3

11

7

11

2

14

Example

Index 0.4

Index

1A_Ch0(31)

(a) Improper fractions :

0.4 Fractions

(a) Which of the following are improper fractions?

1219

, 132

, 85

, 1210

, 34

, 117

, 5

12 ,

76

(b) Express the improper fractions in (a) as mixed numbers.

1219

, 34

, 5

12

(b)5

12can be written as ,

52

234

can be written as , 31

1

1219

can be written as . 127

1 Key Concept 0.4.2

Equal Fractions

Index

1A_Ch0(32)

C)

0.4 Fractions

1. When we multiply the numerator and the denominator

of a fraction by the same non-zero number, the value

of the faction remains the same.

E.g. 168

8 4

84

4 2

42

2 1

21

2

222

22

2. When the numerator and the denominator of a

fraction have no common factor except 1, the fraction

is said to be in its simplest form.

Example

Index 0.4

Index

1A_Ch0(33)

0.4 Fractions

Match the equal fractions.

53

72

65

127

4212

3621

2420

159

15

9

35

33

5

3

24

20

46

45

6

5

36

21

312

37

12

7

42

12

67

62

7

2

Index

1A_Ch0(34)


Reduce fractions.

0.4 Fractions

Reduce to its simplest form.10542

10542

6

15

2

5

52

Key Concept 0.4.3

Comparing Fractions

Index

1A_Ch0(35)

D)

0.4 Fractions

‧ Fractions can be compared when they are expressed

with the same denominators. To do this, we need to

know the L.C.M. of their original denominators.

Example

Index 0.4

Index

1A_Ch0(36)

The L.C.M. of 3 and 5 is 15.

0.4 Fractions

Compare and .51

31

153

3531

51

155

5351

31

，

∴

155

is greater than .153

∴31

is greater than .51

Index

1A_Ch0(37)


Compare fractions.

0.4 Fractions

Arrange the fractions , and in ascending

order of value.21

32

83

122121

21

，

8382

32

，

3833

83

Arrange these in ascending order of value, we have , i.e. .

24

16 ,

24

12 ,

24

9

3

2 ,

2

1 ,

8

3

12

24

16

24

9

24

The L.C.M. of 2, 3 and 8 is 24.

∴ The fractions , and are in ascending order of value.3

2

2

183

Key Concept 0.4.4

Index

1A_Ch0(38)

0.4 Fractions

Arrange the fractions , and in descending order of value.52

21

43

102101

21

，

5453

43

，

4542

52

Arrange these in descending order of value, we have ,i.e. .

20

8 ,

20

10 ,

20

15

5

2 ,

2

1 ,

4

3

10

20

15

20

8

20

The L.C.M. of 2, 4 and 5 is 20.

∴ The fractions , and are in descending order of value.52

21

43

Key Concept 0.4.4

Arithmetic Operations with Fractions

Index

1A_Ch0(39)

E)

0.4 Fractions

1. When adding or subtracting fractions, we often start

by changing the denominators of these fractions into

the same numbers first.

Example

Arithmetic Operations with Fractions

Index

1A_Ch0(40)

E)

0.4 Fractions

2. When we multiply or divide fractions, we often

change the mixed numbers into improper fractions

first and then look for factors to cancel from the

numerators and denominators.

Example

Index 0.4

Index

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0.4 Fractions

Calculate .83

21

43

83

21

43

83

84

86

8346

87

Index

1A_Ch0(42)


Expressions involving fractions.

0.4 Fractions

Calculate . 53

21

1

53

21

1 106

105

1010

106510

109

Index

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0.4 Fractions

Calculate . 31

154

241

4

3

11

5

42

4

14

3

4

5

14

4

17

6080

60168

60255

6080168255

60

167

6047

2 Key Concept 0.4.5

125

8

Index

1A_Ch0(44)

0.4 Fractions

(a)53

183

58

83

53

(b)52

285

128

310

Calculate :

(a) (b)53

183

52

28

2

3

31

3

Index

1A_Ch0(45)



0.4 Fractions

Calculate . 32

31

41

32

2

32

31

41

32

2 23

31

41

38

2

1

21

32

634

61

Index

1A_Ch0(46)


Everyday applications.

0.4 Fractions

Mr Chan earns $15 000 a month. If he spends of his in

come and saves the rest, how much does he save in a mon

th?

54

The amount he saves = $15 000 )54

1(

= $15 00051

= $3 000

Key Concept 0.4.6

Choosing Appropriate Measuring Tool

Index

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A)


‧ When we measure a quantity, we have to choose an

appropriate measuring tool to achieve a particular

purpose.

Example

Index 0.5

Index

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Which of the following is an appropriate measuring tool for

measuring the length of a monitor screen?

A. Mechanical scale

B. Thermometer

C. Ruler

D. Syringe

C


Key Concept 0.5.1

Choosing Appropriate Unit for Measurement

Index

1A_Ch0(49)

B)


‧ When we measure a quantity, we have to choose an

appropriate unit so that other people can understand

the result easily.

Example

Index 0.5

What unit and quantity would you use to tell others

(a) for how long has Aaron sung this song?

(b) how much does the fish weigh?

Index

Key Concept 0.5.2

1A_Ch0(50)


(a) Aaron has sung this song for 3 min , rather than 180 s

or 0.05 h.

(b) The weight of the fish is 1.5 kg , rather than 1 500 g or

0.001 5 tonne.

Documents

1A_Ch0(1). 0.3Multiples and Factors A Multiples B Factors C Prime Numbers and Prime Factors Index 1A_Ch0(2) D Index Notation