1A_Ch1(1)

1.1 The Concept and Applications of Directed Numbers

A The Applications of Directed

Numbers

B Ordering of Directed Numbers

on the Number Line

Index

1A_Ch1(2)

1.2 Addition and Subtraction of Directed Numbers

Index

1A_Ch1(3)

A Addition of Directed Numbers on a Vertical Number Line

Subtraction of Directed Numbers on a Number Line

B

Addition and Subtraction of Directed Numbers Using a Calculator

C

Addition and Subtraction of Directed Numbers by Removing Brackets

D

1.3 Multiplication and Division of Directed Numbers

Index

1A_Ch1(4)

A Multiplication of Directed Numbers

Division of Directed NumbersB

Multiplication and Division of Directed Numbers Using a Calculator

C

Mixed Operations of Directed Numbers Using a Calculator

D

The Applications of Directed Numbers

1. A number which carries a positive (+) sign or a

negative (–) sign is called a directed number.

2. The ‘+’ sign attached to a positive number can be

omitted but a negative number must carry the ‘–’

sign.

Index

A)

1A_Ch1(5)1.1 The Concept and Applications of Directed Numbers

Index 1.1

Example

Answer the following questions. Use positive numbers to represent

increases in temperature and negative numbers to represent

decreases in temperature.

Index

1A_Ch1(6)1.1 The Concept and Applications of Directed Numbers

(a) An increase of 5°C in temperature

(b) A decrease of 2°C in temperature

(c) An increase of 8°C in temperature

(a) +5°C

(b) –2°C

(c) +8°C Key Concept 1.1.1

Ordering of Directed Numbers on the Number Line

1. A number line is a straight line with directed numbers

marked on it in a certain order.

Index

B)

1A_Ch1(7)1.1 The Concept and Applications of Directed Numbers

2. On a vertical number line, the values of

the directed numbers increase from

bottom to top.

3. On a horizontal number line, the

values of the directed numbers

increase from left to right.

Index 1.1

Example

increasing

inc

rea

sin

g

Arrange the following numbers in descending order and mark

them on the number line below.

Index

1A_Ch1(8)1.1 The Concept and Applications of Directed Numbers

+3, –2, +5, +10, –3, 0

–4 –1 +1 +4 +6–3 0 +3 +5 +10–2

+10, +5, +3, 0, –2, –3

On the horizontal number line given below, find the directed

numbers represented by the letters A, B, C, D and E.

Index

1A_Ch1(9)1.1 The Concept and Applications of Directed Numbers

–8 –7 A B –4 –3 –2 C D +1 +2 +3 E +5

A = B =

C = D =

–6

0 E = +4–1

–5

Key Concept 1.1.2

Addition of Directed Numbers on a Vertical Number Line

On a vertical number line,

1. if we add a positive number ‘+a’ to a given number,

we move up ‘a’ units from the given number to

obtain the sum;

2. if we add a negative number ‘–b’ to a given number,

we move down ‘b’ units from the given number to

obtain the sum.

Index

A)

1A_Ch1(10)

1.2 Addition and Subtraction of Directed Numbers

Index 1.2

Example

Find the sum of each of the following.

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(11)

(a) (–2) + 5 (b) (–3) + 5 (c) (–4) + 5

(a) +3+2+1

0–1–2–3–4–5

+(+5)

(–2) + 5

= +3

(b) +3+2+1

0–1–2–3–4–5

+(+5)

(c) +3+2+1

0–1–2–3–4–5

+(+5)

(–3) + 5

= +2

(–4) + 5

= +1

Find the sum of each of the following.

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(12)

(a) 0 + (–4) (b) 1 + (–4) (c) –1 + (–4)

(a) +10

–1–2–3–4–5

+(–4)

0 + (–4) = –4

(b) +10

–1–2–3–4–5

1 + (–4) = –3

(c) +10

–1–2–3–4–5

–1 + (–4) = –5

+(–4)

+(–4)

Index

Use a vertical number line to find the sum of each of the

following.

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(13)

(a) (–4) + 5 (b) (–7) + 2

(c) 7 + (–3) (d) 4 + (–5)

(a)+3+2+1

0–1–2–3–4

+(+5)

(–4) + 5

= +1

(b)0

–1–2–3–4–5–6–7

+(+2)

(–7) + 2

= –5

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(14)

+(–3)

7 + (–3) = +4

(c)+7+6+5+4+3+2+1

0–1–2

+(–5)

4 + (–5) = –1

(d)+4

+3

+2

+1

0

–1

–2

Fulfill Exercise Objective

Addition and subtraction using a number line.

Index

With the help of a vertical number line, find the sum of

(+1) + (–4) + (+5).

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(15)

+2+1

0–1–2–3

+(–4)

With the help of the vertical number line,

(+1) + (–4) = –3

∴ (+1) + (–4) + (+5)

= (–3) + (+5)

= +2

Fulfill Exercise Objective

Addition and subtraction using a number line.

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(16)

Yesterday John borrowed $5 from his

classmate and $7 from his brother. This

morning his mother gave him $13. Use

directed numbers to find out how much John

has after he pays back the borrowed money.

+2+1

0–1–2–3–4–5–6–7–8 –9

–10–11–12–13

The amount that John had after borrowing money

= $[(–5) + (–7)]

+(–7)

= – $12

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(17)

+2+1

0–1–2–3–4–5–6–7–8 –9

–10–11–12–13

The amount that he has now

= $[(–12) + 13]

= +$1

Fulfill Exercise Objective

Real-life applications.

+(+13)

Key Concept 1.2.1

Back to Question

Subtraction of Directed Numbers on a Number Line

On a vertical number line,

1. if we subtract a positive number ‘+a’ to a given number,

we move down ‘a’ units from the given number to

obtain the difference;

2. if we subtract a negative number ‘–b’ to a given

number, we move up ‘b’ units from the given number to

obtain the difference.

Index

B)

1A_Ch1(18)

1.2 Addition and Subtraction of Directed Numbers

Subtraction of Directed Numbers on a Number Line

Note : In general,

Index

B)

1A_Ch1(19)

1.2 Addition and Subtraction of Directed Numbers

Example

Index 1.2

Subtract (+) Add (–)=

Subtract (–) Add (+)=

Index

1A_Ch1(20)

1.2 Addition and Subtraction of Directed Numbers

Find the difference of each of the following.

(a) 1 – (+3) (b) 1 – (–3) (c) (–1) – (–3)

(a) +3+2+1

0–1–2–3

–(+3)

1 – (+3) = –2

(b) +4+3+2+1

0–1–2

–(–3)

1 – (–3) = +4

(c) +3+2+1

0–1–2–3

–(–3)

(–1) – (–3) = +2

Index

Use a vertical number line to find the difference of each

of the following.

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(21)

(a) 4 – (+5) (b) (–4) –

(+6)

(c) 2 – (–3) (d) (–6) – (–

4)(a)+4+3+2+1

0–1–2–3

–(+5)

4 – (+5)

= –1

(–4) – (+6)

= –10

(b) 0–1–2–3–4–5–6–7–8–9

–10

–(+6)

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(22)

(c)+6+5+4+3+2+1

0–1

–(–3)

2 – (–3) = +5

(d)+1

0–1–2–3–4–5–6

–(–4)

(–6) – (–4) = –2

Fulfill Exercise Objective

Addition and subtraction using a number line.

Back to Question

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(23)

Jane has $3 more than Winnie while Winnie has

$7 less than May. Does Jane have more or less

money than May? By how much?

+4+3+2+1

0–1–2–3–4–5

The amount by which Jane has more than May

= $[(+3) – (+7)]

= –$4

i.e. Jane has $4 less than May.–(+7)

Fulfill Exercise Objective

Real-life applications.

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(24)

On a certain day in Beijing, the temperature in the morning

was 5°C. It was expected to drop to –3°C at midnight.

(a) By how many degrees was the temperature expected

to drop?

(b) If the temperature at midnight was 2°C higher than

expected, what was the actual drop in temperature?

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(25)

(a) The expected drop in temperature

= [5 – (–3)] °C

+9+8+7+6+5+4+3+2+1

0

–(–3)

= 8°C

(b) The actual drop in temperature

= [8 – (+2)] °C

= 6°C

+9+8+7+6+5

–(+2)

Fulfill Exercise Objective

Real-life applications.

Key Concept 1.2.2

Back to Question

Addition and Subtraction of Directed Numbers Using

a Calculator

1. To input a positive number, we just press the key

corresponding to the numerical value of the number.

Index

C)

1A_Ch1(26)

1.2 Addition and Subtraction of Directed Numbers

2. To input a negative number, first press the key ,

then press the key(s) corresponding to its numerical

value.

(–)

Index

1A_Ch1(27)

1.2 Addition and Subtraction of Directed Numbers

C) Use a calculator to express the following directed numbers.

Directed number Keying Sequence

256 2 5 6

–1 820 (–) 1 8 2 0

13

– (–) 1 3abc

Example

Index 1.2

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(28)

Use a calculator to evaluate each of the following.

(a) –21 – (–60) (b) 34 – (+16)

(c) –5.2 + 9.3 (d) )52

(31

(a) –21 – (–60)

KeyingSequence

(–)

(–)

–

EXE Answer 39.

–∴ 21 – (–60) = 39

2 1

6 0

Keying Sequence

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(29)

Keying Sequence 34 –

Answer 18. ∴ 34 – (+16) = 18

(b) 34 – (+16)

EXE

(–) 5.2 +

–∴ 5.2 + 9.3 = 4.1

(c) –5.2 + 9.3

EXE9.3

Answer 4.1

Back to Question

16

Index

1.2 Addition and Subtraction of Directed Numbers 1A_Ch1(30)

(d) )5

2(

3

1

KeyingSequence

(–)

(–)

1

2

+

EXE

abc 3

5abc

Answer -11 15∴ )

52

(31 =

1511

Fulfill Exercise Objective

Addition and subtraction using a

calculator.

Key Concept 1.2.3

Back to Question

Addition and Subtraction of Directed Numbers by

Removing Brackets

‧ Rules for removing brackets attached to directed numbers

Index

D)

1A_Ch1(31)

1.2 Addition and Subtraction of Directed Numbers

+ (+) = +

+ (–) = –

– (–) = +

– (+) = –

Example

Index 1.2

Index

1A_Ch1(32)

1.2 Addition and Subtraction of Directed Numbers

Find the values of the following.

(a) 14 + (+25) (b) –14 + (–25)

(c) 14 – (+25) (d) –14 – (–25)

(a) 14 + (+25) = 14 + 25

= 39

(b) –14 + (–25) = –14 – 25

= –39

(c) 14 – (+25) = 14 – 25

= –11

(d) –14 – (–25) = –14 + 25

= 11 Key Concept 1.2.4

Multiplication of Directed Numbers

Index

A)

1A_Ch1(33)

1.3 Multiplication and Division of Directed Numbers

‧ For positive numbers +a, +b and negative numbers

–a, –b,

(+a) × (+b) = +(a × b)

(–a) × (–b) = +(a × b)

(–a) × (+b) = –(a × b)

(+a) × (–b) = –(a × b)

Example

Index 1.3

Find the value of each of the following.

(a) 4 x 3 =

Index

1A_Ch1(34)

1.3 Multiplication and Division of Directed Numbers

(a) 4 × 3 (b) (–4) × 3 (c) 4 × (–3)

(b) (–4) × 3 =

(c) 4 × (–3) =

12

–12

–12

Find the value of each of the following.

(a) (–1) × (–5) =

Index

1A_Ch1(35)

1.3 Multiplication and Division of Directed Numbers

(a) (–1) × (–5) (b) (–2) × (–5)

(b) (–2) × (–5) =

5

10

+ (+) = + – (–) = +

+ (–) = – – (+) = –

(a) (+9) × (–6)

Index

1A_Ch1(36)

1.3 Multiplication and Division of Directed Numbers

(a) (+9) × (–6) (b) (–5) × (–7) ×

(–2)

Find the value of each of the following.

= –(9 × 6)

= –54

(b) (–5) × (–7) × (–2)= +(5 × 7) × (–2)

= (+35) × (–2)

= –(35 × 2)

= –70Fulfill Exercise Objective

Multiplication and division

without using a calculator. Key Concept 1.3.1

Division of Directed Numbers

Index

B)

1A_Ch1(37)

1.3 Multiplication and Division of Directed Numbers

‧ For positive numbers +a, +b and negative numbers

–a, –b,

Example

Index 1.3

=(+a)

(+b)=

(–a)

(+b)

=(–a)

(–b)=

(+a)

(–b)

a

b+( )

a

b–( )

a

b+( )

a

b–( )

Find the value of each of the following.

Index

1A_Ch1(38)

1.3 Multiplication and Division of Directed Numbers

(a)3

18

(b)3

18

(c)3

18

(d)3

18

(a)3

18

318

= = +6 (b)3

18

318

= = –6

(c)3

18

318

= = +6 (d)3

18

318

= = –6

Index

1A_Ch1(39)

1.3 Multiplication and Division of Directed Numbers

Find the value of each of the following.

(a) (+42) ÷ (–7)

(c) (–57) ÷ (–3)

(b) (–48) ÷ (+6)

(d) (+12) ÷ (+2) ÷ (–3)

(a) (+42) ÷ (–7)= –(42 ÷ 7)

= –6

(b) (–48) ÷ (+6) = –(48 ÷ 6)

= –8

Index

1A_Ch1(40)

1.3 Multiplication and Division of Directed Numbers

Fulfill Exercise Objective

Multiplication and division without using a calculator.

(c) (–57) ÷ (–3) = +(57 ÷ 3)

= +19

(d) (+12) ÷ (+2) ÷ (–3)= +(12 ÷ 2) ÷ (–3)

= (+6) ÷ (–3)

= –(6 ÷ 3)

= –2

Key Concept 1.3.2

Back to Question

Multiplication and Division of Directed Numbers

Using a Calculator

Index

C)

1A_Ch1(41)

1.3 Multiplication and Division of Directed Numbers

‧ Calculator can be used to multiply and divide directed

numbers by pressing the buttons and .× ÷

Index

1A_Ch1(42)

10

C) Use a calculator to evaluate each of the following.

Expression

10 × 11 × 11

(–)(–5) × 3 5 × 3

÷ 4

1.3 Multiplication and Division of Directed Numbers

Keying Sequence

EXE

EXE

20 ÷ (–4) (–)20 EXE

Example

Index 1.3

Index

1A_Ch1(43)

Use a calculator to evaluate each of the following.

(a) (–14) × 12 ÷ (–8) (b) (–50) × 9 ÷ 15 × 0

(c))85.1)(25.0(

)9.4)(37(

(a) (–14) × 12 ÷ (–8)

1.3 Multiplication and Division of Directed Numbers

KeyingSequence

(–)

÷

14 12

8

×

EXE Answer 21.

∴ (–14) × 12 ÷ (–8) = 21

(–)

Keying Sequence

Index

1A_Ch1(44)

(b) (–50) × 9 ÷ 15 × 0

1.3 Multiplication and Division of Directed Numbers

(–) 50

∴ (–50) × 9 ÷ 15 × 0 = 0

EXE

9

Answer 0.

÷×

15 × 0

Back to Question

Index

1A_Ch1(45)

1.3 Multiplication and Division of Directed Numbers

)85.1)(25.0()9.4)(37(

(c)

392Answer 392.

Keying Sequence

(–) 37

EXE

4.9 ÷×

0.25 ÷ (–) 1.85

∴ = )85.1)(25.0()9.4)(37(

Fulfill Exercise Objective

Multiplication and division using a calculator. Key Concept 1.3.3

Back to Question

Mixed Operations of Directed Numbers Using a

Calculator

Index

D)

1A_Ch1(46)

1.3 Multiplication and Division of Directed Numbers

‧ Calculator can be used to evaluate an expression which

may involve addition, subtraction, multiplication and

division.

Example

Index 1.3

Index

1A_Ch1(47)

Use a calculator to evaluate each of the following.

(a) 14 ÷ (3 + 4) (b) (–3) × (5 + 2)

1.3 Multiplication and Division of Directed Numbers

(a) 14 ÷ (3 + 4)

Answer 2.

Keying Sequence

14

EXE

÷ ( 3

+ 4 )

∴ 14 ÷ (3 + 4) = 2

Index

1A_Ch1(48)

(b) (–3) × (5 + 2)

1.3 Multiplication and Division of Directed Numbers

Answer –21.

∴ (–3) × (5 + 2) = –21

Keying Sequence

(–) 3

EXE

(× 5

+ 2 )

Back to Question

Index

1A_Ch1(49)

Use a calculator to evaluate each of the following.

(a) 0 × (–15) ÷ (10 + 5) (b) (–28) × 7 ÷ [13 + (–13)]

(a) 0 × (–15) ÷ (10 + 5)

1.3 Multiplication and Division of Directed Numbers

∴ 0 × (–15) ÷ (10 + 5) = 0Answer 0.

Keying Sequence

(–)0

EXE

15× ÷ (

10 + 5 )

Keying Sequence

Index

1A_Ch1(50)

(b) (–28) × 7 ÷ [13 + (–13)]

1.3 Multiplication and Division of Directed Numbers

∴ (–28) × 7 ÷ [13 + (–13)] is meaningless.

Answer MATH ERROR

(–)

EXE

28 × ÷ (

+ )

7

13 (–) 13

Fulfill Exercise Objective

Mixed operations using a calculator.

Back to Question

Index

1A_Ch1(51)

1.3 Multiplication and Division of Directed Numbers

(a) If Siu Ming answered all the questions in the test

and got 6 correct answers, find his final score.

(b) If the final score of Tai Kwong was –9 marks and

he only got 1 correct answer, how many of his

answers were wrong?

There are 10 multiple choice questions in

a test. 3 marks will be given for a correct

answer, –2 marks for a wrong answer

and no marks if the question is

unanswered.

Soln

Soln

Index

1A_Ch1(52)

1.3 Multiplication and Division of Directed Numbers

(a) The total score obtained for the 6 correct answers

= 6 × 3 marks

= 18 marks

The total score obtained for the wrong answers

= (10 – 6) × (–2) marks

= –8 marks

∴ Siu Ming’s final score = [18 + (–8)] marks

= 10 marks

Back to Question

Index

1A_Ch1(53)

1.3 Multiplication and Division of Directed Numbers

(b) The score obtained for 1 correct answer

= 1 × 3 marks

= 3 marks

Since Tai Kwong’s final score was –9 marks, the total

score obtained for his wrong answers

= [(–9) – 3] marks

= –12 marks

∴ The number of wrong answers = (–12) ÷ (–2)

=

212

= 6

Fulfill Exercise Objective

Real-life applications. Key Concept 1.3.4

Back to Question