1A_Ch1(1)
1A_Ch1(2)
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
1.2 Addition and Subtraction of
Directed Numbers
1A_Ch1(3)
A Addition of Directed Numbers on a
Vertical Number Line
B Subtraction of Directed Numbers on a
Number Line
C Addition and Subtraction of Directed
Numbers Using a Calculator
D Addition and Subtraction of Directed
Numbers by Removing Brackets
Index
1A_Ch1(4)
1.3 Multiplication and Division of
Directed Numbers
A Multiplication of Directed Numbers
B Division of Directed Numbers
C Multiplication and Division of Directed
Numbers Using a Calculator
D Mixed Operations of Directed Numbers
Using a Calculator
Index
1.1 The Concept and Applications of Directed Numbers
1A_Ch1(5)
 Example
A)
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 1.1
Index
1.1 The Concept and Applications of Directed Numbers
1A_Ch1(6)
Answer the following questions. Use positive numbers to represent
increases in temperature and negative numbers to represent
decreases in temperature.
(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
Index
1.1 The Concept and Applications of Directed Numbers
1A_Ch1(7)
 Example
B)
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.
3. On a horizontal number line, the
values of the directed numbers
increase from left to right.
increasing
2. On a vertical number line, the values of
the directed numbers increase from
bottom to top.
increasing
 Index 1.1
Index
1.1 The Concept and Applications of Directed Numbers
1A_Ch1(8)
Arrange the following numbers in descending order and mark
them on the number line below.
+3, –2, +5, +10, –3, 0
+10, +5, +3, 0, –2, –3
–4 –3 –2 –1
0
+1
+3 +4 +5 +6
+10
Index
1.1 The Concept and Applications of Directed Numbers
1A_Ch1(9)
On the horizontal number line given below, find the directed
numbers represented by the letters A, B, C, D and E.
–8 –7 A B –4 –3 –2 C D +1 +2 +3 E +5
A=
–6
B=
–5
C=
–1
D=
0
E = +4
 Key Concept 1.1.2
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(10)
 Example
A) 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 1.2
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(11)
Find the sum of each of the following.
(a) (–2) + 5
(b) (–3) + 5
(c) (–4) + 5
(a)
(b)
(c)
+3
+2
+1
0
–1
–2
–3
–4
–5
+(+5)
(–2) + 5
= +3
+3
+2
+1
0
–1
–2
–3
–4
–5
+(+5)
(–3) + 5
= +2
+3
+2
+1
0
–1
–2
–3
–4
–5
+(+5)
(–4) + 5
= +1
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(12)
Find the sum of each of the following.
(a) 0 + (–4)
(b) 1 + (–4)
(c) –1 + (–4)
(a)
(b)
(c)
+1
0
–1
–2
–3
–4
–5
+(–4)
0 + (–4) = –4
+1
0
–1
–2
–3
–4
–5
+(–4)
1 + (–4) = –3
+1
0
–1
–2
–3
–4
–5
+(–4)
–1 + (–4) = –5
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(13)
Use a vertical number line to find the sum of each of the
following.
(a) (–4) + 5
(b) (–7) + 2
(c) 7 + (–3)
(d)
(a)
(b)
+3
+2
+1
0
–1
–2
–3
–4
(–4) + 5
= +1
+(+5)
4 + (–5)
0
–1
–2
–3
–4
–5
–6
–7
(–7) + 2
= –5
+(+2)
Index
1.2 Addition and Subtraction of Directed Numbers
(c)
+7
+6
+5
+4
+3
+2
+1
0
–1
–2
(d)
+(–3)
7 + (–3) = +4
1A_Ch1(14)
+4
+3
+(–5)
+2
+1
0
–1
–2
4 + (–5) = –1
Fulfill Exercise Objective
 Addition and subtraction using a number line.
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(15)
With the help of a vertical number line, find the sum of
(+1) + (–4) + (+5).
With the help of the vertical number line,
(+1) + (–4) = –3
∴
(+1) + (–4) + (+5)
= (–3) + (+5)
= +2
+2
+1
0
–1
–2
–3
+(–4)
Fulfill Exercise Objective
 Addition and subtraction using a number line.
Index
1.2 Addition and Subtraction of Directed Numbers
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.
The amount that John had after borrowing money
= $[(–5) + (–7)]
= – $12
1A_Ch1(16)
+2
+1
0
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
–12
–13
+(–7)
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(17)
 Back to Question
The amount that he has now
= $[(–12) + 13]
= +$1
+(+13)
+2
+1
0
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
–11
–12
–13
Fulfill Exercise Objective
 Real-life applications.
 Key Concept 1.2.1
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(18)
B) 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
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(19)
 Example
B) Subtraction of Directed Numbers on a Number Line
Note : In general,
Subtract (+)
=
Add (–)
Subtract (–)
=
Add (+)
 Index 1.2
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(20)
Find the difference of each of the following.
(a) 1 – (+3)
(b) 1 – (–3)
(c) (–1) – (–3)
(a)
(b)
(c)
+3
+2
+1
0
–1
–2
–3
–(+3)
1 – (+3) = –2
+4
+3
+2
+1
0
–1
–2
–(–3)
1 – (–3) = +4
+3
+2
+1
0
–1
–2
–3
–(–3)
(–1) – (–3) = +2
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(21)
Use a vertical number line to find the difference of each
of the following.
(a) 4 – (+5)
(b) (–4) – (+6)
(c) 2 – (–3)
(d)
(a)
(b)
+4
+3
+2
+1
0
–1
–2
–3
4 – (+5)
= –1
–(+5)
(–6) – (–4)
0
–1
–2
–3
–4
–5
–6
–7
–8
–9
–10
(–4) – (+6)
= –10
–(+6)
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(22)
 Back to Question
(c)
+6
+5
+4
+3
+2
+1
0
–1
(d)
–(–3)
2 – (–3) = +5
+1
0
–1
–2
–3
–4
–5
–6
–(–4)
(–6) – (–4) = –2
Fulfill Exercise Objective
 Addition and subtraction using a number line.
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?
The amount by which Jane has more than May
= $[(+3) – (+7)]
= –$4
i.e. Jane has $4 less than May.
Fulfill Exercise Objective
+4
+3
+2
+1
0
–1
–2
–3
–4
–5
–(+7)
 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)
 Back to Question
(a) The expected drop in temperature
= [5 – (–3)] °C
= 8°C
+9
+8
+7
+6
+5
+4
+3
+2
+1
0
–(–3)
(b) The actual drop in temperature
= [8 – (+2)] °C
= 6°C
Fulfill Exercise Objective
 Real-life applications.
+9
+8
+7
+6
+5
–(+2)
 Key Concept 1.2.2
Index
1.2 Addition and Subtraction of Directed Numbers
C)
1A_Ch1(26)
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.
2. To input a negative number, first press the key (–) ,
then press the key(s) corresponding to its numerical
value.
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(27)
 Example
C) Use a calculator to express the following directed numbers.
Directed number
256
Keying Sequence
2
5
6
–1 820
(–)
1
8
2
– 1
3
(–)
1
ab c
3
 Index 1.2
0
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)
(d)  1  ( 2 )
3
5
(c) –5.2 + 9.3
(a) –21 – (–60)
Keying
Sequence
(–)
2
1
–
(–)
6
0
EXE
Answer
39.
∴ –21 – (–60) = 39
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(29)
 Back to Question
(b) 34 – (+16)
Keying Sequence
34
–
16
∴ 34 – (+16) = 18
EXE
Answer
18.
(c) –5.2 + 9.3
Keying Sequence
∴ –5.2 + 9.3 =
4.1
(–)
5.2
+
9.3
EXE
Answer
4.1
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(30)
 Back to Question
1
3
2
5
(d)   ( )
Keying
Sequence
(–)
1
ab c
3
+
(–)
2
ab c
5
EXE
1
2
11
∴   ( ) = 
3
5
15
Answer
-11 15
Fulfill Exercise Objective
 Addition and subtraction using a calculator.
 Key Concept 1.2.3
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(31)
 Example
D) Addition and Subtraction of Directed Numbers by
Removing Brackets
‧ Rules for removing brackets attached to directed numbers
+ (+) = +
– (–) = +
+ (–) = –
– (+) = –
 Index 1.2
Index
1.2 Addition and Subtraction of Directed Numbers
1A_Ch1(32)
Find the values of the following.
(a) 14 + (+25)
(b) –14 + (–25)
(c) 14 – (+25)
(d) –14 – (–25)
(a) 14 + (+25) = 14 + 25
(b) –14 + (–25) = –14 – 25
= 39
(c) 14 – (+25) = 14 – 25
= –11
= –39
(d) –14 – (–25) = –14 + 25
= 11
 Key Concept 1.2.4
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(33)
 Example
A)
Multiplication 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)
 Index 1.3
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(34)
Find the value of each of the following.
(a) 4 × 3
(b) (–4) × 3
(c) 4 × (–3)
(a) 4 x 3 = 12
(b) (–4) × 3 = –12
(c) 4 × (–3) = –12
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(35)
Find the value of each of the following.
(a) (–1) × (–5)
(a) (–1) × (–5) = 5
(b) (–2) × (–5)
+ (+) = + – (–) = +
+ (–) = – – (+) = –
(b) (–2) × (–5) = 10
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(36)
Find the value of each of the following.
(a) (+9) × (–6)
(b) (–5) × (–7) × (–2)
(a) (+9) × (–6) = –(9 × 6)
= –54
(b) (–5) × (–7) × (–2) = +(5 × 7) × (–2)
= (+35) × (–2)
= –(35 × 2)
Fulfill Exercise Objective
 Multiplication and division
without using a calculator.
= –70
 Key Concept 1.3.1
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(37)
 Example
B)
Division of Directed Numbers
‧
For positive numbers +a, +b and negative numbers
–a, –b,
(+a)
a
= +( )
(+b)
b
(–a)
a
= –( )
(+b)
b
(–a)
a
= +( )
(–b)
b
(+a)
a
= –( )
(–b)
b
 Index 1.3
Index
1.3 Multiplication and Division of Directed Numbers
Find the value of each of the following.
 18
 18
 18
(a)
(b)
(c)
3
3
3
1A_Ch1(38)
 18
(d)
3
 18
18 

(a)
=    = +6
3
3
 18
18
(b)
=    = –6
3
3
 18
18 

(c)
=    = +6
3
3
 18
18
(d)
=    = –6
3
3
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(39)
Find the value of each of the following.
(a) (+42) ÷ (–7)
(b) (–48) ÷ (+6)
(c) (–57) ÷ (–3)
(d) (+12) ÷ (+2) ÷ (–3)
(a) (+42) ÷ (–7) = –(42 ÷ 7)
= –6
(b) (–48) ÷ (+6) = –(48 ÷ 6)
= –8
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(40)
 Back to Question
(c) (–57) ÷ (–3) = +(57 ÷ 3)
= +19
(d) (+12) ÷ (+2) ÷ (–3) = +(12 ÷ 2) ÷ (–3)
= (+6) ÷ (–3)
= –(6 ÷ 3)
= –2
Fulfill Exercise Objective
 Multiplication and division without using a calculator.
 Key Concept 1.3.2
Index
1.3 Multiplication and Division of Directed Numbers
C)
1A_Ch1(41)
Multiplication and Division of Directed Numbers
Using a Calculator
‧
Calculator can be used to multiply and divide directed
numbers by pressing the buttons
×
and
÷ .
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(42)
 Example
C) Use a calculator to evaluate each of the following.
Keying Sequence
Expression
10 × 11
10
×
11
EXE
(–5) × 3
(–)
5
×
3
EXE
20 ÷ (–4)
20
÷
(–)
4
EXE
 Index 1.3
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(43)
Use a calculator to evaluate each of the following.
(a) (–14) × 12 ÷ (–8)
(c)
(b) (–50) × 9 ÷ 15 × 0
(37)(4.9)
(0.25)(1.85)
(a) (–14) × 12 ÷ (–8)
Keying
Sequence
(–)
14
×
12
÷
(–)
8
EXE
∴ (–14) × 12 ÷ (–8) =
Answer
21.
21
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(44)
 Back to Question
(b) (–50) × 9 ÷ 15 × 0
Keying
Sequence
(–)
50
×
9
15
×
0
EXE
÷
Answer
∴ (–50) × 9 ÷ 15 × 0 =
0.
0
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(45)
 Back to Question
(c)
(37)(4.9)
(0.25)(1.85)
Keying
Sequence
(–)
37
×
0.25
÷
(–)
(37)(4.9)
∴
= 392
(0.25)(1.85)
Fulfill Exercise Objective
 Multiplication and division using a calculator.
4.9
÷
1.85 EXE
Answer
392.
 Key Concept 1.3.3
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(46)
 Example
D)
Mixed Operations of Directed Numbers Using a
Calculator
‧
Calculator can be used to evaluate an expression which
may involve addition, subtraction, multiplication and
division.
 Index 1.3
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(47)
Use a calculator to evaluate each of the following.
(a) 14 ÷ (3 + 4)
(b) (–3) × (5 + 2)
(a) 14 ÷ (3 + 4)
Keying
Sequence
14
÷
(
3
+
4
)
EXE
∴ 14 ÷ (3 + 4) =
2
Answer
2.
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(48)
 Back to Question
(b) (–3) × (5 + 2)
Keying
Sequence
(–)
3
×
(
+
2
)
EXE
5
Answer
–21.
∴ (–3) × (5 + 2) = –21
Index
1.3 Multiplication and Division of Directed Numbers
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)
Keying
Sequence
0
×
(–)
15
÷
10
+
5
)
EXE
∴ 0 × (–15) ÷ (10 + 5) =
0
Answer
(
0.
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(50)
 Back to Question
(b) (–28) × 7 ÷ [13 + (–13)]
Keying
Sequence
(–)
28
×
7
÷
(
13
+
(–)
13
)
EXE
Answer
MATH ERROR
∴ (–28) × 7 ÷ [13 + (–13)] is meaningless.
Fulfill Exercise Objective
 Mixed operations using a calculator.
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(51)
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.
(a) If Siu Ming answered all the questions in the test
and got 6 correct answers, find his final score.  Soln
(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?
 Soln
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(52)
 Back to Question
(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
Index
1.3 Multiplication and Division of Directed Numbers
1A_Ch1(53)
 Back to Question
(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)
Fulfill Exercise Objective
 Real-life applications.
12 

=  
2
= 6
 Key Concept 1.3.4
Index