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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