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1A_Ch1(1)
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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.
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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.
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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
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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
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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
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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 (+)=
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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
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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)
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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
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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.
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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?
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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
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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.
(–)
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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
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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
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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
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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
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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
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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
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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
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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
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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
+ (+) = + – (–) = +
+ (–) = – – (+) = –
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(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
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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–( )
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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
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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
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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
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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 .× ÷
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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
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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
(–)
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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
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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
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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
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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
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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 )
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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 )
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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.
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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
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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
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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
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