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

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


(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)


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


+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


–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)


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


(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.


(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


+(–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).


+2+1

0–1–2–3

+(–4)

With the help of the vertical number line,

(+1) + (–4) = –3

∴ (+1) + (–4) + (+5)

= (–3) + (+5)

= +2



Index


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


+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


Real-life applications.

+(+13)

Key Concept 1.2.1

Back to Question


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)



Note : In general,

Index

B)

1A_Ch1(19)


Example

Index 1.2

Subtract (+) Add (–)=

Subtract (–) Add (+)=

Index

1A_Ch1(20)


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.


(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


(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



Back to Question

Index


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)



Index


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


(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)



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)


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

then press the key(s) corresponding to its numerical

value.

(–)

Index

1A_Ch1(27)


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


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


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


(d) )5

2(

3

1

KeyingSequence

(–)

(–)

1

2

+

EXE

abc 3

5abc

Answer -11 15∴ )

52

(31 =

1511


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)


+ (+) = +

+ (–) = –

– (–) = +

– (+) = –

Example

Index 1.2

Index

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

= 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)


‧ 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)


(a) 4 × 3 (b) (–4) × 3 (c) 4 × (–3)

(b) (–4) × 3 =

(c) 4 × (–3) =

12

–12

–12


(a) (–1) × (–5) =

Index

1A_Ch1(35)


(a) (–1) × (–5) (b) (–2) × (–5)

(b) (–2) × (–5) =

5

10

+ (+) = + – (–) = +

+ (–) = – – (+) = –

(a) (+9) × (–6)

Index

1A_Ch1(36)


(a) (+9) × (–6) (b) (–5) × (–7) ×

(–2)


= –(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)


‧ 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–( )


Index

1A_Ch1(38)


(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)



(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)



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)


‧ 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


Keying Sequence

EXE

EXE

20 ÷ (–4) (–)20 EXE

Example

Index 1.3

Index

1A_Ch1(43)


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

(c))85.1)(25.0(

)9.4)(37(

(a) (–14) × 12 ÷ (–8)


KeyingSequence

(–)

÷

14 12

8

×

EXE Answer 21.

∴ (–14) × 12 ÷ (–8) = 21

(–)

Keying Sequence

Index

1A_Ch1(44)

(b) (–50) × 9 ÷ 15 × 0


(–) 50

∴ (–50) × 9 ÷ 15 × 0 = 0

EXE

9

Answer 0.

÷×

15 × 0

Back to Question

Index

1A_Ch1(45)


)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(


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)


‧ Calculator can be used to evaluate an expression which

may involve addition, subtraction, multiplication and

division.

Example

Index 1.3

Index

1A_Ch1(47)


(a) 14 ÷ (3 + 4) (b) (–3) × (5 + 2)


(a) 14 ÷ (3 + 4)

Answer 2.

Keying Sequence

14

EXE

÷ ( 3

+ 4 )

∴ 14 ÷ (3 + 4) = 2

Index

1A_Ch1(48)

(b) (–3) × (5 + 2)


Answer –21.

∴ (–3) × (5 + 2) = –21

Keying Sequence

(–) 3

EXE

(× 5

+ 2 )

Back to Question

Index

1A_Ch1(49)


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

(a) 0 × (–15) ÷ (10 + 5)


∴ 0 × (–15) ÷ (10 + 5) = 0Answer 0.

Keying Sequence

(–)0

EXE

15× ÷ (

10 + 5 )

Keying Sequence

Index

1A_Ch1(50)

(b) (–28) × 7 ÷ [13 + (–13)]


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

Answer MATH ERROR

(–)

EXE

28 × ÷ (

+ )

7

13 (–) 13


Mixed operations using a calculator.

Back to Question

Index

1A_Ch1(51)


(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)


(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)


(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


Real-life applications. Key Concept 1.3.4

Back to Question

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

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