1a_ch0(1). 0.3multiples and factors a multiples b factors c prime numbers and prime factors index...
TRANSCRIPT
1A_Ch0(1)
0.3 Multiples and Factors
A Multiples
B Factors
C Prime Numbers and
Prime Factors
Index
1A_Ch0(2)
D Index Notation
0.4 Fractions
A Introduction
B Proper and Improper
Fractions
C Equal Fractions
Index
1A_Ch0(3)
D Comparing Fractions
E Arithmetic Operations with
Fractions
0.5 Choosing Appropriate Measuring Tools and Units
A Choosing Appropriate
Measuring Tool
B Choosing Appropriate Unit for
Measurement
Index
1A_Ch0(4)
Numbers
0.1 Numbers
Index
Example
1A_Ch0(5)
1. Natural numbers are the numbers we used in counting
and they are 1, 2, 3, 4, 5, 6, ... .
2. The first five whole numbers are 0, 1, 2, 3 and 4.
3. The whole numbers 0, 2, 4, 6 and 8 are examples of
even numbers. They are divisible by the number 2.
4. The whole numbers 1, 3, 5, 7, 9 are examples of odd
numbers. When these numbers are divided by 2, there
is always a remainder 1.
From numbers 0 to 8, list all the
(a) natural numbers, (b) whole
numbers,
(c) even numbers, (d) odd numbers.
Key Concept 0.1.1
Index
1A_Ch0(6)0.1 Numbers
(a) Natural numbers : 1, 2, 3, 4, 5, 6, 7, 8
(b) Whole numbers : 0, 1, 2, 3, 4, 5, 6, 7, 8
(c) Even numbers : 2, 4, 6, 8
(d) Odd numbers : 1, 3, 5, 7
The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)
Index
Example
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1. In 4 + 7 = 11, the number 11 is called the sum of 4 and 7.
2. In 11 – 7 = 4, the number 4 is called the difference
between 11 and 7.
3. In 4 × 7 = 28, the number 28 is called the product of 4
and 7.
4. In 28 ÷ 7 = 4, the number 4 is called the quotient and 7
is called the divisor.
Find (a) 13 + 8 (b) 92 – 9
(c) 36 × 4 (d) 56 ÷ 8
Index
1A_Ch0(8)
(a) 13 + 8 = 21
(b) 92 – 9 = 83
(c) 36 × 4 = 144
(d) 56 ÷ 8 = 7
0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)
Find 5 + 4 × 2.
5 + 4 × 2
Index
1A_Ch0(9)
= 5 + 8
= 13
Fulfill Exercise Objective
+, –, × and ÷
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Find 250 × 5 ÷
25.
250 × 5 ÷ 25
Index
1A_Ch0(10)
= 1 250 ÷ 25
= 50
Fulfill Exercise Objective
+, –, × and ÷
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Key Concept 0.2.1
Brackets
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Index
Example
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1. Brackets are used to indicate the priority of operations.
2. We should always do the operations within the brackets
first.
3. If more than one pair of brackets are used, the general
rule is to use ( ) for the first arithmetic operation, then
[ ] and then { }.
Find (a) (20 – 8) ÷ (5 – 3) (b) 3 × (8 + 3) – (4 – 2)
Index
1A_Ch0(12)
(b) 3 × (8 + 3) – (4 – 2) = 3 × 11 – 2
= 33 – 2
= 31
(a) (20 – 8) ÷ (5 – 3) = 12 ÷ 2
= 6
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Find 6 × {100 ÷ [(6 – 2) × 5] – 5}.
6 × {100 ÷ [(6 – 2) × 5] – 5}
Index
1A_Ch0(13)
= 6 × {100 ÷ [4 × 5] – 5}
= 6 × {100 ÷ 20 – 5}
= 6 × {5 – 5}
= 6 × 0
= 0
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Fulfill Exercise Objective
Expressions involving brackets.
Key Concept 0.2.2
Several verbs used to describe the arithmetic operations
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Index
Example
1A_Ch0(14)
Divide 20 by 4,or 20 is divided by 4
Multiply 7 by 3,or 7 times 3
Subtract 6 from 10,or 10 minus 6
Add 5 to 2,or 2 plus 5
Arithmetic operationsTerms and descriptions
2 + 5
10 – 6
7 × 3
20 ÷ 4
Index
1A_Ch0(15)
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Write down the result of each of the following.
(a) Find the difference when 7 is subtracted from 18.
(b) Find the product of 6 and 12.
(c) When 100 is divided by 5, find the quotient.
(a) 18 – 7 = 11
(b) 6 × 12 = 72
(c) 100 ÷ 5 = 20 Key Concept 0.2.3
Multiples
0.3 Multiples and Factors
Index
Example
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‧ When we multiply a number by the natural numbers
1, 2, 3, 4 and so on, we get multiples of that number.
E.g. The first 4 multiples of 6 are : 6, 12, 18 and 24.
A)
Index 0.3
List the first four multiples of 5.
Index
0.3 Multiples and Factors 1A_Ch0(17)
5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20
∴
∴ The first four multiples of 5 are 5, 10, 15 and 20.
5 10 1515 20
5 × 1 5 × 2 5 × 3 5 × 4
Index
1A_Ch0(18)
Write down the first 5 multiples of each of the following numbers.
0.3 Multiples and Factors
15
12
9
6
Multiples
6 × 1 , 6 × 2 , 6 × 3 , 6 × 4 , 6 × 5
9 × 1 , 9 × 2 , 9 × 3 , 9 × 4 , 9 × 5
12 × 1 , 12 × 2 , 12 × 3 , 12 × 4 , 12 × 5
15 × 1 , 15 × 2 , 15 × 3 , 15 × 4 , 15 × 5
6 , 12 , 18 , 24 , 30
9 , 18 , 27 , 36 , 45
12 , 24 , 36 , 48 , 60
15 , 30 , 45 , 60 , 75
Key Concept 0.3.1
Factors
0.3 Multiples and Factors
Index
1A_Ch0(19)
1. When a given number is expressed as a product of
two or more natural numbers, then each of these
natural numbers is a factor of the given number.
2. In general, the method of division can be used to test
whether a number is a factor of another number.
E.g. Since 48 is divisible by 4, 4 is a factor of 48.
B)
Note :
0.3 Multiples and Factors
Index
Example
1A_Ch0(20)
B)
i. All numbers are divisible by 1, therefore 1 is a
factor of any number.
ii. All even numbers are divisible by 2, therefore 2 is
a factor of any even number.
iii. Any number (except 0) is divisible by itself,
therefore any number is a factor of itself.
Index 0.3
Determine whether 5 is a factor of 30.
Since 30 ÷ 5 = 6, we say 30 is divisible by 5.
Therefore 5 is a factor of 30.
Index
0.3 Multiples and Factors 1A_Ch0(21)
Index
1A_Ch0(22)
Write down all the factors of each of the following numbers.
32
22
12
4
Factors
1 × 4 , 2 × 2 , 4 × 1
1 × 12 , 2 × 6 , 3 × 4 , 4 × 3 , 12 × 1
1 × 22 , 2 × 11 , 11 × 2 , 22 × 1
1 × 32 , 2 × 16 , 4 × 8 , 8 × 4 , 16 × 2 , 32 × 1
1 , 2 , 4
1 , 2 , 3 , 4 , 6 , 12
1 , 2 , 11 , 22
1 , 2 , 4 , 8 , 16 , 32
0.3 Multiples and Factors
Key Concept 0.3.2
Prime Numbers and Prime Factors
0.3 Multiples and Factors
Index
1A_Ch0(23)
1. A prime number is a natural number (other than 1)
which is not divisible by any natural number except
1 and itself.
2. Consider the factors of 24, 2 and 3 are prime
numbers and they are therefore called prime factors
of 24.
C)
Example
Index 0.3
Index
1A_Ch0(24)
List all the prime numbers
(a) from 80 to 150, (b) from 200 to
250.
(a) The prime numbers from 80 to 150 :
83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149
(b) The prime numbers from 200 to 250 :
211, 223, 227, 229, 233, 239, 241
0.3 Multiples and Factors
Express 120 as a product of prime factors.
120
Index
1A_Ch0(25)
Fulfill Exercise Objective
Prime factors.
2 120
2 60
2 30
3 15
5
0.3 Multiples and Factors
= 2 × 60
= 2 × 2 ×
30= 2 × 2 × 2 × 15
= 2 × 2 × 2 × 3 × 5
Key Concept 0.3.3
Index Notation
0.3 Multiples and Factors
Index
1A_Ch0(26)
1. When a number is multiplied by itself several times,
we can express the product using the index notation.
2. Consider the index notation 72, 73, 74 and 75. The
number 7 is called the base and the numbers 2, 3, 4
and 5 are each called the index.
D)
Example
Index 0.3
Index
1A_Ch0(27)
Using index notation, express each of the following numbers as a
product of prime factors.
0.3 Multiples and Factors
(a) 50 (b) 132
(c) 180 (d) 225
(a) 50 (b) 132= 2 × 52 = 22 × 3 × 11
(c) 180 (d) 225= 22 × 32 × 5 = 32 × 52
Key Concept 0.3.4
Introduction
0.4 Fractions
Index
Example
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A)
‧ The number or is called a fraction in which 4
is called the denominator and 1 or 3 is called the
numerator of the fraction.
41
43
Index 0.4
Index
1A_Ch0(29)
8 triangles are shaded.
0.4 Fractions
Shade of the triangles below. How many triangles have you
shaded?
41
Key Concept 0.4.1
Proper and Improper Fractions
Index
1A_Ch0(30)
B)
0.4 Fractions
1. Examples of proper fractions are : , and , w
here the numerator of the fraction is smaller than the
denominator.
2
1
4
1
5
2
2. Examples of improper fractions are : , and ,
where the numerator of the fraction is greater than or
equal to the denominator.
3
3
3
4
11
12
3. Examples of mixed numbers are : , and ,
where the fraction is written as a sum of a whole nu
mber and a proper fraction.
3
11
7
11
2
14
Example
Index 0.4
Index
1A_Ch0(31)
(a) Improper fractions :
0.4 Fractions
(a) Which of the following are improper fractions?
1219
, 132
, 85
, 1210
, 34
, 117
, 5
12 ,
76
(b) Express the improper fractions in (a) as mixed numbers.
1219
, 34
, 5
12
(b)5
12can be written as ,
52
234
can be written as , 31
1
1219
can be written as . 127
1 Key Concept 0.4.2
Equal Fractions
Index
1A_Ch0(32)
C)
0.4 Fractions
1. When we multiply the numerator and the denominator
of a fraction by the same non-zero number, the value
of the faction remains the same.
E.g. 168
8 4
84
4 2
42
2 1
21
2
222
22
2. When the numerator and the denominator of a
fraction have no common factor except 1, the fraction
is said to be in its simplest form.
Example
Index 0.4
Index
1A_Ch0(33)
0.4 Fractions
Match the equal fractions.
53
72
65
127
4212
3621
2420
159
15
9
35
33
5
3
24
20
46
45
6
5
36
21
312
37
12
7
42
12
67
62
7
2
Index
1A_Ch0(34)
Fulfill Exercise Objective
Reduce fractions.
0.4 Fractions
Reduce to its simplest form.10542
10542
6
15
2
5
52
Key Concept 0.4.3
Comparing Fractions
Index
1A_Ch0(35)
D)
0.4 Fractions
‧ Fractions can be compared when they are expressed
with the same denominators. To do this, we need to
know the L.C.M. of their original denominators.
Example
Index 0.4
Index
1A_Ch0(36)
The L.C.M. of 3 and 5 is 15.
0.4 Fractions
Compare and .51
31
153
3531
51
155
5351
31
,
∴
155
is greater than .153
∴31
is greater than .51
Index
1A_Ch0(37)
Fulfill Exercise Objective
Compare fractions.
0.4 Fractions
Arrange the fractions , and in ascending
order of value.21
32
83
122121
21
,
8382
32
,
3833
83
Arrange these in ascending order of value, we have , i.e. .
24
16 ,
24
12 ,
24
9
3
2 ,
2
1 ,
8
3
12
24
16
24
9
24
The L.C.M. of 2, 3 and 8 is 24.
∴ The fractions , and are in ascending order of value.3
2
2
183
Key Concept 0.4.4
Index
1A_Ch0(38)
0.4 Fractions
Arrange the fractions , and in descending order of value.52
21
43
102101
21
,
5453
43
,
4542
52
Arrange these in descending order of value, we have ,i.e. .
20
8 ,
20
10 ,
20
15
5
2 ,
2
1 ,
4
3
10
20
15
20
8
20
The L.C.M. of 2, 4 and 5 is 20.
∴ The fractions , and are in descending order of value.52
21
43
Key Concept 0.4.4
Arithmetic Operations with Fractions
Index
1A_Ch0(39)
E)
0.4 Fractions
1. When adding or subtracting fractions, we often start
by changing the denominators of these fractions into
the same numbers first.
Example
Arithmetic Operations with Fractions
Index
1A_Ch0(40)
E)
0.4 Fractions
2. When we multiply or divide fractions, we often
change the mixed numbers into improper fractions
first and then look for factors to cancel from the
numerators and denominators.
Example
Index 0.4
Index
1A_Ch0(41)
0.4 Fractions
Calculate .83
21
43
83
21
43
83
84
86
8346
87
Index
1A_Ch0(42)
Fulfill Exercise Objective
Expressions involving fractions.
0.4 Fractions
Calculate . 53
21
1
53
21
1 106
105
1010
106510
109
Index
1A_Ch0(43)
Fulfill Exercise Objective
Expressions involving fractions.
0.4 Fractions
Calculate . 31
154
241
4
3
11
5
42
4
14
3
4
5
14
4
17
6080
60168
60255
6080168255
60
167
6047
2 Key Concept 0.4.5
125
8
Index
1A_Ch0(44)
0.4 Fractions
(a)53
183
58
83
53
(b)52
285
128
310
Calculate :
(a) (b)53
183
52
28
2
3
31
3
Index
1A_Ch0(45)
Fulfill Exercise Objective
Expressions involving fractions.
0.4 Fractions
Calculate . 32
31
41
32
2
32
31
41
32
2 23
31
41
38
2
1
21
32
634
61
Index
1A_Ch0(46)
Fulfill Exercise Objective
Everyday applications.
0.4 Fractions
Mr Chan earns $15 000 a month. If he spends of his in
come and saves the rest, how much does he save in a mon
th?
54
The amount he saves = $15 000 )54
1(
= $15 00051
= $3 000
Key Concept 0.4.6
Choosing Appropriate Measuring Tool
Index
1A_Ch0(47)
A)
0.5 Choosing Appropriate Measuring Tools and Units
‧ When we measure a quantity, we have to choose an
appropriate measuring tool to achieve a particular
purpose.
Example
Index 0.5
Index
1A_Ch0(48)
Which of the following is an appropriate measuring tool for
measuring the length of a monitor screen?
A. Mechanical scale
B. Thermometer
C. Ruler
D. Syringe
C
0.5 Choosing Appropriate Measuring Tools and Units
Key Concept 0.5.1
Choosing Appropriate Unit for Measurement
Index
1A_Ch0(49)
B)
0.5 Choosing Appropriate Measuring Tools and Units
‧ When we measure a quantity, we have to choose an
appropriate unit so that other people can understand
the result easily.
Example
Index 0.5
What unit and quantity would you use to tell others
(a) for how long has Aaron sung this song?
(b) how much does the fish weigh?
Index
Key Concept 0.5.2
1A_Ch0(50)
0.5 Choosing Appropriate Measuring Tools and Units
(a) Aaron has sung this song for 3 min , rather than 180 s
or 0.05 h.
(b) The weight of the fish is 1.5 kg , rather than 1 500 g or
0.001 5 tonne.