Transcript Document

1A_Ch0(1)
0.3 Multiples and Factors
1A_Ch0(2)
A Multiples
B Factors
C Prime Numbers and
Prime Factors
D Index Notation
Index
1A_Ch0(3)
0.4 Fractions
A Introduction
B Proper and Improper
Fractions
C Equal Fractions
D Comparing Fractions
E Arithmetic Operations with
Fractions
Index
1A_Ch0(4)
0.5
Choosing Appropriate
Measuring Tools and Units
A Choosing Appropriate
Measuring Tool
B Choosing Appropriate Unit
for Measurement
Index
0.1 Numbers
1A_Ch0(5)
 Example
Numbers
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.
Index
0.1 Numbers
1A_Ch0(6)
From numbers 0 to 8, list all the
(a) natural numbers,
(b) whole numbers,
(c) even numbers,
(d) odd 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
 Key Concept 0.1.1
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)
1A_Ch0(7)
 Example
The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
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.
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, x, ÷)
Find
(a) 13 + 8
(b) 92 – 9
(c) 36 × 4
(d) 56 ÷ 8
1A_Ch0(8)
(a) 13 + 8 = 21
(b) 92 – 9 = 83
(c) 36 × 4 = 144
(d) 56 ÷ 8 = 7
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(9)
Find 5 + 4 × 2.
5+4×2 =5+8
= 13
Fulfill Exercise Objective
 +, –, × and ÷
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(10)
Find 250 × 5 ÷ 25.
250 × 5 ÷ 25 = 1 250 ÷ 25
= 50
Fulfill Exercise Objective
 +, –, × and ÷
 Key Concept 0.2.1
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(11)
 Example
Brackets
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 { }.
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
Find
(a) (20 – 8) ÷ (5 – 3)
1A_Ch0(12)
(b) 3 × (8 + 3) – (4 – 2)
(a) (20 – 8) ÷ (5 – 3) = 12 ÷ 2
= 6
(b) 3 × (8 + 3) – (4 – 2) = 3 × 11 – 2
= 33 – 2
= 31
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(13)
Find 6 × {100 ÷ [(6 – 2) × 5] – 5}.
6 × {100 ÷ [(6 – 2) × 5] – 5}
= 6 × {100 ÷ [4 × 5] – 5}
= 6 × {100 ÷ 20 – 5}
= 6 × {5 – 5}
Fulfill Exercise Objective
=6×0
 Expressions involving brackets.
= 0
 Key Concept 0.2.2
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(14)
 Example
Several verbs used to describe the arithmetic operations
Terms and descriptions
Arithmetic operations
Add 5 to 2,
or 2 plus 5
2+5
Subtract 6 from 10,
or 10 minus 6
10 – 6
Multiply 7 by 3,
or 7 times 3
7×3
Divide 20 by 4,
or 20 is divided by 4
20 ÷ 4
Index
0.2 The Four Fundamental Arithmetic Operations (+, –, ×, ÷)
1A_Ch0(15)
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
Index
0.3 Multiples and Factors
1A_Ch0(16)
 Example
A)
Multiples
‧
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.
 Index 0.3
Index
0.3 Multiples and Factors
1A_Ch0(17)
List the first four multiples of 5.
5 × 1 = 5, 5 × 2 = 10, 5 × 3 = 15, 5 × 4 = 20
5
10
15
20
5×1
5×2
5×3
5×4
∴
∴ The first four multiples of 5 are 5, 10, 15 and 20.
Index
0.3 Multiples and Factors
1A_Ch0(18)
Write down the first 5 multiples of each of the following numbers.
Multiples
6
6 × 1 , 6 × 62 ,, 12
6 × ,3 18
, 6, ×24
4 ,, 630× 5
9
9 × 1 , 9 × 92 ,, 18
9 × ,3 27
, 9, ×36
4 ,, 945× 5
12
12 × 1 , 1212× 2, 24
, 12, ×363 , 48
12 ×, 460, 12 × 5
15
15 × 1 , 1515× 2, 30
, 15, ×453 , 60
15 ×, 475, 15 × 5
 Key Concept 0.3.1
Index
0.3 Multiples and Factors
B)
1A_Ch0(19)
Factors
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.
Index
0.3 Multiples and Factors
1A_Ch0(20)
 Example
B)
Note :
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
Index
0.3 Multiples and Factors
1A_Ch0(21)
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(22)
Write down all the factors of each of the following numbers.
Factors
4
1 × 4 , 2 × 2 , 4 × 11 , 2 , 4
12
1 × 12 , 2 × 16 , 32× ,4 3, 4, ×43 ,, 12
1
6 ,× 12
22
1 × 22 , 2 × 11 , 111, ×22 ,, 22
11 ×, 122
32
1 × 32 , 2 × 116, ,24 ,× 48 , 88× ,4 16
, 16, ×32
2 , 32 × 1
 Key Concept 0.3.2
Index
0.3 Multiples and Factors
1A_Ch0(23)
 Example
C)
Prime Numbers and Prime Factors
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.
 Index 0.3
Index
0.3 Multiples and Factors
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
Index
0.3 Multiples and Factors
1A_Ch0(25)
Express 120 as a product of prime factors.
120 = 2 × 60
= 2 × 2 × 30
= 2 × 2 × 2 × 15
= 2×2×2×3×5
2
120
2 60
2 30
3 15
5
Fulfill Exercise Objective
 Prime factors.
 Key Concept 0.3.3
Index
0.3 Multiples and Factors
1A_Ch0(26)
 Example
D)
Index Notation
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.
 Index 0.3
Index
0.3 Multiples and Factors
1A_Ch0(27)
Using index notation, express each of the following numbers as a
product of prime factors.
(a) 50
(b) 132
(c) 180
(d) 225
(a) 50 = 2 × 52
(b) 132 = 22 × 3 × 11
(c) 180 = 22 × 32 × 5
(d) 225 = 32 × 52
 Key Concept 0.3.4
Index
0.4 Fractions
1A_Ch0(28)
 Example
A)
Introduction
‧
1
3
or is called a fraction in which 4
4
4
is called the denominator and 1 or 3 is called the
The number
numerator of the fraction.
 Index 0.4
Index
0.4 Fractions
1A_Ch0(29)
1
of the triangles below. How many triangles have you
4
shaded?
Shade
8 triangles are shaded.
 Key Concept 0.4.1
Index
0.4 Fractions
1A_Ch0(30)
 Example
B)
Proper and Improper Fractions
1. Examples of proper fractions are :
1
,
2
1
4
and
2
,
5
where the numerator of the fraction is smaller than
the denominator.
2. Examples of improper fractions are : 3 , 4 and 12 ,
3
3
11
where the numerator of the fraction is greater than or
equal to the denominator.
3. Examples of mixed numbers are : 1
1
3
,1
1
and 4
7
where the fraction is written as a sum of a whole
1
,
2
number and a proper fraction.
 Index 0.4
Index
0.4 Fractions
1A_Ch0(31)
(a) Which of the following are improper fractions?
6 12 7 4 10 5 2 19
,
,
, ,
, ,
,
7 5 11 3 12 8 13 12
(b) Express the improper fractions in (a) as mixed numbers.
(a) Improper fractions :
(b)
12 4 19
, ,
5 3 12
12
4
can be written as 2 2 ,
can be written as 1 1 ,
5
5 3
3
19
 Key Concept 0.4.2
can be written as 1 7 .
12
12
Index
0.4 Fractions
1A_Ch0(32)
 Example
C)
Equal 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.
1 1 2
2 2 2
4 4 2
8

 
 

2
2
2 2 2
4 4
8 8
16
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.
 Index 0.4
Index
0.4 Fractions
1A_Ch0(33)
Match the equal fractions.
3
5
12
42
2 2  6 12


7 7  6 42
2
7
21
36
7 7  3 21


12 12  3 36
5
6
20
24
5 5  4 20


6 6  4 24
7
12
9
15
3 3 3 9


5 5  3 15
Index
0.4 Fractions
1A_Ch0(34)
42
Reduce
to its simplest form.
105
6 2
42
105
15

2
5
5
Fulfill Exercise Objective
 Reduce fractions.
 Key Concept 0.4.3
Index
0.4 Fractions
1A_Ch0(35)
 Example
D)
Comparing 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.
 Index 0.4
Index
0.4 Fractions
1A_Ch0(36)
1
1
Compare
and .
3
5
The L.C.M. of 3 and 5 is 15.
1 1 5 5
1 1 3 3



， 
3 3  5 15
5 5  3 15
∴
∴
5
3
is greater than
.
15
15
1
1
is greater than
.
3
5
Index
0.4 Fractions
1A_Ch0(37)
Arrange the fractions 1 , 2 and 3 in ascending
2 3
8
order of value.
9
1 112 12 2 2  8 16
3 3 3




， 
， 
2 2 12 24 3 3  8 24
8 8  3 24
Arrange these in ascending order of value, we have
∴ The fractions
The L.C.M.
of 2, 3 and 8
is 24.
9 12 16
, i.e. 3 , 1 , 2 .
, ,
24 24 24
8 2 3
2
3 1
,
and
are in ascending order of value.
3
8 2
Fulfill Exercise Objective
 Key Concept 0.4.4
 Compare fractions.
Index
0.4 Fractions
1A_Ch0(38)
Arrange the fractions 1 , 3 and 2 in descending order of value.
2 4
5
1 110 10 3 3  5 15
2 2 4 8




， 
， 
2 2 10 20 4 4  5 20
5 5  4 20
Arrange these in descending order of value, we have
∴ The fractions
The L.C.M.
of 2, 4 and 5
is 20.
15 10 8
3 1 2
,i.e. , , .
, ,
20 20 20
4 2 5
3 1
2
,
and
are in descending order of value.
4 2
5
 Key Concept 0.4.4
Index
0.4 Fractions
1A_Ch0(39)
 Example
E)
Arithmetic Operations with Fractions
1. When adding or subtracting fractions, we often start
by changing the denominators of these fractions into
the same numbers first.
Index
0.4 Fractions
1A_Ch0(40)
 Example
E)
Arithmetic Operations with 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.
 Index 0.4
Index
0.4 Fractions
1A_Ch0(41)
3 1 3
Calculate   .
4 2 8
3 1 3 6 4 3
    
4 2 8 8 8 8
6 43

8
7

8
Index
0.4 Fractions
1A_Ch0(42)
Calculate 1  1  3 .
2 5
1
1 3 10 5 6
   
2 5 10 10 10

10  5  6
10
9

10
Fulfill Exercise Objective
 Expressions involving fractions.
Index
0.4 Fractions
1A_Ch0(43)
Calculate 4 1  2 4  11 .
4
5
3
1
4 1 17 14 4
4  2 1   
4
5
3 4 5 3
255 168 80



60 60 60
255  168  80
60
167

60
47
 2
60

Fulfill Exercise Objective
 Expressions involving fractions.
 Key Concept 0.4.5
Index
0.4 Fractions
1A_Ch0(44)
Calculate :
(a) 3  1 3
8 5
(a)
3 3 3 8
1  
8 5 8 5
3

5
(b) 8  2
2
5
(b) 8  2
2
12
 8
5
5
2
5
 8
12 3
10

3
1
 3
3
Index
0.4 Fractions
1A_Ch0(45)
Calculate 2 2  1  1  2 .
3 4 3 3
2
2 1 1 2 8 1 1 3
2       
3 4 3 3 3 413 2
2 1
 
3 2
43

6
1

6
Fulfill Exercise Objective
 Expressions involving fractions.
Index
0.4 Fractions
1A_Ch0(46)
4
Mr Chan earns $15 000 a month. If he spends
of his
5
income and saves the rest, how much does he save in a
month?
The amount he saves = $15 000  (1  4 )
5
1
= $15 000 
5
= $3 000
Fulfill Exercise Objective
 Everyday applications.
 Key Concept 0.4.6
Index
0.5 Choosing Appropriate Measuring Tools and Units
1A_Ch0(47)
 Example
A)
Choosing Appropriate Measuring Tool
‧
When we measure a quantity, we have to choose an
appropriate measuring tool to achieve a particular
purpose.
 Index 0.5
Index
0.5 Choosing Appropriate Measuring Tools and Units
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
 Key Concept 0.5.1
Index
0.5 Choosing Appropriate Measuring Tools and Units
1A_Ch0(49)
 Example
B)
Choosing Appropriate Unit for Measurement
‧
When we measure a quantity, we have to choose an
appropriate unit so that other people can understand
the result easily.
 Index 0.5
Index
0.5 Choosing Appropriate Measuring Tools and Units
1A_Ch0(50)
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?
(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.
 Key Concept 0.5.2
Index