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BCRB296038 ISBN 978-616-541-320-6
9 786165 413206
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© Pelangi Publishing (Thailand) Co., Ltd. 2019 All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means without permission of Pelangi Publishing (Thailand) Co., Ltd. 2019
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ISBN 978-616-541-320-6 First Published 2562
Contents Chapter 1 Factors
B Division of decimals.............................. 48 1
C Performing combined operations on decimals.......................................... 54
A Factors of a whole number................... 2
D Solving word problems involving combined operations on decimals........ 56
B Prime numbers and prime factors........ 7 C Factorization of a whole number.......... 9
Chapter 4
D Highest common factor (HCF).............. 12 E
Lowest common multiple (LCM)........... 16
F
Solving word problems involving HCF and LCM...................................... 20
Chapter 2 Fractions
Percentage
A Finding the percentage of a number out of another.......................... 62 B Finding profit or loss as a percentage........................................... 65 C Percentage change.............................. 73
25
D Solving word problems involving percentages.......................................... 76
A Comparing and ordering fractions........ 26 B Addition and subtraction of fractions.... 31
Chapter 5
C Combined operations of fractions......... 35 D Solving word problems involving fractions................................................ 38
Chapter 3 Decimals
61
Ratios and proportions
81
A Ratios .................................................. 82 B Proportions........................................... 87 C Solving word problems involving ratios and proportions........................... 91
44
A Relationship between fractions and decimals........................................ 45
II
Chapter 6 Volume
Chapter 9
94
Polygons
A Volumes of shapes made up of cuboids................................................. 95
A Interior angles of polygons................... 141 B Perimeter of a polygon......................... 149
B Solving word problems involving volumes of shapes made up of cuboids................................................. 99
Chapter 7 Circles
140
C Area of a polygon................................. 151 D Solving word problems involving perimeter and area of a polygon.......... 154
Chapter 10
103
A Parts of a circle..................................... 104
Three-dimensional shapes
B Drawing circles..................................... 106
A Types and properties of three-dimensional shapes.................... 160
C Circumference of a circle...................... 108
B Nets of 3-D shapes............................... 163
D Area of a circle..................................... 111 E
Chapter 11
Solving word problems involving circumference and area of a circle....... 114
Chapter 8 Triangles
159
Patterns
169
A Geometric patterns and number patterns................................................ 170
118
B Solving problems involving patterns............................................... 175
A Types and properties of triangles......... 119 B Drawing triangles.................................. 123
Chapter 12
C Interior angles of a triangle................... 128 D Perimeter of a triangle.......................... 130
Pie charts
E
Area of a triangle.................................. 132
A Reading pie charts............................... 180
F
Solving word problems involving perimeter and area of a triangle........... 135
III
179
Special features in this book
Provides practical activities to enhance students’ interest, knowledge and experience in learning mathematics.
Encourages students to recall and list down what they know about the topic.
Provides an activity that engages in the application of knowledge of scientists, mathematicians and engineers.
Provides the learning outcomes of the lesson and encourages students to share what they want to learn in the topic.
Challenges students with questions that promote higher thinking skills.
4C’s of the 21st Century 4C’s The Skills are Communication Sharing thoughts, questions, ideas and solutions.
Helps students to identify what they have learned at the end of the lesson.
Collaboration Working together to reach a goal. Putting talent, expertise and smarts to work. Provides exercises to reinforce students’ grasp of mathematical concepts.
Critical Thinking Looking at problems in a new way and linking learning across subjects and disciplines. Creativity Trying new approaches to get things done equals innovation and invention.
IV
1
Chapter
Factors Today, Irene and Jess played badminton in school. Irene plays badminton every 4 days and Jess every 5 days. When will they play badminton on the same day again?
What do I know about factors? 1. 2. 3. What else do I want to know about factors? 1. What are factors, prime numbers, prime factors and factorization? 2. How do I factorize whole numbers? 3. How do I find the highest common factor (HCF) and lowest common multiple (LCM) of a set of numbers? 4. How do I solve word problems involving HCF and LCM? 5.
A Factors of a whole number
Determining if a given number is a factor of a whole number Let’s examine these division problems.
12 ÷ 3 = 4 18 ÷ 9 = 2 15 ÷ 4 = 3 R 3
12 is divisible by 3, so 3 is a factor of 12. 18 is divisible by 9, so 9 is a factor of 18. 15 is indivisible by 4, so 4 is not a factor of 15.
A factor of a whole number is a number that can divide the whole number exactly, that is without leaving any remainder. Determine if 8 is a factor for these numbers. 48
71
96
Working: 48 ÷ 8 = 6 This is an exact division. Therefore, 8 is a factor of 48. 71 ÷ 8 = 8 R 7 This is not an exact division. Therefore, 8 is not a factor of 71. 96 ÷ 8 = 12 This is an exact division. Therefore, 8 is a factor of 96. Answer: 8 is a factor of 48 and 96.
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Listing the factors of a whole number List all the factors of 18.
18 ÷ 1 = 18
18 ÷ 2 = 9
18 ÷ 3 = 6
18 ÷ 6 = 3
18 ÷ 9 = 2
18 ÷ 18 = 1
18 is divisible by 1, 2, 3, 6, 9 and 18. Therefore, the factors of 18 are 1, 2, 3, 6, 9 and 18. List all the factors of 42. Working: 42 ÷ 1 = 42
42 ÷ 2 = 21
42 ÷ 3 = 14
42 ÷ 6 = 7
42 ÷ 7 = 6
42 ÷ 14 = 3
42 ÷ 21 = 2
42 ÷ 42 = 1
Answer: The factors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. List all the factors of 175. Working: 175 ÷ 1 = 175
175 ÷ 5 = 35
175 ÷ 7 = 25
175 ÷ 25 = 7
175 ÷ 35 = 5
175 ÷ 175 = 1
Answer: The factors of 175 are 1, 5, 7, 25, 35 and 175.
M AT H T I P S ✹ 1 is a factor of all whole numbers. ✹ Every number is a factor of itself.
Factors
3
Using factors to find products When solving a multiplication problem, we can use the factors of the numbers being multiplied and the commutative, associative and distributive properties to make the calculations easier.
Find the product of 12 × 15. Working: Method 1:
12 × 15 = 12 × (3 × 5)
= (12 × 3) × 5
= 36 × 5
= 180
Method 2:
12 × 15 = (6 × 2) × 15
= 6 × (2 × 15)
= 6 × 30
= 180
Method 3:
Here, 15 is rewritten as the multiplication of its factors.
12 × 15 = (6 × 2) × (3 × 5)
= (6 × 3) × (2 × 5)
= 18 × 10
= 180
Here, 12 is rewritten as the multiplication of its factors.
Here, 12 and 15 are rewritten as the multiplications of their factors.
Answer: 12 × 15 = 180
Find the product of 24 × 18 using the factors of 24 and 18.
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Using factors to find quotients When solving a division problem, we can use the factors of the divisor to make the calculations easier.
Find the quotient of 455 ÷ 35. Working: 35 = 5 × 7 5 455
7 9 1
1 3 Answer: 455 ÷ 35 = 13
Here, we find the factors of the divisor, 35. Then, we divide 455 by the factors continuously using short division. 455 ÷ 5 = 91 91 ÷ 7 = 13 The final answer is 13.
Find the quotient of 576 ÷ 12. Working: 12 = 6 × 2 6 576
2 9 6
4 8 Answer: 576 ÷ 12 = 48
Here, we find the factors of the divisor, 12. Then, we divide 576 by the factors continuously using short division. 576 ÷ 6 = 96 96 ÷ 2 = 48 The final answer is 48.
Always check your answer by multiplying the answer with the divisor.
Factors
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1. Tick (✓) if the given number is a factor of the number in the brackets. (a) 3 (26)
(b) 5 (140)
(c) 15 (135)
(d) 21 (105)
2. List all the factors of the following numbers. (a) 8 (b) 13 (c) 74 (d) 99 3. Find the products using the factors of the numbers being multiplied. (a) 14 × 36 =
(b) 150 × 48 =
Working: Working:
Answer: Answer: 4. Find the quotients using the factors of the divisors. (a) 168 ÷ 42 =
(b) 552 ÷ 24 =
Working: Working:
Answer: Answer:
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B Prime numbers and prime factors
Prime numbers
Each of the numbers, 2, 3 and 5 has only two factors, that is 1 and the number itself.
Factors of 2: 1 and 2 Factors of 3: 1 and 3 Factors of 5: 1 and 5
Therefore, 2, 3 and 5 are called prime numbers.
A prime number is a whole number that has only two factors, that is 1 and itself. Determine if these numbers are prime numbers.
6
7
14
23
Working: 1×6=6 2×3=6 The factors of 6 are 1, 2, 3 and 6. So, 6 is not a prime number. 1×7=7 The factors of 7 are 1 and 7. So, 7 is a prime number. 1 × 14 = 14 2 × 7 = 14 The factors of 14 are 1, 2, 7 and 14. So, 14 is not a prime number. 1 × 23 = 23 The factors of 23 are 1 and 23. So, 23 is a prime number. Answer: 7 and 23 are prime numbers.
M AT H T I P S ✹ 1 is not a prime number because it has only 1 factor. ✹ All prime numbers except 2 are odd numbers.
Factors
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Prime factors The factors of 18 are
1
2
3
6
9
18
Of the six factors, 2 and 3 are prime numbers. So, 2 and 3 are known as the prime factors of 18. If a factor of a whole number is also a prime number, it is called a prime factor of the whole number. List the prime factors of 75. Working: The factors of 75 are 1, 3 , 5 , 15, 25 and 75. Out of the six factors, 3 and 5 are prime factors. Answer: The prime factors of 75 are 3 and 5.
1. Circle the prime numbers.
15
16
21
23
24
31
40
53
69
77
83
97
2. List all the prime numbers between 1 and 20. 3. List the prime factors of the following numbers. (a) 46 (b) 62 (c) 98
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C Factorization of a whole number
Prime factorization
A whole number can be written as a product of its factors (not including 1 and itself). For example: 18 = 2 × 9
or
18 = 3 × 6
Some of these factors can be factorized further, e.g. 9 = 3 × 3, 6 = 2 × 3.
When the factors cannot be factorized further, we will be left with the prime factors: 18 = 2 × 3 × 3. When a whole number is written as a product of its prime factors, it is called the prime factorization of the whole number. Write down the prime factorization of 20. Working: 20 = 2 × 10 = 2 × 2 × 5 Or
20 = 4 × 5
= 2 × 2 × 5
10 is not a prime factor. All factors are prime factors. 4 is not a prime factor. All factors are prime factors.
So, the prime factorization of 20 is 2 × 2 × 5.
The above prime factorization can also be written using exponents : 20 = 22 × 5. 22 means 2 × 2 (2 multiplied by itself) and it is read as two to the power of two.
Factors
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A number that is multiplied by itself several times can be written in the exponential form as shown below : 3 × 3 × 3 = 33; 33 is read as three to the power of three. 3 × 3 × 3 × 3 = 34; 34 is read as three to the power of four.
Finding prime factorization (a) Finding prime factorization using division One method of finding the prime factorization of a whole number is by dividing the number repeatedly by prime numbers. Find the prime factorization of 48 using division. Working:
2 48 2 24 2 12 2 6 3
The division is started using the smallest prime number, in this case 2 as the divisor.
Answer: The prime factorization of 48 is 2 × 2 × 2 × 2 × 3 or 24 × 3. The following are the steps for finding the prime factorization of a whole number by division: 1. Divide the whole number by the smallest prime number that can divide it. 2. If the quotient is not a prime number, keep dividing by the smallest prime numbers until a quotient that is a prime number is obtained. 3. Write down the prime factorization as a multiplication sentence with every divisor and the last quotient as the prime factors.
Is it possible to prime factorize a prime number? Why?
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BCRB296038 ISBN 978-616-541-320-6
9 786165 413206
6