Download 1 - SAP Education

ABOUT THIS BOOK
As a seasoned secondary level Mathematics teacher who has taught for more than
10 years, I have been exposed to, and therefore am very familiar with, the changes
in the secondary mathematics curriculum over the years. It is evident that our
curriculum focuses on 4 fundamental and important themes: Algebra, Numbers,
Geometry and Statistics.
While schools differ in their pace and the sequence in which the topics are being
taught, their objective remains largely similar: to ensure their students are
ready to take on the demands of the upper secondary mathematics
syllabus and be ready for national examinations.
It is this same desire to help my students that this series of books is conceived.
I have written three books to cover all the topics in the entire lower secondary
mathematics course: Algebra, Numbers, Functions and Graphs, Geometry
and Statistics.
The examples and questions are paced to help students understand
concepts, recognize common steps, see through typical trick questions
and apply mathematical knowledge taught and learned.
While the progressive practices ensure familiarity and exposure to a
wide variety of question types, there are opportunities to exercise higher
order thinking through the challenging questions provided in every topic.
Detailed solutions are included to guide and help students through
critical thinking processes.
While my students will certainly benefit more from my personal teaching of these
questions, I am sure many others will also benefit from working through the same
questions systematically and studying the solutions diligently.
Alex Lim
CONTENTS
Numbers, Functions and Graphs
1
Primes and Prime Factorisation
1
2
Highest Common Factor (HCF) and
Lowest Common Multiple (LCM)
9
3
Squares, Cubes, Square roots and Cube roots
27
4
Operations on Integers
38
5
Operations on Rational Numbers
62
6
Real numbers, Using calculator and Number line
83
7
Approximation and Estimation
98
8
Ratio, Rate and Speed
112
9
Percentage
131
10 Functions and Graphs (I)
144
11
200
Functions and Graphs (II)
12 Map Scales
242
13 Direct and Inverse Proportion
260
14 Set Language and Notation
277
Solutions
S1 - S87
1
Primes and Prime
Factorisation
A prime number is an integer bigger than 1 which has only two
factors, 1 and the number itself.
Examples : 2, 3, 5, 7, 11, 13, 17, 19, …
A composite number is an integer bigger than 1 which has more
than 2 factors.
Examples : 4, 6, 8, 9, 10, 12, 14, 15, 16, …
Example 1
Express 42 as a product of its prime factors.
Solutions:
Method 1 (Factor tree)
Method 2 (Ladder method)
42
2
6
×
7
×
3
×
2
3
7
Divide by the
smallest prime
number, in order.
7
42
21
7
1
Thus 42 = 2 × 3 × 7
Example 2
Express 105 as a product of its prime factors.
Solutions:
Method 1 (Factor tree)
105
5
35
×
3
×
7
×
3
Thus 105 = 3 × 5 × 7
© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre
1
Topic 1
Method 2 (Ladder method)
3
5
7
105
35
7
1
Let’s Practise Prime factors
1
Express the following numbers as a product of their prime factors.
a
45
b 74
c
110
d 128
e
146
f
g
320
h 4345
i
7115
j
9600
l
50 400
k 11 350
© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre
2
Topic 1
258
Let’s Practise Mixed practice
2
Find the largest multiple of 25 that is less than 427.
3
Given that x is a prime number and y is a composite number, state “True” or “False”
for each statement.
(a) 7y is a prime number.
(b) x + x + x is a composite number.
4
Find the sum of the first three prime numbers that end with the digit 1.
5
Find a possible pair of prime numbers such that their sum is 90.
© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre
3
Topic 1
6
Find a possible pair of prime numbers such that their difference is 2.
7
Find the smallest number that is divisible by all prime numbers between 1 and 11.
8
Express, in index notation, 150 as a product of its prime factors, using a factor tree.
9
Express, in index notation, 156 as a product of its prime factors, using a factor tree.
10
Express, in index notation, 432 as a product of its prime factors, using a factor tree.
© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre
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Topic 1
11
Express, in index notation, 896 as a product of its prime factors, using ‘ladder
method’.
12
Express, in index notation, 2376 as a product of its prime factors, using ‘ladder
method’.
13
Express, in index notation, 3432 as a product of its prime factors, using ‘ladder
method.
14
Find the prime factorisation of 245.
15
List the prime factors of 2100.
© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre
5
Topic 1