Grade 6 Unit 1 and 2

SARASAS AFFILIATED SCHOOLS
Mathematics Grade 6

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No part of this publication may be reproduced, stored in
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or otherwise, without the prior permission of the
publishers.

Foreword:
By Dr. Chamrat Nongmak

When talking about bilingual education in private elementary and secondary schools in Thailand,
discussion invariably tends to focus on Sarasas Affiliated Schools, a large group of private schools that was
formerly owned and administered by the late Mr. Peboon Yongkamol. Mr. Peboon, a well-known and
respected educator, was a leading pioneer of the bilingual education concept here in Thailand. These
schools provide a benchmark and a showcase for bilingual education in Thailand. Many school proprietors,
directors, and managers make it a point to visit Sarasas Affiliated Schools seeking advice and guidance
from a successful leader in the field of bilingual education.

Sarasas programs for teaching students both in English and in Thai have been carefully planned
and developed step by step. One of the toughest challenges Sarasas has had to face has been finding the
necessary texts and materials for teaching Thai subjects in English. This is because a bilingual school, like
any other Thai school, needs government approval to operate and must teach the curriculum mandated
by the Ministry of Education. Unfortunately, the materials needed for teaching Thai subjects in English
are not readily available. Publishers do little to support this need because the concept is still fairly new
and the demand is small. This has forced Sarasas to create its own texts and materials.

Sarasas has been more fortunate though, than most other schools struggling to develop their own
bilingual program. It already has bilingual education experience, a qualified staff and the resources
necessary to produce its own high-quality textbooks and materials. The staff, of course, is the most
important ingredient. Sarasas’ large, well-seasoned staff of highly competent, well qualified and
dedicated teaching professionals, both Thai and native speakers of English, gives it a unique advantage
over others attempting to produce these much-needed materials.

Even so, it has not been an easy task. The materials have been produced through the careful
process of research and development. Revision or total change is done when those materials are proven
to be inadequate. The people involved in developing these textbooks or workbooks are classroom
teachers. They are the ones who will have to teach with the same materials they have helped to create.
Knowing this gives them a greater incentive to do it right.

The teams of classroom teachers responsible for classroom material development operate under
the close supervision of the Sarasas Board of Executives.

One of the roles of the board is to assure proper quality control by collecting and carefully
evaluating the actual work done by students using the new materials. They make certain that the students
are able to understand the material and use them effectively. One of the things they check for, is whether
the level of English used in the books is appropriate for the students’ level of English. The board also makes
certain that the content of the material is in accordance with the approved syllabus. The Mathematics
book has been developed through this process.

This book is not only useful for teachers but also for parents and guardians. Its purpose is to
encourage and support students in their quest for knowledge and understanding of Mathematics, as well
as stimulate their interest and improve their skills. Each unit provides an introduction to the material, a
list of new words the student will encounter, problem solving examples to help the student better
understand how to do the work, oral exercises, and a review of what was learned.

This new Maths material will provide students with a greater international perspective and
understanding of Mathematics and Mathematics terminology which will benefit them greatly in later life.
This said, I would like to take this opportunity to thank those responsible for developing this material and
offer my sincere congratulations for a job well done.

Chamrat Nongmak, PhD.

Former Deputy Permanent Secretary
The Ministry of Education

PREFACE

This Mathematics textbook has been developed for bilingual learners based
on the B.E. 2560 (A.D. 2017) revised version of Thailand’s Basic Education Core
Curriculum B.E. 2551 (A.D. 2008). The content of this book follows the Basic
Mathematics for Grade 6 Book written by The Institute of Academic Development.

This textbook aims to enhance students’ abilities needed in the 21st century.
These include analytical skills, problem solving skills, creativity and collaboration
skills.

At the beginning of each unit of this textbook, a useful vocabulary list is
provided to learners. Examples demonstrate to students how to apply the
information they have learnt to solve related problems. Exercises and activities
follow that allow for immediate practice. At the end of each unit, a summary of the
key learning concepts along with a revision exercise is provided to help students
consolidate what they have learnt.

Author,
Sarasas Affiliated Schools

March, 2022











Unit 1 FACTORS AND MULTIPLES

Eratosthenes of Cyrene (276 BC - 195/194 BC)

Attributes factors and multiples was a Greek
polymath: a mathematician, geographer, poet,

astronomer, and music theorist. His work is comparable

to what is now known as the study of geography, and he introduced some of the
terminology still used today.

He is best known forbeing the first person to calculate the circumference of the Earth,
which he did by using the extensive survey results he could access in his role at the
Library; his calculation was remarkably accurate. He was also the first to calculate
the tilt of the Earth's axis, once again with remarkable accuracy. Additionally, he may
have accurately calculated the distance from the Earth to the Sun and invented the
leap day.

Mathematical Terms

composite number จำนวนตวั ประกอบ
divisible หำรลงตวั
divisor ตวั หำร
factorization กำรแยกตวั ประกอบ
factor ห.ร.ม.
highest common factor (HCF) ค.ร.น.
least common multiple (LCM) จำนวนเฉพำะ
จำนวนเตม็ ทไ่ี ม่ตดิ ลบ
prime number
whole number

Mathematics 6 1

Factors, Multiples and Divisors

Factors

Counting numbers, which are multiplied together to give the product are called
factors.

When we find the factors of two or more numbers, and then some factors are the
same ("common"), then they are the "common factors".

Factors of 10 : 1, 2, 5, 10
Factors of 15 : 1, 3, 5, 15

common factors

Multiples

The multiples of a number are the numbers you get when you add a number to
itself repeatedly. We can also say that the products of multiplication are the

multiples of the multipliers and multiplicand.

Multiples of 5 : +5 +5 +5 +5
Multiples of 10 :
5, 10, 15, 20, 25,…

+10 10 +10 +10

10, 20, 30, 40, 50,…

Note: Do not confuse factors from the multiples.

2 Mathematics 6

Divisors

Divisor can be any number with which you want to divide another number
(dividend). A factor however is a divisor that divides the number and leaves no

remainder. So all factors of a number are its divisor but not all divisors will be

factors.

6÷1=6 ×6=6
6÷2=3 ×3=6
6÷3=2 ×2=6
6÷6=1 ×1=6

divisor factors

Example 1 56 = 1 × 56
56 = 2 × 28
List all the factors of 45 and 56. 56 = 4 × 14
56 = 7 × 8
Solution 56 = 8 × 7
56 = 56 × 1
45 = 1 × 45
45 = 3 × 15 Therefore, 1, 2, 4, 7, 8
45 = 5 × 9 and 56 are the factors
45 = 9 × 5
45 = 45 × 1 of 56.

Therefore, 1, 3, 5, 9 and
45 are the factors of 45.

Mathematics 6 3

Example 2

List the first five multiples of 2 and 3.

First 5 multiples of 2 : +2 +2 +2 +2

2, 4, 6, 8, 10

+3 +3 +3 +3

First 5 multiples of 3 : 3, 6, 9, 12, 15

Therefore, the first five multiples of 2 are 2, 4, 6, 8 and 10; the first five
multiples of 3 are 3, 6, 9, 12 and 15.

Example 3

List all the possible divisors of 48 and 54

Solution 54 ÷ 1 = 54

48 ÷ 1 = 48

48 ÷ 2 = 24 54 ÷ 2 = 27

48 ÷ 3 = 16 54 ÷ 3 = 18

48 ÷ 4 = 12 54 ÷ 6 = 9

48 ÷ 6 = 8 54 ÷ 9 = 6

48 ÷ 8 = 6 54 ÷ 18 = 3

48 ÷ 12 = 4 54 ÷ 27 = 2

48 ÷ 16 = 3 54 ÷ 54 = 1

48 ÷ 24 = 2

48 ÷ 48 = 1 Therefore, the divisors of 54 are 1,
2, 3, 6, 9, 18, 27 and 54.

Therefore, the divisors of 48 are 1,
2, 3, 4, 6, 8, 12, 16, 24 and 48.

4 Mathematics 6

Example 4

Determine the common factors of the following sets of number:

a) 12 and 40
b) 27, 36 and 42

a) Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Therefore, the common factors of 12 and 40 are 1, 2 and 4.

b) Factors of 27: 1, 3, 9, 27
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

Therefore, the common factors of 27, 36 and 42 are 1 and 3 only.

Determine the first 5 common multiples of 4 and 8.

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

Therefore, the first five common multiples of 4 and 8 are 8, 16, 24, 32
and 40.

Mathematics 6 5

1. List 5 factors of each of the following numbers.
a. 100
b. 144
c. 325

2. List the first 5 multiples of each of the following numbers.
a. 7
b. 12
c. 25

3. List the first 5 divisors of each of the following numbers.
a. 78
b. 216
c. 425

4. Which of the following numbers below are the factors of 4?
3, 8, 16, 28, 32, 50

5. Which of the following numbers below are the multiples of 8?
1, 8, 13, 24, 27, 35, 48, 96

6. Which whole numbers from 20 to 40 have 7 as their factor?

6 Mathematics 6

a) Which of the whole numbers below are the possible divisor of 168?
2, 3, 4, 5, 6, 7, 8, 21, 32, 42, 84,

b) Find the common factors of the following pairs of whole numbers. Use
the space provided below.

a. 45 and 63 b. 36 and 84 c. 49 and 77

The common The common The common
factors of 45 and 63 factors of 36 and 84 factors of 49 and 77
are are are

c) Find the first 5 common multiples of the following pairs of whole
numbers. Use the space provided below.

a. 6 and 9 b. 10 and 15 c. 8 and 12

The first 5 common The common The common
multiples of 6 and 9 factors of 36 and 84 factors of 49 and 77
are are are

Mathematics 6 7

Prime Numbers and Factorization

Prime numbers

A prime number is a whole number that has a value greater than 1 and can only be
divided exactly by 1 and itself. In other words, a prime number has only two factors:

1 and itself.

2×1=2 7×1=7

3×1=3 11 × 1 = 11

5×1=5 13 × 1 = 13

2, 3, 5, 7, 11 and 13 are examples of prime numbers. Each number has onlytwo
factors.

4×1=4 9×1=9

2×2=4 3×3=9

4 has factors of 1, 2 9 has factors of 1, 3
and 4. and 9.

4 and 9 have more than 2 factors.

Therefore, 4 and 9 are not prime numbers.

Note:
1 is not a prime number because it has only one factor and that is itself.2 is
the only even prime number.

Let’s Try! Prime Number Not a Prime Number

Put check in the right box.

51
129

8 Mathematics 6

Prime Factorization

Prime factorization is finding which prime numbers multiply together to make the
original number. These prime numbers are called prime factors.

You can find the prime factors of a number by using a factor tree and by dividing
the number into smaller parts or by repeated division.

36 36
18 18

36 = × × × 36 = × × ×

Use factor tree to express 315 as the products of its prime factors.

Answer: 315 = × × ×

Mathematics 6 9

Use the repeated division to express 312 as the products of its prime factors.

Answer: 312 = × × × ×

1. Color the boxes that contain prime numbers.

10 Mathematics 6

a) Use the repeated division to express each of the following numbers as
products of its prime factors. Show the works in the space provided.

a. 42 b. 180 c. 335

b) Use the factor tree to express each of the following numbers as products
of its prime factors. Show the works in the space provided.

a. 100 b. 143 c. 336

c) What is the highest and the lowest prime factor of 390?

Mathematics 6 11

Highest Common Factor (HCF)

The highest common factor or HCF is the largest number that can divide
exactlytwo or more numbers.
We can find the HCF of two or more numbers by listing all their factors and
comparing the common factors.
To are some ways of finding the HCF, by listing the factors, by prime factorizationor
by repeated division.
Always ensure that all the factors are written down before comparing the common
factors.

Determine the highest common factor (HCF) of 18 and 36.

Method 1: By listing the factors
18 : 1, 2, 3, 6, 9, 18
36 : 1, 2, 3, 4, 6, 9, 12, 18, 36

Therefore, 18 is the highest common factor of 18 and 36.
Method 2: By prime factorization

18 = 2 × 3 × 3
36 = 2 × 2 × 3 × 3
HCF : 2 × 3 × 3 = 18
Therefore, 18 is the highest common factor of 18 and 36.
Method 3: By repeated division

2 18 36
3 9 18
336

32
2 × 3 × 3 = 18. ∴ 18 is the highest common factor 18 and 36.

12 Mathematics 6

Example 9

Find the highest common factor of 30 and 45, using prime factorization.

Solution

30 = 2 × 3 × 5
45 = 3 × 3 × 5

HCF : 3 × 5 = 15
Therefore, the highest common factor of 30 and 45 is 15.

Example 10

Find the HCF of 24, 48 and 60 using the repeated division.

Solution

2 24 48 60
2 12 24 30
3 6 12 15

245

HCF = 2 × 2 × 3 = 12
Therefore, the HCF of 24, 48 and 60 is 12.

Example 11

Find the highest common factor of 20, 30 and 40 by “listing the factors”.

Solution

20 : 1, 2, 4, 5, 10, 20
30 : 1, 2, 3, 5, 6, 10, 15, 30
40 : 1, 2, 4, 5, 8, 10, 20, 40

The common factors are 1, 2, 5 and 10. 10 is the highest.
Therefore, highest common factor of 20, 30 and 40 is 10.

Mathematics 6 13

1. Find the highest common factor of each of the following sets of number by
“repeated division”.

a. b.

14, 28, 35 35, 49, 56

HCF: HCF:
c. d.

48, 72, 96 36, 90, 150

HCF: HCF:
2. Determine the HCF of the following sets of numbers by “prime factorization”.

a. 16, 24, 40

HCF:
b.

14 Mathematics 6

c.

d.
256, 298
_

HCF:
a) Find the highest common factor of each of the following sets of number

by “listing the factors”.
a.

28, 50

HCF:
b.

24, 30, 36

HCF:
c.

144, 156, 180

HCF:

Mathematics 6 15

Least Common Multiple (LCM)

Least Common Multiple (LCM) is also referred as the Lowest Common Multiple.

We may recall that common factors are the factors which are the same for the sets of
numbers. We can conclude that the least common multiple is the lowest common
factor of the given set of numbers.

We can find the LCM of a set of numbers by “listing multiples and comparing the
common multiples” or by “repeated division”.

Example 12

Determine the least common multiple or LCM of 6 and 8.

Solution

Method 1: By listing and comparing

Multiples of 6: 6, 12, 18, 24, 30, …
Multiples of 8: 8, 16, 24, 32, 40, …

The lowest common factor is 24.
Therefore, 24 is the LCM of 6 and 8.

Method 2: By repeated division

268 When a number is not exactly
234
divisible, write the number itself
32 below the line.

2 × 2 × 3 × 2 = 24 When we cannot divide the

numbers by a common factor, we
discontinue dividing the numbers.

Therefore, the least common multiple of 6 and 8 is 24.

16 Mathematics 6

1. Find the least common multiple of the following sets of numbers by the “listing
method”.
a.
6 and 30

LCM:

b. 2, 4 and 7

LCM:
c. 3, 15 and 25

LCM:
d. 15, 30 and 60

LCM:

Mathematics 6 17

2. Find the least common multiple of the following sets of numbers by the
“repeated division”.

a. b. 14 and 25
35, 40 and 55

5 35 40 55

27 11

LCM: × × × × × LCM:
LCM: LCM:

c. d.
15, 24 and 32 28, 56 and 70

LCM: LCM:
LCM: LCM:

Extra Challenge

1. What is the highest common factor and the least common multiple of 25, 42
and 81?

18 Mathematics 6

Word Problems

Example 13

There is a piece of paper which measures 36 centimeters by
48 centimeters. Wong wants to cut the paper into equal-

sized squares to fold into paper cranes, without having
any remaining paper. Find the greatest length of each
side of each paper.

Solution

Wong does not want any paper left, so we have to find the greatest length
of one side of each paper.

Therefore, we need to find the highest common factor (HCF) of 36 and 48.

2 36 48

2×2×3=12 18 24
9 12
34

The highest common factor of 36 and 48 is 12.
Therefore, the greatest length of one side of each paper is 12 cm.

Example 14

Bananas cost 30 baht per kilogram and mangoes cost
50 baht per kilogram. Ken paid an equal amount of
money to buy each fruit. He also paid the smallest
amount of money for them. How much did Ken pay for

each fruit and how many kilograms of each fruit did he
buy?

Mathematics 6 19

Solution

Ken paid an equal and the smallest amount of money for each fruit.

Therefore, we need to find the least common multiple (LCM) of 30 and 50.
Then, we can divide the amount of money by the cost of each fruit to find
the number of kilograms of each fruit.

2×3×5×5=150 30 50
15 25

The least common multiple of 30 and 50 is 150.

Therefore, Ken paid 150 baht for each fruit.

Then,
the number of kilograms of bananas: 150 ÷ 30 = 5 kilograms
the number of kilograms of mangoes: 150 ÷ 50 = 3 kilograms

Therefore, Ken paid 150 baht for each fruit. He bought 5 kilograms of
bananas and 3 kilograms of mangoes.

Find the highest number which divides both 167 and 95 leaving 5 as
remainder.

Each number leaves 5 as a remainder. Therefore,

167 − 5 = 162 and 95 − 5 = 90

We are looking for the highest number that can divide exactly 162 and 90.
Therefore, we are finding the HCF of 162 and 90.

Factors of 162: 1, 2, 3, 6, 9, 18, 27, 54,81, 162

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

The HCF is 18. Therefore, 18 is the required number.

20 Mathematics 6

Exercise 1E

Solve the following problems below. Use the space provided.
1. A piece of cardboard measured 88 centimeters by 121 centimeters. Nitikorn cut

the cardboard into squares of equal size without leaving any remainder. What
was the length of each square?

__________________________
__________________________
__________________________
__________________________
__________________________

2. Somchai wanted to buy apples and oranges. Apples cost 35 baht per kilogram,
whileoranges cost 40 baht per kilogram. If Somchai wanted to pay an equal
amount of money and the least amount of money for each fruit, how much would
Somchai pay for each fruit? How many kilograms of each fruit would she get?

_______________________
_______________________
_______________________
_______________________
_______________________

Mathematics 6 21

3. Two ropes have the lengths of 32 meters and 56 meters, respectively. John wantsto
cut each rope into equal pieces of the longest length. What is the possible lengthof
each piece of rope?

____________________
____________________
____________________
____________________

4. Find the largest number that divides 92 and 74 leaving 2 as remainder.

5. Two neon lights are turned on at the same time. One blinks every 4 seconds and
the other blinks every 6 seconds. In 60 seconds, how many times will they blink at
the same time?

_______________________
_______________________
_______________________
_______________________
_______________________

22 Mathematics 6

ary

✓ Counting numbers, which are multiplied together to give the product are
called factors.

✓ When we find the factors of two or more numbers, and then some factors are
the same ("common"), then they are the "common factors".

✓ The multiples of a number are the numbers you get when you add a numberto
itself repeatedly.

✓ Divisor can be any number with which you want to divide another number
(dividend).

✓ All factors of a number are its divisor but not all divisors will be factors.
✓ A prime number has only two factors: 1 and itself.
✓ 1 is not a prime number because it has only one factor and that is itself.
✓ 2 is the only even prime number.
✓ Prime factorization is finding which prime numbers multiply together to make

the original number. These prime numbers are called prime factors.
✓ The highest common factor or HCF is the largest number that can divide

exactly two or more numbers.
✓ The least common multiple or LCM is the lowest common factor of the given

set of numbers.

Mathematics 6 23

M A S T E R Y P RA C T IC E

Circle the letter of the correct answer.

1. The following numbers are all 6. List the prime factorization of 24.
of 5.
a. 2 × 2 × 2 × 3
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, … b. 3 × 2 × 4
c. 3 × 8
a. numbers d. 12 × 2
b. factors
c. divisors 7. The HCF of two numbers is 8 while
d. multiples their LCM is 144. Find the other
number if one of it is 16.
2. How many factor pairs are there
for the number 28? a. 108
b. 96
a. 1 c. 72
b. 2 d. 36
c. 3 8. Find the greatest 4-digit number
d. 4
which is divisible by 15, 25, 40
3. Which of the following is a prime and 75.
a. 600
number? b. 9000
a. 383 c. 9600
b. 72 d. 9400

c. 54 9. The greatest length of the scale
that can measurer exactly 30 cm,
d. 211
90 cm, 120 cm and 135 cm
4. What is the least common lengths is….
multiple of 8 and 10?
a. 5 cm
a. 2 b. 10 cm
b. 5 c. 15 cm
c. 40 d. 30 cm
d. 80 10.When a number is divided by 893

5. What is the highest common the remainder is 193. What will be
factor of 42 and 72?
the remainder when it isdivided
a. 2 by 47?
b. 3 a. 19
c. 6 b. 5
d. 9 c. 33
d. 23

24 Mathematics 6

Get the prime factors of the following number using factor tree.

1. 54 2. 81 3. 124 4. 264

Solve for the LCM and HCF of the following numbers. You can use any method
taught in the unit to find the answer.

5. 36 and 49 6. 72 and 124 7. 57 and 133

LCM: _______________ LCM: _______________ LCM: _______________

8. 36 and 49 9. 72 and 124 10. 57 and 133

HCF: _______________ HCF: _______________ HCF: _______________

Mathematics 6 25

At the end of the activity the students will be able to:
1. Create counting wheels which will help them revise skip counting and factors.
2. Write down factors and multiples of a number.
3. Recycle materials which are already at home to create an interactive activity.

Materials

1. paper plate 2. pegs/ clothes-pin

3. marker/ coloured pencil 4. coloured paper

Directions:
1. The students will cut coloured paper and glue it on the center of the paper
plate. The students will also cut small circles and glue it to the pages.

SKIP COUNTING
The students will use coloured pencil and select any number and write it on the
paper plate. Then, the students will write the first twelve multiples on the
clothespin of that selected number.

FACTORS & MULTIPLES
For factors, the students will select a number from the numbers below and
write the factors of each number on the clothespin.

36 48 24 16 58
42 44 18 22 65
42 28 33 12 56

26 Mathematics 6

Unit 2 FRACTIONS

The word fraction comes from the Latin word “fractio” which means
to break. To understand how fractions have developed into theformwe

recognize, we’ll have to step back even further in timeto discover what
the first number system were like.

From as early as 1800 BC, the Egyptians were writing fractions. Their number system
was a base 10 idea (a little but like ours now) so they had separate symbols for 1, 10,
100, 1000, 10,000, 100,000, 1,000,000. The ancient Egyptian writing system wasall in

pictures which were called hieroglyphs and in the same way, they had picturesfor the
numbers:

In India fractions were written very much like we do now, with one number (the
numerator) above another (the denominator), but without a line.

For example: 2
3

It was the Arabs who added the line (sometimes drawn horizontally, sometimes on
a slant) which we now use to separate the numerator and denominator: 23.

Mathematical Terms

Numerator ตวั เศษ Convert แปลง
Quotient ผลหำร
Denominator ตวั สว่ น Simplify ทำใหง้ ่ำยขน้ึ
Reciprocal กำรกลบั เศษสว่ น
Fraction เศษส่วน Proper fraction เศษสว่ นแท้
Improper Fraction เศษสว่ นเกนิ
Mixed Fraction จำนวนคละ

Difference ผลตำ่ ง

Product ผลคูณ

Mathematics 6 27

Comparing and Ordering Fractions

In 6th grade, students are expected to compare two fractions that have either the
same numerator or the same denominator. When comparing fractions, it is

important for students to understand the value of the fraction and consider both
the numerator and denominator and their relationship to each other.

Remember

“The smaller the denominator, the larger the fraction.”
“The larger the denominator, the smaller the fraction.”

Same numerator, In this case, we have the same numerator but different
different denominator denominator. The denominators are three and six.
Thirds are larger than sixths. The rule for comparing
2
6 here is that the fraction with the smaller denominator is
larger.
2
3

Different numerator, For comparing fractions with the same
same denominator
denominators, it becomes easier to
determine the greater or the smaller
3 fraction. After checking if the
8 denominators are the same, we can

7 simply look for the fraction with the
8 bigger numerator.

Different numerator, For comparing fractions with unlike
same denominator denominators, we need to convert them to like
denominators, for which we should find the
3 Least Common Multiple (LCM) of the
4 denominators. When the denominators are
made the same, we can compare the fractions
5 easily.
6

28 Mathematics 6

3×3 9 Note:
4 × 3 = 12
5 × 2 10 When you multiply a value
6 × 2 = 12
by one, the value doesn’t

change. In this case, 3 and 2
3 2

are both equivalent to one.

Multiplying these fractions

doesn’t change the value of

the original fractions. After

getting the LCM and both

fractions have the same

Repeat for the other denominator, we can say
fraction. Since the LCM is 12,
that 5 is greater than 34.
we have to multiply 2/2 to 6
the given fraction.

Compare the fractions, and write <, > or = between them.

1.) 1 10 2.) 12 9 3.) 10 10 4.) 1 12
2 10 12 12 10 11 2 12

5.) 2 11 6.) 6 6 7.) 5 5 8.) 6 7
2 11 7 10 9 96 11

9.) 3 6 10.) 3 1 11.) 1 1 12.) 9 10
3 12 9 12 2 2 11 10

13.) 1 1 14.) 1 2 15.) 2 2 16.) 7 1
5 12 2 32 9 10 2

Mathematics 6 29

Addition of Fractions

To add fractions there are 3 simple steps:

Step 1: Make sure the bottom numbers (the denominators) are the same.
Step 2: Add the top numbers (the numerators), put that answer over the
denominator
Step 3: Simplify the fraction (if needed).

Example 1:

1+ 1
44

Step 1: The bottom numbers (the denominators) are the same. Go straight to

step 2.
Step 2: Add the top numbers and put the answer over the same denominator.

1 1 1+1 2
4+4= 4 =4

Step 3: Simplify the fraction.

21
4=2

Example 2: 11
3 + 6 =?

Step 1: The bottom numbers are different. We need to make them the same

beforewe can continue, because we can’t add them like that. The number

6 is twice as big as 3, so to make the bottom numbers the same we can
multiply the top and bottom of the first fraction by 2, like this:

X 2 Remember!

12 You multiply both top and bottom
3=6
by the same amount, to keep the value of
the fraction the same.

X2
Now that the fraction has the same bottom number (“6”), we can go to step 2.

30 Mathematics 6

Step 2: Add the top numbers and put them over the same denominator.
Step 3:
2 1 2+1 3
6+6= 6 =6

Simplify the fraction:

31
6=2

A rhyme to help you remember!

♫ "If adding or subtracting is your aim,
The bottom numbers must be the same!
♫ "Change the bottom using multiply or divide,
But the same to the top must be applied,
♫ "And don't forget to simplify,
Before its time to say good bye"

1. 2 1 2. 3 1
5 +4= 10 + 5 =

3. 1 1 4. 1 2
5 +3= 2 +4=

5. 2 2 6. 8 3
10 + 4 = 10 + 5 =

Mathematics 6 31

3. 4 3 4. 1 1
12 + 12 = 3 +2=

5. 5 7 6. 4 9
10 + 10 = 7 +7=

Subtraction of Fractions

There are 3 simple steps to subtract fractions.

Step 1: Make sure the bottom numbers (the denominator) are the same
Step 2: subtract the top numbers (the numerator). Put the answer over the same
denominator.

Step 3: Simplify the fraction (If needed).

Example 1:

31
4 - 4 =?

Step 1: The bottom numbers are already the same. Go straight to step 2.
Step 2: Subtract the top numbers and put the answer over the same denominator.

3 1 3-1 2
4 - 4 = 4 =4

Step 3: Simplify the fraction.

2= 1

42

32 Mathematics 6

Example 2:

1 + 3 =?
2 6

Step 1: The bottom numbers are different. To make the bottom numbers the same,multiply

the top and bottom of the first fraction by 3.

X3 The bottom numbers now

13 are the same, so we can go
2=6 to step 2.

X3

Step 2: Subtract the top numbers and put the answer over the same denominator.

3 1 3-1 2
6 - 6 = 6 =6

Step 3: Simplify the fraction.

21
6=3

Find the difference.

1. 1 1 2. 5 2
2 − 9= 7 − 3=

3. 1 1 4. 7 − 3
2 − 4= 8 8=

Mathematics 6 33

5. 5 3 6. 1 1
9 − 9= 4 − 7=

7. 2 1 8. 1 2
3 − 12 = 3 − 10 =

Multiplication of Fractions

There are 3 simple steps to multiply fractions.

Step 1: Multiply the top numbers (the numerator)

Step 2: Multiply the bottom numbers (the denominator)
Step 3: Simplify the fraction (If needed).

Example: 12
2×5=

Step 1: Multiply the top numbers:

1 2 1×2
2 × 5 = 1 =2

Step 2: Multiply the bottom numbers:

1 2 1×2 2
2 × 5 = 2 × 5 = 10

Step 3: Simplify the fraction:

21
10 = 5

♫ "Multiplying fractions: no big problem,Top
times top over bottom times bottom. "And

don't forget to simplify,
Before it's time to say goodbye" ♫

34 Mathematics 6

What about multiplying fractions and whole number?

Make a whole number a fraction, by putting it over 1.

4
4 is also 1

Then continue as before

Example:

2
9 × 3=?

3
Make 3 into 1

2 36 2
9 × 1 = 9 or 3

1. 8 × 1 2. 3
5 4= 3 × 9=

3. 1 4. 4 × 7
2 × 12 = 6 6=

5. 9 6. 6
7 × 6= 6 × 4=

Mathematics 6 35

Division of Fractions

There are 3 simple steps to division fractions.
Step 1: Turn the second fraction (the one you want to divide by) upside down.
Step 2: Multiply the first fraction by that reciprocal.
Step 3: Simplify the fraction (If needed).

Example:
11
2 ÷ 6 =?

Step 1: turn the second fraction upside down (it becomes a reciprocal):
16
6 becomes 1

Step 2: Multiply the first fraction by that reciprocal (top and bottom):
1 6 1×6 6
2 × 1 = 2×1=2

Step 3: Simplify the fraction:

6
2 =3
To help you remember!

♫ "Dividing fractions, as easy as pie,

Flip the second fraction, then multiply.
And don't forget to simplify,

before it's time to say goodbye" ♫

What about division with fraction and whole number?

Make the whole number a fraction, by putting it over 1.

5
5 is also 1

Example:

2 ÷ 5=?
3

Make 5 into 5

1

36 Mathematics 6

Step 1: turn the second fraction upside down (the reciprocal):
51
1 becomes 5

Step 2: Multiply the first fraction by that reciprocal:
2 1 2×1 2
3 × 5 = 3 × 5 = 15

Step 3: Simplify the fraction:

The fraction is already as simple as it can be.
Answer = 2
15

1. 1 2. 98
3÷ 4 = 12 ÷ 9 =

3. 2 4. 3÷ 5
9 ÷ 10 = 10 =

5. 78 6. 33
9 ÷ 10 = 4 ÷ 6=

7. 1 8. 2
8÷ 8 = 12 ÷ 10 =

Mathematics 6 37

Remember!
There are three types of fractions:

Proper Fractions – the numerator is less than the denominator.

Examples: 7 86 3 1 3 21
9 10 8 7 2 5 3 4

Improper Fractions – the numerator is greater than or equal to the
denominator.

Examples: 9 10 87 25 46
78 63 13 46

Mixed Fractions – A whole number and a proper fraction together.

Examples:

5 1 4 3 6 2 3 1 1 1
2 4 6 7 4

Proper fraction smaller number
larger number
Proper

Fractions

Improper larger number
Fractions smaller number

Mixed whole number smaller number
Fractions larger number

38 Mathematics 6

Converting Improper Fractions to Mixed Fractions

There are 3 simple steps to convert an improper fraction to a mixed fraction.
Step 1: Divide the numerator with the denominator.
Step 2: Write down the whole number answer.
Step 3: Then write the remainder on top of the denominator.

Example: Convert 11 to a mixed fraction.
4

Step 1: Divide!

11 ÷ 4 = 2 with a remainder of 3

Write down the 2 and then write down the remainder (3) above the denominator

(4). 3
4
Answer: 2

That example can be written like this:

11 = 11 ÷ 4 = 2 r 3
4

3
24

1. 26 = 2. 13
6 4=

3. 9 4. 7 =
2= 3

Mathematics 6 39

Converting Mixed Fractions to Improper Fractions

There are 3 simple steps to convert a mixed fraction to an Improper fraction.

Step 1: Multiply the whole number part by the fraction’s denominator.

Step 2: Add that to the numerator.

Step 3: Then write the result on top of the denominator.

Example: Convert 3 2 to an improper fraction.
5

Multiply the whole number part by the denominator:

3 × 5 = 15

Add that to the numerator:

15 + 2 = 17

Then write that result above the denominator: 17
5

Second method:

3 2 3 × 5 + 2 = 17
5 5

Convert mixed fractions to improper fractions.

1. 3 2. 6 =
27 = 29

3. 9 = 4. 3 =
3 11 42

40 Mathematics 6