HBMT 3203

Table of Contents
Course Guide ix-xvi
Topic 1 Whole Numbers 1
1.1 Pedagogical Content Knowledge 2
1.1.1 Whole Numbers Computation 2
1.1.2 Estimation and Mental Computation 3
1.1.3 Computational Procedure 4
1.2 Major Mathematical Skills for Whole Numbers 5
1.3 Teaching and Learning Activities 6
1.3.1 Basic Operations of Whole Number 6
1.3.2 Estimation and Mental Computation 15
Summary 18
Key Terms 18
References 18
Topic 2 Fractions 20
2.1.1 Types of Fractions 22
2.1.2 Equivalent Fractions 23
2.1.3 Simplifying Fractions 25
2.2 Major Mathematical Skills for Fractions 26
2.3.1 Improper Fractions 27
2.3.2 Mixed Numbers 29
2.3.3 Addition of Fractions 31
2.3.4 Subtraction of Fractions 33
2.3.5 Multiplication of Fractions 35
2.3.6 Division of Fractions 37
Summary 39
Key Terms 40
References 40
Topic 3 Decimals 41
3.1.1 Meanings of Decimals 43
3.1.2 Decimal Fractions 43
3.1.3 Extension of Base-10 Place Value System 45
3.1.4 Decimal Place 46

i v X TABLE OF CONTENTS
3.2 Major Mathematical Skills for Decimals 47
3.3.1 Decimal Numbers 49
3.3.2 Converting Fractions to Decimal Numbers 51
and Vice Versa
3.3.3 Addition of Decimal Numbers 53
3.3.4 Subtraction of Decimal Numbers 55
3.3.5 Multiplication of Decimal Numbers 57
3.3.6 Division of Decimal Numbers 59
Summary 61
Key Terms 62
References 62
Topic 4 Money 63
4.1.1 Teaching Children About Money 64
4.1.2 Teaching Money Concepts 65
4.1.3 Using Coins to Model Decimals 67
4.2 Major Mathematical Skills for Money 68
4.3.1 Basic Operations on Money 69
4.3.2 Problem Solving on Money 73
Summary 76
Key Terms 76
References 77
Topic 5 Percentages 78
5.1.1 Meaning and Notation of Percent 79
5.1.2 Teaching Aids in Learning Percent 81
5.1.3 Fraction and Decimal Equivalents 82
5.2 Major Mathematical Skills for Percentage 83
5.3.1 Meaning and Notation of Percent 84
5.3.2 Fraction and Decimal Equivalents 87
Summary 93
Key Terms 94
References 94

TABLE OF CONTENTS W v
Topic 6 Time 95
6.1.1 History of Time 97
6.1.2 Time Zones 98
6.1.3 Telling the Time Correctly 99
6.1.4 24-hour System 101
6.2 Major Mathematical Skills for Time 103
6.3.1 Time in the 24-hour System 104
6.3.2 Converting Time in Fractions and Decimals 107
6.3.3 Year, Decade, Century and Millennium 109
6.3.4 Basic Operations Involving Time 111
6.3.5 Duration of an Event 113
6.3.6 Problem Solving Involving Time 115
Summary 117
Key Terms 117
References 118
Topic 7 Length, Mass and Volume of Liquids 119
7.1.1 Historical Note on Measurement 122
7.1.2 The Basic Principles of Measurement 123
7.1.3 The Meanings of Length, Mass and Volume of Liquids 124
7.1.4 Units of Length, Mass and Volume of Liquids 126
7.2 Major Mathematical Skills for Measurement in
Year 5 and Year 6 127
7.3 Teaching And Learning Activities 129
7.3.1 Length 129
7.3.2 Basic Operations on Length 131
7.3.3 Mass 133
7.3.4 Problem Solving Involving Mass 135
7.3.5 Volume of Liquids 137
7.3.6 Problem Solving Involving Volume of Liquids 139
Summary 141
Key Terms 142
References 142
Topic 8 Shape and Space 143
8.1.1 Geometric Formulas 144
8.1.2 Perimeter and Area 145
8.1.3 Volume 147

v i X TABLE OF CONTENTS
8.2 Major Mathematical Skills for Shapes 148
8.3.1 Finding Perimeter 150
8.3.2 Finding Area 154
8.3.3 Finding Volume 157
Summary 159
Key Terms 160
References 160
Topic 9 Averages 161
9.1.1 Teaching Averages 162
9.1.2 Measures of Central Tendency 163
9.2 Major Mathematical Skills for Averages 166
9.3.1 Meaning of Average 167
9.3.2 Calculating Average 171
Summary 175
Key Terms 175
References 176
Topic 10 Data Handling 177
10.1.1 Statistical Measures 179
10.1.2 Collecting, Recording, Organising and Interpreting 181
Data
10.1.3 Methods of Organising Data 183
10.1.4 Types of Graphs 186
10.2 Major Mathematical Skills for Data Handling in Year 5 190
and Year 6
10.3.1 Average 192
10.3.2 Organising and Interpreting Data 194
10.3.3 Pie Chart 196
10.3.4 Problem Solving 198
Summary 200
Key Terms 201
References 201

Topic
1
 Whole
Numbers
LEARNING OUTCOMES
By the end of this topic, you should be able to:
1. Explain the importance of developing number sense for whole
numbers to 1,000,000 in KBSR Mathematics;
2. List the major mathematical skills and basic pedagogical content
knowledge related to whole numbers to 1,000,000;
3. Show how to use the vocabulary related to addition, subtraction,
multiplication and division of whole numbers correctly;
knowledge related to addition, subtraction, multiplication and division
of whole numbers in the range of 1,000,000; and
5. Plan basic teaching and learning activities for whole numbers,
addition, subtraction, multiplication and division of whole numbers in
the range of 1,000,000.
 INTRODUCTION
Welcome to the first topic of Teaching of Elementary Mathematics Part IV. What
is your expectation of this topic? Well, this topic has been designed to assist you
in teaching whole numbers to primary school pupils in Years Five and Six.
For hundreds of years, computational skills with paper-and-pencil algorithms have
been viewed as an essential component of children’s mathematical achievement.
However, calculators are now readily available to relieve the burden of
computation, but the ability to use algorithms is still considered essential. In An
Agenda for Action (NCTM, 2000, p. 6), the NCTM standards support the
decreased emphasis on performing paper-and-pencil calculations with numbers
more than two digits. Most of the operations in this topic will cover the content
area of whole numbers to 1,000,000 in KBSR Mathematics.

2  TOPIC 1 WHOLE NUMBERS
PEDAGOGICAL CONTENT KNOWLEDGE
Computation with whole numbers continues to be the focus of KBSR Mathematics.
Thus, when you observe a classroom mathematics lesson, there is a high probability
you will find a lesson related to computation being taught.
The National Council of Teachers of Mathematics (NCTM) emphasises the
importance of computational fluency, that is, “having efficient and accurate
methods for computing” (NCTM, 2000, pg. 152). Computational fluency includes
children being able to flexibly choose computational methods, understand these
methods, explain these methods, and produce answers accurately and efficiently.
1.1.1 Whole Numbers Computation
A common but rather narrow view of whole numbers computation is that it is a
sequence of steps to arrive at an answer. These sequence or step-by-step
procedures are commonly referred to as algorithms. Tell your pupils, that there are
three important points that need to be emphasised when teachers talk about whole
numbers computation.
(a) Computation is much broader than using just standard paper-and-pencil
algorithms. It should also include estimation, mental computation, and the
use of a calculator. Estimation and mental computation often make better
use of good number sense and place-value concepts.
(b) Children should be allowed ample time and opportunity to create and use
their own algorithms. The following shows a child’s procedure for
subtracting (Cochran, Barson, & Davis, 1970):
64
- 28
- 4
+40
36
1.1
ACTIVITY 1.1
Talk to children in your classroom about the algorithms they use to
solve problems. Describe these algorithms.

TOPIC 1 WHOLE NUMBERS  3
What is the child doing? His thinking could be as follows: “4 minus 8 is -4,
60 minus 20 is 40. -4 plus 40 is 36”!
This child’s method might not make sense to all or most children, however,
it did make sense to that child, which makes it a powerful and effective
method for him at that moment.
(c) There is no one correct algorithm. Computational procedures may be altered
depending on the situation. There are many algorithms that are efficient and
meaningful. For this reason, teachers should be familiar with some of the
more common alternative algorithms.
Alternative algorithms may help children develop flexible mathematical
thinking and may also serve as reinforcement, enrichment, and remedial
objectives.
1.1.2 Estimation and Mental Computation
Estimation and mental computation skills should be developed along with paper-and-
pencil computation because these help children to spot unreasonable answers.
Teachers should also provide various sources for computational creativity for
children.
(a) Mental Computation
Sometimes, we need to do mental computation to estimate the quantity or
volume. Mental computation involves finding an exact answer without the
aid of paper and pencil, calculators, or any other device. Mental computation
can enhance understanding of numeration, number properties, and
operations and promote problem solving and flexible thinking (Reys, 1985;
Reys and Reys, 1990).
When children compute mentally, they will develop their own strategies
and, in the process, develop good number sense. Good number sense helps
pupils use strategies effectively. Teachers should explain to the children how
to do mental computation. You should also encourage children to share and
explain how they did a problem in their heads. Children often can learn new
strategies by hearing their classmates’ explanations.
Mental computation is often employed even when a calculator is used. For
example, when adding 1,350, 785, 448, and 1,150, a child with good number
sense will mentally compute “1,350 plus 1,150” and key in 2,500 into the
calculator before entering the other numbers (Sowder, 1990).

(b) Estimation
You should know that estimation involves finding an approximate answer.
Estimation may also employ mental computation, but the end result is only
an approximate answer. Teachers should ensure that children are aware of
the difference between Mental Computation and Estimation.
Reys (1986) describes four strategies for whole number computational
estimation. They are the front-end strategy, rounding strategy, clustering
strategy, and compatible number strategy. The definition of each strategy is
as follows:
(i) Front-end strategy
The front-end strategy focuses on the left-most or highest place-value
digits. For example, for children using this strategy they would
estimate the difference between 542 and 238 by subtracting the front-end
digits, 5 and 2, and estimate the answer as 300.
(ii) Rounding strategy
Children using this rounding strategy would round 542 to 500 and 238
to 200 and estimate the difference between the numbers as 300.
(iii) Clustering strategy
The clustering strategy is used when a set of numbers is close to each
other in value. For example, to find the sum of 170 + 290 + 230,
children would first add 170 and 230 to get 400, and then they can
estimate the sum of 400 + 290, so it’s about 700.
(iv) Compatible number strategy
When using the compatible number strategy, children adjust the
numbers so that they are easier to work with. For example, to estimate
the answer for 332 , they would note that 333 is close to 332 and is
divisible by 3, and that would give an estimated answer of 111.
1.1.3 Computational Procedure
When teachers engage their children in the four number operations of addition,
subtraction, multiplication and division, it is important that they pay special
attention to the following points:
(a) Use models for computation
Concrete models, such as bundled sticks and base-ten blocks help children
to visualise the problem.
(b) Use estimation and mental computation
These strategies help children to determine if their answers are reasonable.

(c) Develop bridging algorithms to connect problems, models, estimation
and symbols
Bridging algorithms help children connect manipulative materials with
symbols in order to make sense of the symbolic representation.
(d) Develop time-tested algorithms
These algorithms can be developed meaningfully through the use of
mathematical language and models.
(e) The teacher poses story problems set in real-world contexts.
Children are able to determine the reasonableness of their answers when
story problems are based in familiar and real-world contexts.
SELF-CHECK 1.1
1. Explain the three important points that need to be emphasised
when teaching whole number computations.
2. Explain Reys’ four strategies for whole number computational
estimation.
MAJOR MATHEMATICAL SKILLS FOR
WHOLE NUMBERS
1.2
The introduction of the basics of whole number skills will help children to learn
higher mathematical skills more effectively. Teachers should note that before
children learn to name and write numbers they will already have developed
considerable number sense.
The major mathematical skills to be mastered by your pupil when studying the
topic of whole numbers are as follows:
(a) Name and write numbers up to 1,000,000.
(b) Determine the place value of the digits in any whole number up to
1,000,000.
(c) Compare value of numbers up to 1,000,000.
(d) Round off numbers to the nearest tens, hundreds, thousands, ten thousands
and hundred thousands.
(e) Add any two to four numbers to 1,000,000.
(f) Subtract one number from a bigger number less than 1,000,000.
(g) Subtract successively from a bigger number less than 1,000,000.

(h) Solve addition and subtraction problems.
(i) Multiply up to five digit numbers with a one-digit number, a two-digit
number, 10, 100 and 1,000.
(j) Divide numbers up to six digits by a one-digit number, a two-digit number,
10, 100 and 1,000.
(k) Solve problems involving multiplication and division.
(l) Calculate mixed operations of whole numbers involving multiplication and
division.
(m) Solve problems involving mixed operations of division and multiplication.
TEACHING AND LEARNING ACTIVITIES
1.3
There are a few activities that can be carried out with pupils for better
understanding about this topic.
1.3.1 Basic Operations of Whole Number
Now, let us look at a few activities to learn the basic operations of whole numbers
in class.
ACTIVITY 1.2
Learning Outcome:
 To practise the algorithms of addition.
Materials:
 Clean writing papers; and
 Task Sheet as below
Procedures:
1. Divide the class into groups of four.
2. Give each pair some clean writing paper and a Task Sheet.

3. Each pupil in the group takes turn to fill in numerals from 0 to 9
randomly on the Task Sheet.
4. The teacher gives the instruction for addition by saying,
Find the sum of any three three-digit numbers.
5. Each pupil identifies three three-digit numbers by reading the
numerals from the square from left to right, right to left, top to
bottom, bottom to top or even diagonally.
Each pupil in the group checks the calculation of their peers using
the calculator.
Example: 841 + 859 + 768 = 2,469
8 6 7
4 5 3
1 0 9
6. The winner for this round is the pupil with the highest sum and is
awarded 5 points.
7. Pupils in the group repeat steps (5) and (6) when the teacher gives
the instruction for the next addition.
8. The teacher summarises the lesson on addition.

In subsequent sections, some examples are provided for pupils to practise the
algorithms of addition, subtraction multiplication and division. The next section
discusses subtraction using the calculator and estimation of the product of two
numbers. Let us look at Activity 1.3 first.
ACTIVITY 1.3
Learning Outcome:
 To practise the algorithms of addition.
 To increase the understanding of place value.
Materials:
 10 cards numbered 0 through 9
 Task Sheet as below
Procedures:
2. Give each pair some clean writing paper and a Task Sheet.
3. Each pupil in the group takes turns to draw a card and announces
the number on it. All players in the group write this number in one
of the addend boxes on the Task Sheet. Once a number has been
written on the Task Sheet, it cannot be moved or changed.
4. Replace the card and shuffle the cards.
5. Repeat steps (3) and (4) until all addend boxes are filled.
6. Pupils will compute their respective sum.
7. The winner is the pupil with the greatest sum and is awarded 5
points.
8. Repeat steps (3) through (7) until the teacher stops the game.

Learning Outcomes:
 To practise subtraction using the calculator.
 To practise the algorithms of subtraction.
 To increase the understanding of place value.
Materials:
 Calculator
 Clean writing papers
Procedures:
1. Pupils play this game in pairs.
2. Give each pair a calculator and some clean writing paper.
3. Throw a dice to decide who should start first.
4. Pupil A chooses three different single-digit numbers. For example:
1, 2, and 4.
5. Enter the selected digits into the calculator in order to create the
largest number possible.
6. Enter “-“
7. Next, enter the same three selected digits to create the smallest
number possible followed by the “=” sign.
Example: The largest number created from the three single-digit
numbers is 421.
The smallest number created from the three single-digit
numbers is 124.
421
- 124
297
ACTIVITY 1.4

8. Repeat steps (5) through (7) with the digits 2, 7 and 9 (derived from
the first subtraction) as shown below.
ACTIVITY 1
421 972 963
- 124 - 279 - 369
297 693 594
954
- 459
495
9. Pupil B will have to write out all the algorithms of the subtractions
and Pupil A will check it.
10. If Pupil B had carried out all the subtractions correctly, the answer
will eventually yield the magic number 495!
11. Pupil B repeats steps (4) through (8).
12. The game continues until the teacher instructs the the pupils to stop.
13. The teacher summarises the lesson on subtraction.

ACTIVITY 1.5
Learning Outcomes:
 To estimate the product of two numbers.
 To practise the algorithms of multiplication.
Materials:
 Calculator
 Task Sheet as given
Procedures:
2. Give each group some clean writing paper, a calculator and a Task
Sheet.
3. Working in their group pupils will discuss the best strategy to fill
in the missing numbers in the boxes.
4. Pupils will compute the algorithm of multiplication and fill in the
blank boxes.
5. The winner is the group who obtained the correct answer in the
shortest time.
6. Members of the winning group will explain to the class their
strategy and also the algorithm of multiplication.
7. Teacher summarises the lesson on multiplication.

TASK SHEET
ACTIVITY 1
1. Use only the numbers 4, 5, 6, 7, 8 and 9 to make
 The largest possible product
X
 The smallest possible product
X
2. Use your calculator to help you find the missing number.
X
8 6
2
1 9 2
+ 5 9

ACTIVITY 1.6
Learning Outcome:
 Using calculators to develop number sense involving division.
Materials:
 Task Sheet
 Four calculators
Procedures:
2. Provide each group some clean writing papers, a Task Sheet and
four calculators.
3. Teacher explains the rules and starts the game.
4. Pupils will compete against members of their own group.
5. Pupils will use the calculator to determine a reasonable dividend
and divisor.
6. The winner is the one in the group with the dividend and divisor
that results in a quotient closest to the target number.
Example: Target Number = 6,438
Entered into the calculator: 32,195 5
Followed by = (within 5 sec.) :
Display shows “6,439”
7. The winner will explain to the group members his strategy in
determining a reasonable answer.
8. The teacher summarises the lesson on division.

TASK SHEET
Target Numbers
446 815 845 490
6,438 654 8,523 6,658
29,881 31,455 44,467 51,118
 Pick a target number and circle it.
 Enter any number into your calculator.
 Press the key.
 Enter another number that you think will give you a product close to
the target number.
 Press the “=” key to determine your answer.
 How close are you to the target number?

1.3.2 Estimation and Mental Computation
Below are the activities you can use to teach your pupils about estimation and
mental computation.
ACTIVITY 1.7
Learning Outcomes:
 To recognise patterns in whole number operations.
 To practise estimation and computation of whole numbers.
Materials:
 Calculator
Procedures:
2. Ask each member of the group to choose a two-digit number.
3. Using the calculator ask them to multiply their numbers by 99.
4. Pupils in their group record and compare their results.
5. Ask them if they can see a pattern or relationship in their answers.
6. In their groups pupils will write a statement describing their
pattern.
7. Ask pupils to predict the results of multiplying 5 other numbers by
99.
8. Repeat steps (2) through (7) but this time multiply the numbers by
999.
9. Ask pupils to compare results obtained from multiplication by 99
and 999 and write statements describing the pattern
- The same as the one for two-digit numbers x 99.
- Different from the two-digit numbers x 999.

ACTIVITY 1.8
Learning Outcome:
 To practise estimation and computation of whole numbers.
Materials:
 Calculator
 Task Sheet
Procedures:
2. Give each group some clean writing paper, a calculator and a Task
Sheet.
3. In their groups, ask pupils to discuss the best strategy to fill in the
missing numbers.
4. Pupils will compute the algorithm of division and fill in the blank
boxes.
5. The winner is the group that arrives at the correct answer in the
shortest time.
6. Members of the winning group will explain to the class their
strategy and also the algorithm of division.
7. Teacher summarises the lesson on division.

TASK SHEET
1. Use only the numbers 4, 5, 6, 7, 8 and 9 to make
 The largest possible answer
)
 The smallest possible answer
)
2. Use your calculator to help you find the missing number.
5 R 2
8 ) 6
0 7
8 ) 2 8

In this topic, we have learned :
 To explain the importance of developing number sense for whole numbers to
1,000,000 in KBSR Mathematics.
 The major mathematical skills and basic pedagogical content knowledge
related to whole numbers to 1,000,000.
 How to use the vocabulary related to addition, subtraction, multiplication and
division of whole numbers correctly.
 The major mathematical skills and basic pedagogical content knowledge
related to addition, subtraction, multiplication and division of whole numbers
in the range of 1,000,000.
 To plan basic teaching and learning activities for whole numbers, as well as
the addition, subtraction, multiplication and division of whole numbers in the
range of 1,000,000.
Addition
Division
Multiplication
Place value
Subtraction
Whole numbers
Hatfield, M. M., Edwards, N. T., & Bitter, G. G. (1993). Mathematics methods for
the elementary and middle school. Needham Heights, MA: Allyn & Bacon.
Kennedy, L. M., & Tipps, S. (2000). Guiding children’s learning of mathematics.
US: Allyn &Wadsworth.
Rucker, W. E., & Dilley, C.A. (1981). Heath mathematics. Washington, DC:
Heath and Company.

Tucker, B. F., & Weaver, T. L. (2006). Teaching mathematics to all children.
Ohio: Merill Prentice Hall.
Vance, J. H., & Cathcart, W. G. (2006). Learning mathematics in elementary and
middle schools. Ohio: Merrill Prentice Hall.

Topic
2
 Fractions
LEARNING OUTCOMES
1. Use vocabulary related to fractions correctly as required by the Year 5
and Year 6 KBSR Mathematics Syllabus;
knowledge related to fractions;
3. Use the vocabulary related to addition, subtraction, multiplication and
division of fractions correctly;
of fractions; and
5. Plan basic teaching and learning activities for addition, subtraction,
multiplication and division of fractions.
 INTRODUCTION
Hello, and welcome to the topic on fractions. The basis of mathematics is the study of
fractions, yet it is among the most difficult topics for school-going children. They often
get confused when learning the concept of fractions as many of them have difficulty
recognising when two fractions are equal, putting fractions in order by size, and
understanding that the symbol for a fraction represents a single number. Pupils also
rarely have the opportunity to understand fractions before they are asked to perform
operations on them such as addition or subtraction (Cramer, Behr, Post, & Lesh, 1997).
For that reason, we should provide opportunities for children to learn and understand
fractions meaningfully. We could use physical materials and other representations to
help children develop their understanding of the concept of fractions. The three
commonly used representations are area models (e.g., fraction circles, paper folding,
geo-boards), linear models (e.g., fraction strips, Cuisenaire rods, number lines), and

TOPIC 2 FRACTIONS  21
discrete models (e.g., counters, sets). We introduced these representations to our pupils
in Year 3 and Year 4. It would be useful to show them again these representations to
reaffirm their understanding about fractions.
In order to start teaching fractions in Year 5 and Year 6, it is important for us to have an
overview of the mathematical skills pupils need in order to understand the concept of
improper fractions and mixed numbers. It is also important to acquire the mathematical
skills involved in adding, subtracting, multiplying and dividing fractions.
At the beginning of this topic, we will learn about the pedagogical content
knowledge of fractions such as the meanings of proper fractions, improper
fractions and mixed numbers. In the second part of this topic, we will look at the
major mathematical skills for fractions in Year 5 and Year 6. Before we finish this
topic we will learn how to plan and implement basic teaching and learning
activities for addition, subtraction, multiplication and division of fractions.
ACTIVITY 2.1
Can you think of five reasons why fractions exist in our life? List down
the reasons before comparing them with the person next to you.
2.1
Do you know how fractions came to be used? When human beings started to
count things, they used whole numbers. However, as they realised that things do
not always exist as complete wholes, they invented numbers that represented “a
whole divided into equal parts”. In fact, fractions were invented to supplement the
gap found in between whole numbers.
We have discussed the meanings of fractions comprehensively in the Year 3. We
have seen that there are three interpretations of fractions:
(a) Fractions as parts of a whole unit;
(b) Fractions as parts of a collection of objects; and
(c) Fractions as division of whole numbers.
In fact, it is important for us to provide opportunities for our children to
differentiate these three interpretations in order to understand fractions better. In
the following section, we will look at the pedagogical content knowledge of
fractions such as the types of fractions; namely, proper fractions, improper
fractions and mixed numbers.

 TOPIC 2 FRACTIONS
22
2.1.1 Types of Fractions
You can introduce the meaning of fraction to teach them the types of fractions. A
fraction is a rational number which can be expressed as a division of numbers in
the form of
p , where p and q are integers and q ≠ 0. The number p is called the
q
4   and 7 8
numerator and q is called the denominator. For example, 4 5
5
7   .
8
Let us look at the different types of fractions in the next section.
(a) Proper Fractions
A proper fraction is a fraction where its numerator is less than the
denominator.
, 123
24
, 7
7
, 3
2
1
For example : ,....
245
, 13
15
, 5
4
, 1
4
1
4
1
2
3
4
(b) Improper Fractions
An improper fraction is a fraction where its numerator is equal to or
greater than the denominator.
, 523
24
For example : ,....
245
, 33
15
, 15
7
, 9
4
4
, 5
4

4
4
5
4
(c) Mixed Numbers
A mixed number consists of an integer (except 0) and a proper fraction.
, 122 133
24
For example: ,....
245
, 22 13
15
, 5 2
7
, 3 2
4
1 3 
11
2
1 3
4
Pupils should have ample opportunity to identify and represent the different types
of fractions as well as to name and write them down in symbols and words.
2.1.2 Equivalent Fractions
Similar to whole numbers, fractions too have various terms and names. For
example,
, 4
6
8
, 3
4
, 2
2
1 and
5 all represent the same amount. They are called
10
equivalent fractions. In other words, fractions with identical values are called
equivalent fractions.

24
5
1  2
 3
 4

1 and
, 4
6
, 2
2
5 are equivalent fractions.
Note that to find an equivalent fraction, we multiply or divide both the numerator
and the denominator by the same number. For example:
(i) Multiplying both numerator and denominator by the same number.
3
6
1 1 
3

2 3
2


Therefore,
1 and
2
3 are equivalent fractions.
6
(ii) Dividing both numerator and denominator by the same number.
1
3
5 5 
5

15 5
15


Therefore,
5
and
15
1
are equivalent fractions.
3
Use models to verify the generalisation:
1
2
2
4
3
6
6
12
Equivalent Fractions
Since,
10
8
6
4
2
Therefore,
8
, 3
4
10

2.1.3 Simplifying Fractions
Now we move on to simplifying fractions. Remind your pupils that the ability to
change a fraction to its equivalent fraction is an important skill that is required to
understand the characteristics of fractions and to master other skills concerning
basic operations of fractions. We should provide various activities for our pupils
to master this skill. These activities should involve all the three stages of learning:
concrete, spatial concrete and abstract.
A fraction with its numerator and denominator without any common factors
(except 1) is said to be in its simplest form. For example:
, 7
7
15
, 3
3
, 5
4
1
, 2
4
and
2 and
, 5
10
, 2
4
9 . Conversely, ,
25
15
, 4
6
7 are not in their simplest form
28
because their numerators and denominators have common factors. The process of
changing a fraction to its simplest form is called simplifying a fraction.
Simplifying should be thought of as a process of renaming and not cancellation.
In the example below,
4 and
8
2 are renamed or simplified to
4
1 .
2
1
2
4 2 
2

4 2
2
4
4 
2
8 2
8

 


1 is the simplified form of
2
2 and
4
4 .
8
As a teacher you need to tell your pupils that before they can master the skill of
simplifying fractions, they must first understand the concept of proper fractions,
improper fractions, mixed numbers and equivalent fractions.
SELF-CHECK 2.1
1. Describe briefly with examples the three types of fractions.
2. Explain the two ways of finding equivalent fractions for a given
fraction.
3. What is meant by simplifying a fraction?

26
FRACTIONS
2.2
A systematic conceptual development of fractions will be very helpful for our
pupils to learn this topic effectively. It would be advisable for teachers to
introduce the topic in a less stressful manner. It is important for us to provide
opportunities for our pupils to understand improper fractions and mixed numbers
meaningfully. We should use physical materials and other representations to help
our children develop their understanding of these concepts. We should also
provide opportunities for our children to acquire mathematical skills involved in
adding, subtracting, multiplying and dividing fractions.
The major mathematical skills to be mastered by pupils studying the topic of
fractions in Year 5 and Year 6 are as follows:
(a) Name and write improper fractions with denominators up to 10.
(b) Compare the value of the two improper fractions.
(c) Name and write mixed numbers with denominators up to 10.
(d) Convert improper fractions to mixed numbers and vice versa.
(e) Add two mixed numbers with the same denominators of up to 10.
(f) Add two mixed numbers with different denominators of up to 10.
(g) Solve problems involving addition of mixed numbers.
(h) Subtract two mixed numbers with the same denominators of up to 10.
(i) Subtract two mixed numbers with different denominators of up to 10.
(j) Solve problems involving subtraction of mixed numbers.
(k) Multiply any proper fraction with a whole number up to 1,000.
(l) Add three mixed numbers with the same denominators of up to 10.
(m) Add three mixed numbers with different denominators of up to 10.
(n) Subtract three mixed numbers with the same denominators of up to 10.
(o) Subtract three mixed numbers with different denominators of up to 10.
(p) Solve problems involving addition and subtraction of fractions.
(q) Multiply any mixed numbers with a whole number up to 1,000.
(r) Divide fractions with a whole number and a fraction.
(s) Solve problems involving multiplication and division of fractions.

ACTIVITY 2.3
2.3
Now let us look at several activities that could help pupils not only to understand
improper fractions and mixed numbers, but also to acquire the mathematical skills
involved in adding, subtracting, multiplying and dividing fractions.
2.3.1 Improper Fractions
ACTIVITY 2.2
Learning Outcomes:
 To write the improper fractions shown by the shaded parts.
 To write the improper fractions in words.
 To compare the value of the two improper fractions.
Materials:
 Task Cards
 Answer Sheets
Procedure:
1. Divide the class into groups of six pupils and give each pupil an
Answer Sheet.
2. Ask pupils to write their name on the Answer Sheet.
3. Six Task Cards are shuffled and put face down in a stack at the centre.
4. Each player begins by drawing a card from the stack.
5. The player writes all the answers to the questions in the card drawn on
the Answer Sheet.
6. After a period of time (to be determined by the teacher), each pupil in
the group exchanges the card with the pupil on their left in clockwise
direction.
7. Pupils are asked to repeat steps (5 and 6) until all the pupils in the
group have answered questions in all the cards.
8. The winner is the pupil that has the most number of correct answers.
9. Teacher summarises the lesson by recalling the basic facts of improper
fractions.

28
Example of an Answer Sheet:
Name :________________________ Class :______________________
Card A Card B Card C
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
Card D Card E Card F
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
Example of a Task Card:
Card A
1. Write the improper fractions of the shaded parts.
=
2. Write in words.
5 =
4
3. Circle the larger improper fraction.
7
4
9
4
ACTIVITY 2.3
1. Work with a friend in class to prepare five more Task Cards.
2. There should be three questions in each card.
3. Make sure your cards are based on the learning outcomes of Activity
2.2.

2.3.2 Mixed Numbers
ACTIVITY 2.4
Learning Outcomes:
 To write the mixed numbers shown by the shaded parts
 To convert improper fractions to mixed numbers
 To convert mixed numbers to improper fractions
Materials:
 30 different Flash Cards
 Clean writing paper
Procedure:
1. Divide the class into groups of three pupils and give each group a
clean writing sheet.
2. Instruct the pupils to write their names on the clean paper.
3. Flash Cards are shuffled and put face down in a stack at the centre.
4. Player A begins by drawing a card from the stack. He shows the
card to Player B.
5. Player B then reads out the answers within the stipulated time
(decided by the teacher).
6. Player C writes the points obtained by Player B below his name.
Each correct answer is awarded one point (a maximum of 3 points
for each Flash Card).
7. Players repeat steps (4 and 5) until 10 cards have been drawn by
Player A.
8. Players now change roles. Player B draws the cards, Player C reads
out answers and Player A keeps the score.
9. Repeat steps (3 through 6) until all the players have had the
opportunity to read the 10 Flash Cards shown to them.
10. The winner in the group is the student that has the most number of
points.
11. Teacher summarises the lesson on the basic facts of mixed
numbers.

30
Example of a Flash Card:
Flash Card 1
1. Write the mixed number shown by the shaded parts.
2. Convert this improper fraction to a mixed number.
15 =
4
3. Convert this mixed number to an improper fraction.
3 3 =
7
ACTIVITY 2.5
1. Work with three friends in class to prepare another 29 Flash
Cards.
2. There should be three questions in each Flash Card.
3. Make sure your cards are based on the learning outcomes of
Activity 2.4.

2.3.3 Addition of Fractions
ACTIVITY 2.6
Learning Outcomes:
 To add two mixed numbers
 To add three mixed numbers
 To solve problems involving addition of mixed numbers.
Materials:
 Task Sheets
 Colour pencils
Procedure:
1. Divide the class into groups of four to six pupils. Provide each
group with a different colour pencil and a clean writing sheet.
2. The teacher sets up five stations in the classroom. A Task Sheet is
placed at each station.
3. Instruct the pupils to work together to solve the questions in the
Task Sheet at each station.
4. Each group will spend 10 minutes at each station.
5. At the end of 10 minutes, the groups will have to move on to the
next station in the clockwise direction.
6. At the end of 50 minutes, the teacher collects the answer papers.
7. The group with the highest score (highest number of correct
answers) is the winner.
8. The teacher summarises the lesson on how to add mixed numbers
with the same denominators and different denominators.

32
Example of a Task Sheet:
STATION 1
1. Add the following two mixed numbers. Express your answers in the
simplest form.
(a)  
3 3
4
2 3
4
4 2
13
(b)  
3
5
2. Add the following three mixed numbers. Express your answers in the
simplest form.
(a)   
2 1
5
2 2
5
13
5
3 3
1 2
2 1
(b)   
4
3
2
3. Encik Ahmad sold
3 3 kg of prawns to Mr. Chong and
7
2 2 kg of
5
prawns to Mr. Samuel. Find the total mass of prawns sold by Encik
Ahmad.
The total mass of prawns sold is kg.
ACTIVITY 2.7
Work with two of your friends to prepare another four Task Sheets for
the other stations. There should be three questions in each sheet. Make
sure your sheets are based on the learning outcomes of Activity 2.6.

2.3.4 Subtraction of Fractions
ACTIVITY 2.8
Learning Outcomes:
 To subtract two mixed numbers
 To subtract three mixed numbers
 To solve problems involving subtraction of mixed numbers
Materials:
 Activity Cards
 Colour pencils
Procedure:
1. Divide the class into groups of four pupils. Provide each group
with a different colour pencil and a clean writing sheet
2. A set of 12 Activity Cards are shuffled and put face down in a
stack at the centre.
3. When the teacher signals, pupils will begin solving the questions in
the first Activity Card drawn.
4. Once they are done with the first Card, they may continue with the
next Activity Card.
5. At the end of 10 minutes, the groups will stop and hand their
answer paper to the teacher.
6. The group with the highest score is the winner.
7. The teacher summarises the lesson on how to subtract mixed
numbers with the same denominators and different denominators.

34
Example of an Activity Card:
1. Subtract the following two mixed numbers. Express your answers in
the simplest form.
(a)  
2 3
4
4 1
4
2 2
4 3
(b)  
3
5
2. Find the difference of the following mixed numbers. Express your
answers in the simplest form.
(a)   
11
7
2 2
7
4 4
7
2 3
1 2
5 1
(b)   
4
3
2
3. A container holds
6 3 litres of water. Abu Bakar pours
8
2 2 litres of
5
water from the container into a jug while his brother Arshad pours
3
1 2 litres of water from the container into a bottle. How much water,
in fractions, is left in the container?
The amount of water left is litres.
ACTIVITY 2.9
Prepare another 11 Activity Cards for the group. There should be three
questions in each card.
Make sure your cards are based on the learning outcomes of Activity
2.8.

2.3.5 Multiplication of Fractions
ACTIVITY 2.10
Learning Outcomes:
 To multiply proper fractions with whole numbers
 To multiply mixed numbers with whole numbers
 To solve problems involving multiplication of mixed numbers
Materials:
 Exercise Sheets
 Colour pencils
Procedure:
1. Divide the class into groups of two pupils. Give each group a
different colour pencil.
2. Give each group an Exercise Sheet with five questions.
3. The group that finishes fastest with all correct answers is the
winner.
4. The teacher summarises the lesson on how to multiply fractions
with whole numbers.

36
Example of an Exercise Sheet:
1. Solve the following multiplication
1
(a)  32 
4
3
(b)  200 
5
2. Solve the following multiplication
4 4
(a)  28 
7
5 1
(b)  400 
4
3. There are 440 apples in a box.
3 of the apples are green apples.
4
The remaining apples are red. How many red apples are there in
the box?
There are red apples in the box.
4. Muthu drinks
1 3 litres of water a day. How much water in litres,
4
will he drink in two weeks?
Muthu drinks litres of water in two weeks.
5. Shalwani spends
1 3 hours watching television in a day. How much time
4
does she spend watching television in three weeks?
Shalwani spends hours watching television in three weeks.

2.3.6 Division of Fractions
ACTIVITY 2.11
Learning Outcomes:
 To divide fractions with whole numbers
 To divide fractions with fractions
 To solve problems involving division of fractions
Materials:
 Division Worksheets
 Colour pencils
Procedure:
1. Divide the class into 10 groups. Give each group a Division
Worksheet, clean writing paper and a colour pencil.
2. Instruct the groups to answer all the questions in the Divison
Worksheet.
3. The groups write their answers on the clean writing paper.
4. After a period of time (to be determined by the teacher), the
teacher instructs the groups to exchange the Division Worksheets.
5. Repeat steps 2 to 4.
6. Once all the 10 Division Worksheets have been answered, teacher
collects the papers and corrects the answers.
8. The teacher summarises the lesson on how to divide fractions with
fractions and with whole numbers.

38
Example of a Division Worksheet:
WORKSHEET 1
1. Solve the following division of fractions.
1
1
(a)  
28
4
9
3
(b)  
25
5
2. Solve the following division of fractions.
2 3
(a)  33 
4
3
13
(b)  
10
5
3. A company wants to donate RM
2 3 million equally to eight charities.
4
How much money will each charity receive?
Each charity receives RM
million.
4. The total length of 7 similar ropes is
10 1 m. Find the length of one
2
rope.
The length of one rope is
m.
ACTIVITY 2.12
Prepare another nine Division Worksheets for the group. There should
be four questions in each worksheet.
Make sure your worksheets are based on the learning outcomes of
Activity 2.11.

 The three commonly used representations for fractions are area models (e.g.,
fraction circles, paper folding, geo-boards), linear models (e.g., fraction strips,
Cuisenaire rods, number lines), and discrete models (e.g., counters, sets).
 The three interpretations for fractions are (i) fractions as parts of a unit whole,
(ii) fractions as parts of a collection of objects, and (iii) fractions as division of
whole numbers.
 It is important to provide opportunities for our children to differentiate these
three interpretations so that they can understand fractions better.
 A fraction is a rational number which can be expressed as a division of
numbers in the form of , where p and q are integers and q ≠ 0. The number
p is called the numerator and q is called the denominator.
 Pupils in Year 5 and Year 6 should be able to identify proper fractions,
improper fractions and mixed numbers. They should be able to simplify the
given fractions into its simplest form.
 A proper fraction is a fraction where its numerator is less than the
denominator.
 An improper fraction is a fraction where its numerator is equal to or greater
than the denominator.
 A mixed number consists of an integer (except 0) and a proper fraction.
 Fractions with identical values are called equivalent fractions.
 The process of changing a fraction to its simplest form is called simplifying a
fraction.
 Pupils should be able to acquire the mathematical skills involved in adding,
subtracting, multiplying and dividing fractions.
 Pupils should also be able to solve daily life problems involving basic
operations on fractions.
p
q

40
 Story problems are set in real-life situations. Children are able to determine
the reasonableness of their answers when story problems are based on familiar
contexts.
Addition
Denominator
Division
Fraction
Half
Multiplication
Numerator
Quarter
Share
Subtraction
Whole
Anne Toh. (2007). Resos pembelajaran masteri: Mathematics year 3. Petaling
Jaya: Pearson Malaysia.
Bahagian Pendidikan Guru (1998). Konsep dan aktiviti pengajaran dan
pembelajaran matematik: Pecahan. Kuala Lumpur: Dewan Bahasa dan
Pustaka.
Nur Alia bt. Abd. Rahman, Nandhini (2008). Siri intensif: Mathematics KBSR
year 5. Kuala Lumpur: Penerbitan Fargoes.
Nur Alia bt. Abd. Rahman & Nandhini (2008). Siri intensif: Mathematics KBSR
Ng S.F. (2002). Mathematics in action workbook 2B (Part 1). Singapore: Pearson
Education Asia.
Peter C. et al. (2002). Maths spotlight activity sheet 1. Oxford: Heinemann
Educational Publishers.
Sunny Yee & Lau P.H. (2007). A problem solving approach : Mathematics year
3. Subang Jaya: Andaman Publication.

Topic
3
 Decimals
LEARNING OUTCOMES
1. Use the vocabulary related to decimals correctly as required by the
Year 5 and Year 6 KBSR Mathematics Syllabus;
2. Relate major mathematical skills and basic pedagogical content
knowledge related to decimals;
3. Use the vocabulary related to addition, subtraction, multiplication and
division of decimals correctly;
4. Use major mathematical skills and basic pedagogical content
of decimals; and
5. Plan basic teaching and learning activities for the addition, subtraction,
multiplication and division of decimals.
 INTRODUCTION
Do you know the meaning of the word “decimal”? It means "based on 10" (from
Latin decima: a tenth part). We sometimes say "decimal" when we mean anything
to do with our numbering system, but a "decimal number" usually means there is a
decimal point. The word “decimal” is used so loosely that most uses of it are really
wrong. Properly speaking, since the "deci-" in the word means "ten", any number
written in a base-ten system (that is, with each digit worth ten times as much as the
one next to it) can be called a "decimal number". When we write "123", the 3 is
worth 3 ones, the 2 is worth 2 tens, and the 1 is worth a ten of tens, or a hundred.
That is decimal.
A decimal fraction is a special form of fraction where the denominator is in the base-ten,
or a power of ten. A decimal fraction, also called a decimal, is a number with a
decimal point in it, like 1.23. The decimal point separates the whole number from the

42  TOPIC 3 DECIMALS
fractional part of a number. Generally speaking, any number with a decimal point in it
would be commonly called a decimal, not just a number less than 1.
Hopefully the explanation did not confuse you. What about young children? If
children are taught the wrong concepts of decimals then working with decimals is
going to be a dreadful experience for them. For that reason, we must provide
opportunities for our children to learn and understand decimals meaningfully.
We can use physical materials and other representations to help our children
develop their understanding of the concept of decimals. Since decimal numbers
are closely related to fractions, the three commonly used representations for
fractions, namely the area models (e.g., fraction circles, paper folding, geo-boards),
linear models (e.g., fraction strips, Cuisenaire rods, number lines), and
discrete models (e.g., counters, sets) can be used again to teach the concept of
decimals. It would be useful to show pupils these representations to reaffirm their
understanding about decimals.
In order to teach decimals in Years 5 and 6, it is important for us to have an
overview of the mathematical skills involved in changing fractions and mixed
numbers to decimals and vice versa. It is also important to acquire the
mathematical skills involved in adding, subtracting, multiplying and dividing
decimals.
At the beginning of this topic, we will explore the pedagogical content knowledge
of decimals such as the basic interpretation of decimals, and then ways to
represent and read decimals. In the second part of this topic, we will look at the
major mathematical skills for decimals in Years 5 and 6. Before we end this topic,
we will learn how to plan and implement basic teaching and learning activities for
addition, subtraction, multiplication and division of decimals.
ACTIVITY 3.1
Write your answers for these two questions and compare them with
the person sitting next to you.
1. What are decimals?
2. Why is it necessary for us to learn about decimals?
3.1
Before you teach your pupils decimals, you should talk to them about the number
system. The modern number system originated in India. Other cultures discovered

TOPIC 3 DECIMALS  43
a few features of this number system but the system, in its entirety, was compiled
in India, where it attained coherence and completion. By the 9th century, this
complete number system had existed in India but several of its ideas were
transmitted to China and the Islamic world before that time.
A straightforward decimal system, where 11 is expressed as ten-one and 23 as
two-ten-three, is found in the Chinese and Vietnamese languages. The Japanese,
Korean, and Thai languages imported the Chinese decimal system while many
other languages with a decimal system have special words for the numbers
between 10 and 20, and decades. Incan languages such as Quechua and Aymara
have an almost straightforward decimal system, in which 11 is expressed as ten
with one and 23 as two-ten with three.
3.1.1 Meanings of Decimals
The decimal (base-ten or sometimes denary) numeral system has ten as its base.
It is the most widely used numeral system, perhaps because humans have ten
digits over both hands. Ten is the number which is the count of fingers on both
hands. In many languages the word digit or its translation is also the anatomical
term referring to fingers and toes.
In English, decimal means tenth, decimate means reduce by a tenth, and denary
means the unit of ten. The symbols for the digits in common use around the globe
today are called Arabic numerals by Europeans and Indian numerals by Arabs,
the two groups' terms both referring to the culture from which they learned the
system.
Decimal notation is the writing of numbers in the base 10 numeral system, which
uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3,
4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are
often used with a decimal separator (decimal point) which indicates the start of a
fractional part.
The decimal system is a positional numeral system; it has positions for units, tens,
hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to
be used with that digit - each position has a value ten times that of the position to
its right.
3.1.2 Decimal Fractions
A decimal fraction is a special form of fraction where the denominator is in the
base ten, or a power of ten. Decimal fractions are commonly expressed without a
denominator, the decimal separator being inserted into the numerator (with

leading zeros added if needed), at the position from the right corresponding to the
power of ten of the denominator. Examples:
0.0008
8    
0.083 8
10000
0.83 83
1000
0.8 83
100
10
In English-speaking and many Asian countries, a period (.) is used as the decimal
separator; in many other languages, a comma (,) is used (e.g. in France and
Germany)
The part from the decimal separator (decimal point) to the right is the fractional
part; if considered as a separate number, a zero is often written in front (example:
0.23). Trailing zeros after the decimal point are not necessary, although in
science, engineering and statistics they can be retained to indicate a required
precision or to show a level of confidence in the accuracy of the number. Whereas
0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a
measurement with an error of up to 1 part in one thousand (±0.001), while 0.08
suggests a measurement with an error of up to 1 part in one hundred.
The integer part or integral part of a decimal fraction is the part to the left of the
decimal separator (decimal point). Decimal fractions can be expressed as fractions
by converting the digits after the decimal separator to fractions in the base ten or
power of ten. Example:
2 34
100
2.34  2  3      2  34

100
4
100
2 30
100
4
100
10
Fractions with denominators of base-ten, or power of ten, can be directly
expressed as decimal fractions. Decimal fractions are confined to tenths,
hundredths, thousandths and other powers of ten. Examples:
3.008
2.23 3 8
1 5   
1000
1.5 2 23
100
10

3.1.3 Extension of Base-10 Place Value System
To help pupils understand the meaning of decimal fractions and its relationship to
place value, you should first give opportunities to them to see and investigate the
pattern that exists between place values in whole numbers. This is because the
concept of place value in whole numbers is the basis of decimal fractions. In fact,
decimal fractions should be introduced as an extension of base-10 place value.
In the base-10 place value, it is clear that the digit in every place value is 10 times
more than the digit which is to the right of it. In other words, the digit in every
place value is 1
of the digit which is to the left of it.
10
Thousands Hundreds Tens Units
1000 100 10 1
1 1 1 1
The digit in the hundreds
place value is
1 of the digit
10
in the thousands place value.
The digit in the units place
value is
1 of the digit in the
10
tens place value.
By exploring the pattern that exists in the place value system, you should extend
the procedure to identify the place value to the right of units. All the place values
to the right of units represent decimal parts (parts of the number which is less than
one). To show the separation between the decimal part and the whole number
part, a decimal point (decimal separator) is placed after the place value of units.
Thousands Hundreds Tens Units Tenths Hundredths Thousandths
1000 100 10 1
1
10
1
100
1
1000
1000 100 10 1 0.1 0.01 0.001
From the extended place value system above, you can see that there is a symmetry
between the place values. The centre of symmetry is the place value of units. With

the help of this extended place value system, pupils would be able to explain
every place value and appreciate the meaning of decimal fractions.
Apart from that, it would be easier to read decimal fractions if pupils could write
the decimal fractions according to the extended place value system. The digits
before the decimal point will be read according to the place value, whereas the
digits after the decimal point will be read as the digits themselves.
For example:
6.5 is read as “six point five”
12.34 is read as “twelve point three, four”
45.005 is read as “forty five point zero, zero, five”
235.237 is read as “ two hundred and thirty five point two, three,
seven”
One or more than one Less than One
Thousands Hundreds Tens Units Tenths Hundredths Thousandths
1000 100 10 1
1
10
1
100
1
1000
1000 100 10 1 0.1 0.01 0.001
6 • 5
1 2 • 3 4
4 5 • 0 0 5
2 3 5 • 2 3 7
For the decimal fraction, 235.237, the first digit after the decimal point, 2, is the
tenths digit, the second digit, 3 is the hundredths digit, and the third digit, 7 is the
thousandths digit.
3.1.4 Decimal Place
Now, how are you going to teach pupils to count the number of decimal places.
The decimal place (d.p.) for decimal fractions is counted by adding the number of
digits after the decimal point. For example:
2 3 5 . 2 3 7 has 3 decimal places
1 digit + 1 digit + 1 digit = 3 (3 digits after the decimal point)

SELF-CHECK 3.1
1. Though the meaning of decimal number is accepted by all, the
symbol (the way the decimal separator is used) still varies. List
down the various symbols used for decimal numbers.
2. How can you show the place value of hundredths with the help of
a diagram of a concrete model ?
DECIMALS
3.2
A systematic conceptual development of decimals will be helpful for your pupils
to learn this topic efficiently and effectively. It would be beneficial to introduce
this topic in a meaningful way. In order to provide opportunities for your pupils to
develop their understanding of decimal numbers in a less stressful manner, you
should use models such as decimal squares, square grids, number lines, base-ten
blocks. You should also provide opportunities for your pupils to acquire
mathematical skills involved in adding, subtracting, multiplying and dividing
decimal numbers. Pupils should be exposed to real life contexts that apply
practical usage of decimals.
The major mathematical skills to be mastered by pupils studying decimals in Year
5 and Year 6 are as follows:
(a) Name and write decimal numbers to three decimal places.
(b) Recognise the place value of thousandths.
(c) Convert fractions of thousandths to decimal numbers and vice versa.
(d) Round off decimal numbers to the nearest:
(i) tenth,
(ii) hundredth.
(e) Add any two to four decimal numbers up to three decimal places involving:
(i) decimal numbers and decimal number
(ii) whole numbers and decimal numbers
(f) Solve problems involving the addition of decimal numbers.

(g) Subtract a decimal number from another decimal number up to three
decimal places.
(h) Subtract successively any two decimal numbers up to three decimal places.
(i) Solve problems involving subtraction of decimal places.
(j) Multiply any decimal number up to three decimal places with:
(i) a one-digit number,
(ii) a two-digit number,
(iii) 10, 100 and 1000.
(k) Solve problems involving multiplication of decimal numbers.
(l) Divide a whole number by:
(i) 10
(ii) 100
(iii) 1000
(m) Divide a whole number by:
(i) a one-digit number
(ii) a two-digit number
(n) Divide a decimal number of three decimal places by:
(i) a one-digit number
(ii) a two-digit whole number
(iii) 10
(iv) 100
(o) Solve problems involving division of decimal numbers.
(p) Add and subtract three to four decimal numbers of up to 3 decimal places
involving:
(i) decimal numbers only
(ii) whole numbers and decimal numbers
(q) Solve problems involving addition and subtraction of decimal numbers.

3.3
Let us look at a few activities for pupils to develop their understanding of decimal
numbers and major mathematical skills for decimals.
3.3.1 Decimal Numbers
ACTIVITY 3.2
Learning Outcomes:
 To write the decimal that represents the shaded parts
 To write the decimal numbers in words
 To write the place value of the underlined digits
 To compare the value of the two decimal numbers
Materials:
 Task Cards
 Answer Sheets
Procedure:
1. Divide the class into groups of six pupils. Each student is given an
Answer Sheet.
2. Ask pupils to write their names on the Answer Sheet.
3. Shuffle Six Task Cards and place them face down in the centre.
4. Each player begins by drawing a card from the stack.
5. The player writes all the answers to the questions in the Task Card
drawn on the Answer Sheet.
6. After a period of time (to be determined by the teacher), the pupils
(in their groups) exchange the cards with the pupil on their left in
clockwise direction.
7. Pupils repeat steps (5 and 6) until all the members of the group
have answered the questions in all the cards.
8. The winner is the pupil that has the most number of correct
answers.
9. The teacher summarises the lesson on the basic facts of decimal
numbers.

Example of an Answer Sheet:
Name :________________________ Class :______________________
Card A Card B Card C
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
4.________________ 4.________________ 4.________________
Card D Card E Card F
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
4.________________ 4.________________ 4.________________
Example of a Task Card:
Card A
1. Write the shaded part in decimals.
2. Write in words.
1.408 =
3. Write the place value of the underlined digit.
8.354 =
4. Circle the decimal with the largest value.
27.357 27.537 27.753 27. 375

ACTIVITY 3.3
Work with your colleagues or cousemates to prepare another five
Task Cards. There should be four questions in each card. Make sure
your cards are based on the learning outcomes of Activity 3.2.
3.3.2 Converting Fractions to Decimal Numbers and
Vice Versa
ACTIVITY 3.4
Learning Outcomes:
 To convert fractions to decimal numbers
 To convert decimal numbers to fractions
 To round off decimal numbers to the nearest tenth
 To round off decimal numbers to the nearest hundredth
Materials:
 30 different Flash Cards
Procedure:
1. Divide the class into groups of three pupils and give each group a
clean writing sheet.
2. Ask pupils to shuffle the Flash Cards and place them face down in
a stack at the centre.
3. Player A begins by drawing a card from the stack. He shows the
card to Player B.
4. Player B then reads out the answers within the stipulated time
(decided by the teacher).
5. Player C writes the points obtained by Player B below his name.
Each correct answer is awarded one point (a maximum of 4 points
for each Flash Card).

7. Steps 4 and 5 are repeated until 10 cards have been drawn by Player A.
8. Players now change roles. Player B draws the cards, Player C reads out
answers and Player A keeps the score.
9. Steps (3 through 6) are repeated until all the players have the
opportunity to read 10 Flash Cards shown to them.
10. The winner in the group is the pupil that has the highest score.
11. The teacher summarises the lesson on the basic facts of decimal
numbers.
Example of a Flash Card:
Flash Card 1
1. Convert this decimal number to a fraction.
0.083 =
2. Convert this fraction to a decimal number.
154
1000
=
3. Round off the decimal number to the nearest tenth.
3.628 =
4. Round off the decimal number to the nearest hundredth.
15.589 =
ACTIVITY 3.5
Work with a few colleagues or cousemates to prepare another 29 Flash
Cards. There should be four questions in each Flash Card. Make sure
your cards are based on the learning outcomes of Activity 3.4.

3.3.3 Addition of Decimal Numbers
ACTIVITY 3.6
Learning Outcomes:
 To add two to four decimal numbers up to three decimal places
 To add two to four decimal numbers involving whole numbers and
decimal numbers
 To solve problems involving the addition of decimal numbers
Materials:
 Task Sheets
 Colour pencils
Procedure:
1. Divide the class into groups of four to six pupils. Give each group
a different colour pencil and a clean writing sheet.
2. The teacher sets up five stations in the classroom. A Task Sheet is
placed at each station.
3. The teacher instructs pupils to solve the questions in the Task
Sheet at each station.
4. Each group will spend 10 minutes at each station.
5. At the end of 10 minutes, the groups will have to move on to the
next station in a clockwise direction.
6. At the end of 50 minutes, teacher will collect the answer papers.
7. The group with the highest score (highest number of correct
answers) is the winner.
8. Teacher summarises the lesson on how to add decimal numbers up
to three decimal places.

Example of a Task Sheet:
STATION 1
1. Add the following decimal numbers. Express your answers in three
decimal places.
(a) 1.724 + 3.055 =
(b) 9.2 + 2.32 + 0.535 =
(c) 6.07 + 5.234 + 2.5 + 0.56 =
2. Add the following whole numbers and decimal numbers. Express
your answers in three decimal places.
(a) 6 + 3.652 =
(b) 2.345 + 7 + 4.78 =
(c) 4.534 + 2.43 + 6.8 + 8 =
3. The length of ribbon A is 21.43m. Ribbon B is 3.26m longer than
ribbon A. What is the total length of the two ribbons?
The total length of the two ribbons is
ACTIVITY 3.7
Work with two of your friends to prepare another four Task Sheets for
the other stations. There should be three questions in each sheet.
Make sure your sheets are based on the learning outcomes of Activity
3.6.

3.3.4 Subtraction of Decimal Numbers
ACTIVITY 3.8
Learning Outcomes:
 To subtract two decimal numbers up to three decimal places
 To subtract successively any two decimal numbers up to three
decimal places
 To solve problems involving subtraction of decimal numbers
Materials:
 Activity Cards
 Colour pencils
Procedure:
1. Divide the class into groups of four pupils. Give each group a
different colour pencil and a clean writing sheet.
2. Ask pupils to shuffle a set of 12 Activity Cards and place them
face down in a stack at the centre.
3. Teacher instructs pupils to draw an Activity Card and begin
solving the questions on the first Card drawn.
4. Once they have answered the questions on the first Card, they may
continue with the next Activity Card.
5. At the end of 10 minutes, the groups will stop and hand their
answer papers to the teacher.
7. The teacher summarises the lesson on how to subtract decimal
numbers up to three decimal places.

Example of an Activity Card:
1. Subtract the following two decimal numbers. Express your answers in
three decimal places.
(a) 7.34 – 3.567 =
(b) 23. 6 – 11. 782 =
2. Carry out the subtraction of the following decimal numbers. Express
your answers in three decimal places.
(a) 6.7 – 1.24 – 3.007 =
(b) 50.23 – 15.14 – 12.224 =
3. A fence measuring 12.47m needs to be painted. If 7.029m of the fence
has been painted, how many metres more need to be painted?
more need to be painted.
ACTIVITY 3.9
Work in pairs to prepare another 11 Activity Cards for the group.
There should be three questions in each card.
Make sure your cards are based on the learning outcomes of Activity
3.8.
ACTIVITY 2.4

3.3.5 Multiplication of Decimal Numbers
ACTIVITY 3.10
Learning Outcomes:
 To multiply decimal numbers with one-digit whole numbers
 To multiply decimal numbers with two-digit whole numbers
 To solve problems involving multiplication of decimal numbers
Materials:
 Exercise Sheets
 Colour pencils
Procedure:
1. Divide the class into pairs (two pupils in each group).
2. Give each group a different colour pencil.
3. Provide each group with an Exercise Sheet containing five
questions each.
4. The group that finishes fastest with all correct answers will be the
winner.
5. The teacher summarises the lesson on how to multiply whole
numbers with decimal numbers.
ACTIVITY 3.10

Example of an Exercise Sheet:
1. Solve the following multiplication problems.
(a) 6.42  7 =
(b) 3.456  15 =
2. Solve the following multiplication problems.
(a) 2.34  10 =
(b) 0.346  100 =
3. Mr. Lee bought 6 pieces of iron rods. The length of each iron rod is
4.56m. Find the total length of the iron rods.
Total length of the iron rods is
4. A box of grapes weighs 7.2 kg. A box of oranges weighs 3 times the
mass of the box of grapes. What is the mass of the box of oranges?
The mass of the box of oranges is
5. A packet of green apples weighs 3.402 kg. What is the total weight of
100 packets of green apples?
The total mass of 100 packets of green apples is

3.3.6 Division of Decimal Numbers
ACTIVITY 3.11
Learning Outcomes:
 To divide decimal numbers with 10, 100, 1000
 To divide decimal numbers with one-digit numbers
 To divide decimal numbers with two-digit whole numbers
 To solve problems involving the division of decimal numbers
Materials:
 Division Worksheets
 Colour pencils
Procedure:
1. Divide the class into 10 groups. Give each group a Division
Worksheet, clean writing paper and a colour pencil.
2. Teacher instructs the groups to answer all the questions in the
Divison Worksheet.
3. The group answers on the clean writing paper provided.
4. After a period of time (to be determined by the teacher), the
teacher instructs the groups to exchange the Division Worksheets.
5. Repeat Steps 2 to 4.
6. Once all the 10 Division Worksheets have been answered, the
teacher collects the answer papers and corrects the answers.
8. The teacher summarises the lesson on how to divide decimal
numbers with whole numbers.

Example of a Division Worksheet:
WORKSHEET 1
1. Solve the following division problems.
(a) 921  100 =
(b) 8652  1000 =
2. Solve the following division problems.
(a) 44.272  8 =
(b) 18.324  12 =
3. Puan Rohana pours 3.26 litres of syrup equally into 5 bottles. What
is the volume of syrup in each bottle?
The volume of syrup in each bottle is
4. Mrs. Rama put 31.85 kg of prawns equally into 7 boxes. What is the
mass of prawns in each box?
The mass of prawns in each box is
ACTIVITY 3.12
Prepare another nine Division Worksheets for the groups. There should
be four questions in each worksheet.
Make sure your worksheets are based on the learning outcomes of
Activity 3.11.
ACTIVITY 3.12

 The three commonly used representations for fractions namely the area
models (e.g., fraction circles, paper folding, geo-boards), linear models (e.g.,
fraction strips, Cuisenaire rods, number lines), and discrete models (e.g.,
counters, sets) can be also used to explain the concept of decimals.
 The decimal (base-ten or sometimes denary) numeral system has ten as its
base.
 Decimal notation is the writing of numbers in the base-10 numeral system,
which uses various symbols (called digits) for no more than ten distinct values
(0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large.
 A decimal fraction is a special form of fraction where the denominator is in
the base-ten, or a power of ten.
 The integer part or integral part of a decimal fraction is the part to the left of
the decimal separator (decimal point).
 Decimal fractions can be expressed as fractions by converting the digits after
the decimal separator to fractions in the base ten or power of ten.
 All the place values to the of right of units represent decimal parts (parts of
the number which are less than one).
 The separation between the decimal part and the whole number part is shown
by a decimal point placed after the place value of units.
 The digits before the decimal point will be read according to the place value,
whereas the digits after the decimal point will be read as the digits themselves.
 The decimal place (d.p.) for decimal fractions is counted by adding the
number of digits after the decimal point.

Addition
Decimal
Decimal fractions
Decimal point
Decimal place
Decimal separator
Integral part
Subtraction
Multiplication
Division
Anne Toh. (2007). Resos pembelajaran masteri: Mathematics year 3. Petaling
Jaya: Pearson Malaysia.
Bahagian Pendidikan Guru (1998). Konsep dan aktiviti pengajaran dan
pembelajaran matematik: Perpuluhan dan peratus. Kuala Lumpur: Dewan
Bahasa dan Pustaka.
Nur Alia bt. Abd. Rahman & Nandhini (2008). Siri intensif: Mathematics KBSR
Nur Alia bt. Abd. Rahman & Nandhini (2008). Siri intensif : Mathematics KBSR
year 6. Kuala Lumpur. Penerbitan Fargoes.
Ng S.F. (2002). Mathematics in action workbook 2B (Part 1). Singapore: Pearson
Education Asia.
Peter Clarke et al. (2002). Maths spotlight activity sheets 1. Oxford: Heinemann
Educational Publishers.
Sunny Yee & Lau P.H. (2007). A problem solving approach: Mathematics year 3.
Subang Jaya: Andaman Publication.

Topic
4
 Money
LEARNING OUTCOMES
1. Demonstrate to your pupils how to use the vocabulary related to
money correctly for the topic of Money in the KBSR Mathematics
Syllabus;
2. Illustrate the major mathematical skills and basic pedagogical content
knowledge related to the addition and subtraction of money up to the
value of RM 10,000,000;
3. Illustrate the major mathematical skills and basic pedagogical content
knowledge related to the multiplication and division of money up to
the value of RM 10,000,000;
4. Plan basic teaching and learning activities for the topic of Money up to
a value of RM 10,000,000; and
5. Plan basic teaching and learning activities to help pupils solve daily
problems related to money.
 INTRODUCTION
The lifelong benefits of teaching children good money habits make it well worth
the effort. Children who are not taught these lessons face the consequences for a
lifetime. Some parents do not teach children about money because they think they
should not talk about money with children, do not have the time, or think they do
not have enough money.
ACTIVITY 4.1
Most people have strong feelings and opinions about money, based
on childhood experiences and the values and beliefs of their families.
Do you agree with the above statement? Discuss the truth of this
statement with your coursemates.

64  TOPIC 4 MONEY
4.1
Teaching children about money is more than preparing them for employment or
teaching them to save some of the money they earn. It includes helping them
understand the positive and negative meanings of money. For example, children
need to learn that while it is nice to show someone love by buying a gift, it is just
as important to show love through actions and words. Teachers and children
should talk about their feelings, values, attitudes, and beliefs about money. This
helps children understand the issues that may occur due to money and that
compromises are often necessary to deal with them.
ACTIVITY 4.2
1. How do you create an open environment to discuss money
issues?
2. How do you respond to the effects of advertising and peer
pressure on our children's requests for things?
4.1.1 Teaching Children About Money
How do you teach your children the topic on money? When teaching children
about money, teachers need to make an effort to think from the children's point of
view, not from adults’ point of view. For instance, a young child may ask his or
her parents how much money they make, but what they really want to know is not
how much their parents earn, but why they cannot have certain toys or why their
family cannot go for holidays overseas. It is important for teachers to use
examples or activities that match the child's stage of development, not necessarily
the child's actual age in years.
It is also important for teachers to communicate with children about money
matters in very concrete terms. Children want to know how to operate in the adult
world. Any time money is earned, moved, spent, donated, shared, borrowed or
saved provides an opportunity for teachers to teach children how the money world
works and what thoughts and feelings go into making money decisions.
Children should be introduced to the origin of money; the barter system, the use
of objects to represent money and the use of coins and notes in various
denominations. Children learn mainly through observation and example;
participation in discussions and group decision making; direct teaching through
planned experiences; and by making their own decisions. Through observation,
children learn a great deal more than teachers realise. Teachers can add to this

TOPIC 4 MONEY  65
experiential learning through intentionally planned learning activities. As you
teach children about money they can learn about:
1. Responsibility;
2. Family values and attitudes;
3. Decision-making;
4. Comparison-shopping;
5. Setting goals and priorities; and
6. Managing money outside the home.
Let us learn about teaching money concepts in the next section. Enjoy!
4.1.2 Teaching Money Concepts
The right focus
I was browsing through the chapter on
Money in the Mathematics Year One
textbook that my children are using in
school when it suddenly struck me that we
may not be teaching our children the right
values about money.
Almost all the problem-solving questions
in the textbook focus on buying things and
totalling up the amount spent.
Why can’t the writers ask better
questions, for instance, those which revolve
around saving money and using it wisely?
Questions pitched from this angle would
help to inculcate good values and teach our
children to be money-savvy at the same
time.
I think the present focus imparts
unhealthy values about money to our
children from Year One.
Something is not quite right here.
H.C. FOO
(Source: Sunday STAR, 30 March 2008)
What do you think about the truth as expressed by H.C. FOO? Do you think there
is a need to educate children on the concepts of earning, saving, borrowing and
sharing, besides spending?
These financial concepts of earning, spending, saving, borrowing, and sharing are
generic money concepts. Earning refers to how children receive money. Spending
refers to the way children decide to use their money. Saving refers to money that
the children set aside for some future use. Borrowing means that money can be
obtained for use in the present but must be paid back in the future with an
additional cost. Sharing means both the idea of sharing what we have with those
who are less fortunate and obligations such as paying taxes which are required of

everybody. By providing children with intentional learning experiences related to
these financial concepts we can provide children practical skills and knowledge
and a perspective on money based upon values and beliefs. Among the benefits of
teaching these concepts are:
Earning teaches:
(a) Financial independence
(b) Work standards and habits
(c) How to evaluate job alternatives
(d) Relationship of money, time, skills and energy
Spending teaches:
(a) Difference and balance between wants and needs
(b) Opportunities for comparing alternatives
(c) Making decisions and taking responsibility for them
(d) Keeping records
Borrowing teaches:
(a) Cost of borrowing
(b) Borrowed money needs to be paid back
(c) When it is appropriate to borrow
(d) Consequences of buying now and paying later
(e) Structure of borrowing
(f) The idea of credit limits
Sharing teaches:
(a) Good feelings for giver and receiver
(b) Helps other people
(c) Doesn't always require public recognition
(d) Obligations to give money to certain organisations, i.e. taxes to the
government
(e) Giving of yourself rather than giving money or gifts

Saving teaches:
(a) How to get what you want or need by saving for it
(b) Planning and delayed gratification
(c) Interrelationship of spending and earning
(d) Different purposes of planned and regular saving
(Source: Sharon M. Danes and Tammy Dunrud, 2002. University of Minnesota)
Now, let us look at some mathematical skills, beginning with how to model
decimals using coins.
4.1.3 Using Coins to Model Decimals
Do you know how to model decimals? Some teachers use coins to model
decimals. Recording amounts in Ringgit and sen does involve decimal fractions,
but care must be taken on how the children see the connection between the sen
and the fractional part of a decimal number.
For example, children do not readily relate RM75.25 to RM75 and 25 hundredths
of a Ringgit or 10sen to one-tenth of a Ringgit. If money is used as a model for
decimals, children need to think of 10 sen and 1 sen as fractional parts of a
Ringgit.
It is common to find in supermarket advertisements the use of incorrect decimal
notations. For example, the price of an item may be indicated as .75 sen. The
assumption is that .75sen means the same as RM0.75. In fact .75 sen means 75
hundredths of a sen! It is important to provide children with opportunities to
practise recording money correctly.
SELF-CHECK 4.1
1. Explain with examples, the meaning of the following
statement:
“When teaching children about money, teachers need to make
an effort to think from children's point of view, not from
adults’ point of view ”.
2. State the benefits of teaching the concept of saving and earning
money.

MONEY
4.2
The introduction of money usually follows instruction on the basics of fraction
and decimal skills. Teachers should note that various basics of fraction and
decimal skills are prerequisite skills for the topic of money.
The major mathematical skills to be mastered by pupils studying the topic of
money are as follows:
(a) Read and write the value of money in ringgit and sen up to RM10 million.
(b) Add money in ringgit and sen up to RM10 million.
(c) Subtract money in ringgit and sen within the range of RM10 million.
(d) Multiply money in ringgit and sen with a whole number, fraction or decimal
with products within RM 10 million.
(e) Divide money in ringgit and sen with the dividend up to RM10 million.
(f) Perform mixed operations of multiplication and division involving money in
ringgit and sen up to RM10 million.
(g) Solve problems in real context involving money in ringgit and sen up to RM
10 million.
(h) Perform mixed operations with money up to a value of RM10 million.
4.3
Below are several activities for pupils to understand basic operations on money.
They also can acquire the major mathematical skills involved in adding,
subtracting, multiplying and dividing money.

4.3.1 Basic Operations on Money
ACTIVITY 4.3
Learning Outcome:
 To practise the basic operations on money.
Materials:
 A deck of cards comprising sets of question cards and answers.
Example:
RM 1 642 000
- RM 871 420
RM 167 234 X 23 =
RM 770 580 RM 3 846 382
Procedures:
1. Prepare cards comprising sets of question cards and answers.
2. Place the answer cards (grey cards) in a circle on the floor.
3. Instruct the children to march around the circle of answer cards on
the floor, chanting this rhyme:
Basic operations, ‘round we go,
Not too fast and not too slow.
We won’t run and we won’t hop,
We are almost there, it’s time to stop.
4. When the rhyme finishes, the teacher will hold up a question card
(white card) and ask them to work out the answer to the question.
5. The child who is standing by the card with the answer to the
question, picks up the answer card and shows it to the rest of the
children.
6. Instruct the children to check his or her answer. Is she or he
correct?
7. Repeat the procedure several times or until all the answer cards
have been picked up.
8. The child with the most answer cards wins and is awarded a prize.

ACTIVITY 4.4
Learning Outcome:
 To practise the basic operations on money
Materials:
 Four lists of questions on mixed operations with money. Some of the
questions may be repeated on each list.
 Answers to the questions.
Example:
List 1
1. RM 328 200 + RM 6 720 X 15 =
2. RM 564 000 ÷ 40 + RM 484 120 =
3. RM 1 875 223 – RM 956 600 ÷ 20 =
4. RM 12 875 X 12 + RM 840 280 =
5. RM 840 280 ÷ 20 – RM 9 027 =
6. RM 2 411 610 – RM 21 140 X 22 =

List 2
1. RM345,225 + RM2,550 X 24 =
2. RM564,000 ÷ 40 + RM484,120 =
3. RM528,500 – RM225,000 ÷ 20 =
4. RM56,780 X 12 + RM450,228 =
5. RM840,280 ÷ 20 – RM9,027 =
6. RM2,667,345 – RM18,246 X 32 =
Procedures:
1. Prepare four lists of questions on mixed operations with money.
Some of the questions may repeated on each list.
2. Prepare 24 cards, each containing an answer for each of the 24
questions. Tape these cards to the walls around the classroom.
3. Divide the children into four teams.
4. Give one list to each team. (You might want to provide a copy of
the list for every member of the team).
5. Ask the children to calculate the answers to the questions on their
list.
6. Ask the team members to search for the answer cards taped on the
walls of the classroom.
7. The first team to correctly calculate the answers to all the questions
in their list and collect all the answer cards wins and will be
awarded a prize.

ACTIVITY 4.5
Learning Outcome:
 To practice the basic operations on money.
Materials:
 A deck of cards comprising sets of question cards and answers.
Example:
RM328,100 ÷ 25
+ RM532,590 =
RM545,714
Procedures:
1. Prepare cards comprising sets of question cards and answer cards.
The questions on mixed operations should involve money in
ringgit and sen up to ten million Ringgit.
2. Hand a card to each child. Some of the children will get question
cards and some will get answer cards.
3. Get the children holding the card with the question to calculate its
answer.
4. Ask the children to find their partner holding the card showing the
answer to the question.
5. If there is an odd number of children in the class, you should take a
card and participate so that everyone has a partner.
6. Have the partners stand together so that everyone can see the
other’s card. Ask the children to check everyone’s calculation.
Are the partners matched correctly?
7. Hand out a Task Sheet containing ten questions on mixed
operations with money up to ten million Ringgit and have the
children work out the answers to reinforce their understanding of
mixed operations with money.

4.3.2 Problem Solving on Money
ACTIVITY 4.6
Learning Outcomes:
 To practise the basic operations on money.
 To solve daily problems involving money.
Materials:
 Sets of cards
Procedures:
1. Instruct the children to form groups of three.
2. Make three sets of the Game Cards and cut out the cards.
3. Give each group a set of the cards.
4. Shuffle the cards and spread them out face down on the table.
5. Ask the children to take turns to choose two cards and place them
face up on the table.
6. If the cards show a word problem and its matching calculation,
give the child time to solve the problem. If the pupil can give the
correct answer, the child keeps both the cards.
7. If the cards that the child chose do not show a word problem and
its matching calculation or the child offers an incorrect answer to
the problem, the cards are replaced in their original position on the
table.
8. When all the cards have been chosen, the children will count how
many cards they have. The winner is the child with the most
number of cards.
ACTIVITY 1

A Proton Iswara costs
RM26,754. A Waja
costs RM65,467. How
much cheaper is the
Proton Iswara than the
Waja?
RM65,467
- RM26,754
12 girls bought a gold
chain as a wedding
present for a friend.
Each paid RM725.
What was the cost of
the gold chain?
RM725
x 12
The usual price of a
luxurious car is
RM236,789. Its sale
price is RM199,888.
How much is the
difference between the
sale price and the
usual price?
RM236,789
- RM199,888
8 brothers and sisters
shared an inheritance
of RM3,465,000
equally. How much
money does each of
the siblings receive?
RM3,465,000 ÷ 8
=
Pn Salmah bought a
refrigerator and a
stove. The refrigerator
cost RM2,225. The
stove cost RM4,355
more than the
refrigerator. How much
did she spend
altogether?
RM2,225
RM2,225
+ RM4,355
A single-storey house
costs RM93,888. A
double-storey
bungalow costs 6
times as much as the
single-storey house.
Find the cost of the
double-storey
bungalow.
RM93,888
x 6
Dr Chen donated
RM121,000 to Rumah
Charis and
RM324,500 to Rumah
Chaya. He had
RM3,500,000 left. How
much money did he
have at the beginning?
RM121,000
RM324,500
+ RM3,500,000
Mustafa has
RM345,000 as
savings. He has 5
times as much money
as his brother. How
much money does his
brother have?
RM345,000
÷ 5
Suhaimee has
RM55,345 in his
savings. His mother
gave him some more
money. He now has
RM115,300. How
much money did his
mother give him?
RM115,300
- RM55,345
Syarikat Jefa donated
RM125,700 and
RM67,000 to two relief
funds. What is Syarikat
Jefa’s total donation ?
RM125,700
+ RM67,000

GAME CARDS
ACTIVITY 4.7
Learning Outcome:
 To solve daily problems involving money.
Materials:
 Sets of catalogues
Procedures:
1. Instruct pupils to form groups of four.
2. Give each pupil in the group a different catalogue.
3. Tell each group that its the newspaper’s 10th Anniversary. In
conjuction with their anniversary celebration, they are carrying out
some charity work.
4. The publisher of the newspaper has generously donated
RM250,000 to the school. The money will be used to further equip
the school resoure centre.
5. Each person in the group is to study the catalogue provided to him
or her.
6. The person is to write the name and cost of one or two items that
he or she feels would be of use to the school resource centre.
7. Using the round robin format of the cooperative learning
technique, members of the group will discuss each item chosen
and why it was chosen. One member of the group serves as a
recorder.
8. The group will have to come out with a final list of items to be
purchased. The group may need to make adjustments to keep the
total cost below RM250,000.
9. Prepare a bulletin-board to display the list of items presented by
the groups. Displays help pupils to recap what they have learned
and it is also a means of seeing the practical applications of
mathematics.

 Teaching children about money is more than preparing them for employment
or teaching them to save some of the money they earn. It includes helping
them understand the positive and negative aspects of money.
 Teachers and children should talk about their feelings, values, attitudes and
beliefs about money.
 When teaching children about money, teachers need to make an effort to think
from the children's point of view, not the adults’ point of view.
 As you teach children about money they can learn about responsibility; family
values and attitudes; decision-making; comparison-shopping; setting goals and
priorities; and managing money outside the home.
 The financial concepts of earning, spending, saving, borrowing, and sharing
are generic money concepts.
 Some benefits of providing intentional learning experiences related to these
financial concepts are children’s mastery of practical skills and knowledge, as
well as a perspective about money based upon values and beliefs.
 Recording amounts in Ringgit and sen does involve decimal fractions, but care
must be taken on how children see the connection between the sen and the
fractional part of a decimal number.
 It is important to give children contextual examples on the use of money.
Coin
Money
Note
Value

Hatfield, M. H., Edwards, N. T., & Bitter, G. G. (1993). Mathematics methods for
the elementary and middle school. Needham Heights, MA: Allyn & Bacon.
Kennedy, L. M., & Tipps, S. (2000). Guiding children’s learning of mathematics.
US: Allyn &Wadsworth.
Rucker, W. E., & Dilley, C. A. (1981). Heath mathematics. Washington, DC:
Heath and Company.
Tucker, B. F., & Weaver, T. L. (2006). Teaching mathematics to all children.
Ohio: Merrill Prentice Hall.
Vance, J. H., & Cathcart, W. G. (2006). Learning mathematics in elementary and
middle schools. , Ohio: Merrill Prentice Hall.

Topic
5
 Percentages
LEARNING OUTCOMES
1. Demonstrate the importance of developing the basics of fraction and
decimal skills as prerequisites to the learning of percentages;
2. Use the vocabulary related to percentages correctly;
knowledge related to percentages; and
4. Plan basic teaching and learning activities for percentages.
 INTRODUCTION
Basically, percentages are used in many everyday situations. Children probably
already know a bit about percentages. They are exposed to percentages when they
go shopping with their parents. Shops use percentages in sales. Banks use them
for loan rates. Schools use percentages in their forecast of examination results.
Unfortunately, they are also often incorrectly used. For example, a store advertises
prices reduced by 100%, rather than 50%; an interest rate of .03%, rather than 3%;
and a school reports the number of straight A’s pupils increased by 200%, which
is correct, but a little misleading, since the number of pupils that scored straight
A’s went up from 1 to 3!
ACTIVITY 5.1
Visit the Math Forum website:
http://mathforum.org/dr.math/tocs/fractions.middle.html
Find out the frequently asked questions about percentages in the
website.

TOPIC 5 PERCENTAGES  79
The introduction of percentages usually follows instruction after the mastery of
basic fraction and decimal mathematical skills.
Teachers should note that various fraction and decimal skills are prerequisite
skills for learning percentages. For example, to solve a percentage problem, the
pupil must be able to convert a percentage into a fraction or a decimal as shown
below:
For example,
26
100
= 26%, and 45% = 45
100
= 0.45
5.1.1 Meaning and Notation of Percent
Figure 5.1: The various sales discount signs that we often see in shopping centres.
[Source: http://www.bbc.co.uk]
5.1
ACTIVITY 5.2
1. Change the following percents to decimals: ½ %, ⅘ %, ⅝ %.
2. Develop an instructional sequence to teach pupils how to
change percents like ½ %, ⅘ %, or ⅝ % to decimals.

HBMT 3203

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HBMT 3203