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
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;
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.
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.

Name :________________________ Class :______________________
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.________________
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:
Procedure:
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
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.
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.
3. The teacher instructs pupils to solve the questions in the Task
Sheet at each station.
next station in a clockwise direction.
6. At the end of 50 minutes, teacher will collect the answer papers.
8. Teacher summarises the lesson on how to add decimal numbers up
to three decimal places.

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.
answer papers to the teacher.
7. The teacher summarises the lesson on how to subtract decimal
numbers up to three decimal places.

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.
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

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
2. Teacher instructs the groups to answer all the questions in the
Divison Worksheet.
3. The group answers on the clean writing paper provided.
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
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
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
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;
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.
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
US: Allyn &Wadsworth.
Rucker, W. E., & Dilley, C. A. (1981). Heath mathematics. Washington, DC:
Heath and Company.
Ohio: Merrill Prentice Hall.
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.

 TOPIC 5 PERCENTAGES
80
First of all, let us look at the meaning of the term “percent”. The term percent
means “parts per hundred.” It expresses a relationship between some number and
100. The symbol % indicates a denominator of 100. For example, 25% is an
expression of the ratio between the number 25 and 100 and means 25 parts of 100,
or 25 out of 100.
When an item is sold for RM 100, the cost is the base to which the discount is
applied. A 20% off the cost price is the rate of discount, and RM 20 is the amount
of discount, or percentage. The table below illustrates some ways percent is used
and it helps to clarify the confusion about the term per cent and percentage.
Table 5.1: Common Uses of Percent
Rate Base Percentage
Sales 25% off
retail price
of plasma
TV
RM3,200
retail price
RM800
reduction in
price
Service
tax
10%
service tax
RM250
purchased
RM25 in
service tax
charged
Increase
in tax
4% raise in
property
tax
Property tax
of a
RM360,000
house
Increased
RM14,400
in property
taxes
As a teacher, you must make it clear that per cent indicates the rate (of discount,
and taxes), whereas percentage indicates the amount, or quantity (of discount
and taxes). Note that the base and percentage always represent numbers that refer
to the same units, and per cent is the rate by which percentage compares with the
base. However in the Year 5 and Year 6 textbooks, percentage is represented with
the symbol ‘%’ and is called ‘percent’.
Another point of confusion arises when a given rate is applied to different bases.
Consider the result when a RM 50 book is increased by 20%. An increase of 20%
raises the price of the book to RM 60. After a year, the price of the book is
reduced by 20%. Will the price of the book be the same as it was a year ago? In
both cases, the percent is the same; an increase of 20% and a year later a reduction
of 20%. Try calculating it and check if the price of the book a year later is the
same as the price before the 20% increment?.

5.1.2 Teaching Aids in Learning Per Cent
A key idea in mathematics is that numbers can be represented in many ways. A
rational number can be expressed as a fraction, a decimal, or a percent.
The content readiness children need before they are introduced to per cent is an
understanding of both common and decimal fractions. The pedagogical readiness
required is an understanding of the teaching aids they will use.
During introductory and developmental activities each whole unit or set should be
one that is easily subdivided into 100 parts. It is easier for children to understand
the meaning of per cent when they deal with portions of the 100 parts of a unit. As
an example, teachers are encouraged to use the 10-by-10 grid to represent per
cent as shown in Figure 5.2:
Figure 5.2: Using a 10-by-10 Grid to Represent Percents
Source: http://www.bbc.co.uk
This large square is made up of 100 small parts.
10 parts are yellow.
So 10% of the large square is yellow.
40 parts are red.
So 40% of the large square is red.
50 parts are brown.
So 50% of the large square is brown.
Other than the 10-by-10 grid, teachers can also use the Cuisenaire materials.

82
5.1.3 Fraction and Decimal Equivalents
As children show percent on a 10-by-10 grid and reflect on the language they use
to describe their representations, the fraction and decimal names of the numbers
will become apparent.
Example
7% = 7
100
= 0.07
Because 7% (seven per cent) means 7 out of 100, it is seven-hundredths, which is
written as
in fraction notation and 0.07 in decimal notation.
(a) Decimals as percent
Writing a decimal as a per cent involves finding an equivalent decimal in
hundredths. For example,
For example, eight-tenths = eighty percent or
0.8 = 0.8 x 100% = 80%
Children find that to change a decimal to percent, one needs only to multiply
by 100, which means “moving” the decimal point two places to the right.
For example, 0.33 = 33% and 1.2 = 120%.
To express a percent as a decimal, the opposite rule applies. For example,
62.5% = 0.625 and 225% = 2.25.
(b) Fractions as percent
Children who have mastered the meaning of percent as “parts per hundred”
should not have much problem expressing fractions as percent.
For example, 29
100
= 29%.
Children can be challenged to apply this understanding to find ways of
writing a fraction whose denominator is other than 100 as a percent.
For example, 4
5
= 80
100
= 80%.

As a teacher you can tell the pupils that a basic method is to find an
equivalent fraction having a denominator of 100. Another method is to write
the fraction in decimal and then multiply this number by 100.
For example, 4
5
= 0.8 = 80%.
PERCENTAGE
Now, we move on to the major mathematical skill for percentage. Remember that
various basic fraction and decimal skills are prerequisites for learning percentage.
The major mathematical skills to be mastered by pupils when studying the topic
of percentage are as follows:
 Name and write the symbol for percentage.
 State and convert fraction of hundredths to percentage and vice versa.
Example
26
100
= 26% and 45% = 45
100
 Convert proper fractions with the denominations of 2, 4, 5, 10, 20, 25 and 50
to percentage.
Example
4
5
= 80
100
= 80%.
 Convert percentage to decimal number and fraction in its simplest form.
= 0.05 = ½
5.2
SELF-CHECK 5.1
1. Explain the meaning of percent and percentage.
2. Using a suitable teaching aid, explain how you can introduce the
topic on Percentages.
5
100

84
 Convert mixed numbers to percentage.
Example
1
1
2
= 3
2
= 150
100
= 150%.
 Convert decimal numbers of value more than 1 to percentage.
 Find the value for a given percentage of a quality
 Finding values of percentage of a quantity.
 Solve problems in real context involving relationships between percentage,
fractions and decimals.
We move on to the teaching and learning activities in the following section.
5.3
Let us look at a few activities to develop pupils’ understanding of percentage and
master the major mathematical skills for percentage.
5.3.1 Meaning and Notation of Per Cent
ACTIVITY 5.3
Learning Outcome:
 To name and write the notion of per cent.
 To state fraction of hundredths in percentage
Materials:
 10 x 10 Grid.

Procedures:
1. Display a 10 x 10 grid. Ask the children to verify that there are 100
equal squares on the grid.
2. Shade one square and ask a pupil to name the shaded square. .
[one hundredth].
3. Ask for a volunteer to come to the board to write a numeral to name
the shaded square. [ Accept either 1
ACTIVITY 1
100
or 0.01 ].
4. Tell the children that 1
100
can also be named 1 per cent.
5. Explain to the children that percent means per hundred, or out of
hundred.
6. Explain to the children that the symbol % expresses a denominator
of 100. As such, the name of 1 of the small square can be written as
1% and read as one per cent.
7. Ask fo a volunteer to count the number of shaded squares in the
diagram above.
8. Ask the volunteer to come to the board to write a numeral to name
shaded squares. [ 40
100
].
9. Ask the volunteer to express the shaded squares in per cent. [ 40%].
10. Give out the Task Sheet and instruct the children to complete it.

86
TASK SHEET
1. Study the 10-by-10 grid below and fill in the blanks.
ACTIVITY 1
Now 20 parts have been coloured green. 20 out of the 100 is _____,
so ____ % of the square is green.
There are ____ parts not shaded. ____ out of 100 is ____%, so
_____ of the square is not shaded.
What happens if you add up the percentages for the blue, green and
unshaded parts?
____ + ____ + ____ = _____
So, the whole square is equal ______ .
2. Study the picture below and fill in the blanks.

5.3.2 Fraction and Decimal Equivalents
Learning Outcomes:
 To convert proper fractions with denominators of 2, 4, 5, 10, 20, 25
and 50 to percent.
 To convert mixed numbers to percentage
Materials:
Procedures:
1. Display 10 magnetic chips, 4 green and 6 blue on a magnetic
board.
2. Ask for a volunteer to come forward to count the number of
coloured magnetic chips. [10]
3. Ask the children,
“What part of the set of magnetic chips is green?” [ 4/10]
“Can anyone tell what percent of the chips is green?” [ 40% ]
If a child gives the answer as 40%, ask for an explaination of how
it was determined.
If no answer is given, ask, “What must we do to change 4
10
to a
fraction with a denominator of 100?” [Multiply both numerator
and denominator by 10].
Ask, “Why do we do this ?” [ 40
100
is equivalent to 40% ]
ACTIVITY 5.4

88
4. Next, add another 10 green magnetic chips to the magnetic board.
ACTIVITY 1
5. Ask for a volunteer to come forward to count the number of
coloured magnetic chips. [20]
6. Ask the children, “Are the green chips in this set still 40% of the
set?”
“What part of the set of magnetic chips is green?” [ 14
20
]
“Can anyone tell what percent of the chips is green?” [ 70% ]
If a child gives the answer as 70%, ask for an explaination of how it
was determined.
If no answer is given, ask, “What must we do to change 14
20
to a
fraction with a denominator of 100?” [Multiply both numerator and
denominator by 5].
Ask, “Why do we do this ?” [ 14 5
x
20 5
= 70
100
is equivalent to 70% ]
8. Repeat steps (4) through (7) with more examples.
9. Handout Task Sheet and ask pupils to complete it.

TASK SHEET
State the percentage of the shaded region in the diagram below:
ACTIVITY 1
________ % of the figure is shaded.
Convert the following fractions to percentages.
(a)
3
5

(b)
1
2

(c)
3
4

(d)
3
25

(e)
7
1
10

(f)
2
3
5

(g)
1
5
4

(h)
7
6
20


90
ACTIVITY 5.5
Learning Outcomes:
 To convert percentage to decimal number and vice versa
 To convert decimal numbers of values more than 1 to percentages
Materials:
Procedures:
1. Display a 10 x 1 grid on the board.
2. Have a volunteer come forward to count the number of boxes on
the grid. [10]
“What decimal represents the shaded part of the grid ?” [ 0.3 ]
“Can anyone tell what per cent of the grid is shaded?” [ 30% ]
If a child gives the answer as 30%, ask for an explanation on how
it was determined.
If no answer is given, ask, “What decimal fraction represents the
shaded part of the grid?” [ ]
“Can anyone change the fraction to a decimal?” [ 0.3 ]
“What must we do to change a decimal to per cent?” [Multiply by
100].
Ask, “What per cent is 0.3 ?” [ 0.3 x 100 is equivalent to 30% ]
4. Next, show another strip of 10 x 2 grid on the board.
5. Ask for a volunteer to come forward to count the number of boxes
on the grid. [20]
6. Ask the children, “What decimal represents the shaded part of the
grid ?” [ 0.4]
“Can anyone tell what percent of the grid is shaded?” [ 40% ]

8. Ask for an explanation on how it was determined. [Multiply 0.4 by
100 ]
9. Repeat steps (4) through (7) with other examples.
10. Hand out the Task Sheet and ask pupils to complete it.
TASK SHEET
ACTIVITY 1
1. Convert the following decimal to percentage.
(a) 0.4 =
(b) 0.7 = (c) 0.6 = (d) 0.9 =
(e) 0.53 =
(f) 0.78 = (g) 0.13 = (h) 0.66 =
2. Convert the following decimal to percentage.
(a) 1.5 =
(b) 3.1 = (c) 2.7 = (d) 9.1 =
(e) 5.01 =
(f) 1.99 = (g) 3.14 = (h) 8.08 =

92
ACTIVITY 5.6
Learning Outcome:
 To practise the fraction and decimal equivalent of per cent.
Materials:
 A deck of cards comprising 13 numbers in 4 equivalent forms.
Example:
50% ½ 0.5
Procedures:
1. Two, three or four players can play this game. The objective of the
game is to lay all your cards down.
2. Begin by dealing seven cards to each player. The remainder of the
pack is placed face down on the table.
3. Next, the top card from the deck is placed face up near the pack to
begin the discard pile.
4. The first player may either draw the top card from the face down
pile or pick up the top card on the discard pile. The player must
then discard a card, and the turn goes to the next player.
5. When one player has accumulated three cards of equivalent value,
these are laid face up on the table.
6. The player who has the fourth equivalent value for the set may lay
that card face up on the table in front of himself or herself. The
player next to the one who laid down the three equivalent cards
continues the play.
7. When the pack is gone, the discard pile is turned over and becomes
the pack.

8. The first player to lay all his or her cards down wins that hand. Each
player receives 5 points for every card laid down and loses 5 points
for every card still held.
9. The game is over when one player has 100 points or the teacher
gives the instruction to stop playing the game.
Percentage Fraction
10%
20%
25%
50%
75%
 The term per cent means “parts per hundred.” It expresses a relationship
between some number and 100.
 The symbol % indicates a denominator of 100.
 Percent indicates the rate (of discount and taxes), whereas percentage
indicates the amount or quantity (of discount and taxes).

94
 During introductory and developmental activities on per cent, materials used
should be one that is easily subdivided into 100 parts.
 To change a decimal to percent, one needs only to multiply by 100, which means
“moving” the decimal point two places to the right. To express a percent as a
decimal, the opposite rule applies.
 To change a fraction to percent, a basic method is to find an equivalent
fraction having a denominator of 100.
 Another method is to write the fraction in decimal and then multiply this
number by 100.
Percent Percentage
the elementary and middle School. Needham Heights, MA.: Allyn & Bacon.
US: Allyn & Wadsworth.
Rucker, W. E., & Dilley, C. A. (1981). Heath mathematics. Washington DC:
Heath and Company.
middle Schools. Ohio: Merrill Prentice Hall.

Topic
6
 Time
LEARNING OUTCOMES
1. Use vocabulary related to time correctly as required by the Year 5
and Year 6 KBSR Mathematics Syllabus ;
2. Apply the major mathematical skills and basic pedagogical content
knowledge related to time;
3. Use the vocabulary related to addition, subtraction, multiplication
and division of time correctly;
knowledge related to addition, subtraction, multiplication and
division of time; and
5. Plan basic teaching and learning activities of time for Years 5 and 6.
 INTRODUCTION
Throughout history, people have sought out various ways to measure time.
Timekeeping has been an important part of all cultures throughout the centuries.
How did people first tell time? People first told time by looking at the sun as
it crossed the sky. When the sun was directly overhead in the sky, it was the
middle of the day, or noon. When the sun was close to the horizon, it was either
early morning (sunrise) or late evening (sunset).
The history of clocks is very interesting, and there have been many elaborate
types of clocks developed over the centuries. The word clock was first used in the
14th century (about 700 years ago). It comes from the Latin word for bell
"clocca".

96  TOPIC 6 TIME
The oldest type of clock was a sundial, also called a sun clock. Sundials used the
sun to tell the time. The shadow of the sun pointed to a number on a circular disk
that showed you the time. In the picture below, the shadow created by the sun
points to 9, so it is nine o'clock. Since sundials depend on the sun, they can only
be used to tell the time during the day.
SUNDIAL WATER CLOCK PENDULUM CLOCK
Figure 6.1: Types of clocks
A water clock was made of two containers of water, one higher than the other.
Water travelled from the higher container to the lower container through a tube
connecting the containers. The containers had marks showing the water level, and
the marks told the time. Water clocks worked better than sundials because they
told the time at night as well as during the day. They were also more accurate than
sundials.
The first practical clock was driven by a pendulum. The pendulum swings left and
right, and as it swings, it turns a wheel with teeth. The turning wheel turns the
hour and minute hands on the clock. One problem with pendulum clocks is that
they stopped running after a while and had to be restarted.
Quartz crystal clocks were then invented. Quartz is a type of crystal that looks like
glass. When you apply voltage, or electricity, and pressure, the quartz crystal
vibrates or oscillates at a very constant frequency or rate. The vibration moves the
clock's hands very precisely.

TOPIC 6 TIME  97
6.1
In this subtopic, we will be looking at how to teach pupils how to tell time.
ACTIVITY 6.1
Search the Internet for information about how man started to tell time.
State two reasons for time to be taught as one of the important topics
in the Years 5 and 6 KBSR Mathematics syllabus.
6.1.1 History of Time
The Greeks divided the year into 12 parts that are called months. They divided
each month into 30 parts that are called days. Their year had a total of 360 days or
12 times 30 days (12 x 30 = 360).
The Egyptians and Babylonians decided to divide the day from sunrise to sunset
into 12 parts that are called hours. They also divided the night, the time from
sunset to sunrise, into 12 hours. This system of measuring time was not very
accurate because the length of an hour changed depending on the time of year.
Somebody finally figured out that by dividing the whole day into 24 hours of
equal length (12 hours of the day plus 12 hours of the night), the time could be
measured more accurately.
The hour is divided into 60 minutes, and each minute is divided into 60 seconds.
The idea of dividing the hour and minutes into 60 parts comes from the Sumerian
sexagesimal system, which is based on the number 60. This system was developed
about 4,000 years ago.
As we know, a clock only shows 12 hours at a time, and the hour hand must go
around the clock twice to measure 24 hours, or a complete day. To tell the first 12
hours of the day (from midnight to noon) apart from the second 12 hours of the
day (from noon to midnight), we use these terms:
AM – Ante meridiem, from the Latin term for "before noon"
PM – Post meridiem, from the Latin term for "after noon"

98  TOPIC 6 TIME
ACTIVITY 6.2
Visit the Math Forum website:
http://mathforum.org/dr.math/tocs/time.middle.html
Find out why day and night are divided into 12 parts.
6.1.2 Time Zones
Because the Earth turns, it is daytime on one side of the world and night time on
the other side. In 1884, delegates from 25 countries met and agreed to divide the
world into time zones. If you draw a line around the middle of the Earth, it is a
circle (equator). The delegates divided the 360 degrees of the circle into 24 zones,
each 15 degrees apart (24 x 15 = 360). They decided to start counting from
Greenwich (pronounced GREN-ich), England, which is 0 degrees longitude. To
see the standard time zones of the world, refer to the Figure 6.2 below.
Figure 6.2: Time zones
Source: http://www.arcytech.org/jaya/clock/images/time_zones.jpg

TOPIC 6 TIME  99
6.1.3 Telling the Time Correctly
Clocks and watches have both a big hand to tell the minutes and a small hand to
tell the hour. Look at the picture below. The hour hand is pointing to the 1, and
the minute hand is pointing to the 12 (or 0 minutes). It is exactly one o'clock.
One way to write one o'clock is 1.00. Another way to write it is 1:00. The symbol
: is called a colon. It separates the hours from the minutes. The number on the left
side of the colon tells the hour and the number on the right side tells the minutes.
To tell the time, we look at the hour hand first and then the minute hand.
In the picture above, the hour hand is pointing to the number 1, and the minute
hand is pointing to the number 15 (look at the outside of the clock), so it is one-fifteen,
or 1:15. Notice that the hour hand is not pointing exactly at the 1, but has
moved a little closer to the 2. As the minute hand moves all the way around the
clock, the hour hand moves from one hour to the next.
You can divide an hour, which is 60 minutes long, into four parts. The parts are
divided by the 0, 15, 30, and 45 minute marks as shown in the picture below. Each
of the four parts is called a quarter. In the table below, you will learn ways to say
the time using the word "quarter".

100  TOPIC 6 TIME
O'clock
quarter to quarter past
half past
When the number of minutes is greater than 30, instead of saying the number of
minutes after the hour, you can say the number of minutes before the next hour, or
the number of minutes to the next hour. The following table shows different ways
to say the time, including using the word "quarter" and the word "to".
Table 6.1: Different Ways to Say the Time
Time Ways to Tell the Time
6:00 Six o'clock
2:15 Two-fifteen
Quarter past two
5:30 Five-thirty
Half past five
8:45 Eight-forty-five
Quarter to nine
3:50 Three-fifty
Ten to four
7:11 Seven-eleven
Eleven minutes past seven
11:48 Eleven-forty-eight
Twelve minutes to twelve
12:00 Twelve o'clock
Noon (middle of the day)
Midnight (middle of the night)

TOPIC 6 TIME  101
ACTIVITY 6.3
Look through the last few years of the Arithmetic Teacher or other
journals of teaching Mathematics in Primary Schools. Read an article
on the teaching and learning of time that is relevant to the Year 5 and
Year 6 KBSR Mathematics Syllabus. Discuss your article with your
coursemates and tutor.
6.1.4 24-Hour System
A 24-hour system is used for international time readings. The times of arrivals and
departures of airplanes, international trains and ships are read in the form of the
24-hour clock instead of the 12-hour clock. The international time system uses 4
digits to indicate time, the first 2 digits indicate hours while the last two digits
indicate minutes.
For example:
12-Hour Clock 24-Hour Clock
5.30 am 0530
8.15 pm 2015
The time-line below can be used to show the relationship between the 12-hour
system and the 24-hour system. It is similar to the number line used in the number
system except that in the time-line we have 60 divisions to represent the minutes
in an hour.
12-hour System
mid-night morning (a.m.) noon
0 1 2 3 4 5 6 7 8 9 10 11 12
0000 0100 0200 0300 0400 0500 0600 0700 0800 0900 1000 1100 1200
24-hour System

12-hour System
noon afternoon (p.m.) mid-night
0 1 2 3 4 5 6 7 8 9 10 11 12
1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400
24-hour System
To convert the 12-hour system to the 24-hour system, we do the following:
(i) 1.45 a.m. = 0145 we add 0 to make 4 digits and we read as 01, 45.
(ii) 5.48 p.m. = 1748 we add 12 to the hours if it is after noon and we read
it as 17, 48.
To convert the 24-hour system to the 12-hour system, we do the following:
(i) 0045 = 0.45 am The first two digits (less than 12) indicates morning
(a.m.). We put a dot (.) after the first two digits to indicate hours and
minutes.
(ii) 1535 = 3.15 pm The first two digits (more than 12) indicates afternoon
(p.m.). For hours more than 12, we subtract 12 from the given hour (15 – 12
= 3 hours)
(iii) 2345 = 11.45 pm Here again, the first two digits are more than 12, so
we subtract 12 from 23 (23 – 12 = 11hours). It indicates (p.m.) in this case it
is night.
SELF-CHECK 6.1
1. Explain the difference between the 12-hour system and the 24-
hour system.
2. Using a suitable teaching aid, explain how you would convert
2145 into the 12-hour system.

TIME
Our pupils will learn the topic of time effectively if we plan the lesson
systematically. A well organised conceptual development of time will help our
pupils to understand the concept of time better. Though pupils have been exposed
to time before, it is still our responsibility as teachers of Year 5 and Year 6 to
provide adequate opportunities for our pupils to explore and have practical
experience of time. We should use physical materials and other representations to
help our children develop their understanding of time.
The major mathematical skills related to time to be mastered by Year 5 and Year
6 pupils are as follows:
(a) Time in the 24-hour system
(i) Read and write time in hours and minutes in the 24-hour system;
(ii) Convert time from the 24-hour system to the 12-hour system and vice
versa; and
(iii) Solve real life problems involving time in the 24-hour system.
(b) Convert time in fractions and decimals;
(i) Convert time in fractions and decimals of a minute to seconds; and
(ii) Convert time in fractions and decimals of an hour to minutes and to
seconds.
(c). Year, Decade, Century and Millennium
(i) Convert time involving year and decade;
(ii) Convert time involving year and century;
(iii) Convert time involving year, decade, century and millennium; and
(iv) Solve real problems involving year, decade, century and millennium.
(d) Operations
(i) Add and subtract time involving hours, minutes and seconds;
(ii) Multiply and divide time involving hours, minutes and seconds; and
(iii) Solve real problems involving addition, subtraction, multiplication and
division of time.
6.2

(e) Calculate the duration of an event
(i) Calculate the duration of an event involving hours, minutes and
seconds;
(ii) Calculate the duration of an event involving days and hours;
(iii) Calculate the duration of an event involving months, years and dates;
(iv) Determine the start or end time of an event from a given duration of
time; and
(v) Solve problems involving time duration in fractions and/or decimals of
hours, minutes and seconds.
6.3
In this subtopic, we demonstrate to you the teaching and learning activities for the
topic of time that can be used in the classroom. Pupils can master major
mathematical skills involving time by carrying out these activities.
6.3.1 Time in the 24-hour System
ACTIVITY 6.4
Learning Outcomes:
 To write the time in words
 To write the time in numerals
 To convert the time from the 24-hour system to the 12-hour system
and vice versa
Materials:
 Task Cards
 Answer Sheets

Procedure:
1. Divide the class into groups of five pupils and give each
pupil an Answer Sheet.
2. Instruct pupils to write their name on the Answer Sheet.
3. Shuffle five Task Cards and place them face down in a stack
at the centre.
5. Ask the player to write all the answers to the questions in the
card drawn on the Answer Sheet.
pupils in the group exchange the card with the pupil on their
left in clockwise direction.
7. Pupils repeat steps (5 and 6) until all of them in the group
have answered the questions in all the cards.
answers.
9. Teacher summarises the lesson on the vocabulary related to
time.
Example of an Answer Sheet :
Name :________________________ Class :______________________
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
Card D Card E
1.________________ 1.________________
2.________________ 2.________________
3.________________ 3.________________

Card A
1. Write the time in words.
0932 hrs =
2. Write the time in numerals.
Seventeen twenty-four hours =
3. Convert the time from the 24-hour system to the 12-hour system.
1352 hrs =
4. Convert the time from the 12-hour system to the 24-hour system
7. 30 a.m. =
ACTIVITY 6.5
Work with your friend in class to prepare four more Task Cards.
There should be four questions in each card.
6.4.

6.3.2 Converting Time in Fractions and Decimals
ACTIVITY 6.6
Learning Outcomes:
 To convert time in fractions and decimals of a minute to seconds
 To convert time in fractions and decimals of an hour to minutes and
to seconds
 To convert time in fractions and decimals of a day to hours, minutes
and seconds
Materials:
Procedure:
2. Shuffle the Flash Cards and place them face down in a stack at the
centre.
3. Instruct Player A to begin by drawing a card from the stack and
showing the card to Player B.
4. Instruct Player B to read the answers to the questions in the card
within the stipulated time (decided by the teacher).
5. Instruct Player C to write the points below Player B’s name. Each
correct answer is awarded one point (a maximum of 4 points for
each Flash Card).
6. Players repeat steps (4 and 5) until 10 cards are drawn by Player
A.
opportunity to read the 10 Flash Cards shown to them.
8. The winner in the group is the pupil that has the most number of
points.
9. Teacher summarises the lesson on the basic facts about units of
time.

Flash Card 1
1. Convert the following time to seconds.
0.2 minute = seconds
2. Convert the following time to minutes.
3 hour = minutes
5
3. Convert the following time to hours.
0.5 day = hours
4. Convert the following time to hours, minutes and seconds.
0.48 day = hours minutes seconds
ACTIVITY 6.7
Work with a few friends of yours in class to prepare 29 more Flash
Cards.
There should be four questions in each Flash Card.
6.6.

6.3.3 Year, Decade, Century and Millennium
ACTIVITY 6.8
Learning Outcomes:
 To convert units of time from century to years and vice versa
 To convert units of time from century to decades and vice versa
 To convert units of time from millennium to years and vice versa
 To convert units of time from millennium to decades and vice versa
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 paper.
3. The teacher instructs pupils to answer the questions in the Task
5. At the end of 10 minutes, the groups will move on to the next
station in the clockwise direction.
8. Teacher summarises the lesson on how to convert units of time
from century and millennium to years and decades and vice versa.

STATION 1
1. Convert the following centuries to years.
(a) 6 centuries = years
(b)
2 centuries = years
5
2. Convert the following years to centuries
(a) 175 years = centuries
(b) 800 years = centuries
3. Convert the following decades to centuries and vice versa.
(a) 5 centuries = decades
(b) 150 decades = centuries
4. Convert the following millennium to centuries.
(a) 7 millennium = centuries
(b) 50 centuries = millennium
ACTIVITY 6.9
Work with two of your friends to prepare four more Task Sheets for
the other stations. There should be four questions in each sheet. Make
sure your sheets are based on the learning outcomes of Activity 6.8.

6.3.4 Basic Operations Involving Time
ACTIVITY 6.10
Learning Outcomes:
 To add time in hours, minutes and seconds
 To subtract time in hours, minutes and seconds
 To multiply time in hours, minutes and seconds
 To divide time in hours, minutes and seconds
Materials:
 Activity Cards
 Colour pencils
Procedure:
1. Divide the class into groups of four pupils and give each group a
different colour pencil and a clean writing paper.
2. Instruct pupils to shuffle a set of 12 Activity Cards and place them
face down in a stack at the centre.
3. Teacher signals to the pupils to begin answering the questions in
the first Activity Card drawn.
4. Once they have completed the first Card, they continue with the
next Activity Card.
answer paper to the teacher.
7. Teacher summarises the lesson on how to add, subtract, multiply
and divide time in hours, minutes and seconds.

1. Add the following time in hours, minutes and seconds.
(a) 3 hrs 40 min 30 s (b) 2 hrs 35 min 20 s
+ 4 hrs 35 min 35 s + 5 hrs 35 min 40 s
2. Subtract the following time in hours, minutes and seconds.
- 4 hrs 35 min 35 s - 5 hrs 35 min 40 s
3. Multiply the following time in hours, minutes and seconds.
 5  3
4. Divide the following time in hours, minutes and seconds.
(a) 6 18 hrs 24 min 30 s (b) 8 20 hrs 42 min 32 s
ACTIVITY 6.11
Work in pairs to prepare eleven more Activity Cards for the group.
There should be four questions in each card. Make sure your cards are
based on the learning outcomes of Activity 6.10.

6.3.5 Duration of an Event
ACTIVITY 6.12
Learning Outcomes:
 To calculate the duration of an event involving hours, minutes and
seconds
 To calculate the duration of an event involving days and hours
 To determine the start or end time of an event from a given duration
of time
 To calculate the duration of an event in months, years and dates
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. Give each group an Exercise Sheet with four questions each.
4. The group that finishes first with all correct answers will be the
winner.
5. Teacher summarises the lesson on how to find the duration of an
event.
ACTIVITY 1

1. Find the duration of the following events.
From To Duration
(a) 1335 hrs 1945 hrs
(b) 11.30 a.m. 3.45 p.m.
Starting Time Ending Time Duration
(a) 0900 hrs,
5 April
1100 hrs,
12 April
(b) 6.30 a.m.,
15 November
3.30 p.m.,
17 November
3. Calculate the starting or ending time of the following events.
Starting Time Ending Time Duration
(a) 0900 hrs,
5 April
1 hour
15 minutes
(b) 3.30 p.m.,
17 November
4 days
3 hours
(a) From July 2013 to September 2014
= _________ years ______ months
(b) From 0730 hrs, 20 June 2013 till 1740 hrs, 21 June 2014
= _________ day ______hours ______ minutes

6.3.6 Problem Solving Involving Time
ACTIVITY 6.13
Learning Outcomes:
 To solve problems involving duration of time in fractions and/or
decimals of hours, minutes and seconds
 To solve problems involving computations of duration of time
Materials:
 Time worksheets
 Colour pencils
Procedure:
1. Divide the class into ten groups and give each group a Time
2. The teacher instructs the groups to answer all the questions in the
Time Worksheet.
teacher instructs the groups to exchange the Time Worksheets.
5. Repeat Steps 2 to 4.
6. Once all the 10 Time Worksheets have been answered, the teacher
collects the answer papers and corrects the answer papers.
8. Teacher summarises the lesson on how to solve problems
involving duration of time.

Example of a Time Worksheet:
TIME WORKSHEET 1
1. A drawing competition started at 1425 hrs and ended at 1645 hrs.
Calculate the duration of the competition.
The duration of the competition is _____________.
2. Mrs. Chong spent
1 day to bake a cake and
8
1 day to sew a
4
dress. How long did she take to complete the work altogether?
She took ___________ to complete the work altogether.
3. Sharipah works in Ipoh General Hospital as a nurse. She works
for
1 of a day. How many hours does she work?
3
Sharipah works_________ hours.
4. Meng Choo was posted to Sabah on 2 August 2008. Then, she
was transferred to Perak on 1 July 2012. Find the duration, in
years and months, of her stay in Sabah.
The duration of her stay in Sabah is ____________.
ACTIVITY 6.14
Prepare nine more Time Worksheets for the group. There should be
four questions in each worksheet. Make sure your worksheets are
based on the learning outcomes of Activity 6.13

 Timekeeping has been an important part of all cultures throughout the
centuries. The history of clocks is very long, and many different types of
clocks have been invented over the centuries.
 The first method people used to tell the time was by looking at the sun as it
crossed the sky. The oldest type of clock was a sundial, also called a sun
clock. Water clocks worked better than sundials because they told the time at
night as well as during the day.
 The first practical clock was driven by a pendulum. One problem with
pendulum clocks was that they stopped running after a while and had to be
restarted. Quartz crystal clocks were invented in 1920.
 In Year 5 and Year 6, pupils need to know how to read and write time using
the 24-hour system; convert time in fractions and decimals to hours, minutes
and seconds; add, subtract, multiply and divide time; calculate the duration of
an event; and finally solve problems involving duration of time.
 Pedagogical Content Knowledge for this topic is divided into history of time,
time zones, saying time correctly and the 24-hour system. This knowledge
would equip us with some added information for the teaching and learning of
time.
 It is important to provide our pupils opportunities to explore and have
practical experiences with the concept of time; using physical materials and
other representations to help them develop their understanding of it.
24-hour system
Analog clock
Ante meridiem
Digital clock
Hour hand
Minute hand
Post meridiem
Time zones

Ng, S.F. (2002). Mathematics in education workbook 2B (Part 1). Singapore:
Pearson Education Asia.
Nur Alia bt. Abd. Rahman & Nandhini. (2008). Mathematics KBSR Year 5, siri
intensif. Kuala Lumpur: Penerbitan Fargoes.
Nur Alia bt. Abd. Rahman & Nandhini. (2008). Mathematics KBSR Year 6, Siri
Intensif. Kuala Lumpur. Penerbitan Fargoes.
Peter, C. et al. (2002). Maths spotlight activity sheets 1. Oxford: Heinemann
Reys, R. E., Suydam, M. N., & Lindquist, M. M. (1989). Helping children learn
mathematics. New Jersey: Prentice Hall.
Smith, K. J. (2001). The nature of mathematics. US: Thomson Learning.
Sunny Yee & Ng, K. H. (2007). A problem solving approach : Mathematics year

Topic
7
 Length, Mass
and Volume
of Liquids
LEARNING OUTCOMES
1. Use the vocabulary related to length, mass and volume of liquids
correctly as required by the Year 5 and Year 6 KBSR Mathematics
Syllabus;
2. Relate the major mathematical skills and basic pedagogical content
knowledge related to the length, mass and volume of liquids;
division involving length, mass and volume of liquids correctly;
of length, mass and volume of liquids; and
5. Plan basic teaching and learning activities for addition, subtraction,
multiplication and division involving length, mass and volume of
liquids.
 INTRODUCTION
Welcome to a new topic on Length, Mass and Volume of Liquids. I am sure you
will agree with me that measurement problems, such as arithmetic problems, are
encountered in many different situations in our daily lives. One of the reasons to
include measurement in KBSR mathematics is to enable children to work with its
many practical applications in real life situations. It is important for children to
have opportunities to learn more about measurement. Knowing how children tend
to think about measurement helps teachers to guide children's discovery of the
principles of measurement.

120  TOPIC 7 LENGTH, MASS AND VOLUME OF LIQUIDS
One of the earliest measuring tools invented by man was used to weigh things.
Primitive societies needed fundamental measures for daily jobs (for example,
constructing homes of an appropriate size and shape, fashioning clothes, or
bartering food or raw materials). As man evolved, measurement units became
more and more complex. For more sophisticated jobs, it was necessary not only to
weigh and measure complex things - it was also necessary to do it accurately time
after time and in different places.
The need for a single worldwide coordinated measurement system was recognized
over 300 years ago. Measures for mass were to be derived from the unit of length.
The metric unit of mass, called the “gram” was defined as the mass of one cubic
centimetre of water. The name Le Systeme International d’Units (International
System of Units), with the international abbreviation SI, was adopted for this
modernised metric system.
Children can use unconventional items like paper clips to measure lengths, seeds
to measure mass and glass containers to measure volume of liquids. However,
they need to understand that identical standard units must be used when
uniformity in measuring is required. In Year 5 and Year 6, our pupils would have
to learn the relationship between centimetres, metres and kilometres, the
relationship between kilograms and grams, as well as to estimate the volume of
liquids in litres. It is important that our pupils master these concepts and
relationships in order to extend their skills to cover addition, subtraction,
multiplication and division of units of length, mass and volume of liquids.
In the first part of this topic, we will learn about the pedagogical content
knowledge of measurement such as the historical notes, the vocabulary, the basic
principles, units, and relationship between units of measurement. In the second
part of the topic, we will look at the major mathematical skills of measurements
for Year 5 and Year 6. Before we conclude this topic, we will learn how to plan
and carry out innovative activities to teach the topic of measurement of length,
mass and volume of liquids.
ACTIVITY 7.1
Think of five reasons why measurement plays an important role in
our lives. List the reasons before you compare them with your
partner.

TOPIC 7 LENGTH, MASS AND VOLUME OF LIQUIDS  121
Important information regarding the content and pedagogical aspects for teaching
measurement covers the following aspects:
(a) Historical notes on length, mass and volume of liquids;
(b) The basic principles of measurement;
(c) The meanings of length, mass and volume of liquids; and
(d) Units of length, mass and volume of liquids.
Figure 7.1: Some human-referenced units of measurement
7.1
ACTIVITY 7.2
Figure 7.1 shows human-referenced units of measurement. List
down four more of such units of measurement that were used in the
olden days.
You may refer to the following URL :
http://www-history.mcs.st-andrews.ac.uk/
HistTopics/Measurement.html

7.1.1 Historical Note on Measurement
Measurement has been important ever since man settled from his nomadic
lifestyle and started using building materials, occupying land and trading with his
neighbours. As society become more technologically orientated much higher
accuracies of measurement are now required in an increasingly diverse set of
fields, from micro-electronics to interplanetary ranging.
Ancient measurement of length was based on the human body (refer to Figure
7.2). There were many different measurement systems developed in early times,
most of them only being used in a small locality. One which gained a certain
universal nature was that of the Egyptian cubit developed around 3000 BC. Based
on the human body, it was taken to be the length of an arm from the elbow to the
extended fingertips. A traditional tale tells the story of Henry I (1100-1135) who
decreed that the yardstick should be "the distance from the tip of the King's nose
to the end of his outstretched thumb".
The cubit
(finger tip to elbow)
The Yardstick
(Henry I – thumb to nose)
Figure 7.2: Ancient length measurements
It had long been realised that a universal standard of measurement was needed,
and that it should be a natural constant. The need for a single worldwide
coordinated measurement system was recognised over 300 years ago. In 1790, the
National Assembly of France requested the French Academy of Sciences to
“deduce an invariable standard for all the measures and all the weights.” The
Commission that was appointed created a system that was, at once, simple and
scientific. Measures for mass were to be derived from the unit of length.
Furthermore, the larger and smaller version of each unit was to be created by
multiplying and dividing the basic units by 10 and its power. The metric unit of
mass, called the “gram” was defined as the mass of one cubic centimetre of water.
The name Le Systeme International d’Units (International System of Units), with
the international abbreviation SI, was adopted for this modernised metric system.

Figure 7.3: Triple beam balances
(Instruments to measure mass)
Figure 7.4: Measuring cylinders
(Instruments to measure volume of liquids)
7.1.2 The Basic Principles of Measurement
Understanding the following basic principles will definitely help us to teach this
topic effectively. The four basic principles underlying the measurement of length,
mass and volume of liquids are as follows:
(a) Comparison principle – This principle deals with comparing and ordering
of objects by a specific attribute. It involves using suitable vocabulary to
describe and compare:
(i) Length such as short, shorter, tall, taller, long, longer, high, higher,
deep, deeper, wide, wider, width, depth, height, etc.
(ii) Mass such as heavy, heavier, light, lighter, etc.
(iii) Volume of liquids such as big volume, bigger volume, small volume,
smaller volume

(b) Transitivity principle – This principle involves comparing and ordering of
three or more objects using appropriate language, e.g. If A is heavier than B
and C is heavier than A, then C must be heavier than B, etc.
(c) Conservation principle – This principle states that the length, mass or the
volume of an object does not change even when the position or the
orientation of the object is changed.
(d) Measuring principle – This principle refers to the fact that measurement
involves stating how many of a given unit match the attribute (e.g. length,
mass or volume) of an object. For example, when measuring the mass of a
rod, stating the number of kilograms that can be used to weigh it.
One other point to note is that there are some conceptual differences between
counting and measuring. For instance, when counting the number of pupils in the
classroom, the result must be a whole number, i.e. the quantity is discrete.
However, when measuring the height of pupils, the result can take on values other
than whole numbers, for example, 129.3 cm, etc. Such quantities are called
continuous quantities. The number line model can be used to help your pupils to
visualise the continuous number scales used in measuring length, mass and
volume.
7.1.3 The Meanings of Length, Mass and Volume of
Liquids
Let us look at the meaning of measurement in broad terms. It is associating
numbers with physical quantities and so the earliest forms of measurement
constituted the first steps towards mathematics. Once “associating numbers with
physical objects” was carried out, it became possible to compare the objects by
comparing the associated numbers. This led to the development of methods of
working with numbers.
(a) Length
Now, let us take a look at the formal definition of length. The length of an
object refers to the number of standard units (e.g. centimetres) which can be
laid in a straight line along or beside the object.
Length of a coloured ribbon = 6 units

In other words, “length” is the distance between any two points (locations)
measured along a straight line. Two lengths can be compared directly by putting
them side by side, with one end of each length aligned. In fact, lengths can be
measured indirectly by comparing each length with a third length and that third
length is a measuring instrument such as a ruler or scale.
(b) Mass
Do you know how to introduce mass to primary school pupils? For primary
school pupils, the concept of mass can be described as the general
“heaviness” of an object. Mass is one of the least common forms of
measurement used for comparing objects in everyday situations. In fact, it
has been found that the concept of mass is quite difficult for children to
grasp because mass cannot be seen but has to be held and felt. In other
words, the mass of two objects cannot be compared by just seeing them
together. Moreover, the mass of an object may not be proportional to its
size. A big piece of cotton wool may be lighter than a small piece of metal.
Therefore, it is important for us to establish in the minds of children that “a
smaller sized object may not necessarily be lighter than a bigger sized
object” and vice versa.
Scientifically, the terms weight and mass have different meanings. Mass is
the measure of the amount of matter in an object whereas weight is the
gravitational force (g) acting on that mass. For example, a boy of mass 20 kg
has a weight of 200 N (taking g = 10 ms 2 ). However, these two terms are
used to mean the same thing. Nevertheless, it is normal to refer to the
“weighing of an object” as a process to find its mass.
(c) Volume of liquids
Volume is literally the “amount of space filled” by an object. But on a
practical level, we often want to know about its capacity, how much does a
container hold? So, we often measure volume as the number of units it takes
to “fill the object”. Figure 7.5 shows a container and a rock. The space that
the container surrounds (and is occupied by air) and the space that the rock
takes up (and is occupied by elements such as oxygen, silicon and
aluminium) are both called volume.

Volume Volume
Container Rock
Figure 7.5: Two meanings of volume
The concept of volume is tricky. Two objects (like the container and the rock)
might occupy the same volume but might contain totally different amounts of
matter. Children often confuse the amount of matter, which we call mass, with the
space occupied, which we now know is volume. Thus children tell us that a
“heavy” object has more volume than a “light” object even though the latter may
actually occupy more space. Indeed, volume is so oversimplified in primary
schools that many Year 6 pupils think of volume as length width height, no
matter what the shape of the object. Others assume that volume is length cubed.
Misconceptions such as these are a result of a curriculum that emphasises
memorisation of formulas without giving attention to the conceptual foundations
of volume.
7.1.4 Units of Length, Mass and Volume of Liquids
Now, let us look at how to teach children units of length, mass and volume of
liquid. If children are simply told to measure length in a unit like an inch, they
develop very little understanding of the basic concept of measurement. Children
need the opportunity to understand these basic concepts of measurement. These
basic concepts of measurements include:
(a) Appropriate units
Use units of measurement appropriate to the thing being measured. Units
that work for measuring the length of your car porch may not work for
measuring the length of your notebook. Units used to measure the mass of a
book may not work for measuring the mass of a bus. Similarly, units used to
measure the volume of liquid medicine consumed by a sick child may not
serve well for measuring the volume of water in a swimming pool.

(b) Non-standard units of measurement
Non-standard unit for measurement is any arbitrary measure used as a unit.
Some common examples are:
(i) Length – body parts such as span, foot, pace and arm length, paper
clips;
(ii) Mass – objects such as beans, thumb tacks and rubber seeds;
(iii) Volume of liquids – containers such as cups, mugs, bottles and
tumblers.
(c) Standard Units of measurement
A standard unit for measurement is any fixed measure that has been
accepted as a standard internationally. Some examples include:
(i) Yards, miles, feet, inches, metres and kilometres;
(ii) Ounces, pounds, grams and kilograms; and
(iii) Pints, gallons, litres and cubic metres.
Units such as the yard, mile, inch, ounce, pound, pint and gallon are known
as Imperial units, whereas the metre, kilometre, gram, kilogram, litre and
cubic metre are known as Metric units. However, in the Malaysian school
curriculum, only metric units are taught.
SELF-CHECK 7.1
1. Describe briefly the four basic principles of measurement.
2. Explain the difference between discrete quantities and
continuous quantities.
MEASUREMENT IN YEAR 5 AND YEAR 6
7.2
Pupils will learn this topic of measurement effectively if we plan the lessons
systematically. A well organised conceptual development of length, mass and
volume of liquid is essential for our pupils to understand these concepts . It would
be advisable to introduce this topic in a less stressful manner. Remember to
provide opportunities for pupils to understand the meanings of length, mass and
volume of liquid and their respective units. Physical materials and other
representations should be used to help children develop their understanding of
these concepts.

measurement in Year 5 and Year 6 are as follows:
(a) Measuring Length
(i) Convert units of length metres to kilometres and vice versa;
(ii) Add and subtract units of length involving metres and kilometres;
(iii) Multiply and divide units of length involving metres and kilometres;
and
(iv) Solve problems in real context involving computation of units of
length.
(b) Comparing Mass
(i) Convert units of mass from fractions and decimals of kilogram to
grams and vice versa;
(ii) Add and subtract units of mass involving grams and kilograms;
(iii) Multiply and divide units of mass involving grams and kilograms; and
(iv) Solve problems in real context involving computation of units of mass.
(c) Comparing Volume of Liquids
(i) Convert units of volume involving fractions and decimals of litres to
millilitres and vice versa;
(ii) Add and subtract units of volume involving millilitres and litres;
(iii) Multiply and divide units of volume involving millilitres and litres;
and
(iv) Solve problems in real context involving computation of units of
volume of liquids.
Next, we move on to the teaching and learning activities on length, mass and
volume of a liquids. Let us consider Activity 7.3 first.

7.3
7.3.1 Length
ACTIVITY 7.3
Learning Outcomes:
 To convert metre to kilometre and vice versa; and
 To convert units of length from fractions and decimals of
kilometres to metres and vice versa.
Materials:
 Task Cards; and
 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. Ask them to shuffle Six Task Cards and place them face down in
a stack at the centre.
4. Ask each player to begin by drawing a card from the stack.
5. Ask the players to write all the answers to the questions in the
pupils in the group exchange the card with the pupil on their left
in clockwise direction.
7. Ask the pupils to repeat steps (5 and 6) until all the pupils in the
group have answered questions in all the cards.
answers.
9. Teacher summarises the lesson on the basic facts of length.

Name :________________________ Class :______________________
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
Card D Card E Card E
1.________________ 1.________________ 1.________________
2.________________ 2.________________ 2.________________
3.________________ 3.________________ 3.________________
Task Card A
1. Convert metres to kilometres.
8492 m = _______________ km
2. Convert kilometres to metres.
7,125 km = ______________ m
3. Calculate the fraction of length.
2 of 27 km = ___________ m
9
ACTIVITY 7.4
Work with a friend in class to prepare five more Task Cards.
7.3.

7.3.2 Basic Operations on Length
ACTIVITY 7.5
Learning Outcomes:
 To add units of length in metres and kilometres;
 To subtract units of length in metres and kilometres;
 To multiply units of length in metres and kilometres; and
 To divide units of length in metres and kilometres.
Materials:
 30 different Flash Cards; and
 Clean writing papers.
Procedure:
clean writing paper.
2. Ask the pupils to write their names on the clean paper given.
centre.
4. Asks Player A to begin by drawing a card from the stack. He shows
the card to Player B.
5. Asks Player B to do the calculations and read out the answers
within the stipulated time (decided by the teacher).
6. Asks Player C to write the points below Player B’s name. Each
each Flash Card).
7. Ask the Players to repeat steps (4 and 5) until 10 cards are drawn by
Player A.
opportunity to read and complete the questions on all 10 Flash
Cards shown to them.
9. The winner in the group is the pupil that has the most number of
points.
10. Teacher summarises the lesson on the basic operations on length .

Flash Card 1
1. Add the following:
1.8 km + 870 m = ________ km
2. Subtract the following:
4.82 km – 1 293 m = ________ m
3. Multiply the following:
2.34 km  4 = _______ m
4. Divide the following:
4 992  8 = _________ km
ACTIVITY 7.6
Work with three friends of yours in class 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 7.5.

7.3.3 Mass
ACTIVITY 7.7
Learning Outcomes:
 To convert units of mass from fractions and decimals of a kilogram
to grams and vice versa;
 To add and subtract units of mass in grams and kilograms: and
 To multiply and divide units of mass in grams and kilograms.
Materials:
 Task Sheets;
 Colour pencils.
Procedure:
1. Divide the class into groups of four to six pupils and give each
group a different colour pencil and a clean writing paper.
2. The teacher sets up five stations in the classroom and places a
Task Sheet at each station.
3. The teacher instructs pupils to solve the questions in the Task
8. Teacher summarises the lesson on the basic facts of mass and how
to do basic operations on mass.
ACTIVITY 1

STATION 1
1. Convert the following to gram:
(a)
2 kg = _______ g
5
(b) 0.64 kg = ______ g
2. Convert the following to kilogram:
(a) 250 g = ________ kg
(b) 8 015 g = ________ kg
3. Add and subtract the following:
(a) 7.27 kg + 1 025 g = _________ kg
(b) 0.137 kg – 55 g = ___________ g
4. Multiply and divide the following:
(a) 6.32 g  100 = ________ kg
(b) 654  100 = _________ g
ACTIVITY 7.8
Work with two of your friends to prepare another four Task Sheets
for the other stations. There should be four questions in each sheet.
Make sure your sheets are based on the learning outcomes of
Activity 7.7.

7.3.4 Problem Solving Involving Mass
ACTIVITY 7.9
Learning Outcomes:
 To solve problems involving conversion of units of mass in
fractions and decimals; and
 To solve problems involving computation of mass.
Materials:
 Activity Cards;
 Colour pencils.
Procedure:
2. Shuffle a set of 12 Activity Cards and place them face down in a
3. Signal to the pupils to 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. Ask the groups to stop and hand their answer paper to the teacher
at the end of 10 minutes.
7. Teacher summarises the lesson on how to solve problems in real
contexts involving computation of mass.
ACTIVITY 1

Activity Card 1
1. The combined mass of a watermelon and a comb of bananas is
4.8 kg. If the mass of the watermelon is 2.2 kg, what is the mass
of the bananas?
The mass of the banana is _____________ kg.
2. The mass of a book is 430 g. Find the total mass of 7 similar
books in kg.
The mass of the books is _________ kg.
3. En. Mahmud filled 8.4 kg of prawns into 4 containers. What is
the mass of prawns (in grams) in each container ?
The mass of prawns in each container is _________ g.
4. Box A weighs 5 kg. The mass of Box B is
2 1 times the mass
5
of Box A. Find the mass, in kg, of Box B.
The mass of Box B is ___________ kg.
ACTIVITY 7.10
Prepare 11 more Activity Cards for the group. There should be four
Make sure your cards are based on the learning outcomes of Activity 7.9

7.3.5 Volume of Liquids
ACTIVITY 7.11
Learning Outcomes:
 To convert units of volume involving fractions and decimals of
litres to millilitres and vice versa;
 To add and subtract units of volume in litres and millilitres; and
 To multiply and divide units of volume in litres and millilitres.
Materials:
 Exercise Sheets; and
 Colour pencils.
Procedure:
1. Divide the class into groups of two pupils and give each group a
different colour pencil.
2. Give each group an Exercise Sheet with four questions.
3. Instruct them to answer the questions in the Exercise Sheet.
3. The group that finishes first with all correct answers is the winner.
4. Teacher summarises the lesson on the basic facts of volume of
liquids and how to do basic operations on volume of liquids.

Exercise Sheet A
1. Convert the following to millilitres:
(a)
1 litres = ______ millilitres.
4
(b)
3 litres = ______ millilitres.
5
2. Convert the following to litres:
(a) 1 008 millilitres = ________ litres.
(b) 555 millilitres = ________ litres.
3. Add and subtract the following:
(a) 5.723 litres (b) 17.536 litres
+ 2.758 litres - 9.043 litres
4. Multiply and divide the following:
(a) 7.424 litres (b) 3 49.623 millilitres
 5
ACTIVITY 7.12
Prepare 10 more Exercise Sheets for the group activity. There should
be four questions in each card.
Make sure your Exercise Sheets are based on the learning outcomes of
Activity 7.11.

7.3.6 Problem Solving Involving Volume of Liquids
ACTIVITY 7.13
Learning Outcomes:
 To solve problems involving conversion of units of volume in
fractions and decimals; and
 To solve problems involving computation of volume of liquids.
Materials:
 Volume Worksheets;
 Colour pencils.
Procedure:
1. Divide the class into ten groups and give each group a Volume
2. The teacher instructs the groups to answer all the questions in the
Volume Worksheet.
teacher instructs the groups to exchange the Volume Worksheets.
5. Repeat steps 2 to 4.
6. Once all the 10 Volume Worksheets have been answered, teacher
collects the answer papers and corrects the answer papers.
contexts involving computation of units of volume of liquids.

Example of a Volume Worksheet:
VOLUME WORKSHEET 1
1 A container has a capacity of 2.75 litres. How many millilitres of
water does Mary need to fill up the container?
Mary needs ________ of water to fill up the container.
2. Mrs. Chong needs 25.35 litres of water to clean the floor every
day. How much water does she need in a week?
She needs ________ of water in a week.
3. Miss Siew bought 4.25 litres of soy sauce. She used the soy
sauce to cook food in her restaurant and has 745 millilitres of soy
sauce left. Find the volume of soy sauce that she used.
She used __________ of soy sauce.
4. Ahmad has a bottle of orange juice. He pours the juice equally
into 20 glasses. Each glass contains 50 millilitres of juice. What
is the volume of orange juice, in millilitres, contained in the
bottle?
The bottle contains ________ millilitres of orange juice.
ACTIVITY 7.14
Prepare nine more Volume Worksheets for the group. There should
Activity 7.13.

 Including measurements in KBSR mathematics is important because it has
many practical applications in real life situations.
 Ancient measurement of length was based on the human body.
 The four basic principles underlying the measurement of length, mass and
volume of liquids are comparison principle, transitivity principle, conservation
principle and measuring principle.
 Measuring quantities are continuous quantities whereas counting quantities are
discrete quantities.
 The length of an object refers to the number of standard units (e.g.
centimetres) which can be laid in a straight line along or beside the object. In
other words, length is the distance between any two points (locations)
measured along a straight line.
 The concept of mass can be described as the general heaviness of an object.
Scientifically, the terms weight and mass have different meanings. Mass is the
measure of the amount of matter in an object whereas weight is the
gravitational force acting on that mass.
 Volume is literally the amount of space filled by an object. But on a practical
level, we often want to know about capacity, how much does a container
hold? So, we often measure volume as the number of units it takes to fill the
object.
 These basic concepts of measurements include appropriate units, non-standard
units of measurements and standard units of measurement.
 Units such as the yard, mile, inch, ounce, pound, pint and gallon are known as
Imperial units, whereas the metre, kilometre, gram, kilogram, litre and cubic
metre are known as Metric units.

Addition
Capacity
Continuous quantities
Discrete quantities
Division
Imperial units
Litre
Mass
Metric units
Multiplication
Subtraction
Weight
pembelajaran matematik: Ukuran. Kuala Lumpur: Dewan Bahasa dan
Pustaka.
Nur Alia Abd. Rahman & Nandhini (2008). Siri intensif : Mathematics KBSR year
5. Kuala Lumpur: Penerbitan Fargoes.
Nur Alia Abd. Rahman & Nandhini (2008). Siri Intensif: Mathematics KBSR year
Ng, S.F. (2002). Mathematics in action workbook 2B (Part 1). Singapore: Pearson
Education Asia.
Peter, C. et al. (2002). Maths spotlight activity sheets 1. Oxford: Heinemann
Sunny Yee & Ng, K. H. (2007). A problem solving approach: Mathematics Year
Sunny Yee & Lau, P.H. (2007). A problem solving approach: Mathematics Year

Topic
8
 Shape and
Space
LEARNING OUTCOMES
1. Explain the importance of developing the basics of measurements as
preskills to the learning of perimeter and area;
2. Show how to use the vocabulary related to perimeter, area and
volume of solids correctly;
knowledge related to perimeter, area and volume of solids; and
4. Plan basic teaching and learning activities for perimeter, area and
volume of solids.
 INTRODUCTION
Children typically enjoy learning the topics in geometry because they can relate
what they learn to what they explore in the real world. Learning about geometric
properties and shapes helps them to make sense of their environment as they
become more capable of describing their world. As a result, they find the subject
interesting and therefore, are motivated to learn it.

144  TOPIC 8 SHAPE AND SPACE
ACTIVITY 8.1
“Geometry offers pupils an aspect of mathematical thinking that is
different from, but connected to, the world of numbers ... Some pupils’
capabilities with geometric and spatial concepts exceed their numerical
skills. Building on these strengths foster enthusiasm for mathematics
and provides a context in which to develop number and other
mathematical concepts.” (NCTM, 2000, p.97).
Discuss the truth of this statement in our Malaysian context.
8.1
Two Dutch teachers, Dina and Pierre van Hiele, have developed the van Hiele
model for children to learn geometry. The van Hieles were concerned about the
difficulties their pupils were having with geometry (Geddes & Fortunato, 1993).
They conducted research aimed at understanding children’s level of geometric
thinking to determine the kinds of instructions that could best help children learn
geometry. A brief description of their work can be found in the OUM module,
“HBMT2103 Teaching of Mathematics for Year Two Primary School”.
Thus, it is important for teachers to assess the thinking of the children in their
classes based on the van Hiele levels and use this information to plan instruction
on shape and space that is suitable and relevant to the children’s level of thinking.
ACTIVITY 8.2
Examine the chapter on Shape and Space in a textbook and
describe it in relation to the van Hiele levels.
8.1.1 Geometric Formulas
Formulas for area, perimeter, volume and surface area are introduced in Year 5 and 6.
While formulas are necessary and useful tools for measuring, they should not take the
place of careful development of the attributes and the measuring process.
One skill that needs to be developed in children learning about perimeter and area
is that of making the correct choice of formula when calculating perimeter and
area. Equally important is that children need to see how the formulas are derived.
This is to enable children to build understanding of the meaning of the perimeter
and area formulas.

TOPIC 8 SHAPE AND SPACE  145
8.1.2 Perimeter and Area
Let us now discuss how to teach perimeter and area.
(a) Perimeter
The perimeter of a shape is the distance all the way round its edges.
Perimeter is measured using the same unit as in the measurement of length
such as centimetres, feet or metres. The measurements needed to calculate
perimeter depends on the shape. For a rectangle you will need to know the
length and width of the shape. (It is usual to call the longest side the length
and the shortest the width or breadth.)
Example:
The diagram below represents a pen for Badrul's goats. How much netting
does he need to go round the plot? All measurements are in metres.
Tell the pupils that in a rectangle the opposite sides are equal, so to work out
the perimeter of Badrul’s pen, you just need to know the length and width.
Here the length is 5 m and the width is 4 m.
Method 1
Length = 5 m and width = 4 m
Perimeter = 5 + 4 + 5 + 4 = 18 m
Method 2
Because opposite sides are equal you can also work out the perimeter in this
way: double the length, double the width, then add the results together.
(5 x 2) + (4 x 2) = 10 + 8 = 18 m
Method 3
Add the length and width then double it.
5 + 4 = 9 m
9 m x 2 = 18 m

All three methods will give you the same answer. From the above example we
can show children how the formula for the perimeter of a rectangle is derived.
(b) Area
The area of a shape is the amount of surface enclosed in a plane. We do not
actually measure area by measuring the length. In most cases, we measure
some combination of lengths and use them in a formula to calculate the area.
As such, the teaching and learning of area consist of two parts. The first part
consists of developing the concepts of area and unit of area. The second part
consists of the development of the area formulas.
The concept of area should be developed first by making gross comparisons
of areas of different shapes. Comparisons of area are more complex than
comparisons of length. When comparing areas, we must take into account
length, width, and shape.
When the following shapes are compared, children may have problems
deciding which has a bigger area because one shape is longer and the other
is wider. Hence, this forces the child to think beyond one dimension.
A
B
One way to check the comparison is to cut shape A into two parts and
rearrange them on top of shape B. Then it can be easily seen that B has a
bigger area than A.
A
B
The unit of measurement for areas is called square units. If you use metres to
make your measurement, the area will be measured in square metres (m²). If
centimetres are used, the area will be in square centimetres (cm²). Children
need to know that the symbol m2 is read “square meter” and not “meter
square”.

Children can become familiar with square centimetres (cm2) by using
centimetre graph paper (Horak & Horak, 1982). One task is to find the
approximate area of their hand by tracing it on a graph paper and counting
the unit squares. Children can also be asked to count the number of square
centimetres enclosing specific rectangles and polygons. Let us go through
the following example in the class.
Example:
This square measures 1cm long and 1cm wide. It is 1 square centimetre (cm²).
A rectangle drawn on the 1 cm² paper below is 3 cm long and 2 cm wide.
Count the number of 1 cm
squares. There are 6 squares.
So the area of the rectangle is 6
cm².
8.1.3 Volume
Volume is a measure of the amount of space inside a three-dimensional region, or
the amount of space occupied by a three-dimensional object. It is measured in
cubic units such as cubic centimetres (cm³) or cubic metres (m³). The Imperial
system uses units such as cubic feet (ft³). One cubic centimetre (cm3) is the
measure of a cube having an edge with a length of 1 cm.
To introduce the concept of volume, you might hold up two solid rectangular
prisms and ask which is “bigger”. The discussion should lead to the question of
which occupies more space. Two empty shoe boxes, one of which fits within the
other, can be used for direct comparison of volume.
You can also show samples of objects made up of units of cubic centimetres (cm3)
and have the children count the number of unit cubes it contains to determine its
volume.

Each of the following diagrams represents a shape made from unit cubes.
SELF-CHECK 8.1
1. Why is the comparison of area more complex than comparison of
different length?
2. Explain how you would introduce the concept of volume and its
unit of measurement.
SHAPES
8.2
shapes are as follows:
(a) Measure the perimeter of the following composite 2-D shapes:
(i) Square and square
(ii) Rectangle and rectangle
(iii) Triangle and triangle
(iv) Square and rectangle
(v) Square and triangle
(vi) Rectangle and triangle
(b) Calculate the perimeter of the following composite 2-D shapes:
(iii) Triangle and triangle
(iv) Square and rectangle
(v) Square and triangle

(vi) Rectangle and triangle
(c) Solve problems involving perimeters of composite 2-D shapes.
(d) Measure the area of the following composite 2-D shapes:
(iii) Square and rectangle
(e) Calculate the area of the following composite 2-D shapes:
(iii) Square and rectangle
(f) Solve problems involving areas of composite 2-D shapes.
(g) Measure the volume of the following composite 3-D shapes:
(i) A cube and another cube
(ii) A cuboid and another cuboid
(iii) A cube and a cuboid
(h) Calculate the volume of the following composite 3-D shapes:
(i) A cube and another cube
(ii) A cuboid and another cuboid
(iii) A cube and a cuboid
(i) Solve problems involving volume of composite 3-D shapes.
(j) Find the perimeter of a 2-D composite shape of two or more quadrilaterals
and triangles.
(k) Find the area of a 2-D composite shape of two or more quadrilaterals and
triangles.
(l) Solve problems in real contexts involving calculation of perimeter and area
of 2-D shapes.

(m) Find the surface area of a 3-D composite shape of two or more cubes and
cuboids.
(n) Find the volume of a 3-D composite shape of two or more cubes and
cuboids.
(o) Solve problems in real contexts involving calculations of surface areas and
volumes of 3-D shapes
8.3
In this section, we demonstrate to you some teaching and learning activities that
can be used in the classroom to teach the topic Space and Shape.
8.3.1 Finding the Perimeter
ACTIVITY 8.3
Learning Outcome:
 To develop the concept of perimeter.
Materials:
 A variety of large regular and irregular shapes taped on the floor of
the classroom.
Procedures:
2. Give each group some clean writing paper.
3. Tape a variety of large regular and irregular shapes on the floor
of the classroom.

4. Instruct the pupils in their team walk around the edge of each of the
shapes on the floor.
5. Ask pupils to keep a record of the number of steps they take as they walk
along each of the edges of the shape.
6. Have pupils post their “walk around” numbers for each of the shapes
using each of the sides as an addend, for example 6 + 6 + 6 steps for an
equilateral triangle.
7. Ask the pupils to look for patterns in the measurement of each side, for
example all of the sides of a square are of the same length and that a
rectangle has two long sides and two short sides.
8. Ask pupils to write a sentence using words instead of numbers for the
perimeter of each figure.
9. Teacher summarises the lesson and introduces the concept that perimeter
is the measure of the distance around a closed figure.

ACTIVITY 8.4
Learning Outcome:
 To reinforce the concept of perimeter.
Materials:
 Graph paper (cm square); and
 Strings, ruler, pins.
Procedures:
2. Give each group some graph papers, strings, ruler and pins.
3. Cut a string 14 cm long and ask the children
“How many different rectangles can you make with a perimeter of
14 cm?”
4. Using the graph paper, string and pins, demonstrate to the class
how you can make a rectangle with a perimeter of 14 cm. Remind
the children to keep the sides (in cm) a whole number.
5. Ask the children to explore other rectangles with a perimeter of 14
cm.
P = 14
6. Repeat steps (3) through (5) for rectangles with a perimeter of 20
cm. What about rectangles with a perimeter of 13 cm?
7. The teacher summarises the lesson on finding perimeter.

ACTIVITY 8.5
Learning Outcome:
 To practise finding perimeters.
Materials:
 Squares of sides 2 cm; and
Procedures:
1. Dive the class into groups of four.
2. Give each group some squares and some clean writing paper.
3. Take four squares of sides 2 cm. Ask the children,
“How many figures can you make by putting the squares side by
side?”
Figure 1, Figure 2 and Figure 3 are some possible examples.
Figure 1 Figure 2 Figure 3
4. Allow the children to explore and draw as many figures as
possible.
5. For each of the figures drawn, ask the children to calculate its
perimeter.
Example:
Perimeter of Figure 1 = 20 cm
6. Repeat steps (3) through (5) for five, six and eight squares.
7. The teacher summarises the lesson on finding perimeter.

8.3.2 Finding Area
ACTIVITY 8.6
Learning Outcome:
 To develop the concept of area of a rectangle.
Materials:
 Rectangular cards.
Procedures:
1. Display a rectangle (15 cm by 6 cm).
2. Ask a pupil to come forward to measure the length and the width
of the rectangle. Label the rectangle.
6 cm
15 cm
3. Ask another pupil to seperate the rectangle into 1 cm squares, as
shown below:
15 cm

4. Have another pupil count the number of squares in the rectangle.
5. Discuss with the pupils that each of the square is 1 cm2. Since a total of
90 squares were used to cover the rectangle, the area of the rectangle is
90 cm2.
6. Point out to the pupils that the length is the same as the number of
squares in one row, and the width is the same as the number of rows of
squares.
7. So, instead of counting the number of squares, the area of the rectangle
can be found by multiplying the length and the width (or breath) of the
rectangle.
The area of a rectangle = Length x Width
8. Ask the pupils to work on more examples of calculating area of
rectangles.

ACTIVITY 8.7
Learning Outcome:
 To reinforce the concept of area.
Materials:
 A deck of cards, some showing a rectangle with its sides labeled and
the others showing the product of the two sides.
Example:
12 cm
32 cm
Area = 12 x 32 cm2
1.6 m
0.8 m
Area = 1.6 x 0.8 m2
Procedures:
1. Prepare cards, some showing a rectangle with its sides labeled and
others showing the product of the two sides.
2. Hand a card to each child.
3. Have the children holding the card with the rectangle calculate its
area.
4. Have the children find their partners holding the card showing the
calculation of its area.
5. If there is an odd number of children, you should take a card and
participate so that everyone has a partner.
6. Have the partners stand together so that everyone can see each
other’s card. Have the children check everyone’s calculation of the
area. Are the partners correctly paired?
7. Hand out a Task Sheet and have the children work out the area of
the rectangles.

8.3.3 Finding Volume
ACTIVITY 8.8
Learning Outcome:
 To introduce the concept of volume.
Materials:
 A variety of boxes of different sizes and shapes.
Procedures:
1. Choose two boxes from the collection of boxes of different sizes and
shapes.
2. Ask the pupils which will hold the most.
3. Repeat this process with several pairs of boxes. If the pupils are
unable to decide which of the pair of boxes is bigger, set the pair of
boxes aside.
4. Pick out one of the pairs of boxes for which the pupils were unable
to identify the bigger box.
5. Conduct a brainstorming session, asking pupils to think of ways to
decide which box is bigger. Remind them that the bigger box is the
one that would hold more.
6. Write every suggestion on the chalkboard, regardless of how
reasonable or how practical it is. Then ask the pupils to decide which
methods are most reasonable.
7. Then, try some of the suggested methods to see if it works to
determine the bigger box.

ACTIVITY 8.9
Learning Outcome:
 To introduce the formula for calculating the volume of a cuboid.
Materials:
 Boxes of fixed volume; and
 Unit cm3 cubes.
Procedures:
2. Give each group some unit cm3 cubes and a box (cuboid) of fixed
volume.
Example: A small box measuring 15 cm by 5 cm by 2 cm.
3. Have each group fill their cuboid with cubes to see how many unit
cm3 cubes are needed. The number of cubes needed to fill the
cuboid is the volume of the cuboid.
4. Conduct a brainstorming session, asking pupils to think of ways to
calculate the volume of a cuboid instead of counting the cubes
needed to fill the cuboid.
5. Write every suggestion on the chalkboard, regardless of how
reasonable or how practical it is. Then ask the pupils to decide
which method is the most reasonable.
6. Using the following example, the teacher leads the pupil to derive
the formula for calculating the volume of a cube and cuboid.
Example
Each of these two cuboids has the same volume, 8 cm³, and the
same dimensions: length 4 cm, width 2 cm, height 1 cm.
The volume of the first can be found by counting the unit cubes.
The volume of the second is found using the rule:
Volume of a cuboid = length x width x height

The dimensions of a cube are all the same, so the rule for finding the
volume is:
Volume of a cube = length x length x length = length³
Hand out a Task Sheet and have the pupils work out the volume of
cubes and cuboids.
SELF-CHECK 8.2
1. Design a teaching activity to introduce the standard measurement of
area.
2. Write a lesson plan to introduce the formula for calculating the
perimeter of a rectangle.
 It is important for teachers to assess the geometrical thinking of the children in
their classroom based on Dina and Pierre van Hiele’s levels of thinking. This
information can then be used to plan instruction on shape and space that is
suitable and relevant to the children’s level of thinking.
 While formulae are necessary and useful tools for calculating perimeter and
area, they should not take the place of careful development of these attributes
and the activities and processes that lead to the development of the formulas.
 The area of a shape is the amount of surface enclosed in a plane. We do not
actually measure area as in measuring length. In most cases, we measure some
combination of lengths and use them in a formula to calculate the area.
 The teaching and learning of area consist of two parts. The first part consists
of developing the concepts of area and unit of area. The second part consists
of the development of the area formulas.
 To introduce the concept of volume, hold up two solid rectangular prisms and
ask which is “bigger”. The discussion should lead to the question of which
occupies more space.

Area
Perimeter
Quadrilaterals
Triangles
Volume
Rucker, W. E., & Dilley, C. A. (1981). Heath mathematics. Washington, DC:
Heath and Company.
middle schools. Ohio: Merrill Prentice Hall.

Topic
9
Averages
LEARNING OUTCOMES
1. Explain that average is a measure of central tendency and it
describes what is “typical” of a set of data;
2. Use the vocabulary related to averages correctly;
knowledge related to averages; and
4. Plan basic teaching and learning activities for averages.
 INTRODUCTION
Data are all around us. Indeed, sometimes there is so much data that children can
become overwhelmed. Beside examining a set of data by looking at graphs and
tables, it is often convenient to describe a set of data by choosing a single number
that indicates where the data in the set are centred or concentrated.
The number most commonly used to characterise a set of data is the arithmetic
mean, frequently called the average.
ACTIVITY 9.1
Examine the following set of data for three teachers. All of them claim
that his or her class scored better than the other two classes.
Cikgu Ahmad: 40, 62, 85, 99, 99
Miss Lee: 20, 84, 85, 98, 98
Mr Sivanesan: 59, 59, 78, 89, 100
Are these teachers correct in their assertions? Discuss it.

 TOPIC 9 AVERAGES
162
9.1
The average value is a number that is typical for a set of data. The average is like
the middle point of a set of numbers. Finding the average helps you do
calculations and also makes it possible to compare sets of numbers.
Figure 9.1: Weekly shopping
For example, you might spend between RM20 and RM90 a week on shopping
(refer to Figure 9.1). Finding the average amount you have spent per week will
help you plan your month's spending. The average weekly expenditure gives you
an idea of whether you are spending more or less than you plan to.
ACTIVITY 9.2
Calculating the mean temperature.
Choose your town and select a full five day weather
forecast. Use the data to calculate the mean
temperature.
Now, we move on to learn on how to conduct teaching of averages.
9.1.1 Teaching Averages
A good introduction might be to have pupils engage in a discussion about “being
average” (Paull, 1990). What does it mean to be “an average pupil”? Is being
average something good or bad?

TOPIC 9 AVERAGES  163
An understanding of average can be developed through using concrete materials
and visual manipulation (Rubenstein, 1989). For example, using interlocking
cubes, ask pupils to build two towers.
Figure 9.2: Interlocking cubes
Build a tower with seven cubes and another with five cubes (see Figure 9.2). They
can now discuss what they would have to do to make both towers the same height,
using only the cubes they have used to construct the towers. As a teacher, you
advise the pupils, to make both towers the same height, they have to find the total
number of interlocking cubes used in building both towers. Next, the pupils will
have to divide the total number of cubes by two. By doing the calculation, the
pupils will understand the concept of average and also the method of calculating
averages.
After several examples with two towers of interlocking cubes, pupils can then use
the same strategy in determining the average heights of three or four towers.
As an extension to the above activity, pupils can attempt to apply the process and
discuss a situation in which the cubes cannot be equally shared. Allow the pupils
to use their own language in their discussion, but the end result should be an
understanding that the average is simply one number that describes or
characterizes all the numbers in the data set.
Once the pupils understand the concept, provide them with more activities that
reinforces their understanding of averages.
9.1.2 Measures of Central Tendency
Measures of central tendency describe what is “typical” in a set of data. There are
three types or measures of central tendency. They are the arithmetical average
(mean), mode and median.

164
(a) The Average
This section illustrates how to teach your pupils the average. Do you know
what is the arithmetical average? The arithmetical average is the most
commonly used measure of central tendency. It is calculated by dividing the
sum of a set of numbers by the number of numbers in the set.
When people talk about the average of something, like average price,
average wage or average height, they are usually talking about the mean
value.
The mean value of the weekly spending shown in the graph as indicated in
Figure 9.1 is RM46.
The calculation is as follows:
Mean spending = RM20 + RM40 + RM30 + RM90 + RM50
----------------------------------------------------
5
= RM230
---------
5
= RM46
Can you see that the value is located about the middle of the five different
amounts shown?
The average can be useful for comparing things. For example you can find
the average height for the pupils in your class. When you compare the
averages of two classes you are comparing the average height of the pupils
in the two classes.
Sometimes averages may give a false impression of the figures. In that case
the average is said to be distorted.
Example: The mean annual salary earned in a pharmaceutical store is
RM42,200. You might like the idea of working for the store!
But let us look at the figures:
Employee 1 earns RM8,000 per annum

Employee 4 earns RM 8,000 per annum
The Manager of the store earns RM175,000 per annum
Because the Manager earns a lot more than the employees, his/her salary
raises the mean salary. Let us do the calculation:
The total of the wages :
RM8,000 + RM11,000 + RM9,000 + RM8,000 + RM175,000 =
RM211,000
Then divide the total amount by 5, the number of people:
RM211,000 ÷ 5 = RM42,200
The average salary is RM42,200. But most of the staff earn a lot less than
this. Most employees earn less than the mean salary. For this reason we say
that the mean is distorted.
(b) The Mode
A second measure of central tendency is the mode. The concept of the mode,
but not the term, is introduced informally in a child’s early school
experience. When a child makes statements like:
“April has the most number of public holidays.”
“Most members of the class like white coffee.”
“Blue seems to be their favourite colour.”
The mode is the name of another type of average. The mode is the most
common item in a set of data. It is the number or thing that appears most
often. Sometimes one or two values in a data set can distort the “typical”
value described by the average as in the example mentioned above. In this
case, the mode is the preferred measure of central tendency.
Example: The mode for the annual salary for the staff of the pharmaceutical
store is RM8,000. That is because two out of five of the employees earn
RM8,000 per annum.
(c) The Median
Another measure of the central tendency is the median. The median is the
middle number in a set of numbers recorded in ascending manner. It is the
mid-point when the numbers are written out in order.
The concept of the median can be easily modelled. In the example above,
arrange the salary in ascending order:
First put the numbers in order. This makes it easier to find the median.

166
RM8,000; RM8,000; RM9,000; RM11,000; RM175,000
You can now see that RM9,000 is the middle number. It is half way along
the list. So the median of this set of data is RM9,000.
The mean is RM42,200. This is misleading as it is much higher than most of
the employees’ salaries.
The median value is the middle one in the list. The median salary is
RM9,000. This is a good indication of the general level of the staff salaries.
SELF-CHECK 9.1
1. Explain the meaning of average.
2. Using an example, explain why average may not be a good
measure of central tendency.
AVERAGES
9.2
The major mathematical skills to be mastered by pupils studying the topic on
averages are as follows:
(a) Describe the meaning of average.
(b) State the average of two or three quantities.
(c) Determine the formula for average.
(d) Calculate the average using a formula.
(e) Calculate the average of up to five numbers.
(f) Solve problems in real life situations involving average.

9.3
This section highlights the teaching and learning activities for you to conduct a
lesson on the topic of average in the classroom.
9.3.1 Meaning of Average
ACTIVITIY 9.3
Learning Outcome:
 To describe the meaning of average.
Materials:
 Interlocking blocks
Procedures:
1. In a place visible to all pupils, and using 15 interlocking blocks
arrange five stacks of blocks as illustrated below.
2. Discuss with the children the number of blocks in each stack.
3. Demonstrate the meaning of average by having a pupil remove
enough blocks from the tallest stack to put atop the shortest stack
so that both stacks match the middle stack.
4. Have another pupil do the same with the second and fourth stacks.

168
“Are all the five stacks of the same height?” [Yes]
“How many blocks are there in each stack?”[3]
“Does each stack have the same number of blocks?”[Yes]
6. The teacher explains by saying,
Each stack has three blocks; three is the average of the set of blocks.
7. Repeat step (1) through (6) using different number of stacks and
different number of blocks in each stack.
ACTIVITY 9.4
Learning Outcome:
 To investigate the addition-division process for determining average.
Materials:
 Task Sheet; and
 Interlocking blocks.
Procedures:
1. Using 15 interlocking blocks arrange five stacks of blocks as
illustrated below (as in Activity 1).
2. Have the children demonstrate the meaning of average by having a
pupil remove enough blocks from the tallest stack to put atop the
shortest stack so that both stacks match the middle stack.

3. Have another student do the same with the second and fourth stacks.
4. The teacher explains the concept of average by saying,
“In the example, average means having all the highs and low
stacks evened out until all the stacks are of the same height.
Each stack has three blocks and three is the average of the set
of blocks.”
5. Instruct the children to form groups of four.
6. Instruct the children to discuss how to determine the average for the
blocks without shifting blocks from one stack to another.
7. Using the children’s thinking as the basis for discussion, guide the
children learn the addition-division process of determining the
average.
8. The teacher explains the steps involved in finding average by saying,
“The average for the blocks is determined by adding 1 + 2 + 3 + 4 +
5 = 15; 15 5 = 3.”
9. Then teacher introduces the formula for finding average:
10. Give the Task Sheets and ask students to complete it

170
TASK SHEET
The average is calculated by adding up the item values and dividing it by the number of
items.
1. Calculate the average of the following numbers.
(a) 132, 246 and 174
(b) 1345, 1080, 1605
and 1830
(c) 156, 145, 556, 3352 and
4488
(d) 14.3, 9.68, and 8.7
(e) 20.36, 13.6, 22.44
and 45.6
(f) 23.4, 7.4, 46.1, 18.3 and
5.6
2. Calculate the average of the following quantities.
(a) 45 kg, 48 kg, 52 kg,
and 43 kg
(b) RM675, RM725,
RM750, and
RM775
(c) 900 mℓ, 950 mℓ,
1050 mℓ and
1200 mℓ
(d) 13.5 m, 6.3 m,
14 m and 84.1 m
(e) 19.6 cm3 , 600 cm3 ,
198 cm3 and
129.8 cm3
(f) 86.6 ℓ, 43 ℓ, 51.3 ℓ,
61 ℓ and 44.6 ℓ.

9.3.2 Calculating Average
Learning Outcome:
 To practise calculating averages.
Materials:
 A deck of cards comprising sets of numbers and answers; and
Example:
Procedures:
1. Prepare cards, some with sets of numbers and some with the
average of the sets of numbers.
2. Hand a card to each child.
3. Ask the children who are holding the cards with the sets of
numbers to calculate the average.
4. Ask the children to find their partners who are holding the
calculated averages.
5. If there is an odd number of children, you should take a card and
participate so that everyone has a partner.
6. Have the partners stand together so that everyone can see each
other’s cards. Have the children check everyone’s calculation of
the average. Are the partners correctly paired?
7. Distribute the Task Sheets and have the children work out the
answer.
4.66, 9, 0.12, and
13.5
8604, 777, 20, 1639
and 535
2315 6.82
ACTIVITIY 9.5

172
ACTIVITY 9.5
TASK SHEET
The average is calculated by adding up the item values and dividing it by
the number of items.
1. Calculate the average of the following numbers.
(a) 79, 105, 211,
234 and 81
(b) 100, 2000,
250,139 and
1331
(c) 7511, 1380,
4, 22, and 28
2. Solve the following problems.
(a) The sum of five numbers is 78 140. What is their average?
(b) The average of four numbers is 265.7. What is the sum of the
four numbers?
(c) The average of 55, 219, 7, 77 and X is 134. Find the value of X.
(d) Pak Hanif bought 4 watermelons. The masses of the watermelons
were 4.45 kg, 3.2 kg, 5.6 kg and 3.85 kg. What is the average
mass of the watermelons?
(e) The KL Monorail transported a total of 23 568 passengers from
KL Central to Bukit Bintang station over three days. What is the
average number of passengers transported in a day over the three
day period?

ACTIVITIY 9.6
Learning Outcome:
Materials:
 A deck of cards comprising sets of numbers and answers.
a Example:
830, 1000 and
960
930
24.3 and 10.7
17.5
Procedures:
1. Prepare cards, some with sets of numbers and some with the
average of the sets of numbers.
2. Place the answer cards (grey cards) in a circle on the floor.
3. Have the children march around the circle of answer cards on the
floor, chanting this rhyme:
Marching, marching, ‘round we go,
Not too fast and not too slow.
We won’t run and we won’t hop,
We’re almost there, it’s time to stop.
4. When the rhyme is finished, hold up a question card and ask the
children to find the average of the set of numbers shown on the
card.
5. The child who is standing by the answer card with the correct
calculated average, picks up the answer card and shows it to the
rest of the children.
6. Have the children 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.

174
ACTIVITY 9.7
Learning Outcome:
Materials:
 Four lists of questions on finding averages. Some of the questions
may be the same on each list.
 Answers to the averages.
Example:
List 1
Calculate the average of the following numbers.
1. 212, 108, 124 and 176
2. 16, 315, 4, 1986 and 24
3. 63, 147, 8, 192, and 10
4. 14.3, 16.76, 9.6 and 8.7
5. 12.8, 509 and 200.6
6. 13.02, 3.8, 5.22 and 14
List 2
Calculate the average of the following numbers.
1. 346, 15 and 1307
2. 16, 315, 4, 1986 and 24
3. 25, 125, 5, 25, and 225
4. 15.4, 34.7 and 75.9
5. 18, 120.9 and 221.1
6. 13.02, 3.8, 5.22 and 14

Procedures:
1. Prepare four lists of questions on finding averages. Some of the
questions may be the same on each list.
2. Prepare 24 cards, each showing the calculated average for each
question on each list. Tape these cards to the walls of the classroom.
3. Divide the children into four teams.
4. Give one of the lists to each of the team. (You might want to provide a
copy of the list for every member of the team).
5. Have the children calculate the averages for the numbers on their list.
6. Have the team members search for the answer cards taped on the walls
 A set of data can be described by choosing a single number that indicates
where the data in the set are centred or concentrated. The number most
commonly used to characterise a set of data is the arithmetic mean, frequently
called the average.
 The understanding of average can be developed through using concrete
materials and visual manipulation like using the interlocking cubes.
 Three types or measures of central tendency can be calculated. They are the
arithmetical average (mean), mode and median.
Distorted
Average
Mean (Average)
Median
Mode
of the classroom.
7. The first team to correctly calculate the average and collect all the
answer cards wins.

176
Rucker, W. E., & Dilley, C. A. (1981). Heath mathematics. Washington DC:
Heath and Company.
middle Schools. Ohio: Merrill Prentice Hall.

Topic
10
 Data
Handling
LEARNING OUTCOMES
1. Use vocabulary related to data handling correctly as required by the
Year 5 and Year 6 KBSR Mathematics Syllabus;
knowledge related to data handling;
3. Use the vocabulary related to data organisation in graphs correctly;
knowledge related to data organisation in graphs; and
5. Plan basic teaching and learning activities for data handling and data
organisation in graphs.
 INTRODUCTION
Most of the important decision making of modern society is based on statistics,
graphs and probability. In politics, advertising and economics, samples are
organised, survey questions developed, answers sought, results tabulated and
organised and predictions displayed with averages and graphs to show
distributions, relationships and trends of the data collected before decisions are
made. What will be the next flavour of cakes manufactured? Where will the land
for the next supermarket be bought? Data handling has become an important
aspect of life for many people today.
Graphs and statistics are indispensable to comprehending the raw data on which
decision making is based. A mass of data is incomprehensible. Averages supply a
framework with which to describe what happens. Graphs supply a visual way of
presenting the range of alternatives available and indicating where the density of

178  TOPIC 10 DATA HANDLING
interest lies. The forms of graph that are commonly used are bar graphs,
histograms, picture graphs, line graphs and pie charts.
Statistics within the primary school is predominantly the study of procedures for
collecting, recording, organising and interpreting data. Data handling is
introduced in primary schools in the belief that it is crucial for children to begin
study of the concepts and processes in statistics, graphs and probability as early as
possible. The difficulty lies in the lack of knowledge of what aspects of data
handling are suitable for primary children. Many primary school teachers have
little preparation for teaching data handling and little experience of it being taught
to them. By reading and applying what is written in this topic, it is expected that
teachers will be able to:
(a) Show pupils that statistics and graphs are part of mathematical activities in
their daily lives;
(b) Show pupils the connections between statistics and graphs to basic numbers
and space concepts; and
(c) Allow pupils to conduct simple statistical investigations and graphical
presentations.
ACTIVITY 10.1
Can you think of reasons why data handling exists in our lives? List
down the reasons before you could compare them with your partner.
10.1
Important information regarding the content and pedagogical aspects for teaching
data handling covers the following aspects:
(a) Statistical measures such as range, mode, median and mean;
(b) Collecting, recording, organising and interpreting data;
(c) Statistical procedures on organising data such as tables, charts and diagrams;
and
(d) Types of graphs used to visualise data.

TOPIC 10 DATA HANDLING  179
Mean: 60.07 inches
Median: 62.50 inches
Range: 42 inches
Variance: 117.681
Standard deviation: 10.85 inches
Minimum: 36 inches
Maximum: 78 inches
First quartile: 51.63 inches
Third quartile: 67.38 inches
Count: 58 bears
Sum: 3438.1 inches
2
1
0
Black Bears
3 4 5 6 7 8
Frequency
Length in Inches
Figure 10.1: Histogram showing the statistics of Black Bears
ACTIVITY 10.2
Figure 10.1 above shows an example of how a histogram can be used to
visualize data on black bears. List down four other graphical
representations and show how they differ from one another.
10.1.1 Statistical Measures
Computational statistics is a large and complex branch of mathematics with
significance for the social as well as physical and biological sciences. However, in
primary schools, pupils will be exposed only to the simplest of descriptive
statistics. The statistical measures studied in Year 5 and Year 6 are range, mean,
mode and median.
(a) Range
In a list of data, range is the difference between the greatest and the least
value. Consider the following results (out of 20) in a mathematics test for
two groups of students (the BLUE and the RED):
The BLUE scores: 6, 8, 10, 10, 5, 6, 11, 8, 11, 6, 7
The RED scores: 7, 9, 12, 14, 7, 9, 9, 5, 16, 9, 13

The range for the BLUE group is 11 – 5 = 6, while the range for the RED
group is 16 – 5 = 11.
(b) Mean
Mean is the average of the scores. To calculate it, the scores are added and
the result is divided by the number of scores. In the example above, the
mean for the BLUE group is
6 + 8 + 10 + 10 + 5 + 6 + 11+ 8 + 11 + 6 + 7 = 88, 88 divided by 11 is 8.
While the mean for the RED group is
7 + 9 + 12 + 14 + 7 + 9 + 9 + 5 + 16 + 9 + 13 = 110, 110 divided by 11 is 10.
(c) Mode
Mode is the most commonly occurring score. In the example above, the
mode for the BLUE group is 6, while the mode for the RED group is 9.
(d) Median
Median is the middle score when the scores are arranged in ascending order.
In the above example, there are 11 scores altogether, therefore the median is
the sixth score when the scores are arranged in ascending order.
BLUE: 5, 6, 6, 6, 7, 8, 8, 10, 10, 11, 11
RED: 5, 7, 7, 9, 9, 9, 9, 12, 13, 14, 16
Hence, the median for the BLUE group is 8 and the median for the RED
group is 9.
Note: If there is an even number of scores (say 10), then the median is
halfway between the half “score” and the next score (example: half way
between the 5th and the 6th score in ascending order). For example, for scores
5, 9, 3, 8, 6, 4, 6, 3
The arrangement of the scores in ascending order is
3, 3, 4, 5, 6, 6, 8, 9
And the fourth score is 5 and the fifth score is 6.
This means that the median is 5 + 6 = 11 divided by 2, and that will be 5.5.

10.1.2 Collecting, Recording, Organising and
Interpreting Data
Data handling can be a valuable aid in decision making. A commonly used format
to investigate problems (Thompson et al; 1976) is stated in the following 5 steps:
(a) Recognise and clearly formulate a problem;
(b) Collect relevant data;
(c) Organise the data appropriately;
(d) Analyse and interpret the data; and
(e) Relate the statistics obtained from the data to the original problem.
The five step format in using data to make decisions can be illustrated with the
example adapted from Thompson et al (1976).
(a) A group of children wished to send a representative to a softball throwing
contest. Three children volunteered. Each volunteer was asked to make five
throws which were measured with a trundle wheel to the nearest metre. The
results were:
Table 10.1: Result of softball throwing contest
Volunteers Their 5 throws ( to the nearest metre)
Shahar 28, 23, 22, 24, 27
Bala 24, 23, 27, 24, 27
Tony 23, 27, 29, 18, 26
(b) To help comprehend these results, the children tallied them into a frequency
table and graphed them onto bar graphs. They then calculated the mean,
median and range for each volunteer. The tables and the bar graphs are
shown below:
Table 10.2: The frequency
Distance of
throw (m)
18 19 20 21 22 23 24 25 26 27 28 29
Shahar 1 1 1 1 1
Bala 1 2 2
Tony 1 1 1 1 1

Bar Graph: Shahar
18 19 20 21 22 23 24 25 26 27 28 29
F
r
e
q
u
e
n
c
y
3
2
1
0
Length of throw (m)
Bar Graphs: Bala
18 19 20 21 22 23 24 25 26 27 28 29
F
r
e
q
u
e
n
c
y
3
2
1
0
Length of throw (m)
Bar Graphs: Tony
18 19 20 21 22 23 24 25 26 27 28 29
F
r
e
q
u
e
n
c
y
3
2
1
0
Length of throw (m)
Figure 10.2: Statistical measures
Next, the three statistical measures, mean, median an range are calculated and
tabulated in the table below.

Table 10.3: Three Statistical Measures
Mean Median Range
Shahar 24.8 24 6
Bala 25.0 24 4
Tony 24.6 26 11
Based on the frequency table, bar graphs and the statistical measures
constructed, ask your students the following questions.
(c) Who would be the best representative? Why?
Who is the most consistent? Why?
Who has the longest throw?
(d) What should be our criteria for selecting the best representative?
Who has the best typical throw?
How do we define typical?
Is consistency important?
Should we have measured more or less than five throws?
Should bad throws be excluded?
Is anything important lost in rounding to the nearest metre?
(e) Would it make it easier if we tallied the throws into sections, say 15-19, 20-
24, 25-29 etc.?
ACTIVITY 10.3
Write your answers for these two questions and compare them with your
partner next to you.
1. What are statistical measures?
2. Why is it necessary for children to know how to collect, record,
organise and interpret data?
10.1.3 Methods of Organising Data
The appropriate methods of organising data that seem suitable for the primary
years are interpreting and constructing simple tables, charts and diagrams that are
commonly used in everyday life to display information. The basis of this
component is the organisation of raw data into collections. This means
determining the extent of the possible outcomes, forming these into categories and

organising the data under these categories. The techniques that may have to be
used in this process are combinatorial counting (to determine all the possible
outcomes) and tallying (to organise the data under the categories). Let us begin
this section by introducing to you the tables.
(a) Tables
(i) The simple table
An example of this simple table is the table of contents on a cereal packet.
It consists of words and figures in two columns (refer to Figure 10.3).
Oats Meal Cereal: Average contents per serving:
Vitamin C 25 mg
Iron 27 mg
Niacin 11 mg
Riboflavin 38 mg
Figure 10.3: Table of contents on a cereal packet
(ii) The regular table
The regular table is the matrix style table where there are more than
two columns (more than column of data). The everyday example is the
bus timetable. It is useful when comparing, for example, results from
one year to another or between different people. Another common
example of this table is in advertisements where prices at different
shops are compared (refer to Table 10.4).
Table 10.4: Materials Collected by the Children in 6 Orkid
Bakar Muthu Chong Mary Rokiah
Bottle tops 5 8 7 6 2
Cotton reels 9 3 5 2 8
Egg Cartons 5 7 2 9 3
Plastic spoons 3 5 8 3 7
(b) Charts
Charts are less regular in terms of rows and columns. They attempt to
display information more visually, to relate the display to what actually
occurs. As such, we have the road maps and bus routes of transport and the
time lines of history.

(i) The strip map
This may be the bus route of an area or the time line of a history topic.
A line is drawn and on this line are marked references to major
features (refer to Figure 10.4).
Bus Route:
Ipoh Tapah Bidor Sungkai
The Rule of King Willhem:
Coronation The
Great
War
Birth of
Prince
Henry
Birth of
Prince
Derek
Death of
the Duke
Figure 10.4: Bus route of an area or time line of a history topic
(ii) The branch map
This is a combination of strip maps, involving branching as in a tree.
The most straight forward examples are the road maps or genealogy
diagrams (family tree of parents, grandparents etc.). The skill of
following directions from a map is an important life skill that our
children must master. An example of a family tree is shown below.
Kamal Baharuddin and Fauziah Hamid
Kassim Fauziah m Ahmad Karim m Rokiah Kamsiah
Siti Yusuf Kamarul
Figure 10.5: Kamal Baharuddin’s family tree

(a) Diagrams
These are visual ways to represent membership in different sets and subsets.
A Venn diagram and a Carroll diagram could be considered the most
favourable diagrams used to show the relationship between the members of
a given group of objects.
(i) Venn Diagram: An example of a Venn diagram for flowers in terms
of red and scented.
Neither red nor scented
Red
Flowers
Scented
flowers
Red and scented
Figure 10.6: Venn Diagram
(ii) Carroll Diagram: An example of a Carroll diagram for flowers in
terms of red and scented.
Red Not Red
Scented Red and scented
flowers
Not red and scented
flowers
Not Scented Red and not scented
flowers
Not red and not scented
flowers
Figure 10.7: Carroll diagram
10.1.4 Types of Graphs
The importance of graphs in primary schools arises from two simple ideas
(a) A picture is worth a thousand words; and
(b) Mathematics is a study of relationships.

Graphs are not in the syllabus to give light relief to the numerical activities. Their
purpose is to improve communication and understanding, especially for children
of lower ability. However, we can all gain insight to complicated statistical
information if it is displayed in a graphical manner. Obviously, knowing how to
draw graphs and to draw inferences from them are valuable skills to acquire.
Bar graphs, picture graphs, line graphs, circle graphs and scatter graphs, can all be
used to visualise data. These various forms of graphs are commonly seen in real
life – in magazines, newspapers, textbooks and advertisements. The objective in
using a graph is to visually present information in a form which enables it to be
assimilated “at a glance” as compared to a list of numbers.
Graphs are yet further examples of representing information in such a way that
patterns are evident or worthwhile seeking. If particular patterns emerge, time and
time again we can conclude that, indeed, some generalisation can be made about
the circumstances we are representing. Hypothesis can be formulated and tested
and a visual display made of the results. Concepts are more clearly understood as
a consequence and fundamental principals are consolidated.
(i) Bar graphs
Bar graphs facilitate comparisons of quantities. Bar graphs can be vertical as
well as horizontal (columns as well as rows). They can also be in the form of
blocks, or bar lines. The following are examples of bar graphs (Figure 10.8):
Cats
Dogs
Fish
Birds
0 5 10 15 20 25
(a) Horizontal Bar Graph: Types of pets children have

Bus Car Bicycle Motorcycle
25
20
15
10
5
(b) Vertical Bar Graph: Types of vehicles children use to go to school
Figure 10.8 (a) & (b): Bar graphs
(ii) Picture Graphs
Picture graphs can also facilitate comparisons of quantities just like bar
graphs. They can easily be updated. Picture graphs are also called
pictographs and isotypes. An example of a picture graph is shown below.
KEY: represents RM 100
Class A
Class B
Class C
Class D
Figure 10.9: Picture Graph Money accumulated for classroom projects

(iii) Line Graphs
Line graphs can be used for comparisons and for expressing allocations of
resources, but they seem particularly useful for communicating trends. Here
is an example of a line graph.
40oC
30oC
20oC
10oC
Mon Tue Wed Thurs Fri
Figure 10.10: Line graph maximum temperatures during the week
(iv) Circle Graphs
Circle graphs (also known as pie charts) are used to picture the totality of a
quantity and to indicate how portions of the totality are allocated. Here is a
circle graph indicating how one college student spent his budget.
College Costs
Room and Board
Entertainment
Clothing
Miscellaneous
Figure 10.11: Circle graph: Kamaruddin’s budget
(v) Scatter graphs
Scatter graphs are similar to line graphs which show the relationship between
two different sets of data. The scatter graph is made for data which is not in
sequence (in terms of the horizontal axis) and is unsuitable for a line graph.
Here is a scatter graph which shows that mass is related to height.

50 kg 100 kg 150 kg 200 kg
200 cm
150 cm
100 cm
50 cm
Figure 10.12: Scatter graph weight and height of students
SELF-CHECK 10.1
1. Describe briefly the three methods of organising data.
2. Explain the five types of graphs with the help of visual
representations.
DATA HANDLING IN YEAR 5 AND
YEAR 6
10.2
Our students will learn the topic of data handling effectively if we plan the
lessons systematically. A well organised conceptual development of statistical
measures, collecting, recording, organising and interpreting of data will be very
helpful for our students to understand these concepts better. It is recommended to
instruct this topic within a problem solving environment and in a less stressful
manner. Remember to provide opportunities for our students to differentiate the
different types of graphs and when they are best used.

The major mathematical skills to be mastered by pupils studying the topic of data
handling in Year 5 and Year 6 are as follows:
(a) Average
(i) Describe the meaning of average;
(ii) State the average of two, three, four or five quantities;
(iii) Calculate the average using a formula; and
(iv) Solve problems in real life situations.
(b) Data Collection
(i) Collect data;
(ii) Process data; and
(iii) Analyse data.
(c) Pictograph
(i) Identify pictograph which represents one or more than one unit;
(ii) Extract information from a pictograph; and
(iii) Construct a pictograph.
(d) Bar Charts
(i) Identify characteristics of a bar chart;
(ii) Extract information from a bar chart; and
(ii) Construct a bar chart.
10.3
This section begins by describing the teaching and learning activities for you to
conduct a lesson on data handling. Let us do Activity 10.4 first. Enjoy!

10.3.1 Average
ACTIVITY 10.4
Learning Outcomes:
 To state the average of two, three, four or five quantities
 To calculate the average using a formula
Materials:
 Task Cards
 Answer Sheets
Procedure:
1. Divide the class into groups of five students and give each student
an Answer Sheet.
2. Ask the students to write their name on the Answer Sheet.
3. Shuffle the Five Task Cards and place them face down in a stack at
the centre.
4. Instruct each player to begin by drawing a card from the stack.
5. Instruct the player to write all the answers to the questions in the
6. After a period of time (to be determined by the teacher), the pupils
in the group exchange cards with the pupil on their left in a
clockwise direction.
7. Pupils repeat steps (5 and 6) until everyone has answered the
questions in all the cards.
8. The pupil with the most number of correct answers, wins.
9. Teacher summarises the lesson on the meaning of average.

Name :___________________ Class :__________
1._____ 1._____ 1._____
2._____ 2._____ 2._____
3._____ 3._____ 3._____
Card D Card E
1._____ 1._____
2._____ 2._____
3._____ 3._____
Task Card A
1. Calculate the average of 264 and 246.
Average = _______________
2. Calculate the average of RM273, RM264 and RM 252.
Average = RM ___________
3. Find the average of 4.2 km, 5.1 km, 4900 m and 5 km.
Average = ___________ km
ACTIVITY 10.5
Work with your friend in class to prepare four other Task Cards.
Make sure your cards are based on the learning outcomes of
Activity 10.4.

10.3.2 Organising and Interpreting Data
ACTIVITY 10.6
Learning Outcomes:
 To recognise frequency, mode, range, average, minimum and
maximum value from a bar graph; and
 To find the frequency, mode, range, average, minimum and
maximum value from a given bar graph.
Materials:
 30 different Flash Cards; and
Procedure:
1. Divide the class into groups of three students and give each group a
clean writing paper.
2. Ask the students to write their names on the clean paper given.
centre.
4. Instruct Player A to begin by drawing a card from the stack. He
shows the card to Player B.
5. Instruct Player B to read the answers within the stipulated time
6. Instruct Player C to write the points below Player B’s name. Each
each Flash Card).
7. Players repeat steps (4 and 5) until all 10 cards have been drawn by
Player A.
8. Repeat steps (3 through 6) until all the players have the opportunity
to read all 10 Flash Cards shown to them.
9. The winner is the group of students that has the most number of
points.
10. Teacher summarises the lesson on how to find the frequency,
mode, range, average, minimum and maximum value from a given
bar graph.

Flash Card 1
Mass of fish caught in kg
Mon
150
100
50
Tue Wed Thurs Days
1. What is the most common amount of fish caught?
Answer: __________ kg
2. What is the mass of fish caught on Monday?
Answer: __________ kg
3. Find the average mass of fish caught in the four days.
Answer: __________ kg
4. What is the minimum mass of fish caught?
Answer: __________ kg
5. What is the maximum mass of fish caught?
Answer: __________ kg
6. Find the range between the maximum and the minimum mass of fish
caught.
Answer: __________ kg
ACTIVITY 10.7
Work with three friends of yours in class to prepare twenty-nine other
Flash Cards. There should be six questions in each Flash Card. Make
sure your cards are based on the learning outcomes of Activity 10.6.

10.3.3 Pie Chart
ACTIVITY 10.8
Learning Outcomes:
 To recognise frequency, mode, range, average, minimum and
maximum value from a pie chart; and
 To find the frequency, mode, range, average, minimum and
maximum value from a given pie chart.
Materials:
 Task Sheets;
 Colour pencils.
Procedure:
1. Divide the class into groups of four to six students. Give each
group a different colour pencil and a clean writing paper.
2. The teacher sets up five stations in the classroom and places a Task
3. The teacher instructs students to solve the questions in the Task
8. Teacher summarises the lesson on how to find the frequency,
mode, range, average, minimum and maximum value from a given
pie chart.

STATION 1
The pie chart below shows the colours of 1,000 marbles owned by Gopal.
White
19%
Blue
Black
5%
Red
25%
Green
19%
1. What is the percentage of blue marbles?
Answer:___________
2. What is the most common colour of the marbles?
Answer:___________
3. Calculate the range.
Answer:___________
4. Find the average percentage of the different colours of marbles owned by
Gopal.
Answer:___________
ACTIVITY 10.9
Work with two of your friends to prepare four other Task Sheets for the
other stations. There should be four questions in each sheet.
Make sure your sheets are based on the learning outcomes of Activity
10.8.

10.3.4 Problem Solving
ACTIVITY 10.10
Learning Outcomes:
 To solve problems involving average; and
 To solve problems involving graphs.
Materials:
 Activity Cards;
 Colour pencils.
Procedure:
2. Shuffle a set of 12 Activity Cards and place them face down in a
3. Teacher signals to the students to begin solving the questions in the
first Activity Card drawn.
4. Once they have completed the first Card, they may continue with
the next Activity Card.
answer papers to the teacher.
contexts involving averages and graphs.
ACTIVITY 10.10
ACTIVITY 1

Activity Card 1
1. The total score of Ali, Babu and Chin in a mathematics test is 260. The
average score of Ali and Chin is 85. Find Babu’s score.
2. The average mass of four pupils is 22.9kg. Ali joins the group and the
average mass of the pupils is now 23.6 kg. What is Ali’s mass in kg?
Questions 3 and 4 are based on the bar graph below.
Amount of money saved by four students
Suzy
150
100
50
Samy Sarah Samsul Girl
Money (RM)
3. What is the percentage of money saved by Sarah?
4. What is the difference between the amount of money saved by Samy
and Samsul?
ACTIVITY 10.11
Prepare 11 other Activity Cards for the group. There should be four
10.10.

 Most of the important decision-making carried out in modern society is based
on statistics, graphs and probabilities.
 Graphs and statistics are indispensable to comprehending the raw data on
which decision-making is based.
 Statistics within the primary school is predominantly the study of procedures
for the collection, recording, organisation and interpretation of data.
 Many primary school teachers have little preparation for teaching data
handling and little experience of it being taught to them.
 In a list of data, range is the difference between the greatest and the least
value. Mean is the average of the scores. Mode is the most commonly
occurring score. Median is the middle score when the scores have been
arranged in an ascending order.
 A commonly used format to investigate problems in data handling are the
following 5 steps:
– Recognise and clearly formulate a problem;
– Collect relevant data;
– Organise the data appropriately;
– Analyse and interpret the data; and
– Relate the statistics obtained from the data to the original problem.
 The appropriate methods of organising data that seem suitable for the primary
years are interpreting and constructing simple tables, charts and diagrams
that are commonly used in everyday life to display information.
 Bar graphs, picture graphs, line graphs, circle graphs and scatter graphs,
can all be used to picture data. These various forms of graphs are commonly
seen in the real world – in magazines, newspapers, textbooks and
advertisements.

Chart
Diagram
Graph
Mean
Median
Mode
Probability
Range
Raw data
Statistics
Table
Bahagian Pendidikan Guru. (1998). Konsep dan aktiviti pengajaran dan
pembelajaran matematik: Ukuran. Kuala Lumpur: Dewan Bahasa dan
Pustaka.
Burrows, D., & Cooper, T. (1987). Statistics, graphs and probability in the primary
school (trial materials). Queensland, Australia: Carseldine Campus.
Nur Alia Abd. Rahman & Nandhini. (2008). Siri intensif: Mathematics KBSR year
Nur Alia Abd Rahman & Nandhini. (2008). Siri intensif : Mathematics KBSR year
Ng, S.F. (2002). Mathematics in action workbook 2B (Part 1). Singapore: Pearson
Education Asia.
Clarke, P. et al. (2002). Maths spotlight activity sheets 1. Oxford: Heinemann

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