HBMT 1203 Mathematic

Topic
1
Numbers
0 to 10
LEARNING OUTCOMES
By the end of this topic, you should be able to:
1. Recognise the major mathematical skills of whole numbers from 0 to
10;
2. Identify the pedagogical content knowledge of pre-number
concepts, early numbers and place value of numbers from 0 to 10;
3. Plan teaching and learning activities for pre-number concepts and
early numbers from 0 to 10; and
4. Determine and learn the strategies for teaching and learning
numbers in order to achieve Âactive learningÊ in the classroom.
INTRODUCTION
Beginning number concepts are much more complex than we realise. Just because
children can say the words ÂoneÊ, ÂtwoÊ, ÂthreeÊ and so on, does not mean that they
can count the numbers. We want children to think about what they are counting.
Children can count numbers if they understand the words Âhow manyÊ. As
teachers, we do not teach numerals in isolation with the quantity they represent
because numerals are symbols that have meaning for children only when they are
introduced as labels of quantities. In order to start teaching numbers effectively, it
is important for you to have an overview of the mathematical skills of whole
numbers. At the beginning of this topic, you will learn about the history of
various numeration systems and basic number concepts such as the meanings of
ÂnumberÊ, ÂnumeralÊ and ÂdigitÊ. You will also learn about the stages of conceptual
development for whole numbers including pre-number concepts and early
numbers. Children learn to recognise and write numerals as they learn to develop
early number concepts. In the second part of this topic, you will learn more about
the strategies for the teaching and learning of numbers through a few samples of

TOPIC 1 NUMBERS 0 TO 10
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teaching and learning activities. You are also encouraged to hold discussions
with your tutor and classmates. Some suggested activities for discussion are also
given.
PEDAGOGICAL CONTENT KNOWLEDGE OF
WHOLE NUMBERS: NUMBERS 0 TO 10
1.1
In this section, we will be focusing on the major mathematical skills for pre-number
concepts and whole numbers 0 to 10 as follows:
(a) Determine pre-number concepts;
(b) Compare the values of whole numbers 1 to 10;
(c) Recognise and name whole numbers 0 to 10;
(d) Count, read and write whole numbers 0 to 10;
(e) Determine the base-10 place value for each digit 0 to 10 ; and
(f) Arrange whole numbers 1 to 10 in ascending and descending order.
1.1.1 Pre-number Concepts
The development of number concepts for children in kindergarten begins with
pre-number concepts and emphasises on developing number sense the ability
to deal meaningfully with whole number ideas as opposed to memorising
(Troutman, 2003).
At this level, children are guided to interact with sets of things. As they interact,
they sort, compare, make observations, see connections, tell, discuss ideas, ask
and answer questions, draw pictures, write as well as build strategies. They begin
to form and organise cognitive understanding. In short, children will have to
learn the prerequisite skills needed as stated below:
(a) Develop classification abilities by their physical attributes;
(b) Compare the quantities of two sets of objects using one-to-one matching;
(c) Determine quantitative relationships including Âas many asÊ, Âmore thanÊ
and Âless thanÊ;
(d) Arrange objects into a sequence according to size (small to big), length
(short to long), height (short to tall) or width (thin to thick) and vice versa;
and

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(e) Recognise repeating patterns and create patterns by copying repeating
patterns using objects such as blocks, beads, etc.
1.1.2 Early Numbers
Mathematics starts with the counting of numbers. There are no historical records
of the first uses of numbers, their names and their symbols. Various symbols are
used to represent numbers based on their numeration systems. A numeration
system consists of a set of symbols and the rules for combining the symbols.
Different early numeration systems appeared to have originated from tallying.
Ancient people measured things by drawing on cave walls, bricks, pottery or
pieces of tree trunks to record their properties. At that time, ÂnumbersÊ were
represented by using simple Âtally marksÊ (/). Some numeration systems
including our present day system are shown in Table 1.1.
Table 1.1: Early Number Representations
Today 1 2 3 4 5 6 7 8 9
Ancient
Egypt
Babylon
Mayan . . . . . . . . . .
.
. .
. . .
. . . .
About 5000 years ago, people in places of ancient civilisations began to use
different symbols to represent numbers for counting. They created various
numeration systems. For example, the Egyptian numeration system used picture
symbols called hieroglyphics as illustrated in Figure 1.1.

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Figure 1.1: Egyptian hieroglyphics
This is a base-10 system where each symbol represents a power of 10.
What number is represented by the following illustration?
2(10 000) + 1000 + 3(100) + 4(10) + 6 = 21 346
Try writing the following numbers in hieroglyphics:
(a) 245
(b) 1 869 234
On the other hand, the Babylonians used a base-60 system consisting of only two
symbols as given below.
one ten
As such, the number 45 is represented as follows:
4(10) + 5 = 45
For numbers larger than 60, base-60 is used to represent numbers in the
Babylonian Numeration System.
Have fun computing the following illustrations:
(a)

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(b)
Apart from the nine symbols in Table 1.1, the Mayan Numeration System consists
of 20 symbols altogether and is a base-20 system, as shown in Figure 1.2.
Figure 1.2: Mayan numerals
The following illustration depicts clearly the unique vertical place value format of
the Mayan Numeration System, see Figure 1.3.
Figure 1.3: Mayan number chart
Source: Mayan number chart from http://en.wikipedia.org/wiki/Maya_numerals
What number is represented thus?
12 + 7(20) + 0(20.18) + 14(20.18.20)
= 12 + 140 + 0 + 100800 = 100952

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Simple addition can be carried out by combining two or more sets of symbols as
shown in the examples given below. Try computing these operations using
Hindu-Arabic numerals.
(a)
(b)
Solutions:
(a) 6 + 8 = 14
(b) {7 + 0(20) + 14(20.18) + 1(20.18.20)} + {14 + 0(20) + 3(20.18) + 2(20.18.20)} + {1
+ 1(20) + 17(20.18) + 3(20.18.20)}
= 7 + 0 + 5040 + 7200 + 14 + 0 + 1080 + 14400 + 1 + 20 + 6120 + 21600}
= 55482
The complexities of the above examples and illustrations of the various ancient
numeration systems discussed in this section should help you to realise why they
are no longer in use today. Table 1.2 shows some other famous historical
numeration systems used to this day including the Roman Numeration System,
Greek Numeration System and our Hindu-Arabic Numeration System.
Table 1.2: Famous Number Representations
Roman
200 B.C. I II III IV V VI VII VIII IX
Greek 500
B.C.
z
Hindu-
Arabic 500
A.D.
1 2 3 4 5 6 7 8 9
Hindu-
Arabic 976
A.D.
l

7 8 9

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Along with the development of numbers, mathematics was further developed by
famous mathematicians. The numeration system used today is based on the
Hindu-Arabic numeration system. Can you explain why the Hindu-Arabic
numeration system is being used today?
At this point, you should have a clearer picture about the difference between a
ÂnumberÊ, a ÂnumeralÊ and a ÂdigitÊ. The terms ÂnumberÊ, ÂnumeralÊ and ÂdigitÊ are
all different. A number is an abstract idea that addresses the question, Âhow
manyÊ and means Ârelated to quantityÊ, whereas a numeral is a symbol for
representing a number that we can see, write or touch. Thus, numerals are names
for numbers. A ÂdigitÊ refers to the type of numerals used in a numeration system.
For example, our present numeration system is made up of only 10 different
digits, that is, 0 to 9.
SAMPLES OF TEACHING AND LEARNING
ACTIVITIES
1.2
In this section, you will read about some samples of teaching and learning
activities that you can implement in your classroom.
1.2.1 Teaching Pre-number Concepts
There are many pre-number concepts that children must acquire in order to
develop good number sense. These are as follows:
(a) Classify and sort things in terms of properties (e.g. colour, shape, size, etc.);
(b) Compare two sets and find out whether one set has Âas many asÊ, Âmore
thanÊ, or Âless thanÊ the other set;
(c) Learn the concepts of Âone moreÊ and Âone lessÊ.
(d) Order sets of objects according to a sequence according to size, length,
height or width; and
(e) Recognise and copy repeating patterns using objects such as blocks, beads,
etc.
Now, let us look at some activities that you can do with your pupils.

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Activity 1: Classifying Things by Their Properties
Learning Outcomes:
By the end of this activity, your pupils should be able to:
(a) Classify things by their general and specific properties.
Materials:
Sets of toys;
Sets of pattern blocks (various shapes, colour, size, etc.); and
Plastic containers or boxes.
Procedure:
(a) Classify Objects by Their General Properties
Teacher asks children to work in groups of five and distributes four types of
toys (e.g. car, train, boat and aeroplane) to each group.
Teacher says: „LetÊs work together, look at the toys.‰
Teacher asks: „Which are the toys that can fly? Which one can sail in the
sea? Which is the longest vehicle? Which is the smallest vehicle? Which is
the fastest vehicle? Which is the slowest vehicle?‰
Children respond to questions asked.
In this activity, children should be asked why they chose that specific object
and not the others. Teacher listens to childrenÊs responses.
(b) Classify Objects by Their Specific Properties
Teacher distributes a set of pattern blocks with different shapes, sizes and
colours to each group, see Figure 1.4.

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Figure 1.4: Pattern blocks
(i) Teacher says: „Firstly, classify these objects by their shapes.‰
„Put the objects into the boxes: A, B, C and D according to their
shapes.‰ (e.g. circle, triangle, rectangle and rhombus, see Figure 1.5
(a).
Figure 1.5 (a): Pattern blocks and containers
(ii) Teacher says: „Secondly, classify these objects by their sizes.‰
„Put the objects into the boxes: A, B and C according to their sizes.‰
(e.g. small size in box A, medium size in box B and large size in box C
with respect to their shapes, see Figure 1.5 (b).
Figure 1.5 (b): Pattern blocks and containers

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(iii) Teacher says: „Lastly, classify these objects by their colours.‰
„Put the objects into the boxes: A, B, C, D, E and F according to their
colours‰. (e.g. orange, blue, yellow, red, green and purple, see Figure
1.5 (c).
Figure 1.5 (c): Pattern blocks and containers
At this stage, children will recognise that shape is the first property to consider,
followed by size and colour. Children should be encouraged to find as many
properties as they can when classifying objects.
You can also try some other activities with the children such as classifying objects
by their texture (smooth, rough and fuzzy) or by their size (short and long), etc.
to prepare them to learn about putting objects into a sequence, that is, the skill of
ordering or seriation, which is more difficult than comparing since it involves
making many decisions.
For example, when ordering three drinking straws of different lengths from short
to long, the middle one must be longer than the one before it, but shorter than the
one after it.
Next, in Activity 2, your pupils will be asked to find the relationship between two
sets of black and white objects. Let us now take a look at Activity 2.

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Activity 2: Finding the Relationship between Two Sets of Objects
Learning Outcomes:
(a) Match items on a one-to-one matching basis;
(b) Understand and master the concept of Âas many asÊ, Âmore thanÊ and Âless
thanÊ; and
(c) Compare the number of objects between two sets.
Materials:
Picture cards (A, B, C and D);
Erasers; and
Pencils, etc.
Procedure:
(i) One-to-One Matching Correspondence
Children are presented with two picture cards, (Card A and Card B)
consisting of the same number of objects.
Teacher demonstrates how the relationship of Âas many asÊ can be
introduced using a one-to-one matching basis as follows, see Figure 1.6 (a):
Figure 1.6 (a): One-to-one matching correspondence
Teacher asks: „Are there as many moons as stars? Why?‰
(ii) As Many As, More and Less
Teacher takes out a star from Card B and asks, „Are there as many moons
as stars now? Why? How can you tell? etc.‰ See example in Figure 1.6 (b).

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Figure 1.6 (b): One-to-one matching correspondence
Teacher guides the children to build the concept of ÂmoreÊ and ÂlessÊ. For
example, which card has more moons? Which card has fewer stars?
(iii) More Than, Less Than
The children are presented with another two picture cards (Card C and
Card D) with different numbers of objects. Teacher guides the children to
compare the number of objects between the two sets and introduces the
concept of Âmore thanÊ and Âless thanÊ.
Teacher says: „Can you match each marble in Card C one-to-one with a
marble in Card D? Why?‰
Teacher says: „Children, we can say that Card C has more marbles than
Card D, or, Card D has less marbles than Card C‰.
In addition, teacher can ask her pupils to do a group activity as follows:
Teacher says: „Sit together with your friends in a group‰. „Everybody, show all
the erasers and pencils you have to your friends‰. „Can you compare the number
of objects and tell your friends using the words, Âmore thanÊ or Âless thanÊ?‰
Pupils should be able to respond as such: „I have more erasers than you but, I
have fewer pencils than you‰, „You have more erasers than me‰, etc.
Do try and think of other appropriate activities you can plan and implement to
help children to acquire pre-number experience or concepts essential for
developing good number sense prior to learning whole numbers.
ACTIVITY 1.1
Which of the pupilsÊ learning activities do you like the most? Explain.

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1.2.2 Teaching Early Numbers
This section elaborates on the activities which you can implement with your
pupils to help them understand the concept of early numbers.
Activity 3: Name Numbers and Recognise Numerals 1 to 10
Learning Outcomes:
By the end of this activity, pupils should be able to:
(a) Name and recognise numerals 1 to 5.
Materials:
Picture cards (0 to 5);
Number cards (1 to 5); and
PowerPoint slides.
Procedure:
(a) Clap and Count
Teacher claps and counts 1 to 5. Teacher and pupils clap and count a series
of claps together. ÂClapÊ, say ÂoneÊ. ÂClapÊ, ÂClapÊ, say ÂoneÊ, ÂtwoÊ.
Teacher asks pupils to clap twice and count one, two; Clap four times and
count one, two, three, four, etc. Pupils respond accordingly. Do the same
until number 5 is done.
(b) Slide Show
Teacher displays a series of PowerPoint slides one by one as shown in
Figure 1.7. The numerals come out after the objects.
Figure 1.7: Picture numeral cards

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Teacher asks: „How many balls are there in this slide?‰ and says, „Let us
count together.‰
Teacher points to the balls and asks pupils to count one by one. Then, point
to the numeral and say the number name. Guide pupils to respond (e.g.
„There is one ball‰, „There are two balls‰, etc.). Repeat with different
numbers and different pictures of objects.
(c) Class Activity
(i) Teacher shows a picture card and asks pupils to stick the correct
number card beside it on the white board. e.g.:
Teacher says: „Look at the picture. How many clocks are there?‰
Pupils respond accordingly. Then teacher asks a pupil to choose the
correct number card and stick it beside the picture card on the white
board.
Teacher repeats the steps until the fifth picture card is used. At the
end, teacher asks pupils to arrange the picture cards in ascending
order (1 to 5) and then asks them to count accordingly.
(ii) Teacher shows a number card and asks the pupils to stick the correct
picture card beside it on the white board. e.g.:
Teacher says: „Look at the card. What is the number written on the
card?‰

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Pupils respond accordingly. Then teacher asks a pupil to choose the
correct picture card and stick it beside the number card on the white
board.
Teacher repeats the steps until the fifth numeral card is done. At the
end, teacher asks pupils to arrange the number cards in ascending or
descending order (e.g. 1 to 5 or 5 to 1) before asking them to count in
sequence and at random.
(d) Group Activity
Pupils sit in groups of five. Teacher distributes five picture cards of objects
and five corresponding numeral cards (1 to 5).
Teacher says: „Choose a pupil in your group. Put up the number five card
in his/her left hand and the correct picture card on his/her right hand.
Help him/her to get the correct answer.‰
Teacher asks the group to choose another pupil to do the same for the rest
of the cards. Repeat for all the numbers 1 to 5.
Teacher distributes a worksheet.
Teacher says: „LetÊs sing a song about busy people together.‰ (refer to
Appendix 1)
Activity 4: Read and Write Numbers, 1 to 10
Learning Outcomes:
(a) Read and write numbers from 1 to 10.
Materials:
Picture cards;
Cut-out number cards (1 5);
Number names (name cards, one to five); and
Plasticine.

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Procedure:
(i) Numbers 1 to 5
Teacher shows the picture cards with numbers, 1 to 5 in sequence. Pupils
count the objects in the picture card, point to the number and say the
number name out loud. e.g.:
Teacher sticks the picture card on the writing board. Repeat this activity for
all the picture and number cards, that is, until the fifth card is done.
(ii) Technique of Writing Numbers
Teacher demonstrates in sequence the technique of writing numerals, 1 to 5.
Firstly, teacher writes the number Â1Ê on the writing board step by step as
follows: e.g.:
1
Teacher writes the number in the air followed by the pupils. Repeat until
number 5 is done.
Repeat until the pupils are able to write numbers in the correct way.
(iii) Plasticine Numerals
Teacher distributes some plasticine to pupils and says: „Let us build the
numerals with plasticine for numbers 1 to 5. Arrange your numbers in
sequence.‰

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(iv) Cut-out Number Card
Teacher gives pupils the cut-out number cards, 1 to 5. Then, teacher asks
them to trace the shape of each number on a piece of paper. e.g.:
Teacher distributes Worksheet 1 (refer to Appendix 2).
Note: This strategy can also be used to teach the writing of numbers, from 6
to 10.
Can you write these numbers in the correct way?
Activity 5: The Concept of Zero
Learning Outcomes:
(a) Understand the concept of ÂzeroÊ or ÂnothingÊ; and
(b) Determine, name and write the number zero.
Materials:
Picture cards; and
Three boxes and five balls (Given to each group).
Procedure:
(i) Teacher shows three picture cards.

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Teacher asks: „How many rabbits are there in Cage A, B and C?‰
Pupils respond: „There is one rabbit in Cage B, two rabbits in Cage C and
no rabbits in Cage A.‰
Teacher introduces the number Â0Ê to represent Âno rabbitsÊ or ÂnothingÊ.
(ii) Teacher distributes some balls into three boxes.
Teacher asks: How many balls are there in Box A, Box B and Box C
respectively?‰
Teacher guides pupils to determine the concept of ÂzeroÊ or ÂnothingÊ
according to the number of balls in Box B.
Teacher reads and writes the digit Â0‰ (zero), followed by pupils.
Activity 6: Count On (Ascending) and Count Back (Descending) in Ones, from 1
to 10
Learning Outcomes:
(a) Count on in ones from 1 to 10;
(b) Count back in ones from 10 to 1; and
(c) Determine the base-10 place value for each digit from 1 to 10.
Materials:
Number cards (1 10);
Picture cards; and
PowerPoint slides.

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Procedure:
(a) Picture Cards
(i) Ascending Order
Teacher flashes picture cards and the corresponding number cards in
ascending order, (i.e. 1 to 10). Pupils count the objects in the picture
cards and say the numbers. Teacher sticks the cards on the whiteboard
in sequence. e.g.:
Continue until the 10th picture card is done.
Pupils are asked to count on in ones from 1 to 10. The activity is
repeated.
(ii) Descending Order
Teacher flashes picture cards and the corresponding number cards in
descending order, (i.e. 10 to 1). Pupils count the objects in the picture
cards and say the numbers. Teacher sticks the cards on the whiteboard
in sequence. e.g.:

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Continue until the first picture card is done.
Pupils are asked to count back in ones from 10 to 1. The activity is
repeated.
(b) Slide Show
(i) Ascending Order
Pupils are presented a series of slides (PowerPoint presentation):
Teacher asks pupils to count and say the number name, e.g. „one‰.
Teacher clicks a button to show the second stage and asks pupils to
count and say the number.

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Continue until the 10th stage. Repeat until the pupils are able to count
on in ones from 1 to 10.
(ii) Descending Order
Teacher repeats the process as above but in descending order (i.e. 10
to 1).
Teacher presents another slide show, see Figure 1.8:
Figure 1.8: Number ladder
(c) Teacher Distributes a Worksheet
(i) Jump on the Number Blocks
Teacher asks pupils to sing the ÂNumbers Up and DownÊ song while
jumping on the number blocks around the pond, that is, counting on
or counting back again and again!
„Let us sing the ÂNumbers Up and DownÊ song together‰ (see Figure
1.9).
Figure 1.9: Number blocks

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(ii) Arranging Pupils in Sequence
Teacher selects two groups of 10 pupils and gives each group a set of
number cards, 1 to 10, see Figure 1.10. Teacher asks them to stand in
front of the class in groups. Teacher asks both groups to arrange
themselves in order. The group that finishes first is the winner. The
losing group is asked to count on and count back the numbers in ones.
Repeat the game.
Figure 1.10: Number cards
(iii) Going Up and Down the Stairs
Pupils are asked to count on in ones while going up the stairs and
count back in ones while going down the stairs.
As a mathematics teacher, you have to generate as many ideas as possible
about the teaching and learning of whole numbers. There is no „one best
way‰ to teach whole numbers.
As we know, the goal for children working on this topic is to go beyond
simply counting from one to 10 and recognising numerals. The emphasis here
is developing number sense, number relationships and the facility with
counting.
The samples of teaching and learning activities in this topic will help you to
understand basic number skills associated with childrenÊs early learning of
mathematics.
They need to acquire ongoing experiences resulting from these activities in
order to develop consistency and accuracy with counting skills.

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Ascending order
Descending order
Digit
Early numbers
Number
Numeral
One-to-one matching correspondence
Pre-number Concepts
Seriation
Whole numbers
1. Describe the chronological development of numbers from ancient civilisation
until now. Present your answer in a mind map.
2. Teaching number concepts using concrete materials can help pupils learn
more effectively. Explain.
1. Pupils might have difficulties in understanding the meaning of 0 and 10
compared to the numbers 1 to 9. Explain.
2. Learning outcomes: At the end of the lesson, pupils will be able to count
numbers in ascending order (1 to 9) and descending order (9 to 1) either
through:
(a) Picture cards first and number cards later; or
(b) Number cards first and picture cards later.
Suggest the best strategy that can be used in the teaching and learning
process of numbers according to the above learning outcomes.

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APPENDIX
Busy People
One busy person sweeping the floor
Two busy people closing the door
Three busy people washing babyÊs socks
Four busy people lifting the rocks
Five busy people washing the bowls
Six busy people stirring ÂdodolÊ
Seven busy people chasing the mouse
Eight busy people painting the house
Nine busy people sewing clothes
Resource: Pusat Perkembangan Kurikulum
Numbers Up and Down
I'm learning how to count,
From zero up to ten.
I start from zero every time
And I count back down again.
Zero, one, two, three,
Four and five, I say.
Six, seven, eight and nine,
Now I'm at ten ~ Hooray!
But, I'm not finished, no not yet,
I got right up to ten.
Now I must count from ten back down,
To get to zero again!
Ten, nine, eight, seven,
Six and five, I say.
Four, three, two, one,
I'm back at zero ~ Hooray!
Resource: Mary Flynn's Songs 4 Teachers

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WORKSHEET
How many seeds are there in each apple?
Count and write the numbers.

Topic
2
Addition
within 10 and
Place Value
LEARNING OUTCOMES
1. Identify the major mathematical skills related to addition within 10
and place value;
2. Recognise the pedagogical content knowledge related to addition
within 10 and place value; and
3. Plan teaching and learning activities for addition within 10 and
introduction to the place value concept.
INTRODUCTION
Adding is a quick and efficient way of counting. Sometimes we notice that
adding and counting are alike, but adding is faster than counting. You will also
see that addition is more powerful than mere counting. It has its own special
vocabulary or words, and is easy to learn because only a few simple rules are
used in the addition of whole numbers. When teaching addition to young pupils,
it is important that you recognise the meaningful learning processes which can be
acquired through real life experiences. The activities in this topic are designed as
an introduction to addition. It provides the kind of practice that most young
children need. What do children need to know in addition? Children do not gain
understanding of addition just by working with symbols such as Â+Ê and Â=Ê. You
have to present the concept of addition through real-world experiences because
symbols will only be meaningful when they are associated with these
experiences. Young children must be able to see the connection between the
process of addition and the world they live in. They need to learn that certain
symbols and words such as ÂaddÊ, ÂsumÊ, ÂtotalÊ and ÂequalÊ are used as tools in
everyday life.

TOPIC 2 ADDITION WITHIN 10 AND PLACE VALUE
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This topic is divided into two main sections. The first section deals with
pedagogical skills pertaining to addition within 10 and includes an introduction
to the concept of place-value. The second section provides some samples of
teaching and learning activities for addition within 10. You will find that by
reading the input in this topic, you will be able to teach addition to young pupils
more effectively and meaningfully.
PEDAGOGICAL SKILLS OF ADDITION
WITHIN 10
2.1
In this section, we will discuss further the pedagogical skills of addition within
10. This section will look into the concept of 'more than', teaching and learning
addition through addition stories, acting out stories to go with equations, number
bonds up to 10, reading and writing addition equations and finally reinforcement
activities.
2.1.1 The Concept of ‘More Than’
It is important for pupils to understand and use the vocabulary of comparing and
arranging numbers or quantities before learning about addition. We can start by
comparing two numbers. For example, a teacher gives four oranges (or any other
concrete object) each to two pupils. The teacher then gives another orange to one
of the pupils and asks them to count the number of oranges each of them has.
Teacher: How many oranges do you have? Who has more oranges?
Teacher introduces the concept of Âmore thanÊ, Âand one moreÊ as well as Âadd one
moreÊ for addition by referring to the example above. The pupils are guided to
say the following sentences to reinforce their understanding of addition with
respect to the above concept.
e.g.: Five oranges are more than four oranges. Five is more than four.
Four and one more is five.
Four add one more is five.
Teacher repeats with other numbers using different picture cards or counters and
pupils practise using the sentence structures given above.

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2.1.2 Teaching and Learning Addition Through
Addition Stories
Initially, addition can be introduced through story problems that children can act
out. Early story situations should be simple and straightforward. Here is an
example of a simple story problem for teaching addition with two addends:
Salmah has three balls. Her mother bought two more balls for her. How many
balls does Salmah have altogether?
At this stage, children have to make connections between the real world and the
process of addition by interpreting the addition stories. Children must read and
write the equations that describe the process they are working with. The concept
of ÂadditionÊ should be introduced using real things or concrete objects. At the
same time, they have to read and write the equations using common words, such
as ÂandÊ, ÂmakeÊ, as well as ÂequalsÊ as shown in Figure 2.1:
Figure 2.1: Acting out addition stories
However, you have to study effective ways in which your pupils can act out the
stories. Based on the situations given, pupils can act out the stories in different
ways as follows:
(a) Act out stories using real things as counters such as marbles, ice-cream
sticks, top-up cards, etc.;
(b) Act out stories using counters and counting boards (e.g. trees, oceans.
roads, beaches, etc.);
(c) Act out stories using models such as counting blocks; and
(d) Act out stories using imagination (without real things).
Figure 2.2 shows some appropriate teaching aids for teaching and learning
addition.

29
Figure 2.2: Acting out addition stories using appropriate teaching aids
2.1.3 Acting Out Stories to go with Equations
Figure 2.3 suggests a way for acting out stories to go with equations using the
ÂplusÊ and ÂequalÊ signs:
Figure 2.3: Flowchart for ÂActing out stories to go with equationsÊ

30
After pupils are able to write equations according to teacher-directed stories, they
can begin writing equations independently using suitable materials (refer to
Figure 2.2). Here are some examples of how to use the materials.
Example 1: Counting Board (e.g. Aquarium)
I have two clown fish in my aquarium. My mother bought three goldfish
yesterday. How many fish do I have altogether? See Figure 2.4.
2 clown fish and 3 gold fish make 5 fish altogether.
2 + 3 = 5
Figure 2.4: Story problem
ACTIVITY 2.1
Use the above example to show that 2 + 3 = 3 + 2 = 5.
2.1.4 Number Bonds Up to 10
Activity 1: Count On and Count Back in Ones, from 1 to 10
There are three boys playing football. Then another
boy joins them. How many boys are playing football
altogether? See Figure 2.5.
3 + 1 = 4
Figure 2.5: Count on: Using an Abacus
Teachers can also use number cards as a number line. The teacher reads or writes
the story problem and then begins a discussion with pupils on how to use the
number line to answer the question as in the example shown in Figure 2.6:

31
„Four pupils and three pupils are seven pupils‰
„Four plus three equals seven‰
4 + 3 = 7
Figure 2.6: Count on: Aligning number cards to form a number line
Teachers are encouraged to teach the addition of two addends within 5 first,
followed by addition within 6 until 10. Pupils need to be ÂimmersedÊ in the
activities and go through the experience several times. By repeating the tasks,
pupils will learn the different number combinations for bonds up to 10 efficiently.
Activity 2: Count On and Count Back in Ones, from 1 to 10
The activities on number bonds provide opportunities for teachers to apply a
variety of addition strategies. The objective of these activities is to recognise the
addition of pairs of numbers up to 10. You can start by asking your pupils to
build a tower of 10 cubes and then break it into two towers, for example, a tower
of four cubes and a tower of six cubes, (refer Figure 2.7) or any pairs of numbers
adding up to 10.
Example:
Figure 2.7: Number towers
Guide pupils to produce addition pairs up to 10, e.g. 4 + 6 = 10 or 6 + 4 = 10.
Repeat with other pairs of numbers. Ask pupils what patterns they can see before
getting them to produce all the possible pairs that add up to 10. Record each
addition pair in a table as shown in Table 2.1:

32
Table 2.1: Sample Table for ÂAddition ActivityÊ: Addition Pairs Up to 10
After Breaking Height of Tower Before into Two Towers
Breaking into Two Towers
(Cubes)
Height of First
Tower (Cubes)
Height of Second
Tower (Cubes)
10 0 10
10 1 9
10 2 8
10 3
10 4
10 5
10 6
10 7
10 8
10 9
10 10
Discuss the results with pupils and ask them to practise saying the number bonds
repeatedly to facilitate instant and spontaneous recall in order to master the basic
facts of addition up to 10.
To develop the skill, the teacher should first break the tower of 10 cubes into two
parts. Show one part of the tower and hide the other. Then, ask pupils to state the
height of the hidden tower. To extend the skill, you may progressively ask the
pupils to learn how to add other pairs of numbers, such as 9, 8, 7 and so on.
ACTIVITY 2.2
What is the Âcommutative law in additionÊ? How do you introduce this
concept to your pupils? Explain clearly the strategy used for the teaching
and learning of the commutative law in addition.
2.1.5 Reading and Writing Addition Equations
As we know, there are two common methods of writing the addition of numbers,
either horizontally or vertically, as shown below:

33
(a) Adding horizontally, in row form (i.e. Writing and counting numbers from
left to right).
Example: 4 + 5 = 9
The activities discussed above are mostly based on this method, which are
suitable for adding two single numbers.
(b) Adding vertically, in column form (i.e. Writing and counting numbers from
top to bottom).
Example: 3
+ 4
7
This method is suitable for finding a sum of two or more large numbers
because putting large numbers in columns makes the process of adding
easier compared to putting them in a row.
ACTIVITY 2.3
Numbers are most easily added by placing them in columns. Describe
how you can create suitable teaching aids to enhance the addition of two
addends using this method.
2.1.6 Reinforcement Activities
To be an effective mathematics teacher, you are encouraged to plan small group
or individual activities as reinforcement activities for addition within 10. Here are
some examples of learning activities that you can do with your pupils.
(a) Number Shapes
Have pupils take turns rolling a number cube to see how many counters
they have to place on their number shapes. Then they fill in the remaining
spaces with counters of different colours. Finally, they describe the number
combinations formed, as illustrated in Figure 2.8. Repeat with different
number shapes.

34
Figure 2.8: Number shapes
(b) Number Trains
Let pupils fill their number-train outlines (e.g. 7, 8 or 9) with connecting
cubes of two different colours. Ask them to describe the number
combinations formed. See Figure 2.9.
Figure 2.9: Number train
In addition, pupils can also describe the number combination formed as Âthree
plus three plus two equals eightÊ, that is (3 + 3 + 2 = 8).
PLACE VALUE
2.2
This section teaches you how to introduce the place-value concept to your pupils.
2.2.1 Counting from 11 to 20
Pupils will be able to read, write and count numbers up to 20 through the same
activities as for learning numbers up to 10 covered in Topic 1. Similar teaching
aids and methods can be used. The only difference is that we should now have
more counters, say, at least 20. In this section, we will not be focusing on counting
numbers from 11 to 20 because it would just be repeating the process of counting
numbers from 1 to 10. You are, however, encouraged to have some references on
the strategies of teaching and learning numbers from 11 to 20.

35
ACTIVITY 2.4
Describe a strategy you would use for the teaching and learning of
ÂCounting from 11 to 20Ê.
2.2.2 Teaching and Learning about Place Value
The concept of place value is not easily understood by pupils. Although they can
read and write numbers up to 20 or beyond, it does not mean that they know
about the different values for each numeral in two-digit numbers. We are lucky
because our number system requires us to learn only 10 different numerals.
Pupils can easily learn how to write any number, no matter how large it is. Once
pupils have discovered the patterns in the number system, the task of writing
two-digit numbers and beyond is simplified enormously. They will encounter the
same sequence of numerals, 0 to 9 over and over again. However, many pupils
do not understand that numbers are constructed by organising quantities into
groups of tens and ones, and the numerals change in value depending on their
position in a number.
In this section, you will be introduced to the concept of place value by forming
and counting groups, recognising patterns in the number system and organising
groups into tens and ones. The place-value concept can be taught in kindergarten
in order to help pupils count large numbers in a meaningful way.
You can start teaching place value by asking pupils to form and count
manipulative materials, such as counting cubes, ice-cream sticks, beans and cups,
etc. For example, ask pupils to count and group the connected cubes from 1 to 10
placed either in a row or horizontally as shown in Figure 2.10.
Figure 2.10: Connected cubes placed horizontally
You can now introduce the concept of place value of ones and tens (10 ones) to
your pupils. The following steps can be used to demonstrate the relationship
between the numbers (11 to 19), tens and ones. The cubes can also be arranged in
a column or vertically as shown below. Here, you are encouraged to use the
enquiry method to help pupils familiarise themselves with the place-value of tens
and ones illustrated as follows:

36
Example:
Teacher asks: What number is 10 and one more? See Figure 2.11 (Pupils should
respond with 11).
Can you show me using the connecting cubes?
The above step is repeated for numbers 12, 13, , 20.
Figure 2.11: Connected cubes placed vertically
In order to make your lesson more effective, you should use place-value boards
or charts to help pupils organise their counters into tens and ones. A place-value
board is a piece of thick paper or soft-board that is divided into two parts of
different colours. The size of the board depends on the size of the counters used.
An example of the place-value board is given in Figure 2.12:
Figure 2.12: Place value board

37
The repetition of the pattern for numbers 12 to 19 and 20 will make your pupils
understand better and be more familiar with the concept of place value. They will
be able to learn about counting numbers from 11 to 20 or beyond more
meaningfully. At the same time, you can also relate the place-value concept to the
addition process. For example, 1 tens and 2 ones make 12, which means 10 and
two more make 12.
ACTIVITY 2.5
In groups of four, create some reinforcement activities for teaching
numbers 11 to 20 using the place-value method. Describe clearly how you
will conduct the activities using suitable Âhands-onÊ teaching aids.
ACTIVITIES
2.3
This section provides some samples of teaching and learning activities you can
carry out with your pupils to enhance their knowledge of addition within 10 and
the place-value concept.
Activity 1: Adding Using Patterns
Learning Outcomes:
At the end of this activity, your pupils should be able to:
(a) Add two numbers up to 10 using patterns;
(b) Read and write equations for addition of numbers using common words;
and
(c) Read and write equations for addition of numbers using symbols and signs.
Materials:
Picture cards; and
PowerPoint slides.

38
Procedure:
(a) Adding Using Patterns (in Rows)
(i) Teacher divides the class into 5 groups of 6 pupils, and gives 10
oranges to each group. Teacher then asks each group to count the
oranges, see Figure 2.13.
Teacher says: „Can you arrange the oranges so that you can count
more easily?‰ Discuss with your friends.
Teacher says: „Now, take a look at this picture card.‰
Figure 2.13: Picture card: Addition using patterns
(ii) Teacher says: „Can you see the pattern? Let us count in groups of
fives instead of counting on in ones.‰
For example: Five and five equals ten, or 5 + 5 = 10
(iii) Teacher says: „Now, let us look at another pattern. How many eggs
are there in the picture given below (see Figure 2.14)?‰
Figure 2.14: Picture card: Addition using patterns (in rows)
(iv) Teacher says: „Did you count every egg to find out how many there
are altogether? Or did you manage to see the pattern and count along
one row first to get 4, and then add with another row of 4 to make 8
eggs altogether?‰
„Well done, if you have done so!‰
Let your pupils add using different patterns of different numbers of objects
with the help of PowerPoint slides. Guide your pupils to read and write
equations of addition of numbers in words, symbols and signs (You may
discuss how to write the story-board of your PowerPoint presentation).

39
(b) Adding Using Patterns (in Columns)
(i) Teacher says: „Let us look at the pictures and try to recognise the
patterns (see Figure 2.15). Discuss with your friends.‰
Figure 2.15: Picture cards
(ii) Teacher discusses the patterns with pupils. For example, teacher
shows the third picture [Picture (c)] and tells that it can be divided
into two parts, namely, the top and bottom parts as shown in Figure
2.16:
Figure 2.16: Picture card: Addition using patterns (in columns)
(iii) This is a way of showing how to teach addition using columns by the
inquiry-discovery method. As a conclusion, the teacher explains to
the pupils that arranging the objects in patterns will make it easier to
add them. Using columns to add also makes the addition of large
numbers easier and faster.
(c) Teacher distributes a worksheet on addition using patterns (in rows or in
columns).

40
Activity 2: Addition within the Highest Total of 10
Learning Outcomes:
(a) Add using fingers;
(b) Add by combining two groups of objects; and
(c) Solve simple problems involving addition within 10.
Materials:
Fingers;
Counting board (tree);
Picture cards;
Number cards;
Counters;
Storybooks;
Apples; and
Other concrete objects, etc.
Procedure:
(a) Addition Using Fingers
(i) Initially, use fingers to practise adding two numbers as a method of
working out the addition of two groups of objects, see Figure 2.17.
e.g.:
Figure 2.17: Finger addition

41
(b) Addition of Two Groups of Objects
(i) Teacher puts three green apples on the right side of the tree and
another four red apples on the left side. Teacher asks pupils to count
the number of green apples and red apples respectively.
(ii) Teacher asks: „How many green apples are there? How many red
apples are there?‰
(iii) Teacher tells and asks: „Put all the apples at the centre of the tree.
Count on in ones together. How many apples are there altogether?‰
(iv) Teacher guides them to say and write the mathematical sentence as
shown: „Three apples and four apples make seven apples‰.
(v) Repeat with different numbers of apples or objects. Introduce the
concept of plus and equals in a mathematical sentence.
e.g. „There are two green apples and three red apples in Box A.‰
„There are five apples altogether.‰
„Two plus three equals five.‰
(vi) Teacher sticks the picture cards on the whiteboard. Encourage pupils
to add by counting on in ones (e.g. 4 ... 5, 6 ,7) and guide them to say
that „Four plus three equals seven‰ (see Figure 2.18).
Figure 2.18: Picture card: Addition of two groups of objects
(vii) Introduce the symbols for representing „plus‰ and „equals‰ in a
number sentence. Ask them to stick the correct number cards below
the picture cards to form an addition equation as above. Repeat this
step using different numbers.
(c) Problem Solving in Addition
(i) Teacher shows three balls in the box and asks pupils to put in some
more balls to make it 10 balls altogether.

42
(ii) Teacher asks: „How many balls do you need to make up 10? How did
you get the answer?‰
Let them discuss in groups using some counters. Ask them to explain
how they came up with their answers.
(iii) Repeat the above steps with different pairs of numbers.
(iv) Teacher discusses the following problem with the pupils.
Sarah has to read six story books this semester. If she has finished
reading four books, how many more story books has she got to
read?
(v) Teacher asks them to discuss the answer in groups. Encourage them
to work with models or counters and let them come up with their own
ideas for solving the problem. For example:
(Note: They can also use mental calculation to solve the problem.)
Activity 3: Reinforcement Activity (Game)
Learning Outcomes:
(a) Complete the addition table given; and
(b) Add two numbers shown at the toss of two dices up to a highest total of 10.
Materials:
Laminated Chart (Addition Table Table 1.2);
Two dices for each group; and
Crayons or colour pencils.

43
Procedure:
(i) Teacher guides pupils to complete the addition table given. (Print out the
table in A4 size paper and laminate it). You can also use the table to explain
the additive identity (i.e. A + 0 = 0 + A = A).
Table 2.2: Adding Squares
+ 0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
Instructions for Game:
(i) Toss two dices at one go. Add the numbers obtained and check your answer
from the table.
(ii) Colour the numbers 10 in green (Table 2.2). List down all the pairs adding
up to 10.
(iii) Colour the numbers totalling 9 in red. List down all pairs adding up to 9.
(iv) Continue with other pairs of numbers using different colours for different
sums.

44
Activity 4: Place Value and Ordering
Learning Outcomes:
(a) Read and write numerals from 0 to 20;
(b) Explain the value represented by each digit in a two-digit number; and
(c) Use vocabulary for comparing and ordering numbers up to 20.
Materials:
Connecting cubes;
Counting board;
Place-value block/frame; and
Counters.
Procedure:
(a) Groups of Tens
(i) Teacher divides the class into 6 groups of 5 pupils each. Teacher
distributes some connecting cubes (say, at least 40 cubes) to each
group.
(ii) Teacher asks the following questions and pupils are required to
answer them using the connecting cubes:
What number is one more than 6?, 8?, and 9? 11?, 17? and 19?
What number comes after 5?, 7?, and 9? 12?, 16? and 19?
Which number is more, 7 or 9?, 3 or 7?, 14 or 11? etc.
e.g.: 14 is more than 11 as shown in Figure 2.19.
Figure 2.19: Representing numbers using connecting cubes

45
16 is one more than a number. What is that number?
Repeat the above steps with different numbers.
(b) Place Value and Ordering
(i) Teacher introduces a place-value block and asks pupils to count
beginning with number 1 by putting a counter into the first column
(see Figure 2.20 (a). Teacher asks them to put one more counter on the
board in that order. Repeat until number 9 is obtained. Teacher then
introduces the concept of „ones‰.
1 ones represents 1
2 ones represent 2, ..., 9 ones represent 9
Figure 2.20 (a): Representing numbers with place-value block and counters
(ii) Teacher asks: „What is the number after 10? How do you represent
number 11 on the place-value block?‰
Teacher introduces the concept of „tens‰ and „ones‰ as follows, see
Figure 2.20 (b):

46
Figure 2.20 (b): Representing numbers with place-value block and counters
(iii) Teacher asks pupils to put the correct number of counters into the
correct column to represent the numbers 11, 12, etc. until 20.
(iv) Teacher asks pupils to complete Table 1.3.
Table 2.3: Place Value
Number Tens Ones Number Tens Ones
11 1 3
12 9
13 17
16 14 4
19 1 8
20
15 1
(v) Teacher distributes a worksheet to reinforce the concept of place value
learnt.
A teacher should know his/her pupilsÊ levels of proficiency when applying
strategies to solve problems related to addition.
Problem solving related to addition depends on pupilsÊ ability to work based
on their counting skills.

47
At an early stage, it is enough if they could work using counting all or
counting on.
However, you have to guide and encourage them to work by seeing the
relationship or answer by knowing and mastering the number combinations
or number bonds.
Adding
Addition
Equation
Place Value
Sum
Plus
1. An effective way to teach addition is to ask pupils to act out the stories in
real life using their imagination (without real things) and their own ideas.
Elaborate using one example.
2. Describe clearly how you would teach addition up to 10 involving zero
using real materials.
3. Counting numbers from 11 to 20 should be taught after pupils are
introduced to the concept of place value. Give your comments on this.
Based on the following learning outcome, „At the end of the lesson, pupils will be
able to count numbers from 11 to 20 using place-value blocks‰, suggest the best
strategy or method that can be used in the teaching and learning process to
achieve this learning outcome.

48
APPENDIX
WORKSHEET
(a) Count and add.
(i)
(ii)
(b) Count and add.
(c) Draw the correct number of fish on each plate and complete the equation.

49
(d) Match the following.
(e) Match the following (Read and add).

Topic
3
Subtraction
within 10
LEARNING OUTCOMES
1. Recognise the major mathematical skills pertaining to subtraction
within 10;
2. Identify the pedagogical content knowledge pertaining to subtraction
within 10; and
3. Plan teaching and learning activities for subtraction within 10.
INTRODUCTION
This topic will provide you with the instruction and practice you need to
understand about subtraction. Beginning with the comprehension of basic skills
in subtraction, this topic will cover various strategies for teaching and learning
subtraction. The step-by-step approach used in this topic will make it easy for
you to understand the ideas about teaching and learning subtraction especially at
kindergarten level. As in all other topics, some examples of teaching-learning
activities are also given. They include several classroom activities incorporating
the use of concrete materials and a variety of methods such as inquiry-discovery,
demonstration, simulation, etc. The inquiry-discovery method comprises
activities such as planning, investigating, analysing and discovering. It is very
important that pupils take an active part in the teaching-learning activities
because by doing mathematics, they will learn more meaningfully and
effectively.

TOPIC 3 SUBTRACTION WITHIN 10
51
PEDAGOGICAL SKILLS OF SUBTRACTION
WITHIN 10
3.1
Subtraction in simple words means taking away. When you take objects away
from a group, the mathematical term for this process is known as ÂsubtractionÊ or
ÂsubtractingÊ. It is all about separating a large group of things into smaller groups
of things. Besides taking away, some other common terms or vocabulary that
also indicate subtraction are ÂremainderÊ or Âwhat is leftÊ, Âcounting backÊ and
Âfinding the differenceÊ. Subtraction is also involved when phrases or questions
such as ÂHow many more?Ê, ÂWhat is the amount to be added?Ê, as well as ÂHow
many remain?Ê etc., are used.
There are at least three ways to illustrate the meaning of subtraction as listed
below:
(a) Subtraction as counting back;
(b) Subtraction as taking away; and
(c) Subtraction as the difference.
You will be shown how to teach subtraction contextually according to each of the
meanings of subtraction mentioned above. In addition, you also have to know
about other important parts related to the teaching and learning of subtraction
such as teaching materials, the relationship of subtraction with addition and
pairs of basic subtraction facts.
3.1.1 Subtraction as Counting Back
Subtraction is the reverse of addition. Counting on in ones is simply counting by
ones or moving forward between numbers one at a time. As counting on is a
reliable but slow way of adding, counting back is the reverse and is thus a slow
but reliable way of subtracting. Initially, subtraction within 10 as counting back
can be introduced by counting backwards either from 5 to 0 or from 10 to 0, that
is 5, 4, 3, 2, 1, 0 or 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.
Take a look at Figure 3.1. For example, a teacher can give out number cards of 0
to 5 to six pupils and ask them to come out to the front and hold up their cards.
Get the pupils to arrange themselves in ascending order and ask who should
come first if the numbers are to be counted backwards from 5 to 0.

52
Figure 3.1: Count on and count back using number cards
Ask pupils to count backwards from 5 to 0. Repeat with counting backwards,
starting with any other number less than 5, for example starting from 4 or 3, etc.
Next, ask pupils to try doing the same thing without using number cards.
Then, guide the pupils to compare the difference between counting onwards and
counting backwards. At this stage, do not introduce the words subtract or minus
yet. Just use common words such as Âone lessÊ and ÂbeforeÊ as shown below:
„In the sequence of numbers between 0 to 5, what is the number before 5?,
before 4?‰ and so on.
„4 is one less than 5‰, „3 is one less than 4‰, „2 is one less than 3‰, etc.
Let them try to count backwards from 10 to 0, 9 to 0 and so on. At this stage,
pupils should also be able to arrange the numbers in descending order from 10 to
0.
Subtraction can also be done by counting back using a ruler as a number line.
Here is an example of how to count back using a ruler in order to solve the
subtraction problem given:
Sally has 7 sweets. She wants to give 3 to her friend. How many can she keep
for herself (see Figure 3.2)?
Answer: The result is 4. So Sally can keep 4 sweets for herself.

53
Figure 3.2: Counting back using a ruler
Suggest a teaching and learning activity to demonstrate subtraction as the
process of counting back using a calendar.
3.1.2 Subtraction as Taking Away
Subtraction facts are the numbers we get when we take one or more objects from
a group of objects, or the answer we get when we take one number from another.
First, let us look at the following steps for finding the six basic subtraction facts
illustrated in Figure 3.3 (a), (b) and (c).
For example, we start off with a group of six oranges.
(a) Put the oranges in a row, to make it easier to see what we are doing (see
Figure 3.3 (a).
Figure 3.3 (a): One group of six oranges
(b) Separate them into two groups, see Figure 3.3 (b): (Separating, in actual fact,
is a way of subtracting).
Figure 3.3 (b): Two groups of oranges
ACTIVITY 3.1

54
The numbers in the boxes tell us how many members are in each group. We
can describe the ÂsubtractionÊ process using common words like below:
Six take away one leaves five, or
Taking one from six leaves five.
(c) Repeat by working with groups of two and four oranges, as illustrated in
Figure 3.3 (c):
Figure 3.3 (c): Another two groups of oranges
Six take away two leaves four, or
Taking two from six leaves four.
(d) Repeat with other possible combinations of two groups of oranges, i.e. three
and three, four and two, as well as five and one in that order. At this stage,
you may also introduce subtracting terms, such as, minus, in order to teach
pupils to read and write the subtraction equations or mathematical
sentences given below:
Six take away one leaves five.
Six minus one equals five.
6 – 1 = 5
Six take away two leaves four.
Six minus two equals four.
6 2 = 4
3.1.3 Subtraction as the Difference
Sometimes you need to count on to find the difference between two numbers. For
example, if you have to answer 10 questions as practice but you have just
finished six only, you can find the number of remaining questions to be
answered in this way:
„I have finished six questions. To find out how many more questions I need to
answer in order to finish all the 10 questions, I can count on in ones starting from
7‰.
„7 + 1 = 8, 8 + 1 = 9, and 9 + 1 = 10‰, meaning 7 + 1 + 1 + 1 = 10

55
By using a ruler as a number line, you can find that the difference between 10
and 6 is 4 by counting on in ones as illustrated in Figure 3.4:
Figure 3.4: Number line
The difference is thus 4 questions. This means that you need to do 4 more
questions to finish off. It is now obvious that by counting on from seven to 10, six
plus four gives 10. Pupils can be guided to state that the difference between 10
and six is four, i.e.
Â6 + 4 = 10Ê is the same as Â10 6 = 4Ê.
This may be the case with your pupils because they were probably right to think
that counting on was much easier than subtracting. However, this was only
because the numbers were small. A real-life example is counting change. For
example if we gave RM1 (or ten 10 sen) to the cashier at the shop counter, and
the price of the things that you bought was only 60 sen, usually, the cashier will
give you back 40 sen as your change by counting on in 10 sen. The cashier will
normally say: „70 sen, 80 sen, 90 sen, RM 1. Here is the change, 40 sen.‰
What do you think of this way of doing subtraction? Is this a correct way to do
subtraction? Do you have other ideas?
3.1.4 Pairs of Subtraction Facts
We usually get two subtraction facts from each addition fact. Pupils have learnt
that adding two numbers together in any order gives the same result. However,
you have to encourage them to find out the results when they do subtraction.
Here is a way in which they can discover related subtraction facts. Pupils are
asked to work in groups.
(a) Give out seven rings to each group. Ask them to arrange the rings in a row
and separate them into two groups, i.e. a group of 3 rings and a group of 4
rings respectively, as illustrated below:

56
Let them read and write the addition fact depicted in the diagram above:
3 + 4 = 7
(b) Using the above addition fact, guide them to work out the subtraction facts
below:
(i) First subtraction fact:
7 3 = 4
(ii) Second subtraction fact:
7 4 = 3
Note: The order of the numbers to be subtracted is important!
ACTIVITY 3.2
Try listing out other fact families such as for the addition fact, 3 + 5 = 8.
3.1.5 Subtraction Using Models
Another way to do subtraction is to use any type of counters or teaching
materials as models to set up the problem.

57
Here are some examples:
(a) Subtract 3 from 8 using Counters
(i) Set up 8 counters as 8 units like below.
(ii) Subtract 3 units by crossing out three counters as shown.
(iii) Then, count the units that are left. The answer is 5 units. Ask pupils
to write down the subtraction equation as follows:
8 3 = 5
(iv) You are also encouraged to use another model such as illustrated
below:
Say: 8 take away 3 leaves 5
(b) Subtract 2 from 7 using Counting Board and Counters
(i) Story problem:
There are seven apples on a tree. Two of them fall down to the
ground. How many apples are left on the tree? (See Figure 3.5)

58
Figure 3.5: Story problem that can be used in teaching subtraction
(ii) First, ask them to stick on seven green counters on the tree. Then
colour two of them in red and pull them down from the tree. Put
them on the ground.
(You may like to make your counters from either soft paper or manila
card. Explain your choice.)
(iii) Write down the subtraction equation and find the answer:
7 2 = 5
Say: Taking away two from seven leaves five.
Answer: There are five mangosteens left on the tree.
(c) Subtract 4 from 9 using an Abacus and Counting Chips
Figure 3.6: Sample subtraction of 4 from 9

59
(i) Ask pupils to put on 9 counting chips in the first column of the
abacus. Then pull out 4 chips (either one by one or all at once), refer
to Figure 3.6.
(ii) Ask pupils to count and say how many chips are left.
(iii) Guide your pupils to write and read the subtraction equation as
follows:
Taking four from nine leaves five
9 4 = 5
(iv) Repeat the activity with different numbers of chips.
3.1.6 Number Sentences for Subtraction
We can write subtraction equations in rows or columns. Most of the examples in
this topic thus far have focused on writing equation in rows. Subtraction in a
column requires us to put the number we are subtracting from at the top and the
number we are going to subtract at the bottom. Make sure the numbers are lined
up exactly below each other in the column. Take a look at the following example
in Figure 3.7:
Figure 3.7: Number sentences for subtraction
ACTIVITY 3.3
What happens to the signs: Â-Ê and Â=Ê when you write down the ÂrowÊ
equation into a ÂcolumnÊ equation? Explain the process that occurs.

60
ACTIVITIES
Some samples of teaching-learning activities that you can implement to help
guide young children to understand and build the concept of subtraction in order
to acquire the skill are included in this section.
Activity 1: Working Out ÂOne Less ThanÊ
Learning Outcomes:
(a) Use Âone less thanÊ to compare two numbers within 10; and
(b) Count back in ones from 10 to 0.
Materials:
10 balloons;
11 number cards (0 10);
String; and
Worksheet 1.
Procedure:
(a) Get 10 balloons and hang them in a row or horizontal line. Initially, stack
the 11 number cards, numbered 0 - 10 in sequence, with the card numbered
10 at the top followed by the card numbered 9 below and so on, with the
card numbered 0 at the bottom of the pile. Hook the stack of number cards
on the extreme right as shown in Figure 3.8.
Figure 3.8: Ten balloons in a row
(i) Get one pupil to count the balloons and say the number out loud.
(ii) Ask another pupil to pick and burst any one of the balloons, count
the remaining balloons and say „9‰. Then, take out the card
numbered 10 to show the card numbered 9 underneath.
3.2

61
(iii) Teacher asks the pupils:
„How many balloons are left?‰ (9)
„Are there more or less balloons now compared to before?‰ (less)
„How many less?‰ (1 less)
(iv) Teacher explains that 9 is Âone less thanÊ 10.
(v) Continue doing the activity until the last balloon is pricked.
(b) Ask pupils to count back in ones, starting with any number up to 10 e.g.
You can start with number 8 or 7 and so on.
(c) Get the 11 number cards and ask pupils to arrange the cards in sequence
again. Practise using the phrase Âone less thanÊ to compare two numbers
within 10 e.g. Start from number 10 and say. Â9 is one less than 10Ê, 8 is one
less than 9, etc.
(d) Teacher distributes Worksheet 1 (refer to Appendix).
Activity 2: Subtracting Sums by Finding the Difference
Learning Outcomes:
(a) Use Âless thanÊ and Âmore thanÊ to compare two numbers; and
(b) Find the difference of two numbers.
Materials:
Table (worksheet);
Balls;
PowerPoint slides; and
Plain paper.
Procedure:
(a) Start with a story problem (PowerPoint slides).
1st Slide:
Salleh has 5 balls, while Salmah has 3 balls. Who has more balls?
What is the difference?

62
2nd Slide:
Show the following illustration, see Figure 3.9 (a).
Figure 3.9 (a): Finding the difference
Teacher asks: „Who has more balls?‰
„How many are there?‰
„Which one is more, 3 or 5?‰
„Which one is less, 3 or 5?‰
(At this stage, the teacher just wants to introduce the concept of Âone-to-one
matchingÊ and it is not necessary for pupils to answer the questions yet if
they are unable to do so).
(b) Teacher asks them to show how they arrived at the answer using the
materials given.
(i) Step 1:
Distribute some counters and a piece of plain paper to each group.
(ii) Step 2:
Guide them to work out the Âone-to-one matchingÊ correspondence
using the materials given as shown in Figure 3.9 (b).

63
Figure 3.9 (b): One-to-one matching correspondence
(iii) Step 3:
„How many balls have no match?‰
Teacher now introduces the concept of difference and relates this to
the words, more and less.
e.g. 5 is more than 3.
3 is less than 5.
The difference between 5 and 3 is 2.
(iv) Step 4:
Teacher guides pupils to compare two numbers by using the words
more and less before finding the difference using Table 3.1 given.
e.g. Compare the numbers 4 and 6.
Which is more?
Which is less?
What is the difference?
Teacher then asks pupils to write the numbers in the correct space in
the table before finding the difference.
For example, write 6 in the ÂmoreÊ column, 4 in the ÂlessÊ column and
2 as the difference in the space provided (see Table 3.1).

64
Table 3.1: Sample of a Table that can be Used for
Recording the Difference between Two Numbers
(v) Step 5:
Teacher gets pupils to do the same for the other numbers in Table 3.1
above and asks them to record the answers in the table given.
(c) Group activity:
Give a set of number cards numbering 1 to 10 to each group. Ask them to
play the game as follows:
(i) Step 1: Teacher gives the instructions on how to play the game.
(ii) Step 2: Teacher says: „Listen, choose two numbers with a difference
of 1. Whoever gets the correct answer first is the winner. Check your
answers together.‰
(iii) Step 3: Repeat the game using other numbers with differences of 2, 3,
etc.

65
(iv) Step 4: Teacher asks them to find out all possible pairs of numbers in
their groups using the number cards and record the results in Table
3.2.
Table 3.2: Subtraction Pairs
Difference List Down All Possible Pairs
1 e.g.
10 - 9
2 10 - 8 9 - 7
3 10 - 7 9 - 6
4
5
6
7
8
9
10
(v) Step 5: Check all the answers together.
(d) Closure: (You may teach subtraction involving zero in the next lesson!).
Teacher: „What is the answer of 5 0? 4 4? 7 0?‰
ACTIVITY 3.4
Suggest two suitable teaching and learning activities for this statement:
„Subtracting zero from a number does not change the value of the number‰.
Activity 3: Subtracting by Taking Away
Learning Outcomes:
(a) Subtract by taking away; and
(b) Use subtraction to solve word problems.

66
Materials:
Counting boards;
Counters; and
Plasticine.
Procedure:
(i) Initially, use fingers to practise taking away as a method for working out
the subtraction process, see Figure 3.10. e.g.:
Figure 3.10: Subtracting with fingers
(ii) Teacher shows a story problem on a question card.
Aida has 8 apples. She gives 3 of them to Sharifah.
How many apples are left?
Get two pupils to come in front and act out the story. They will act as Aida
and Sharifah, respectively. The others are asked to solve the problem by
observing the action shown.
(iii) Teacher shows the subtraction process using a counting board and some
counters, see Figure 3.11.

67
Figure 3.11: Subtraction using a counting board and counters
(iv) Teacher shows another story problem with a different context.
There are 6 players on the field. 2 of them take a rest.
How many players are left on the field?
Ask pupils to act out the story using a counting board and some plasticine
or encourage them to role play in the class, see Figure 3.12.
Figure 3.12: Sample subtraction of 4 from 9
(v) Teacher asks them to solve the story problem in groups.
„Write the subtraction equations on the card given. Present your answers
in front of the class‰.
(vi) Do a quick mental-recall of the activity in the class. This will help pupils to
work fast and accurately.
e.g. 8 take away 4?
10 take away 5? Take away 4? Take away 6?
What take away 5 leaves 3? Leaves 2? Leaves 5?
(vii) Distribute Worksheet 2 (refer to Appendix).
Can you think of another suitable activity like the above?

68
Activity 4: Predicting the Missing Part
Learning Outcomes:
(a) Predict the missing part in a subtraction problem; and
(b) Relate the subtraction problem to the addition process.
Materials:
Connecting cubes;
Number lines;
Beads; and
Cups.
Procedure:
(a) Teacher puts several connecting cubes (or counters) on a number line.
e.g. 8 connecting cubes.
(b) Teacher then keeps any 3 of the cubes behind her/him, while the pupils
predict how many cubes are hidden.
(c) Teacher guides the pupils to get the answer as follows:
(i) How many connecting cubes are there at first? (8)
(ii) How many connecting cubes are there left now? (5)
(iii) How many connecting cubes are hidden?
Let pupils brainstorm to get some suggestions from them.

69
(d) Teacher shows a way to solve the problem as shown below:
(i) „We have 5 cubes left. How many more cubes do we need to make 8
cubes?‰
(ii) Teacher adds 3 red cubes one by one on the number line and asks
pupils to count on in-ones from 5 to 8. „Start at 5, then 6, 7 and 8‰.
(iii) „We have added 3 red cubes which represents the number of cubes
hidden‰. „We thus write the subtraction equation as 8 3 = 5‰.
(iv) „We can also write down the addition equation as 5 + ? = 8, to find
the number of cubes hidden‰.
(e) Ask them to work out the game in groups. You are encouraged to let them
work out another game, e.g. Âbeads and cupÊ.
(i) First, count the number of beads given to pupils.
(ii) Put some of the beads into the cup. Take out 3 beads and ask pupils
to predict the number of beads (hidden) under the cup.
(f) Let pupils do other examples to reinforce the skill learnt.
ACTIVITY 3.5
Create another game as an enrichment activity for the subtraction process.

70
You need to pay attention when teaching the meanings of subtraction
because conceptual understanding of this operation will help students learn
the topic more efficiently.
The concrete materials used can help pupils master the subtraction algorithms
better.
The samples of teaching and learning activities for subtraction provided in
this topic are to motivate you to collect a set of good teaching-learning
activities for subtraction.
The more activities you know of, the more creative and innovative you will
be when planning your mathematics lessons.
Counting back
Difference
Fact family
Subtraction
Subtraction fact
Taking away
1. Define the term ÂfactÊ.
2. Subtraction can be defined as Âtake awayÊ. Explain this meaning of
subtraction with the help of a suitable teaching and learning activity using
concrete materials.
3. Addition is the reverse of the subtraction process. Explain addition as the
reverse of the Âtake awayÊ process with the help of a suitable teaching and
learning activity using concrete materials.

71
Explain the statements below with the help of a suitable teaching and learning
activity using concrete materials:
(a) ÂThe differenceÊ.
(b) The order of the numbers in a subtraction problem is important.
(c) You can subtract only one number at a time, but you can add more than
one number at one go.
APPENDICES
WORKSHEET 1
Answer all questions.
1. Write the number which is one less than the one given in the space
provided.
2. Colour the number which is less.
3.

72
4. Fill in the blanks starting with the biggest number for each row of numbers.
WORKSHEET 2
1. 6 take away 4 leaves
8 take away 4 leaves
2. Complete the subtraction sentences below:
3. Circle the objects which have to be taken away. Write down the subtraction
sentences.

73
4. 4 3 = ________ 7 1 = ________
6 3 = ________ 9 7 = ________
10 3 = _______ 10 2 = _______
5. Colour two pairs of numbers that give the same answer.
6. Circle the correct answers.
(a)
(g)
(b)
(h)
(c)
(i)
(d)
(i)
(e)
(k)
(f)
(l)

Topic
4
Numbers to
100 and Place
Value
LEARNING OUTCOMES
1. Explain how to say, count, read and write numbers to 100;
2. Demonstrate how to count in tens and ones;
3. Describe how to arrange numbers to 100 count on and count back;
and
4. Explain the concept of place value of numbers to 100.
INTRODUCTION
You need to recall what was discussed in Topic 1 in order to understand this
topic better. After mastering numbers 1 to 10, children should now learn how to
say numbers up to 100 progressively. For example, you have to teach them to
understand, count and write numbers from 10 to 20 before getting them to count
in tens and ones until 100. To ensure that your pupils know how to say numbers
to 100 either in words or in symbols correctly, it is essential to stress on the
correct pronunciation of the names of numbers up to 100. The next step is to teach
pupils to read and write numbers to 100 in words as well as in symbols neatly
and correctly. Then, let pupils arrange numbers to 100 in sequence either by
counting on (in ascending order), or counting back (in descending order), using
various methods. Last but not least, teach pupils to recognise place value, first
discussed in Topic 2. The place-value concept of tens and ones is introduced for
counting numbers up to 100, especially when larger numbers are involved. Pupils
can do regrouping with numbers from 10 onwards e.g. ten ones is the same as
one tens and zero ones; eleven ones can be regrouped as one tens and one ones,
and so on and so forth. In conclusion, the most important thing to remember
when teaching kindergarten and elementary Mathematics is to make the teaching

TOPIC 4 NUMBERS TO 100 AND PLACE VALUE
75
and learning process as interesting and as fun as possible. The samples given in
the following section will help you to teach Mathematics more effectively and
meaningfully to the young ones.
SAY AND COUNT NUMBERS TO 100
4.1
This section will further discuss how to say and count numbers to 100.
4.1.1 Say Numbers to 100
In general, parents or guardians normally feel so proud or are thrilled when they
hear their children say numbers written in words or symbols flawlessly for the
first time. With this in mind, it is thus the responsibility of parents or guardians
and teachers especially, to guide them to pronounce the names of numbers up to
100 correctly.
There are a lot of ways to encourage pupils to practise saying the numbers. One
effective way is by using picture-number cards that have numbers in words
and/or symbols on them, or number charts. For example, you can easily use
number charts in the form of 10 X 10 grids made from manila cards (or other
suitable material) like the one in Table 4.1:
Table 4.1: Number Chart (Numbers 1 to 100)
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Using the 10 X 10 grid shown above, cover some numbers and let the pupils say
the numbers occupying the covered spots. Alternatively, you may also jumble up
the sequence of the numbers by putting the numbers at the wrong places and

76
then ask the pupils to rearrange them in order before getting them to say the
numbers.
Some sample teaching-learning activities to reinforce the skill of counting
numbers up to 100 are discussed here.
Activity 1: Say the Number Names
Learning Outcome:
By the end of this activity, the pupils should be able to:
(a) Pronounce the names of numbers up to 100 correctly.
Materials:
10 pieces of manila cards (size 15 cm by 20cm) per group;
Colour pencils; and
Books or magazines with page numbers.
Procedure:
In general, there are five steps, which are:
(i) Divide pupils into two groups. Ask them to make five picture number
cards with numbers written in symbols by drawing some pictures/objects
for different numerals (numbers up to 100) allocated to each group and
another five drawings for cards with numbers written in words. Ask them
to give the finished products to you to be checked for accuracy before
giving them back the respective cards.
(ii) Once they are ready, you can start the activity of „Saying number names‰.
Tell them to make sure that all the drawings can only be revealed one by
one by their own group members. The first group (Group 1) will show one
of their picture numeral cards, for example, the card with the numeral „99‰
written on it. The other group (Group 2) will have to say the number
Âninety-nineÊ out loudly and clearly. Award two points if the second group
can say it correctly.
(iii) Next, the second group takes turns to show a picture number card with the
number written in words e.g. Âsixty-fourÊ and ask the other group to say the
number on the card loudly and clearly. Award two points to Group 1 if
they can say the number name correctly. Continue doing this until all the
drawings have been shown.

77
(iv) Another way to let your pupils practise saying numbers to 100 is by
showing them the page numbers from various kinds of books or
magazines. Just randomly flip through one page at a time and then ask the
pupils to say what number is on the next page. This activity can be carried
out in pairs or groups.
(v) Finally, distribute Worksheet 1 to your pupils to reinforce the skill of saying
numbers to 100.
4.1.2 Count Numbers to 100
It is natural for pupils to use their fingers when they first start counting and if
that is not enough, some will even continue to count using their toes which can
be rather awkward. However, when counting larger numbers such as numbers
more than 20, other more suitable manipulatives (e.g. counters) are required.
The fun way to teach pupils to count is by using counting objects such as beads,
beans, nuts, marbles, etc. Fill up a jar with beads, beans, nuts or marbles and pour
them out onto a mat or table cloth. Then, ask the pupils to count them in different
ways other than in ones. For example, get the pupils to group the beads into
groups of ÂfivesÊ or ÂtensÊ. Counting in tens means adding ten to the previous
number in the sequence each time, for instance, 10, 20, 30, 40, 50, 60, 70, 80, 90 and
100.
Finally, help the pupils make some conclusions. When counting on in tens, the
numbers create a pattern. All the numbers end with zero and the first digits are
the same as when you count from 1 to 9, that is, (1, 2, 3, 4, 5, etc.).
Once the pupils have discovered the patterns in the number system, the task of
writing numerals of two digits and beyond is simplified enormously. They will
encounter the same sequence of numerals, 0 to 9 over and over again. However,
at this stage, many pupils do not know yet that numbers are constructed by
organising quantities into groups of tens and ones, and that the digits in
numerals change value depending on their positions in a number, thereby giving
rise to the concept of place value in our number system.
Activity 2: Count Numbers to 100
Learning Outcome:
(a) Count numbers to 100.

78
Materials:
Picture cards of bicycles, aeroplanes, flowers, motorcycles, etc.;
Manila cards with pictures;
Colour pencils; and
Objects (Beads or beans or nuts or marbles, etc.).
Procedure:
In general, there are three steps, which are:
(i) Show pupils the pictures of bicycles, aeroplanes, flowers, motorcycles, etc.
Ask them to count the number of objects on the cards.
(ii) Ask them to colour the pictures on the manila cards and then count how
many objects there are on each card.
(iii) Distribute Worksheet 2 to the pupils.
READ AND WRITE NUMBERS TO 100
This section will guide you through some relevant activities on reading and
writing numbers to 100. It is useful to revise the correct techniques of writing 0 to
9 taught in Topic 1 earlier.
4.2.1 Read and Write Numbers to 100
First of all, you need to revise or teach the pupils the correct way of writing the
numbers as shown in Figure 4.1.
Figure 4.1: Correct way for writing numbers
4.2

79
Write down the numbers randomly on a piece of manila card or on a sheet of
paper. Ask the pupils to read the numerals. Next, do the reverse, that is, get them
to write down the numbers, in words, randomly on the manila card or on the
sheet of paper. Then, ask the pupils to read the numbers in word form.
Activity 3: Read and Write Numbers to 100
Learning Outcome:
(a) Read and write numbers to 100 correctly.
Materials:
Manila card or a sheet of paper; and
Pencils.
Procedure:
(i) Ask the pupils to fill in the empty boxes in Table 4.2:
Table 4.2: Drawing and Writing numbers
Read Draw and Write the Numerals Write the Numbers in Words
20
55 Fifty-five
67
77 Seventy-seven
18
29
98 Ninety-eight
(ii) Distribute Worksheet 3 to your pupils.

80
ARRANGE NUMBERS TO 100 IN ORDER
(ASCENDING OR DESCENDING ORDER)
4.3
This section will focus on „arranging the numbers to 100‰ in ascending or
descending order.
4.3.1 Arrange Numbers to 100 in Order
In general, there are two ways in arranging numbers to 100 in order, which are:
(a) Arrange Numbers to 100 in Ascending Order (Count On)
ÂCount onÊ order means arranging the numbers in ascending order. You can
start at any number as long as the sequence of the numbers is in order. The
same thing goes with the gap or the difference in value between the
numbers. You can have any value for the difference as long as it is the same
throughout the whole number sequence.
(b) Arrange Numbers to 100 in Descending Order (Count Back)
ÂCount backÊ order means arranging the numbers in descending order. You
can again start at any number as long as the sequence of the numbers is in
order. The same thing goes with the gap or the difference in value between
the numbers. You can have any value for the differences as long as it is the
same throughout the whole number sequence.
Activity 4: Count On and Count Back in Ones using a Number Ladder or
Number Chart Up to 100 (Snakes and Ladders Game)
Learning Outcome:
(a) Count on and count back in ones to 100.
Materials:
Dice;
Markers; and
Number ladder game (Snakes and Ladders Game).

81
Procedure:
(i) Several pupils can participate in this game at the same time. Each of them
will be given a marker. Players take turns to roll the dice.
(ii) After taking turns to throw the dice, the players have to move their markers
according to the number rolled. For example, if the first player rolls a 5, he
will have to move his marker along five squares until it reaches the fifth
square. If it happens that at the fifth square there is a ladder pointing to
square number 23, then the player will have to climb up the ladder to end
on the square number 23.
(iii) On the other hand, if the marker lands on a square with a snake slithering
down, the player will have to follow suit and slide down the snake to
wherever it should be. e.g. If the marker reaches, say, square number 46
showing a snake slithering down to square number 14, the player must
follow the snake and place his/her marker on square number 14.
(iv) The winner is the first player to reach the number 100.
(v) Distribute Worksheet 4 to your pupils.
PLACE VALUE OF NUMBERS TO 100
4.4
When objects are placed in order, we use ordinal numbers to tell their position.
Ordinal numbers are similar to the numbers that you have learned before. The
pupils need to understand the ordinality of numbers to enable them to position
items in a set. If 10 pupils ran a race, we would say that the pupil who ran the
fastest was in first place, the next pupil was in second place, and so on until the
last runner. Here, we are actually arranging the winners in order. In short, the
first 10 ordinal numbers are listed as: first, second, third, fourth, fifth, sixth,
seventh, eighth, ninth and tenth.
4.4.1 Place Value of Numbers to 100
Place value is used within number systems to allow a digit to carry a different
value based on its position, that is, the place it occupies has a value. The concept
of place value is very important when applied to basic mathematical operations.
The skill of regrouping numbers in tens and ones is very important to help
develop the concept of place value at the early stage for numbers to 100.

82
In our present number system, place value works in the same way for all whole
numbers no matter how big the number is. Numbers, such as Â84Ê, have two
digits. Each digit is at a different place value. For instance, the left digit, Â8Ê is at
the tens place. It tells you that there are 8 tens in this number. The last digit on the
right is in the ones place, that is, 4 ones in this example. Therefore, there are 8
tens plus 4 ones in the number 84, as illustrated below:
Activity 5: Ordinal Numbers and Place Value of Tens and Ones
Learning Outcomes:
(a) Label pupils in a row from left to right using ordinal numbers such as, first,
second, third, etc; and
(b) Identify the place value of tens and ones for two-digit numbers up to 100.
Materials:
Word cards (Ordinal numbers: first, second, ... tenth);
Ten pupils;
Number cards (two-digit numbers up to 100); and
Place value chart/mat.
Procedure:
The four steps in this procedure are:
(i) Ask 10 pupils to line up from left to right in front of the class. Then ask
another pupil to determine which pupil is in third position from the left
side? Label the pupilÊs position using the correct ordinal card. Do the same
with other positions, e.g. the sixth from pupilsÊ left, etc.
(ii) Repeat the activity by asking pupils to label various positions of the pupils
from the right side using the appropriate ordinal number cards.

83
(iii) Show pupils how to identify the place value for each digit in a two-digit
number. Ask pupils to fill in the place value for numbers up to 100 given in
the place-value chart or place value mat below:
Place Value
Number
Tens Ones
98 9 8
29 2 9
64 ? ?
75
13
60
(iv) Distribute Worksheet 5.
Familiarise yourself with numerals and numbers in words by saying them
loud and clear.
Know how to read and write numbers in words and in symbols
spontaneously.
Know how to arrange the numbers to 100 in ascending or descending order.
The skill of regrouping by tens and ones is an important process to understand
the concept of counting and place value.
Ascending
Count back
Count on
Descending
Ordinal Numbers

84
What other concrete objects can you use as base-10 materials in teaching the
concept of place value? How would you use the materials to show ones, tens and
hundreds?
Consider the following scenario:
LetÊs say one of your pupils knows how to count using concrete materials and
can clearly count out loud e.g.„one, two and three, etc.‰. When you ask her:
„How many objects are there?‰, she immediately starts to count them all over
again.
Discuss based on the above scenario.
What do you know about her understanding of counting? What do you think is
the next step in her learning? How might you enable her to achieve this?
APPENDICES
WORKSHEET 1
1. (a) Say the numbers given on the door of each house.
44 34 66 70 98
22 10 33 50 79

85
(b) Say the numbers written on the manila cards.
Fifty-eight Ninety-six
Sixty-one
Eighty-two
One
hundred
Twenty-seven
WORKSHEET 2
1. Count the heart-shaped beads. Write the numerals in the boxes provided.
(a)
(b)

86
(c)
(d)
2. Fill in the boxes with the correct numbers.
(a)
(b)

87
(c)
(d)
(e)
(f)

88
WORKSHEET 3
1. Write the missing numerals or words.
(a) thirty-four =
(k) =
(b) sixty-nine =
(l) =
(c) thirteen =
(m) =
(d) forty =
(n) =
(e) ninetythree =
(o) =
(f) thirtyeight =
(p) =
(g) forty-four =
(q) =
(h) thirty-seven =
(r) =
(i) thirty =
(s) =
(j) sixteen =
(t) =
99
87
77
70
61
35
11
79
80
36

89
WORKSHEET 4
1. Fill in the missing numbers in the boxes/spaces below. (Count on/count
back).
Number Patterns
(a) (i)
(ii)
(b) (i) 21, 31, 41, __, __, 71, __, __
(ii) 80, 70, 60, 50, __, __, 20, __
(c) (i)
(ii)
(iii)
(iv)
(d) Now try to write your own number patterns.
(i) __, __, __, __, __, __, __, __, __, __,
(ii) __, __, __, __, __, __, __, __, __, __
(e) (i) Between 51, _____, 53
(ii) Just after 1, 2, _____
(iii) Just before _____, 5, 6

90
(iv) Just before and after _____, 74, _____
(v) In the middle of 98, _____, 96
(f) Order each group of numbers from smallest to largest.
(i) 37, 11, 90 _____, _____, _____
(ii) 26, 12, 82 _____, _____, _____
(iii) 83, 59, 95 _____, _____, _____
(iv) 97, 0, 15 _____, _____, _____
(g) Order each group of numbers from largest to smallest.
(i) 74, 42, 47 _____, _____, _____
(ii) 39, 74, 91 _____, _____, _____
(iii) 28, 82, 49 _____, _____, _____
(iv) 27, 1, 80 _____, _____, _____
WORKSHEET 5
(a) What is the position of the yellow car from the right?
(b) What is the position of the yellow car from the left?
(c) What is the position of the red car from the right?
(d) What is the position of the red car from the left?
(e) Which car is in the first position from the left?
(f) Which car is in the last position from the left?
(g) Which cars are in the first three positions from the right?
(h) Which cars are in the last two positions from the right?
(i) Which car is in the middle?
(j) What is the position of the purple car from the left?

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(k) What is the position of the purple car from the right?
(l) Which car is in the fifth position from the right?
(m) Which car is in the second position from the right?

Topic
5
Addition
within 18
LEARNING OUTCOMES
1. Describe how to add one more, two more and beyond to a number
for addition within 18;
2. Explain how to add numbers by combining two groups of objects for
addition within 18;
3. Explain how to add numbers by counting on for addition within 18;
and
4. Demonstrate how to write number bonds for addition within 18.
INTRODUCTION
Previously, in Topic 2, addition within 10 was introduced whereby pupils
learned the concept of Âone moreÊ either by counting all or counting on. Number
bonds up to 10 were also highlighted. Here, the discussion is further extended to
include addition within 18 and covers number bonds up to 18. A sound
knowledge of number bonds, or basic facts of addition, is a must to enable pupils
to apply them when adding bigger numbers to go beyond totals of 18. The
process of addition is usually taught with the help of suitable teaching aids and
concrete manipulatives such as counters, number lines, picture cards, etc. As in
other chapters, some samples of teaching and learning activities for addition
within 18 are provided to show how pupils can be helped to acquire this basic
concept effectively.

TOPIC 5 ADDITION WITHIN 18
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ADDING ‘ONE MORE’ TO A NUMBER
In this section, we will discuss further the concept of adding 'one more' to a number.
5.1.1 The Concept of ‘One More’
In order to approach the concept of addition as Âone moreÊ than a number, a
variety of methods can be used. For instance, if you want the pupils to learn what
is one more than 16, you can try the ones suggested below.
(a) Use suitable counters such as beads, beans, nuts or marbles, etc. to add one
more to a number. Ask pupils to first count how many beads are in a jar
and then ask them how many beads will there be if one more bead is
added. For example, if there are 16 beads in the jar initially, how many
beads will there be if one more bead is added?
Encourage them to first say ÂOne more than 16 is 17Ê or Â17 is one more than
16Ê and then show them how to write the mathematical sentence for the
addition operation as in Figure 5.1:
Figure 5.1: Adding one more to a number using counters
(b) Next, you can also use a number line. Addition on a number line
corresponds to moving to the right along the markings on a number line.
The number line below is marked with ticks at equal distance intervals of 1
unit. To add one more to 16, first move 16 units from 0 and then move 1
more unit to finally end up at 17. The sum of 16 + 1 which is equal to 17 is
shown in Figure 5.2. The addition operation that corresponds to the
situation acted out on the number line is represented as 16 + 1 = 17.
Figure 5.2: Adding one more to a number using a number line
5.1

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(c) Another way is to use number cards, see Figure 5.3. For example, first show
the number card 16 to the pupils.
Then, ask the pupils what number card is supposed to come out next if you
add one more to the number 16.
Get them to write the mathematical sentence for this operation, that is, 16
add one equals 17.
Figure 5.3: Adding one more to a number using number card
(d) The concept of addition can be modelled using other concrete and
manipulative materials. Addition can be done by counting on or by counting
all as shown in Figure 5.4.
(i) Finding one more than a number.
e.g. 1 more than 10 is ___. (Ask pupils to get the answer by counting
on).
(ii) Finding the total by counting all the objects.
e.g. ____ is 1 more than 13. (Ask pupils to get the answer by counting
all the objects).
Figure 5.4: Adding one more to a number using concrete materials
Activity 1: Adding One More to a Number
Learning Outcomes:
(a) Add one more to numbers up to 18; and
(b) Write the mathematical sentence for addition within 18.

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Materials:
Counters e.g. beads, beans, nuts, marbles, etc.;
Numeral cards (e.g. 1, 10, 11, 12, 13, 14, 15 16, 17, 18); and
Symbol cards (+, = ).
Procedure:
(i) Fill a plate with 11 marbles. Then ask the pupils what is the total number of
marbles if you add one more marble to the plate. Guide pupils to say
ÂtwelveÊ.
(ii) Ask one of the pupils to represent the operation with a mathematical
sentence using the respective numeral cards and symbol cards e.g. 11 + 1 =
12.
(iii) Get the pupils to write the mathematical sentence for the addition
performed.
(iv) Repeat the above steps using other quantities of numbers up to 17 in order
to get a highest total of 18 e.g. 12 + 1 = 13, 17 + 1 = 18, etc.
(v) Distribute Worksheet 1 to your pupils.
ADDING TWO OR MORE TO A NUMBER
5.2
The above activities described in Section 5.1 can be repeated to develop the
addition of two or more to a number. Let us now take a look at the addition of
two or more to numbers up to a highest total of 18.
5.2.1 Adding More than One to Numbers up to a
Highest Total of 18
In order to approach the addition of more than one (e.g. 2, 3 etc.) to a number,
you can use the following suggested methods.
(a) Use beads, beans, nuts, marbles, etc. to add two more to a number. Ask the
pupils to count how many beads are in the jar and then ask them how
many beads will there be if you add two more beads.

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For example, if there are 16 beads in the jar, how many beads will there be
altogether if you add two more?
Encourage the pupils to say Âtwo more than 16 is 18Ê or Â18 is two more
than 16Ê and then ask them to write the mathematical sentence for the
addition operation (see Figure 5.5):
Figure 5.5: Adding two more to a number using counters
(b) You can also use a number line. As mentioned earlier, addition
corresponds to moving to the right along the markings on a number line.
First, move 16 units from 0 and then move 2 more units to finally arrive at
18. The sum of adding two more to 16 is shown in Figure 5.6. The addition
sentence that corresponds to the situation is 16 + 2 = 18.
Figure 5.6: Adding two more to a number using a number line
(c) You can also use number cards, see Figure 5.7. For example, show the
number card 16 to the pupils and then ask the pupils what number card is
supposed to come out next if you add two more to the number. Ask pupils
to write the mathematical sentence for the operation performed.
Figure 5.7: Adding two more to a number using number cards

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(d) Use the counting on and counting all techniques to add two more to a
number, see Figure 5.8. Model the concept of addition using other concrete
and manipulative materials as before.
(i) Finding two more than a number by counting on.
(ii) Finding totals by counting all the objects.
____ is 2 more than 13.
Figure 5.8: Adding two more to a number using concrete materials
Activity 2: Adding More than One to a Number
Learning Outcomes:
(a) Add more than one (e.g. 2 or more) to a number with a highest total of 18;
and
(b) Write the corresponding mathematical sentences for addition within 18.
Materials:
Counting objects (e.g. beads, beans, nuts, marbles, etc.)
Procedure:
In general, there are three steps, which are:
(i) Fill a plate with 11 marbles. Then ask the pupils what is the total number of
marbles if three more marbles are added to the plate.
(ii) Get pupils to write the corresponding mathematical sentence for the above
operation e.g. 11 + 3 = 14.
(iii) Repeat the above steps using different quantities to be added to get a
highest total of 18.

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5.2.2 Adding by Combining Two Groups of Objects
Apart from the above approaches to addition described in the previous sections,
the meaning of addition can also be developed simply as the process of
combining two groups or sets of objects as follows.
(a) Combining two groups or sets of objects; see Figure 5.9.
Figure 5.9: Adding by combining two sets of objects
(b) Combining two numbers using a number line; see Figure 5.10.
Figure 5.10: Adding by combining two numbers on a number line
Activity 3: Addition of Two Numbers
Learning Outcome:
(a) Add two numbers by combining two sets of objects.
Materials:
Counting objects e.g. beads, beans, nuts, marbles, shells, pencils, etc.

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Procedure:
In general, there are six steps, which are:
(i) Choose two pupils to come in front. Give eight pencils to one pupil and
nine pencils to the other.
(ii) Ask the pupils to put (combine) the two sets of pencils together and count
how many pencils there are altogether.
(iii) Get the pupils to write the mathematical sentence in words and then in
symbols for the addition operation performed.
e.g. Eight plus nine equals seventeen
8 + 9 = 17
(iv) Repeat the activity using different quantities for the sets of objects to be
added or combined to get a highest total of 18.
(v) The above steps can be repeated using different counting objects.
(vi) Distribute Worksheet 2 to the pupils.
Activity 4: Combining Two Numbers to Derive Basic Facts of Addition
Learning Outcome:
(a) List down all the possible combinations of any two single digit numbers for
deriving basic facts of addition.
Materials:
Sets of numeral cards (0 to 18); and
Sets of 10 flash cards per group.
Procedure:
In general, there are six steps, which are:
(i) Teacher asks pupils to work in groups and give each group a set of
numeral cards from 0 to 18.
(ii) First, ask a pupil to choose any two-digit number card within 18 and say
aloud the number e.g. the Number 12 before pasting the number card on
the board.

100
(iii) Ask the class what two single digit numbers when added together give a
total of 12, e.g. 7 + 5, etc. Then, get the pupil to write the mathematical
sentence for the basic fact of addition derived e.g. 7+ 5 = 12 on a flash card
and show it to the whole class.
(iv) Get the pupils from each group to choose any 2 cards that sum up to 12
other than 7 and 5 mentioned above e.g. 8 + 4; 3 + 9; 11 + 1, 12 + 0; etc. and
hold them up for everyone to see.
(v) Ask each group of pupils to jot down all the possible combinations for
getting a total of 12 on the flash cards provided.
(vi) Repeat with other numbers until the pupils are familiar with all possible
combinations comprising two single digit numbers for deriving various
basic facts of addition.
Activity 5: Lucky Throws
Learning Outcome:
(a) Total up the two numbers shown on the faces of two tossed dice.
Materials:
Two dices with different numbers (1 to 6 and 7 to 12) on their faces.
Procedure:
(i) Prepare two dice, one which has the numbers 1 to 6 on its faces and another
with numbers 7 to 12 on its faces.
(ii) Give the two dice to one pupil.
(iii) Ask the pupil to toss the dice on the table simultaneously and say out loud
the sum of the numbers shown on the faces of the tossed dice.
(iv) Ask their friends to check the accuracy of the answer.
(v) Get pupils to work in pairs. Pupils take turns at tossing the dice and
checking the answers.

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Activity 6: Number Wheel
Learning Outcome:
(a) Add two numbers shown on the number wheel up to a highest total of 18.
Materials:
Number wheel; and
Sets of number cards with single digit and two digit numbers from 0 to 18.
Procedure:
(i) Give two number cards to a pupil and ask the pupil to paste the two cards
along the diameter of the wheel. The sum of the two numbers should be
within 18 e.g. numbers 7 and 5.
(ii) The two numbers should be put diagonally on a straight line as shown in
Figure 5.11.
Figure 5.11: Adding by combining two numbers on a number wheel
(iii) Get pupils to call out the sum by combining the two numbers displayed on
the wheel.
(iv) Ask another pupil to pick any two cards and fill up the remaining spaces
on the wheel and get the pupils to add up the numbers.
(v) Repeat the activity by getting the pupils to work in groups. Give each
group a set of number cards. Pupils take turns to pick and paste the cards
whilst the rest of the members add and check the answers.

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5.2.3 Adding Two Numbers by Counting On
It is not easy to answer questions or solve word problems on addition especially
when ordinal numbers are involved such as in the following examples.
(a) The original height of a plant is 7cm. If it increased its height by 5cm, what
is its present height? Refer to Figure 5.12.
(b) This year is ArifÊs fifth birthday. What will his age be in 10 yearsÊ time?
The first question can be easily answered with the help of a count on model.
Number lines can be used to solve this problem.
Figure 5.12: Adding two numbers by counting on a number line
The use of a combination of two models for adding, such as combining two
groups and counting on, is an important step in developing the understanding of
the addition operation among pupils. At this stage, pupils know how to find the
total of two groups without having to count all the objects one by one all over
again. Pupils who have not really understood the technique will count one by
one all over again when trying to find the answer for the combination of two
groups of objects.
5.2.4 Adding Two Numbers According to Place Value
Conventional algorithms involve adding digits according to their place value.
Basically, there are two ways to do the addition of two numbers, which are:
(a) Horizontal Form
This is commonly used when adding single digit numbers.
(b) Vertical Form
This is commonly used when adding two or more digit numbers.

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Pupils can manipulate a place value model to build sets of ten to find the sum of
two whole numbers less than 10. It is important for children to remember the
basic facts of addition derived from simple combinations of any two single digit
numbers before moving on to perform the addition of two-digit and single digit
numbers after having learnt about place value. You are encouraged to start with
simple sums like the one below:
In order to perform the addition given according to place value, first add 5 to 3
and then 10 to 0 to get 8 and 10 respectively as follows:
In short, the addition in the above example was carried out by first adding the
ones before adding the tens, that is, according to place value.
5.2.5 Families of Addition Facts
Addition facts for different sums can be organised into families. Basically, as
described earlier, addition involves combining two groups into one bigger group.
Conversely, a separation activity is used for splitting a single group into two
subsets or smaller groups. The advantage of this activity is that it can be used to
introduce the families of addition facts for a number.
For instance, let us take the number 6. Pupils can discover many patterns as well
as find the combinations for getting a total of six and organise them into a table
(see Table 5.1).
Table 5.1: Family of Addition Facts for Six
0 + 6 = 6
1 + 5 = 6
2 + 4 = 6
3 + 3 = 6
4 + 2 = 6
5 + 1 = 6
6 + 0 = 6

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As can be seen from Table 5.1, there is an addition fact in the form of 6 + 0 = 6, or
0 + 6 = 0. Zero is called the identity element for addition. This is true for all whole
numbers. Other addition facts occur in the form of pairs like 2 + 4 = 6 and 4 + 2 =
6.
This is again true for all whole numbers and the operation of addition of all
numbers is said to be commutative, that is, reversing the order of the numbers to
be added does not affect the result. These two properties are important for
learning the basic facts of addition and for learning more advanced mathematics.
Developing the facts for sums greater than 10 is also important. This is done after
many activities with sums of less than 10 and equal to 10 are introduced. By then,
pupils should be able to organise what they already know accordingly:
(a) Know the basic facts and can put them under families, where each fact in
the family gives the same total.
(b) Know that the family facts for 10 are important and can build 10s.
(c) Able to use place value to build numbers that are equal to and greater than
10.
The three skills above can be used together to help pupils learn facts for sums
greater than 10 as illustrated in the following example:
What is 6 plus 5 equal to?
Figure 5.13 shows how pupils can be helped to develop facts for sums greater
than 10.
Figure 5.13: Adding two numbers by building a set of ten
(a) Use counting chips or loose objects to represent 6 and 5.
(b) Then build a set of 10 e.g. 6 + 4 = 10.

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(c) Look at the results and write an addition fact for the action performed e.g,
10+1= 11.
(d) Justify the actions undertaken.
The above process or steps can be explained thus:
This is because in the family of addition facts for 10, to make a set of 10 with
a set of 6, a set of 4 is needed, which has to come from the set of 5. The
family of addition facts for 5 tells us that if you take 4 to make 10 with 6,
there will be 1 left, resulting in a ten and one, such that 10 + 1 = 11.
Actually, describing this process in words is more difficult than demonstrating it
with objects as illustrated in the figure above. Experience working with families of
addition facts for numbers and place value will enable children to pick up visual
and sensory impressions and put these concepts together to develop a clearer
understanding of the whole addition process. The following developmental
activity illustrates how children can use place value and fact families to find sums
greater than 10 by using facts for sums less than or equal to 10.
Activity 7: Family Facts of Addition
Learning Outcome:
(a) Use families of addition facts for sums less than or equal to 10 to find
addition facts for sums greater than 10.
Materials:
Unifix cubes (red and blue).
Procedure:
In general, there are seven steps, which are:
(i) Ask pupils to first take out 8 loose blue cubes and then 6 loose red cubes.
(ii) Next, put them all in a row, see Figure 5.14 (a).
(iii) Join the blue cubes together to make a bar of 8;
(iv) Ask them to make a ten-bar by combining with two red cubes as in Figure
5.14 (b).
(v) Record the respective mathematical sentence for each step, as illustrated In
Figure 5.14 (c).

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Figure 5.14 (a): Adding two single digit numbers by using cubes or blocks
Figure 5.14 (b): Adding two numbers by using the family facts for six and ten
Figure 5.14 (c): Adding two numbers using families of addition facts
(vi) Do the same with other quantities of coloured cubes e.g. 7 + 5, 9 + 3, etc. to
find sums greater than ten by building sets of tens and using relevant family
facts of addition.
(vii) Emphasise that 7 + 5 = 7 + 3 + 2 = 10 +2 = 12. Get pupils to record the whole
process using appropriate mathematical sentences for each step involved.
Activity 8: Build a Ten
Learning Outcome:
(a) Build a 10 for addition within 18.
Materials:
Counters; and
Worksheet.
Procedure:
(i) Give pupils two sets of counters.
(ii) Ask them to count the number of counters in each set.
(iii) Let them write the numerals for each set in the blanks provided in the
worksheet, see Figure 5.15.
(iv) Then, ask pupils to build a set of ten.
(v) Write the new numerals and find the sum.

107
Figure 5.15: Adding two numbers by building a ten
Another important property for addition is the associative property of addition.
This property helps us to rewrite sums in terms of relevant facts that represent
sums greater than 10 to make them easier to learn.
Any sum that is greater than 10 can be found by rewriting one of the numbers so
that a fact for 10 is obvious. Then, the expression involving 10 plus another
number is simply a place value expression that can be written directly. This
shows the importance of the associative property in making the regrouping
process possible (see Table 5.2).
Table 5.2: Associative Property of Addition
Sum
Rewriting the Sum in
Terms of Relevant
Facts
Use of
Associative Property
Simplification
7 + 6 7 + (3 + 3) (7 + 3) + 3 10 + 3 = 13
8 + 7 8 + (2 + 5) (8 + 2) + 5 10 + 5 = 15
9 + 5 9 + (1 + 4) (9 + 1) + 4 10 + 4 = 14
Separation activities can also be used to relate the addition operation to the
subtraction operation, especially to emphasise addition as the inverse of subtraction.
5.2.6 Writing Mathematical Sentences for Addition
It is essential to teach pupils how to write the mathematical sentence for the
addition operation carried out both in words and in symbols. Provide enough
practice on writing the relevant mathematical sentences for addition to help
pupils master the skill of addition within 18. For example, when adding a set of
eight objects to a set of seven objects, the addition process can be recorded as
such:
Eight plus seven is equal to fifteen OR 8 + 7 = 15.

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NUMBER BONDS UP TO 18
A number bond is a pair of numbers making up a particular total. Pairs making a
10 such as 1 + 9, 2 + 8, 3 + 7, 4 + 6, 5 + 5, etc. were previously discussed in Topic
2. To help pupils master the skill of adding within 18, it is useful to teach them
number bonds up to 18 which actually correspond to all the basic facts of
addition within 18. Word problems can be used to introduce number bonds. e.g.
Abu has 8 balloons and Osu has 7. How many balloons are there altogether?
8 + 7 = 15
Number Bonds Corresponding to 15
The following Table 5.3 shows all number bonds that make up a total of 15.
Table 5.3: Number Bonds of 15
0 + 15 = 15
1 + 14 = 15
2 + 13 = 15
3 + 12 = 15
4 + 11 = 15
5 + 10 = 15
6 + 9 = 15
7 + 8 = 15
8 + 7 = 15
9 + 6 = 15
10 + 5 = 15
11 + 4 = 15
12 + 3 = 15
13 + 2 = 15
14 + 1 = 15
15 + 0 = 15
5.3

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Magic Square
Figure 5.16: A 3 x 3 magic square with a magic sum of 15
Another example that shows number bonds of 15 is a 3 by 3 magic square
consisting of a square array of numbers such that the sum along each column, row
and diagonal is the same and is equal to 15, see Figure 5.16. This common value is
called the „magic sum‰. The order of a magic square is simply the number of rows
(and columns) in the square.
Try to find the missing entries in the magic square provided. Have fun!
8 3 (a)
(b) 5 (c )
6 7 (d)
Solutions:
(a) 4;
(b) 1;
(c) 9; and
(d) 2.

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5.3.1 Number Bonds to 18
The following Table 5.4 shows number bonds corresponding to 18.
Table 5.4: Number Bonds of 18
0 + 18 = 18
1 + 17 = 18
2 + 16 = 18
3 + 15 = 18
4 + 14 = 18
5 + 13 = 18
6 + 12 = 18
7 +11 = 18
8 + 10 = 18
9 + 9 = 18
10 + 8 = 18
11 + 7 = 18
12 + 6 = 18
13 + 5 = 18
14 + 4 = 18
15 + 3 = 18
16 + 2 = 18
17 + 1 = 18
18 + 0 = 18
The addition operation concept can be explained by carrying out combination
and counting on activities with the numbers involved.
Developing the facts for sums greater than 10 is important. This can be done
by learning up the basic facts of addition and organising them into fact
families, where each pair of numbers in a family gives the same sum. Also,
the putting of the addition facts into families facilitates the learning of
mathematical sentences for addition.

111
Pupils should also know how to build 10s and recognise that the fact family
for 10 is very important and useful in performing addition especially when
carrying out the combination of two groups of objects.
The extremely important point for learning facts with sums greater than 10 is
using place value concepts to build numbers that are 10 or greater.
Associative property
Commutative operation
Identity element
Magic square
Number bond
The number line is a significant tool that can be used as a teaching aid in various
explorations in more advanced mathematics. Plan a strategy to show how to
apply this tool in teaching the concept of addition.
APPENDICES
WORKSHEET 1
1. Underline the correct numbers.
(a) One more than 11 is (12, 13).
(b) One more than (17, 14) is 18.
(c) (15, 17) is one more than 16.
(d) What is one more than 10? (13, 11).
(e) One more than (11, 14) is 15.
(f) (12, 14) is one more than 13.

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(g) (10, 13) is one more than 11.
(h) One more than (16, 17) is 18.
2. Fill in the blanks with the correct numbers (words).
(a) One more than twelve is __________________.
(b) One more than ______________ is thirteen.
(c) ______________ is one more than sixteen.
(d) What is one more than fifteen? ______________.
(e) One more than seventeen is ________________.
(f) ______________ is one more than seventeen.
(g) ______________ is one more than eleven.
(h) One more than ______________ is fourteen.
WORKSHEET 2
1. Write the missing numbers in the table.
Add Sum Write in Words
5 + 6 11
6 + __ 12 Twelve
__ + 11 15 Fifteen
12 + 4 Sixteen
8 + 5 13
9 + __ Eighteen
__ +13 Seventeen
12 + 0
12 + 5

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2. Write the correct numbers in the boxes.
(a)
(b)
(c)
3. Match the following.

Topic
6
Subtraction
within 18
LEARNING OUTCOMES
1. Identify Âone lessÊ or Âmore than one lessÊ than a number given;
2. Explain the differences between two numbers;
3. Subtract two numbers by taking away;
4. Subtract by finding the difference between numbers of objects in two
groups;
5. Write mathematical sentences for subtraction; and
6. Count back in steps from any number.
INTRODUCTION
Subtraction and addition are closely related such that they undo each other and
for that reason they are called inverse operations.
There are several types of subtraction situations. They include the idea of taking
away, the additive principle (what is needed), the comparative situation
(comparison of two sets), the partitioning concept (separating a set of objects into
parts) and the incremental aspect (involving decrease) to illustrate subtraction.
The most intuitive idea for subtraction is taking away. Reinforcing the take away
interpretation requires the presence of both addition and subtraction situations at
the same time. All five types of situations occur in real life and pupils must
explore them for themselves. Attention must be focused on the basic idea of
subtraction and how it relates to each of these situations.

TOPIC 6 SUBSTRACTION WITHIN 18
115
As place value concepts are developed, pupils should also learn basic arithmetic
facts for sums greater than 10 by building sets of 10. If not, they will encounter
difficulties when they begin to learn the procedures for computing with large
numbers.
THE CONCEPT OF ‘LESS THAN’
6.1
For subtraction involving numbers greater than 10, pupils should manipulate
loose objects, such as blocks, tiles or counting chips, so that they can participate
actively in the regrouping process. Suppose we want to subtract 5 from 13, we
can do so in the following way:
Represent 13 as one ten bar
and three loose blocks -------------
Subtract 3 to get a ten bar
Or one tens --------------------------
Exchange the one tens
for ten ones, and then
subtract 2 more ones --------------
End up with 8 ones ---------------
When pupils have had experience working with loose objects and begin to
appreciate the regrouping process, they can work with visual models on a
worksheet. The models on the worksheet must lead them through the reasoning
that is necessary for finding the missing number in a subtraction mathematical
sentence. For example:
Find the missing number in the mathematical sentence: 13 5 =
(a) Put out one 10 and 3 loose blocks for representing 13
(b) Subtract to get a ten i.e. (13 3 = 10)
(c) Subtract 2 more to get to subtract 5 in all. That gives 8.
The regrouping process is extremely important. Pupils must learn that when they
add facts with sums greater than 10, they have to regroup. Similarly, they must
also learn that some subtraction situations require regrouping too. The activity
represented either physically or pictorially, should relate addition to subtraction.
This practical tool is superior to counting forwards or backwards to find a sum or

116
difference. It is a beginning towards building the facility to do mental arithmetic
as illustrated as follows.
Break tens down and always
use families of relevant facts
For example: When I add, I sometimes build tens.
What is 12 5?
To subtract 5 from 12, I have to think that: 12 is 1 set of ten and 2 singles.
Removing 2 singles is easy and that gets me to 10.
There are 3 more to be removed and that is easy too because I know my
family of facts for 10.
I break down the ten (regroup) and remove 3 more. This leaves 7.
Thus, 12 5 = 7.
6.1.1 Patterns for Subtraction
While practising subtraction with regrouping, pupils can be prompted to notice
helpful patterns that involve families of facts.
If I subtract 5 from 12, that is easy because 6 + 6 = 12 and I have subtracted one
less than 6 which is 5. Therefore we must have 7 remaining.
Sometimes a subtraction sentence can be translated into an easier subtraction
sentence as below:
6.1.2 ‘One Less’ Than a Number
Subtracting one or more from a number to show Âone or more lessÊ than a number
can be done easily using a number line as shown in Figure 6.1 and Figure 6.2.

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Figure 6.1: Subtracting 1 from 15 on a number line is 1 less than 15, that is 14
Figure 6.2: Subtracting 2 from 17 on a number line is 2 less from 17, that is 15
6.1.3 ‘Take Away’
Several types of problem situations fall into the subtraction category.
Unfortunately, these problem situations are not as intuitive as problem situations
for addition. This is because many subtraction problems sound like addition.
Teaching subtraction requires extra attention, thus there is a need to get used to
all types of subtraction situations.
In general, subtraction of whole numbers applies to two kinds of situations. The
first is called Âtake- awayÊ. This is the easiest and most natural interpretation of
subtraction for pupils to learn. It is easy to represent the situation with objects,
and it is a natural extension of the combination interpretation for addition. Have
pupils characterise situations by describing and drawing rather than have them
write the appropriate mathematical sentence.
When reinforcing the Âtake awayÊ interpretation for subtraction, present
equivalent addition and subtraction situations at the same time.
For example: A bowler with 10 pins knocks down 8 of them. From 10 pins, the
bowler takes away 8 pins. There are now 2 pins left. The subtraction is thus
represented as 10 8 = 2.
The following terminologies are used with subtraction:

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The ÂminuendÊ is the number from which another number is being subtracted.
The ÂsubtrahendÊ is the number being subtracted.
The ÂdifferenceÊ is the result of subtracting the subtrahend from the minuend.
Related Sentences for Subtraction
Subtraction is often defined in terms of addition.
e.g. 5 2 is that number which when added to 2 gives 5.
Thus the subtraction sentence can be written as:
5 2 = 3 (Taking away 2 from 5 gives 3)
There is a related addition sentence for this situation:
5 = 3 + 2 (Putting back the 2 gives 5 again)
In fact, we know that the answers we find for subtractions are correct only
because of the related addition operations.
Subtracting on a Number Line
Subtraction also corresponds to moving distances on a number line. The number
line below is marked with tick marks at equal distances of 1 unit. To perform this
operation, we first move 10 units to the right from 0 and then to the left 8 units, to
end up at 2.
The subtraction that corresponds to this situation is written as 10 8 = 2 (see
Figure 6.3)
Figure 6.3: Subtracting on a number line

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Revision of the ÂTake AwayÊ Situation
Table 6.1 explain the revision of the 'take away' situation in detail.
Table 6.1: Revision of the 'Take Away'
Situation Physical Objects Mathematical Sentence
There are 3 birds sitting on a fence. Soon 2 more birds fly onto the fence. How many birds
are on the fence now?
Pick out 3 counters, then
2 more. Ask: „How many
3 + 2 = 5
counters in all?‰
There are 5 birds sitting on a fence. Three of them fly away. How many birds are still
sitting on the fence?
Put out 5 counters, then
remove 3. Ask: „How
5 3 = 2
many are left?‰
If pupils perform this activity often enough, they will naturally be able to construct
concepts that relate to subtraction and addition.
Pupils are ready to move on to a pictorial representation of the operation by covering up
or marking out pictures of objects.
Here are six cherries;
mark out 3 to show how
many the crows ate. Tell
us how many are left.
6 3 = 3
6.1.4 ‘Difference’ between Two Groups of Objects
Besides the take away idea, there are many other types of subtraction situations
as to how to know the difference between two groups of objects.
Sometimes we can use the additive situation to focus on what is needed. We can
ask „How many must be added to what I already have to obtain a certain
amount?
For example: Dani has 3 stamps, but needs a total of 5 to mail his letters. How
many more stamps does he need?

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The comparative situation is matching objects in two groups on a one-to-one
basis. A strong foundation, based on the one-to-one matching test for the
comparison of two sets, can be used to help children with this application.
Questions that could be asked include „Which is more?‰ to „How many more‰ to
provide pupils with a procedure to solve problems of this type.
The partitioning situation involves separating or partitioning a set of objects into
parts. It is extremely important for pupils to draw pictures and diagrams in this
case. For example: Here are 5 cars. If 3 of them are blue and the rest are green,
how many are green?
The last type of subtraction situation, known as the incremental situation,
involves a decrease in quantity. All measurements require this type, for example,
when situations such as losing weight, shortening the length of a pair of pants or
the temperature drops when it is cold, etc. are involved.
The following are some samples of teaching-learning activities denoting the
various types of subtraction situations.
Activity 1 4: Representing Different Subtraction Situations
Learning Outcomes:
By the end of the activities, pupils should be able to:
(a) Represent different subtraction situations such as taking away, additive,
comparative and partitioning types; and
(b) Write the appropriate mathematical sentence for the subtraction performed.
Materials:
Groovy Boards with dots; and
Rubber band.
Figure 6.4: The board with 2 dots on one side

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Procedure: Situation 1 (Take Away)
(i) Teacher shows board with 2 dots on one side of the rubber band and 3 on
the other side (see Figure 6.4).
(ii) Ask pupils to show the operation of 2 + 3 = 5.
(iii) Teacher shows how to represent this operation as subtraction.
(iv) Teacher asks: „How many dots are there altogether? If 2 dots are covered,
how many dots are left?‰
(v) Let pupils think and say together: 5 2 = 3
That is: Five take away two gives three
Figure 6.5: The board with 5 dots
Procedure: Situation 2 (Additive)
(i) Teacher shows board with 2 dots and tells that there are 5 in all.
(ii) Ask pupils how many more do they need to get to 5.
(iii) Let them put the rubber band so that they can see 2 on one side.
(iv) Ask them how many more do they need to have all the dots on the board.
(v) Let pupils think and say together: 5 2 = 3 (additive).
Figure 6.6: The boards with 2 dots and 3 dots separately

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Procedure: Situation 3 (Comparative)
(i) Teacher shows board that have 5 dots and 2 dots separately. The boards
need to be put on a one-to-one matching basis.
(ii) Ask pupils to examine the lengths of the 2 dots board and the 5 dots board
(see Figure 6.7). Ask pupils how much longer the 5 dot-board is compared
to the 2 dot- board.
(iii) Ask them to compare and say together: 5 2 = 3 (comparative).
(iv) Both boards can be matched on a one-to-one basis.
Figure 6.7: The boards with 2 dots and 5 dots separately
Procedure: Situation 4 (Partitioning)
(i) Teacher shows 5 dots in all. Show the pupils that 2 dots are on one side of
the rubber band (see Figure 6.8).
(ii) Ask them how many are on the other side.
(iii) Let them put the rubber band in place and think of the subtraction as: 5 2
= 3 (partitioning).
Figure 6.8: The boards with 2 dots on one side with rubber band

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Exercises on finding the difference between two groups of objects.
1. The difference between 7 and 3 is 4:
There are 4 more circles in the row of 7 compared to the row of 3.
2. Find the difference between these numbers:
Between 8 and 5, the difference is _____
3. Possible pairs of numbers with a difference that is equal to a given number.
15 6 = 16 4 =
12 6 = 13 3 =
14 7 = 18 6 =
13 5 = 18 4 =
15 9 = 15 8 =
6.1.5 Writing Mathematical Sentences for Subtraction
The following equation with a missing addend can be considered as a special
mathematical sentence for subtraction. For example, take a look at the following
equation:
7 +
= 18

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It is referred to as a missing addend sentence because it is an addition sentence in
which one of the addends is not known. Constructing this new concept should
just be a matter of helping pupils to reorganise what they already know. Using
relevant families of facts can help pupils to master the missing addend idea with
ease.
Activity 5: Writing the Subtraction Sentence with the Help of a Number Line
Learning Outcome:
At the end of the activity, pupils should be able to:
(a) Write a mathematical sentence for subtraction within 18 using a number
line.
Materials:
Number line Worksheet.
Procedure:
(i) Ask pupils to get a partner. Find the missing number in each subtraction
sentence.
13 7 =
10 4 =
(ii) Start at 13, then go back 7 spaces.
End up at 6: 13 7 =
(iii) Start at 10, then go back 4 spaces.
End up at 6: 10 4 =

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(iv) Story problem that will fit both sentences.
Danial, the hopping cricket, must hop on the number line to show 13 7.
He will start at 13 and hop back 7 spaces. What must Fakri do if he must
start at 10 and then go back to meet Danial?
COUNTING BACK
A type of subtraction situation, known as the incremental situation, involves a
decrease in quantity. Counting back is also an activity of counting numbers in
descending order. It is also the inverse of counting on in the addition concept. In
real life, we are not always dealing with concrete objects that can be counted. All
measurements require this type of subtraction such as the calculation of weight
loss, the drop in temperature when it is cold and the shortening of the length of a
pair of pants. Pupils should realise that they need to subtract in these instances by
exploring the various situations.
Examples of Subtraction in Real Life
Example 1: Pupils can explore what happens to the length of a chain or the height
of a tower when the number of links or cubes is decreased.
HumairaÊs weight is 20 kg. Three weeks later she loses 5 kg. What is her weight
now?
Example 2: What is the number that is 3 less than 12, see Figure 6.9.
Figure 6.9: Subtracting on a number line
6.2

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SUBTRACTION SQUARES (Enrichment Activity)
The following activity can be used to provide subtraction computation practice at
many levels (i.e. involving single and multiple digits and subtraction beyond 18)
in different and interesting formats. This enrichment activity can be used in
different forms and can serve as one of the following:
Group activity Independent activity
Cooperative activity Abstract procedure
Learning Outcome:
At the end of the activity, pupils should be able to
(a) Perform subtraction involving single and multiple digits in different and
interesting formats.
Materials:
Worksheet Subtraction squares (Arithmagons).
Procedure:
(i) Ask pupils to perform subtractions along horizontal, vertical and diagonal
lines for each of the subtraction square (arithmagon) provided in the
Worksheet (Steps 1 4).
17 16 = 1 12 8 = 4
16 8 = 8 17 12 = 5
6 4 = 2 6 2 = 4
8 2 = 6 8 4 = 4
Along Diagonal Lines

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4 2 = 2 6 4 = 2
Along Vertical and Horizontal lines
2 2 = 0
Along Diagonal Lines
(ii) The above steps can be repeated using other numbers to provide more
practice on subtraction.
(iii) A master copy of the template (Blank Subtraction Squares or Arithmagons)
is provided for you to make photocopies. Prepare enough copies of the
template and distribute to pupils.
Master Copy of the Blank Subtraction Squares (Arithmagon Template)

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Try to find the answer for each question and state what is the best type of
subtraction to use to solve each problem.
(a) Rolando had 12 crayons and bought 6 more.
(i) How many crayons does he have?
(ii) He broke 6 crayons. How many crayons does he still have?
(b) A mail carrier had 18 letters to deliver. He delivered 12 letters to the first
house, 3 letters to the second house and 2 letters to the last house on the
same street. How many more letters does he still have to deliver?
(c) Timmy earned 9 stars for good behaviour last week and 7 stars this week
but then he lost 10 stars for fighting. How many stars does he still have?
Several types of problem situations fall under the subtraction category.
All five types of subtraction situations (take away, additive, comparative,
partitioning and incremental) make use of the minus or subtraction symbol
() for recording the operation.
To perform subtraction involving numbers greater than 10, pupils should be
provided with the opportunity to manipulate loose objects so that they can
participate meaningfully in the regrouping process.
While practising subtraction with regrouping, pupils can be prompted to note
down helpful patterns involving families of relevant facts.
Additive
Comparative
Difference
Incremental
Partitioning
Take away

129
In your discussion groups, try to differentiate between the 5 types of subtraction
that have been discussed in this topic.
Supposing your pupils have difficulty in deciding that a problem requires
subtraction in order to be solved and then have trouble writing the appropriate
subtraction sentence, explain how can you tackle this problem.
APPENDIX
WORKSHEET
1. Draw lines to match the following.

130
Tick (ÂÊ) the answer in the correct boxes.
2. Complete the table below. Subtract, find the remainder and write the
missing numbers in words in the correct columns.
Subtract Remainder Write in Words
18 8 10
__ 3 thirteen
__ 0 15
12 2 ten
18 9 9
17 __ eleven
__ 5 thirteen
13 3 10
14 3 eleven

Topic
7
Money
LEARNING OUTCOMES
1. Explain the concept of money in the form of notes and coins in
Malaysian currency to your pupils;
2. Represent the value of Malaysian money in symbols, viz. ÂRMÊ and
ÂsenÊ;
3. Differentiate between the value of various denominations of money;
4. Exchange notes and coins for a given value of money within RM10;
5. Add and subtract money in the form of coins and notes within RM10;
and
6. Solve simple problems involving money within RM10 in real-life
situations.
INTRODUCTION
Children gain a lot of mathematical knowledge informally through their
experience with money in their daily lives. Learning how to count and use, as
well as identify, coins are important basic skills related to money to be acquired
at an early age.
Spending money is an interesting way to reinforce your pupilsÊ basic
mathematical skills such as addition, subtraction, multiplication and division
besides other language skills such as reading comprehension. Understanding
how to exchange money, plus knowing how much change you will receive when
spending money and purchasing items, are crucial.

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However, pupilsÊ classroom learning experience is also important to help them
clarify their misconceptions regarding money. One of the first concepts that
children must understand is the value of coins and notes. In this topic, you will
learn how you can help children to formalise their conceptual understanding of
money.
RECOGNISING NOTES AND COINS
Money, both notes and coins, have its own size, shape and colour. In the
Malaysian currency, different denominations of notes and coins are used in
everyday life. It is important that children learn to recognise the various notes
(named Ringgit) and coins (named Sen) in our currency.
First, introduce your pupils to the sen, and then the ringgit. Use the following
printable pages to assist in early money identification. Pupils may use the sheets
as a colouring activity involving money including coins and notes or merely as a
money reference sheet. Refer to Table 7.1 and Table 7.2 for Malaysian coins and
notes in various denominations.
Table 7.1: Malaysian Coins in Various Denominations
Value Front Back
1 sen
5 sen
10 sen
7.1

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20 sen
50 sen
Table 7.2: Malaysian Notes in Various Denominations
Value Front Back
RM 1
RM 2
RM 5
RM 10

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7.1.1 Recognise the Symbols for Money
Before 1993, the symbols of Â$Ê and Â¢Ê were used to denote a certain value for
money e.g. 20¢, 75¢, $5.00, $9.50, etc. The Malaysian currency was officially
changed to the Ringgit (RM) in place of the dollar ($), while ÂsenÊ replaced cents
(¢) e.g. 20 sen, 75 sen, RM5.00, RM9.50, etc.
Some samples of teaching and learning activities for ÂMoneyÊ are described in
this topic.
The following activity allows children to learn to recognise and write money
using the correct symbols.
Activity 1: Recognise and Write Symbols and Words for Money
Learning Outcome:
By the end of the activity, pupils should be able to:
(a) Name and write the value of money in symbols and in words.
Materials:
Pencils;
Crayons;
Coins; and
Paper.
Procedure:
(i) Ask pupils to trace the pattern of the coin. Put a piece of paper over the
coin, rub gently and then show it to the class
(ii) Trace/draw coins and notes of various denominations.
(iii) After that, ask them to cut and paste the money on the Worksheet given.
(iv) Label the traced money or drawings using symbols and words in Table 7.3.

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Table 7.3: Money Tracing/Drawing
Traced Money/Drawing Symbol Words
10 sen ten sen
50 sen fifty sen
RM 10.00 ten ringgit
SELF-CHECK 7.1
How do blind people know the value of coins and notes? Jot down your
ideas.
7.1.2 The History of Money
Money did not exist in the olden days initially. Prior to using money to trade,
bartering was the only way goods were exchanged. However, the barter system
was not very efficient as trading animals for other goods proved inconvenient.
Money acts as an intermediary for market goods, which may be exchanged for
other goods. Throughout history, money has taken many different forms,
including scarce metals. Today, the majority of the types of money exchanged
takes no physical form and only exists as bytes and bits in a computer's memory.
Go to the following suggested links for learning about the history of money in
general and Malaysian money specifically. Then view any home page for lessons
to learn about money skills.
http://library.thinkquest.org/28718/history.html
http://moneymuseum.bnm.gov.my/index.php?ch=8

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The symbol of the Âcoin treeÊ (see Figure 7.1) logo is an adaptation from the
Âpohon pitisÊ (the coin tree) made from a tin bearing 13 coins issued by Sultan
Muhammad IV of the State of Kelantan in 1903 when it was still a tributary state
of Siam.
Figure 7.1: The coin tree
Source: http://moneymuseum.bnm.gov.my
(a) History of Money
Thousands of years ago, money did not exist. There were no stores, markets
or other places to spend money. People got their food, clothing and shelter
from the land around them. For many years, people lived independently or
on their own and had little or no contact with others who lived far away.
In the early days, people exchanged goods by bartering for what they
needed. For example, a person might trade animal skins for fresh fish or
trade vegetables and grains for meat. Trading one item for another is called
bartering. Under this arrangement, goods were exchanged for other goods.
As the years went by, bartering became very popular. Markets were created
where people could trade goods and people began to depend on getting
things from others but this was not always convenient.

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Sometimes bartering can get kind of tricky. Let us say you go to the market
with 50 fish and need to bring home animal skins and grains. However, the
person trading animal skins does not need any fresh fish and wants other
items for his animal skins. You could try to trade your fish for whatever he
wants, but that would take a lot of time and effort. Instead, people
developed another solution. They began to use special items, like tokens,
that everyone agreed upon which had a certain value. You could trade your
fish for tokens, and then you could use tokens to buy animal skins and
grains. The animal skin merchant could use your tokens to buy whatever he
needed. The token system was a great improvement, because everyone
could use the tokens to get exactly what they needed.
All around the world, people developed trading systems like these. Not all
of them used tokens though. Salt, shells, barley, feathers and tea leaves
were used in exchange for other goods. In most cultures, precious metals
like gold and silver were also highly valued. Many people began trading
goods for bits of gold or silver. It is hard to tell the value of a lump of metal
just by looking at it, so merchants began weighing the gold and silver
pieces. In many places, the metal was cut into circular discs and the weight
was stamped on the discs so everyone would know its value. These
stamped discs were the earliest coins!
As the trading industry grew over the years, many countries decided to
make official money. Governments made coins out of precious metals like
gold and silver, and everyone agreed on the value of each one. The
introduction of official coins made the buying and selling of goods within a
country much easier.
Through the ages, money has become not only a medium of exchange but
also a unit and store of value. With time, money in the form of paper
currency or notes was introduced as this was convenient to be issued.
For years, most countries used only coins for their money. Coins last a long
time and are easy to use. However, the citizens of China used another kind
of money. The Chinese government made its money out of paper because
precious metals were very rare in China. Moreover, paper bills are very
light and easy to carry. After a number of years, other countries began to
make paper money like the Chinese. These bills became very popular and
made things much easier when buying expensive items. Can you imagine
trying to buy a one hundred dollar item with coins? That would take a lot
of coins indeed!

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Remember!
Too many coins can become rather heavy for us to carry around so it is wise
that we resort to using paper money or notes instead.
Over the years, countries continued to develop new kinds of coins and bills,
and we now have other ways to exchange money too. We can write
cheques, use credit cards and transfer money electronically through
automatic teller machines (ATMs), see Figure 7.2 for example. Despite these
new technologies, the basics of our money system still remain the same.
What kind of changes in the use of money do you think we will see in the
next 100 years?
Figure 7.2: Automatic Teller Machine (ATM)
ACTIVITY 7.1
Do you know why we use the ÂRinggitÊ in Malaysia, ÂYenÊ in Japan and
ÂDollarÊ in America? Why do we have to change our money when we go
to Japan or other countries?
(b) The Barter System
Have you ever wondered what it would be like and what we would do
without money? To give children some idea about the barter system,
engage them actively in a discussion about trading and fair trade.
Arrange the children in groups of six. Give each group 12 index cards, a
bottle of glue, scissors and magazines or departmental sales brochures,
supermarket advertisements, newspaper cuttings, etc. Ask the children to
cut and paste food items on six cards and clothing items on the remaining
six cards. Have the children shuffle the cards and place them face down.
Each group member then draws two cards. Children can barter (conduct

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139
fair trade) within their groups so that each member finishes with one food
item and one clothing item. Bring the pupils together to talk about their
experience with the bartering process. Stress on the fairness and efficiency
of this type of trading.
ACTIVITY 7.2
Can you summarise the history of the flow of money since the barter
system? Discuss.
The following activities can be used to help pupils to learn to recognise and
identify Malaysian currency in different ways.
Activity 2: Recognising Malaysian Coins and Notes
Learning Outcomes:
(a) Recognise and say the name of various denominations of Malaysian coins
e.g. one sen, five sen and ten sen; and
(b) Recognise and say the name of various denominations of Malaysian notes
e.g. one ringgit, five ringgit and ten ringgit.
Materials:
Lyrics sheet: „Ten Little Sen‰;
Magnetic board; and
Specimen money (coins and notes) e.g. one sen, five sen and ten sen, one
ringgit, five ringgit and ten ringgit.
Procedure:
(i) Teach the children the song, „Ten Little Sen‰ (sung to the tune of „Ten
Little Indians‰.
(ii) As the class sings, place a magnetic sen on the board for each sen
mentioned in the song.
(iii) When five sen is reached, put a five sen coin on the board. Similarly, when
ten sen is reached, paste a ten sen coin on the board.

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140
(iv) Repeat the song, this time round, get the pupils to show the respective
money as they sing along.
Lyrics:
One little, two little, three little sen;
Four little, five little, six little sen;
Seven little, eight little, nine little sen;
Ten little sen make a ten sen.
(v) Ask pupils to name the places where they can see people use money? An
example is given in Figure 7.3.
(vi) Extension Activity: Replace the word ÂsenÊ with ÂRinggitÊ to enable the
pupils to recognise and say the name of Malaysian notes up to ten ringgit
in a similar manner.
Figure 7.3: People use money in hypermarkets to buy groceries
Activity 3: Coin Patterns
Learning Outcomes:
(a) Recognise and trace the patterns of various Malaysian coins on to a sheet of
paper; and
(b) Identify the features of Malaysian coins.
Materials:
Clean sheet of paper;
Pencils/crayons; and
Malaysian coins of various denominations.

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Procedure:
(i) Teacher guides pupils to say aloud various denominations of Malaysian
coins.
(ii) Ask pupils to arrange and make a pattern using the coins.
(iii) Ask them to put a sheet of paper over the coins and get them to trace or rub
gently on the surfaces of the coin to obtain the coin pattern with crayons.
(iv) Ask pupils to show and talk about the features of each coin traced in their
small groups.
(v) Guide pupils to summarise the features of various denominations of the
Malaysian coins discussed.
(vi) Attach a reference sheet of the coin patterns for pupils to refer to Table 7.1.
Activity 4: Identifying Malaysian Coins
Learning Outcomes:
(a) Identify coins and their values; and
(b) Write the amount of money based on a certain value of money given.
Materials:
Coloured construction paper (Size A4);
Pencil;
Scissors; and
Specimen coins.
Procedure:
(a) Instructions/Practice: Ask pupils the following.
(i) How many of you think you can count money really well and how
many think you could improve with some practice?
(ii) When it comes to counting money, bills are probably the easiest
things to count. Why? (Possible response: The amount is written on
them).

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(iii) How do we tell the coins apart? (Possible response: By their features:
size, thickness and pictures).
(iv) List the names of the coins and their values.
(v) Now, as a class, determine the value of the money each time it is
traded.
(vi) How much money did you use at the end of the activity?
(b) Model:
(i) Demonstrate how to count money with the help of models (specimen
money). Always start with the bill or coin of greatest value and work
down to the bill or coin with the least value.
(c) Guided Practice:
(i) Give the class fake bills and coins to cut out and keep in an envelope
throughout the activity. (Bills were copied on to green construction
paper and coins were copied on to yellow construction paper).
(ii) Tell them to count how much change they have and record their
answers on their blank sheets of paper.
(iii) Continue with other questions. (See ÂProcess QuestionsÊ next). Choose
volunteers to demonstrate and explain how they got their answers.
Process Questions:
How much change do you have? (RM3.28)
How much is 2 ringgit, 20 sen, 10 sen, 5 sen and 2 sen? (RM2.37)
How much is 1 five ringgit, 1 ringgit, 3 ten sen and 1 sen?
(RM6.31)
How much is 1 five ringgit, 1 two ringgit, 2 ten sen, 1 twenty sen,
and 5 one sen? (RM 7.45)
How much is 1 five ringgit, 1 two ringgit, 2 one ringgit and 9 ten
sen? (RM9.90)
You want to buy a candy bar for 45 sen. You have 3 ten sen, a five
sen and 3 sen. Do you have enough money? Why? (No, only
RM0.38)
You have four coins that add up to RM 0.46. Which coins do you
have?

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(d) Closure:
Discuss the importance of knowing the value of different money.
(e) Evaluation/Checking for Understanding:
(i) Listen to pupilsÊ responses during the Âwhole-classÊ questioning
process.
(ii) Have pupils record their answers to the class questions on the paper
provided. Check the responses together.
7.1.3 Values of Coins and Notes
The different denominations of coins that we use daily are shown here, for
example, one sen, five sen, ten sen, twenty sen and fifty sen (See Figure 7.4). Do
you remember the features of each coin? Can you describe them?
Figure 7.4: Different denominations of Malaysian coins
Various denominations of notes normally used by children include one ringgit,
two ringgit, five ringgit and ten ringgit as illustrated in Table 7.2 earlier.
Likewise, do you remember the features of each of these notes? Can you describe
them?
7.1.4 Counting Money (Ringgit and Sen)
Lessons on ÂCounting money using coinsÊ will teach pupils to learn about the
value of money. As mentioned earlier, learning how to count, use and identify
coins is an important basic skill to be learnt at an early age. Printable worksheets
and sample lessons will help your pupils master the skill of counting money with
coins, whether they are just beginning to learn to count coins, or if they need
additional practice to do so.

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Here is a fun way to reinforce money skills:
(a) Take out a manila folder.
(b) Open it up.
(c) On the left side, draw a tree or paste a photocopy of it.
(d) On the right side, have four library book pockets (or as many as you need
to teach your concept).
(e) Fill up the tree with little pieces of soft Velcro everywhere (on all parts of
the branches) and take one sen, five sen, ten sen and twenty sen (cut-outs
from a coin and coloured accordingly).
(f) Glue them to small pictures of apples printed on construction paper.
Laminate and place hard Velcro on the back of the ÂapplesÊ. Scatter the coins
everywhere so that all the one sen can be separated from the ten sen, etc.
such that the tree is now filled with ÂapplesÊ (i.e. all the scattered coins).
(g) On the pockets, label the values of each denomination of coins: e.g. 1 sen, 5
sen, 10 sen and 20 sen.
(h) Give each pupil one of these folders and ask them to ÂpickÊ the apples from
the tree and place them in the correct pocket. For example, if a child ÂpicksÊ
a one sen, he or she places it in the pocket that is labelled Â1 senÊ.
(i) When they are done, it is very easy to check their work because all you
have to do is to empty out the pockets and make sure each coin is matched
with its appropriate value.
(j) Get pupils to count the money in the respective pockets and write the value
using the correct symbols.
(k) Repeat the above steps and make another money tree using ringgit in place
of sen.
(l) Ask pupils to count the money by combining various pockets containing
ringgit and sen to let them practise counting money in ringgit and sen.
It has been shown that pupils who have worked with this activity all loved to
ÂpickÊ the apples from the tree. As a point of interest, the money tree idea can be
modified for teaching other concepts, for instance, addition and subtraction facts,
telling the time, picture-word identification, phonics activities, etc. In fact, you

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can choose any concept to ÂpickÊ from the tree as suggested by Ms. Jany Mederos,
in ÂBeginning Teacher, 1st and 2nd grade Autistic, Miami, FLÊ.
7.1.5 Exchanging Notes and Coins
Another money skill to be learnt by pupils involves ÂMaking changeÊ. The
following activity illustrates how children can be taught to exchange notes and
coins when performing buying and selling activities.
Activity 5: Making Change
Learning Outcomes:
(a) Count the correct change in sen; and
(b) Exchange notes (ringgit) and coins (sen).
Materials:
Activity sheet (containing pictured items);
Construction paper;
Scissors;
Specimen money (one ringgit and sen of various denominations);
Pen markers/coloured pencils/crayons; and
Glue.
Procedure:
(i) Give groups of children an Activity Sheet, scissors and Specimen money
(Notes and coins).
(ii) Ask children to cut out the cards on the activity sheet and then fill in a price
of 99 sen or less for each item shown.
(iii) Ask each group member to select an item to buy, pretend to pay for it with
one ringgit notes and then use the play money to show how much change
they will receive.
(iv) Group members can check each otherÊs work. Let the children paste their
items and coins on construction paper after each transaction.

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ADDING AND SUBTRACTING MONEY
7.2
Different amounts of money may be written in several ways. Coins may be
written with the ÂsenÊ symbol and the ringgit can be written with the ringgit
symbol (ÂRMÊ). Adding money that is expressed in these forms just involves
adding the amounts and placing the proper symbols on the answer.
One way to add money is to count the coins and then the notes on the price tag
for each item pictured. Often money is written as a decimal with the ringgit to
the left of the decimal point and the sen to the right of the decimal point. For
example, five ringgit and eighty-seven sen is written as RM5.87.
Money amounts are added the same way as decimals are added. Remember to
put the RM sign before the answer. Similarly, money amounts are subtracted the
same way as decimals are subtracted. Again, remember to put the RM sign
before the answer. In short, subtracting and adding money are just like
subtracting and adding other decimal numbers. Always line up the decimal
points when subtracting and adding decimals.
Let children practise adding and subtracting money in the following way:
„Let us pretend you have ten ringgit to buy vegetables. Practise buying items
from a grocery store and making change using decimal numbers. Use your skills
with decimals to find the answers to these questions. Remember to put the
decimal point in the proper place in your answer‰.
Grocery List
Grape juice RM 2.00
Pickles RM 1.80
Banana 85 sen
Did you know?
The word, money, comes from the Latin word,
ÂMonetaÊ. ÂMonetaÊ was the name of the place
in ancient Rome where money was made.
Activity 6: Shopping with Money
Learning Outcomes:
(a) Have hands-on experience of buying and selling with money within RM10;
(b) Add and subtract money appropriately;
(c) Solve word problems involving the addition and subtraction of money
within RM10; and

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(d) Create word problems involving the addition and subtraction of money
within RM10.
Materials:
Jotter book;
Shopping Cards;
Play Money;
BuyersÊ Worksheet; and
SellersÊ Worksheet.
Procedure:
(a) Each pupil is to prepare his play money worth of RM5 a day before.
(b) Ask pupils to recall the algorithms for the addition and subtraction of
money.
(c) Emphasise the importance of the ÂRMÊ sign, the decimal pointÂ.Ê, and the
alignment of the signs and numbers.
(d) Split the class into two groups and start briefing the pupils on the activity.
(e) Pupils work in pairs. Assign them as ÂSellersÊ or ÂBuyersÊ. Each pair of
ÂSellersÊ and ÂBuyersÊ can only trade within their group.
(f) Each pair of pupils is to combine their play money to total up to RM10.
(g) First, brief the ÂBuyersÊ on their roles.
(i) They will go on a shopping spree to buy three items using their
money. Once they buy an item, they get to keep the item card.
(ii) The items bought and the amounts spent have to be recorded on their
worksheet.
(iii) At the end of the shopping spree, they have to total up their
expenditure and find out how much they have left.
(iv) Then, they have to count their play money to tally with their
worksheet.
(h) Next, brief the ÂSellersÊ on their roles.
(i) Each pair of pupils is to display their goods along with the price tags
(matching colour).

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(ii) They will sell their goods to whichever ÂBuyersÊ who are interested.
When the item is sold, the price tag is turned over.
(iii) After each sale, they have to record the item sold and the amount
spent on their worksheet.
(iv) At the end of the shopping spree, they have to total up their earnings
with the RM10 they have at the start of the activity to find the amount
of money in their possession.
(v) Then, they have to count their play money to tally with their
worksheet.
(i) Tell pupils that they may need to use their jotter books to do the working
when calculating the change.
(j) Give out the worksheets to the respective pairs.
(k) Distribute the shopping cards to the ÂSellersÊ.
(l) Carry out the activity.
(m) Collect the worksheets and shopping cards.
Activity 7: Money in the Bank
Learning Outcome:
(a) Use coins to practise their addition and subtraction skills involving money.
Materials:
Small container with a lid;
Basic art supplies (scissors, construction paper and crayons);
A pair of dice; and
12 ten sen per pupil.
Procedure:
(i) Teacher holds up some coins and asks pupils to identify them. Review the
monetary value of each combination of coins.
(ii) Ask pupils to make their own piggy banks out of the small containers. Give
one to each pupil and let them decorate their piggy bank with the art
supplies.

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(iii) Put three ten sen inside each pupil's bank and place 9 more outside their
banks. Ask them to determine the value of their coins.
(iv) Roll the dice to determine how much money they will have in the bank.
Pupils will then put in or take out the appropriate number of sen from their
bank. (eg. Roll a six and they will put in three more sen). Ask them to count
the number of remaining sen outside the bank.
(v) Conduct several trials. Allow them to take turns to roll the dice.
(vi) When they are done, collect all the sen and allow your pupils to take home
their banks.
Assessment:
Observe pupils as they place the appropriate amount of money in their
banks.
7.2.1 Adding Coins and Notes (Worksheet)
What can you buy with the money in the box? Use the following worksheet (see
Table 7.4) and ask your pupils to add up the sums of money on the left before
matching the correct amount with the price of the items on the right.
Table 7.4: Matching Activities
RM 7, 10 sen, 20 sen, 20 sen
(40 sen)
RM 1, RM 2, RM 5, 10 sen, 20 sen
(RM 8.30)

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5 sen, 10 sen, 10 sen
(RM 7.50)
(25 sen)
7.2.2 Subtracting Coins and Notes
The following worksheet (see Table 7.5), can be used to teach the skill of
subtracting coins and notes.
Table 7.5: Subtracting Coins
I Have I Buy Money Left
45 sen
_______ sen
RM 5, 50 sen
RM 3.50 sen
RM _______

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RM 1, RM 5
RM 2
RM _______
7.2.3 Finding Balance within RM10.00
The following worksheets provide practice for finding the change or balance
within RM10.00.
Write down the amount of change you will get back in Table 7.6.
Table 7.6: Working Out the Change
Item I Gave My Change
85 sen
50 sen, 20 sen, 20 sen _______ sen
RM 3.50
RM 5 RM _______

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RM 1.20
RM 1.50 sen _______ sen
Name _____________________________ Date ___________________
Buying Food and Getting Change
Show your pupils the following food with their respective prices.
40 sen 5 sen 35 sen
40 sen 5 sen
Draw an X on the change received after buying the items pictured, see Table
7.7.

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Table 7.7: Getting the Change
Purchase These Items Pay Change Got Back
1.
100 sen
2.
25 sen
3.
50 sen
4.
25 sen
5.
75 sen
USING MONEY (SPENDING MONEY AND
CONSUMER MATHEMATICS LESSONS)
7.3
Allow pupils to learn and practise their money spending skills by using various
worksheets, lesson plans, lessons, activities and exercises on spending money.
Spending money is an interesting way to reinforce pupilsÊ basic mathematical
skills such as addition, subtraction, multiplication, division and other skills
including reading comprehension. Therefore, it is important to let children learn
practical consumer mathematical skills including buying and bartering for goods
or services.
Children are not born with Âmoney senseÊ. They learn by what they see, hear and
experience and parents have a very strong influence on all of these. Childhood is
the appropriate time to learn about money management, when parents are able

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to provide them with learning experiences that will benefit them in years to
come.
Family councils are an excellent way to help children learn how to manage
money. This can be done by helping them understand what money means, how
to make wise and satisfying choices, how to use money to get things important to
them and how to have money on hand for daily needs as well as for emergencies
and future needs.
Children need to have money of their own to learn how to manage it. An
allowance is a better teaching method than simply giving children money upon
their request. An allowance for children should be a set amount, paid out
regularly and not tied to regular tasks required of the child. When deciding on
the amount of an allowance, discuss what items would be covered. The amount
should be large enough so that the child has money to manage with no strings
attached.
Money should not be used as a means to discipline, such as an incentive for good
grades or as a reward for doing household tasks. If money is used in this manner,
a child will get the idea that everyone and everything has a price tag. In addition,
money should not be used to buy love or as a substitute for companionship.
Suggestions: Using Your Own Money
(a) What do you spend your money on?
(b) Keep an account book for a week to find out what you spent on.
(c) How much do you save each week?
(d) Where do you put your savings?
(e) What are you saving your money for?
Date Money In Date Money Out
Record
how much
you receive
Record
what you buy,
what it costs

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A dash of humour will certainly make your lessons more interesting. Some jokes
on money are given below:
Jokes
Where do you find money?
In the dictionary under M
Where do fish put their money?
In the riverbank
Which is the richest insect in the world?
The centipede
7.3.1 Changing Money
Changing money is a skill that needs to be reinforced amongst children. Let us
take a look at the following scenario/situation:
Nina and her friends save money for charity.
Nina has 132 sen.
The bank changes them for her.
She gets: 1 one ringgit, 2 one sen and 3 ten sen
We write: RM1.32
Ringgit sen
We put a decimal point between the ringgit and the sen when writing out the
value for the amount of money saved.
7.3.2 Lessons on Counting and Making Change
One of the more difficult but basic money skills is understanding how to make
change and knowing how much change you will receive when spending money
and purchasing an item or items. Use these worksheets and lessons to help your
pupils learn how to make change. Practice is available with coins and notes.
Learn to make change for a ringgit. Teach by creating your own money
Worksheets and Interactive lessons.

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Please note that making change requires more advanced money skills. Prior to
using these lessons, pupils should have mastered other more basic money skills,
including identifying coins and bills in the main Âcounting money lessonsÊ
category.
Enrichment Activity:
(a) Fund Raiser
Allow the pupils to decide on how to raise money and the cause.
Discuss: Where will the money go to?
(Perhaps having a charity function or a class trip, for example.)
(b) Discuss how pupils earn money. How many people receive an allowance?
Create a class graph on the amounts of allowance received. Analyse the
results. What do pupils do with their allowance?
(c) Distribute a Worksheet.
Pupils should be familiar with money right from preschool days. They
should be taught to recognise and know the values of various coins and notes
of the Malaysian currency.
Classroom experience should aim at helping pupils clear their misconceptions
in order to formalise their daily experiences involving money.
Activities like trading games including buying and selling are effective in
helping pupils to understand the concept of money.
Various active learning experiences such as games and hands-on activities with
manipulation are important to help pupils consolidate their understanding of
money.
Allowance
Budget
Coins
Currency
Earn
Purchase
Savings
Spend

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Try to find how money became a historical document of society when it was first
issued. Have a class discussion with your tutor.
Good nation development comes from good financial management and people
who are highly-skilled. How can this kind of people be developed in the
classroom for the future?

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APPENDIX
WORKSHEET (ENRICHMENT EXERCISE)
Money in Hand Things Bought Balance

Topic
8
Teaching the
Measurement
of Time
LEARNING OUTCOMES
1. Explain the historical development of measuring time and the
calendar;
2. Explain the major mathematical skills for teaching the measurement
of time;
3. Recognise the pedagogical content knowledge for teaching the
measurement of time;
4. Identify four difficulties in teaching the measurement of time; and
5. Plan teaching and learning activities for the measurement of time.
INTRODUCTION
This topic will give you some ideas about teaching the measurement of time to
young children. Besides basic comprehension skills in telling time, this topic also
covers the history of telling time as well as the strategy of teaching and learning
the measurement of time. Specifically, the major mathematical skills related to
the measurement of time are as stated below:
(a) Tell the time and events of the day;
(b) Name the days of the week;
(c) Name the months of the year; and
(d) Read and write the time.

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160
You will find that by following this topic with its step by step approach, it will be
easy to learn the ideas of teaching the measurement of time especially to
kindergarten or pre-school pupils. There are many examples of teaching
activities in this topic. They have been designed for application in the classroom
using concrete materials through practical methods including the inquiry-discovery
method, demonstration, simulation, etc. The inquiry-discovery method
covers activities such as planning, investigating, analysing and discovering. The
activities in this chapter will guide you as to how to create a good and conducive
classroom environment in order to teach the topic of ÂTimeÊ more efficiently. A
discussion of the history of measuring time, the concept of events of a day, telling
time, the days of the week, the months of the year and the calendar, etc. is
included. In addition to hands-on materials such as the wall clock, models of
clocks and the calendar, you are also encouraged to find some other materials
like online calendars or tasks from websites to make your teaching more
interesting, meaningful and enjoyable. There are many types of tasks which are
suitable as consolidation activities such as games, songs, etc. and samples of
worksheets are provided as well. Completing this topic will make you more
confident in teaching the measurement of time.
HISTORY OF MEASUREMENT OF TIME
8.1
In this section, we will further discuss the historical development of measuring
time and calendar in further detail.
8.1.1 Historical Development of Measuring Time
Prehistoric man came up with a very primitive method of measuring time by
simple observation of the stars, changes of the seasons, plus day and night. It was
necessary for them to plan their nomadic activity, farming, sacred feasts, etc.
Before clocks and watches existed, the earliest measurements of time were made
using the sundial (see Figure 8.1), the hourglass, the sand clock, the wax clock
and the water clock.
In early times, the forerunners to the sundial were poles and sticks, as well as,
larger objects such as pyramids and other tall structures. Later, the more formal
sundial was invented. It was generally a round disk marked with the hours like a
clock. However, the ancient Egyptian sundial clock (3100 B.C.) shape was quite
different from the Chinese sundial (1100 B.C.).

TOPIC 8 TEACHING THE MEASUREMENT OF TIME 161
Figure 8.1: The sundial
Source: Mok (2005)
The hourglass used in England 1200 years ago was made up of two rounded
glass bulbs connected by a narrow neck of glass between them. When the
hourglass was turned upside down, a measured amount of sand particles
streamed through from the top to the bottom bulb of the glass (see Figure 8.3).
Figure 8.2: The sand clock
Source: Mok (2005)
Another ancient tool for measuring time was the water clock or the clepsydra
shown in Figure 8.3, found during the Roman civilisation (200BC). It was an
evenly marked container with a spout in which water dripped out. As the water
dripped out of the container, one could note what time it was by looking at the
water level against the markings.

162
Figure 8.3: The clepsydra
Source: Mok (2005)
The more advanced clocks known as mechanical clocks, which used weights or
springs, came into existence in the 1300s. At first, they had no faces and no hour
or minute hands, rather they struck a bell every hour. Later, clocks with the hour,
and then the minute hands, began to appear. In the 1400s, clocks with their hands
controlled by a coiled spring were made. With this discovery, smaller clocks and
later watches were made.
In 1656, Christian Huygens invented clocks which used weights and a swinging
pendulum, known as the pendulum clock. These clocks were much more
accurate than previous clocks. Then, in 1761, John Harrison finally succeeded at
inventing a small clock accurate enough to be used for navigation at sea. This
tiny pocket watch lasted only five to six weeks. In the early 1800s, Eli Terry
developed machines, patterns, and techniques that produced clock parts that
were exactly alike. This drove the price of clocks way down low and allowed
common people to own at least one time-keeping device.
8.1.2 Historical Development of the Calendar
The oldest calendar in history was designed by the Egyptians around 4000BC. It
had only 360 days, based on observation of the movement of the sun. It was later
modified and improved by the Romans. The Roman calendar (46BC) contained
365 days in a year with one day added for February every four years. Each
month contained 30 or 31 days except for February, with 28 days in a normal
year and 29 days for leap years.

People of China and Arabia also devised their own calendars which were based
on the movement of the moon. Both calendars were divided into 12 months with
every month containing either 29 or 30 days. The Arabic calendar or Islamic
calendar (i.e. Taqwim Hijrah begins with the month of Muharram as the first
month and ends with the 12th month, Zulhijjah).
In 1852, Pope Gregory XIII redesigned the Roman calendar to become a new
calendar known as the Gregorian calendar. One year was divided into 52 weeks.
Each week contained seven days beginning from Sunday and ending on
Saturday. There are 12 months in one year starting from January and ending in
December. This calendar is the one that is most popular and is generally accepted
as the official calendar throughout the world today. Figure 8.4 explains the
chronological development of calendars starting from Ancient Egypt in 3100BC
until Europe in 1582AD.
Figure 8.4: Chronological development of calendars
Source: Mok (2005)
TEACHING THE MEASUREMENT OF TIME
8.2
This section will further discuss the teaching measurement of time including time
of the day, telling time, time duration, days of the week, months of the year and,
finally, the difficulties in teaching the measurement of time.
8.2.1 Time of the Day
Children start learning about time by telling the time of the day i.e. day time and
night time. This can be done by relating various phrases denoting time into their
daily routines.

164
MORNING I wake up in the morning.
Norli goes to school in the morning.
NOON It is noon, I am in school.
Norli learns mathematics in school at noon.
AFTERNOON I have my lunch in the afternoon.
Norli goes home in the afternoon.
EVENING I am playing football in the evening.
Norli goes to the garden in the evening.
NIGHT I do my homework at night.
Norli watches television at night.
MIDNIGHT We will be fast asleep at midnight.
To reinforce the usage of the correct time phrases mentioned above, take some
pictures and stick them on the board in sequence, starting from day time to night
time. You are encouraged to use appropriate pictures to illustrate the events
happening at that time according to the time given (see Figure 8.5).
Figure 8.5: Different times of a day

Guide your pupils to carry out a discussion according to the events in the
pictures shown. When introducing them to telling the time of the day, include an
appropriate analogue clock although they have not been taught how to tell the
time yet. Figure 8.5 shows a sample of a teaching-learning material for teaching
the skill of telling the time of a day to young children.
SELF-CHECK 8.1
List some activities or events of the day that children always do. Explain
how to conduct an effective lesson on teaching your pupils how to tell
time.
8.2.2 Telling Time
How do we start to teach kindergarten or young children to tell the time? Firstly,
let them look back at the pictures used for teaching them how to tell the time of
the day used in the previous lesson. Then teach them how to say the time shown
on the clock face given in the pictures. Since they are able to count from 1 to 12,
children should have no difficulty telling the time although they are usually not
used to telling the time yet at this stage.
Next, introduce the minute hand and hour hand on a clock face, see Figure 8.6.
Figure 8.6: Clock face: Minute and hour hands
Have them count the minutes on the clock in 5s and show them that every time
the minute hand goes one complete round from the number Â12Ê to Â12Ê, the hour
hand moves on to the next number. Then point out that every time the hour hand
moves to another number, the minute hand is on the number 12. When the

166
minute hand is on the number 12, it is called o'clock (of the clock) and we read
the time as the number that the hour hand is pointing to, such as, 7 oÊclock, for
example. They build their understanding of the measurement of time by
repeated reference to the clock, using the position of the hands for hours and
minutes as shown in Figure 8.7.
Figure 8.7: Telling the time
In addition, you may put in the digital time together with the analogue time in
your teaching material (i.e. using appropriate picture cards); see Figure 8.8.
Relate how to tell the time with a specific Âtime of the dayÊ, for example, 4 oÊclock
in the evening, see Figure 8.6.
Figure 8.8: Timeline
Figure 8.9: Picture card with clock face

Teaching the measurement of time requires repetitive hands-on experimentation.
There are many types or multiple clock face manipulative for use in the classroom.
The better ones have the hands geared together so that rotating the minute hand
one full revolution causes the hour hand to move from one hour to the next.
Children should answer questions about the time indicated on the clock, and
record their responses, starting with the hour values. They should also be taught
how to set the clocks according to the time given by turning the hands of the clock.
Discuss different timepieces (e.g. clock, watch, timer, hourglass. Guide them to
recognise the types of timepieces they have at home. Let children design a chart
that displays their findings. Then, show some of the earliest instruments for
measuring time using a PowerPoint presentation on the sundial, candle clock,
sand clock, etc.)
SELF-CHECK 8.2
Teaching time using a circular clock face with the hour and minute
hands will make it easier for children to learn the measurement of time.
Discuss.
8.2.3 Time Duration
Figure 8.10: Time piece the hourglass
Have you seen the above instrument before? Can you think of its connection to
this topic?
Time duration is a difficult concept to teach because the circumstances vary so
much from situation to situation. There are, however, several aspects of children's
lives at school and at home in which elapsed time is important. Initially, you do
not need to state the duration of time in minutes or seconds specifically. At this
time, just let them know about the passing of time and compare among elapsed

168
times for short term events and longer timed events (in which case, you may teach
them to state the period of time in hours). Events in daily life that help children to
understand the concept of time duration include the following:
(a) Elapsed time for:
(i) Eating (e.g. Fried rice, pizza, doughnut);
(ii) A movie/video/television show;
(iii) A football game (or other games);
(iv) Running around the field (and other distances);
(v) A nap; and
(vi) Various classes at school.
(b) Longer time for:
(i) A baby to be born;
(ii) A chick to be hatched; and
(iii) Bean plants to grow a metre high.
Can you give some more appropriate examples of both types of time duration?
Ask your children to estimate the time duration or period of time in hours.
8.2.4 Days of the Week
ACTIVITY 8.1
Find attractive online calendars from any website which are suitable to be
used when introducing this section in class. Do a slide presentation and
discuss with your classmates.
The days of the week starts with Sunday, Monday, right till Saturday. You are
encouraged to introduce the days of the week using a calendar. For more
interesting presentations, click on any attractive calendars on the Web e.g.
Calendar for the year 2012 (United States), Utusan Malaysia Online, etc.

Then, have the children name the days of the week on the calendar and sing a Â7
days of the weekÊ song. The teacher can make large name cards for the days of
the week and have children hold them up when they recite the days of the week.
The children can also parade with the cards along with the music and sing the
ÂDays of the weekÊ songs too (See Figure 8.11).
Figure 8.11: Example of cards sing along with the music
Source: Hummingbird Education Resource
The important vocabulary that you have to use in teaching the days of the week
include these words:
The following examples show how we use these words in daily life situations.

170
(a) In statement sentences as below:
(b) In question form such as:
(i) Today is Thursday. What day was yesterday?
Answer: Yesterday was Wednesday.
(ii) Today is Wednesday. What day is tomorrow?
Answer: Tomorrow is Thursday.
(iii) What day comes after Friday?
Answer: Saturday comes after Friday.
(iv) What day comes before Saturday?
Answer: Friday comes before Saturday.
SELF-CHECK 8.3
Produce a suitable worksheet that can be used for teaching the days of
the week.

8.2.5 Months of the Year
Figure 8.12: Example of a calendar
Source: http://manhunt.wikia.com
Find a suitable online calendar which can be used when introducing this section
in the class, see Figure 8.12. Make use of slide presentations.
There are 12 months in a year i.e. January, February, right till December. Since
children are able to count from 1 to 12, there should be no difficulties in
arranging all the months in sequence. However, they may face some difficulties
in the spelling of the words.
You may start your lesson by asking your children about the special days
occurring in each month of the year. Let us look at some festive celebrations in
Malaysia.

172
Some other appropriate celebrations include MotherÊs Day, FatherÊs Day and
ChildrenÊs Day. Find out which months they fall in.
Singing varieties of songs of the Months of the Year will help children to master
the learning of the months in a year easier.
For example, do the actions for the song „Macarena‰ as you sing or chant the
months in a year. Forming a square, repeat the song four times, see Figure 8.12.
This is definitely a class favourite!
Figure 8.13: Songs lyrics
Source: Hummingbird Education Resource
Generally, pupils are able to arrange all or most of the months in sequence from
January to December, but they may also have difficulties in arranging some of
the months. Thus, have them arrange a few months in sequence first and increase
gradually or progressively as an exercise like the one described.

Example:
1. Arrange the months in sequence:
2. Then, by using the questioning technique, ask them some questions as
follows:
(a) What is the month after March? September? July?
(b) What is the month before March? September? July?
(c) May comes before _____________.
(d) October comes after _____________, and so on.
(e) How many days are there in January? March? October?
(f) Which month/s contain/s 30 days? 31 days? 28 days? 29 days?
3. Guide pupils to read and write the months using word cards, for example:
4. Finally, ask pupils to state their birthday (or any other celebrations day)
and show the day on the calendar. Get them to recognise the date today!
5. Distribute Worksheet 1.
8.2.6 Difficulties in Teaching the Measurement of
Time
You ought to know some aspects about the measurement of time which make it
difficult to be learnt amongst young children. This is because:
(a) Time is an abstract concept;
(b) Time is measured using a mixture of non-decimal systems such as base 12
and base 60 systems and when extended to include days, months and years,
base 4, 7, 365 and 28, 29, 30 as well as 31 systems are involved;
(c) Time is measured indirectly by the movement of the sun, the hands on a
clock face, the changing of digits in a display, the changing seasons, etc.

174
(d) Clocks come in all sorts of styles and designs some with all 12 numerals
shown, (some use Roman numerals), others with only 12, 3, 6 and 9
numerals shown, and still others with no numerals at all on the clock face.
ACTIVITY 8.2
Search any suitable website to find various games which are suitable to be
used in the teaching and learning of the measurement of time in the
classroom. Discuss your findings with your classmates.
SAMPLE LESSONS OR ACTIVITIES FOR
TEACHING THE MEASUREMENT OF TIME
8.3
In this section, we will discuss further the sample lessons or activities for
teaching the measurement of time.
Lesson 1: Telling Time in Hours
Vocabulary:
Hour, o'clock, minute hand, hour hand
Learning Outcomes:
(a) Recognise the elements of a clock and explain their functions; and
(b) Tell time to the hour.
Materials:
TeacherÊs transparency of a demo clock;
PupilsÊ individual clocks (cardboard/paper plate clock faces) with arrow
cards;
Numeral cards (1 12);
Time index cards (1 oÊclock 12 oÊclock); and
PowerPoint presentations on the earliest instruments for measuring time.

Procedure:
(i) Set Induction
Discuss different timepieces (e.g. clock, watch, timer, hourglass). Guide
them to recognise the types of timepieces they have at home. Then show
them various types of the earliest instruments for measuring time using a
PowerPoint presentation e.g. sundial, candle clock, sand clock, etc. Use
suitable websites. Tell them how prehistoric man measured time.
(ii) Show the Big Demo Clock
Ask pupils how many big numbers are on the clock? Have pupils point to
the hour hand. Tell them that when the hour hand moves from one number
to the next, one hour has passed. Get pupils to discuss what they can do in
an hour.
(iii) Have Pupils Point to the Minute Hand
Tell them that when the minute hand moves from one tick mark to the
next, one minute has passed. Ask them what they can do in a minute.
(iv) Review
Review that the minute (long-blue) hand points to the 12, while the hour
(short-red) hand indicates the hour (i.e. 1, 4, 8, etc). Guide them to read the
time for each time shown in the o'clock form. Later, let them write the
time in words and symbols.
e.g. „The minute hand points to 12‰. „The hour hand points to 1‰.
„It is 1 oÊclock (or one oÊclock)‰.
(v) Team or Group Work
Divide the classroom into teams. Have each group of pupils make a
paper plate clock face. Using a brass paper fastener, attach a tag board
or construction paper hands to the centre of the plate. These clocks can
then be used in various reinforcement activities. For example, as the
teacher calls out a specific time, the pupils show the correct time on
their clocks (adapted to a team game).

176
Give each pupil a worksheet (clock faces without the hour and minute
hands) and let them draw in the minute and hour hands to show the
correct time. Can you suggest some exercises to be included in the
worksheet? Jot down your questions and answers.
(vi) Simulation Activity
Write times to the hour from 1 o'clock to 12 oÊclock on index cards and a
number from 1 to 12 on a tag-board square. Place the numbers 1 to 12 in a
large circle to form a clock-face. Children sit around the clock formed. Give
12 children a time card each to be kept facedown. Two volunteers are
needed to stand in the centre of the clock to be the hour hand (hold a long-blue
arrow card) and the minute hand (hold short-red arrow card)
respectively.
(vii) Ask Who Wants to be the Minute Hand and the Hour Hand
Show 1 oÊclock. Where should the minute hand point to? Where
should the hour hand point to? Both ÂminuteÊ and ÂhourÊ hand pupils
have to point to the correct number on the tag-board. Check the
answer!
Repeat the activity until all the children have a turn to show the time.
(viii) Give out the second worksheet.
(ix) Closure
Have an open discussion on the importance of spending time effectively in
their daily life.
Lesson 2: Concept of Time and Period: Day Time and Night Time
Vocabulary:
Morning, afternoon, noon, evening, night, midnight.
Learning Outcomes:
(a) Name the various parts of the day; and
(b) Recognise the various parts of the day based on different daily activities.

Materials:
PowerPoint slides (pictures related to daily activities);
Picture cards (activities of a day);
Flash cards (written ÂmorningÊ, ÂafternoonÊ, etc.); and
Clock.
Procedure:
(i) Set induction:
Open discussion on what pupils do as a routine everyday during the day
time and night time (e.g. in the morning, afternoon, noon, evening and
night).
(ii) Teacher shows a series of pictures related to daily activities using
PowerPoint slides. Guide pupils to state what they see in the pictures, for
example: Waking up in the morning, going to school, Âplaying footballÊ,
etc.)
(iii) Place a series of pictures related to daily activities in sequence.
(iv) By referring to the pictures shown, introduce words such as morning,
afternoon, noon, evening, night and midnight, see Figure 8.14. Guide
pupils to place the flash cards just below the relevant picture.
Figure 8.14: Word cards
Guide pupils to read clearly all the events of a day by referring to the
pictures given.
(v) Guide the pupils to tell their own daily activities.
What do you do in the morning? In the afternoon? At night?
Can you tell the time? (Just let them think about the time).

178
(vi) Group activity:
Teacher gives a different set of daily activities to each group. Ask them to
arrange the events in sequence. Let them present and read the sentences
related to the pictures.
(vii) Let them do some exercises in a worksheet.
(viii) Closure:
Open discussion on the disadvantages of wasting time for everyone.
Lesson 3: Calculate the Duration of Time from a Calendar
Vocabulary:
Calendar
Learning Outcomes:
(a) Read information from the calendar given; and
(b) Calculate the numbers of days in a week, the numbers of months in a year,
and the numbers of days in a year.
Materials:
Calendar;
Activity cards;
Flash cards (day cards);
PowerPoint slides; and
Worksheet.
Procedure:
(i) Set induction:
Show a calendar in the slides. Have them look at the calendar. Referring to
the calendar, begin by asking questions:
What do you see in the calendar?
What is the purpose of a calendar?
Do you have any calendars at home? What types of calendars do you
have?

(ii) Lead a discussion and guide pupils to tell the information in the calendar
(e.g. Show a calendar for the month of January 2012). Introduce the concept
of ÂweekÊ and ÂmonthÊ.
A week starts from Sunday to Saturday. How many days are there in a
week?
How many weeks are there in a month? How many days are there in
the month of January?
(iii) Show the complete calendar for the year 2012. Give them a piece of the
calendar (or photocopy of the calendar in A4 paper) each.
How many months are there in a year?
Can you count the number of days in the year 2012? Tell me how.
(iv) Group activity (groups of four five persons):
Give them the seven-day name cards (Sunday, Monday, , Saturday). Ask
them to arrange the days of a week using the day cards in sequence.
How many days are there in a week? (Seven days).
How many days are there from Sunday to Wednesday?
How many days are there from Wednesday to Friday?
Guide them to count the number of days using their day cards.
There are four days from Sunday to Wednesday.
There are three days from Wednesday to Friday.
(v) Repeat with other sequence of days in a week. Let them record their
answers in Table 8.1.

180
Table 8.1: Number of Days
From To Number of Days
Sunday Wednesday
Monday Thursday
Wednesday Friday
Thursday Saturday
Tuesday Saturday
(vi) Have a quiz among the groups to ensure they have mastered the skills that
they learnt today.
(vii) Distribute a worksheet as homework. Have them complete their own
calendar.
Example:
(viii) Closure:
Lead a discussion about the importance of calendars in our daily life.
Conceptual understanding on the meanings of the measurement of time is
very important because it will help pupils to learn this topic more efficiently
and meaningfully.
Pupils may easily learn to tell the time by reading the numerals on the clock,
tell the days of a week on the calendar, and so on, but the important thing is
how they will understand and use the concept of the measurement of time in
their daily life.
You need to pay attention to these aspects when you are planning the lessons.

Gregorian Calendar
Hour Hand
Minute Hand
OÊclock
1. List down the pupilsÊ prior knowledge for learning the measurement of
time.
2. List down two difficulties in teaching the measurement of time.
You are asked to plan a lesson according to the learning outcomes below:
(a) Recognise the days of a week
(b) Read and write the days of the week
Write your lesson plan. (You have to focus on the set induction, development
and consolidation aspects).

182
APPENDIX
WORKSHEET
1. Fill in the blanks with the correct days of the week.
2. Write the correct answer in each box.
(a) Tomorrow is Friday. Today is
(b) The second month of the year is
(c) Yesterday was Saturday. Today is
(d) The month that comes before May is

(e) The month that comes after December is
(f) The fifth month of the year is
(g)
is the month of my birthday.

Topic
9
Three
Dimensional
Shapes
(3D Solids)
LEARNING OUTCOMES
1. Discuss the importance of geometry in solving daily life problems;
2. Explain why the teaching of geometry should be introduced at
primary school level;
3. List down childrenÊs levels of understanding and learning geometry;
and
4. Plan instructional strategies and teaching-learning activities
pertaining to geometry for kindergarten or pre-school and early
primary school children.
INTRODUCTION
We live in the world of three dimensional (3D) shapes or solids. Everything
around us is in the form of solids such as the house we live in, the garden, the
trees, the cars, the fruits and the furniture we use. The round shape of an apple
we consume, the cylindrical shape of a pencil we use to write with and the
cubical shape of the thick books that we read are all examples of 3D shapes
around us. Most of the objects around us are three dimensional solids and occur
as either regular or irregular solids.

TOPIC 9 THREE DIMENSIONAL SHAPES (3D SOLIDS)
185
SHAPES OF THE WORLD
9.1
Our world is made up of three dimensional (3D) shapes or solids with
dimensions of length, breadth (width) and thickness (depth). Solids have either
flat or curved surfaces. Cornflakes containers, classrooms, tables, cupboards and
match boxes of cubical or cuboidal shapes are examples of solids with flat
surfaces. Other 3D solids of spherical shapes with curved surfaces include globes,
tennis balls, footballs, cones and cylinders whilst examples of 3D solids of oval
shape are eggs or rugby balls.
We have to understand more about shapes around our world. Shapes and figures
change when looked at from different perspectives. As newborns open their eyes,
they would either be looking directly at their mothersÊ faces as a plane or two-dimensional
(2D) shape or be directly exposed to 3D solids from the front view.
Slowly as they grow older, children will develop more advanced geometric
thinking and better understand the concept of geometry dealing with solids,
shapes and space applicable in the world they live in.
SELF-CHECK 9.1
How do architects, engineers or designers interpret the graphic
drawings of 2D shapes in 3D models of houses, apartments, cars,
aeroplanes, ships and tankers? Discuss.
WHAT IS GEOMETRY?
9.2
Geometry is a branch of Mathematics. It is the study of angles and shapes formed
by the relationship of lines, surfaces and solids in space as defined in LongmanÊs
Dictionary.
Geometry is the exploration or investigation of space or discovery of patterns
and the relationship of shape, size and position or place in space. These are
observed in and derived from the immediate environment and the much wider
world, both natural and Âman-madeÊ.
The teaching of geometry is the development of experiences, skills and processes
for children to enable them to operate and understand their world or
environment better. It is thus essential that children learn about geometry and its
wide applications in real life well so as to be better equipped for the future.

186
ACTIVITY 9.1
Think of some examples of real-life applications of geometry. Discuss the
importance of geometry in real life.
THE TEACHING OF GEOMETRY
9.3
The purpose of teaching geometry in primary and secondary schools is to help
children acquire knowledge, provide basic concepts of geometry and critical
geometric thinking that will improve the childrenÊs ability to manipulate their 3D
environment. Since geometry is a branch of mathematics, it should be integrated
within the Mathematics syllabus for KBSR and KBSM. The teaching of geometry
should be done early at pre-school or primary level to be continued to secondary
school and higher level education.
The following are various reasons why geometry should be taught in schools:
(a) Problem solving, the ability to solve daily life problems, is an important
skill to be mastered by all children. Learning mathematics and geometry
will prepare children to solve problems they face or are confronted with
everyday in real life as stated by Tom Cooper (1986).
(b) Solving geometry problems involve the manipulation of shapes and visual
imagery within a geometric framework. A strong foundation in Geometry
is thus necessary.
(c) Learning about geometry and its applications to real life provides the basic
knowledge and geometric understanding vital for application in future
careers especially in the technical and vocational areas. Understanding
geometry is essential in the fields of navigation and exploration. Geometry
comprises important elements or essential knowledge for astronauts, pilots,
sea navigators, architects, engineers, mathematicians, carpenters, interior
decorators, models and fashion designers.
ACTIVITY 9.2
How do children learn geometry? Why should children learn geometry in
stages? Explain.

187
THE TEACHING AND LEARNING OF
GEOMETRY
9.4
When preparing experimental tasks or planning learning activities for young
children, teachers have to take special consideration of the childÊs intellectual
development as a frame of references.
According to Jean PiagetÊs Model (1964-1967), a childÊs experience, living
environment and biological maturation will all influence his or her development
of geometric thinking and understanding, which are:
(a) Level of development of mathematical abilities and understanding of
geometric concept; and
(b) Level of conceptual development within geometry, and perception of
geometric properties and relationships within oneÊs environment.
Children should thus be learning the concepts in stages.
9.4.1 The Learning of Geometry
Integration of Jean PiagetÊs research, Van Hiele's Model and other research
findings will be the basis for designing instructional tasks and learning
experiences for young children. There is no one universal theory in designing the
teaching strategies or learning activities of geometry. Thomas Fox (2000)
suggested that instructional tasks should be in line with the childrenÊs ability or
their level of reasoning. Hannibal (1999) suggested the importance of language,
vocabulary and description in helping childrenÊs development of defining and
categorising features of shapes.
9.4.2 Van Hiele’s Model of Learning Geometry
Pierre van Hiele and Dina van Hiele-Degolf are two Dutch educators who
provided guidance in designing the instruction and curriculum of geometry. The
Van HielesÊ work which began in 1959 attracted a lot of attention especially in the
Soviet Union (Hoffer, 1983).
Today, the Van Hiele theory has become the most influential factor in designing
the curriculum for geometry worldwide. Pierre and Dina van Hiele believed that
children should learn geometry in five levels or stages. Learning activities must
be in progressive stages and avoid gaps resulting in confusion. Hypothesis

188
showed that when children miss certain stages of these experiences, they would
face obstructions in their progress in understanding geometric concepts.
According to research by the Van Hieles, teachers or educators must provide
children in elementary or primary school with at least the first three stages in the
process of learning geometry. Instruction and learning activities should be well
planned according to the pupilsÊ level of geometric thought. Pupils will
successfully learn geometry when they are given the opportunity of a good
learning environment and the right experiences according to Andria Troutman et
al. (2003).
Van HieleÊs five levels of geometric thought are explained in Table 9.1:
Table 9.1: Van HieleÊs Five Levels of Geometric Thought
Level Description
Level One
Visualisation
This is the basic level where children recognise figures by looking at
their appearance. They are able to identify the shapes of two
dimensions or three dimensions through visualisation. Their ability
to identify shapes is basic depending on the sense of sight or feeling
without understanding the geometric properties of each figure.
Level Two
Analysis
At this level, children are able to classify or group depending on the
characteristics of shapes or figures but they cannot visualise the
interrelationship between them.
Level Three
Informal
Deduction
After undergoing the first two levels, visualisation and analysis,
children are able to establish or see interrelationships between
figures. They are able to derive relationships among figures
followed by simple proofs but not with complete understanding.
Level Four
Deduction
PupilsÊ mental thinking and geometric thinking develop
significantly. They can understand the significance of deduction, the
role of postulates, theorems and proofs. They are able to write
proofs with understanding.
Level Five
Rigour
Pupils are now able to make abstract deductions and understand
how to work in axiomatic systems and even non-Euclidean
geometry can be understood at this level.

189
TEACHING STRATEGIES OF GEOMETRY
Although we live in a three dimensional (3D) environment, learning about
geometry can be very difficult and confusing to young children. This topic will
provide kindergarten or preschool and year one primary school teachers the
overview of teaching geometry to children four to seven years of age.
Since there is no universal theory for the teaching of geometry, this topic will
guide and expose teachers to a few research findings and models as a knowledge
base in designing instructional teaching experiences and tasks for different age
groups.
Before designing or planning teaching and learning activities for young children,
teachers have to know their pupilsÊ background experiences, living environment
and their levels of thinking. The childrenÊs intellectual development and their
levels of understanding will be the basis or the framework in designing
geometric thinking. Learning geometry is even more critical and very important
since it provides tools for critical thinking and analysis for problem solving in
real world situations.
The teaching of geometry to young children can be formal within the
mathematics classroom or during informal activities at the canteen, playground
or other outdoor venues. The introduction of simple geometry concepts can be
within their environment or the childÊs natural settings. It can be integrated while
they are playing in the playground, painting, acting in a drama, story-telling or
having a puppet show. Tom Cooper (1986) suggested the following teaching
approaches in line with the levels of Van HielesÊ Model, see Figure 9.1.
Figure 9.1: Approaches for teaching geometry
9.5

190
KINDERGARTEN OR PRE-SCHOOL
GEOMETRY
Children at the early age of four to seven years old are unable to visualise shapes
or solids. To them, shapes or solids Âlook alikeÊ or Âare similarÊ and confusing. The
spatial concepts of solids and space are still undeveloped in young children of
three, four and five years of age. They cannot visualise shapes from different
perspectives or do not possess visual imagery as yet.
The way children learn about geometry is through exposure or appropriate
learning activities and experience. Teachers have to plan and choose appropriate
learning activities and suitable materials for relevant tasks to develop childrenÊs
understanding of topology, simple Euclidean concepts and discovery of the
properties of shapes, solids and space. Andria Troutman (2003) suggested that
activities for kindergarten or preschool children should be of three kinds (see
Table 9.2):
Table 9.2: Andria Troutman's Three Kinds of Activities
for Kindergarten or Pre-school Children
Activity Description
Refine topology
ideas
ChildrenÊs experience of indoor and outdoor activities will
enhance childrenÊs understanding about topology, space and
directions. Such activities help children to use relative
prepositions and vocabulary such as enclosed boundary, inside,
outside, adjacent, beside, between, from above, under, bottom,
etc. Well-planned activities will help children to use suitable
prepositions, whether written or oral, to describe where the object
is located in space. These activities will facilitate the development
of childrenÊs spatial sense.
Extended geometric
knowledge of
simple Euclidean
and topological
ideas
It is the study of shapes, size, direction, parallelism, perpendicular
lines and angles. They can visualise and differentiate between
shapes of triangles, rectangles, squares and trapeziums.
Discover properties
and relationships of
geometric figures
Children learn about three dimensional shapes by seeing or
through observation (visualisation). They will observe the
properties of the shape and study how they behave. Suitable
materials and appropriate activities such as matching, sorting,
fitting and altering shapes allow children to discover
relationships and properties of shapes.
9.6

191
Children have not developed the ability to understand the conservation of
length, area, volume and proximity at this stage. Through appropriate drawing
and painting activities, children can refine many properties of geometric figures.
Drawings of young children show that mountains, trees, houses, and cars are
smaller than flowers or themselves. Spatial relationship among objects within
space is not established among young children yet but they begin to structure
and order in space.
During the process of a childÊs development, these three types of activities should
be integrated with the respective geometric concept and relevant manipulative
materials. Besides the commercially made or manufactured materials, it is
important that teachers make geometry materials that will provide for a better and
wider range of teaching aids to enhance their pupilsÊ understanding.
TEACHING AND LEARNING ACTIVITIES
FOR PRE-SCHOOL GEOMETRY
9.7
A wide range of teaching and learning activities can be used for teaching
Geometry at pre-school level. Various learning outcomes to be achieved pertaining
to the learning of pre-school Geometry include:
(a) Identifying shapes using the surface area and exploring the relevant solids;
(b) Matching and labelling each shape and solid through discovery;
(c) Identifying similarities and differences between shapes and solids; and
(d) Using the correct vocabulary and language to describe shapes and solids
during activities.
Some samples of teaching and learning activities suitable for teaching Geometry
to pre-school children are described below.
Activity 1: Identifying and Matching Shapes and Solids
Learning Outcome:
(a) Match and label each shape and solids given.

192
Materials:
Sets of playing blocks as shown in Figure 9.2;
Pencils;
Sets of coloured pencils or crayons; and
A4 paper.
Procedure:
(i) Divide the children into small groups of four to five pupils each.
(ii) Each group will get a set of playing blocks, a set of coloured pencils or
crayons, a piece of A4 paper and pencils for each pupil.
(iii) Get pupils to match the surface area of the solids (blocks) to the respective
template or hole in the circular box (see Figure 9.2). Teacher will facilitate
the activities and give instructions to guide them while doing the activity.
This activity should take about 10 15 minutes.
(iv) Children in their groups will insert the blocks into the appropriate hole and
each pupil should be given the opportunity to explore and discover on their
own.
Figure 9.2: Set of playing blocks
(v) When they are done, check the childrenÊs findings. Point to one of the holes
and ask the pupils to choose or select the suitable or appropriate block from
the pile, see Figure 9.3 (a).

193
Figure 9.3 (a): Set of shapes of templates and playing blocks
Check to confirm the pupilsÊ understanding. Repeat with other solids and
let them try to fit into the respective hole of the shapes, see Figure 9.3 (b).
Figure 9.3 (b): Set of playing blocks and shapes of templates
(vi) Guide pupils to label and identify the shapes and solids given using their
pencils and the piece of A4 paper.
Activity 2: Visualise Shapes and Solids
Learning Outcomes:
(a) Identify and label each shape and solid given; and
(b) Use the correct vocabulary and language to describe the relationship
between 2D shapes and 3D solids.
Materials:
Set of trace blocks;
Set of solids;
A4 paper;
Pencils;
Coloured pencils/crayons; and
Names of shapes and solids (word cards).

194
Procedure:
(i) Let the children trace the surface area/shape of the blocks on the piece of
A4 paper, match and then compare with that on the circular box, see Figure
9.4 (a).
Figure 9.4 (a): Set of playing blocks and shapes of surface area
(ii) Introduce appropriate vocabulary and guide pupils to label each shape and
solid with the correct geometrical terms (word cards). Use relevant
language to describe the relationship between the 2D shapes and 3D solids.
(iii) Using arrows, match or pair the shapes with the correct solids. The colour
clue for respective pairs of shape and solid will guide the children to pair
them up. Ask them to look at the similarities between them. Encourage
them to use the right vocabulary and language in their descriptions, see
Figure 9.4 (b).
Figure 9.4 (b): Matching activity: Match the correct surface area to the playing block
(iv) Match the shape and solids by colouring the correct pairs with the same
colour, see Figure 9.4 (c).
Figure 9.4 (c): Matching activity: Colour the correct pairs

195
At this level, introduce new vocabulary and use correct language to
describe each solid and the respective surface area. Encourage them to use
the new vocabulary and work in groups, only then can they freely
communicate with each other during the play activities.
Activity 3: Naming and Labelling of Solids
Learning Outcome:
(a) Name and label each solid given.
Materials:
Models of 3D solids; and
Names of 3D solids.
Procedure:
(i) Naming and labelling of solids:
Before this activity, teachers must introduce the correct geometrical terms
and vocabulary for every solid using flash cards or suitable teaching aids.
Teacher describes and explains to the young children simple procedures for
identifying and naming the solids. The features of solids will give the solid
its name and this can be used to identify them, see Figure 9.4 (d). Numbers
can be introduced to describe the geometric features. For instance, a cube
has 6 equal surfaces and a cuboid has 3 pairs of equal surfaces. Then, let
them ÂdiscoverÊ the ideas and features for themselves. Check their
understanding using worksheets or handouts.

196
Figure 9.4 (d): Matching activity: Match solid to name
(ii) Naming of solids: Rearrange the letters (or spelling) to form the name.
Ask the children to rearrange the letters to form the names of the solids from
the flash cards. Each flash card bears a single solid and scrambled letters. (see
Figure 9.5).
Figure 9.5: Naming of solids
Activity 4: Relationship and Properties of Shapes and Solids
Learning Outcome:
(a) Group given shapes and solids according to their similarities and differences.

197
Materials:
Models of 3D solids; and
Cut-outs of 2D shapes.
Procedure:
(a) Grouping shapes and solids according to their similarities and differences.
Promote group discussion:
Children in their groups will discuss similarities between different solids.
They will have to group the solids according to their similar properties and
how they behave. Each solid has certain properties of its own.
For example, circular prisms can roll, possess a circular surface, with or
without edges and do not have any vertices. Prisms have flat surfaces,
edges plus vertices and can stand still on their surfaces. Teachers will guide
children to classify them into different groups according to their similarities
and differences (See Table 9.3).
Allow children or give them time to discuss and improve their oral
communication, so they are able to use suitable language and vocabulary to
describe the relationship and properties of shapes and solids.
Table 9.3: Grouping Activity: 2D Shapes and 3D Solids
Types Circular/Oval
Cylinders
Triangular
Prisms
Quadrilateral
Prisms
Polygonal
Prisms
2D
Shapes
3D
Solids

198
FOR ELEMENTARY GEOMETRY
9.8
Some teaching and learning activities for Elementary Geometry are discussed
here.
9.8.1 Learning Areas for Elementary Geometry
Skills related to the learning of Elementary Geometry generally extend from what
is learnt in pre-school and include the following:
(a) Naming, labelling and using the correct vocabulary for describing each 3D
solid;
(b) Describing features or parts of solids including classifying and grouping
shapes according to similarities and differences; and
(c) Ability to assemble and explain types of shapes used to build models and
relate models to solids in real life.
9.8.2 Elementary Geometry: Early Experience of
Space and Shape
With respect to the learning of Elementary Geometry, three levels of the Van
Hiele model discussed earlier are emphasised:
(a) Level One (Visualisation)
Identifying 3D shapes by intuitive understanding of symmetry and
perspective. At this level, children are guided to identify geometric figures
through various activities that enable them to visualise shapes, name them,
use correct vocabulary and differentiate them from other shapes.
(b) Level Two (Analysis)
Analysing the attributes of geometric figures and introducing simple
concepts of proximity, separation, direction, size, length, line, enclosure,
properties and spatial ideas.
(c) Level Three (Informal Deduction)
ChildrenÊs ability to compare geometric figures covers visualising
similarities and differences, developing basic concepts on spatial ideas and

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simple Euclidean concepts as well as the transformation approach on
extended concepts of geometry.
Teachers have to develop effective and creative teaching materials to
enhance pupilsÊ learning and understanding of geometric concepts. Models
or manipulative teaching materials as illustrated below are required to
promote the mental reasoning mentioned above (See Figure 9.6).
Figure 9.6: Set of 3D models or manipulative
9.8.3 Cycle for Teaching Elementary Geometry
Figure 9.7 shows the cycle for teaching 3D solids.
Figure 9.7: Cycle for teaching elementary geometry

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9.8.4 Early Geometry Activities
In general, activities for learning early Geometry at primary level are mostly
extensions of pre-school activities and are concentrated on achieving the
following learning outcomes:
(a) Identifying and naming solids;
(b) Making skeletons and opaque solids from straw and play dough (coloured
plasticine);
(c) Recognising and describing parts of solids; and
(d) Building models of solids.
Some practical activities are described in the following section.
Activity 1: Naming of Solids
Learning Outcome:
(i) Identify and name solids.
Materials:
Blocks of solids/3D shapes;
Pictures of solids/3D shapes;
Play dough (multi-coloured);
Chart; and
Sets of flash cards (Vocabulary/Name cards).
Procedure:
The following activities are suitable for group activities.
(i) Identifying and naming solids:
This activity is in line with Level One (Visualisation) of the Van Hiele
model.
Teacher will give one flash card and a set of solids to every group and ask
the children to choose an appropriate solid from the pile. Then, get them to
stick the flash card on the board and display the chosen solid.
(ii) Repeat the activity until all the solids have been identified and named.

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(iii) Each group will be given a chart as shown in Table 9.4 and a set of picture
cards of various solids; pupils have to match and name the appropriate
solids accordingly.
Table 9.4: Set of 3D Models or Manipulative
Cube Cuboid Tetrahedron Pyramid Cylinder Cone Sphere
At this level, children are guided to identify geometric figures through the
visualisation of 3D shapes.
(iv) Identifying and naming each figure:
Encourage discussion among the children. Use correct vocabulary and
suitable language to describe the features and properties of solids. Guide
them to identify or differentiate the solids from the other shapes.
Activity 2: Analysing Similarities within a Group of Solids
Learning Outcomes:
(a) Identify the properties of groups of solids e.g. cubes, cuboids, pyramids,
cones, cylinders, spheres;
(b) Make solids from play dough; and
(c) Compare the differences within and between groups of solids e.g.
triangular and quadrilateral prisms, etc.
Materials:
Blocks of solids/3D shapes (cubes, cuboids, pyramids, cones, cylinders,
tetrahedrons, prisms, spheres);
Pictures of solids/3D shapes (cubes, cuboids, pyramids, cones, cylinders,
Play dough (multi-coloured);
Chart;
Sets of flash cards;
Vocabulary (Name cards/word cards); and
Worksheet.

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Procedure:
(i) Visualisation: (Properties of Solids)
Teacher guides pupils to visualise and explore the properties of a set of
solids one by one and record the properties in a chart.
Introduce appropriate vocabulary including face, edge, corner, etc. to
describe the properties of solids.
Repeat the steps for each solid in the set comprising cube, cuboid, pyramid,
tetrahedron, cone, cylinder and sphere. (See Table 9.5 for a more detailed
description of the procedure outlined in step (i)).
Table 9.5: Procedure for Step (a)
Level Teacher Pupils
Visualisation
of shapes or
figures
1. Teacher shows a cube
2. Introduce appropriate
vocabulary to describe the
properties of a cube.
3. Ask pupils to count and record
the number of faces, corners
and edges of a cube and write a
summary.
4. Teacher shows a cuboid next
and repeats steps 2 and 3.
5. Repeat the activity until all the
solids in the set have been
explored in sequence.
Note: Pictures of 3D solids can
be used in place of 3D models.
1. Pick out a cube and other
similar solids out of the pile.
same shape or size
similar size
(bigger or smaller size)
2. Label the corner, face and edge
of the cube accordingly.
3. Record a summary of the
properties of the cube e.g.:
6 flat faces
8 corners
12 edges
4. Pick out a cuboid and repeat
the steps above for the solid
chosen.
5. Repeat each step as above for
each solid displayed in
sequence.

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(ii) Learning experience: Making solids from play dough
Group activity:
Using 3D solids as samples, guide pupils to produce various solids as listed
below:
Cube;
Cuboid;
Triangular Prism;
Quadrilateral Prism;
Pyramid; and
Tetrahedron.
Guide pupils to explore the properties of each solid made.
(iii) Comparing differences within and between groups of solids.
Using the solids made in Step (ii) or other models, guide pupils to compare
the similarities and differences within and between groups of solids.
Begin with comparing triangular prisms with quadrilateral prisms.
Ask pupils to record the properties of both solids by analysing the surface
area (2D shapes) and the respective 3D solids as in Table 9.6.

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Table 9.6: Comparison Chart
Triangular Prisms Quadrilateral Prisms
2D
Shapes
3 sides
3 edges
3 corners
4 sides
4 edges
4 corners
3D
Solids
5 flat surfaces
6 vertices
6 flat surfaces
8 corners or
vertices
sets of 3 pairs
(surfaces)
12 edges
(iv) Repeat the above step and compare different groups of solids e.g. cubes
with cuboids, pyramids with tetrahedrons, etc.
(v) Distribute worksheet to pupils.
Activity 3: Building Models and Nets of Solids from 2D Shapes
Learning Outcome:
(a) Build models and nets of solids (cuboids and cubes) from 2D shapes.
Materials:
Blocks of solids/3D shapes (e.g. cubes, cuboids, pyramids, cones, cylinders,
Objects: pencil box;
Cardboard;
A4 paper (multi-coloured);
Pencils; and
Worksheet.

205
Procedure:
(a) Always guide young children on how to relate solids of three dimensions
(3D) to shapes of two dimensions (2D). The visualisation of cuboids can be
attained by using a pencil box to represent the cuboid. Let pupils rotate,
visualise and trace the shapes from different perspectives by looking at the
cuboid from different orientation or from different elevations i.e. top, front
and side, see Figure 9.8 (a).
Step 1: Visualisation
Figure 9.8 (a): Views of a cuboid from different perspectives
Step 2: Trace surfaces of a cuboid
(i) Trace the surfaces of the cuboid from the top and bottom to get two
similar faces.
Cut the traced shapes and slide the two pieces over one another to
check if they are congruent and similar, see Figure 9.8 (b).
Figure 9.8 (b): Top and bottom views of the cuboid

206
(ii) Trace the surfaces from the left and right sides of the cuboid.
Cut the traced shapes and slide the two pieces over one another to
check if they are congruent and similar, see Figure 9.8 (c).
Figure 9.8 (c): Left and right side views of the cuboid
(iii) Trace the surfaces from the front and back elevation (looking from
the front and back/behind).
Cut the traced shapes and slide the two pieces over one another to see
if they are congruent and similar, see Figure 9.8 (d).
Figure 9.8 (d): Front and back views of the cuboid
Step 3: Match and relate the traced surfaces for geometric reasoning
Figure 9.8 (e) shows how to trace the surfaces of the cuboid.

207
Figure 9.8 (e): Traced surfaces of the cuboid
Step 4: Attach the traced surfaces on to the cuboid/pencil box and make a
net of a cuboid
The cuboid/the pencil box has six surfaces (3 pairs of similar surfaces),
refer Figure 9.8 (f)).
Figure 9.8 (f): Net of the cuboid
(b) Using the same procedure as in (a), produce nets of a cube like the ones
shown in Figure 9.9:
Figure 9.9: Nets for the cube

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Activity 4: Drawing 3D Solids
Learning Outcomes:
(a) Draw 3D solids with the help of essential tools like graph paper and GSP;
and
(b) Identify the 3D solids drawn using GSP.
Materials:
PowerPoint slides (3D shapes: prisms);
Graph paper/square dot paper/isometric dot paper; and
Computer software, GeometerÊs Sketchpad (GSP).
Procedure:
(a) Identifying 3D shapes by intuitive understanding of symmetry and
perspective.
Let pupils look at some animated pictures of 3D solid from a PowerPoint
presentation (e.g. cubes, cuboids, triangular prisms, pyramids, tetrahedrons).
Have an open discussion about symmetry and perspective to guide pupils to
identify and name the 3D shapes shown in the slides.
The drawing of solids will be easier for young children with the help of
graph paper, square dot paper or isometric dot paper and computer
software, such as the GeometerÊs Sketchpad (GSP). The GSP is often used
as a tool to draw regular and irregular prims of 3D solids. Figure 9.10
shows some examples of solids drawn using the GSP. Teach pupils how to
draw the figures one by one.

209
Figure 9.10: Examples of regular and irregular prisms drawn using GSP
(b) Ask pupils to identify the 3D solids drawn.
Activity 5: Build Skeletons for 3D Solids
Learning Outcomes:
(a) Build skeletons for 3D solids (e.g. cubes, cuboids, pyramids, tetrahedrons,
triangular prisms); and
(b) Make a summary of the properties for each skeleton of the 3D solids made.
Materials:
Play dough or plasticine; and
Drinking straws.
Procedure:
(i) After the drawing of solids, a suitable follow-up activity will be the the
making of skeletons of 3D shapes using drinking straws and play dough as
illustrated in Figure 9.11. All properties of solids (corners/vertices, edges
and flat surfaces will be discussed here).

210
Figure 9.11: Examples of skeletons of 3D solids
(ii) Ask pupils to name each of the skeletons of 3D solids and make a summary
of the properties for all the skeletons constructed.
Activity 6: Building Opaque Models of Solids
Learning Outcomes:
(a) Make solids from play dough;
(b) Build models using a combination of various solids; and
(c) Recognise and describe parts of solids.
Materials:
Play dough or plasticine (multi-coloured);
3D blocks or solids (wooden/plastic);
Picture card chart of 3D solids (regular and irregular prisms, non-prisms);
and
Flash cards (name cards/word cards).
Procedure:
(i) Distribute some play dough or plasticine and a picture card chart/guide to
each group of pupils. Let pupils make models of 3D solids using play
dough or plasticine based on the picture card chart.

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(ii) Using the 3D models constructed or ready-made 3D blocks (wooden/
plastic), make models like the ones displayed in Figure 9.12.
(iii) Creative play: Allow pupils to indulge in free-play and make models of
their own using the play dough or plasticine.
Figure 9.12: Examples of opaque models of 3D shapes
(iv) Ask pupils to label and describe the parts of the solids constructed using
appropriate vocabulary, language, name and word cards such as cube,
cuboid, triangular prism, vertex, edge, face, etc.
In this topic, the theory and approaches of learning geometry are highlighted.
The teaching and learning of geometry should be aligned to pupilsÊ levels of
thinking.
In planning instruction or teaching and learning activities, teachers have to
consider the childrenÊs logical thinking, their levels or stages of learning
geometry, their biological maturation as well as their living environment.
A few research findings and models of teaching geometry are integrated
within this model as reference and as a basic framework for teachers when
designing instruction or teaching and learning activities.
During the earlier part of the topic, discussion focused on the importance of
learning geometry and the use of geometric concepts to solve real life
problems.
The concepts of geometry and spatial sense, incorporated with numerical
literacy, should be introduced to children at an early age.

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Teaching strategies of three dimensional (3D) shapes or solids for young
children are suggested under various sub-headings.
Examples of teaching and learning activities for geometry given in this topic,
take into account the theory and childrenÊs levels of thought, starting from
kindergarten or pre-school and extending to early primary school.
Apex
Area
Boundary
Capacity
Cone
Corner
Cube
Cuboid
Cylinder
Edge
Oval
Polygon
Polyhedron
Prism
Solid
Sphere
Symmetry
Tessellate
Tetrahedron
Three dimensional (3D)
Vertex
Volume
List three levels of teaching geometry for early primary or pre-school and suggest
a suitable learning activity for each level.
Pupils learn the concept of geometry while playing with and building models
using the three dimensional solids. Think of a strategy to teach Euclidean
Geometry to young children through play.

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APPENDIX
WORKSHEET
1. Draw lines to match each 3D shape with the correct description.

Topic
10
Two
Dimensional
Shapes (2D)
LEARNING OUTCOMES
1. Explain the shape and space in daily life;
2. Identify shapes using the correct vocabulary related to 2D shapes;
3. Classify two dimensional (2D) shapes; and
4. Describe the teaching and learning of shape activities.
INTRODUCTION
Look at things around you. How do they appear or look like? Almost everything
around you is in the form of solids or in three-dimensional form, but topologically,
they can be described as two dimensional shapes. What does an apple, an ice-cream
cone, a star fruit and a ball look like to small children? Are children able to
name and relate logically between three dimensional solids and two dimensional
shapes?
SHAPE AND SPACE IN DAILY LIFE
10.1
Understanding the environment we live in is very important as we live in an
environment made up of shape and space. Take a look at the things around you
many objects around are either in the form of two dimensional shapes or three
dimensional solids. Some pertinent questions come to mind. How do young
children see things surrounding them? How do they develop geometric thinking
and mental reasoning about shapes? How does the human mind, or thinking,
change and make connections between 3D solids and 2D shapes? Things like
cauliflower, cabbage and broccoli are usually spherical in shape like that of a ball,

TOPIC 10 TWO DIMENSIONAL SHAPES (2D)
215
see Figure 10.1 (i.e. 3D in shape) but at certain angles, they may look like the face
of a clock (i.e. 2D in shape)!
Figure 10.1: Spherical 3D solids can be viewed as circular 2D shapes
SPATIAL SENSE
How do we deal with the space around us? To understand more about spatial
sense, let us consider a few situations. How are bags and luggage arranged in the
compartments of an aeroplane to accommodate all baggage checked in? How do
we ensure that we can walk into a laboratory or classroom without stumbling
over instruments and furniture? Why do drivers position their cars on the right
lane of the road in order to avoid tragic accidents? Why is it we cannot simply
drive on the road without getting a driving licence first? One has to apply spatial
sense to be safe on the roads. To acquire a driving licence, we have to undergo
several tests and practise driving for hours under the supervision of experts
before we can drive independently and safely on the roads.
SELF-CHECK 10.1
Spatial sense is defined as an intuition about shape and the relationship
among shapes, including our ability to mentally visualise objects and spatial
relationships by turning things around in our minds. It is about our feeling of
geometric aspects of objects and shapes that appear within our surroundings
or our living environment.
(Walle and Lovin, 2006)
10.2
1. What is meant by geometric spatial sense?
2. Why is geometric spatial sense important for understanding our
environment?

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Spatial sense is spatial perception or spatial visualisation that helps students
to understand the relationship between objects and their locations in three
dimensional worlds.
(Kennedy and Tipps, 2006)
Geometry has several applications in real life. Spatial sense is spatial visualisation
or spatial perception that helps children in understanding their world.
Furthermore, spatial sense is an imaginary visualisation of object orientation in
our minds. People with good spatial sense are able to analyse, using their
geometric reasoning and ideas to appreciate nature, space exploration, home
decoration, architecture, art and design. It promotes creativity in art and design.
One is also able to imitate and transfer a bouquet of flowers into 2D shapes, see
Figure 10.2.
Figure 10.2: Bouquet of flowers
ACTIVITY 10.1
Do your pupils/young children have spatial sense? Is spatial sense innate
in children or do we have to teach spatial sense in the classroom? Discuss.
Next, carry out the following task. Take a look at the pictures shown in Figure
10.3.

217
Figure 10.3: Pictures of objects
Show these pictures to your pupils. Ask them to arrange the pictures in
ascending order according to their size in real life. Discuss and look out for
your pupilsÊ spatial reasoning when doing the arranging.
The answer: (picture frame door Eiffel Tower)
If their arrangement is as such, how did they know that Eiffel Tower is the
biggest/ tallest among the three things?
Have they ever visited Eiffel Tower?
Did they use their spatial sense when arranging the pictures in ascending
Get your children to look closely at the picture of the rabbit and ask them
whether it is possible that the size of the rose (flower) can be bigger than
the rabbit in reality.
GEOMETRIC THINKING
order?
10.3
Geometry recognition is part of the primary mathematics curriculum. The aim of
introducing two dimensional shapes in the primary school curriculum is to
develop the pupilsÊ reasoning and spatial sense with respect to geometry since
geometric practical applications are very useful in everyday life. Most of the
mathematics primary curriculum incorporates number systems and numerical
thinking as a foundation into the teaching of geometry. The development of the
human mind on geometric concepts and reasoning of solids and shapes
encompasses two basic areas, (see Table 10.1).

218
Table 10.1: Two Basic Areas of Human Mind on Geometric Concept
Basic Area Description
Visual spatial
thinking
This happens on the right hemisphere of the brain that is associated
with literature and can occur unconsciously without one being aware
of it. It can operate holistically and intuitively, with more than one
thing at a time and is literally called simultaneous processing.
Verbal logical
thinking
This lies on the left hemisphere of the brain consisting of continuous
processing and one is always aware of it. It operates sequentially and
logically and is related to language or symbols and numbers.
Gardner proposed that the multiple intelligence of spatial ability can be
developed through experience. Children are able to explain and demonstrate
their discoveries after seeing how things work and observing their properties.
The levels of thought, or childrenÊs thinking, is the basis for the instructional
activities at primary school level.
The Van Hiele Theory: Levels of Geometric Thinking
Figure 10.4 explains the level of geometric thinking according to Van Hiele.
Figure 10.4: Van HieleÊs levels of geometric thinking
According to Pablo Picasso, „Observation is the most significant element of my
life, but not just any kind of observation‰. This means that certain observations or
the way we look at things will form a visual image that can be used in the study
of mathematics and its applications.

219
GEOMETRIC SYSTEMS
Children learn geometry at primary level which can be divided into four separate
geometrical systems as suggested by many mathematicians (Kennedy and Tipps,
2000), as described in Table 10.2.
Table 10.2: Four Separate Geometrical Systems
Geometrical
System Description
Euclidean
geometry
Euclidean geometry is the geometry of shapes and objects in a plane (2D)
or in space (3D). It is about the properties or the characteristics of objects,
points and lines, circles and spheres, triangles, polygons, pyramids,
cylinders, cones and other solids. Shapes have properties including
similarities and congruence, length of sides, number of parallel sides,
lines or rotational symmetry.
Coordinate
geometry
It is about the location of shapes on coordinate or grid systems.
Coordinate geometry ranges from simple to complex uses that define the
location of an object on plane coordinates of the vertical and horizontal
axes for 2D shapes or the positioning of objects on grid systems for three
dimensional spaces. Complex uses of coordinate geometry include the
location of vessels in the Pacific Ocean or the location of a travellerÊs
camp at the Antarctic or the grid location of Mount Everest.
Transformation
geometry
Transformation geometry is about geometry in motion. It describes the
movement of shapes or objects in a plane or in space. Objects or shapes in
motion can be transformed by flipping (reflection), sliding or gliding
(translation), and turning (rotation) or a combination of these
transformations in many different ways. For example, during an
aircraftÊs landing or departing, it slides on the runway, flips and turns in
the sky or exhibits a combination of movements in different ways.
Topological
geometry
Topological geometry describes the locations of objects and their
relations in space or the recognition of objects in the environment.
Children view everything and their perceptions relative to their standing
positions or locations in space. It focuses on the development of the
childrenÊs mental understanding, the use of extensive vocabulary, giving
descriptions of objects in space, as well as the size and position of objects
within their perspectives. The use of vocabulary to describe the locations
of objects in space include words such as: far-near, high-low, big-small,
above-below, inside-outside or in front, in between, front and behind, etc.
10.4

220
ACTIVITY 10.2
Group discussion:
The teaching of geometry covers four areas of geometric systems and
must be aligned to the Van HieleÊs theory of levels of thought. In groups,
discuss how to integrate the teaching of the concept of geometry with any
two areas underlined.
GEOMETRIC CONTENT
The Primary Mathematics curriculum touches on the simple concept of geometric
systems. The Geometric content for primary schools focuses more on
Visualisation and Euclidean Geometry, as well as Van HieleÊs theory of childrenÊs
thinking. The sequence of teaching geometric content and the teaching of
concepts at primary school level is as illustrated in Figure 10.5.
Figure 10.5: Sequence of teaching geometry in primary schools
10.5

221
Visualisation covers the recognition of shapes in the environment, classification,
sorting and naming of shapes. Euclidean geometry is the study of shapes and
their properties. Advance concepts of topological geometry, Euclidean Geometry,
coordinate and transformation geometry will be taught at secondary school or at
higher levels.
The Geometric Content includes:
(a) Identifying shapes sorting, classification and grouping;
(b) Knowing and naming shapes (vocabulary):
(i) Triangle (types of triangles);
(ii) Rectangle (quadrilaterals);
(iii) Polygon; and
(iv) Circle and ellipse.
(c) Identifying geometric properties of shapes;
(d) Classification and grouping; and
(e) Shapes in the environment.
THE TEACHING AND LEARNING OF
SHAPES
10.6
The learning of shapes is the second stage for children learning about geometry.
The teaching and learning of geometry should be associated with the childrenÊs
levels of thinking and the four areas of geometric systems described earlier. As a
teacher, we have to understand our pupilsÊ levels of thinking and mental
reasoning before teaching them the concept of two dimensional shapes. Both
hemispheres of our pupilsÊ minds must be stimulated. Exposure and experience
through investigation and discovery will promote pupilsÊ learning.
Some suggested teaching-learning activities for helping children to develop or
consolidate ideas and further understanding of geometrical concepts are
highlighted. There should be a progressive development of activities and a
proper sequence for introducing concepts, starting from basic geometric concepts
to the highest level of geometric problem solving.

222
Learning Activities
The following are some suitable learning activities:
(a) Contextual learning children look around and observe the environment
plus describe in words what they have seen.
(b) Explore and experiment with shapes (visual images) in order to gain insight
into the properties and their uses.
(c) Analyse shapes informally, observe size and position in order to make
inferences; then refine and extend knowledge that develop from various
learning activities.
Introduction of three dimensional shapes must be done earlier or before the
teaching of 2D shapes. The concept of two dimensional shapes can be developed
from three dimensional shapes. Shapes that can be introduced to pre-school or
early primary level include those easier concepts that are commonly found
within their environment.
Figure 10.6: Different views of the surface area of faces of a cuboid
A teacher should always guide young children on how to relate solids of three
dimensions (3D) to shapes of two dimensions (2D). As described previously in
Topic 9, for the visualisation of cuboids, pencil boxes can be used to represent
cuboids. Pupils can be encouraged to rotate, visualise and trace the shapes from
different perspectives by looking at them from different orientations or from
different sides or elevations (see Figure 10.6).

223
10.7
Geometric thinking and spatial reasoning can be developed through formal or
informal activities. A good instructional activity includes good planning,
appropriate activities and a variety of selection of effective teaching materials.
The learning of geometric concepts can be incorporated into the childrenÊs
activities such as playing activities, discussion, role play, music activities, dramas
as well as art and design activities while they are actively involved. Teaching and
learning activities suggested here cover the four areas mentioned earlier and can
be modified to teach 2D shapes to pre-school and early primary school children.
Part I: Euclidean Geometry
10.7.1 Identifying Shapes – Sorting, Classification and
Grouping
Shape is generally defined as Âspace within an enclosed boundaryÊ.
Shapes are drawn on a flat surface called a plane. Two dimensional plane
geometry is about shapes like lines, circles and triangles. As such, shapes can be
enclosed by straight or curved lines. Shapes enclosed by only straight lines are
called polygons. Other shapes are known as non-polygons.
The activities described here are geared towards achieving the following learning
outcomes with respect to 2D shapes:
(a) Identify 2D shapes, i.e. figures with closed boundaries;
(b) Sort and classify 2D shapes;
(c) Discover features and properties of 2D shapes;
(d) Identify similarities and differences of shapes between groups; and
(e) Use correct vocabulary and language while doing activities.
Activity 1.1: Identifying 2D Shapes
Learning Outcomes:
(a) Identify and colour 2D shapes i.e. figures with closed boundaries; and
(b) Count how many figures there are with closed and open boundaries.

224
Materials:
Exercise sheet: Set of shapes (open and closed figures); and
Colour pencils.
Procedure:
(i) Distribute the Exercise sheet (see Figure 10.7), containing examples of
various shapes to the pupils.
(ii) Ask pupils to colour the shapes with closed boundaries and count how
many shapes there are with closed boundaries and how many there are
without.
Figure 10.7: Set of shapes
This activity is an early introduction to the concept of shapes whereby shapes are
identified as figures that have closed boundaries. Children have to colour all the
shapes with closed boundaries and leave out those with open boundaries. This
activity serves to give a clear picture to young children about 2D shapes.
ACTIVITY 10.3
Play the game of Look Around.
Children who can spot the most number of 2D shapes will be the winner.

225
Activity 1.2: Sort and Group 2D Shapes
Learning Outcomes:
(a) Sort, group and classify shapes to discover their features and properties.
Materials:
Set of 2D shapes/Cut-outs of 2D shapes;
A4 paper/manila cards; and
Vocabulary (Word cards).
Procedure:
(i) Sort and group the shapes. Place the sorted shapes under the respective
categories on the pieces of A4 paper/manila cards provided. Use common
features for grouping the shapes into various categories: Triangles,
Quadrilaterals/Rectangles, Polygons and Non-polygons/Enclosed
boundaries (see Figure 10.8).
(ii) Discuss what is interesting about each group?
Figure 10.8: Examples of 2D shapes
(iii) Ask simple questions and guide pupils to describe in simple words the
common features used for sorting and grouping shapes:
How many groups of shapes are there?
What are the special features of the figures or shapes in each group?
What are the common features within each group?
Is there any difference between the groups?

226
(iv) Say in simple sentences or describe clearly the special features of each
group.
(v) Encourage the children to say in simple words what they understand about
the interesting features of the shapes.
(vi) Ask pupils to look for similarities and differences within and between the
groups.
(vii) Introduce simple geometric words and correct vocabulary to help pupils to
describe the features and propertries of the shapes.
(viii) Check the answers for the grouping of 2D shapes: For example as shown in
Figure 10.9:
Figure 10.9: Grouping of shapes under different categories
(ix) Distribute worksheet to reinforce the concept learnt.
10.7.2 Knowing and Naming Shapes (Vocabulary)
Introduce different types of shapes and let pupils look for features and properties
to identify the various groups of shapes. At this level, only simple geometry is
used and the shapes shown here are to be considered as extra knowledge for
teachers.

227
Activity 2.1: Identifying Polygons
Polygons are two dimensional flat surfaces with length and breadth or width.
Polygons have special names based on the number of angles and the number of
sides (straight edges) that enclose them. A triangle is a polygon with the least
number of points and sides to form a closed boundary, followed by quadrilaterals,
pentagons, hexagons, heptagons, octagons, nonagons, decagons, dodecagons, etc.
Other figures known as non-polygons also have enclosed boundaries but with
sides that are circular and elliptical. Here, various examples of polygons and a few
non-polygons are illustrated in Table 10.3 (to be used as teacherÊs notes).
Table 10.3: Examples of Polygons and Non-Polygons
Triangle
3 points
3 sides
Quadrilateral
4 points
4 sides
Pentagon
5 points
5 sides
Hexagon
6 points
6 sides
Heptagon
7 points
7 sides
Octagon
8 points
8 sides
Nonagon
9 points, 9
sides
Decagon
10 points, 10
sides
Hendecagon/Undecagon
11 points, 11 sides
Dodecagon
12 points
Non-polygons:
Circle, Ellipse
and other
shapes
Enclosed
boundaries

228
Most of the polygons shown in Table 10.3 are irregular plane figures or polygons
where all the sides and angles are not of equal measure. Regular polygons have
sides that are all equal in length and angles that are all equal in measure. Figure
10.10 shows an example of a regular and irregular hexagon.
Figure 10.10: Regular and irregular hexagons
ACTIVITY 10.4
Stop and Have Fun! Activity: Shapes of games
Take your pupils outside and play these games:
(a) Shape jumping
(b) Track to the moon
(c) Play ting-ting

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1. Shape jumping
Lay out a few sets of
shapes (coloured,
hard cardboard) on
the floor.
Call out the shapes
and jump to land on
the shape called out
e.g. triangles,
quadrilaterals,
polygons, curved
shapes.
2. Track to the moon
Lay out a few sets of
plane figures
(coloured, hard
cardboard) on the
floor.
Call out the shapes
and Jump and sing
the Jumping song:
Names of shapes.
3. Play ting-ting
Number and shapes
Jump in steps.
Count the number of
sides, corners and
angles.
Discuss special
features of regular
rectangles (squares).
Activity 3: Vocabulary for Naming Shapes
This activity is to enhance childrenÊs understanding on different types of shapes
for each group and the differences between various classes of shapes. Provide the
opportunity for pupils to look for special features and properties as well as learn
the names of shapes.
Learning Outcomes:
(a) Identify different types of shapes (triangle, quadrilateral, polygon, etc.);
(b) Name shapes;
(c) Discover features and properties of shapes;
(d) Identify special features and properties of each group of shapes;
(e) Identify similarities and differences of shapes between groups; and
(f) Use correct vocabulary and language while doing activities.

230
Materials:
Geoboard (safety nail pegs), see Figure 10.11;
Coloured rubber bands (see Figure 10.11); and
Vocabulary cards: (Names of shapes; properties of shapes).
Figure 10.11: Geoboard and rubber bands
Procedure:
(i) Take a rubber band and form shapes using the pegs on the geoboard.
(ii) Name and label the shapes made using appropriate vocabulary cards.
(iii) Identify the features or properties of the shapes labelled.
Activity 3.1: Triangles
Learning Outcomes:
(a) Discover features and properties of triangles;
(b) Identify special features and properties of triangles; and
(c) Use correct vocabulary and language to describe triangles.
Materials:
Rubber bands;
Geoboard;
Grid paper; and
Vocabulary cards: (names of shapes, labels of properties).

231
Procedure:
(i) Ask pupils to use three points and a rubber band to form a triangle.
(ii) Then, ask them to form different types of triangles see Figure 10.12 (a) -
and draw the shapes on grid paper.
(iii) Guide them to look for properties of triangles, classify and describe the
triangles made.
Figure 10.12 (a): Examples of triangles
A. TeacherÊs Instructions:
A triangle is a shape with three points (see Figure 10.12 (b)), three corners and
three straight sides.

232
Figure 10.12 (b): Examples of triangles
B. TeacherÊs notes: Extra information on types of triangles.
Types of Triangles
These are triangles, each of them has three straight sides, three corners or three
vertices but they are all different. There are many types of triangles e.g.
equilateral, right angle triangle, isosceles, acute, obtuse or scalene triangle (see
Figure 10.12 (c)).
At this level of visualisation, we want young children to be able to use their
senses to observe differences and similarities between the shapes within a group.
It may seem difficult to explain the different types of triangles but at this level,
children only have to understand that there exists special properties for triangles
and that there are many different types of triangles (see Figure 10.12 (d)).

233
Figure 10.12 (c): Types of triangles
Figure 10.12 (d): Classification of triangles
Activity 3.2: Matching Triangles
Learning Outcomes:
(a) Identify special features and properties of different types of triangles; and
(b) Use correct vocabulary and language to describe the types of triangles.
Materials:
Exercise sheet Matching activity; and
Vocabulary cards (Names of shapes, labels of properties).

234
Procedure:
(i) Can you see any differences between those four triangles in the upper row?
Look at them closely look for similarities and differences between them.
(ii) Try to match those on the upper row by drawing arrows to those shown in
Figure 10.13.
Figure 10.13: Matching activity: Types of triangles
Activity 3.3: Quadrilaterals
Learning Outcomes:
(a) Discover features and properties of quadrilaterals;
(b) Identify special features and properties of quadrilaterals; and
(c) Use correct vocabulary and language to describe quadrilaterals.

235
Materials:
Rubber bands;
Geoboard;
Grid paper; and
Vocabulary cards: (names of shapes, labels of properties).
Procedure:
(i) Ask pupils to use 4 points and a rubber band to form a quadrilateral.
(ii) Then, ask them to form different types of quadrilaterals see Figure 10.14
(a) - and draw the shapes on grid paper.
(iii) Guide them to look for properties of quadrilaterals, name and describe the
quadrilaterals made, (see Figure 10.14 (b) and Figure 10.14 (c)).
Figure 10.14 (a): Examples of quadrilaterals

236
A. TeacherÊs Instructions:
Figure 10.14 (b): How to make the quadrilaterals
Figure 10.14 (c): Examples of quadrilaterals shapes
Activity 3.4: Polygons
Learning outcomes:
(a) Discover features and properties of polygons;
(b) Identify special features and properties of polygons; and
(c) Use correct vocabulary and language to describe polygons.
Materials:
Rubber bands;
Geoboard;
Grid paper; and
Vocabulary cards (names of shapes, labels of properties).

237
Procedure:
(i) Ask pupils to use five points and a rubber band to form a pentagon.
(ii) Then, ask them to form different types of polygons using six pegs and
more, see Figure 10.15 and draw the shapes on grid paper.
(iii) Guide them to look for properties of polygons, name and describe the
polygons made.
Figure 10.15: Examples of polygons
Activity 3.5: Curved Shapes: Circle and Ellipse
Learning Outcomes:
(a) Discover features and properties of non-polygons or curved shapes e.g.
circle, ellipse;
(b) Identify special features and properties of non-polygons or curved shapes;
and
(c) Use correct vocabulary and language to describe non-polygons or curved
shapes.
Materials:
Picture card: Set of pictures of curved shapes;
Vocabulary cards: Names of shapes (circle, semicircle, ellipse); and
Vocabulary cards: Features/properties (crescent/lunar/cloud/heart shape).

238
Procedure:
(i) Ask pupils to look at the set of curved shapes in the picture card
distributed. Explore and observe shapes through visual images to gain
insight to their properties. Look for similarities and differences (see Figure
10.16).
(ii) Guide them to look for properties of non-polygons, name and describe the
non-polygons shown. Introduce names, vocabulary or the language of
geometry for describing non-polygons or curved shapes shown.
(iii) Count the total number of faces, the number of similar faces and slowly
introduce names and properties of the curved shapes.
Figure 10.16: Examples of non-polygons or curved shapes
Shapes in the Environment
ACTIVITY 10.5
Colour and print shapes of the environment
Use any 3D solids or objects from the environment. Cut and colour the
cross-sections or longitudinal sections of objects e.g. (star fruits, pears,
banana stems, potatoes, leaves). Then, print them on to a piece of drawing
paper. Use your creativity. Are they shapes of the environment?
The learning of geometric concepts will be easier if pupils are actively involved in
the fun learning process using appropriate teaching materials from the
environment, such as the use of common materials from the childrenÊs
environment like potatoes, star fruits, pears or banana stems as materials for
drawing and painting shapes (see Figure 10.17).

239
Cross-sectional or longitudinal cuttings of:
Figure 10.17: Cross-sectional or longitudinal cuttings of objects
Part II: Transformation Geometry
The following activity allows children to have some fun with shapes, with respect
to motion geometry, where the learning of geometric concepts is incorporated
into childrenÊs play.
Activity 4.1: Fun with Shapes: The Most Powerful Spinning Propeller (Motion
Geometry)
Learning Outcomes:
(a) Construct propellers of different shapes: semi-circular shape, crescent
(lunar) shape, heart shape and arrow shape; and
(b) Discover which shape forms the most powerful spinning propellers.
Materials:
Instruction cards for making different shaped propellers;
Plastic glass;
Manila card;
Coloured paper/Fancy cards;
Straws;
Pencils;
Pins; and
Plasticine.

240
Procedure:
(i) Divide the children into four groups of five.
(ii) Ask each group to prepare different shapes of propellers as follows:
Group 1 : semi circular shaped propellers;
Group 2 : lunar shaped propellers;
Group 3 : heart shaped propellers; and
Group 4 : arrow shaped propellers.
(iii) Using their finished products, ask pupils to blow on to the propellers to see
which propellers will spin the fastest.
Group 1
Figure 10.18 (a), Figure 10.18 (b), Figure 10.18 (c) and Figure 10.18 (d) explain the
steps in making the semi-circular, lunar, heart and arrow shaped propellers for
each of the four groups.
Figure 10.18 (a): The making of semi-circular propellers

241
Group 2
Figure 10.18 (b): The making of lunar-shaped propellers
Group 3
Figure 10.18 (c): The making of heart-shaped propellers
Group 4
Figure 10.18 (d): The making of arrow-shaped propellers

242
Finished Products
Figure 10.18 (e) shows the finished products of the different shaped propellers
from each group.
Figure 10.18 (e): Different shaped propellers
Part III: Coordinate Geometry
An activity about coordinate geometry dealing with the location of places or
destinations is described next.
Activity 5.1: Location of Places
Learning Outcomes:
(a) Explain the location and state the direction of places from a picture map.
Materials:
Picture map; and
A4 paper.
Procedure:
(i) Distribute the picture map (see Figure 10.19) to your pupils.
(ii) Ask them to study the map and explain the location of the places stated in
the map to a tourist who lands at KLIA.
(iii) Discuss in detail the location and direction of those places.

243
Figure 10.19: Picture map
Part IV: Topological Geometry
The last area of geometric concept to be discussed concerns topology and touches
on the location of objects according to a relative standing position in space.
Activity 6.1: Location of Shapes
Learning Outcomes:
(a) Describe the locations and state the positions of objects with respect to a
relative standing position from the picture provided.
Materials:
Picture card;
A4 paper; and
Vocabulary cards: (positional words e.g. behind in front, left right, far
near, beside adjacent, etc.).

244
Procedure:
(i) Ask pupils to colour the shapes according to the colour scheme given i.e.
(triangles green, quadrilaterals red, polygons light blue, circles dark
blue, and ellipses yellow) or any scheme of their choice in the picture card
as shown in Figure 10.20.
(ii) Count the number of figures for every group of shapes that appears in the
picture.
(iii) Topology concept: Discuss the site or the location of objects in the picture
from the girlÊs standing position i.e. behind in front, left right, far near,
beside adjacent, etc.
(iv) Ask pupils to describe and state the positions of objects using appropriate
vocabulary.
Figure 10.20: Picture card
This topic of two dimensional shapes discusses spatial sense in detail, how we
understand our world, childrenÊs level of thinking and the teaching of
geometry concepts within four geometric systems, which are:
(a) Euclidean Geometry;
(b) Transformation Geometry;

245
(c) Coordinate Geometry; and
(d) Topological Geometry.
Examples and activities suggested are within the primary school curriculum
prescribed especially for pre-school and early primary school levels.
Teachers are encouraged to develop good lesson plans, creative and effective
teaching activities to suit pupilsÊ interest and their ability to understand the
geometric aspects of their surrounding and the environment.
Acute Angle
Apex
Base
Boundary
Circle
Cone
Corner
Edge
Equilateral Triangle
Hexagon
Isosceles Triangle
Rectangle
Scalene Triangle
Sphere
Square
Symmetry
Tessellate
Triangle
Two Dimensional
Vertex (Vertices (p))
As an early childhood mathematics school teacher, you have to plan teaching and
learning activities that covers the teaching of geometric concepts for the four
geometric systems. Suggest suitable teaching and learning activities that can be
carried out to enable the children to acquire the geometric concepts discussed.
Teaching activities and teaching materials should be creative and effective based
on respective learning outcomes.

246
The teaching of geometric concepts has to be aligned to childrenÊs levels of
thought, pupilsÊ experience and their geometric reasoning ability. List three
learning activities that can suit the criteria mentioned.
APPENDIX
WORKSHEET
1. Count the shapes. Fill in the blanks with the correct numbers:
(a) There are _____ circles.
(b) There are _____ rectangles.
(c) There are _____ squares.
(d) There are _____ triangles.
(e) There are _____ stars.

HBMT 1203 Mathematic

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HBMT 1203 Mathematic