Explanatory Notes to

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Mathematics Education Section Curriculum Development Institute Education Bureau

2018

Explanatory Notes to

Primary Mathematics Curriculum

(Key Stage 1)

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(Blank Page)

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Foreword

To keep abreast of the ongoing renewal of school curriculum at primary and secondary levels, the revised Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 6) (2017) and its supplements setting out the learning content at each key stage have been prepared by the Curriculum Development Council and released in late 2017.

Among these documents, the Supplement to Mathematics Education Key Learning Area Curriculum Guide: Learning Content of Primary Mathematics (2017) (hereafter referred to as “Supplement”) aims at elucidating in detail the learning targets and content of the revised primary Mathematics curriculum.

In the Supplement, the Learning Objectives of the primary Mathematics curriculum are grouped under different Learning Units in the form of a table. The notes in the “Remarks”

column of the table provide supplementary information about the Learning Objectives.

The explanatory notes in this booklet aim at further explicating:

1. the requirements of the Learning Objectives of primary Mathematics curriculum;

2. the strategies suggested for the teaching of primary Mathematics curriculum;

3. the connections and structures among different Learning Units of primary Mathematics curriculum; and

4. the curriculum articulation between the primary Mathematics and the junior secondary Mathematics.

Teachers may refer to the “Remarks” column and the suggested lesson time of each Learning Unit in the Supplement, with the explanatory notes in this booklet being a supplementary reference, for planning the breadth and depth of treatment in learning and teaching. Teachers are advised to teach the content of the primary Mathematics as a connected body of mathematical knowledge and develop in students the capability for using mathematics to solve problems, reason and communicate. Furthermore, it should be noted that the ordering of the Learning Units and Learning Objectives in the Supplement does not represent a prescribed sequence of learning and teaching. Teachers may arrange the learning content in any logical sequences which take account of the needs of their students.

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Comments and suggestions on this booklet are most welcomed. They may be sent to:

Chief Curriculum Development Officer (Mathematics) Curriculum Development Institute

Education Bureau

4/F, Kowloon Government Offices 405 Nathan Road, Kowloon Fax: 3426 9265

E-mail: ccdoma@edb.gov.hk

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Learning Unit Learning Objective Time

Number Strand 1N1

Numbers to 20

1. recognise numbers 1-20

2. perform counting onwards and counting backwards 3. recognise the concepts of ordinal numbers and

cardinal numbers

4. recognise the odd and even numbers up to 20

5. recognise the decomposition and composition of numbers 2-18

13.5

Explanatory Notes:

Students are required to read and write the numerals representing the numbers 1-20, to count a group of not more than 20 objects, and to take out a prescribed quantity (not more than 20) of objects from a group of objects.

When performing counting onwards from a number, students have to start from the given number to a larger number in equal steps. Conversely, when performing counting backwards from a number, they have to start from the given number to a smaller number in equal steps.

In this learning unit, students are only required to perform counting by 1.

Students should recognise the concepts of ordinal numbers and cardinal numbers, which mean using numerals to indicate respectively the sequence of events or the order of the arrangement of objects and the quantities of objects. Although teachers may use the terms

“ordinal numbers” or “cardinal numbers” in their explanation, the curriculum does not require students to use these terms.

Students are required to compare the quantities of two groups of objects (Groups A and B), e.g. by using the method of one-to-one correspondence or by counting the numbers of objects in each group, and use sentences “Group A has more objects than Group B” and “Group B has less objects than Group A” to present the result of comparison. Besides, students should recognise the magnitude of numbers (5 is greater than 3) through the result of comparing the quantities of two groups of objects (a group of 5 books has more books than a group of 3 books). In Primary One, students are not required to use the symbols “>” and “<”to record the comparison result, and they are also not required to explicate the result such as Group A has 2 books less than that of Group B, as it is the learning content of Learning Unit 1N2 “Basic

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addition and subtraction”.

Teachers may make use of those activities that are related to students' daily experience as examples to let students recognise the odd and even numbers up to 20, e.g. to discuss the situation when a class of students walking hand in hand in groups of 2.

Recognising the composition and decomposition of numbers not only helps students learn the concepts of addition and subtraction in Learning Unit 1N2 “Basic addition and subtraction”, but also constitutes the foundation knowledge for performing addition with carrying and subtraction with borrowing in the future. The Learning Objective 5 only requires students to compose a number from two 1-digit numbers and decompose a number into two 1-digit numbers. For example, 4 and 8 make 12 (4+8 = 12); and 12 is 4 and 8 (12 = 4+8).

Teachers should arrange hands-on activities to let students familiarise with the composition and decomposition of numbers 2-18 and use various daily life examples to demonstrate their applications. For example, students make use of the fact that 12 is 4 and 8 to understand if 8 dots out of 12 dots are covered, only 4 dots can be seen (i.e. 12 minus 8 equals 4; 12 8 = 4).

Students are required to present the result verbally first and then use text and symbols “+”,

“–” and “=” to record them, in which the terms “addition sign”, “subtraction sign” and “equal sign” are required, but the symbols “+”, “” and “=” here are not used for calculation purpose.

Examples of vocabularies to be learnt: more than , less than , greater than, smaller than, counting onwards, counting backwards, even number, odd number, the first, the second, the third, ... , the twentieth, plus, minus, equals, addition sign, subtraction sign, equal sign, etc.

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Learning Unit Learning Objective Time

Basic addition and subtraction

1. understand the basic concepts of addition and subtraction

2. perform addition and subtraction of numbers within 18 verbally

3. recognise the concept of 0

4. understand the relation between addition and subtraction

5. recognise the commutative property of addition

13.5

In Learning Unit 1N1 "Number to 20", students recognised the decomposition and composition of numbers 2-18. In this learning unit, students are required to use the concepts of decomposition and composition to learn the basic concepts of addition and subtraction.

The concepts associated with addition include merging and adding, while those with subtraction include taking away and comparing, etc. Teachers should use daily life examples to explain the various concepts associated with addition and subtraction to students. Students are required to recognise that the terms “sum” and “difference” denote respectively the results of addition and subtraction.

In this learning unit, students are only required to solve the problems presented mainly by diagrams. Teachers should let students verbally solve the problems before recording their work in horizontal form. The related problems only involved addition and subtraction of those decomposition and composition number pairs within 18 which students learnt in 1N1.

That means operations such as 12+5 and 17–5 are not required. In this learning unit students are not required to learn the column form of addition and subtraction, which is the learning content of Learning Unit 1N4 “Addition and subtraction (I)”.

The concept of 0 is relatively abstract. Teachers should use daily life examples of subtraction to explain the concept of 0 to students. For example, there are 4 books on the desk. If all books are taken away and no books are left on the desk, it means there are 0 books on the desk. Apart from the concept of 0, students are required to recognise the properties of 0.

Teachers should use concrete examples to let students recognise that 0 plus any number or any number plus 0 equals that number; and, any number minus 0 is that number while any number minus itself is 0.

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Students are required to recognise the relation between addition and subtraction in this learning unit. This provides the foundation for recognising how to verify the result of subtraction by using addition in the future. As the objective is not for drilling of calculation, the numbers involved in the related exploratory activities should be within 10, e.g.

2+3 = 5 5–2 = 3 5–3 = 2

Teachers may make use of students’ daily life experience to let them discover the facts such as 5+4 = 4+5. That is to say, commutative property holds for addition. For example, no matter whether mother first gives you 5 books and then gives you another 4 books, or mother first gives you 4 books and then gives you another 5 books, the total number of books you get from mother is the same. Teachers may also provide students with examples with contexts to enable them to discover that commutative property does not hold for subtraction. For example, if there are 5 books on the desk and Ann takes 2 away, then 3 books were left, whereas if there are 2 books on the desk, Ann cannot take 5 from it. It means 5–2 does not equal 2–5.

Although teachers may use the term “commutative property” in their explanation, the curriculum does not require students to use this term. This is because the curriculum focuses on students' ability to make use of the properties of operations in a flexible way during calculations, rather than on their recollection or the rote learning of the names of properties.

Examples of vocabularies to be learnt: total, take away, remaining, addition, subtraction, sum, difference, horizontal form, etc.

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Learning Unit Learning Objective Time

Numbers to 100

1. recognise numbers 21-100

2. recognise the concepts of the units place and the tens place

3. compare the magnitude of numbers 4. perform counting in groups of 2, 5 and 10 5. estimate the quantity of objects

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Similar to the learning of numbers within 20, students are required to recognise the numerals representing the numbers from 21 to 100 by counting, reading and writing.

Students are required to recognise the meaning of the numerals in the units place and the tens place. For example, in the numeral “24”, “2” is in the tens place and stands for 20 while “4”

is in the units place and stands for 4. Teachers do not need to introduce the term “place value”.

In Learning Unit 1N1 “Numbers within 20”, students recognised that 1,3,5,7,9, … 19 are odd numbers, and 2,4,6,8, … 20 are even numbers. At this stage, teachers can let students recognise 0 is an even number by counting backwards from some even numbers, e.g.

8,6,4,2,0, and also recognise how to determine whether a 2-digit number is an odd number or an even number by checking the numeral in its units place. If it denotes an odd number, this 2-digit number is an odd number. If it denotes an even number, this 2-digit number is an even number.

Students should extend the strategy for comparing the magnitude of two numbers to comparing the magnitude of a group of numbers, arranging them in ascending order or descending order. The symbols “<” and “>” will be introduced in Learning Unit 2N1 “3-digit numbers”

Students are required to count the number of objects of quantity less than 100 in groups of 2, 5 and 10, in which calculation is not required. Students can use this counting method to count the value of a group of coins and the time that the minute hand shows on a clock in the future. After students have gained sufficient experience in counting and mastered the skills of counting, teachers should teach students how to estimate the numbers of objects by, for

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example, setting up some reference quantities for estimation. The quantity to be used for estimation should be less than 100.

Examples of vocabularies to be learnt: units place, tens place, the largest, the smallest, estimation, etc.

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Learning Unit Learning Objective Time

Addition and subtraction (I)

1. perform addition of two numbers 2. perform addition of three numbers

3. recognise the associative property of addition 4. perform subtraction of two numbers

5. recognise the column form of addition and subtraction 6. solve simple problems

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In this learning unit, students are required to perform addition of two numbers, addition of three numbers, and subtraction of two numbers, in which the numbers involved are up to two digits. Addition includes carrying, but subtraction doesn’t include borrowing. At this stage, students have not recognised the concept of hundreds place and the results of the addition should therefore be less than 100.

In this learning unit, students are required to recognise the column form of both addition and subtraction. Teachers should let students understand why column form should be in place when calculating addition and subtraction by using the concepts of the units place and the tens place. While performing the addition in column form, teachers can let students use some marks to indicate the number carrying to the tens column. However, this is for helping those students who need to remember the number carrying and not a method that all students must adopt.

In Learning Unit 1N2 “Basic addition and subtraction”, students recognised the relation between addition and subtraction. In this unit, they will recognise how to use this relation to verify the result of subtraction by addition.

Students should recognise that the conventional order of performing addition of three numbers goes from left to right, e.g. 10+6+2 = 16+2 = 18. Teachers may provide students with examples with contexts to enable them to discover the associative property of addition, and design some concrete examples to let students understand that using the associative property of addition can speed up the calculations, so as to motivate their learning interest.

Although teachers may use the term “associative property” in their explanation, the curriculum does not require students to use this term.

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In Learning Unit 1N2 “Basic addition and subtraction”, students solved the problems presented mainly by diagrams. In this learning unit, the problem type will be extended to simple word problems, but it only involves addition or subtraction of two numbers. Problems involving addition of three numbers are tackled in Learning Unit 2N2 “Addition and subtraction (II)”. At this stage, students are not required to explain their calculation with statements.

Examples of vocabularies to be learnt: 1-digit number, 2-digit number, in place, carrying, column form, etc.

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Learning Unit Learning Objective Time

Measures Strand 1M1

Length and distance (I)

1. recognise the concepts of length and distance 2. compare intuitively the lengths of objects and

compare intuitively the distances between objects 3. compare directly the lengths of objects and compare

directly the distances between objects

4. compare the lengths of objects and compare the distances between objects in improvised units

3.5

In this learning unit, students are required to recognise the concepts of length and distance by intuitive comparison, direct comparison and comparison in improvised units. Students are required to understand that they can obtain the comparison results by observation or by tools.

Teachers should introduce the concepts of length and distance in some contexts that students are familiar with, e.g. lining up from the tallest to the shortest.

In Learning Objective 2, students are required to compare the length of objects intuitively.

Teachers should select objects with significant differences in lengths to help students to achieve the learning objective. Students are required to recognise the distance between two objects is the straight distance between them and they should distinguish between distance and route.

Teachers should guide students to recognise the points to note when comparing the lengths of objects directly. For example, the end points of two objects must be aligned.

Students are required to choose appropriate improvised units for taking measurements in particular situations. They are also required to compare the lengths of curve or some curvy parts of objects, such as ropes. Teachers should not ask students to arrange many objects by length at the same time.

Examples of vocabularies to be learnt: length, distance, long, short, tall, wide, narrow, thick, thin, far and near, etc.

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Learning Unit Learning Objective Time

Money (I)

1. recognise the coins in circulation in Hong Kong 2. recognise the notation of marked prices from price

Learning Unit Learning Objective Time

Length and distance (II)

1. recognise centimetre (cm)

2. measure and compare the lengths of objects, and measure and compare the distances between objects in centimetre

3. estimate the result of measurements with ever-ready rulers

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Students have learnt how to compare the lengths of objects and the distances between objects by intuitive comparison, direct comparison and comparison in improvised units. Teachers should help students understand the need for using standard unit through activities.

After introducing the standard unit centimetre (cm), teacher should help students recognise centimetre through various activities. Teachers should teach students the skill on using rulers, e.g. aligning one end of the object with the mark “0” on the ruler, noting the marking of starting point and end point of the object on the ruler, etc. Students are required to select appropriate tools for taking measurement in particular situations.

Students are required to estimate the result of measurements with ever-ready rulers, such as thumb width, length of foot and span, and understand the points to note when using ever-ready rulers. After students have acquired the experience of measurements, teachers should teach them how to estimate the result of measurements and encourage them to estimate the result before measuring.

Examples of vocabularies to be learnt: centimetre (cm), centimetre ruler, tape measure, ever-ready ruler, etc.

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Learning Unit Learning Objective Time

Time (I)

1. tell time to the hour and half hour 2. recognise hour (h)

3. measure and compare the time intervals in hour 4. solve simple problems related to time intervals 5. recognise that there are seven days in a week and the

names of the days of the week

6. recognise that there are 12 months in a year and the names of the months

7. recognise calendars

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In preparing students to grasp the contents of telling time and time intervals, teachers should help students recognise the clock face of an analogue clock through their life experience first.

Students are only required to tell time from analogue clocks using “…o’clock” and “half past…”. Drawing hour hands and minute hands to indicate time is not required. Students are not required to tell time such as “a little after half past nine” and “nearly nine o’clock”.

After introducing the concepts of 1 hour, 2 hours, ... and 12 hours, teachers should help students establish the sense of one hour through activities, e.g. asking students to record the number of pages of a book they can read in one hour.

Students are required to use clocks or stopwatches to measure and compare the time intervals in hours. Each time interval in the measurement must be a whole number, and is not more than 12 hours. Teachers should help students recognise the concepts of quicker and slower by time intervals.

Students are only required to solve the following two types of simple problems: finding the finishing time from the starting time and time interval; finding the time interval from the starting time and finishing time. All time intervals involved must be whole numbers, and not more than 12 hours. For example, Eric goes to bed at half past nine and wakes up after 10 hours. When does he wake up?

Students are required to recognise that there are seven days in a week and the names of the

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days of the week. Teachers do not need to stress whether Monday or Sunday is the first day of the week. Students are required to recognise that there are 12 months in a year and the names of the months. They also need to know that there are about 4 weeks in a month.

Students are required to recognise calendars and get information on dates and the days of the week from a calendar.

Examples of vocabularies to be learnt: …o’clock, half past…, hour hand, minute hand, hour, year, month, day, the days of the week, etc.

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Learning Unit Learning Objective Time

Shape and Space Strand 1S1

3-D shapes (I)

1. recognise the intuitive concepts of prisms, cylinders, pyramids, cones and spheres

6

At the primary level, the starting point for learning the content in Shape and Space Strand is recognising 3-D shapes, as the objects that students come across in their daily life are 3-D objects.

Teachers should provide opportunities for students to touch, stack up, roll and observe real objects or models of some 3-D shapes, in order to help students build up the intuitive concepts of prisms, cylinders, pyramids, cones and spheres.

To enable students to identify 3-D shapes intuitively from their 2-D representation, teachers may provide pictures of the real objects for students to outline the shape of these 3-D shapes, so as to help them connect the 3-D shapes with their 2-D representations.

Teachers should avoid using oblique 3-D shapes such as oblique prisms and oblique cones as examples and need not require students to make a 3-D shapes or recognise the names of different prisms and pyramids. These are respectively the learning contents of Learning Units 5S2 “3-D shapes (III)” and 2S1 “3-D shapes (II)”.

Examples of vocabularies to be learnt: 3-D shape, prism, pyramid, cylinder, cone, sphere, etc.

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Learning Unit Learning Objective Time

2-D shapes

1. recognise the intuitive concepts of points, straight lines and curves

2. draw and make straight lines and curves

3. recognise the basic concepts of triangles, quadrilaterals, pentagons, hexagons and circles

4. draw and make triangles, quadrilaterals, pentagons, hexagons and circles

5. form 2-D shapes by triangles, quadrilaterals, pentagons, hexagons and circles

10

Students are only required to recognise the intuitive concepts of points, straight lines and curves and to identify straight lines and curves intuitively. Teachers may arrange hands-on activities for students to consolidate their conception of straight lines and curves, e.g. the activity of arranging cubes one by one.

Students should recognise that in mathematics, a point has no size. Dots and crosses are daily life symbols commonly used to represent points. The symbols drawn may be different in size, but they only represent the locations of the points which have no size. In mathematics, there is only one straight line passing through two points, however, there are many different curves passing through these two points. Students should recognise that a line has no breadth. Even though students may draw lines with different breadths passing through two fixed points, they represent the same straight line in mathematics. All these mathematical knowledge will be useful for students to grasp the concepts of distance, line segment, centre, and radius, etc.

Teachers may let students use different methods to draw and make straight lines and curves, e.g. tracing the outline of daily life items such as a cup, a book, etc. to draw curves and straight lines, and to use cotton strings to make straight lines and curves.

Students should recognise the basic concepts of triangles, quadrilaterals, pentagons, hexagons and circles. For example, quadrilaterals are formed with four straight lines connected end to end on a plane (the concept of line segments is the learning content of Learning Unit 2S4

“Quadrilaterals (II)”). At the primary level, students are only required to deal with simple shapes. They are not required to recognise the quadrilaterals like . Students may identify

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the type of a 2-D shape by counting its number of sides. The shapes can be convex polygons or concave polygons, but teachers do not have to introduce the concepts of convex polygons and concave polygons to students. Teachers may let students use different methods to draw, different tools or different materials to make 2-D shapes.

Teachers may let students freely create 2-D shapes or request them to form certain 2-D shapes. Apart from letting students know that the type of a shape remains the same even though its orientation changes, students may also appreciate the beauty of geometric shapes as well.

Examples of vocabularies to be learnt: point, straight line, curve, 2-D shape, triangle, quadrilateral, pentagon, hexagon, circle, etc.

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Learning Unit Learning Objective Time

Directions and positions (I)

1. use “over”, “under”, “left”, “right”, “in front of”,

“behind” and “between” to describe relative positions of objects with respect to the observer’s point of view

3.5

In this learning unit, students are required to use “over”, “under”, “left”, “right”, “in front of”, “behind” and “between” to describe relative positions of objects with respect to the observer’s point of view. Teachers should provide students with hands-on activities to enable them to use these terms to describe the relative positions of objects around them.

Students are also required to use “over”, “under”, “left”, “right”, “in front of”, “behind” and

“between” to describe the relative positions of objects in the pictures:

 relative position left and right

Students are only required to describe the position of an object in a picture from their own point of view by using phrases “ on the left” or “ on the right”.

 relative position over and under, in front of and behind

Students are required to use the sentence “ is over/under/in front of/ behind ” to describe the relative positions of objects.

As this is a learning unit in Primary One, solving problems involving complicated relative positions is not required. The information of the picture should be simple.

Examples of vocabularies to be learnt: position, over, under, left, right, in front of, behind, between, etc.

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Learning Unit Learning Objective Time

Further Learning Unit 1F1

Inquiry and investigation

Through various learning activities, discover and construct knowledge, further improve the ability to inquire, communicate, reason and conceptualise mathematical concepts

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This Learning Unit aims at providing students with more opportunities to engage in the activities that avail themselves of discovering and constructing knowledge, further improving their abilities to inquire, communicate, reason and conceptualise mathematical concepts when studying other Learning Units. In other words, this is not an independent and isolated learning unit and the activities may be conducted in different stages of a lesson, such as motivation, development, consolidation or assessment.

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Learning Unit Learning Objective Time

3-digit numbers

1. recognise 3-digit numbers

2. recognise the concept of the hundreds place 3. compare the magnitude of numbers

4. perform counting in groups of 20, 25, 50 and 100 5. estimate the quantity of objects

3.5

In Primary One, students recognised the numbers within 100 and the meaning of the numerals in the units place and in the tens place. In this learning unit, students are required to recognise the concepts of 3-digit numbers and the hundreds place.

Students are required to recognise that ten tens are a hundred and the value represented by the numeral in the hundreds place. Also, they have to read and write the numerals that represent 3-digit numbers by integrating the concepts of units place and tens place, and to count the number (must be a 3-digit number) of objects. For example, when counting the number of cubes: one cube represents a unit, ten cubes form one long rod which represents a ten, and ten long rods form one plate which represents a hundred.

Students are required to recognise how to determine whether a 3-digit number is an odd or even number by checking the numeral in its units place. In this learning unit, students are required to use the symbols “=”, “>” and “<” to express the relation between the magnitude of two numbers. Teachers should let students verbally report the comparison result, e.g. “223 is greater than 124, and 124 is greater than 56”, and then let students use the symbols 223>124>56 to record them.

In Primary One, students recognised how to count the number of objects in groups of 2, 5 and 10. In this learning unit, they are required to perform counting in groups of 20, 25, 50 and 100, in which calculation is not involved. When students have gained sufficient counting experience, teachers should encourage them to estimate before perform counting. The quantity to be used for estimation should be less than 1000.

Examples of vocabularies to be learnt: 3-digit number, hundreds place, greater than, less than, etc.

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Learning Unit Learning Objective Time

Addition and subtraction (II)

1. perform addition of not more than three numbers 2. perform addition by using the commutative and

associative properties of addition

3. perform subtraction of not more than three numbers 4. solve problems

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In Primary One, students performed the addition (including carrying) of not more than three numbers which are of at most two digits. In this learning unit, the addition (including carrying) will be extended to not more than three numbers which are of at most three digits, and the result must be less than 1000.

In Primary One, students recognised the commutative and associative properties of addition.

In this learning unit, teachers may design some concrete examples to let students understand that flexible use of these two properties can speed up the calculations, so as to motivate their learning in mathematics. For example,

399+65+1

= 65+399+1

= 65+400

= 465

Although teachers may use the terms “associative property” and “commutative property” in their explanation, the curriculum does not require students to use these terms.

In this learning unit, subtraction will be extended from subtraction (not including borrowing) of two numbers which are up to two digits in Primary One to subtraction (including borrowing) of not more than three numbers which are at most two digits. Students should recognise that the conventional order of performing subtraction of three numbers goes from left to right, e.g.1072 = 32 = 1.

In Primary One, students recognised that the commutative property does not hold for subtraction. In this learning unit, teachers may provide students with concrete examples to enable them to discover the associative property does not hold for subtraction neither, e.g.

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1072 does not equal 105. Although teachers may use the term “associative property” in their explanation, the curriculum does not require students to use this term.

In this learning unit, students are required to explain their calculation with statements.

Teachers should use contexts related to students’ daily life experience as examples to foster their interest in learning mathematics. Problems involving both “more (less) than” and

“altogether” are tackled in Learning Unit 3N4, e.g. “Andy has 10 pieces of candy and he has 2 pieces less than that of Betty. How many pieces of candy do they have altogether?”

After students have mastered the skills of performing addition with carrying and subtraction with borrowing, teachers should teach them how to estimate the results of calculations so as to let them understand the advantages of estimation and enhance the effectiveness of learning.

Examples of vocabularies to be learnt: borrowing, etc.

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Learning Unit Learning Objective Time

Basic multiplication

1. recognise the basic concept of multiplication 2. understand the multiplication table (0-10) 3. perform basic multiplication

4. recognise the commutative property of multiplication 5. solve problems

11.5

Students are required to recognise that the basic concept of multiplication is repeated addition. Since students have already learnt how to count in groups of 2, 5 and 10 in Primary One, teachers may start with the multiplication of 2, 5 and 10, and let students understand the multiplication table of 2, 5, 10 by means of hands-on counting activities.

A multiple can be written on the left side or right side of the multiplication sign. For example, if we use “2 groups of 3 is 6” to represent the quantity of black dots below, the multiplication can be expressed as “3×2 = 6” or “2×3 = 6”. If the basic concept of multiplication and repeated addition are linked, we have 3+3 = 3×2 or 3+3 = 2×3.

To avoid students’ confusion about the two ways of expressing multiplication, teachers may first select one expression of multiplication to let students recognise the concept of multiplication. After students have recognised the associative property of multiplication, teachers may explain to students that both expressions are correct. That is to say, the multiple can be written on the left side or right side of the multiplication sign.

Teachers may provide students with examples with contexts to enable them to discover the commutative property of multiplication. Although teachers may use the term "commutative property" in their explanation, the curriculum does not require students to use this term.

Students should first compile the multiplication table of 2-10, then compile the multiplication table of 1 and 0. Students are required to recognise that:

 any number times 1 equals that number

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 any number times zero equals zero

Students may use “3×2” or “ 2×3” to represent two groups of three, for example, in solving the following problem:

Each box has 3 pieces of cake, how many pieces of cake are there in 2 boxes?

The expression can be written as “3×2” or “2×3”.

In this learning unit, students are required to recognise that the term “product” denotes the result of multiplication.

Examples of vocabularies to be learnt: multiplication sign, product, multiplication table, etc.

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Learning Unit Learning Objective Time

4-digit numbers

2. recognise the concept of the thousands place

3. perform counting in groups of 200, 250, 500 and 1000

4. compare the magnitude of numbers

2.5

Similar to the learning of 3-digit numbers, students are required to recognise the numerals representing 4-digit numbers by counting, reading and writing. Students should recognise that ten hundreds are a thousand and the value represented by the numeral in the thousands place.

Students are required to count onwards from a designated 3-digit number to a designated 4-digit number, or to count backwards from a designated 4-digit number to a designated 3-digit number, e.g. counting onwards from 990 one by one to 1000, counting backwards by hundreds from 1240 to 540. Students should recognise how to determine whether a 4-digit number is an odd or even number by checking the numeral in its units place. When comparing the magnitude of two numbers, teachers should let students verbally report the comparison result, and then let them use the symbols to record them.

In this learning unit, students are required to perform counting in groups of 200, 250, 500 and 1000, in which calculation is not involved. Students can use this counting method to count a group of notes and coins with a higher total value and to read the measurement on a beaker and a scale. This learning unit should be taught prior to the Learning Unit 2M3 “Money (II)”.

Examples of vocabularies to be learnt: 4-digit number, thousands place, etc.

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Learning Unit Learning Objective Time

Addition and subtraction (III)

1. perform subtraction of two numbers

2. perform mixed operations of addition and subtraction of three numbers

3. solve problems

8.5

In this learning unit, subtraction will be extended to two numbers which are up to three digits, and borrowing is included. In mixed operations of addition and subtraction of three numbers (includes addition of three numbers and subtraction of three numbers), addition and subtraction of numbers more than three digits are not required, which means operations like 567+489−123 = 1056−123 are excluded. Students should recognise that the conventional order from left to right when performing addition or subtraction should also be applied in mixed operations of addition and subtraction of three numbers, such as 72+3 = 5+3 = 8.

This learning unit should be arranged after Learning Unit 2M3 “Money (II)”. Problems related to money are required. When dealing with calculations involving dollars and cents, the calculations should not involve decimals. This is because Primary Two students have not yet recognised the concept of decimals.

e.g. Amy pays 10 dollars for a ruler that costs 3 dollars 80 cents. How much change should the shopkeeper give?

Horizontal form: 10 dollars–3 dollars 80 cents = 6 dollars 20 cents

Primary Two students should solve basic word problems of mixed operations of addition and subtraction. Problems involving both “more (less) than” and “altogether” are tackled in Learning Unit 3N4, for example: “Andy has 10 pieces of candy and he has 2 pieces more

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than that of Betty. How many pieces of candy do they have altogether?”

After students have mastered the skills of performing mixed operations of addition and subtraction, teachers should teach them how to estimate the results of calculations so as to let them understand the advantages of estimation and enhance the effectiveness of learning.

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Learning Unit Learning Objective Time

Basic division

1. recognise the basic concept of division 2. perform basic division

3. recognise the relation between multiplication and division

4. solve problems

11.5

The concepts of division include sharing and grouping. Teachers may introduce the concepts of division by games of distributing things. The examples used should be limited to those that can be solved by “the multiplication table (0-10)”.

In activities of distributing things by division, no matter whether sharing or grouping is involved, situations with remainder and without remainder should be included. In the process of recognising the concept of division, students are required to present verbally the result of division, and then use diagrams, text and symbols to record the result, in order to master the relation between the concept of division and its expression.

Record “8÷2 = 4” in diagram:

Record in text: eight is divided into two equal portions, each portion is four.

Record in symbol：

Students are required to recognise that the term “quotient” denotes the result of division and the concepts of dividend, divisor and remainder. Teachers should discuss with students the reason why the remainder is always smaller than the divisor and use the daily life examples to illustrate how remainders are handled in the problems. Students should recognise that

8÷2 = 4

(read as: 8 divided by 2 equals 4)

7÷3 = 2…1

(read as: 7 divided by 3 equals 2 with remainder 1)

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when 0 is divided by any non-zero number, the result is 0.

Teachers should use concrete examples of division and multiplication to enable students to find out the relation between multiplication and division, so that they can verify the result of division by multiplication.

Teachers may provide students with examples with contexts to enable them to discover that commutative property does not hold for division, e.g. 2÷8 does not equal 8÷2. Although teachers may use the term “commutative property” in their explanation, the curriculum does not require students to use this term.

Examples of vocabularies to be learnt: equal portion, divided by, dividend, divisor, quotient, remainder, division sign, etc.

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Learning Unit Learning Objective Time

Length and distance (III)

1. recognise metre (m)

2. measure and compare the lengths of objects, and measure and compare the distances between objects in metre

3. record the lengths of objects and the distances between objects in appropriate measuring units 4. estimate the result of measurements with ever-ready

rulers

5

Students have learnt how to measure and compare the lengths of objects, and measure and compare the distances between objects in centimetre. Teachers should guide students to understand the need for using a larger unit of length.

After introducing the standard unit metre (m), teacher should help students to recognise metre through various activities. Students are required to recognise that 100 cm equal to 1 m.

Students are required to measure and compare the lengths of objects, and measure and compare the distances between objects in metre by choosing and using appropriate measuring tools, such as metre rulers, tape measures and trundle wheels. Teachers should teach students the skills in using measuring tools and discuss with them on how to record the lengths of objects and the distances between objects in appropriate measuring units through real life examples.

At Key Stage 1, students may use ways such as 110 cm and 1 m 10 cm for recording lengths and distances, and convert 1 m 10 cm to 110 cm. However, problems on converting a single unit to a compound unit are not required. Recording lengths and distances using decimals are dealt with in Learning Unit 4N7 Decimals (I).

Students are required to estimate the result of measurements with ever-ready rulers, such as arm span, length of arm and pace length, and understand the points to note when using the ever-ready rulers. After students have acquired the experience of measurements, teachers should teach them how to estimate the result of measurements and encourage them to estimate the result before measuring.

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Examples of vocabularies to be learnt: metre (m), metre ruler, tape measure, trundle wheel, etc.

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Learning Unit Learning Objective Time

Time (II)

1. tell time to the nearest minute 2. recognise minute (min)

3. measure and compare the time intervals in minutes 4. solve simple problems related to time intervals 5. recognise that there are 24 hours in a day

6. recognise the concepts of morning (a.m.) and afternoon (p.m.)

7. tell time using “morning”, “afternoon”, “noon” and

“midnight”

8. recognise the number of days in each month

9. recognise the numbers of days in a common year and a leap year

10. solve problems related to the numbers of days spent on events

5.5

Students have learnt how to tell time from analogue clocks using “…o’clock” and “half past…”. Teachers should make use of students’ prerequisite knowledge to help them tell time to the nearest minute. Students are required to tell time from analogue clocks and digital clocks. Drawing hour hands and minute hands to indicate time is not required. Teachers should also help students consolidate the concepts of ordering of time, later and earlier.

After introducing the concepts of 1 minute, 2 minutes ... and 60 minutes, teachers should help students establish the sense of one minute through activities. Students should recognise that there are 60 minutes in 1 hour.

Students are required to measure and compare the time intervals in minutes by using clocks or stopwatches. Each time interval in the measurement must be a whole number and is not more than 60 minutes.

Students are only required to solve simple problems on finding the finishing time from the starting time and time interval, and finding the time interval from the starting time and

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finishing time. All time intervals involved must be whole numbers and not more than 60 minutes. For example, Peter agrees to meet with friends at 3:50 in a restaurant but he is 20 minutes late. When does he arrive the restaurant?

Students are required to recognise that there are 24 hours in a day, and the concepts of morning (a.m.) and afternoon (p.m.). They are also required to tell time using “morning”,

“afternoon”, “noon” and “midnight” from analogue clocks and digital clocks by observing the context of the situations. Teachers may let students recognise that the time of noon and midnight can be written as “12:00 noon” and “12:00 midnight” respectively. Students are required to know time is continuous and they should also know that 12:00 midnight is the end of a day as well as the start of the next day.

Students are required to recognise the number of days in each month. By observing the calendars for several consecutive years, students should discover that the number of days in February may not be the same every year. The number of days in a common year and a leap year could be introduced at that time. Given any two of the starting date, finishing date and number of days spent on an event, students are required to find the unknown quantity/date by reading the calendar.

Examples of vocabularies to be learnt: minute, morning, afternoon, noon, midnight, common year and leap year, etc.

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Learning Unit Learning Objective Time

Money (II)

1. recognise the notes in circulation in Hong Kong 2. recognise the notation of marked prices of greater

amounts from price tags

3. recognise the use of money in daily life

5

Students have recognised the coins in circulation in Hong Kong in Primary One. Teachers should help students recognise the notes in circulation in Hong Kong through various activities, e.g. observing the colours, patterns, words and numbers printed on the notes, and comparing the size of the notes, etc. Students should recognise the patterns of notes in circulation, e.g. the notes with same denominations but issued by different note-issuing organisations.

Teachers should help students recognise the notation of marked prices of greater amounts from price tags through examples with context, e.g. reading the marked price such as

“$23.50” as “twenty-three dollars and fifty cents”. Students are required to read marked prices not more than 1000 dollars. Teachers are not required to mention the place value of decimal numbers. Students are not required to write marked prices. However, teachers should write marked prices in the form such as “$23.00”.

Students should recognise the use of money in daily life. It includes counting a group of notes and coins with a total not more than 1000 dollars. However, students are not required to write numerical expressions to show the working. Students are only required to do following exchange:

(1) the exchange involving only 10-dollar, 20-dollar, 50-dollar and 100-dollar notes. For example, one 10-dollar note, two 20-dollar notes and one 50-dollar note can be exchanged for one 100-dollar note;

(2) the exchange involving only 100-dollar, 500-dollar and 1000-dollar notes. For example, one 1000-dollar note can be exchanged for two 500-dollar notes;

Students are not required to do exchange between the notes across group (1) and group (2).

For example, exchanging one 500-dollar note for ten 50-dollar notes is not required.

Problems on giving change are tackled in Learning Unit 2N5 “Addition and subtraction

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(III)”. Problems of decimal operations involving money are tackled in Learning Units 4N8

“Decimals (II)”, 5N4 “Decimals (III)” and 6N1 “Decimals (IV)”.

Examples of vocabularies to be learnt: note and money, etc.

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Learning Unit Learning Objective Time

3-D shapes (II)

1. recognise the concept of faces of a 3-D shape

2. recognise the names of different prisms and pyramids

3

In this learning unit, the concepts of 3-D shapes that students have to recognise will be extended from the intuitive concepts of 3-D shapes in Primary One to faces of 3-D shapes, which is the basic structure of 3-D shapes.

Teachers may arrange hands-on activities, such as rubbing the outline of 3-D shapes on a piece of paper, to enable students to recognise the concept of faces of 3-D shapes, which include the concepts of the bases and lateral faces^ of a prism and a pyramid, the bases and curved surface of a cylinder and a cone and the curved surface of a sphere. However, teachers should avoid using oblique 3-D shapes as examples. Students are required to name different prisms and pyramids, such as “quadrilateral prism” and “pentagonal pyramid”, by the shapes of their bases. In this learning unit, students are not required to further recognise quadrilateral prisms as cubes or cuboids, as this is the learning content of Learning Unit 5S2

“3-D shapes (III)”.

Examples of vocabularies to be learnt: face, base, lateral face, curved surface, square, rectangle, triangular prism, quadrilateral prism, pentagonal prism, hexagonal prism, triangular pyramid, quadrilateral pyramid, pentagonal pyramid, hexagonal pyramid, etc.



Students are required to recognise that the lateral faces of prisms are rectangles or squares. But the concepts and properties of rectangles and squares are the learning content of Learning Unit 2S4 “Quadrilaterals (I)”. In this Learning Unit, students may thus intuitively recognise that the lateral faces of prisms are rectangles or squares, or just describe the lateral faces as quadrilaterals.

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Learning Unit Learning Objective Time

Angles

1. recognise the concept of angles 2. compare the sizes of angles

3. recognise the concepts of right angles and perpendicular lines

4. draw and make perpendicular lines

5. recognise the concepts of acute angles and obtuse angles

6. draw and make angles of different sizes

7

Teachers may introduce the concept of angles to students by making use of the daily life examples, such as turning the hands of a clock face or two sticks.

Students are required to recognise that the size of an angle is the extent of the opening of the two sides of the angle. Students are required to compare the sizes of angles and recognise that the size of an angle is independent of the length of the sides drawn.

Teachers may use the objects in daily life to let students recognise the concept of right angles, e.g. the right angles on book corners, on a piece of paper, on a ruler, etc. Students should know how to use tools such as set squares to identify right angles. They should know that acute angles are smaller than a right angle and obtuse angles are larger than one right angle, but smaller than two right angles. The learning content involving the unit “degrees”, e.g. right angles equal to 90^o, and the concepts of straight angles, round angles and reflex angles are dealt with in Learning Unit 6M1 “Angles (degree)”.

Teachers may let students use different methods to draw and make angles of different sizes, e.g. drawing angles along the corner of a book and a photo frame and making angles on the pin-board and by folding paper etc.

Students recognised the intuitive concept of straight lines in Primary One. In this learning unit, students should recognise that perpendicular lines intersect at right angles, and they should know how to use tools such as set squares to identify perpendicular lines.

Teachers may let students use different tools such as rulers and set squares to draw

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perpendicular lines. Drawing them under the following conditions is required:

 through a given point on the straight line, draw a straight line perpendicular to the given straight line

 through a given point not on the straight line, draw a straight line perpendicular to the given straight line

Students are required to recognise that when given a straight line and a point not lying on the straight line, amongst the lines joining the given point and the points on the given straight line, the line perpendicular to the given line has the shortest length. The length is used to denote the distance from the point to the straight line.

Examples of vocabularies to be learnt: angle, right angle, acute angle, obtuse angle, perpendicular to each other, perpendicular lines, ruler, set square, etc.

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Learning Unit Learning Objective Time

Directions and positions (II)

1. recognise the four main directions: east, south, west and north

2. use a compass to measure directions

2.5

In this learning unit, students are required to recognise the four main directions: east, south, west and north, and their respective short forms “E”, “S”, “W” and “N”. Students are also required to use the sentence “ is to the east/south/west/north of ” to describe relative positions of objects.

Students are required to use the compass to measure the four main directions and walk along the direction indicated by the compass. As a further application to the relative positions left and right of objects learnt in Primary One, request of turning left or right may be involved.

Examples of vocabularies to be learnt: direction, east, south, west, north, compass, etc.

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Learning Unit Learning Objective Time

Quadrilaterals (I)

1. recognise the concept of line segments 2. recognise the concept of quadrilaterals

3. recognise the concepts and basic properties of squares and rectangles

4. draw and make squares and rectangles

9

Students are required to recognise that amongst the lines (shown in the diagram below), joining two given end points, the straight one is called “line segment”. It has the shortest length amongst the lines joining the two given end points. Students are not required to use the term “end point”.

In Learning Unit 1S2 “2-D shapes”, students had a preliminary understanding that a quadrilateral is formed with four straight lines connected end to end on a plane. In this learning unit, students should recognise that these straight lines are line segments. They are the four sides of the quadrilateral. Students are also required to recognise the concepts of opposite sides and adjacent sides of a quadrilateral, which are basic knowledge for their further study of the properties of different quadrilaterals. In this learning unit, students should recognise the basic properties of squares and rectangles.

The basic properties of squares include:

 the four angles are right angles

 the four sides are equal in length The basic properties of rectangles include:

 the four angles are right angles

 the opposite sides are equal in length

Students should know how to identify squares and rectangles according to the above

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properties. In this learning unit, teachers should avoid using squares as examples to describe the properties of rectangles. The inclusion relation between squares and rectangles is dealt with in Primary Four.

Teachers may provide students with different kinds of papers such as dot grid papers, grid papers and blank papers, etc. to draw squares and rectangles of different sizes. Teachers may also provide students with different materials such as cotton strings, straws and geometric strips, etc. to make squares and rectangles of different sizes.

Examples of vocabularies to be learnt: line segment, opposite side, adjacent side, etc.

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Learning Unit Learning Objective Time

Data Handling Strand 2D1

Pictograms

1. recognise pictograms 2. interpret pictograms 3. construct pictograms

3.5

This learning unit is the first topic in Data Handling for students. Before introducing pictograms, teachers should let students recognise the importance of the organisation and representation of data through concrete examples so as to lay the foundation for learning other topics in Data Handling. For example, teachers may show certain number of different 2-D shapes on the blackboard randomly as below.

Then, teachers may ask students to find or compare the number of designated 2-D shapes on the blackboard and let them express the difficulties on telling the results. Teachers should guide students to understand the importance of the organisation and representation of data and introduce some advantages of using pictograms to present data afterwards.

In this learning unit, students are only required to interpret and construct the pictograms using the one-to-one representation. Pictograms in horizontal and vertical forms are required.

Teachers should select some contexts that students are familiar with when asking students to interpret and construct pictograms. There should not be too much data and items involved in a pictogram.

Examples of vocabularies to be learnt: pictogram, title, each…stands for…, item, etc.

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Learning Unit Learning Objective Time

Further Learning Unit 2F1

Inquiry and investigation

Through various learning activities, discover and construct knowledge, further improve the ability to inquire, communicate, reason and conceptualise mathematical concepts

10

This Learning Unit aims at providing students with more opportunities to engage in the activities that avail themselves of discovering and constructing knowledge, further improving their abilities to inquire, communicate, reason and conceptualise mathematical concepts when studying other Learning Units. In other words, this is not an independent and isolated learning unit and the activities may be conducted in different stages of a lesson, such as motivation, development, consolidation or assessment.

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Learning Unit Learning Objective Time

5-digit numbers

2. recognise the concept of the ten thousands place 3. compare the magnitude of numbers

2.5

In Primary Two, students recognised 3-digit numbers and 4-digit numbers, and the meaning of the numeral in the hundreds place and the thousands place. In this learning unit, students are required to recognise 5-digit numbers and the concept of the ten thousands place. Similar to the learning of 3-digit numbers and 4-digit numbers, students are required to recognise the numerals representing 5-digit numbers by counting, reading and writing. They should recognise that ten thousands are one ten thousand and the meaning of the numeral in the ten thousands place.

Students are required to count onwards from a designated 4-digit number or 5-digit number to another designated 5-digit number, or to count backwards, e.g. counting onwards from 9990 one by one to 10000, and counting backwards by thousands from 12400 to 8400, etc.

Students should recognise how to determine whether a 5-digit number is an odd or even number by checking the numeral in its units place.

Students are required to compare the magnitude of numbers and use the symbols “=”, “>”

and “<” to record the result.

Examples of vocabularies to be learnt: 5-digit number, ten thousands place, etc.

Explanatory Notes to