Explanatory Notes to

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Mathematics Education Section Curriculum Development Institute Education Bureau

2020

Explanatory Notes to

Primary Mathematics Curriculum

(Key Stage 2)

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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 4N1

Multiplication (II)

1. perform multiplication

2. perform multiplication by using the commutative and associative properties of multiplication

3. solve problems

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Explanatory Notes:

In Primary Three, students have learnt how to perform the multiplication of a 1-digit number and a 2-digit number and that of a 1-digit number and a 3-digit number. In this learning unit, multiplication will be extended to:

 2-digit number × 2-digit number

 3-digit number × 2-digit number (or 2-digit number × 3-digit number) Multiplication with carrying is required.

The requirements on the learning of multiplication at Key Stage 2 is similar to that at Key Stage 1. Students are required to both master the steps of performing multiplication in column form and understand the principle behind. Teachers should let students know that no matter it is the multiplication of a 2-digit number and a 2-digit number, or that of a 2-digit number and a 3- digit number, or that of a 2-digit number and a multi-digit number, the principle of operations remains the same.

Students have learnt the commutative property and the associative property of multiplication in Primary Two and Primary Three respectively. In this learning unit, teachers may design some concrete examples to facilitate students understand that flexible use of these two properties can speed up some calculations, so as to raise their interest in learning mathematics. For example,

25 × 53 × 4

＝ 53 × 25 × 4

＝ 53 × (25 × 4)

＝ 53 × 100

＝ 5300

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

After students have mastered the skills of performing multiplication, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

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Learning Unit Learning Objective Time

Division (II)

1. perform division

2. recognise the concept of divisibility 3. solve problems

6

At Key Stage 1, the divisors in division are 1-digit numbers. In this learning unit, the division will be extended to:

 2-digit number ÷ 2-digit number

 3-digit number ÷ 2-digit number

Division with borrowing and division involving remainder are required.

The requirements on the learning of division at Key Stage 2 is similar to that at Key Stage 1.

Students are required to both master the steps of performing division in column form and understand the principle behind. Teachers should let students know that no matter it is the division of a 2-digit number divided by a 2-digit number, or that of a 3-digit number divided by a 2-digit number, or that of a multi-digit number divided by a 2-digit number, the principle of operations remains the same.

Students are required to recognise that 0, 1, 2, 3, 4, … are whole numbers* and the concept of divisibility. For example, 24 is divisible by 3 because 24 ÷ 3 = 8. At the primary level, students need to recognise the divisibility tests for 2, 3, 5 and 10, but their proofs are not required by the curriculum. The divisibility tests for 4, 6, 8 and 9 are the learning content at junior secondary level. Learning divisibility helps students to enhance their computation skills. For example, when performing multiplication and division of fractions, students can find the common factors of some numbers in a quicker way.

At Key Stage 1, students determined whether a whole number is odd or even by its units digit.

In this learning unit, students are required to further recognise the concepts of odd and even numbers through the divisibility by 2.

* Directed numbers belong to the learning content of the junior secondary Mathematics curriculum. In the primary Mathematics curriculum, “whole numbers” refers to non-negative integers. Teachers need not introduce the term “positive integers” to primary students.

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After students have mastered the skills of performing division, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

Examples of vocabularies to be learnt: whole number, divisible, even number, odd number, etc.

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Learning Unit Learning Objective Time

Multiples and factors

1. understand the concept of multiples 2. understand the concept of factors

3. understand the relation between factors and multiples 4. recognise the concepts of prime numbers and

composite numbers

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Explanatory Notes: (In this learning unit, numbers refer to positive integers only.)

Teachers should let students understand the concept of multiples through the concept of multiplication. For example, as 5×1=5 (or 1×5=5), 5×2=10, 5×3=15, 5×4=20, the numbers 5, 10, 15, 20 are the multiples of 5. But students are not required to recognise that 0 is a multiple of 5.

Teachers should let students understand the concept of factors through the concept of divisibility. For example, as 20 is divisible by 1 and 5, but not divisible by 3, 1 and 5 are the factors of 20 but 3 is not.

Students should find the multiples of a number by multiplication, such as the first multiple, the second multiple, the third multiple… of the number. They should find all the factors of a number. They should determine whether a given number is a multiple or a factor of another number. Teachers should also let students conclude that 1 is the factor of any numbers while all numbers are multiples of 1. But the concepts of the multiples and factors of 0 are not required by the curriculum.

Students are required to understand that the concepts of multiples and factors are closely related. For example, 8 is a multiple of 2 and 2 is a factor of 8. Teachers should let students grasp the relation between factors and multiples through the concept of divisibility. For example, as 51 is divisible by 3, 3 is a factor of 51and 51 is a multiple of 3.

Students are required to recognise that when a number has only two factors (i.e. 1 and that number), it is a prime number. 1 has only one factor while a composite number has more than two factors. Teachers should explain to students that 1 is neither a prime number nor a

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composite number.

Students should use the concept of divisibility to determine whether a given number within 100 is a prime number. They should find all the prime numbers within 1 to 100 by the Sieve of Eratosthenes.

Examples of vocabularies to be learnt: multiple, factor, prime number, composite number, Sieve of Eratosthenes, etc.

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Learning Unit Learning Objective Time

Common multiples and common factors

1. understand the concepts of common multiples and common factors

2. understand the concepts of the least common multiple and the highest common factor

3. find the least common multiple and the highest common factor of two numbers by listing their multiples and factors

4. find the least common multiple and the highest common factor of two numbers by using short division

7.5

Explanatory Notes: (In this learning unit, numbers refer to positive integers only.)

In this learning unit, students are required to understand the concepts of common factors, common multiples, the highest common factor, and the least common multiple of two numbers.

They are required to recognise the short forms “H.C.F.” and “L.C.M.” for the highest common factor and the least common multiple respectively.

Students should find the L.C.M. and the H.C.F. of two numbers by listing their multiples and factors and by short division. The curriculum does not require students to understand the reasons why short division works or the proof for it. Yet, teachers may consider to use examples to illustrate the principle behind short division according to students’ abilities.

During the discussion of finding the L.C.M. of two numbers by listing their multiples, teachers should let students discover the relation between L.C.M. and other common multiples through observation. For example, L.C.M. of 4 and 6 is 12, and their third common multiple is 36, which is 3 times the L.C.M. However, the curriculum does not require students to recognise the relation between the H.C.F. of two numbers and their other common factors, nor to recognise that the product of two numbers equals the product of their H.C.F. and L.C.M.

Teachers may help students to explore if the greater number of two numbers is the multiple of the smaller number, then the H.C.F. is the smaller number and the L.C.M. is the greater number.

For example, the H.C.F. of 2 and 4 is 2. The L.C.M. of 2 and 4 is 4. Teachers may also consider to discuss with students the case when 1 is the only common factor of two numbers.

Examples of vocabularies to be learnt: common multiple, common factor, the least common multiple, the highest common factor, H.C.F., L.C.M., listing, short division, etc.

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Learning Unit Learning Objective Time

Four arithmetic operations (II)

1. recognise the distributive property of multiplication 2. perform mixed operations of not more than four

numbers

3. perform mixed arithmetic operations of not more than five numbers

4. solve problems

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Students are required to recognise the distributive property of multiplication and its applications. Teachers should let students understand that flexible use of the distributive property of multiplication can speed up some calculations, so as to raise students’ interest in learning mathematics. For example,

23×17+23×13 4 ×(25+8)

= 23×(17+13) = 4×25+4×8

= 23×30 = 100+32

= 690 = 132

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

Students should perform the mixed operations of not more than four numbers, the mixed operations include:

 mixed operations of division and addition

 mixed operations of division and subtraction

 mixed operations of division and multiplication

Students are required to recognise the order of performing mixed operations, including the mixed operations involving brackets. For students with adequate abilities, teachers may discuss with them through concrete examples whether division possess the distributive property.

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Students should also be able to perform mixed arithmetic operations of not more than five numbers. Through concrete examples involving at least two different operations of addition, subtraction, multiplication and division, teachers should let students know how to follow the conventional order of mixed operations in performing calculations with or without brackets.

And it is to let students understand that the flexible use of the properties of addition and multiplication can speed up some calculations. For example:

48×6+42

= 48×(5+1)+42

= 240+48+42

= 240+90

= 330

At the primary level, mixed operations may involve more than one pair of brackets, such as (2+3)  (5+3), but operations involving multiple levels of brackets, such as (4− (2−1))3, are not required.

Students should able to solve problems involving direct proportion by the unitary method. For example, “72 dollars can buy six pencils. How much do eight pencils cost?” or “72 dollars can buy six pencils. How many pencils can be bought with $120?”. Teachers can guide students to find the price of a pen first, and use the result to find the required cost or number of pencils.

The curriculum does not require students to use the term “direct proportion”. Teachers should encourage students to solve complicated problems by parts.

After students have mastered the skills of performing mixed arithmetic operations, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

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Learning Unit Learning Objective Time

Fractions (II)

1. recognise the concepts of proper fractions, improper fractions and mixed numbers

2. perform the interconversions between improper fractions and mixed numbers

3. recognise the concepts of expanding fractions and reducing fractions

4. compare the magnitude of fractions with the same denominators

5. perform addition and subtraction of not more than three fractions with the same denominators

6. perform mixed operations of addition and subtraction of three fractions with the same denominators

7. solve problems

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Students are required to recognise the concepts of proper fractions and improper fractions.

They are required to recognise that when the numerator of an improper fraction is not a multiple of its denominator, the fraction can be written as a combination of a whole number and a proper fraction, which is called a “mixed number”. Thus, students are required to recognise that a mixed number is the sum of a whole number and a proper fraction. Moreover, students should perform the interconversions between improper fractions and mixed numbers (or whole numbers), e.g. the interconversion between 6

2 and 3.

In Primary Three, students have learnt the concept of equivalent fractions not greater than 1.

In this learning unit, teachers should let students discover that when the numerator and denominator of a fraction are multiplied or divided by the same non-zero whole number, the value of the fraction remains unchanged, so as to recognise the concepts of expanding fractions and reducing fraction. Students are also required to recognise “fractions in the lowest terms”

are those fractions with 1 as the H.C.F. of their numerators and denominators, such as, 2 3 ,

5 2

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21 2 .

Students should compare the magnitudes of fractions with the same denominator. They should also be able to compare the magnitudes of fractions and whole numbers, such as comparing the magnitudes of 11

4 , 7 4 ,

3

4 and 2.

Teachers should let students understand the concepts of addition and subtraction of fractions through the concepts of addition and subtraction of whole numbers. Students should perform addition and subtraction of not more than three fractions and mixed operations of addition and subtraction of three fractions with the same denominator. Performing addition, subtraction and their mixed operations for fractions and whole numbers are required, such as 5 + 1

3 , 3–11

4 and 3–1 1 4 +

3

4 . If the result of calculations is a fraction, it should be reduced to the lowest terms. If it is greater than 1, it may be expressed as a fraction in the lowest terms, either as a mixed number or an improper fraction, such as 5

2 and 2 1 2 .

After students have mastered the skills of performing addition, subtraction and mixed operations of addition and subtraction of fractions with the same denominator, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

Examples of vocabularies to be learnt: proper fraction, improper fraction, mixed number, expanding fraction, reducing fraction, fraction in the lowest term, etc.

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Learning Unit Learning Objective Time

Decimals (I)

1. recognise the concept of decimals

2. recognise the concepts of tenth, hundredth, thousandth and ten thousandth places

3. compare the magnitude of decimals

4. recognise the daily life applications of decimals

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Teachers should let students recognise that the decimal point in a decimal divides the decimal into the integer part and the decimal part. Teachers should let students recognise the concepts of tenths, hundredths, thousandths and ten thousandths places of decimals through the concept of fractions. Students are required to recognise the meaning of the numerals in tenths, hundredths, thousandths and ten thousandths places, so as to understand that decimals and fractions are two forms of expressing numbers and the relation between them.

Students are required to recognise that when reading a decimal, the numerals in the decimal part are read one by one from left to right, such as 30.102 being read as “thirty point one zero two". They are also required to recognise that either adding "0" to the end of the decimal or deleting "0" from the end of the decimal does not change the value of the decimal.

Students are required to recognise the classification of decimals. For example, both 0.5 and 24.7 are decimals of 1 decimal place where as 0.15 and 4.07 are decimals of 2 decimal places.

Similarly, 0.135 and 10.315 are decimals of 3 decimal places where as 0.1224 and 20.1224 are decimals of 4 decimal places.

Students should compare the magnitudes of decimals, and compare the magnitudes of whole numbers, fractions (with denominators being factors of 100) and decimals, e.g. comparing the magnitudes of 12.35, 12.205, 120, and comparing the magnitudes of 12.05, 121

4 , 120.

Students are required to recognise the daily life applications of decimals, e.g. the representation of amounts of money, length, capacity, etc. Students should interconvert between units in measurement. For example:

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1.234L = 1234mL ;

23 dollars 50 cents = 23.5dollars .

Complicated interconversions between units are not required. For example:

0.4 km = 40000 cm .

Interconversion between the units of time is dealt with in Learning Unit 6M4 “Speed”.

Examples of vocabularies to be learnt: decimal point, tenths place, hundredths place, thousandths place, ten thousandths place, etc.

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Learning Unit Learning Objective Time

Decimals (II)

1. perform addition and subtraction of not more than three numbers

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

3. solve problems

4.5

Teachers should let students understand the concepts of addition and subtraction of decimals through the concepts of addition and subtraction of whole numbers.

Students should perform addition, subtraction and mixed operations of whole numbers and decimals of at most 2 decimal places. Teachers should let students recognise that the steps of performing addition and subtraction of decimals are similar to those of addition and subtraction of whole numbers. They are required to recognise that the commutative and associative properties of addition for whole numbers also hold for decimals. Teachers should teach students to write the whole numbers as decimals when necessary, such as 4 = 4.00.

The numbers of digits involved in the addition and subtraction of decimals should not exceed that required for the addition and subtraction of whole numbers in Learning Unit 3N4 “Four arithmetic operations (I)”, i.e. four digits. For example, the following addition and subtraction of decimals are not required:

 1.2345 + 5.6

 123.4 + 56.78

 1234 – 5.6

After students have mastered the skills of performing addition, subtraction and mixed operations of addition and subtraction of decimals, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

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(example) (non-example)

Learning Unit Learning Objective Time

Measures Strand 4M1

Perimeter (I)

1. recognise the concept of perimeter

2. measure and compare the perimeters of 2-D shapes 3. recognise and use the formulae for finding the

perimeters of squares and rectangles

6.5

In this learning unit, students have to recognise the concept of perimeter. Teachers may guide students to recognise the concept by using different 2-D shapes as examples and non-examples, such as:

Students have learnt the concept of length and different units of length at Key Stage 1. Teachers may let students measure and compare the perimeters of different simple 2-D shapes (including regular and irregular shapes) for consolidating their recognition on the connection between length and perimeter. However, students are not required to find the perimeters of non-simple 2-D shapes such as:

After students have acquired the experience of making measurements, teachers should teach them how to estimate the results of measurements and encourage them to estimate the result before measuring.

In Learning Objective 3, teachers may guide students to recognise the formulae for finding the perimeters of squares and rectangles through some inquiry activities, such as measuring and recording the lengths of sides and perimeters of some squares and rectangles. Students are required to use the formulae for finding the perimeters of squares and rectangles to solve problems.

Students are required to find the perimeters of 2-D shapes formed by squares and rectangles.

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Although operations of the related problems may involve more than five numbers, teachers should avoid selecting and designing problems with complicated calculations for students to solve.

Examples of vocabularies to be learnt: perimeter, length of a side, etc

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Learning Unit Learning Objective Time

Area (I)

1. recognise the concept of area

2. compare intuitively the areas of 2-D shapes 3. compare directly the areas of 2-D shapes

4. compare the areas of 2-D shapes in improvised units 5. recognise square centimetre (cm²) and square metre

(m²)

6. measure and compare the areas of 2-D shapes in square centimetre and square metre

7. recognise and use the formulae for areas of squares and rectangles

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In this learning unit, students have to recognise the concept of area. Teachers may design learning activities together with daily-life examples to introduce the concept, such as discussion on the sizes of a textbook cover and a desktop, etc. In Learning Objectives 2, students are required to compare the areas of 2-D shapes intuitively. Teachers should select two 2-D shapes which are significantly different in area to enable students to compare their areas by observation.

In Learning Objective 3, students are required to compare the areas of 2-D shapes directly through overlapping the shapes. Teachers may discuss with students the points to note when comparing the areas of 2-D shapes directly. For example, a 2-D shape has to cover another 2- D shape completely.

In this learning unit, students are required to compare the areas of 2-D shapes in improvised units. Students are also required to indirectly compare the areas of 2-D shapes by the comparison with a third 2-D shape. For examples:

 If A and B are of equal area and the area of B is smaller than that of C, the area of A is smaller than that of C.

 If the area of A is greater than that of B and the area of B is greater than that of C, the area of A is greater than that of C.

 If the area of A is smaller than that of B and also smaller than that of C, the area of A is the smallest.

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Teachers may discuss with students the points to note when using the improvised units.

Students are required to choose appropriate improvised units for taking measurements based on actual circumstances. Teachers may guide students to find the areas of regular and irregular shapes using squared paper and understand the need for using standard units through activities.

After introducing the units of area square centimetre (cm²) and square metre (m²), teachers may conduct various activities to enhance students’ recognition on square centimetre and square metre. Students are required to measure and compare the areas of 2-D shapes in square centimetre and square metre. However, interconversion between square centimetre and square metre is not required.

In Learning Objective 7, teachers may guide students to recognise the formulae for areas of squares and rectangles through some inquiry activities, such as counting the number of squares covered by a rectangle. Students are required to use the formulae for areas of squares and rectangles to solve problems. Finding the length of side of a rectangle from its area may be involved.

Students are required to find the areas of 2-D shapes formed by squares and rectangles.

Although operations of the related problems may involve more than five numbers, teachers should avoid selecting and designing problems with complicated calculations for students to solve.

Examples of vocabularies to be learnt: area, square centimetre (cm²), square metre (m²), etc.

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Learning Unit Learning Objective Time

Shape and Space Strand 4S1

Quadrilaterals (III)

1. recognise the concept and properties of rhombuses 2. draw and make rhombuses

3. recognise the relations between different types of quadrilaterals

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Students are required to recognise that a quadrilateral with four equal sides is a rhombus and its opposite sides are parallel. They should identify rhombus according to its properties.

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

Students have recognised the concepts and properties of rectangles, squares and parallelograms and recognised that squares and rectangles are parallelograms at Key Stage 1. In this learning unit, students are required to further recognise the following inclusion relations among the quadrilaterals through the comparison of their properties:

 all squares are rectangles

 all squares, rectangles and rhombuses are parallelogram

 all squares are rhombuses

Teachers may use diagrams such as Venn diagrams or tree diagrams to help students recognise the inclusion relations between different types of quadrilaterals required by the curriculum, but students are not required to construct Venn diagrams and tree diagrams. They are also not required to use the term “inclusion relation”.

Examples of vocabularies to be learnt: rhombus, etc.

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Learning Unit Learning Objective Time

Dissecting and forming shapes

1. dissect a polygon into smaller polygons 2. form a polygon by smaller polygons

3

Students are required to recognise that polygons are formed by connecting at least three line segments end to end on a plane, e.g. triangles, quadrilaterals, pentagons, etc.

In this learning unit, students are required to dissect a polygon into several smaller polygons, and form a polygon by putting several smaller polygons together.

After dissecting a polygon, students should determine the properties of the smaller polygons according to the properties of the original polygon. For example, dissecting a rectangle diagonally will get two right-angled triangles; while dissecting a square diagonally will give two isosceles right-angled triangles. Students are not required to use the term “diagonal”.

Students are required to determine some of the properties of the polygon formed by several polygons according to the properties of the original polygons. For example, two right-angled trapeziums of the same size and shape may form a rectangle.

Examples of vocabularies to be learnt: polygon, dissecting, forming, etc.

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Learning Unit Learning Objective Time

Directions and positions (III)

1. recognise the four directions: southeast, northeast, southwest and northwest

2. use the compass to measure directions

3.5

Students have learnt the four main directions east, south, west and north in Primary Two. In this learning unit, students are required to recognise the four directions “southeast”, “northeast”,

“southwest” and “northwest”, and their respective short forms “SE”, “NE”, “SW” and “NW”.

Students are also required to use the sentence “ is to the southeast/northeast/southwest/northwest of ” to describe the relative positions of objects.

Students are required to find out the above eight directions with a compass and recognise some simple applications of compass, e.g. using the directions indicated on a road map to choose paths. Instructions of turning left or right may be included, e.g. turn left by two right angles.

Examples of vocabularies to be learnt: southeast, northeast, southwest, northwest, etc.

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Learning Unit Learning Objective Time

Data Handling Strand 4D1

Bar charts (II)

1. recognise bar charts of greater frequency counts 2. interpret bar charts of greater frequency counts 3. recognise the concept of approximate values 4. construct bar charts of greater frequency counts

5

Students have learnt the bar charts using one-to-one, one-to-two and one-to-five representations in Primary Three. Teachers may make use of some statistical data with greater frequency counts, e.g. the total number of different types of books in the school library, to guide students to recognise the need for using bar charts of greater frequency counts.

In this learning unit, students are only required to interpret and construct bar charts using one- to-ten, one-to-fifty and one-to-hundred representations. Bar charts in horizontal and vertical forms are required.

For Learning Objective 3, students can round off data to fit the chosen representation and construct bar charts using the one-to-ten, one-to-fifty and one-to-hundred representations after recognising the concept of approximate values. Thus, teachers may make use of examples involving rounding off numbers to the nearest tens or hundreds when teaching this learning objective. For rounding off numbers to the nearest places other than tens or hundreds, it should be taught in Learning Unit 5N1 “Multi-digit numbers”.

Teachers may select some daily life examples or some topics that students are familiar with for them to construct bar charts of greater frequency counts, e.g. the favourite co-curricular activities for Primary Four students. Teachers should guide students to classify data, use frequency tables for recording data, choose the one-to-ten, one-to-fifty or one-to-hundred representations according to the magnitude of the data and round off data to fit the chosen representation. Teachers may remind students the points to note when constructing bar charts, e.g. if “others” is a category for classifying data, its quantity should better be a small proportion of the whole.

Apart from using paper and pencil, teachers may let students use IT to construct bar charts of greater frequency counts.

Examples of vocabularies to be learnt: approximate value, classify, rounding off, etc.

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Learning Unit Learning Objective Time

Further Learning Unit 4F1

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. The time is allocated for students to engage in learning activities from different learning units, for example, activities on enrichment topics, cross learning unit activities, and cross-KLA activities that based on mathematical topics.

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Learning Unit Learning Objective Time

Multi-digit numbers

1. recognise the concept of multi-digit numbers 2. compare the magnitude of numbers

3. use rounding off to obtain approximate value of multi- digit numbers

4. estimate large quantities

3

The concept of multi-digit numbers can be viewed as an extension of the concept of five-digit numbers. Students are required to recognise multi-digit numbers by counting, reading and writing. For example, they should recognise 10 ten thousand equal to 1 hundred thousand, 10 hundred thousand equal to 1 million, 10 million equal to 1 ten million, 10 ten million equal to 1 hundred million, and the meaning of the numerals in the hundred thousands place, millions place, ten millions place and hundred millions place.

Students should perform counting onwards and backwards of multi-digit numbers and to determine whether a multi-digit number is an odd or even number by the test of divisibility of 2. Teachers should let students know that the method of comparing the magnitudes of two multi- digit numbers is similar to that of comparing numbers learnt at Key Stage 1.

In Primary Four, students have learnt the concept of approximation in Learning Unit 4D1 “Bar charts (II)”. In this learning unit, students should recognise to use rounding off to obtain approximate values of multi-digit numbers to the nearest thousands, ten thousands, hundred thousands, millions, ten millions or hundred millions, such as the approximate value of 123456789 equals to 123460000 to the nearest ten thousands.

Students should estimate large quantities, for example, number of words in a book and number of people attending large-scale events, etc. In addition to teaching students how to do estimation, teachers should also let students understand that different methods of estimation may have different degrees of accuracy.

Examples of vocabularies to be learnt: multi-digit number, hundred thousands place, millions place, ten millions place, hundred millions place, rounding off, etc.

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Learning Unit Learning Objective Time

Fractions (III)

1. compare the magnitude of not more than three fractions with different denominators

2. perform addition and subtraction of not more than three fractions with different denominators

3. perform mixed operations of addition and subtraction of three fractions with different denominators

4. solve problems

8.5

Students should compare the magnitudes of not more than three fractions with different denominators. Comparing the magnitudes of fractions and whole numbers is required, such as comparing the magnitudes of 1

12 5 ,

6 7,

4

3and comparing the magnitudes of 11 15 ,

7 6 , 2.

Students should perform addition and subtraction of not more than three fractions with different denominators and perform mixed operations of addition and subtraction of three fractions with different denominators. Addition, subtraction and mixed operations of addition and subtraction of fractions and whole numbers are required, such as 3–11

15 + 5

6 . Teachers should avoid requiring students to perform complicated comparisons and mixed operations of addition and subtraction of fractions with different denominators. When the operations and comparison involving three fractions with different denominators, all given denominators should not exceed 12.

Students are required to recognise that the different denominators should be firstly converted into the same denominator by expanding the fractions, and then perform the addition and subtraction of fractions with the same denominator. Teachers should let students recognise that this common denominator could be any common multiple of these different denominators.

However, finding the L.C.M. of three numbers by listing method or short division is not required by the curriculum. Teachers may teach students to determine the common multiple to be used as the denominator by finding a common multiple of two numbers each time. For

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example, when performing the operation of 13 8 +

5 6 –

2

9 , students can use 24 as the same denominator for 13

8 + 5

6 , as the least common multiple of 8 and 6 is 24, and use 72 as the same denominator for the subsequence subtraction as the least common multiple of 24 and 9 is 72. If the result of calculations is a fraction, it should be reduced to the lowest terms. If it is greater than 1, it may be expressed as a fraction in the lowest terms, either as a mixed number or an improper fraction.

After students have mastered the skills of performing addition, subtraction and mixed operations of addition and subtraction of fractions with different denominators, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

Examples of vocabularies to be learnt: fractions with different denominators, etc.

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Teachers should let students understand the concept of multiplication of fractions through the concept of multiplication of whole numbers.

Students should perform multiplication of not more than three fractions in which whole number can be included.

Teachers should avoid requiring students to perform complicated multiplication of fractions.

Multiplication of three fractions should not involve more than one mixed number.

Teachers should let students revise the conversions of mixed numbers into improper fractions and the conversions of whole numbers into improper fractions with denominator 1, for example 4 = 4

1 , to let students recognise the rules of multiplication of fraction. The result of calculations can be expressed as a mixed number or an improper fraction in the lowest terms. If the result of calculations is a fraction, it should be reduced to the lowest terms. If it is greater than 1, it may be expressed as a fraction in the lowest terms, either as a mixed number or an improper fraction.

After students have mastered the skills of performing multiplication of fractions with different denominators, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

Learning Unit Learning Objective Time

Fractions (IV)

1. perform multiplication of not more than three numbers 2. solve problems

8

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Learning Unit Learning Objective Time

Decimals (III)

1. perform multiplication of a number and 10, 100, 1000 2. perform multiplication of a number and 0.1, 0.01, 0.001 3. perform multiplication of two numbers

4. solve problems

7

Teachers should let students understand the concept of multiplication of decimals through the concepts of multiplication of whole numbers and multiplication of fractions.

Teachers should provide students with concrete examples to facilitate them to discover that multiplying a whole number or a decimal by 10, 100 and 1000, are equivalent to moving the decimal point to the right by one, two and three digits respectively; while multiplying a whole number or a decimal by 0.1, 0.01 and 0.001, are equivalent to moving the decimal point to the left by one, two, and three digits respectively.

Students should perform the following multiplication of decimals:

 a decimal × a whole number (or a whole number × a decimal)

 a decimal × a decimal

Except for the multiplications in Learning Objectives 1 and 2, the number of decimal places of the decimals involved in multiplications should be one or two. Besides, the numbers of digits involved in the multiplication of decimals should not exceed those involved in the multiplication of whole numbers as required in Leaning Unit 4N1 “Multiplication (II)” (i.e. not more than 3-digit × 2-digit). For example, the following multiplication of decimals are not required:

 0.124 × 3.9

 12.4 × 3.42

 12.41 × 2.6

When performing the multiplication of decimals, students should determine the position of the decimal point of the product and round off the result of calculations to the nearest tenth or

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hundredth when necessary. Students are required to use the symbol “≈” to indicate approximately equal.

After students have mastered the skills of performing multiplication of decimals, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

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Learning Unit Learning Objective Time

Fractions (V)

1. further recognise the concept of fractions

2. perform division of not more than three numbers 3. perform mixed arithmetic operations of three

numbers 4. solve problems

9.5

Teachers should let students understand the concept of division of fractions through the concepts of division of whole numbers and multiplication of fractions.

In Primary Three, students have learnt the basic concepts of fraction. In this learning unit, students are required to recognise that fractions can be regarded as the quotients or the ratios of two whole numbers. However, the symbol of ratio “:” needs not be introduced.

Students should perform division of fractions and mixed arithmetic operations of fractions, both include those involving whole numbers, such as the division of a fraction by a whole number and the division of a whole number by a fraction. Teachers should let students recognise that the conventional order of performing mixed arithmetic operations of fractions is the same as that of whole numbers. In this learning unit, the division of fractions involves not more than three numbers while the mixed arithmetic operations of fractions involve three numbers only. Division of three fractions and mixed operations of multiplication and division of three fractions should not involve more than one mixed number.

Students should solve problems of division of fractions, including those situations involving remainders. For example, “Each pen costs 142

5 dollars, how many pens can be bought with 60 dollars? How much is left?”

Students have learnt to solve problems involving direct proportion by the unitary method in the Learning Unit 4N5 “Four arithmetic operations (II)”. In this learning unit, the learning content on this topic will be further extended to deal with cases that involve the operations of

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non-integers. For example, “72 dollars can buy 10 ball pens, how much do 8 ball pens cost?”

Students are not required to use the term “direct proportion”.

Teachers should avoid requiring students to perform complicated division and mixed arithmetic operations of fractions. In this learning unit, if the result of calculations is a fraction, it should be reduced to the lowest terms. If it is greater than 1, it may be expressed as a fraction in the lowest terms, either as a mixed number or an improper fraction.

Problems involving finding the original numbers, such as the following problems, are tackled in Learning Units 5A2 “Simple equations (I)” and 6A1 “Simple equations (II)”:

 if 1

3 of a number is 30, find this number

 if 2

3 of a number is 30, find this number

Problems involving finding the fraction of a number by which it is greater or less than another number, and finding the fractional change of a number when it changes to another number are not required at the primary level. For example:

 by what fraction of 80 is 100 greater than it

 by what fraction of 100 is 80 less than it

 what is the fractional increase when 100 is increased to 120

 what is the fractional decrease when 120 is decreased to 100

After students have mastered the skills of performing division of fractions and mixed arithmetic operations of fractions, teachers should guide students to estimate the results of calculations and let them understand the advantages of estimation, so as to enhance students’ abilities in solving related problems.

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Learning Unit Learning Objective Time

Algebra Strand 5A1

Elementary algebra

1. recognise the use of letters to represent numbers 2. use algebraic expressions to represent the operations of

and relations between quantities that are described in words and involve unknown quantities

6

This learning unit is the first one of the Algebra Strand at the primary level. Teachers may help students to recognise the use of letters to represent numbers through various learning activities, e.g. in the diagram below that if the sum of two numbers in the vertical line is equal to that in the horizontal line, what is the value of x?

3 5 +

x

4

Teachers may guide students to use algebraic expressions to represent the operations of and relations between quantities that are described in words and involve unknown quantities through different examples, e.g. using algebraic expression to represent “the sum of 8 and 2 times of a”. The algebraic expressions in this learning unit should involve only one unknown quantity.

Besides, students are required to recognise the representations, such as:

 3x is 3 × x, x × 3 or x + x + x

 x

3 is x ÷ 3, 1

3 × x or x × 1 3

Teachers may discuss the advantages of using algebraic expression to represent the relations between quantities through different examples, e.g. “John is 2 years older than his sister. What are their ages?” If algebraic expressions are used, the relation between the ages of John and his sister can be clearly shown.

Examples of vocabularies to be learnt: algebraic expression, unknown quantity, etc.

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Learning Unit Learning Objective Time

Algebra Strand 5A2

Simple equations (I)

1. recognise the concept of equations 2. solve simple equations

3. solve problems by using equations

8

Students have already acquired a preliminary recognition to algebraic expressions at this stage.

Teachers may guide students to recognise the concept of equations through various contexts.

For example, teachers may ask students to solve the following problem “I have 15 dollars. How much more money should I add to get 50 dollars?” Teachers may set up an equation accordingly through which teachers could introduce the concepts of equations and the equity of the values on both sides of an equation.

In Learning Objective 2, the types of simple equations that students are required to solve include:

1. x + b = c 2. x – b = c 3. ax = b 4. x

a = b 5. ax + b = c 6. ax – b = c 7. x

a + b = c 8. x

a – b = c where a, b and c are whole numbers, and a is nonzero.

In Learning Objective 2, teachers should use balance to model and explain the process of solving an equation, such as

y + 5 = 9

y + 5 – 5 = 9 – 5

y = 4

Students are required to solve problems by using equations. Teachers should guide students to set equations directly according to the contexts of the problems, so as to avoid student setting the equations such as y = 9 – 5.

y

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In this learning unit, students are required to recognise how to check the answers after solving equations or problems.

Examples of vocabularies to be learnt: equation, solving equation, solution of equation, etc.

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Learning Unit Learning Objective Time

Area (II)

1. recognise the concept of height of triangles and quadrilaterals

2. recognise and use the formulae for finding the areas of parallelograms, triangles and trapeziums 3. find the areas of polygons

8.5

Students have learnt the concept of area and the formulae for areas of squares and rectangles in Primary Four. In this learning unit, teachers may guide students to apply their prerequisite knowledge to investigate the formulae for areas of parallelograms, triangles and trapeziums.

Students should also use the formulae for areas that they have learnt to find the areas of polygons.

Learning Objective 1 aims at broadening student’s recognition on the height of 2-D shapes.

Students have learnt “given a straight line and a point not on the straight line, draw a straight line through that point and perpendicular to the given straight line” in Primary Two. Based on students’ prerequisite knowledge, teachers may guide students to recognise the concept of height of triangles and quadrilaterals with different examples. Students are only required to recognise the corresponding height of each side of triangles and convex quadrilaterals.

However, they are not required to use the term “convex quadrilaterals”.

In Learning Objective 2, teachers may guide students to recognise the formulae for finding the areas of parallelograms, triangles and trapeziums through inquiry activities, such as dissecting and forming shapes. Students are required to use the formulae for finding the areas of parallelograms, triangles and trapeziums to solve problems. Problems on finding the lengths of sides of a triangle or a trapezium from its area should be dealt with in Learning Unit 6A1

“Simple equations (II)”.

In Learning Objective 3, teachers may guide students to make use of the formulae for areas of 2-D shapes that they have learnt to find the areas of polygons in different ways, such as dissecting a polygon or making a new shape from the given one plus additional 2-D shape(s).

Although the operations may involve more than five numbers in this learning unit, teachers

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should avoid selecting and designing problems with complicated calculations for students to solve.

Examples of vocabularies to be learnt: base, height, polygon, etc.

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Learning Unit Learning Objective Time

Volume (I)

1. recognise the concept of volume

2. compare intuitively the volumes of objects 3. recognise cubic centimetre (cm³)

4. measure and compare the volumes of objects in cubic centimeter

5. recognise cubic metre (m3)

6. recognise and use the formulae for finding the volumes of cubes and cuboids

7

In this learning unit, students have to recognise the concept of volume. Teachers may use learning activities together with daily life examples to guide students to recognise the concept, such as discussion on the amount of space that occupied by different types of objects in a box.

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

Teachers should select two objects which are significantly different in volume to enable students to compare their volumes by observation. Teachers may help students understand the need for using standard units through activities.

After introducing cubic centimetre (cm³) and cubic metre (m³) as units of volume, teachers may help students recognise cubic centimetre and cubic metre through various learning activities. Students are required to measure and compare the volumes of objects in cubic centimetre. However, interconversion between cubic centimetre and cubic metre is not required.

In Learning Objective 6, teachers may guide students to recognise formulae for finding the volumes of cubes and cuboids through inquiry activities, such as counting the number of cubes contained in a cuboid. Students are required to use the formulae for finding the volumes of cubes and cuboids to solve problems. However, finding the length of a cube from its volume is not required.

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Students are required to find the volumes of simple 3-D shapes formed by cubes and cuboids.

Although operations of the related problems may involve more than five numbers, teachers should avoid selecting and designing problems involving complicated calculations for students to solve.

Examples of vocabularies to be learnt: volume, cubic centimetre (cm³), cubic metre (m³), etc.

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Learning Unit Learning Objective Time

Circles

1. recognise the concept and basic properties of circles 2. draw circles

2.5

Students are required to recognise the concept and the basic properties of circles, which include the concepts of centre, radius, diameter and circumference of a circle.

The basic properties of circles include:

 all the points on a circle are at equidistant from its centre

 with line segments joining any two end points on a circle, those passing through the centre are the longest

 the length of the diameter is twice of the length of the radius

Teachers may let students use different methods to draw circles, including using compasses.

Students should draw circles of specific radii or diameters. Teachers may let students freely create their own drawings which are formed by circles, and let them appreciate the beauty of geometric shapes.

The concept of circles is a prerequisite knowledge for learning the concept and properties of spheres. This learning unit should be taught prior to Learning Unit 5S2 “3-D shapes (III)”.

Examples of vocabularies to be learnt: centre, radius, diameter, circumference, compass, etc.

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Learning Unit Learning Objective Time

3-D shapes (III)

1. recognise the cross sections of prisms and cylinders 2. recognise the cross-sections of pyramids and cones 3. recognise the concepts of vertices and edges of a 3-D

shape

4. recognise the concepts of cubes and cuboids 5. recognise the nets of cylinders

6. recognise the concept and basic properties of spheres

11

Students have learnt the concepts of the bases and lateral faces of prisms and pyramids, the bases and curved surfaces of cylinders and cones, and the curved surfaces of spheres in Primary Two. In this learning unit, students are required to recognise the concept of cross sections of 3- D shapes. They should recognise that the sizes and shapes of the cross sections of the prisms and the cylinders that are parallel to the bases are the same as that of the bases while the sizes of the cross sections of pyramids and cones that are parallel to their bases are different from that of the bases. Students are not required to use the term “uniform cross sections” to describe cross sections of prisms and cylinders.

Students are required to recognise the concepts of the vertices and edges of 3-D shapes and recognise the concepts of cubes and cuboids in quadrilateral prisms. At primary level, teachers should use cuboids that are not cubes as examples to explain the concepts of cuboids. Teachers should let students participate in activities of making frameworks of cubes and cuboids to enhance their recognition of the concepts of vertices, edges and faces of 3-D shapes. The construction of frameworks of prisms and pyramids is not a must. The relations among the number of edges, the number of vertices and the number of faces of 3-D shapes are not required in the curriculum. Teachers may deal with this content in Enrichment Topic 5E2 “Exploration of 3-D shapes”.

Students are required to recognise the nets of cubes, cuboids and cylinders. They should make the nets of cubes and cuboids.

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Teachers should let students understand the concept and the basic properties of spheres through the concept of circles. The basic properties of spheres include:

 all the points on the sphere are at equidistant from the centre

 all the cross sections of a sphere are circles

Teachers may make use of concrete objects or computer software to help students recognise the basic properties of spheres. Students are not required to recognise the concept of greatest sectional circles.

Examples of vocabularies to be learnt: net, vertex, edge, cross section, centre, cube, cuboid, etc.

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Learning Unit Learning Objective Time

Data Handling Strand 5D1

Bar charts (III)

1. recognise compound bar charts 2. interpret compound bar charts 3. construct compound bar charts

6

Students have recognised bar charts in Primary Three and Primary Four. Teachers may make use of students’ prerequisite knowledge to guide them recognising the limitation of bar charts.

For example, we cannot find the ball game that the boys like most from the bar chart showing the favourite ball games of all students in the school. Then, teachers may guide students to recognise the advantage of using compound bar charts to present data.

In this learning unit, students are only required to interpret and construct compound bar charts using the one-to-thousand, one-to-ten thousand or one-to-hundred thousand representations.

Compound bar charts in horizontal and vertical forms are required.

Students may start to write multi-digit numbers in ways such as 1 million, 2.5 million and 10 million in the strand of Data Handling, but they are required to avoid inappropriate style of writing such as 130 hundred and 3 hundred thousand.

Teachers may ask students to collect data prepared by government departments and related organisations from the media or the Internet, e.g. population by age group and gender. Then, teachers may let students construct compound bar charts after choosing the one-to-thousand, one-to-ten thousand or one-to-hundred thousand representations according to the magnitude of the data and round off the data to fit the chosen representation. Apart from paper and pencil, teachers may let students use IT to construct compound bar charts.

Examples of vocabularies to be learnt: compound bar chart, etc.

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Learning Unit Learning Objective Time

Further Learning Unit 5F1

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. The time is allocated for students to engage in learning activities from different learning units, for example, activities on enrichment topics, cross learning unit activities, and cross-KLA activities that based on mathematical topics.

Explanatory Notes to