MATHEMATICS CURRICULUM 2000

FOR CLASSES K & I-V

GOVERNMENT OF PAKISTAN MINISTRY OF EDUCATION

(CURRICULUM WING) ISLAMABAD

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PREFACE

In pursuance of National Education Policy (1998-2010), a project on Curriculum Reforms (Vision 2010) is in progress. It aims to improve the quality of education through curriculum revision and textbook development. The highest priority has been assigned to the revision of curriculum with a view to update the entire course contents so that Ideology of Pakistan could permeate the thinking of young generation and help them with necessary conviction and ability.

Believing in participatory approach the Ministry of Education requested the provincial governments/Curriculum Bureau to draft need based curricula in all the subjects for classes I through XII. Consequent upon this the Government of the Punjab attempted five initial drafts in Science and Mathematics. The Bureaus of Sindh, N.W.F.P. and Baluchistan furnished their comments on the previous as well as proposed curricula. To synchronize the feedback, the Ministry of Education appointed National Curriculum Development Committees. The panels of the committees were comprised of curriculum developers, subject specialists, educationists, teachers of universities , schools and colleges. The representatives of National Curriculum Bureau and Provincial Curriculum Bureaus were also represented on the panels. The committees thoroughly analyzed and synthesized the comments. Global experiences of curriculum development were also kept in view while revising/ updating the National Curriculum.

In the light of the above considerations, the committees revised the existing National Curriculum in Elementary Science (I-VIII), Physics, Chemistry, Biology (IX-XII), Statistics (XI, XII), Computer Science (IX-XII) and Mathematics (I-XII). The philosophy underlying National Curriculum is Islam and Ideology of Pakistan as set by the Parliament Act X, 1976. The objectives of the National Curriculum are framed in the light of the objectives of the latest National Education Policy (1998-2010). Purposeful learning competencies are suggested in each subject. These aim to provide the learners, skills for continuing education, civilized behaviour and attitude to become useful and peaceful citizens. The objective is also to provide them with the skills for economic development. The curriculum has been made more representative and responsive to the Ideology of Pakistan and social needs. We still believe that curriculum development is a continuous process and can be made more responsive. The Ministry would welcome comments from all concerned. This will help us in making the curriculum more effective and need based.

The Ministry of Education appreciates the contributions of all the Provincial Governments/ Curriculum Bureaux and the National Curriculum Development Committees.

(Dr. Haroona Jatoi) Joint Educational Adviser

Curriculum Wing Ministry of Education

Islamabad

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CONTENT

Objectives .................................................................................................................................................... 1 Syllabus of Physics for Class IX-X............................................................................................................... 2 Expected outcomes .................................................................................................................................... 14 Topic wise weightage ............................................................................................................................... 15 Suggestions for activities .......................................................................................................................... 18 Guidelines for teachers ............................................................................................................................. 22 General Instructions to Authors .................................................................................................................. 23 National Curriculum Review Committee ................................................................................................... 24

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1

Philosophy of Teaching Mathematics at the Primary School Level

It the Mathematics curriculum students develop conceptual understanding from concrete to abstract. Gradually work begins more symbolic as students learn to work with number problems. The mechanics of Mathematics are used to create and solve problems. Logical thinking skills evolve into higher order thinking in problem solving. Young children are given opportunities to learn by doing, using manipulative materials. There is emphasis on Mathematics as it relates to daily life. By the end of primary school, students become proficient in four basic Arithmetic operations on whole numbers and fractions (common as well as decimal). They have also begun to solve various word problems pertaining to the environment around them.

Objectives of Teaching Mathematics Classes I-V

General

1. Satisfy the mathematical needs and interest of the child throughout his development so that he may act effectively in his personal and social life.

2. Develop a disciplined mind and to exercise sound judgement and reasoning in relations to the events of everyday life.

3. Provide the basic groundwork for the understanding of scientific reasoning and calculation.

4. Create favourable attitudes and interest towards Mathematics showing the contribution it has made and a making to our present civilization and culture.

5. Provide opportunities for the guided discovery and creation of patterns.

Specific

1. To acquaint the child with basic knowledge of numbers.

2. To develop appropriate skills of computation in four fundamental operations.

3. To acquaint the child with basic knowledge of two and three dimensional geometrical figures.

4. To develop ability to solve practical problems by the application of fundamental measures (money, time, weight, length…).

5. To make the children understand the presentation of data in visual form.

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Outlines of the Syllabus of Mathematics For Class K & I (Kachi and one)

Contents Scope

Unit I Pre-Number Concepts

Pre-Number Concepts

Classification of two or more objects according to: a. Colour b. Size (big and small, bigger and smaller, biggest and smallest;) c. Length (tall/large/long and short, longer and the longest, shorter and the

shortest, largest and the largest, taller and the tallest). d. Position (above/on and below, far and near, before and after, right and left) e. Capacity (full and empty, less than and more than); f. Mass (heavy and light) equality of mass. g. Identification of objects as more or less and equal by matching. h. Identifying objects which come next in a pattern.

Unit II. Concept of Numbers

Numbers 0-9 a. Introduction of numbers from one to nine by matching them with the objects (establishing correspondence between the objects and the numbers informally)

b. Concept of Zero (0) by matching the collections. c. Reading and writing of whole numbers from 0 to 9 (numerals); d. Demonstration of number sense by practically picking up some objects

(e.g 5 balls. 3 pens etc). e. Comparison of numbers from 1 to 9 (e.g which number comes before

3/after 7 etc). f. Use a number line to represent a whole number.

Numbers upto 100 a. Concept of ten and tens. b. Counting of objects from 10 to 20 and from 21 to 99. c. Concept of digits in a number from 1 to 99 (ones and tens) d. Concept of hundred as 10 tens. e. Counting objects from 1 to 100. f. Reading and writing of natural numbers from 1 to 100 in ascending and

descending order. g. The smallest and greatest numbers comprising 2 digits. h. Identification and writing missing numbers in a sequence from 1 to 100. i. Arranging the numbers (not written in order) in ascending and descending

form. j. Reading and writing numbers randomly upto 100.

Fractions: Concepts of half and quarter with the help of figures without using

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symbols: ½ , ¼.

Unit III. Algebraic Operations.

a. Concept of addition (putting objects together) b. The addition and equality symbols (+ and =). c. Addition of whole numbers with sum upto 9. d. Addition tables with sum upto 9. e. Commutation property of numbers with addition. f. Concept of subtraction (taking away objects from a collection). g. The subtraction symbol ( – ). h. Subtraction of whole numbers with minuend upto 9. i. Addition and subtraction with sum/minuend upto 9. j. Addition of 2-digit numbers (without carrying) k. Subtraction of 2-digit numbers (without borrowing). l. Simple word problems from every day life in addition and subtraction.

Unit IV. Money

a. Recognition of Pakistani currency (one paisa, 50 paisa, 1 rupee and 2 rupees).

b. Recognition of Pakistani notes (5 rupees, 10 rupees. 50 rupees and 100 rupees).

c. Solving word problems from every day life involving addition and subtraction of Pakistani Currency (without carrying and borrowing).

Unit V. Measurement

a. Measurement of lengths of familiar objects using informal units (hands, walking steps, pencils, sticks, pieces of thread etc.

b. Measure and compare capacity of containers using informal units (i.e cups, glass and bowls).

Unit VI. Time

a. Reading and telling time in hours, i.e 1,2 O’ clock etc. b. Knowing the name of the days of the week with their order (i.e which day

comes after or before a particular day).

Unit VII. Geometry

a. Recognition and matching of pictures of objects of similar shapes. b. Recognition of geometrical figures (triangle, rectangle, square and circle). c. Recognition of the objects inside and outside these figures. d. Identification of the position of objects/figures placed in a row (first,

second, third and fourth).

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Outlines of the Syllabus of Mathematics For Class II

Contents Scope

Unit I. Numbers

Numeration and notation of numbers.

a. Writing ordinal numbers from first to twentieth:

b. Concept of hundred.

c. Reading and writing of natural numbers from 101 to 999:

d. Place values of the digits in a number with the concept of zero as a place holder:

e. Concept of 1000 as ten hundreds;

f. Concept of a 3-digit number;

g. The smallest and greatest number comprising three, four – digits.

h. Writing missing numbers in ascending and descending order.

Fractions Concepts of one third, two third and three fourth without using symbols 31 , 3

2 ,

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Unit II. Algebraic Operations

Operations a. Addition of two, one-digit numbers with sum upto 18;

b. Addition of 2 or 3 – digit numbers with and without borrowing, with sum upto 1000;

c. Commutative property of addition.

d. Introduction of parenthesis ( ), associative property of addition;

e. Subtraction of two or three-digit numbers with and without borrowing, with minuend upto 1000.

f. Connection between addition and subtraction;

g. Concept of multiplication as repeated addition;

h. The multiplication symbol (x)

i. Tables of 10, 5, 2, (10x10, 5x10, 2x10)

j. Multiplication of a number by 10, 5, 2, the product being less than 1000.

k. Concept of division (as successive subtraction and as inverse operation of multiplication) with remainder 0.

l. The division symbol (÷).

Unit III. Measurement.

a. Concept of units of measurement:

• Units of length (meter, centimetre)

• Units of capacity of liquids (Litre):

• Units of mass (kilogram):

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b. Solving everyday life word problems involving addition of the above units without carrying and borrowing.

Unit IV. Money

a. Recognition of Pakistani currency notes of 500 and 1000 rupees denominations.

b. Solving everyday life problems involving amounts less than 1000 rupees.

Unit V. Time

a. Concept of half and quarter hour.

b. Telling time in half and quarter hours.

c. Telling number of hours in a day.

d. Telling number of months and their names in solar and lunar calendar year.

Unit VI. Geometry

a. Concept of open and closed plain figures and their boundaries:

b. Vertices and sides of a triangle, rectangle and square;

c. Naming and telling inside and outside of these figures.

d. Identification of inside and outside of a circle.

Unit VII. Information Handling

Reading of picture graph presented in both vertical and horizontal form.

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Outlines of the Syllabus of Mathematics For Class III

Contents Scope

Unit I. Numbers

Natural numbers

a. Reading and writing natural numbers upto 100,000 (1 lac) and place values of the digits in them;

b. The smallest and the greatest 4, 5, 6- digit numbers;

c. Even and odd natural numbers up to 100

d. Reading and writing Roman numbers up to 12;

e. Concept of a dozen;

Common fractions a. Concept of proper common fractions and their symbolic representation.

b. Numerator and denominator of a fraction, (denominators upto 100);

c. Comparison of fractions with equal denominator;

d. Comparison of fractions with equal numerator;

Decimal fractions a. Concept of decimal fractions upto 2 decimal places;

b. Place value of digits in a decimal fraction;

c. Conversion of a decimal fraction into proper common fraction and vice versa (the denominators of common fractions being 2, 4, 5, 10, 20, 25, 50, 100).

Unit II. Algebraic Operations

Operations a. Mental addition and subtraction of numbers upto 50;

b. Addition of two or more natural numbers with and without carrying with sum upto 100,000;

c. Word problems from everyday life involving addition;

d. Subtraction of natural numbers with and without borrowing, with minuend upto 100,000;

e. Rules of addition and subtraction for even and odd numbers;

f. Word problems from everyday life involving subtraction;

g. Multiplication tables upto 10 x 10;

h. Multiplication of two or three-digit numbers by one or two-digit numbers and by 100 and 1000.

i. Multiplication of two natural numbers (upto 3-digits);

j. Word problems from everyday life involving multiplication;

k. Division of two and three-digit natural numbers by one or two-digit numbers with or without remainders.

l. Word problem from everyday life involving division of two and three-digit

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numbers by one or two digit numbers;

Fractions a. Addition and subtraction of proper common fractions with the same denominators (sum being a proper fraction);

b. Addition and subtraction of decimal fractions (upto two decimal places) and the sum bing a proper fraction.

c. Word problems on everyday life involving addition and subtraction of common and decimal fractions.

Unit III. Measurement

Units of measurements.

Operation on Units

a. Units of length kilometre (km), meter (m) centimetre (cm).

b. Units of mass kilogram (kg), gram (g);

c. Units of capacity of liquids liter (lt) mililiter (ml)

d. Tables showing the relationship between units of length, wight and capacity;

e. Mutual conversion of units of measurement;

f. Estimation of length, mass, capacity and checking of the estimate by actual measurement;

g. Addition and subtraction of the units of measurement;

h. Word problems on every day life involving units of measurement.

Unit IV. Time

a. Units of time; Day, month, year, hour, minutes and seconds.

b. Table showing the relationship among the units;

c. Mutual conversion of units of time;

d. Reading and telling time;

e. Reading and telling day and date of a particular month from the lunar and solar calendar.

f. Addition and subtraction of units of time;

g. Word problems from everyday life involving units of time.

Unit V. Money

a. Addition and subtraction of Pakistani currency;

b. Solve problems from everyday life involving Pakistani currency.

Unit VI. Geometry

Concepts and construction.

a. Concepts of point and line (informal), ray, line segment;

b. Drawing and measuring a line segment;

c. Concept of measure of a line segment;

d. Identification of kinds of triangles with respect to sides (equilateral, isosceles and scalene) by measuring their sides;

e. Measuring the sides of a quadrilateral;

f. Identification of rectangle and square by measuring their sides;

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g. Drawing the figure of a triangle, square, rectangle and circle with the help of familiar objects;

h. Center, radius and diameter of a circle;

i. Drawing circle of given radius by compass.

Unit VII. Information Handling

Reading and interpreting a simple bar graph given in horizontal and vertical form.

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Outlines of Syllabus in Mathematics For Class IV

Contents Scope

Unit I. Numbers

Natural numbers a. Reading and writing of natural numbers upto 10000000 (1crore) and place values of the digits.

b. The smallest and the greatest 7,8-digits numbers.

c. Knowing:

10 lac = 1 million 1 crore = 10 million

d. Writing numbers in millions;

e. Identification, reading and writing numbers using urdu numerals.

f. Roman numerals upto 50.

Unit II. Factors and Multiples

Concepts a. Concept of divisibility and factors of natural numbers.

b. Some divisibility tests for 2,3 and 5.

c. Concept of multiples;

d. Concept of prime and composite numbers.

e. Prime factorisation of a composite number.

f. Determining common factors of two or more natural numbers;

g. Concept of the highest common factor (HCF) or the greatest common divisor (GCD);

h. The HCF of two or more natural numbers by prime factorisation, division and common divisor; determining common multiple of two or more natural numbers;

i. Concepts of least common multiple (LCM);

j. Finding the LCM of two or more natural numbers by common multiples, factorisation and division;

k. Word problems from daily life involving HCF and LCM of natural numbers.

Unit III. Common Fractions

Concepts

Operations

Operations

a. Concepts of an improper, mixed and equivalent fraction.

b. Writing a fraction equivalent to a given fraction;

c. Reducing a fraction to its lowest form;

d. Converting an improper fraction into a mixed fraction and vice versa.

e. Comparing common fractions, use of the symbols > (greater than) and < (lesser than);

f. Comparing common fractions by finding equivalent fractions.

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g. Adding two or more common fractions (proper, improper and mixed fractions with the same denominator and with different denominators);

h. Subtracting a fraction from an other fraction (proper, improper, mixed fractions with the same and different denominators);

i. Multiplying two or more common fractions (proper, improper and mixed fractions);

j. The commutative property of addition and multiplication of fractions;

k. The associative property of addition and of multiplication of fractions.

l. Dividing a common fraction by another fraction (proper, improper and mixed fractions);

m. Word problems from everyday life involving addition, subtraction, multiplication and division of common fractions.

Unit IV. Decimal Fractions

Concepts. a. Decimal fractions upto 3 decimal places;

b. Place value of the digits in a decimal fraction;

c. Conversion of a decimal fraction into a common fraction and vice versa (with a denominator in the common fractions as multiple of 2 and 5);

d. Comparing decimal fractions, use of symbols >(greater than) and < (lesser than).

e. Adding two or more decimal fractions (upto three decimal places);

f. Subtraction of a decimal fraction from another (upto three decimal places);

g. Multiplying a decimal fraction by a one digit number;

h. Dividing a decimal fraction by a one digit number (quotient being a decimal fraction upto three decimal places);

i. Word problems from everyday life involving decimal fractions upto three decimal places;

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Unit V. Measurement

Units of measurements.

a. Units of length (km, m, dm, cm, mm) relationship among units of length; b. Mutual conversion of units of length; c. Adding expression involving units of length; d. Subtracting expressions involving units of length; e. Units of mass (quintals, kg, g); f. Table showing relationship among units of mass and their mutual

conversion; g. Adding expressions involving units of mass. h. Subtracting expression involving units of mass. i. Units of capacity (liter, decilitre, millilitre); j. Table showing relationship among units of capacity and their mutual

conversion. k. Word problems from everyday life involving units of length, mass and

capacity.

Unit VI. Geometry

Concepts a. Measuring and drawing a line segment in cm and mm; b. Drawing a circle having radius of a given length. c. Constructing triangle (equilateral, isosceles, scalene) having sides of given

length with the help of compass and scale; d. Concept of perimeter of a geometrical figure. Finding the perimeter of a

triangle with sides of given lengths; e. Concept of angle and its degree measure of an angle to be between

0o and 180o. f. Interior and exterior of an angle informal description; g. Kinds of angles (right angle, acute angle, obtuse angle); h. Measuring an angle with protractor. i. Kinds of triangles with respect to angles (right triangle); j. Constructing an angle of given degree measure with protractor; k. Drawing a line perpendicular to a given line with the help of protractor; l. Concept of vertical and horizontal lines; m. Definition of a square and a rectangle; n. Constructing a square with a side of given length. o. Constructing a rectangle with sides of given length; p. Finding perimeter of a square and a rectangle;

Unit VII. Information Handling.

Reading and interpreting a line graph.

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Outlines of the Syllabus of Mathematics For Class V

Contents Scope

Unit I. Natural Numbers

Natural Numbers. a. Numeration and notation of natural numbers upto 1000000000 (one arab)

b. Reading and writing natural numbers upto 100 crore; knowing that;

100 crore = 1 arab = I billion

1000 million = 1 billion

c. Writing numbers in billions.

d. Roman numerals upto 50.

e. Roman symbols for 100, 500 and 1000 (C. D. M).

Unit II. Fractions.

Common fractions a. Concept of reducible and irreducible common fractions;

b. Reducing a common fraction into an irreducible fraction by the use of H.C.F.

Decimal fractions

Operations

a. Decimal fraction upto 4 decimal places.

b. Converting a decimal fraction upto 4 decimal places into a common fraction.

c. Concept of continued decimal fraction;

d. Determining approximate value of a continued decimal fraction by rounding off a continued decimal fraction to a decimal fraction upto 4 decimal places;

e. Converting a common fraction into a decimal fraction upto 4 decimal places (rounding off if necessary);

f. Adding two decimal fraction which are upto 4 decimal places;

g. Subtracting a decimal fraction from another;

h. Multiplying two decimal fractions (rounding off the product upto 4 decimal places);

i. Dividing a decimal fraction by another decimal fraction (quotient upto 4 decimal places).

j. Solving word problems involving addition, subtraction, multiplication and division of common and decimal fractions.

k. Knowing the brackets ( ) , { } and [ ].

l. Knowing the order of precedence of the algebraic operations in simplification of numeric expressions involving brackets.

m. Simplification of expressions involving addition, subtraction, multiplication and division and brackets.

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Unit III. Unitary Method

a. Calculating the value of many similar things when the value of one of these things is given;

b. Calculating the value of a number of similar things when the value of another number of things is given: (the unitary method).

c. Concept of direct and inverse proportion.

d. Solving every day life problems involving direct and inverse proportion by using unitary method.

Unit IV. Average

a. Concept of average (mean);

b. Finding the average of given numbers;

c. Word problems from everyday life concerning averages.

Unit V. Geometry

Concepts and constructions

a. Concept of adjacent, complementary and supplementary angles;

b. Finding complementary and supplementary angles;

c. Concept of parallel lines;

d. Drawing perpendicular and parallel lines (with set square and scale);

e. Constructing triangles when;

• Two sides and included angle are given.

• Two angles and included side is given.

f. Recognizing kinds of quadrilateral (square, rectangle, parallelogram, rhombus, trapezium and kite).

g. Concept of regions of closed figures;

h. Concept of area of a region. Units of area

i. Finding area of a square and rectangular region and rectangular paths;

j. Word problems from everyday life involving area of a square, rectangular regions and rectangular paths.

k. Concept of solids;

l. Identifying kinds of geometrical solids (cube, cuboids, sphere, cylinder and cone), with respect to their faces, cdges and vertices.

m. Concept of volume and its units;

n. Finding volume of cube and cuboids.

Unit VI. Information Handling

Drawing bar graphs and line graphs representing information related to school life.

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Instructional Objectives (Expected Outcomes I-V)

The students are expected to be able to.

1. Compare objects w.r.t. size, length, mass, direction and quantity,

2. Count things and know the numeration and notation of whole numbers upto One Arab/billion.

3. Apply four basic operations on whole numbers and solve problems pertaining to these operations;

4. Have clear concepts of common and decimal fractions and solve easy problems involving four basic operations on them.

5. Know the commutative and associative properties of addition and multiplication.

6. Understand the necessity of brackets and their kinds and simplify the expressions containing the brackets;

7. Know the importance of average and solve problems on the direct application of the formula for finding the average;

8. Understand the necessity of Unitary Method and get proficiency in solving the problems of direct proportion by using Unitary Method;

9. Recognize the currency coins and notes;

10. Know the units of money, length, mass, capacity and time, and the relationships between them;

11. Apply the four basic operations on the measures and quantities mentioned in no . 10 and solve problems on them;

12. Have clear concept of a point, line segment, ray, angle, triangle, quadrilateral, circle, parallel and perpendicular lines and degree measure of angles;

13. Know the kinds of angles, triangles and quadrilaterals;

14. Measure and draw line segments of given measures;

15. Measure and draw angles of given measures with the help of protractor;

16. Have the concept of area and its units, volume and its units;

17. Find the perimeter and area of rectangles squares and paths;

18. And to know the importance of graph and to read and interpret picture graph, line graph and bar graph.

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TOPIC-WISE WEIGHTAGE & TIME (CLASS-I) UNIT TOPIC WEIGHTAGE PERIODS (35

MINUTES EACH)

I Pre numbers concept 15% 40

II Numbers From 0-9

From 10-100

Fractions

10%

15%

10%

27

41

27

III Algebraic-operations 20% 54

IV Money 10% 27

V Measurement 5% 13

VI Time 5% 14

VII Geometry 10% 27

100% 270 (9 periods a week)

TOPIC-WISE WEIGHTAGE & TIME (CLASS-II) UNIT TOPIC WEIGHTAGE PERIODS (35

MINUTES EACH)

I Numbers Natural

Fractions

25%

10%

68

27

II Algebraic-operations 25% 68

III Measurement 10% 28

IV Money 7 ½ % 20

V Time 7 ½ % 20

VI Geometry 10% 26

VII Information Handling 5% 13

100% 270 (9 periods a week)

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TOPIC-WISE WEIGHTAGE & TIME (CLASS-III) UNIT TOPIC WEIGHTAGE PERIODS (35

MINUTES EACH)

I Numbers Natural Numbers

Common fractions

Decimal fractions

10%

10%

10%

27

27

27

II Operations on Natural Numbers

Common fractions

10%

10%

27

27

III Measurements Concepts

Operations

8%

22

IV Time 6% 16

V Money 6% 16

VI Geometry Concepts

Construction

10%

10%

27

27

VII Information Handling 10% 27

100% 270 (9 periods a week)

TOPIC-WISE WEIGHTAGE & TIME (CLASS-IV) UNIT TOPIC WEIGHTAGE PERIODS (35

MINUTES EACH)

I Numbers 17 ½ % 47

II Factors & Multiples 10% 27

III Common fractions Concepts

Operations

7 ½ %

10%

20

27

IV Decimal fractions Concepts

Operations

7 ½ %

7 ½ %

20

20

V Quantities and Measurements

Units

Operations

10%

7 ½ %

27

20

VI Geometry Concepts

Rectangle

7 ½ %

7 ½ %

20

21

VII Information Handling 7 ½ % 21

100% 270 (9 periods a week)

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TOPIC-WISE WEIGHTAGE & TIME (CLASS-V) UNIT TOPIC WEIGHTAGE PERIODS (35

MINUTES EACH)

I Natural Numbers Numeration and notations

15% 40

II Fractions Common and decimal fraction

Operations on fractions

Brackets

10%

7 ½ %

7 ½ %

27

21

20

III Unitary Method 15% 41

IV Average 7 ½ % 21

V Geometry Concepts Area 22 ½ % 60

VI Information Handling 15% 40

100% 270 (9 periods a week)

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SUGGESTIONS FOR ACTIVITIES

1) COMPARISION

a) Weights

i) Experience of weights by handling different objects should be given.

ii) Guessing the weights (heavy/light) by holding two different objects in two hands at a time should be practiced.

iii) Checking the correctness of the guess by putting them in the two pans of a balance should be made.

b) Sizes

i) Objects different in sizes but of similar shape may compared, e.g. pencils, sticks and fingers of a hand (long and short).

ii) Packing bags of different sizes by copies and books of different sizes (big/small).

iii) Making of children stand side by side and noting their heights (tall/taller tallest).

iv) Containers to be compared by the amount of sand/water they hold (big/small).

c) Distances

i) Putting things in a line and telling as to which one is far from near the first thing.

ii) Standing at some place in a room and guessing as to which wall is near/nearer/nearest.

iii) Checking the correctness of the guess by counting the number of steps taken in reaching the walls.

iv) Measuring the distances between two opposite sides of a table in spans and breadth of fingers.

d) Shapes

i) Distinguishing between the shapes of different objects, e.g. chair, table, almirah, book, inkpot and pen etc.

ii) Recognition of triangular, rectangular and circular frames.

2) COUNTING

i) Counting objects like pebbles, beads, etc.

ii) Counting the number of children of a class on a particular day.

iii) Counting the objects in a picture by putting fingers on them.

iv) Counting things in group by pointing towards them.

v) Counting objects without pointing towards them.

vi) Picking up a number of objects from a heap.

vii) Counting the number of beads in a line of a ball frame.

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3) RECOGNITION OF NUMERALS/NUMBERS

i) Checking the ability of the children to recognize numbers on a flash card.

ii) Asking the students to show a number on the abacus.

iii) Asking the students to tell the place value of a digit on the abacus.

iv) Recognition of odd and even numbers shown in the chart of the following type:

v) Counting the numbers in twos.

4. ADDITION/SUBTRACTION TABLES

i) Concrete objects like pebbles may be used in finding the sum/difference of one digit numbers.

ii) Four facts of addition/subtraction be taught at the same time, e.g., 4 + 9 = 13, 9 + 4 = 13, 13 – 4 = 9, 13 – 9 = 4 (in this way, commutative property of addition and the relationship between addition and subtraction will be clearly understood by the student).

iii) Flash cards may be used to check the mastery of the students in telling the sum/difference of one digit numbers (the habit of counting on fingers etc. should be scrupulously avoided).

5. MULTIPLICATION TABLES

i) Multiplication tables should be taught by the practical addition of the same number of objects so that the children should have clear concept of multiplication. For example, 4 + 4 + 4 + 4 + 4 = 5 x 4; 5 + 5 + 5 + 5 = 4 x 5

ii) Four facts like the following should be taught at the same time: 3 x 7 = 21, 7 x 3 = 21, 21 ÷ 7 = 3, 21 ÷ 3 = 7 (This will help the students in having clear concept of the commutative property of multiplication an the relationship between multiplication and division of numbers).

iii) Flash cards must be used for testing the ability of the students of learning the tables (the automatic response to the stimulus 3 x 5 should be 15 they should not get into the habit of saying 1 x 5 = 5, 2 x 5 = 10 and than answering 3 x 5 = 15, because it defeats the very purpose of learning tables, i.e., saving time and anergy.

iv) Competition can be held between pairs of children by asking them to go from one step to another, telling the products of the numbers in stairs by a certain number

9 6 8 5 1 4 10 2 7 3

8 7 4 1 6 10 3 9 2 5

The child crossing the stair first and telling correct products should be declared the winner.

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6. UNITS OF MEASUREMENT

a) Money

i. Children may be told that there used to be a coin of paisa 1 whose two sides are shown in the textbook.

ii. 50 paisa coin is equal to 50 paisas and one rupee coin is equal to 100 paisas.

iii. Children should be able to recognize 50 paisa coins.

b) Measures of Length

i. The students should be required to count marks of mms on the foot rule to make sure that there are 10 mm in a cm.

ii. The children should be made to count the number of cms on a meter and practically learn that 100 cm = 1 meter.

iii. The students should measure the length and breadth of their textbook in Mathematics in cm and mm.

iv. Children should measure the length and breadth of their classroom with the help of measuring tape.

c) Measures of Mass

i. The students should be shown weighs – 1-gm, 10-gm and 1-kg. They should check by actually weighing 10 gms weight; 10, 10 gms = 100 gm weight; 10, 100gms = 1-Kg.

ii. The children should weighs their books/bags full of books by making use of a balance.

iii. The children should make an estimate of the mass of the things and the check it by weighing them.

e) Units of Time

i. The teacher should prepare a dial of clock and ask the students to tell the numbers written on it.

ii. After fixing a hand in it, the teacher should tell them that the point of this hand moves from one number to the next number in one hour and it is called the hour hand.

iii. The teacher should fix a longer hand on dial and tell the students that the tip of this hand makes the complete revolution in one minute and the time taken by it in going from one mark one mark to the next is called a second. As there are 60 marks on the dial. They may also be asked to tell the number of Sunday, in a particular month.

7. SOLIDS

a. Through various experiments, the teacher should make the students understand that each object occupies some space and, in mathematical language, it is called a solid. Solid can have any shape. However, there are certain solids which have particular and regular shaped and they are called Geometrical solids.

b. The teacher should show the following solids to the students: Cube, Cuboid, and should help the students in recognizing these solids and know their names.

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c. A cuboid may be presented to the students who should tell the teacher that it has:

i. 6 faces, 12 edges, 8 corners and three dimensions (length, breadth and height)

ii. Each face has two dimensions (length and breadth).

iii. Two faces meet at an edge which is called a line segment and it has one dimension (length).

iv. Two edges (line segments) meet at a corner which is called a vertex and it has no length, no breadth and no height.

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GUIDELINES FOR PRIMARY SCHOOL TEACHERS

The teachers of primary school are expected to be very polite, sympathetic and tolerant so as to create a homely atmosphere in the school. They should use play-way methods of teaching and arrange maximum number of activities. The teachers of mathematics must proceed from “concrete to abstract”. The methods adopt in teaching of mathematics should be such as to make Mathematics meaningful to the child by asking them to solve problems directly connected with their own lives. The multiplication tables should taught in the following order:

10, 5; 2, 4, 6; 6, 9; 7

If commutative property of multiplication is made use of in teaching these tables, in the table of 7, the children shall have to learn only one fact, i.e., 7 x 7 = 49. The teacher must make frequent use of flash cards for testing the ability of the students of the immediate response to the stimuli like 5 + 3 = ? , 5 – 3 = ?, 5 x 3 = ? , 15 ÷ 3 = ?

The primary school teachers of Mathematics should not lose sight of the fact that numeration and notation of numbers, the basic operations on whole numbers, common fractions and decimal fractions are to be furnished at this very level.

The task of teachers of Mathematics at the primary school level becomes really difficult as they are expected not only to provide skills in the correct and speedy application of the four fundamental operations of Arithmetic but also to create interest in the children for learning Mathematics. They can succeed in performing their duty efficiently if they are hard working have full grasp over the laws of learning and child psychology.

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GUIDELINES FOR TEXTBOOK WRITERS (PRIMARY)

Textbook writing is an art. As all arts follow certain principles, so does the art of writing textbooks. Undoubtedly, a person attempting to write a textbook does possess sufficient knowledge about the subject but it is the presentation of the subject matter that counts. The most important difficulty that an author has to face is to bring himself to the mental level of the students for whom he is writing the book. He has to arrange the subject matter in a psychologically logical manner. It is possible only if he knows the law of learning and has full control over the language so as to use the vocabulary, which is in keeping with the vocabulary of the students.

The primary schools Textbooks should contain information about the activities that a teacher is to make use of in making his lessons interesting for the children. They should provide sufficient practical and oral work before the students do written work. They should scrupulously avoid giving questions involving unnatural facts and figures.

A textbook writer must realize that he is to cater to the needs of the following three categories of the people:

1. It is to be a necessary tool for regular student.

2. It is to work as a guide for the inexperienced teachers.

3. It is to be helpful for the experienced teachers.

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National Curriculum Review Committee on Mathematics For Classes I-V

1. Dr. Abdul Majeed, Director, Institute of Mathematics, The University of Lahore, I-Km, Raiwind Road, Thokkar Niaz Baig, Lahore.

2. Prof. Bashir Muhammad, Subject Specialist, NWFP Textbook Board, Hayatabad, Peshawar.

3. Prof. Shahid Ali. SSS, Bureau of Curriculum and Extension Centre, Quetta, Balochistan.

4. Mr. Shamshad Muhammad Lodhi, Retd. Prof, Govt Training College of Education, Lahore. 32-C, Tech Society, Lahore.

5. Mr. Khalid Saleem, Associate Professor (Maths), Govt. College, Lahore.

6. Miss Humaira Malik Beacon House School Systems, Head Office, Lahore.

7. Mr. Muhammad Saghir Sheikh, Deputy Director (Maths), Bureau of Curriculum and Extension Wing, Jamshoro Sindh.

8. Dr. R.A. Siyal Deputy Educational Advisor Ministry of Education (Curriculum Wing) Islamabad

9. Mr. Munawar Din Awan, Education Officer, Curriculum Wing, Ministry of Education, Islamabad.