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The Syllabus in the subject of Mathematics has undergone changes from time to time in

accordance with growth of the subject and emerging needs of the society. The present

revised syllabus has been designed in accordance with National Curriculum Framework

2005 and as per guidelines given in the Focus Group on Teaching of Mathematics which is

to meet the emerging needs of all categories of students. For motivating the teacher to

relate the topics to real life problems and other subject areas, greater emphasis has been

laid on applications of various concepts.

The curriculum at Secondary stage primarily aims at enhancing the capacity of students

to employ Mathematics in solving day-to-day life problems and studying the subject as

a separate discipline. It is expected that students should acquire the ability to solve

problems using algebraic methods and apply the knowledge of simple trigonometry to solve

problems of height and distances. Carrying out experiments with numbers and forms of

geometry, framing hypothesis and verifying these with further observations form inherent

part of Mathematics learning at this stage. The proposed curriculum includes the study

of number system, algebra, geometry, trigonometry, mensuration, statistics, graphs and

coordinate geometry, etc.

The teaching of Mathematics should be imparted through activities which may involve the

use of concrete materials, models, patterns, charts, pictures, posters, games, puzzles and

experiments.

Objectives

The broad objectives of teaching of Mathematics at secondary stage are to help the

learners to:

l consolidate the Mathematical knowledge and skills acquired at the upper primary

stage;

l acquire knowledge and understanding, particularly by way of motivation and

visualization, of basic concepts, terms, principles and symbols and underlying

processes and skills;

l develop mastery of basic algebraic skills;

l develop drawing skills;

l feel the flow of reason while proving a result or solving a problem;

l apply the knowledge and skills acquired to solve problems and wherever possible, by

more than one method;

l to develop positive ability to think, analyze and articulate logically;

l to develop awareness of the need for national integration, protection of environment,

observance of small family norms, removal of social barriers, elimination of gender

biases;

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l to develop necessary skills to work with modern technological devices such as

calculators, computers, etc.

l to develop interest in mathematics as a problem-solving tool in various fields for its

beautiful structures and patterns, etc.

l to develop reverence and respect towards great Mathematicians for their contributions

to the field of Mathematics;

l to develop interest in the subject by participating in related competitions;

l to acquaint students with different aspects of Mathematics used in daily life;

l to develop an interest in students to study Mathematics as a discipline.

General Instructions:

l As per CCE guidelines, the syllabus of Mathematics for classes IX and X has been

divided term wise.

l The units specified for each term shall be assessed through both Formative and

Summative Assessments.

l In each term, there will be two Formative Assessments, each carrying 10% weightage.

l The Summative Assessment in term I will carry 30% weightage and the Summative

Asssessment in term II will carry 30% weightage.

l Listed laboratory activities and projects will necessarily be assessed through

formative assessments.

UNIT I: NUMBER SYSTEMS

1. REAL NUMBERS (18 Periods)

1. Review of representation of natural numbers, integers, rational numbers on the

number line. Representation of terminating / non-terminating recurring decimals on

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the number line through successive magnification. Rational numbers as recurring/

terminating decimals.

2. Examples of non-recurring/non-terminating decimals. Existence of non-rational

numbers (irrational numbers) such as 2, 3 and their representation on the number

line. Explaining that every real number is represented by a unique point on the

number line and conversely, viz. every point on the number line represents a unique

real number.

3. Existence of x for a given positive real number x and its representation on the

number line with geometric proof.

4. Definition of nth root of a real number.

5. Recall of laws of exponents with integral powers. Rational exponents with positive

real bases (to be done by particular cases, allowing learner to arrive at the general

laws.)

6. Rationalization (with precise meaning) of real numbers of the type

1

a bx +

and 1

x y +

(and their combinations) where x and y are natural number and

a and b are integers.

UNIT II: ALGEBRA

1. POLYNOMIALS (23) Periods

Definition of a polynomial in one variable, with examples and counter examples.

Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree

of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials,

binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and

State the Remainder Theorem with examples. Statement and proof of the Factor

Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and

of cubic polynomials using the Factor Theorem.

Recall of algebraic expressions and identities. Verification of identities: (x+y+z)2

= x2

+ y2

+ z2

+ 2xy + 2yz + 2zx, (x ± y)3= x3± y3± 3xy (x ± y), x3± y3= (x ± y) (x2 ±

xy + y2

), x3

+ y3 + z3

— 3xyz =

(x + y + z) (x2

+ y2

+z2

— xy — yz — zx) and their use in factorization of polynomials.

UNIT III : GEOMETRY

1. INTRODUCTION TO EUCLID'S GEOMETRY (6) Periods

History - Geometry in India and Euclid's geometry. Euclid's method of

formalizing observed phenomenon into rigorous Mathematics with definitions,

common/obvious notions, axioms/postulates and theorems. The five postulates of

Euclid. Equivalent versions of the fifth postulate. Showing the relationship between

axiom and theorem, for example:

(Axiom) 1. Given two distinct points, there exists one and only one line through

them.

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(Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.

2. LINES AND ANGLES (13) Periods

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so

formed is 180O and the converse.

2. (Prove) If two lines intersect, vertically opposite angles are equal.

3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a

transversal intersects two parallel lines.

4. (Motivate) Lines which are parallel to a given line are parallel.

5. (Prove) The sum of the angles of a triangle is 180O.

6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal

to the sum of the two interior opposite angles.

3. TRIANGLES (20) Periods

1. (Motivate) Two triangles are congruent if any two sides and the included angle of one

triangle is equal to any two sides and the included angle of the other triangle (SAS

Congruence).

2. (Prove) Two triangles are congruent if any two angles and the included side of one

triangle is equal to any two angles and the included side of the other triangle (ASA

Congruence).

3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to

three sides of the other triangle (SSS Congruence).

4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one

triangle are equal (respectively) to the hypotenuse and a side of the other triangle.

(RHS Congruence)

5. (Prove) The angles opposite to equal sides of a triangle are equal.

6. (Motivate) The sides opposite to equal angles of a triangle are equal.

7. (Motivate) Triangle inequalities and relation between ‘angle and facing side'

inequalities in triangles.

UNIT IV: COORDINATE GEOMETRY

COORDINATE GEOMETRY (6) Periods

The Cartesian plane, coordinates of a point, names and terms associated with the

coordinate plane, notations, plotting points in the plane.

UNIT V: MENSURATION

1. AREAS (4) Periods

Area of a triangle using Heron's formula (without proof) and its application in finding

the area of a quadrilateral.

UNIT II: ALGEBRA (Contd.)

2. LINEAR EQUATIONS IN TWO VARIABLES (14) Periods

Recall of linear equations in one variable. Introduction to the equation in two variables.

Focus on linear equations of the type ax+by+c=0. Prove that a linear equation in two

variables has infinitely many solutions and justify their being written as ordered

pairs of real numbers, plotting them and showing that they lie on a line. Graph

of linear equations in two variables. Examples, problems from real life, including

problems on Ratio and Proportion and with algebraic and graphical solutions being

done simultaneously.

UNIT III: GEOMETRY (Contd.)

4. QUADRILATERALS (10) Periods

1. (Prove) The diagonal divides a parallelogram into two congruent triangles.

2. (Motivate) In a parallelogram opposite sides are equal, and conversely.

3. (Motivate) In a parallelogram opposite angles are equal, and conversely.

4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel

and equal.

5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.

6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is

parallel to the third side and in half of it and (motivate) its converse.

5. AREA (7) Periods

Review concept of area, recall area of a rectangle.

1. (Prove) Parallelograms on the same base and between the same parallels have the

same area.

2. (Motivate) Triangles on the same (or equal base) base and between the same parallels

are equal in area.

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6. CIRCLES (15) Periods

Through examples, arrive at definition of circle and related concepts-radius,

circumference, diameter, chord, arc, secant, sector, segment, subtended angle.

1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate)

its converse.

2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord

and conversely, the line drawn through the center of a circle to bisect a chord is

perpendicular to the chord.

3. (Motivate) There is one and only one circle passing through three given non-collinear

points.

4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the

center (or their respective centers) and conversely.

5. (Prove) The angle subtended by an arc at the center is double the angle subtended

by it at any point on the remaining part of the circle.

6. (Motivate) Angles in the same segment of a circle are equal.

7. (Motivate) If a line segment joining two points subtends equal angle at two other

points lying on the same side of the line containing the segment, the four points lie

on a circle.

8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral

is 180° and its converse.

7. CONSTRUCTIONS (10) Periods

1. Construction of bisectors of line segments and angles of measure 60o

, 90o

, 45o

etc.,

equilateral triangles.

2. Construction of a triangle given its base, sum/difference of the other two sides and

one base angle.

3. Construction of a triangle of given perimeter and base angles.

UNIT V: MENSURATION (Contd.)

2. SURFACE AREAS AND VOLUMES (12) Periods

Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and

right circular cylinders/cones.

UNIT VI: STATISTICS (13) Periods

Introduction to Statistics: Collection of data, presentation of data — tabular form,

ungrouped / grouped, bar graphs, histograms (with varying base lengths), frequency

polygons. Mean, median and mode of ungrouped data.

UNIT VII: PROBABILITY (9) Periods

History, Repeated experiments and observed frequency approach to probability.

Focus is on empirical probability. (A large amount of time to be devoted to group

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and to individual activities to motivate the concept; the experiments to be drawn

from real - life situations, and from examples used in the chapter on statistics).

NUMBER SYSTEMS

1. REAL NUMBERS (15) Periods

Euclid’s division lemma, Fundamental Theorem of Arithmetic - statements after

reviewing work done earlier and after illustrating and motivating through examples,

Proofs of irrationality of 2, 3, 5. Decimal representation of rational numbers in

terms of terminating/non-terminating recurring decimals.

UNIT II: ALGEBRA

1. POLYNOMIALS (7) Periods

Zeros of a polynomial. Relationship between zeros and coefficients of quadratic

polynomials. Statement and simple problems on division algorithm for polynomials

with real coefficients.

2. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES (15) Periods

Pair of linear equations in two variables and graphical method of their

solution, consistency/inconsistency.

Algebraic conditions for number of solutions. Solution of a pair of linear equations in

two variables algebraically - by substitution, by elimination and by cross multiplication

method. Simple situational problems. Simple problems on equations reducible to

linear equations.

UNIT III: GEOMETRY

1. TRIANGLES (15) Periods

Definitions, examples, counter examples of similar triangles.

1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two

sides in distinct points, the other two sides are divided in the same ratio.

2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is

parallel to the third side.

3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding

sides are proportional and the triangles are similar.

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4. (Motivate) If the corresponding sides of two triangles are proportional, their

corresponding angles are equal and the two triangles are similar.

5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the

sides including these angles are proportional, the two triangles are similar.

6. (Motivate) If a perpendicular is drawn from the vertex of the right angle of a right

triangle to the hypotenuse, the triangles on each side of the perpendicular are

similar to the whole triangle and to each other.

7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the

squares of their corresponding sides.

8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the

squares on the other two sides.

9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the

other two sides, the angles opposite to the first side is a right angle.

UNIT IV: TRIGONOMETRY

1. INTRODUCTION TO TRIGONOMETRY (10) Periods

Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their

existence (well defined); motivate the ratios whichever are defined at 0o and 90o

.

Values (with proofs) of the trigonometric ratios of 300

, 450 and 600

. Relationships

between the ratios.

2. TRIGONOMETRIC IDENTITIES (15) Periods

Proof and applications of the identity sin2

A + cos2

A = 1. Only simple identities to be

given. Trigonometric ratios of complementary angles.

UNIT V: STATISTICS AND PROBABILITY

1. STATISTICS (18) Periods

Mean, median and mode of grouped data (bimodal situation to be avoided).

Cumulative frequency graph.