MATHEMATICS

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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 daytoday 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 problemsolving 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 / nonterminating recurring decimals on
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the number line through successive magnification. Rational numbers as recurring/
terminating decimals.
2. Examples of nonrecurring/nonterminating decimals. Existence of nonrational
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 conceptsradius,
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 noncollinear
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/nonterminating 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 rightangled 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. 

