FUTURE POINT COACHING CENTER, NEW
DELHI,MOHAN GARDEN
MATHEMATICS
(IX-X)
Session
2019-20
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 are 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:
·
consolidate
the Mathematical knowledge and skills acquired at the upper
primary stage;
·
acquire
knowledge and understanding,
particularly by way of motivation and visualization,
of basic concepts, terms, principles and symbols and underlying processes and skills.
- develop mastery of basic algebraic skills;
- develop drawing skills;
·
feel the flow of
reason while proving a result or solving a problem;
·
apply
the knowledge and skills acquired to solve
problems and wherever possible, by more than one method;
·
to develop
ability to think, analyze and articulate logically;
·
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;
·
to
develop necessary skills to work with modern technological devices and mathematical software's.
·
to
develop interest in mathematics as a problem-solving tool in various
fields for its beautiful structures and patterns, etc.
·
to
develop reverence and respect towards great
Mathematicians for their contributions to the field of Mathematics;
·
to develop
interest in the subject by participating in related competitions;
·
to acquaint
students with different aspects of Mathematics
used in daily life;
·
to develop an
interest in students to study Mathematics
as a discipline.
COURSE STRUCTURE
CLASS –IX
Units
|
Unit
Name
|
Marks
|
I
|
NUMBER SYSTEMS
|
08
|
II
|
ALGEBRA
|
17
|
III
|
COORDINATE GEOMETRY
|
04
|
IV
|
GEOMETRY
|
28
|
V
|
MENSURATION
|
13
|
VI
|
STATISTICS & PROBABILITY
|
10
|
Total
|
80
|
UNIT I: NUMBER SYSTEMS
1.
REAL NUMBERS (16 Periods)
1. Review of representation of natural
numbers, integers, rational numbers on the number line. Representation of
terminating / non-terminating recurring decimals onthe number line through
successive magnification. Rational numbers as
recurring/ terminating decimals. Operations on real numbers.
2. Examples of
non-recurring/non-terminating decimals. Existence of non-rational numbers
(irrational numbers) such as 
, 
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.
, 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.
Definition of nth
root of a real number.
4. Rationalization (with precise
meaning) of real numbers of the type
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.)
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.
and
their use in factorization of polynomials.
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. Explain 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:
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 IV: GEOMETRY
1.
INTRODUCTION TO EUCLID'S GEOMETRY (Not for assessment) (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. (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.
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
equal area.
2. (Motivate) Triangles on the same base
(or equal bases) and between the same parallels are equal in area.
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
1.
AREAS (4) Periods
Area of a triangle using Heron's
formula (without proof) and its application in finding the area of a
quadrilateral.
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 & PROBABILITY
1.
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.
2.
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 groupand 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).
COURSE STRUCTURE CLASS -X
Units
|
Unit Name
|
Marks
|
I
|
NUMBER SYSTEMS
|
06
|
II
|
ALGEBRA
|
20
|
III
|
COORDINATE GEOMETRY
|
06
|
IV
|
GEOMETRY
|
15
|
V
|
TRIGONOMETRY
|
12
|
VI
|
MENSURATION
|
10
|
VII
|
STATISTICS & PROBABILTY
|
11
|
Total
|
80
|
UNIT I: NUMBER SYSTEMS
1.
REAL NUMBER (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 the irrationality of underfoot 2, underoot 3, underoot 5 etc 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.
3.
QUADRATIC EQUATIONS (15) Periods
Standard form of a
quadratic equation ax2 + bx + c =
0, (a ≠ 0). Solutions of quadratic equations (only real roots) by
factorization, and by using quadratic formula. Relationship between
discriminant and nature of roots.
Situational problems
based on quadratic equations related to day to day activities to be
incorporated.
4.
ARITHMETIC PROGRESSIONS (8) Periods
Motivation for studying Arithmetic
Progression Derivation of the nth term and sum of the first n terms
of A.P. and their application in solving daily life problems.
UNIT III: COORDINATE GEOMETRY
1. LINES (In two-dimensions) (14) Periods
Review: Concepts
of coordinate geometry, graphs of linear equations. Distance formula. Section
formula (internal division). Area of a triangle.
UNIT IV: 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.
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.
2.
CIRCLES (8) Periods
Tangent to a circle at, point of
contact
1. (Prove) The tangent at any point of a
circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn
from an external point to a circle are equal.
3.
CONSTRUCTIONS (8) Periods
1. Division of a line segment in a given
ratio (internally).
2. Tangents to a circle from a point
outside it.
3. Construction of a triangle similar to a given triangle.
UNIT V: 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 of the trigonometric ratios of 300,
450 and 600. Relationships between the ratios.
2.
TRIGONOMETRIC IDENTITIES (15) Periods
Proof and applications of
the identity sin2A + cos2A
= 1. Only simple identities to be given.Trigonometric ratios of
complementary angles.
3.
HEIGHTS AND
DISTANCES: Angle of elevation, Angle of Depression. (8) Periods
Simple problems on heights and
distances. Problems should not involve more than two right triangles. Angles of
elevation / depression should be only 30°, 45°, 60°.
UNIT VI: MENSURATION
1.
AREAS RELATED TO CIRCLES (12) Periods
Motivate the area of a
circle; area of sectors and segments
of a circle. Problems based on areas and perimeter
/ circumference of the above said plane figures. (In calculating area of
segment of a circle, problems should be restricted to central angle of 60°, 90°
and 120° only. Plane figures involving triangles, simple quadrilaterals and
circle should be taken.)
2.
SURFACE AREAS AND VOLUMES (12) Periods
1. Surface areas and volumes of
combinations of any two of the following: cubes, cuboids, spheres, hemispheres
and right circular cylinders/cones. Frustum of a cone.
2. Problems involving converting one
type of metallic solid into another and other mixed problems. (Problems with
combination of not more than two different solids be taken).
UNIT VII:
STATISTICS AND PROBABILITY
1.
STATISTICS (18) Periods
Mean, median and mode of grouped data
(bimodal situation to be avoided). Cumulative frequency graph.
2.
PROBABILITY (10) Periods
Classical definition of
probability. Simple problems on finding the probability of an event.
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