
1. Number Systems (Periods  20)



(i) Real Numbers 


Review of representation of natural numbers, integers, rational numbers on the number
line. Representation of terminating / nonterminating recurring decimals, on
the number line through successive magnification. Rational numbers as
recurring/terminating decimals. 


Examples of nonrecurring / non terminating decimals such as v2,
v3, v5 etc. Existence of nonrational numbers (irrational numbers) such as v2, v3 and their representation
on the number line. 


Explaining that every real number is represented by a unique point on the number line and conversely,
every point on the number line represents a unique real number. 


Existence of vx for a given positive real number x (visual proof to be emphasized). 


Definition of nth root of a real number. 


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.) 


Rationalization (with precise meaning) of real numbers of the type (& their combinations) where x and y are natural number and a, b are integers. 





2. Algebra



(i) Polynomials (Periods  25) 


Definition of a
polynomial in one variable, its coefficients, with examples and counter
examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear,
quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and
multiples. Zeros/roots of a polynomial / equation. State and motivate the Remainder
Theorem with examples and analogy to integers. Statement and proof of the
Factor Theorem. Factorization 


of ax^{2} + bx + c, a^{1} 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem. 


Recall of algebraic
expressions and identities. Further identities of the type
(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx, (x y)^{3} = x^{3} y^{3} 3xy (x y). 


x^{3} + y^{3} + z^{3}  3xyz = (x + y + z)
(x^{2} + y^{2} + z^{2}xy  yz  zx) and their use in factorization of. polymonials. Simple
expressions reducible to these polynomials.



(ii) Linear Equations In Two Variables (Periods  12) 


Recall of linear
equations in one variable. Introduction to the equation in two variables. 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 seem to lie on a line. Examples, problems from real life, including
problems on Ratio and Proportion and with algebraic and graphical solutions
being done simultaneously. 


3. Coordinate Geometry



(i) Coordinate Geometry (Periods  9) 


The Cartesian
plane, coordinates of a point, names and terms associated with the coordinate
plane, notations, plotting points in the plane, graph of linear equations as
examples; focus on linear equations of the type ax + by + c =
0 by writing it as y = mx + c and linking with the chapter
on linear equations in two variables. 


4. GEOMETRY



(i) Introduction To Euclid's Geometry (Periods  6) 


History  Euclid
and geometry in India. 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. 


1. Given two distinct points, there exists one and only one line through them. 


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


(ii) Lines And Angles (Periods  10) 


1. (Motivate) If a
ray stands on a line, then the sum of the two adjacent angles so formed is 180^{o} and
the converse. 


2. (Prove) If two
lines intersect, the 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 180^{o}. 


6. (Motivate) If a
side of a triangle is produced, the exterior angle so formed is equal to the
sum of the two interiors opposite angles. 


(iii) Triangles (Periods  20) 


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 Congruene). 


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. 


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. 


(iV) Quadrilaterals (Periods  10) 


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 (motivate) its converse. 


(V) Area 


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 base and between the same parallels are equal in area and
its converse. 


(Vi) Circles (Periods  15) 


Through examples,
arrive at definitions of circle related concepts, radius, circumference,
diameter, chord, arc, 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(s)
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 subtendes 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 the either pair of the opposite angles of a cyclic quadrilateral is 180o and
its converse 


(Vii) Constructions (Periods  10) 


1. Construction of
bisectors of line segments & angles, 60^{o},
90^{o}, 45^{o} angles etc., equilateral triangles. 


2. Construction of
a trangle 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. 


5. Mensuration



(i) Areas (Periods  4) 


Area of a triangle
using Heros formula(without proof) and its application in finding the area of
a quadrilateral 


(ii) Surface Areas And Volumes (Periods  10) 


Surface areas and
volumes of cubes, cuboids, spheres)including hemispheres) and right circular
and right circular cylinders/cones. 


6. Statistics And Probability 


(i) Statistics (Periods  13) 


Introduction to
statistics: Collection of data, Presentation of data tabular form, ungrouped /
grouped, bar graphs, histograms(with varying base lengths), frequency polygons,
qualitative analysis of data to choose the correct form of presentation for the
collected data. Mean, median, mode of ungrouped data. 


(ii) Probability (Periods  12) 


History, Repeated
experiments and observed frequency approach to probability. Focus is on
empirical probability.(A large amount of time to be developed to group and to
individual activities to motivate the concept; the experiment to be drawn from
real  life situations, and from example used in the
chapter on statistics). 



