TRIPURA BOARD OF SECONDARY EDUCATION

SYLLABUS

(effective from 2016)

SUBJECT : MATHEMATICS (Class –IX)

MATHEMATICS

COURSE STRUCTURE

Class IX

HALF YEARLY

One Paper Time: 3 Hours Marks: 90

Unit Title Marks

I Arithmetic 11

II Algebra 24

III Geometry 30

IV Co-Ordinate Geometry 05

V Mensuration 10

Total 80

Internal Assessment 20

Grand Total 100

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Total Page- 10

UNIT-I: ARITHMETIC (18) Periods

1. REAL NUMBERS :

1.1 Review of representation of natural numbers, integers rational numbers on the number line.

Representation of terminating/non terminating recurring decimals on the number line through

successive magnification Rational numbers as recurring/terminating decimals.

1.2 Examples of non-recurring/non-terminating decimals such as √2, √3, √5 etc. Existence of

Non-rational numbers (irrational numbers) such as √2, √3, and their representation on the

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

Number line and conversely every point on the number line represent a unique real number.

1.3 Rational numbers as recurring/terminating decimals.

1.4 Definition of nth

root of a real number, order of irrational members such as √2, 3√4,

5√6 ; like and unlike irrational numbers.

1.5 Recall of laws of exponents with integral powers. Rational exponents with positive real

bases (to be done by particulars cases, allowing learners to arrive at the general laws.)

1.6 Conjugate of irrational numbers. E.g. √3 - √2 is the conjugate to √3 + √2 etc.

1.7 Rationalization (with precise meaning) of real numbers of the type (and their combinations)

1 & 1 , where x and y are natural numbers and a and b are integers (b=0) use of four

A+b√x √x + √y

fundamental rules in irrational numbers.

Note : Emphasis should be given on the following points :

i) Number line is to be reviewed that all real numbers are represented by different points

lying on the number line

ii) Problems related to the use of four fundamental rules on irrational numbers.

UNIT-II: ALGEBRA (23) Periods

2.1. POLYNOMIALS

Definition of a Polynomial in one variable, its co-efficients, with examples and counter

examples, its terms, Zero Polynomial. Degree of a Polynomial. Constant, Linear, Quadratic

and Cubic Polynomials ; Monomials, Binomial, Trinomials. Factors and multiples, Zeros/

Roots of a Polynomial/equation. State and motivate the ‘Remainder Theorem’ with examples

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and analogy to integers. 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.

2.2 Recall and application of algebraic expressions and identifies. Further verification of

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),

x3±y

3=(x±y)(x2 + xy+y

2), x

3+y

3+z

3 – 3xyz=(x+y+z) (x

2+y

2+z

2-xy-yz-zx) and their uses in

factorization of Polynomials. Simple expressions reducible to these Polynomials.

Note : Exercises based on formula of previous classes like (x±y)2, (x+y)

2=(x-y)

2+4xy,(x-y)

2=

(x+y)2- 4xy, and of present class (x+y+z)

2, (x±y)

3, (x

3±y

3),(x+y+z)

3, x

3+y

3+z

3 – 3xyz are to be

dealt with.

UNIT-III: GEOMETRY (36) Periods

3.1 INTRODUCTION TO EUCLID’S GEOMETRY

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) (i) Given two distinct points, there exists one and only one line through them.

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

3.2 LINES AND ANGLES

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

1800

and the converse.

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

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

transversal intersects two parallel lines.

(d) (Motivate) lines which are parallel to a given line are parallel.

(e) (Prove) The sum of the angles of a triangle is 1800

.

(f) (Prove) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of

the two interior opposite angles.

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3.3 TRIANGLES

(a) (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).

(b) (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).

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

sides of the other triangle (SSS congruence)

(d) (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.

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

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

(g) (Motivate) Triangle inequalities and relation between ‘angle and facing side’ inequalities in

triangles.

Note : Items under ‘motivation’ indicates that students will acquire the corresponding concept

and they will be able to apply them in solving problems. There shall be exercises on the above

topics. However, no question will be set from these portions.

UNIT-IV: CO-ORDINATE GEOMETRY (07) Periods

4.1 Co-ordinate Geometry:

Concept of Cartesian plane, co-ordinates of a point, names and terms associated with the Co-

ordinate plane, notations, plotting of points in the plane, graph of linear equations as examples;

focus on linear equations ax +by +c=0 by writing it as y=mx +c, form of an equation, a line

passing origin, equations of Co-ordinate axes and equations of straight lines parallel to Co-

ordinate axes.

UNIT-V: MENSURATION (06) Periods

5.1 AREAS :

Perimeter of a triangle, area of a triangle by using Heron’s formula (without proof) and its

application in finding the area of quadrilateral, perimeter of a quadrilateral, circumference and

area of a circle and their application.

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HALF YEARLY

UNITWISE QUESTION TYPES WITH MARKS DISTRIBUTION

Unit VSA

(1 mark)

SA

(2 marks)

LA-I

(3 marks)

LA-II

(4 marks)

Total Marks

I

Arithmetic

1 - 2 1 11

II

Algebra

1 1 3

3

24

III

Geometry

1 2 3

4

30

IV

Co-ordinate

- 1 1 - 5

V

Mensuration

- 1 - 2

10

No. of

Questions

03 Nos. 05 Nos. 09 Nos. 10 Nos. 27 Nos.

Total marks 03 marks 10 marks 27 marks 40 marks 80 marks

N.B. 1. All questions are compulsory.

2. There is no overall choice in the paper. However internal choice is provided in one question of

three marks in Unit-II and one question of three marks in Unit-III, and also one question of four

marks in Unit –II, two questions of four marks in Unit-III and one question of four marks in Unit-

V.

3. In LA-I and LA-II type of questions total marks may be subdivided into different parts, if

necessary.

4. Use of calculator is not permitted.

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MATHEMATICS CLASS – IX (HALF YEARLY)

Class IX

ANNUAL

One Paper Time: 3 Hours Marks: 90

Unit Title Marks

II Algebra (contd.) 20

III Geometry (contd.) 30

V Mensuration (contd.) 14

VI Statistics and Probability 16

Total (Theory) 80

Internal Assessment 20

Grand Total 100

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UNIT-II: ALGEBRA (Contd.) (14) Periods

2.3 LINEAR EQUATIONS IN TWO VARIABLES:

Recall of linear equations in one variable. Introduction to the equations 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 lie on a line.

Examples, problems from real life, including problems on Ratio and proportions and with

algebraic and graphical solutions being done simultaneously.

UNIT-III: GEOMETRY (Contd.) (10) Periods

3.4 QUADRILATERALS :

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

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

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

d) (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and

equal.

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

f) (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.

3.5 AREA (04) Periods

Review concept of area, recall area of a rectangle.

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

b) (Motivate) Triangles on the same base and between the same parallels are equal in area and its

converse.

3.6 CIRCLES (15) Periods

Through examples, arrive at definitions of circle related concepts, radius, circumference,

diameter, chord, arc, subtended angle.

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

converse.

b) (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.

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

d) (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center(s)

and conversely.

e) (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.

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

g) (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.

h) (Motivate) The sum of the either pair of the opposite angles of a cyclic quadrilateral is 1800

and its converse.

3.7 CONSTRUCTIONS (10) Periods

a) Construction of bisectors of line segments and angles ; construction of angles 600, 30

0, 15

0,

900, 45

0 ……(without protector).

b) Construction of a triangle given its base, sum/difference of the other two sides and one base

angle.

c) Construction of a triangle of given perimeter and base angles.

d) To construct a parallelogram equal in area to a given triangle with one of its angles equal to a

given angle.

e) To construct a triangle equal in area to a given quadrilateral.

UNIT-V: MENSURATION (Contd.) (12) Periods

5.2 SURFACE AREAS AND VOLUMES

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

circular cylinders/cones.

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UNIT-VI: STATISTICS AND PROBABILITY

6. 1. STATISTICS (13) Periods

Introduction to statistics ; Collection of data, Presentation of data – tabular form,

Ungrouped/Grouped, Bar graphs, Histograms (with fixed base lengths), Frequency polygons,

Mean, Medium, mode of ungrouped data.

6.2 PROBABILITY (12) Periods

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 experiments to be drawn from real-life situations, and from

examples used in the chapter on statistics).

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ANNUAL

UNITWISE QUESTION TYPES WITH MARKS DISTRIBUTION

Unit VSA

(1 mark)

SA

(2 marks)

LA-I

(3 marks)

LA-II

(4 marks)

Total

Marks

II

Algebra (contd)

1 1 3 2

20

III

Geometry

(contd)

1 2 3

4

30

V

Mensuration

(contd)

1 1 1 2

14

VI

Statistics and

Probability

- 1 2 2

16

No. of Questions 03 Nos. 05 Nos. 09 Nos. 10 Nos. 27 Nos.

Total marks 03 marks 10 marks 27 marks 40 marks 80marks

N.B. 1. All questions are compulsory.

2. There is no overall choice in the paper. However internal choice is provided in one question of

three marks in Unit-III, and one question of four marks in Unit-II, two questions of four marks

in Unit – III, one question of four marks in Unit-V and one question of four marks in Unit-VI.

3. In LA-I and LA-II type of questions total marks may be subdivided into different parts, if

necessary.

4. Use of calculator is not permitted.

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MATHEMATICS CLASS – IX (ANNUAL)

INTERNAL ASSESMENT FOR CLASS – IX

Internal Assessment will have a weightage of 20 marks as per the following break up :

Year-end evaluation of activities : 10 marks

Evaluation of project work : 05 marks

Continuous assessment : 05 marks

The breakup of 10 marks of activities will be as under :

Complete statement of the objectives of activity : 1 mark

Design or approach to the activity : 2 marks

Actual conduct of the activity : 3 marks

Description /explanation of the

Procedure followed : 3 marks

Result and conclusion : 1 mark

He /she should be asked to maintain a proper activity record for this work done during the

year.

The schools would keep a record of the conduct of this examination for verification. This

assessment will be internal and done preferably by a team of two teachers.

EVALUATION OF PROJECT WORK

Every student will be asked to do one project based on the concepts learnt in the

classroom but as an extension of learning to real life situations. This project work should not be

repetition or extension of laboratory activities but should infuse new elements and could be open

ended and carried out beyond the school working hours.

Five marks weightage could be further split up as under:

Identification and statement of the project : 01 mark

Design of the project : 01 mark

Procedure /processes adopted : 02 marks

Interpretations of results : 01 mark

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CONTINUOUS ASSESSMENT

Continuous assessment will be awarded on the basis of performance of students in their half

yearly and annual examinations. The strategy given below may be used for awarding internal

assessment in Class IX :

(a) Reduce the marks of the half yearly examination to be out of ten.

(b) Reduce the marks of the annual examination to be out of ten.

(c) Add the marks of (a) and (b) above and get the achievement of the learner out of twenty

marks.

(d) Reduce the total in (c) above to the achievement out of five marks.

Some model of Project Works of Class –IX

(1) To divide a line segment into given numbers of equal parts.

(2) Square roots of natural numbers.

(3) Centroid of a triangle.

(4) Incentre of a triangle.

(5) Sides and angles of a triangle.

(6) Circumcenter of a triangle.

(7) Orthocenter of a triangle.

(8) Area of a triangle.

(9) Mid Point theorem.

(10) Chord property of a circle.

Esteemed teachers are requested to include more topics as given above pertaining to the

syllabus.

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