TRIPURA BOARD OF SECONDARY EDUCATION SYLLABUS (effective from 2016) SUBJECT : MATHEMATICS (Class IX)
Total Page- 10 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 Page 1
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 n th 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 Page - 2
and analogy to integers. Statement and proof of the Factor Theorem. Factorization of ax 2 +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), x 3 ±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 180 0 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 180 0. (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. Page 3
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 Coordinate 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 Coordinate 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. Page - 4
MATHEMATICS CLASS IX (HALF YEARLY) 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 1-2 1 11 Arithmetic II 1 1 3 3 24 Algebra III 1 2 3 4 30 Geometry IV - 1 1-5 Co-ordinate V - 1-2 10 Mensuration No. of 03 Nos. 05 Nos. 09 Nos. 10 Nos. 27 Nos. Questions 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. Page - 5
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 Page - 6
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. Page - 7
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 180 0 and its converse. 3.7 CONSTRUCTIONS (10) Periods a) Construction of bisectors of line segments and angles ; construction of angles 60 0, 30 0, 15 0, 90 0, 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. Page 8
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). Page - 9
MATHEMATICS CLASS IX (ANNUAL) 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 1 1 3 2 20 Algebra (contd) III 1 2 3 4 30 Geometry (contd) V 1 1 1 2 14 Mensuration (contd) VI - 1 2 2 16 Statistics and Probability 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. Page 10
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 Evaluation of project work Continuous assessment : 10 marks : 05 marks : 05 marks The breakup of 10 marks of activities will be as under : Complete statement of the objectives of activity Design or approach to the activity Actual conduct of the activity : 1 mark : 2 marks : 3 marks Description /explanation of the Procedure followed Result and conclusion : 3 marks : 1 mark year. He /she should be asked to maintain a proper activity record for this work done during the 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 Design of the project Procedure /processes adopted Interpretations of results : 01 mark : 01 mark : 02 marks : 01 mark Page 11
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. Page 12