# COURSE STRUCTURE CLASS -IX

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1 environment, observance of small family norms, removal of social barriers, elimination of gender biases; mathematical softwares. its beautiful structures and patterns, etc. 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 10 Total 80 UNIT I: NUMBER SYSTEMS 1. REAL NUMBERS (18 Periods) 1. Review of representation of natural numbers, integers, rational numbers on rational 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. th root of a real number. 4. Existence of x for a given positive real number x and its representation on the number line with geometric proof. 50

2 5. Rationalization (with precise meaning) of real numbers of the type 1 and 1 (and their combinations) where x and y are natural a+b x x+ y number and a and b are integers. 6. 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 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 ax 2 and of cubic polynomials using the Factor Theorem. (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) (x 2 xy+y 2 ) 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 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. Prove that a linear equation in two 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. COORDINATE GEOMETRY ± UNIT III: 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. 51

3 UNIT IV: GEOMETRY 1. INTRODUCTION TO EUCLID S GEOMETRY (6) Periods History - Geometry in India and Euclid s geometry. Euclid s method of formalizing 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 180 O 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 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 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. 52

4 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 the same area. 2. (Motivate) Triangles on the same (or equal base) base and between the same parallels are equal in area. 6. CIRCLES (15) Periods 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. 53

5 8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180 o and its converse. 7. CONSTRUCTIONS (10) Periods 1. Construction of bisectors of line segments and angles of measure 60 o, 90 o, 45 o etc., equilateral triangles. 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 2. SURFACE AREAS AND VOLUMES (12) Periods Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and UNIT VI: STATISTICS & PROBABILITY 1. STATISTICS (13) Periods Introduction to Statistics: Collection of data, presentation of data tabular 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 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). 54

6 QUESTIONS PAPER DESIGN CLASS IX Mathematics (Code No. 041) Time : 3 hrs Marks: 80 S. No. Typology of Questions Very Short (VSA) (1 Mark) Short -I (SA) (2 Marks) Short -II (SA) (3 Marks) Long (LA) (4 Marks) Total Marks % Weightage (approx.) 1 Remembering % Simple recall questions, to know principles or theories; Identify, 2 Understanding (Comprehension % to be familiar with meaning and to understand conceptually, interpret, compare, contrast, explain, paraphrase, or interpret information) 3 Application (Use abstract information % in concrete situation, to apply knowledge to new situation; Use given content to interpret a situation, provide an example, or solve a problem) 4 Higher Order Thinking Skills % compare, contrast, or differentiate between different pieces integrate unique pieces of information from variety of sources ) 5 Evaluation % the value or worth of a decision or outcome, or to predict outcomes based on values) Total 6x1=6 6x2 =12 10x3 =30 8x4 = % Note: One of the LA will be to assess the values inherent in the texts. INTERNAL ASSESSMENT 20 Marks 55

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