Topics and Syllabus - CBSE Mathematics Class 9 CBSE (Central Board of Secondary Education)

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. Operations on real numbers.

2. Examples of non-recurring/non-terminating decimals. Existence of nonrational 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.

3. Definition of nth 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.

5. Rationalization (with precise meaning) of real numbers of the type

`1/(a+bsqrtx) " and " 1/(sqrt(x)+sqrt(y))`

 (and their combinations) where x and y are natural 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.)

Concepts :
• Introduction of Real Number  question (58)
• Irrational Numbers  question (55)
• Real Numbers and Their Decimal Expansions  question (14)
• Representing Real Numbers on the Number Line  question (15)
• Operations on Real Numbers  question (82)
• Laws of Exponents for Real Numbers  question (174)

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.

x³+y³+z³-3xyz=(x+y+z) (x²+y²+z²-xy-yz-zx) and their use in factorization of polynomials.

Concepts :
• Remainder Theorem  videos (5) question (42)
• Introduction of Polynomials  question (21)
• Polynomials in One Variable  question (10)
• Zeroes of a Polynomial  question (28)
• Factorisation of Polynomials  question (138)

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 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.

Concepts :
• Introduction of Linear Equations  question (9)
• Solution of a Linear Equation  question (44)
• Graph of a Linear Equation in Two Variables  question (30)
• Equations of Lines Parallel to the X-axis and Y-axis  question (22)
• Concept of Linear Equations in Two Variables  question (3)

Recall of algebraic expressions

Concepts :
• Concept of Algebraic Expressions  question (115)

Recall of algebraic identities. Verification of identities:-

(x+y+z)² = x²+y²+z²+2xy+2yz+2zx

(x±y)³ = x³±y³±3xy (x±y)

x³±y³ = (x±y) (x²±xy+y²)

Concepts :
• Algebraic Identities  question (198)

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.

Concepts :
• Concept for Euclid’S Geometry  question (26)
• Euclid’S Definitions, Axioms and Postulates  question (23)
• Equivalent Versions of Euclid’S Fifth Postulate  question (2)

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.

Concepts :
• Concept to Lines and Angles  question (30)
• Basic Terms and Definitions 
• Intersecting Lines and Non-intersecting Lines  question (22)
• Pairs of Angles  question (61)
• Parallel Lines and a Transversal  question (29)
• Lines Parallel to the Same Line  question (14)
• Angle Sum Property of a Triangle  question (6)

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.

Concepts :
• Concept of Triangles  question (69)
• Congruence of Triangles  question (27)
• Criteria for Congruence of Triangles  question (27)
• Properties of a Triangle  question (99)
• Some More Criteria for Congruence of Triangles  question (7)
• Inequalities in a Triangle  question (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 in half of it and (motivate) its converse.

Concepts :
• Angle Sum Property of a Quadrilateral  question (25)
• Types of Quadrilaterals  question (1)
• Properties of a Parallelogram  question (19)
• Another Condition for a Quadrilateral to Be a Parallelogram  question (41)
• The Mid-point Theorem  question (31)
• Concept of Quadrilaterals  question (17)

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.

Concepts :
• Introduction to Areas of Parallelograms and Triangles  question (50)
• Figures on the Same Base and Between the Same Parallels  question (7)
• Parallelograms on the Same Base and Between the Same Parallels  question (12)
• Triangles on the Same Base and Between the Same Parallels  question (25)

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 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.

8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180o and its converse.

Concepts :
• Circles and Its Related Terms  question (54)
• Angle Subtended by a Chord at a Point  question (9)
• Perpendicular from the Centre to a Chord  question (4)
• Circle Through Three Points  question (7)
• Equal Chords and Their Distances from the Centre  question (18)
• Angle Subtended by an Arc of a Circle  question (40)
• Cyclic Quadrilaterals  question (42)

1. Construction of bisectors of line segments and angles of measure 60°, 90°, 45° 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.

Concepts :
• Introduction of Constructions  question (1)
• Basic Constructions  question (33)
• Some Constructions of Triangles  question (15)

Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral.

Concepts :
• Introduction to Area of a Triangle  question (6)
• Area of a Triangle by Heron’S Formula  question (20)
• Application of Heron’S Formula in Finding Areas of Quadrilaterals  question (39)

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

Concepts :
• Surface Area of a Cuboid and a Cube  question (57)
• Surface Area of a Right Circular Cylinder  question (86)
• Surface Area of a Right Circular Cone  question (47)
• Surface Area of a Sphere  question (55)
• Volume of a Cuboid  question (35)
• Volume of a Cylinder  question (39)
• Volume of a Right Circular Cone  question (39)
• Volume of a Sphere  question (52)

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.

Concepts :
• Introduction of Statistics  videos (1) question (14)
• Collection of Data  question (11)
• Presentation of Data  question (37)
• Graphical Representation of Data  question (63)
• Measures of Central Tendency  question (86)

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).

Concepts :
• Probability - an Experimental Approach  question (43)