CBSE Class 9 Mathematics

Here are the chapter names from the NCERT Class 9 Mathematics book for which we have provided study material on our website. These chapters are there in your NCERT textbook for Class 9 Maths

Chapter 1: Number Systems
Chapter 2: Polynomials
Chapter 3: Coordinate Geometry
Chapter 4: Linear Equations in Two Variables
Chapter 5: Introduction to Euclid’s Geometry
Chapter 6: Lines and Angles
Chapter 7: Triangles
Chapter 8: Quadrilaterals
Chapter 9: Areas of Parallelograms and Triangles
Chapter 10: Circles
Chapter 11: Constructions
Chapter 12: Heron’s Formula
Chapter 13: Surface Areas and Volumes
Chapter 14: Statistics
Chapter 15: Probability

Mathematics for class 9 is considered to be a scoring subject which can help you to improve your marks in Class 9 examination. There are various chapters in Class 9 Maths which is understood properly can help you in long term. 

1. Work with real numbers and consolidate and the concepts of numbers learnt in earlier classes. Some such opportunities could be,

• To observe and discuss real numbers.

• To recall and observe the processes involved in different mathematical concepts studied earlier and find situations in which they come across irrational numbers. For example,finding the length of the diagonal of a square with side, say, 2 units or area of a circle with a given radius etc.

• To observe properties of different types of numbers, such as the denseness of the numbers, by devising different methods based on the knowledge of numbers gained in earlier classes. One of them could be by representing them on the number line.

• In making mental estimations in different situations such as arranging numbers like 2, 2^1/2,2^3/2, 2^5/2, etc. in ascending (or descending) order in a given time frame or telling between which two numbers like, √17, √23, √59, - √2, etc. lie.

• Discussion about how the polynomials are different from algebraic expressions.

2. Discuss proofs of mathematical statements using axioms and postulates. You can refer to various Worksheets for Class 9 Mathematics on regular basis to practice on daily basis which will help you to score more marks.

3. Encourage to play the following games related to geometry. If one group says, if equals are added to equals, then the results are equal. The other group may be encouraged to provide example such as, If a=b, then a+3=b+3, another group may extend it further as a+3+5= b+3+5 and so on and can be used by referring to NCERT Solutions for Class 9 Mathematics.

4. By observing different objects in the surroundings one group may find the similarities and the other group finds differences with reference to different geometrical shapes -lines, rays, angles, parallel lines, perpendicular lines, congruent shapes, non-congruent shapes etc. and justify their findings logically.

5. Discuss in groups about the properties of triangles and construction of geometrical shapes such as triangles, line segment and its bisector, angle and its bisector under different conditions

6. Find and discuss ways to fix position of a point in a plane and different properties related to it.

7. Engage in a survey and discuss about different ways to represent data pictorially such as bar graphs, histograms (with varying base lengths) and frequency polygons.

8. Collect data from their surroundings and calculate central tendencies such as mean, mode or median.

9. Explore the features of solid objects from daily life situations to identify them as cubes, cuboids, cylinders etc and Class 9 Maths Sample Papers.

10. Play games involving throwing a dice, tossing a coin etc. and find their chance of happening.

11. Do a project of collecting situations for different numbers representing probabilities.

Unit-I: NUMBER SYSTEMS (10 Marks)

• REAL NUMBERS: 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.
• Irrational Numbers: Examples of non-recurring/non terminating decimals. Existence of non-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.
• Roots of Real Numbers: Definition of nth root of a real number.
• Rationalization: Rationalization (with precise meaning) of real numbers of the type 1/(a + b√x) and 1/(√x + √y) (and their combinations), where x and y are natural numbers and a and b are integers.
• Laws of Exponents: 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.)

Note for Students: This unit develops a deeper understanding of the real number system, focusing on the representation of irrational numbers and the practical application of exponent laws.

Unit-II: ALGEBRA (20 Marks)

• POLYNOMIALS: 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. Zeroes of a polynomial.
• Remainder and Factor Theorems: 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.
• Algebraic Identities: Recall of algebraic expressions and 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); x³ – y³ = (x – y)(x² + xy + y²); x³ + y³ + z³ – 3xyz = (x + y + z)(x² + y² + z² – xy – yz – zx) and their use in factorization of polynomials.
• LINEAR EQUATIONS IN TWO VARIABLES: 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. Explain 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.

Note for Students: Students will learn the art of factoring polynomials using various theorems and identities, and visualize algebraic solutions through the graphing of linear equations.

Unit-III: COORDINATE GEOMETRY (04 Marks)

• Coordinate Geometry: The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations.

Note for Students: This unit introduces the foundational concepts of the Cartesian plane, enabling students to specify locations and describe spatial relationships using coordinate geometry.

Unit-IV: GEOMETRY (27 Marks)

• 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.
• LINES AND ANGLES: (State without proof) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse. (Prove) If two lines intersect, vertically opposite angles are equal. (State without proof) Lines which are parallel to a given line are parallel.
• TRIANGLES: (State without proof) Two triangles are congruent if any two sides and the included angle of one triangle is equal (respectively) to any two sides and the included angle of the other triangle (SAS Congruence). (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal (respectively) to any two angles and the included side of the other triangle (ASA Congruence). (State without proof) Two triangles are congruent if the three sides of one triangle are equal (respectively) to three sides of the other triangle (SSS Congruence). (State without proof) 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). (Prove) The angles opposite to equal sides of a triangle are equal. (State without proof) The sides opposite to equal angles of a triangle are equal.
• QUADRILATERALS: (Prove) The diagonal divides a parallelogram into two congruent triangles. (State without proof) In a parallelogram opposite sides are equal, and conversely. (State without proof) In a parallelogram opposite angles are equal, and conversely. (State without proof) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. (State without proof) In a parallelogram, the diagonals bisect each other and conversely. (State without proof) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and is half of it and (State without proof) its converse.
• CIRCLES: (Prove) Equal chords of a circle subtend equal angles at the center and (State without proof) its converse. (State without proof) 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. (State without proof) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely. (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. (State without proof) Angles in the same segment of a circle are equal. (State without proof) 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. (State without proof) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

Note for Students: This comprehensive geometry unit focuses on logical reasoning and axiomatic approaches to prove properties of lines, triangles, quadrilaterals, and circles.

Unit-V: MENSURATION (13 Marks)

• AREAS: Area of a triangle using Heron's formula (without proof).
• SURFACE AREAS AND VOLUMES: Surface areas and volumes of spheres (including hemispheres) and right circular cones.

Note for Students: You will learn to calculate the areas of triangles and the surface areas and volumes of essential three-dimensional solid objects like spheres and cones.

Unit-VI: STATISTICS (06 Marks)

• STATISTICS: Bar graphs, Histograms (with varying base lengths), Frequency polygons.

Note for Students: This unit teaches the visual representation of data, allowing you to draw and interpret various types of graphs and frequency polygons.