CBSE Class 9 Syllabus 2027 Mathematics

FOR CBSE CLASS 9 SYLLABUS EXAM PREPARATION

CBSE Class 9 Syllabus
2027
Question Paper ·
Mathematics
EXAM YEAR TYPE SUBJECT

CBSE Class 9 Syllabus 2027 Question Paper Mathematics

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Class IX (2026 – 27)

Introduction:

The Mathematics curriculum for the Secondary stage has been redesigned in alignment with the
National Education Policy 2020 and the National Curriculum Framework for School Education
(NCF – SE) 2023, prioritizing deep conceptual understanding and logical reasoning. The revised m
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syllabus places strong emphasis on developing core mathematical competencies, including
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problem-solving, visualisation, mathematical modelling, mathematical communication,
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computational thinking, and data analytics. The syllabus integrate Indian Knowledge System with
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contemporary mathematical knowledge, highlighting the rich contributions of Indian
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mathematicians to foster a sense of pride and historical context. A deliberate shift from rote
learning to competency-based education ensures that students build deep conceptual
understanding and logical reasoning rather than mere procedural fluency. Greater emphasis has
been laid on the integration of real-life applications and experiential learning, encouraging students
to connect mathematical concepts with everyday situations and cross-disciplinary contexts.
Greater emphasis has been laid on competency based learning outcomes encouraging students
to connect mathematical concepts with everyday situations and inter-disciplinary contexts.

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Continuous and holistic assessment through projects, activities, and investigations forms an

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integral part of the learning process, moving beyond summative examinations.
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At the secondary stage, the curriculum focuses on developing essential global mathematical
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competencies, including mathematical representation through quantities and relations,
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mathematical modelling and algorithm building, and effective mathematical communication. The
study of the number system, algebra, geometry, mensuration, statistics and probability is designed
to build a strong foundation for higher education while enhancing functional life skills. The
curriculum thus aims to build rich mathematical learning frameworks not only for higher academic
pursuits but also for the practical demands of life in a rapidly changing, data-driven world.

Objectives: The broad objectives of teaching Mathematics at the secondary stage are to help the
learners to:
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develop logical thinking, critical reasoning, and a structured approach to problem-solving;
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build the ability to recognise, analyse, and solve diverse problems with confidence and
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communicate mathematical ideas effectively using appropriate language, symbols, and

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representations;
appreciate the beauty, history, and real-life relevance of Mathematics as a discipline;
 connect mathematical concepts to fields such as Science, Technology, Engineering, and
Economics;
 engage in both collaborative and independent mathematical exploration and learning;
 develop habits of precision, accuracy, and logical consistency in mathematical work;
 build confidence to explore, experiment, and grow in mathematical understanding without
fear of failure.
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Curricular Goals (CGs) and Competencies (Cs) from the NCF-SE 2023

CG-1: Understands numbers (natural, whole, integer, rational, irrational, and real), ways of
representing numbers, relationships amongst numbers, and number sets.
C-1.1 Develops understanding of numbers, including the set of real numbers and its properties.

CG-2: Builds deductive and inductive logic to prove theorems related to numbers and their
relationships (such as ‘2 is an irrational number’, a recursion relation for Virahanka numbers,
a formula for the sum of the first n square numbers).
C-2.1 Understanding of powers (radical powers) and exponents.

CG-3: Discovers and proves algebraic identities and models real-life situations in the form
of equations to solve them.
C-3.1 States and proves remainder theorem, factor theorem, and division algorithm.
C-3.2 Models and solves contextualised problems using equations (for example, simultaneous
linear equations in two variables or single polynomial equations), and draws conclusions about a
situation being modelled.
C-3.3 Learns Brahmagupta’s quadratic formula (in both symbolic and poetic form) and its derivation,
and uses it to solve some of the poetic puzzles of Bhaskara as well as modern-day problems.

CG-4: Analyses characteristics and properties of two-dimensional geometric shapes, and
develops mathematical arguments to explain geometric relationships.
C-4.1 Describes relationships including congruence of two-dimensional geometric shapes (such as
lines, angles, triangles) to make and test conjectures and solve problems.
C-4.2 Proves theorems using Euclid’s axioms and postulates for triangles and quadrilaterals, and
applies them to solve geometric problems.
C-4.3 Proves theorems about the geometry of a circle, including its chords, subtended angles,
inscribed polygons, and area in terms of pi.
C-4.4 Understands the irrationality of pi, the best approximations to be discovered over human
history, and the first exact formula (infinite series) for pi given by Madhava.
C-4.5 Specifies locations and describes spatial relationships using coordinate geometry, for
example, plotting a pair of linear equations and graphically finding the solution, or finding the area
of triangle with given coordinates as vertices.
C-4.6 Understands the definitions of the basic trigonometric functions, their history and motivation
(including the introduction of the sin and cos functions by Aryabhata using chords), and their utility
across the sciences.

CG-5: Derives and uses formulae to calculate areas of plane figures, surface area, and
volumes of solid objects.
C-5.1 Visualises, represents, and calculates the area of a triangle using Heron’s formula and its
generalisation to cyclic quadrilaterals given by Brahmagupta’s formula.
C-5.2 Visualises and uses mathematical thinking to discover formulae to calculate surface areas
and volumes of solid objects (cubes, cuboids, spheres, hemispheres, right circular cylinders or
cones, and their combinations).

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CG-6: Analyses and interprets data using statistical concepts (such as measures of central
tendency, standard deviations) and probability.
C-6.1 Applies measures of central tendencies, such as mean, median, and mode.
C-6.2 Applies concepts from probability to solve problems on the likelihood of everyday events.

CG-7: Begins to perceive and appreciate the axiomatic and deductive structure of
Mathematics.
C-7.1 Proves mathematical statements and carries out geometric constructions using stated
assumptions, axioms, postulates, definitions, and mathematics vocabulary.
C-7.2 Visualises and appreciates geometric proofs for algebraic identities and other ‘proofs without
words’.
C-7.3 Proves theorems using Euclid’s axioms and postulates for angles, triangles, quadrilaterals,
circles, area-related theorems for triangles, and parallelograms.
C-7.4 Constructs different geometrical shapes like bisectors of line segments, angles and their
bisectors, triangles, and other polygons, satisfying given constraints.

CG-8: Builds skills, such as visualisation, optimisation, representation, and mathematical
modelling along with their application in daily life.
C-8.1 Models daily-life phenomena and uses representations, such as graphs, tables, and
equations to draw conclusions.
C-8.2 Uses two-dimensional representations of three-dimensional objects to visualise and solve
problems, such as those involving surface area and volume.
C-8.3 Employs optimisation strategies to maximise desired quantities (such as area, volume, or
other output) under given constraints.

CG-9: Develops computational thinking, i.e., deals with complex problems and is able to
break them down into a series of simple problems that can then be solved by suitable
procedures/algorithms.
C-9.1 Decomposes a problem into sub-problems.
C-9.2 Describes and analyses a sequence of instructions being followed.
C-9.3 Analyses similarities and differences among problems to make one solution or procedure
work for multiple problems.
C-9.4 Engages in algorithmic problem-solving to design such solutions.

CG-10: Knows and appreciates important contributions of mathematicians from India and
around the world.
C-10.1 Recognises the important contributions made by mathematicians (Indian and others) in the
field of Mathematics (such as the evolution of numbers, geometry, and algebra).
C-10.2 Recognises modern contributions to Mathematics made in both India and abroad, and
understands the next frontiers and next major open questions in the field of Mathematics.

CG-11: Explores connections of Mathematics with other subjects.
C-11.1 Applies mathematical knowledge and tools to analyse problems or situations in multiple
subjects across Science, Social Science, Visual Arts, Music, Vocational Education, and Sports.

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COURSE STRUCTURE CLASS – lIX
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Units Unit Name Chapter Name Marks

I Number System  Number System 07
II Algebra  Introduction to Polynomials 20
 Sequences and Progressions
 Exploring Algebraic Identities
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
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Linear Equations in Two Variables

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III Coordinate 04
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Coordinate Geometry
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Geometry
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Geometry  Introduction to Euclid’s Geometry:
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Axioms and Postulates
Lines and Angles
 Triangles – Congruence Theorems
 4-gons (Quadrilaterals)
 Circles
V Mensuration  Area and Perimeter 14
 Surface Area and Volume
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VI Statistics and Statistics 10
Probability 
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Introduction to Probability
Total
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Chapter Key Concepts Releva Competencies
Name nt CGs
Unit 1: Number System No. of periods : 12

Number  Introduction to rational The student will be able to:
System numbers  Understand the concept of a rational
 Representation of CG-1, number.
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 Represent rational numbers on the
rational numbers on the C-1.1,

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number line CG-9 number line.

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 Understand the properties of rational
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Density of rational
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 Explain the concept of density of
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numbers
rational numbers.
 Compute decimal representation of
 Decimal representation rational numbers.
of rational numbers  Understand the concept of irrational
 Introduction to irrational numbers.
numbers  Prove the irrationality.
 Proof of irrationality of  Construct the square root spiral.
√2 and √3  Apply computational thinking to
 m represent rational and irrational
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The square root spiral
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numbers through algorithms and visual
models, generate decimal expansions
systematically, and reason about
numbers using step-by-step logical
procedures.

UNIT II: ALGEBRA No. of periods : 66

Introduction  Algebraic expressions The student will be able to:
to  Definition of a  Understand the meaning of an
Polynomials polynomial. CG-3, algebraic expression.
 Degree of a polynomial C-3.2,  Define a polynomial.
 Introduction to linear CG-9  Identify the degree, terms and
polynomials and coefficients of terms in a polynomial.
applications  Model linear growth and decay using
 Exploring linear patterns linear polynomials.
 Modelling linear growth  Explain and identify patterns in linear
and linear decay relationships.
 Linear relationships  Identify the slope and y-intercept of a
 Visualising linear linear equation in two variables.
relationships  Graph a linear equation in two
 Slope and y-intercept of variables.
a line y = ax + b  Use computational thinking to identify
patterns, construct linear expressions,
and systematically represent and
analyse linear relationships using
equations and graphs.

Sequences  Introduction to The student will be able to:
and sequences • Understand the concept of a sequence
Progression  Explicit or general rule CG-11, of numbers.
s of a sequence C-8.1, • Identify the pattern in a sequence and
 Recursive rule of a CG-9 predict the next few terms.
sequence • Determine the recursive and explicit
 Arithmetic Progressions rules for different sequences.
 (AP): nth term, • Obtain the terms of sequence given its
visualising an AP, and recursive and explicit rule.
practical contexts • Identify Arithmetic Progressions (AP).
leading to Aps • Determine the nth term of an AP.
 Sum of the first n • Visualise an AP graphically.
natural numbers • Identify Geometric Progressions (GP).
 Geometric • Determine the nth term of a GP.
Progressions (GP): nth • Visualise a GP graphically.
term, visualising a GP, • Analyse attributes of fractals using GP.
and practical contexts • Solve the Tower of Hanoi puzzle.
leading to GPs • Use computational thinking to identify
patterns, write step-by-step rules, and
 Applications of GP in
model patterns in sequences and
fractals
progressions.
 Tower of Hanoi puzzle

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Exploring  Revisiting algebraic The student will be able to:
Algebraic identities CG-7, • Visualise algebraic identities using
Identities  Visualising identities C-7.2, geometric models.
using geometrical CG-9 • Determine the factors of algebraic
models expressions using identities.
 Factorisation of algebraic • Interpret factors of quadratic
expressions using expressions through geometric
identities models.
 More identities and their • Find simplified versions of rational
applications expressions.
 Visualising factorisation • Use computational thinking strategies,
of quadratic expressions such as decomposition and step-by-
through algebra tiles and step procedures to visualise algebraic
without using algebra identities, factor expressions, and
tiles simplify rational expressions.
 Finding new identities
 Simplifying rational
expressions
Linear  Introduction to linear The student will be able to:
Equations in equations in two  Understand the concept of a linear
Two variables through CG-3, equation in two variables.
Variables practical examples C-3.2,  Graph a pair of linear equations.
 Solution of linear C-8.1,  Solve a pair of linear equations
equation in two variables: CG-9 graphically.
graphical representation  Solve a pair of linear equations
 Slope-intercept form of through the methods of substitution
linear equation in two and elimination.
variables  Determine the nature of solutions of a
 Drawing graphs of linear pair of linear equations.
equations when x and y  Model and solve contextualised
assume only certain problems using a pair of linear
values equations and draw conclusions.
 Pair of linear equations in  Model daily-life phenomena using
two variables representations, such as graphs,
 Graphical method for tables, and equations.
solving a pair of linear  Use computational thinking to
equations in two systematically represent, solve, and
variables interpret pairs of linear equations
 Nature of solutions: through graphs, tables, and step-by-
consistency and step procedures.
inconsistency
 Algebraic methods of
solving a pair of linear
equations: substitution
and elimination method

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UNIT III: COORDINATE GEOMETRY No. of periods : 6

Coordinate  Brief history of
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The student will be able to:
Geometry coordinate geometry  Specify locations and the position of
 The 2-D Cartesian CG-4, one point relative to another point
coordinate system C-4.5, using coordinates.
 Distance between two CG-9  Represent a floor plan on a grid using
points in the 2-D plane coordinates.
 Midpoint of the line-  Compute the distance between two
segment between two points using coordinates.
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points in the 2-D plane  Determine whether three points lie in a
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straight line using coordinates.
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Compute the position of the midpoint
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of a line segment using coordinates.

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angled using coordinates.
Apply computational thinking to model
situations on the coordinate plane and
verify geometric properties through
systematic reasoning.

UNIT IV: GEOMETRY No. of periods : 69

Introduction  History of geometry
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The student will be able to:

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to Euclid’s  Constructing a square • Describe how geometry grew from the
Geometry: with a given side as
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Axioms and
Postulates
described in the
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• Describe contributions of India, Egypt
and Greece to the development of
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 Discovering Euclid’s • Understand the role of definitions,
definitions axioms, and postulates.
 Axioms: Axioms of • Explain that there are elements of
measurement and rules plane geometry (point, line, surface)
for geometric objects for which we have an intuitive sense.
• State the 5 postulates of Euclidean
geometry.
• Define parallelism of straight lines.
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• Explain the construction of a square as
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given in the Sulbasutras.
• Justify simple constructions using the
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axioms.

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Rays and angles
Measures of angles
will be able to:
• Explain the notion of an angle.
 Intersecting lines and CG-7, • Explain the notion of a ray.
angles C-7.1, • Explain that angles are formed
 Pairs of angles C-7.3, between two rays with a common
 Theorems and examples CG-9 starting point.
on intersecting lines • State that a straight angle equals two
 Theorems and examples right angles and measures 180° while
a right angle measures 90°.
on parallel lines
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• Classify angles as acute, right, obtuse,
or reflex.
• Define parallelism.
• State and apply the linear pair theorem
and its converse.
• Follow proof by contradiction in
geometry.
• Prove that vertically opposite angles
are equal.
• Identify corresponding, alternate, and
interior angles.
• Explain transitivity of parallelism.
• Explain why a triangle must have at
least two acute angles; why it cannot
have two obtuse angles, or all three
angles less than 60°
• Apply computational thinking to
analyse geometric ideas by breaking
constructions into ordered steps, using
axioms and postulates as rules, and
justifying geometric results through
logical step-by-step reasoning.

Triangles:  Practical applications of The student will be able to:
Congruence triangles • Explain that a triangle is rigid, unlike a
Theorems  Proofs of conditions of CG-4, quadrilateral.
congruence of triangles C 4.1, • Identify uses of triangle rigidity.
 Theorems on triangles C-7.3 • Explain why triangles give strength and
 Propositions and their stability to structures.
converse • Describe what it means for two
 Problems based on triangles to be congruent.
applications of theorems • Identify correspondence between the
on triangles vertices, sides, and angles of two
congruent triangles.
• Use the SAS congruence axiom.
• Use the SSS congruence condition.
• Use the ASA congruence condition.
• Use the RHS congruence condition.
• Use the AAS congruence condition.
• Prove the basic properties of isosceles
triangles.
• Explain the notion of a proposition.
• Explain the notion of converse of a
proposition.
• Identify the converse of a given
proposition.
• Explain that not all converses are true;
use counter examples to show that
some converses are false.
• Explain why SSA is not, in general, a
valid congruence condition.

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• Identify the situations where SSA is a
valid congruence condition.
• Justify the role of diagram accuracy.

4-gons  Properties of The student will be able to:
(Quadrilater parallelograms • Frame a precise definition of a 4-gon.
als)  Important theorems CG-4, • Prove various characterisations of a
related to parallelograms C-4.2, parallelogram.
and their proof C-7.3 • Prove the midpoint theorem.
 Midpoint theorem and its • Prove a converse of the midpoint
applications theorem.
 Understanding the notion • Prove that the medians of a triangle
of central symmetry in are concurrent and each median is
the context of divided in the ratio 2:1 at the point of
parallelograms concurrence.
• Prove that the 4-gon formed by joining
the midpoints of a given 4-gon is a
parallelogram.
• Find the coordinates of the midpoint of
a line segment given its end points and
find the coordinates of the fourth vertex
of a parallelogram given the other
three.
• Understand reflection and rotation
symmetries of 4-gons.
• Understand how any 4-gon can tile a
plane.
• Practice forming logical converses of
statements and asking questions
guided by converses of theorems.
• Engage in drawing, measurement and
paper manipulation activities to
discover geometric patterns involving
triangles and 4-gons.

Circles  Practical applications The student will be able to:
and uses of circles • State the definition of a circle.
 Definitions related to CG-4, • Explain the meanings of the terms
a circle — centre, C-7.3, ‘chord’, ‘diameter’, ‘radius’, ‘arc’,
diameter, and radius CG-9 ‘segment’, and ‘sector’.
 Chords and the • Explain why there exists a unique
angles they subtend circle through three non-collinear
 Midpoints and points.
perpendicular • Construct the circumcircle and
bisectors of chords circumcentre of a triangle.
 Distance of chords • Describe the location of the
from the centre circumcentre for acute, obtuse, and
 Subtended angles by right-angled triangles.
an arc • Explain what ‘angle subtended by an
 Cyclicity of points arc at the centre’ means.
• Explain why ‘equal chords subtend
equal angles at the centre’.

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• Explain
aangles at the centre are equal’.
• Explain why ‘the line from the centre of
a circle to the midpoint of a chord is
perpendicular to the chord’.
• Explain why ‘a perpendicular from the
centre to a chord bisects the chord’.
• State the relationship between length
of a chord and its distance from the
centre of the circle.
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• Explain why ‘among unequal chords,

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• Explain why ‘the diameter is the
longest chord’.
• Explain why ‘the angle subtended by
an arc at the centre is double the angle
subtended by the arc at any point on
the remaining part of the circle’.
• Explain why ‘angles in the same
segment of a circle are equal’.
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• Explain why ‘the angle in a semicircle
is a right angle’.
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concyclic.
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supplementary opposite angles is
cyclic, and conversely’.
• Explain how circular wheels have
influenced transport, farming, building,
and technology.
• Identify cultural motifs involving circles,
for example, the Dharmachakra,

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• Use computational thinking to break
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chords, angles, and cyclic

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reasoning.

UNIT V: MENSURATION No. of periods : 27

Mensuration  Perimeter of shapes The student will be able to:
: Area and  Perimeter of a circle: • Define perimeter as the length around
Perimeter Introduction to Pi and CG-5, the boundary of any shape.
its irrationality C-5.1, • Explain that the circumference-to-
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 Area of shapes: • List historical approximations to π
rectangles, (from Archimedes, Aryabhata, and Zu
parallelograms, and Chongzhi).
triangles • Compute the circumference of a circle
 Heron’s formula and the length of an arc.
 Squaring a rectangle: • Apply ideas of circle perimeter and arc-
Proof from length to real-world contexts.
Baudhayana’s • Explain why a median of a triangle
Sulbasutras divides it into two triangles of equal
 Area of a circle: area.
derivation • Use Heron’s formula to compute the
 Area of the sector of area of a triangle from its sides.
a circle • Explain the classical problem of
 Brahmagupta’s ‘squaring’ a given shape.
formula for area of a • Explain how ancient civilisations
cyclic 4-gon approximated the area of a circle.
 Heron’s formula as a • Compute the area of a circle using the
special case of formula.
Brahmagupta’s • Explain and use the formula for area of
formula a sector of a circle.
• Solve problems on areas of sectors
and segments of circles.
• State Brahmagupta’s formula for the
area of a cyclic quadrilateral in terms
of its sides.
• Explain why Heron’s formula is a
‘special case’ of Brahmagupta’s
formula.
• Explain the notion of ‘special case’ and
‘generalisation’ in mathematics.
• Use computational thinking to break
down shapes, apply step-by-step
methods to calculate perimeter and
area, recognise patterns across
formulae, and understand
generalisation and special cases in
geometry.

Mensuration  Surface areas and The student will be able to:
: Surface volumes of spheres • Recognise cuboids and cubes in real-
Area and (including hemispheres) CG-5, life situations.
Volume and right circular cones C-5.1, • Compute the surface area and volume
CG-9 of a cuboid.
• Explain how a cube is a ‘special case’
of a cuboid.
• Describe a right circular cylinder using
its radius and height.
• Compute the surface area and volume
of a cylinder.
• Recognise cones in daily life, and
describe them using radius and height.

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• Compute the surface area and volume
of a cone.
• Recognise a pyramid, and identify its
base and apex.
• Compute the surface area and volume
of a pyramid.
• Recognise spheres in real-life
situations.
• Compute the surface area and volume
of a sphere.
• Use computational thinking to
systematically calculate, and compare
surface areas and volumes of 3-D
shapes by varying dimensions and
analysing patterns.

UNIT VI: STATISTICS AND PROBABILITY No. of periods : 24

Statistics  Graphical representation The student will be able to:
of data • Collect, organise, visualise and
 Measures of central CG-6, interpret data to answer a statistical
tendency C-6.1, investigative question.
CG–9 • Compute and apply weighted average
in different settings.
• Read and interpret stacked bar graphs
and 100% stacked bar graphs.
• Apply computational thinking strategies
to analyse real-life data, create
appropriate graphical representations,
and interpret mean, median and mode
for decision-making.

Introduction  Concept of probability The student will be able to:
to and randomness • Understand the concept of
Probability  The probability scale CG-6, randomness.
 Empirical probability: C-6.2, • Describe the likelihood of an event
analysing statistical data CG-9 using the probability scale.
and performing • Estimate the empirical probability of
experiments the occurrence of an event by
 Theoretical probability: analysing statistical data.
sample space and • Define theoretical probability of an
events event.
 Representing probability • Apply the definition of theoretical
through tree diagrams probability to compute the probability
and tables of an event.
• Compute probability of events with the
help of tree diagrams and tables.
• Use computational thinking strategies,
such as pattern recognition and
simulation, to model random
experiments and estimate probabilities.

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MATHEMATICS QUESTION PAPERla DESIGN
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CLASS – IX (2026-27)

Time: 3 Hrs. Max. Marks: 80

%
S. Total
Typology of Questions Weightage
No. Marks
(approx.)
Remembering: Exhibit memory of previously learned
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material by recalling facts, terms, basic concepts, and
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1 43 54
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answers.

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Understanding: Demonstrate understanding of facts
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and ideas by organizing, comparing, translating,
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interpreting, giving descriptions, and stating main ideas

Applying: Solve problems to new situations by 19 24
2
applying acquired knowledge, facts, techniques and
rules in a different way.
Analysing:
Examine and break information into parts by identifying

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motives or causes. Make inferences and find evidence
to support generalizations
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Evaluating:
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Present and defend opinions by making judgments about
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3 information, validity of ideas, or quality of work based on
a set of criteria.
Creating:
Compile information together in a different way by
combining elements in a new pattern or proposing
alternative solutions

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Total 80 100

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c. oINTERNAL ASSESSMENT s e m 20 MARKS
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Portfolio 05 Marks
Lab Practical (Lab activities to be done from the prescribed books) 05 Marks

Prescribed Books:
1. Mathematics - Textbook for class IX - NCERT Publication
2. Guidelines for Mathematics Laboratory in Schools, class IX - CBSE Publication
3. Laboratory Manual - Mathematics, secondary stage - NCERT Publication
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4. Mathematics exemplar problems for class IX, NCERT publication
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 13 of 13