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Grade-wise curriculum

CBSE Grade 9 Mathematics Curriculum and Practice

Chapter-wise Grade 9 Mathematics practice aligned to the CBSE curriculum. Explore topics, skills, worksheets, and assignments for Number System, Introduction to Polynomials, Sequences and Progressions, Exploring Algebraic Identities, Linear Equations in Two Variables, and Coordinate Geometry, and more.

Aligned with CBSE curriculumSkill-wise practiceWorksheets and assignmentsProgress tracking after sign-in

Recommended starting points

1Number System

CBSE Unit I – Number System. Students build fluency with rational and irrational numbers by representing numbers on the number line, finding rational numbers between two numbers, interpreting decimal expansions, proving irrationality of √2 and √3, and constructing the square root spiral through step-by-step visual reasoning.

2Introduction to Polynomials

CBSE Unit II – Algebra. Students begin algebra through expressions and polynomials, learning terms, coefficients and degree, then using linear polynomials to model linear growth, decay, slope, y-intercept and simple real-life relationships with equations, tables and graphs.

3Sequences and Progressions

CBSE Unit II – Algebra. Students study number patterns as sequences, write explicit and recursive rules, identify arithmetic and geometric progressions, visualise AP and GP patterns, connect sums of natural numbers to structure, and explore applications such as fractals and the Tower of Hanoi.

Chapter-wise curriculum

Chapters, topics, and skills

Check what is covered in this grade and where your child can start practicing.

Chapter 1

Chapter 1: Number System

CBSE Unit I – Number System. Students build fluency with rational and irrational numbers by representing numbers on the number line, finding rational numbers between two numbers, interpreting decimal expansions, proving irrationality of √2 and √3, and constructing the square root spiral through step-by-step visual reasoning.

8 topics23 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Rational Numbers and Number Sets

4 skills
Skill: Define rational numbers using p/q formRecognise numbers that can be written as p/q where p and q are integers and q is not zero.
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Skill: Classify numbers as natural, whole, integer or rationalPlace given numbers into the correct number set and explain the inclusion relationship between sets.
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Skill: Identify equivalent rational numbersGenerate and recognise equivalent forms of the same rational number.
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Skill: Compare rational numbers using common denominatorsUse common denominators or cross multiplication to order rational numbers.
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Topic: Number Line Representation

4 skills
Skill: Represent positive rational numbers on a number lineDivide unit intervals correctly to place positive rational numbers.
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Skill: Represent negative rational numbers on a number lineUse direction and equal partitions to place negative rational numbers.
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Skill: Locate rational numbers between consecutive integersIdentify the correct interval for a given rational number.
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Skill: Estimate the position of decimals on a number lineUse decimal place value to approximate a point on the number line.
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Topic: Density of Rational Numbers

3 skills
Skill: Explain density of rational numbersShow that there is always another rational number between any two rational numbers.
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Skill: Find rational numbers between two given rational numbersUse averages or equivalent fractions to generate rational numbers between two values.
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Skill: Generate multiple rational numbers in a given intervalConstruct several rational numbers within a specified range using systematic methods.
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Topic: Decimal Representation

4 skills
Skill: Convert rational numbers to terminating decimalsUse division and denominator factors to identify and compute terminating decimal forms.
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Skill: Convert rational numbers to recurring decimalsPerform long division to identify repeating decimal patterns.
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Skill: Predict terminating or non-terminating decimal expansionUse prime factors of the denominator in lowest form to classify decimal expansions.
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Skill: Convert recurring decimals to rational numbersUse algebraic steps to express recurring decimals in p/q form.
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Topic: Irrational Numbers

3 skills
Skill: Recognise irrational numbers from examples and non-examplesDistinguish irrational numbers from rational numbers using decimal behaviour and known roots.
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Skill: Compare rational and irrational numbersExplain how rational and irrational numbers differ in representation and properties.
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Skill: Approximate irrational square roots on the number lineUse nearby perfect squares and decimals to estimate irrational values.
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Topic: Irrationality Proofs

2 skills
Skill: Prove irrationality of square root of 2Follow contradiction-based reasoning for the irrationality of √2.
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Skill: Prove irrationality of square root of 3Apply contradiction-based reasoning to show √3 is irrational.
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Topic: Square Root Spiral

1 skill
Skill: Construct the square root spiral step by stepUse right triangles and the Pythagorean theorem to construct lengths √2, √3, √4 and beyond.
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Topic: Computational Thinking with Numbers

2 skills
Skill: Design an algorithm to generate decimal expansionsDescribe a step-by-step procedure for obtaining decimal expansions of rational numbers.
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Skill: Use logical steps to classify real numbersApply a decision process to identify whether a number is rational or irrational.
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Chapter 2

Chapter 2: Introduction to Polynomials

CBSE Unit II – Algebra. Students begin algebra through expressions and polynomials, learning terms, coefficients and degree, then using linear polynomials to model linear growth, decay, slope, y-intercept and simple real-life relationships with equations, tables and graphs.

8 topics20 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Algebraic Expressions

3 skills
Skill: Identify variables, constants and terms in expressionsRecognise the building blocks of algebraic expressions.
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Skill: Distinguish algebraic expressions from equationsClassify mathematical statements based on whether they contain an equality relation.
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Skill: Evaluate algebraic expressions for given valuesFind the value of an expression after substituting numerical values.
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Topic: Polynomial Basics

3 skills
Skill: Define a polynomial in one variableRecognise expressions that satisfy the definition of a polynomial.
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Skill: Distinguish polynomials from non-polynomialsUse powers and variables to identify expressions that are not polynomials.
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Skill: Identify terms and coefficients of a polynomialBreak a polynomial into terms and state their coefficients.
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Topic: Polynomial Degree

3 skills
Skill: Determine the degree of a polynomial in one variableIdentify the highest power with non-zero coefficient.
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Skill: Classify polynomials by degreeClassify polynomials as constant, linear, quadratic, cubic or higher degree.
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Skill: Classify polynomials by number of termsIdentify monomials, binomials and trinomials.
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Topic: Linear Polynomials

3 skills
Skill: Recognise linear polynomials in one variableIdentify polynomials of degree one and their coefficients.
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Skill: Model linear growth using a polynomialWrite a linear expression for situations with a constant increase.
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Skill: Model linear decay using a polynomialWrite a linear expression for situations with a constant decrease.
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Topic: Linear Relationships

2 skills
Skill: Identify linear patterns from tablesUse constant differences to recognise a linear relationship.
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Skill: Write a rule for a linear patternConstruct an algebraic rule from a linear sequence or table.
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Topic: Graphs of Linear Relationships

4 skills
Skill: Plot points from a linear relationshipRepresent ordered pairs from a table on the coordinate plane.
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Skill: Interpret slope as rate of changeExplain slope as change in y for each unit change in x.
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Skill: Identify y-intercept from equation or graphRecognise where a line crosses the y-axis and connect it to y = ax + b.
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Skill: Match linear equations with their graphsUse slope and intercept to connect equations with graphs.
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Topic: Applications of Linear Polynomials

1 skill
Skill: Create linear models from daily-life situationsTranslate simple real-world situations into linear polynomial expressions.
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Topic: Computational Thinking in Algebra

1 skill
Skill: Use step-by-step rules to construct linear expressionsDescribe an algorithm for converting a pattern into an algebraic expression.
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Chapter 3

Chapter 3: Sequences and Progressions

CBSE Unit II – Algebra. Students study number patterns as sequences, write explicit and recursive rules, identify arithmetic and geometric progressions, visualise AP and GP patterns, connect sums of natural numbers to structure, and explore applications such as fractals and the Tower of Hanoi.

9 topics23 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Sequence Patterns

3 skills
Skill: Define a sequence using ordered termsRecognise a sequence as an ordered list following a rule or pattern.
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Skill: Identify the next terms of a sequenceUse observed patterns to extend a sequence.
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Skill: Distinguish finite and infinite sequencesClassify sequences based on whether they stop or continue without end.
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Topic: Rules of Sequences

4 skills
Skill: Find an explicit rule from a simple sequenceExpress the nth term directly in terms of n.
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Skill: Find a recursive rule from a sequenceDescribe how each term is obtained from previous term or terms.
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Skill: Generate sequence terms from an explicit ruleCalculate specific terms when the nth-term rule is given.
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Skill: Generate sequence terms from a recursive ruleStart with initial terms and apply the recurrence step by step.
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Topic: Arithmetic Progressions

5 skills
Skill: Identify arithmetic progressions by common differenceRecognise sequences with a constant difference between consecutive terms.
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Skill: Find the common difference of an arithmetic progressionCompute and interpret the common difference in an AP.
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Skill: Find the nth term of an arithmetic progressionUse the first term and common difference to calculate any term.
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Skill: Find a missing term in an arithmetic progressionUse equal differences to determine unknown AP terms.
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Skill: Visualise arithmetic progressions on a graphRepresent AP terms and interpret their linear pattern.
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Topic: Sum Patterns

2 skills
Skill: Derive the sum of first n natural numbers visuallyUse pairing or geometric reasoning to understand the sum formula.
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Skill: Apply the sum of first n natural numbersSolve counting and arrangement problems using the sum of natural numbers.
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Topic: Geometric Progressions

4 skills
Skill: Identify geometric progressions by common ratioRecognise sequences with a constant multiplicative factor.
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Skill: Find the common ratio of a geometric progressionCompute and interpret the common ratio in a GP.
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Skill: Find the nth term of a geometric progressionUse the first term and common ratio to calculate any term.
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Skill: Compare arithmetic and geometric progressionsDistinguish additive and multiplicative growth patterns.
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Topic: Progressions in Context

2 skills
Skill: Model practical situations using arithmetic progressionsRepresent repeated equal increases or decreases using APs.
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Skill: Model practical situations using geometric progressionsRepresent doubling, halving and percentage-growth situations using GPs.
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Topic: Fractals and Recursive Patterns

1 skill
Skill: Analyse geometric progressions in fractal patternsRecognise repeated scaling and multiplicative patterns in simple fractals.
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Topic: Tower of Hanoi

1 skill
Skill: Model Tower of Hanoi moves as a recursive sequenceUse recursion to predict the number of moves for increasing numbers of disks.
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Topic: Computational Thinking with Sequences

1 skill
Skill: Design an algorithm to generate a sequenceDescribe step-by-step instructions for generating terms from a rule.
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Chapter 4

Chapter 4: Exploring Algebraic Identities

CBSE Unit II – Algebra. Students deepen algebraic manipulation by visualising identities with geometric models, applying identities to expand and factor expressions, using algebra tiles for quadratic expressions, discovering new identities and simplifying rational expressions through systematic steps.

8 topics19 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Identity Basics

2 skills
Skill: Distinguish an identity from an equationRecognise statements true for all allowed values versus statements true for specific values.
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Skill: Verify algebraic identities by substitutionTest an identity with multiple values before proving or using it.
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Topic: Standard Identities

3 skills
Skill: Expand square of a binomialUse standard square identities to expand expressions.
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Skill: Expand product of sum and differenceApply (a + b)(a - b) to simplify products.
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Skill: Expand product of binomials using identitiesSelect a suitable identity to expand binomial expressions efficiently.
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Topic: Geometric Models of Identities

2 skills
Skill: Visualise square identities using area modelsConnect algebraic identities with rectangles and squares.
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Skill: Explain difference of squares using a geometric modelUse area rearrangement to understand the difference of squares.
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Topic: Factorisation Using Identities

3 skills
Skill: Factor expressions using common factorsIdentify and remove common algebraic factors.
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Skill: Factor expressions using difference of squaresRecognise and factor expressions of the form A² - B².
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Skill: Factor perfect square trinomialsRecognise trinomials matching square identities.
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Topic: Quadratic Factorisation

3 skills
Skill: Factor simple quadratic expressions by splitting termsUse product-sum reasoning to factor quadratic expressions.
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Skill: Interpret quadratic factors with algebra tilesRepresent quadratic expressions using algebra tiles and identify factors.
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Skill: Factor quadratic expressions without algebra tilesMove from visual models to symbolic factorisation.
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Topic: Creating Identities

2 skills
Skill: Generate new identities from known identitiesSubstitute expressions into known identities to create related identities.
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Skill: Use identities for mental calculationApply algebraic identities to compute numerical products and squares efficiently.
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Topic: Rational Expressions

3 skills
Skill: Identify restrictions in rational expressionsDetermine values that make denominators zero.
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Skill: Simplify rational expressions using factorisationFactor numerator and denominator and cancel common non-zero factors.
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Skill: Check simplification steps for validityAvoid invalid cancellation and justify each simplification step.
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Topic: Computational Thinking in Identities

1 skill
Skill: Decompose complex algebraic simplification into stepsBreak multi-step simplification into expansion, factorisation and cancellation stages.
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Chapter 5

Chapter 5: Linear Equations in Two Variables

CBSE Unit II – Algebra. Students model practical situations using linear equations in two variables, graph equations, interpret slope-intercept form, solve pairs of linear equations graphically and algebraically, and reason about consistency, inconsistency and contextual conclusions.

10 topics21 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Linear Equations Basics

3 skills
Skill: Define a linear equation in two variablesRecognise equations that can be written in the form ax + by + c = 0.
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Skill: Check whether an ordered pair is a solutionSubstitute x and y values to verify whether they satisfy an equation.
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Skill: Generate solutions of a linear equation in two variablesFind multiple ordered pairs that satisfy a given equation.
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Topic: Tables and Graphs

3 skills
Skill: Create a value table for a linear equationChoose x-values and compute matching y-values systematically.
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Skill: Graph a linear equation using plotted pointsPlot solution pairs and join them to form a straight line.
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Skill: Interpret points on and off a lineConnect graph position with whether an ordered pair satisfies the equation.
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Topic: Slope-Intercept Form

3 skills
Skill: Convert simple equations to slope-intercept formRearrange equations to identify slope and y-intercept.
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Skill: Identify slope and intercept from y = mx + cExtract the rate of change and starting value from a linear equation.
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Skill: Sketch a line using slope and interceptUse intercept and slope movement to draw a line efficiently.
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Topic: Restricted Values

1 skill
Skill: Graph linear equations when variables have limited valuesRepresent real contexts where only whole-number or non-negative solutions make sense.
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Topic: Pair of Linear Equations

4 skills
Skill: Recognise a pair of linear equations in two variablesIdentify two simultaneous linear relationships involving the same variables.
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Skill: Solve a pair of linear equations graphicallyUse the intersection point of two lines as the common solution.
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Skill: Determine consistency from graphsClassify pairs by intersecting, parallel or coincident lines.
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Skill: Identify the number of solutions from line positionsConnect graph positions with one solution, no solution or infinitely many solutions.
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Topic: Substitution Method

1 skill
Skill: Solve a pair of linear equations by substitutionSubstitute one expression into another equation to find both variables.
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Topic: Elimination Method

1 skill
Skill: Solve a pair of linear equations by eliminationAdd or subtract equations after matching coefficients to eliminate a variable.
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Topic: Method Selection

1 skill
Skill: Choose substitution or elimination appropriatelySelect an efficient method based on equation structure.
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Topic: Contextual Modelling

3 skills
Skill: Form linear equations from daily-life word problemsIdentify variables and build equations from real situations.
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Skill: Solve contextual problems using equation pairsUse simultaneous equations to answer practical problems and interpret the result.
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Skill: Check whether a solution is meaningful in contextVerify units, constraints and reasonableness of the solution.
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Topic: Computational Thinking with Linear Equations

1 skill
Skill: Represent one situation using tables graphs and equationsTranslate the same relationship across table, graph and equation formats.
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Chapter 6

Chapter 6: Coordinate Geometry

CBSE Unit III – Coordinate Geometry. Students use the Cartesian plane to specify locations, represent floor plans and real situations on grids, compute distance and midpoint using coordinates, test collinearity, and verify simple geometric properties such as right-angled triangles through coordinate reasoning.

7 topics14 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Cartesian Plane Basics

3 skills
Skill: Identify axes origin and quadrantsRecognise the x-axis, y-axis, origin and four quadrants.
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Skill: Plot points using ordered pairsUse x- and y-coordinates to locate points accurately.
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Skill: Write coordinates of plotted pointsDetermine the ordered pair for a point in the plane.
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Topic: Location Modelling

2 skills
Skill: Represent a floor plan on a coordinate gridUse coordinates to describe positions in a practical layout.
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Skill: Describe relative position using coordinatesUse coordinate differences to describe left, right, up and down movement.
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Topic: Distance Formula

3 skills
Skill: Compute horizontal and vertical distances on a gridCalculate distances when points share the same x-coordinate or y-coordinate.
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Skill: Derive distance between two points using right trianglesConnect coordinate differences with the Pythagorean theorem.
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Skill: Apply distance formula to any two pointsCompute distance in the coordinate plane using the distance formula.
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Topic: Collinearity

1 skill
Skill: Check whether three points are collinear using distancesUse segment lengths to determine whether three points lie on one straight line.
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Topic: Midpoint Formula

2 skills
Skill: Find the midpoint of a line segmentAverage coordinates to locate the midpoint between two endpoints.
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Skill: Find an endpoint when midpoint and one endpoint are knownUse midpoint relationships to determine unknown coordinates.
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Topic: Coordinate Geometry Applications

2 skills
Skill: Check whether a triangle is right angled using distancesUse squared distances to test the Pythagorean relationship.
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Skill: Classify simple figures using coordinate distancesUse side lengths and midpoints to identify geometric figures.
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Topic: Computational Thinking on Grids

1 skill
Skill: Design a step-by-step coordinate verification methodUse an ordered procedure to test geometric properties on a coordinate plane.
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Chapter 7

Chapter 7: Introduction to Euclid’s Geometry: Axioms and Postulates

CBSE Unit IV – Geometry. Students understand how geometry developed from practical needs, learn the role of definitions, axioms and postulates, study Euclid’s ideas and parallelism, and connect Indian mathematical heritage through the Sulbasutra construction of a square.

11 topics12 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: History and Need for Geometry

2 skills
Skill: Explain how practical needs led to geometryConnect measurement, construction and land problems with the rise of geometry.
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Skill: Recognise contributions of ancient civilisations to geometryIdentify Indian, Egyptian and Greek contributions to geometric thinking.
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Topic: Indian Geometry Traditions

1 skill
Skill: Describe the Baudhayana Sulbasutra square constructionUnderstand the idea behind constructing a square with a given side.
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Topic: Basic Geometric Terms

1 skill
Skill: Explain intuitive meanings of point line and surfaceDescribe undefined geometric ideas using intuitive language.
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Topic: Definitions and Assumptions

1 skill
Skill: Distinguish definitions axioms and postulatesClassify mathematical statements by their role in reasoning.
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Topic: Euclidean Postulates

1 skill
Skill: State Euclid’s five postulates in simple languageRecall and explain the basic assumptions used in Euclidean geometry.
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Topic: Axioms of Measurement

1 skill
Skill: Use measurement axioms to compare lengthsApply common notions such as equality and addition of equals.
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Topic: Parallel Lines

1 skill
Skill: Define parallel lines using Euclidean ideasExplain what it means for two lines in a plane not to meet.
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Topic: Logical Reasoning

1 skill
Skill: Justify simple geometric conclusions from axiomsBuild short reasoning chains from accepted assumptions.
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Topic: Construction Reasoning

1 skill
Skill: Justify construction steps using postulatesConnect construction actions to geometric assumptions.
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Topic: Mathematical Vocabulary

1 skill
Skill: Use precise geometric vocabulary in explanationsUse terms such as ray, segment, line, parallel and intersect correctly.
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Topic: Proof Foundations

1 skill
Skill: Identify assumptions used in a geometric argumentRecognise which definitions, axioms or postulates support a conclusion.
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Chapter 8

Chapter 8: Lines and Angles

CBSE Unit IV – Geometry. Students learn rays, angle measures and angle types, reason with intersecting and parallel lines, apply linear pair and vertically opposite angle results, identify corresponding, alternate and interior angles, and build step-by-step geometric proof habits.

8 topics19 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Rays and Angles

2 skills
Skill: Define a ray and its endpointRecognise a ray as part of a line with one starting point.
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Skill: Define an angle using two raysExplain how two rays with a common endpoint form an angle.
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Topic: Angle Measures

3 skills
Skill: Measure angles in degreesUse degrees and protractor reasoning to state angle size.
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Skill: Classify angles by sizeIdentify acute, right, obtuse, straight and reflex angles.
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Skill: Use straight angle and right angle factsApply basic angle measures in calculations.
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Topic: Intersecting Lines

4 skills
Skill: Identify adjacent angles around intersecting linesRecognise angle relationships formed when lines meet.
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Skill: Apply the linear pair theoremFind unknown angles using the fact that a linear pair sums to 180°.
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Skill: Apply the converse of the linear pair theoremDecide when two adjacent angles form a straight line.
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Skill: Prove vertically opposite angles are equalUse linear pairs to justify equality of vertically opposite angles.
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Topic: Proof Reasoning

1 skill
Skill: Follow proof by contradiction in angle problemsRecognise a reasoning method that assumes the opposite and reaches an impossibility.
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Topic: Parallel Lines

5 skills
Skill: Identify corresponding angles formed by a transversalRecognise matching-position angles with parallel or non-parallel lines.
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Skill: Identify alternate interior angles formed by a transversalRecognise alternate angles inside two lines cut by a transversal.
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Skill: Identify interior angles on the same side of a transversalRecognise angle pairs inside the lines on the same side of a transversal.
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Skill: Apply angle relationships for parallel linesCalculate missing angles using corresponding, alternate and interior angle properties.
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Skill: Use angle relationships to prove lines are parallelApply converses of parallel-line angle facts.
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Topic: Parallelism Logic

1 skill
Skill: Explain transitivity of parallelismUse the idea that lines parallel to the same line are parallel to each other.
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Topic: Triangle Angle Reasoning

2 skills
Skill: Explain why a triangle must have at least two acute anglesUse angle-sum logic to analyse possible angle types in a triangle.
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Skill: Explain why a triangle cannot have two obtuse anglesUse angle sums to rule out impossible triangle angle combinations.
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Topic: Computational Thinking with Angles

1 skill
Skill: Solve multi-step angle problems systematicallyBreak angle calculations into ordered steps supported by theorems.
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Chapter 9

Chapter 9: Triangles: Congruence Theorems

CBSE Unit IV – Geometry. Students study triangle rigidity and congruence, match corresponding parts, apply SAS, SSS, ASA, AAS and RHS criteria, prove isosceles triangle properties, examine propositions and converses, and use counterexamples to avoid invalid reasoning such as general SSA.

15 topics19 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Triangle Rigidity

1 skill
Skill: Explain triangle rigidity in structuresConnect triangle rigidity with bridges, roofs and frames.
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Topic: Congruence Basics

2 skills
Skill: Define congruent trianglesExplain congruence as exact matching in shape and size.
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Skill: Identify corresponding parts of congruent trianglesPair vertices, sides and angles correctly in congruent triangles.
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Topic: Congruence Notation

1 skill
Skill: Write congruence statements in correct orderUse vertex order to show correct correspondence.
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Topic: SAS Congruence

1 skill
Skill: Apply SAS congruence criterionProve triangles congruent using two sides and the included angle.
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Topic: SSS Congruence

1 skill
Skill: Apply SSS congruence criterionProve triangles congruent using three pairs of equal sides.
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Topic: ASA Congruence

1 skill
Skill: Apply ASA congruence criterionProve triangles congruent using two angles and included side.
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Topic: AAS Congruence

1 skill
Skill: Apply AAS congruence criterionProve triangles congruent using two angles and a non-included side.
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Topic: RHS Congruence

1 skill
Skill: Apply RHS congruence criterionProve right triangles congruent using hypotenuse and one side.
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Topic: Invalid Congruence Patterns

1 skill
Skill: Explain why SSA is not generally validUse examples to show ambiguity in side-side-angle data.
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Topic: Conditional Congruence

1 skill
Skill: Identify cases where SSA becomes validRecognise special conditions where side-side-angle information is enough.
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Topic: Isosceles Triangle Theorems

2 skills
Skill: Prove base angles of an isosceles triangle are equalUse congruence reasoning to establish equal base angles.
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Skill: Prove converse of the isosceles triangle theoremShow that equal angles in a triangle imply equal opposite sides.
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Topic: Using Congruence

1 skill
Skill: Use CPCTC after proving congruenceDraw conclusions about equal sides or angles after congruence is established.
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Topic: Propositions and Converses

3 skills
Skill: Identify the hypothesis and conclusion of a propositionBreak a statement into condition and result.
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Skill: Write the converse of a geometric statementReverse condition and conclusion and judge the meaning.
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Skill: Use counterexamples to disprove false conversesShow that a statement is false by giving one valid exception.
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Topic: Diagram Reasoning

1 skill
Skill: Judge when a diagram may be misleadingUse given information and theorems rather than visual appearance alone.
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Topic: Problem Solving with Triangles

1 skill
Skill: Solve application problems using triangle congruenceUse congruence criteria to solve geometric and real-world problems.
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Chapter 10

Chapter 10: 4-gons (Quadrilaterals)

CBSE Unit IV – Geometry. Students investigate 4-gons with emphasis on parallelograms, prove key characterisations and midpoint theorem results, use medians and coordinate applications, explore symmetry and tiling, and practise writing logical converses of geometric statements.

10 topics20 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Quadrilateral Basics

2 skills
Skill: Define a 4-gon preciselyDescribe a quadrilateral using vertices, sides and non-crossing boundaries.
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Skill: Classify common quadrilaterals by propertiesDistinguish parallelograms, rectangles, rhombi, squares and trapeziums by properties.
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Topic: Parallelogram Properties

3 skills
Skill: Prove opposite sides of a parallelogram are equalUse triangle congruence or parallel-line facts to prove a parallelogram property.
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Skill: Prove opposite angles of a parallelogram are equalJustify equality of opposite angles using parallel-line angle relationships.
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Skill: Prove diagonals of a parallelogram bisect each otherShow each diagonal cuts the other into two equal parts.
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Topic: Parallelogram Characterisations

2 skills
Skill: Use equal opposite sides to identify a parallelogramApply a converse condition to prove a quadrilateral is a parallelogram.
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Skill: Use bisecting diagonals to identify a parallelogramProve a quadrilateral is a parallelogram from diagonal bisection.
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Topic: Midpoint Theorem

4 skills
Skill: State and explain the midpoint theoremConnect midpoints of two sides of a triangle with a segment parallel to the third side.
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Skill: Prove the midpoint theoremUse congruence or parallelogram reasoning to justify the theorem.
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Skill: Apply the midpoint theorem to find lengthsFind unknown side lengths and parallel relationships in triangles.
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Skill: Prove the converse of the midpoint theoremShow that a line through the midpoint parallel to a side bisects the other side.
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Topic: Medians and Centroid

2 skills
Skill: Prove medians of a triangle are concurrentUnderstand why the three medians meet at one point.
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Skill: Apply the 2:1 median division propertyFind segment lengths when medians are divided in the ratio 2:1.
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Topic: Midpoint Quadrilaterals

1 skill
Skill: Prove midpoint quadrilateral is a parallelogramShow that the quadrilateral formed by side midpoints is always a parallelogram.
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Topic: Coordinate Applications

2 skills
Skill: Find the fourth vertex of a parallelogram using coordinatesUse midpoint or diagonal bisection ideas to determine unknown coordinates.
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Skill: Use midpoint coordinates in quadrilateral problemsApply midpoint formula to verify quadrilateral properties.
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Topic: Symmetry of 4-gons

2 skills
Skill: Recognise reflection symmetry in quadrilateralsIdentify mirror lines in special quadrilaterals.
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Skill: Recognise rotational symmetry and central symmetryDescribe turns that map a quadrilateral onto itself.
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Topic: Tiling and Patterns

1 skill
Skill: Explain how any quadrilateral can tile a planeUse angle and side arrangements to understand quadrilateral tiling.
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Topic: Logical Inquiry

1 skill
Skill: Form converses of quadrilateral theoremsPractice turning a theorem into its converse and testing whether it is true.
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Chapter 11

Chapter 11: Circles

CBSE Unit IV – Geometry. Students explore circles through centre, radius, diameter, chords, arcs, sectors and segments, construct circumcircles, prove chord and angle properties, reason about cyclic quadrilaterals, and connect circular forms to culture, design and real-world technology.

10 topics22 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Circle Basics

1 skill
Skill: Define a circle using centre and radiusDescribe a circle as all points at a fixed distance from a fixed point.
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Topic: Circle Vocabulary

1 skill
Skill: Identify radius diameter chord arc sector and segmentUse circle terms correctly in diagrams and problems.
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Topic: Circle Construction

3 skills
Skill: Explain why three non-collinear points determine a unique circleUnderstand the uniqueness of a circle through three non-collinear points.
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Skill: Construct circumcircle of a triangleUse perpendicular bisectors to construct the circle through all three vertices.
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Skill: Locate circumcentre for different triangle typesDescribe whether the circumcentre lies inside, outside or on a triangle.
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Topic: Chords and Central Angles

3 skills
Skill: Explain angle subtended by an arc at the centreConnect an arc with the angle it forms at the circle centre.
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Skill: Prove equal chords subtend equal angles at the centreJustify the relationship between chord equality and central angles.
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Skill: Prove chords subtending equal central angles are equalUse congruence reasoning to prove the converse result.
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Topic: Chords and Perpendiculars

2 skills
Skill: Prove centre-to-midpoint line is perpendicular to a chordShow why the line from centre to chord midpoint is perpendicular.
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Skill: Prove perpendicular from centre bisects a chordUse circle symmetry or congruence to justify chord bisection.
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Topic: Chord Distance

3 skills
Skill: Relate chord length to distance from centreExplain how chord length changes with distance from the centre.
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Skill: Prove equal chords are equidistant from centreJustify the theorem and its converse.
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Skill: Explain why the diameter is the longest chordUse distance from centre or geometry reasoning to compare chords.
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Topic: Angles in Circles

3 skills
Skill: Apply centre-angle and circumference-angle theoremUse the fact that an arc makes double angle at centre compared to circumference.
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Skill: Use angles in the same segment theoremFind or prove equal angles standing on the same arc.
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Skill: Use angle in a semicircle theoremApply the fact that the angle in a semicircle is a right angle.
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Topic: Cyclic Points and Quadrilaterals

3 skills
Skill: Determine whether four points are concyclicUse circle properties to decide whether four points lie on one circle.
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Skill: Apply opposite angle property of cyclic quadrilateralsUse supplementary opposite angles to solve cyclic quadrilateral problems.
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Skill: Use converse of cyclic quadrilateral angle propertyShow concyclicity using supplementary opposite angles.
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Topic: Circle Applications

2 skills
Skill: Explain the use of circular wheels in technologyConnect circle geometry with transport, farming and machines.
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Skill: Identify circular motifs in Indian cultural contextsRecognise circle ideas in symbols such as Dharmachakra and Ashoka Chakra.
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Topic: Computational Thinking with Circles

1 skill
Skill: Solve circle problems using a theorem-selection strategyBreak a circle problem into diagram reading, theorem selection and calculation steps.
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Chapter 12

Chapter 12: Mensuration: Area and Perimeter

CBSE Unit V – Mensuration. Students calculate perimeter and area of plane figures, understand circumference and π, compute arc length, apply Heron’s formula, derive and use circle area and sector/segment ideas, and connect Brahmagupta’s formula with Heron’s formula as a special case.

13 topics23 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Perimeter Concepts

2 skills
Skill: Define perimeter for closed plane shapesMeasure boundary length of polygons and curved shapes.
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Skill: Compute perimeter of composite polygonal shapesAdd required side lengths while avoiding internal boundaries.
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Topic: Circle Perimeter and Pi

3 skills
Skill: Explain constant circumference-to-diameter ratioConnect the circumference of any circle to its diameter.
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Skill: Use historical approximations of piCompare approximate values from Archimedes, Aryabhata and Zu Chongzhi.
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Skill: Calculate circumference of a circleUse radius or diameter to compute circumference.
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Topic: Arc Length

1 skill
Skill: Calculate length of an arc for a given central angleUse fraction of circumference based on central angle.
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Topic: Area of Basic Shapes

3 skills
Skill: Compute area of rectangles and squaresUse length and breadth to calculate area.
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Skill: Compute area of parallelogramsUse base and height to determine area.
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Skill: Compute area of triangles using base and heightUse half of base times height.
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Topic: Area Reasoning

1 skill
Skill: Explain why a median divides a triangle into equal areasUse same base or equal base and common height reasoning.
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Topic: Heron Formula

2 skills
Skill: Use Heron’s formula to find triangle areaCalculate area from three side lengths using semi-perimeter.
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Skill: Check when Heron’s formula can be appliedVerify triangle possibility before using side-length area formula.
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Topic: Historical Area Ideas

2 skills
Skill: Explain the classical problem of squaring a shapeUnderstand the idea of constructing a square equal in area to a given shape.
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Skill: Describe Baudhayana’s rectangle squaring ideaConnect ancient geometric construction with area preservation.
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Topic: Circle Area

2 skills
Skill: Explain the area formula for a circle using rearrangementUnderstand the area formula through visual reasoning.
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Skill: Calculate area of a circleUse radius to compute circular area.
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Topic: Sectors and Segments

2 skills
Skill: Calculate area of a sectorUse central angle as a fraction of full circle area.
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Skill: Calculate area of a segment for simple casesSubtract triangle area from sector area where appropriate.
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Topic: Brahmagupta Formula

2 skills
Skill: State Brahmagupta’s formula for cyclic quadrilateralsRecognise the formula for area of a cyclic quadrilateral from its sides.
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Skill: Use Brahmagupta’s formula in direct problemsApply the formula when four sides of a cyclic quadrilateral are known.
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Topic: Formula Connections

1 skill
Skill: Explain Heron’s formula as a special case of Brahmagupta’s formulaUnderstand special case and generalisation through area formulae.
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Topic: Real-Life Mensuration

1 skill
Skill: Solve area and perimeter problems from daily contextsApply formulas to fencing, flooring, circular tracks and designs.
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Topic: Computational Thinking in Area

1 skill
Skill: Break composite area problems into simpler shapesDecompose complex figures and combine known areas systematically.
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Chapter 13

Chapter 13: Mensuration: Surface Area and Volume

CBSE Unit V – Mensuration. Students visualise three-dimensional solids and compute surface area and volume for cuboids, cubes, cylinders, cones, pyramids, spheres and hemispheres, while comparing how changes in dimensions affect measurements in real-life contexts.

9 topics24 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: 3-D Shape Recognition

1 skill
Skill: Recognise cuboids and cubes in real objectsIdentify box-like solids in daily-life situations.
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Topic: Cuboids and Cubes

3 skills
Skill: Compute surface area of a cuboidUse length, breadth and height to calculate total surface area.
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Skill: Compute volume of a cuboidCalculate space occupied by a cuboid.
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Skill: Explain cube as a special case of cuboidUnderstand how equal dimensions convert a cuboid into a cube.
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Topic: Cylinders

4 skills
Skill: Describe a right circular cylinder using radius and heightIdentify the circular bases and height of a cylinder.
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Skill: Compute curved surface area of a cylinderUse circumference times height to calculate lateral surface area.
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Skill: Compute total surface area of a cylinderCombine curved area with areas of two circular bases.
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Skill: Compute volume of a cylinderUse base area times height for cylindrical volume.
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Topic: Cones

4 skills
Skill: Recognise cones and identify radius height and slant heightIdentify key measurements of a right circular cone.
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Skill: Compute curved surface area of a coneUse radius and slant height to calculate lateral area.
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Skill: Compute total surface area of a coneAdd base area to curved surface area.
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Skill: Compute volume of a coneUse one-third of corresponding cylinder volume.
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Topic: Pyramids

3 skills
Skill: Recognise pyramids and identify base apex and heightDescribe a pyramid using its base and apex.
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Skill: Compute surface area of simple pyramidsAdd base area and triangular side faces in simple cases.
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Skill: Compute volume of simple pyramidsUse one-third of base area times height.
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Topic: Spheres and Hemispheres

6 skills
Skill: Recognise spheres and hemispheres in real objectsIdentify ball-like solids and half-spheres in daily life.
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Skill: Compute surface area of a sphereUse radius to calculate area covering a sphere.
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Skill: Compute volume of a sphereUse radius to calculate space occupied by a sphere.
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Skill: Compute curved surface area of a hemisphereUse half of the surface area of a sphere.
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Skill: Compute total surface area of a hemisphereInclude both curved surface and circular base.
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Skill: Compute volume of a hemisphereUse half of the volume of a sphere.
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Topic: Dimensional Reasoning

1 skill
Skill: Compare surface area and volume when dimensions changeAnalyse how scaling dimensions affects area and volume.
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Topic: Real-Life Solid Problems

1 skill
Skill: Solve container and packaging problems using solidsApply surface area and volume to storage, packaging and construction contexts.
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Topic: Computational Thinking with Solids

1 skill
Skill: Use a step-by-step plan for 3-D mensuration problemsIdentify the solid, list measurements, choose formula and check units.
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Chapter 14

Chapter 14: Statistics

CBSE Unit VI – Statistics and Probability. Students collect, organise and interpret data, choose suitable graphical representations, read stacked and 100% stacked bar graphs, calculate measures of central tendency, and use data to answer practical investigative questions.

9 topics18 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Statistical Questions

1 skill
Skill: Identify a statistical investigative questionDistinguish questions answered by data from non-statistical questions.
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Topic: Data Collection

1 skill
Skill: Collect data suitable for a statistical questionPlan data values needed to answer an investigative question.
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Topic: Data Organisation

2 skills
Skill: Organise raw data into tablesArrange raw observations into a readable table.
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Skill: Create frequency tables from raw dataCount occurrences and summarise data values or categories.
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Topic: Graphical Representation

3 skills
Skill: Choose an appropriate graph for given dataSelect graph types based on categorical or numerical data.
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Skill: Read and interpret bar graphsExtract comparisons and conclusions from bar graphs.
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Skill: Create bar graphs from data tablesRepresent categorical data using labelled bars and suitable scale.
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Topic: Stacked Bar Graphs

2 skills
Skill: Read stacked bar graphsInterpret part-whole relationships in stacked bars.
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Skill: Read 100 percent stacked bar graphsCompare proportional composition across categories.
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Topic: Measures of Central Tendency

4 skills
Skill: Calculate mean of ungrouped dataCompute average by dividing total by number of observations.
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Skill: Calculate median of ungrouped dataOrder data and identify the middle value or average of two middle values.
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Skill: Calculate mode of ungrouped dataIdentify the most frequently occurring value.
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Skill: Compare mean median and mode for the same dataUnderstand how different measures describe the centre of data.
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Topic: Weighted Average

2 skills
Skill: Compute weighted average in simple contextsUse weights to calculate an average when values do not contribute equally.
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Skill: Interpret weighted average in real-life situationsUse weighted average for marks, ratings and mixtures.
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Topic: Data Interpretation

2 skills
Skill: Draw conclusions from statistical graphsAnswer questions and justify conclusions from visual data.
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Skill: Identify misleading features in data displaysCheck scale, labels and representation before trusting a graph.
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Topic: Computational Thinking with Data

1 skill
Skill: Use a data-analysis workflow from question to conclusionFollow a process of asking, collecting, organising, visualising and interpreting.
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Chapter 15

Chapter 15: Introduction to Probability

CBSE Unit VI – Statistics and Probability. Students understand randomness and likelihood, use the probability scale, estimate empirical probability from experiments and data, compute theoretical probability from sample spaces and events, and represent outcomes using tables and tree diagrams.

4 topics6 skillsWorksheets available
Practice questionsWorksheets availableMore practice coming soonProgress tracking after sign-in

Topic: Randomness

1 skill
Skill: Explain randomness using everyday examplesDescribe situations where outcomes cannot be predicted with certainty.
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Topic: Random Experiments

1 skill
Skill: Identify outcomes of a random experimentRecognise all possible results of simple experiments.
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Topic: Probability Scale

2 skills
Skill: Describe likelihood using impossible unlikely even likely and certainPlace events on an informal probability scale.
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Skill: Represent probabilities from 0 to 1Connect impossible with 0 and certain with 1.
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Topic: Empirical Probability

2 skills
Skill: Estimate probability from repeated trialsUse observed frequency to estimate likelihood.
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Skill: Calculate empirical probability from dataDivide number of favourable observed outcomes by total trials.
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Chapter 16

Chapter 16: Mathematics Lab and Inquiry Skills

Internal Assessment – Mathematics Lab and Inquiry Skills. Students practise hands-on construction, modelling, measurement, visual proof, data handling, mathematical communication and investigation skills aligned to the CBSE lab/portfolio component while staying separate from theory chapter practice.

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Number System: Practice 1

Progressive Grade 9 Math practice for Rational Numbers and Number Sets to Number Line Representation.

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Number System: Practice 2

Progressive Grade 9 Math practice for Number Line Representation to Decimal Representation.

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Number System: Practice 3

Progressive Grade 9 Math practice for Decimal Representation to Square Root Spiral.

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