| Unit 1: Number System No. of periods : 12 |
| Number System | - Introduction to rational numbers
- Representation of rational numbers on the number line
- Density of rational numbers and its proof
- Finding rational numbers between any two rational numbers
- Decimal representation of rational numbers
- Introduction to irrational numbers
- Proof of irrationality of √2 and √3
- The square root spiral
| CG-1,
C-1.1,
CG-9 | The student will be able to: - Understand the concept of a rational number.
- Represent rational numbers on the number line.
- Understand the properties of rational numbers.
- Explain the concept of density of rational numbers.
- Compute decimal representation of rational numbers.
- Understand the concept of irrational numbers.
- Prove the irrationality.
- Construct the square root spiral.
- Apply computational thinking to represent rational and irrational numbers. numbers through algorithms and visual models, generate decimal expansions systematically, and reason about numbers using step-by-step logical procedures.
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| UNIT II: ALGEBRA No. of periods : 66 |
| Introduction to Polynomials | - Algebraic expressions
- Definition of a polynomial
- Degree of a polynomial
- Introduction to linear polynomials and applications
- Exploring linear patterns
- Modelling linear growth and linear decay
- Linear relationships
- Visualising linear relationships
- Slope and y-intercept of a line y = ax + b
| CG-3,
C-3.2,
CG-9 | The student will be able to: - Understand the meaning of an algebraic expression.
- Define a polynomial.
- Identify the degree, terms and coefficients of terms in a polynomial.
- Model linear growth and decay using linear polynomials.
- Explain and identify patterns in linear relationships.
- Identify the slope and y-intercept of a linear equation in two variables.
- Graph a linear equation in two variables.
- Use computational thinking to identify patterns, construct linear expressions, and systematically represent and analyse linear relationships using equations and graphs.
|
| Sequences and Progressions | - Introduction to sequences
- Explicit or general rule of a sequence
- Recursive rule of a sequence
- Arithmetic Progressions (AP): nth term, visualising an AP, and practical contexts leading to APs
- Sum of the first n natural numbers
- Geometric Progressions (GP): nth term, visualising a GP, and practical contexts leading to GPs
- Applications of GP in fractals
- Tower of Hanoi puzzle
| CG-11,
C-8.1,
CG-9 | The student will be able to: - Understand the concept of a sequence of numbers.
- Identify the pattern in a sequence and predict the next few terms.
- Determine the recursive and explicit rules for different sequences.
- Obtain the terms of sequence given its recursive and explicit rule.
- Identify Arithmetic Progressions (AP).
- Determine the nth term of an AP.
- Visualise an AP graphically.
- Identify Geometric Progressions (GP).
- Determine the nth term of a GP.
- Visualise a GP graphically.
- Analyse attributes of fractals using GP.
- Solve the Tower of Hanoi puzzle.
- Use computational thinking to identify patterns, write step-by-step rules, and model patterns in sequences and progressions.
|
| Exploring Algebraic Identities | - Revisiting algebraic identities
- Visualising identities using geometrical models
- Factorisation of algebraic expressions using identities
- More identities and their applications
- Visualising factorisation of quadratic expressions through algebra tiles and without using algebra tiles
- Finding new identities
- Simplifying rational expressions
| CG-7,
C-7.2,
CG-9 | The student will be able to: - Visualise algebraic identities using geometric models.
- Determine the factors of algebraic expressions using identities.
- Interpret factors of quadratic expressions through geometric models.
- Find simplified versions of rational expressions.
- Use computational thinking strategies, such as decomposition and step-by-step procedures to visualise algebraic identities, factor expressions, and simplify rational expressions.
|
| Linear Equations in Two Variables | - Introduction to linear equations in two variables through practical examples
- Solution of linear equation in two variables: graphical representation
- Slope-intercept form of linear equation in two variables
- Drawing graphs of linear equations when x and y assume only certain values
- Pair of linear equations in two variables
- Graphical method for solving a pair of linear equations in two variables
- Nature of solutions: consistency and inconsistency
- Algebraic methods of solving a pair of linear equations: substitution and elimination method
| CG-3,
C-3.2,
C-8.1,
CG-9 | The student will be able to: - Understand the concept of a linear equation in two variables.
- Graph a pair of linear equations.
- Solve a pair of linear equations graphically.
- Solve a pair of linear equations through the methods of substitution and elimination.
- Determine the nature of solutions of a pair of linear equations.
- Model and solve contextualised problems using a pair of linear equations and draw conclusions.
- Model daily-life phenomena using representations, such as graphs, tables, and equations.
- Use computational thinking to systematically represent, solve, and interpret pairs of linear equations through graphs, tables, and step-by-step procedures.
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| UNIT III: COORDINATE GEOMETRY No. of periods : 6 |
| Coordinate Geometry | - Brief history of coordinate geometry
- The 2-D Cartesian coordinate system
- Distance between two points in the 2-D plane
- Midpoint of the line-segment between two points in the 2-D plane
| CG-4,
C-4.5,
CG-9 | The student will be able to: - Specify locations and the position of one point relative to another point using coordinates.
- Represent a floor plan on a grid using coordinates.
- Compute the distance between two points using coordinates.
- Determine whether three points lie in a straight line using coordinates.
- Compute the position of the midpoint of a line segment using coordinates.
- Check whether a triangle is right-angled using coordinates.
- Apply computational thinking to model situations on the coordinate plane and verify geometric properties through systematic reasoning.
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| UNIT IV: GEOMETRY No. of periods : 69 |
| Introduction to Euclid’s Geometry: Axioms and Postulates | - History of geometry
- Constructing a square with a given side as described in the Baudhayana’s Sulbasutras
- Discovering Euclid’s Axioms
- Axioms: Axioms of measurement and rules for geometric objects
| CG-7,
C-7.1,
C-7.3 | The student will be able to: - Describe how geometry grew from the practical needs of ancient civilisations.
- Describe contributions of India, Egypt and Greece to the development of geometric ideas.
- Understand the role of definitions, axioms, and postulates.
- Explain that there are elements of plane geometry (point, line, surface) for which we have an intuitive sense.
- State the 5 postulates of Euclidean geometry.
- Define parallelism of straight lines.
- Explain the construction of a square as given in the Sulbasutras.
- Justify simple constructions using the axioms.
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| Lines and Angles | - Rays and angles
- Measures of angles
- Intersecting lines and angles
- Pairs of angles
- Theorems and examples on intersecting lines
- Theorems and examples on parallel lines
| CG-7,
C-7.1,
C-7.3,
CG-9 | The student will be able to: - Explain the notion of an angle.
- Explain the notion of a ray.
- Explain that angles are formed between two rays with a common starting point.
- State that a straight angle equals two right angles and measures 180° while a right angle measures 90°.
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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.
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| Triangles: Congruence Theorems | - Practical applications of triangles
- Proofs of conditions of congruence of triangles
- Theorems on triangles
- Propositions and their converse
- Problems based on applications of theorems on triangles
| CG-4,
C-4.1,
C-7.3 | The student will be able to: - Explain that a triangle is rigid, unlike a quadrilateral.
- Identify uses of triangle rigidity.
- Explain why triangles give strength and stability to structures.
- Describe what it means for two triangles to be congruent.
- Identify correspondence between the 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.
- Identify the situations where SSA is a valid congruence condition.
- Justify the role of diagram accuracy.
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| 4-gons (Quadrilaterals) | - Properties of parallelograms
- Important theorems related to parallelograms and their proof
- Midpoint theorem and its applications
- Understanding the notion of central symmetry in the context of parallelograms
| CG-4,
C-4.2,
C-7.3 | The student will be able to: - Frame a precise definition of a 4-gon.
- Prove various characterisations of a parallelogram.
- Prove the midpoint theorem.
- Prove a converse of the midpoint theorem.
- Prove that the medians of a triangle are concurrent and each median is divided in the ratio 2:1 at the point of 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 and uses of circles
- Definitions related to a circle – centre, diameter, and radius
- Chords and the angles they subtend
- Midpoints and perpendicular bisectors of chords
- Distance of chords from the centre
- Subtended angles by an arc
- Cyclicity of points
| CG-4,
C-7.3,
CG-9 | The student will be able to: - State the definition of a circle.
- Explain the meanings of the terms ‘chord’, ‘diameter’, ‘radius’, ‘arc’, ‘segment’, and ‘sector’.
- Explain why there exists a unique circle through three non-collinear points.
- Construct the circumcircle and circumcentre of a triangle.
- Describe the location of the circumcentre for acute, obtuse, and right-angled triangles.
- Explain what ‘angle subtended by an arc at the centre’ means.
- Explain why ‘equal chords subtend equal angles at the centre’.
|
| | | - Explain why ‘chords that subtend equal angles 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.
- Explain why ‘equal chords are equidistant from the centre (and conversely)’.
- Explain why ‘among unequal chords, the longer chord is closer to the centre’.
- 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’.
- Explain why ‘the angle in a semicircle is a right angle’.
- Determine when four given points are concyclic.
- Explain why ‘a quadrilateral with 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, Ashoka Chakra, Sudarshan Chakra.
- Use computational thinking to break down circle-related problems, apply geometric rules step-by-step, and verify properties of figures such as chords, angles, and cyclic quadrilaterals through systematic reasoning.
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| UNIT V: MENSURATION No. of periods : 27 |
| Mensuration: Area and Perimeter | - Perimeter of shapes
- Perimeter of a circle: Introduction to Pi and its irrationality
- Length of an arc
| CG-5,
C-5.1,
CG-9 | The student will be able to: - Define perimeter as the length around the boundary of any shape.
- Explain that the circumference-to-diameter ratio is constant for all circles.
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| - Area of shapes: rectangles, parallelograms, and triangles
- Heron’s formula
- Squaring a rectangle: Proof from Baudhayana’s Sulbasutras
- Area of a circle: derivation
- Area of the sector of a circle
- Brahmagupta’s formula for area of a cyclic 4-gon
- Heron’s formula as a special case of Brahmagupta’s formula
| | - List historical approximations to π (from Archimedes, Aryabhata, and Zu Chongzhi).
- Compute the circumference of a circle and the length of an arc.
- Apply ideas of circle perimeter and arc-length to real-world contexts.
- Explain why a median of a triangle divides it into two triangles of equal area.
- Use Heron’s formula to compute the area of a triangle from its sides.
- Explain the classical problem of ‘squaring’ a given shape.
- Explain how ancient civilisations approximated the area of a circle.
- Compute the area of a circle using the formula.
- Explain and use the formula for area of 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.
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| Mensuration: Surface Area and Volume | - Surface areas and volumes of spheres (including hemispheres) and right circular cones
| CG-5,
C-5.1,
CG-9 | The student will be able to: - Recognise cuboids and cubes in real-life situations.
- Compute the surface area and volume 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.
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| UNIT VI: STATISTICS AND PROBABILITY No. of periods : 24 |
| Statistics | - Graphical representation of data
- Measures of central tendency
| CG-6,
C-6.1,
CG-9 | The student will be able to: - Collect, organise, visualise and interpret data to answer a statistical investigative question.
- 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.
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| Introduction to Probability | - Concept of probability and randomness
- The probability scale
- Empirical probability: analysing statistical data and performing experiments
- Theoretical probability: sample space and events
- Representing probability through tree diagrams and tables
| CG-6,
C-6.2,
CG-9 | The student will be able to: - Understand the concept of randomness.
- Describe the likelihood of an event using the probability scale.
- Estimate the empirical probability of the occurrence of an event by analysing statistical data.
- Define theoretical probability of an event.
- Apply the definition of theoretical probability to compute the probability of an event.
- Compute probability of events with the help of tree diagrams and tables.
- Use computational thinking strategies, such as pattern recognition, simulation, to model random experiments and estimate probabilities.
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