GEOMETRY

SHAPES & SPATIAL UNDERSTANDING
• Develops and uses vocabulary of spatial relationship (Top, Bottom, On, Under, Inside, Outside, Above,br /> Below, Near, Far, Before, After)

SOLIDS AROUND US
• Collects objects from the surroundings having different sizes and shapes like pebbles, boxes, balls, cones, pipes,
etc.
• Sorts, Classifies and describes the objects on the basis of shapes, and other observable properties.
• Observes and describes the way shapes affect movements like rolling and sliding.
• Sorts 2 - D shapes such as flat objects made of card etc.

Numbers

DEVELOPING A SENSE OF NUMBERNESS, COUNTING AND OPERATIONS OF NUMBERS 1 - 9 AND ZERO
• Observes object and makes collections of objects.
• Arranges the collection of objects in order by
– Matching and
– One to one correspondence
• Counts the number of objects in a collection.
• Makes collection of objects corresponding to a specific number.
• Recognises and speaks numbers from 1 to 9.
• Uses numbers from 1 to 9 in counting and comparison. (Real objects and repeated events like clapping to be used for counting)
• Reads and writes numerals from 1 to 9.
• Adds and subtracts using real objects and pictures.
• Adds and subtracts the numbers using symbols ‘+’ and ‘-’.
• Approaches zero through the subtraction pattern (such as 3 – 1 = 2, 3 – 2 = 1, 3 – 3 = 0).

NUMBERS FROM (10 - 20)

• Forms Number sequence from 10 to 20.
• Counts objects using these numbers.
• Groups objects into a group of 10s and single objects.
• Develops the vocabulary of group of ‘tens’ and ‘ones’.
• Shows the group of tens and ones by drawing.
• Counts the number of tens and ones in a given number.
• Writes the numerals for eleven to nineteen.
• Writes numerals for ten and twenty.
• Compares numbers upto 20.

ADDITION AND SUBTRACTION (UPTO 20)
• Adds and subtracts numbers upto 20.

NUMBERS FROM 21 - 99
• Writes numerals for Twenty-one to Ninety nine.- Groups objects into tens and ones.
• Draws representation for groups of ten and ones.
• Groups a number orally into tens and ones.

MENTAL ARITHMETIC
• Adds two single digit numbers mentally.

Money

• Identifies common currency notes and coins.
• Puts together small amounts of money.

Measurement

LENGTH
• Distinguishes between near, far, thin, thick, longer/taller, shorter, high, low.
• Seriates objects by comparing their length.
• Measures short lengths in terms of non-uniform units (in the context of games e.g. ‘Gilli Danda’ and ‘marblegames’).
• Estimates distance and length, and verifies using nonuniform units (e.g. hand span etc.)

WEIGHT
• Compares between heavy and light objects.

Time

• Distinguishes between events occurring in time using terms -earlier and later.
• Gets the qualitative feel of long & short duration, of school days v/s holidays.
• Narrates the sequence of events in a day.

Data Handling

• Collects, represents and interprets simple data such as measuring the arm length or circumference of the head
using a paper strip.

Patterns

• Describes sequences of simple patterns found in shapes in the surroundings and in numbers, e.g. stamping
activity using fingers and thumb.
• Completes a given sequence of simple patterns found in shapes in the surroundings and in numbers.

Geometry

SHAPES & SPATIAL UNDERSTANDING
3-D and 2-D Shapes
• Observes objects in the environment and gets a qualitative feel for their geometrical attributes.
• Identifies the basic 3-D shapes such as cuboid, cylinder, cone, sphere by their names.
• Traces the 2-D outlines of 3-D objects.
• Observes and identifies these 2-D shapes.
• Identifies 2-D shapes viz., rectangle, square, triangle, circle by their names.
• Describes intuitively the properties of these 2-D shapes.
• Identifies and makes straight lines by folding, straight edged objects, stretched strings and draws free hand
and with a ruler.
• Draws horizontal, vertical and slant lines (free hand).
• Distinguishes between straight and curved lines.
• Identifies objects by observing their shadows.

Numbers

• Reads and writes numerals for numbers up to ninetynine.
• Expands a number with respect to place values.
• Counts and regroups objects into tens and ones.
• Uses the concept of place value in the comparison of numbers.
• Counts in various ways:
– Starting from any number.
– Group counting etc.
• Arranges numbers upto hundred in ascending and descending order.
• Forms the greatest and the smallest two digit numbers with and without repetition of given digits.
• Indicates and identifies the position of an object in a line.

ADDITION AND SUBTRACTION
• Adds and subtracts two digit numbers by drawing representations of tens and ones without and with regrouping.
• Adds zero to a number and subtracts zero from a number.
• Observes the commutative property of addition through patterns.
• Solves addition, subtraction problems presented through pictures and verbal description.
• Describes orally the situations that correspond to the given addition and subtraction facts.
• Estimates the result of addition and subtraction and compares the result with another given number.

PREPARATION FOR MULTIPLICATION AND DIVISION
• Discussion of situations involving repeated addition and situations involving equal sharing.
• Activities of making equal groups.

MENTAL ARITHMETIC
• Adds and subtracts single digit numbers mentally.
• Adds and subtracts multiples of ten mentally.

Money

• Identifies currency - notes and coins.
• Puts together amounts of money not exceeding Rs 50/-.
• Adds and subtracts small amounts of money mentally.
• Transacts an amount using 3-4 notes.

Measurement

LENGTH
• Measures lengths & distances along short & long paths using uniform (non-standard) units, extends to longer
lengths.

WEIGHT
• Compares two or more objects by their weight.
• Appreciates the need for a simple balance.
• Compares weights of given objects using simple balance.

CAPACITY (VOLUME)
• Compares and orders containers in terms of internal volume(capacity).
• Orders given containers as per their capacities on the basis of perception & verifies by pouring out etc.

TIME
• Gets familiar with the days of the week and months of the year.
• Gets a feel for sequence of seasons (varying locally).
• Sequences the events occurring over longer periods in terms of dates/days.

Data Handling

• Collects data through measurement.
• Represents the data followed by discussion (e.g. heights of children).
• Collects and presents the data on birthdays.
• Draws inferences from the data at the appropriate level.

Patterns

• Observes and extends patterns in sequence of shapes and numbers.
• Searches for patterns in different ways of splitting a number.
• Creates block patterns by stamping thumbprints, leaf prints, vegetable prints, etc.
• Creates patterns of regular shapes by stamping.

Geometry

SHAPES & SPATIAL UNDERSTANDING
• Creates shapes through paper folding, paper cutting.
• Identifies 2-D shapes
• Describes the various 2-D shapesby counting their sides, corners and diagonals.
• Makes shapes on the dot-grid using straight lines and curves.
• Creates shapes using tangram pieces.
• Matches the properties of two 2-D shapes by observing their sides and corners (vertices).
• Tiles a given region using a tile of a given shape.
• Distinguishes between shapes that tile and that do not tile.
• Intuitive idea of a map. Reads simple maps (not necessarily scaled)
• Draws some 3D-objects.

Numbers

NUMBER SEQUENCE UPTO 1000
• Reads and writes 3-digit numbers.
• Expands a number w.r.t. place values.
• Counts in different ways - starting from any number.
• Compares numbers.
• Forms greatest and smallest numbers using given digits.

ADDITION AND SUBTRACTION
• Adds and subtracts numbers by writing them vertically in the following two cases:
– without regrouping.
– with regrouping.
• Uses the place value in standard algorithm of addition and subtraction.
• Solves addition and subtraction problems in different situations
presented through pictures and stories.
• Frames problems for addition and subtraction facts.
• Estimates the sum of, and difference between, two given numbers.

MULTIPLICATION
• Explains the meaning of multiplication (as repeated addition).
• Identifies the sign of multiplication.
• Constructs the multiplication tables of 2, 3, 4, 5 and 10
• Uses multiplication facts in situations.
• Multiplies two digit numbers using standard algorithm and Lattice multiplication algorithm.

DIVISION
• Explains the meaning of division from context of equal grouping and sharing.
• Relates division with multiplication.
• Completes division facts:
– by grouping
– by using multiplication tables.

MENTAL ARITHMETIC
• Adds and subtracts single digit numbers and two digit numbers mentally.
• Doubles two digit numbers mentally (result not exceeding two digits).

Money

• Converts Rupee. to Paise using play money.
• Adds and subtracts amounts using column addition, and subtraction without regrouping.
• Makes rate charts and bills.

Measurement

LENGTH
• Appreciates the need for a standard unit.
• Measures length using appropriate standard units of length by choosing between centimetres. and metres.
• Estimates the length of given object in standard units and verifies by measuring.
• Uses a ruler
• Relates centimetre. and metre.

WEIGHT
• Weighs objects using non standard Units.
• Appreciates the conservation of weight.

VOLUME
• Measures and compares the capacity of different containers in terms of non-standard units.
• Appreciates the conservation of volume.

TIME
• Reads a calendar to find a particular day and date.
• Reads the time correct to the hour.
• Sequences the events chronologically.

Data Handling

• Records data using tally marks.
• Collects data and represents in terms of pictograph choosing appropriate scale and unit for display through pictographs.
• Draws conclusions from the data by discussing with the teacher.

Patterns

• Identifies simple symmetrical shapes and patterns.
• Makes patterns and designs from straight lines and other geometrical shapes.
• Identifies patterns in the numerals for odd and even numbers and in adding odd and even numbers.
• Partitions a number in different ways.
• Identifies patterns in his surroundings
• Identifies patterns in multiplication with, and dividing by 10s.

Geometry

SHAPES & SPATIAL UNDERSTANDING
• Draws a circle free hand and with compass.
• Identifies centre, radius and diameter of a circle.
• Uses Tangrams to create different shapes.
• Tiles geometrical shapes: using one or two shapes.
• Chooses a tile among a given number of tiles that can tile a given region both intuitively and experimentally.
• Explores intuitively the area and perimeter of simple shapes.
• Makes 4-faced, 5-faced and 6- faced cubes from given nets especially designed for the same.
• Explores intuitively the reflections through inkblots, paper cutting and paper folding.
• Reads and draws 3-D objects, making use of the familiarity with the conventions used in this.
• Draws intuitively the plan, elevation and side view of simple objects.

Numbers

NUMBERS AND OPERATIONS
• Writes multiplication facts.
• Writes tables upto 10 × 10.
• Multiplies two and three digit numbers using lattice algorithm and the standard (column) algorithm.
• Divides a given number by another number in various ways such as:
– by drawing dots.
– by grouping.
– by using multiplication facts.
– by repeated subtraction.
• Applies the four operations to life situations.
• Frames word problems.
• Estimates sums, differences and products of given numbers.

MENTAL ARITHMETIC

• Adds and subtracts multiples of 10 and 100, mentally.
• Completes multiplication facts by adding partial products, mentally (e.g. 7 × 6 = 5 × 6 + 2 × 6).

FRACTIONAL NUMBERS

• Identifies half, one fourth and three- fourths of a whole.
• Identifies the symbols ½,¼,¾ .
• Explains the meaning of ½,¼ and ¾ .
• Appreciates equivalence of 2/4 and 1/2; and of 2/2, 3/3, 4/4 and 1.

Money

MONEY
• Converts Rupees to Paise.
• Adds and subtracts amounts using column addition and subtraction with regrouping.
• Uses operations to find totals, change, multiple costs and unit cost.
• Estimates roughly the totals and total cost.

Measurement

LENGTH
• Relates metre with centimetre;
• Converts metre into centimetres and vice versa.
• Solves problems involving length and distances.
• Estimates length of an object and distance between two given locations.

WEIGHT
• Weighs objects using a balance and standard units.
• Determines sums and differences of weights.
• Estimates the weight of an object and verifies using a balance.

VOLUME
• Measures volumes of given liquid using containers marked with standard units.
• Determines sums and differences of volumes.
• Estimates the volume of a liquid contained in a vessel and verifies by measuring.

TIME
• Computes the number of weeks in a year.
• Correlates the number of days in a year with the number of days in each month.
• Justifies the reason for the need of a leap year.
• Reads clock time to the nearest hours and minutes.
• Expresses time, using the terms, ‘a.m.’ and ‘p.m.’
• Estimates the duration of familiar events.
• Finds approximate time elapsed by (to the nearest hour) forward counting.
• Computes the number of days between two dates.

Data Handling

• Collects data and represents in the form of bar graphs;
• Draws Inferences by discussing with the teacher.

Patterns
• Identifies patterns in multiplication and division: multiples of 9,
• Casts out nines from a given number to check if it is a multiple of nine.
• Multiplies and divides by 10s, 100s.
• Identifies geometrical patterns based on symmetry.

Geometry

SHAPES & SPATIAL UNDERSTANDING
• Creates shapes through paper folding, paper cutting.
• Identifies 2-D shapes
• Describes the various 2-D shapesby counting their sides, corners and diagonals.
• Makes shapes on the dot-grid using straight lines and curves.
• Creates shapes using tangram pieces.
• Matches the properties of two 2-D shapes by observing their sides and corners (vertices).
• Tiles a given region using a tile of a given shape.
• Distinguishes between shapes that tile and that do not tile.
• Intuitive idea of a map. Reads simple maps (not necessarily scaled)
• Draws some 3D-objects.

Numbers

NUMBER SEQUENCE UPTO 1000
• Reads and writes 3-digit numbers.
• Expands a number w.r.t. place values.
• Counts in different ways - starting from any number.
• Compares numbers.
• Forms greatest and smallest numbers using given digits.

ADDITION AND SUBTRACTION
• Adds and subtracts numbers by writing them vertically in the following two cases:
– without regrouping.
– with regrouping.
• Uses the place value in standard algorithm of addition and subtraction.
• Solves addition and subtraction problems in different situations
presented through pictures and stories.
• Frames problems for addition and subtraction facts.
• Estimates the sum of, and difference between, two given numbers.

MULTIPLICATION
• Explains the meaning of multiplication (as repeated addition).
• Identifies the sign of multiplication.
• Constructs the multiplication tables of 2, 3, 4, 5 and 10
• Uses multiplication facts in situations.
• Multiplies two digit numbers using standard algorithm and Lattice multiplication algorithm.

DIVISION
• Explains the meaning of division from context of equal grouping and sharing.
• Relates division with multiplication.
• Completes division facts:
– by grouping
– by using multiplication tables.

MENTAL ARITHMETIC
• Adds and subtracts single digit numbers and two digit numbers mentally.
• Doubles two digit numbers mentally (result not exceeding two digits).

Money

• Converts Rupee. to Paise using play money.
• Adds and subtracts amounts using column addition, and subtraction without regrouping.
• Makes rate charts and bills.

Measurement

LENGTH
• Appreciates the need for a standard unit.
• Measures length using appropriate standard units of length by choosing between centimetres. and metres.
• Estimates the length of given object in standard units and verifies by measuring.
• Uses a ruler
• Relates centimetre. and metre.

WEIGHT
• Weighs objects using non standard Units.
• Appreciates the conservation of weight.

VOLUME
• Measures and compares the capacity of different containers in terms of non-standard units.
• Appreciates the conservation of volume.

TIME
• Reads a calendar to find a particular day and date.
• Reads the time correct to the hour.
• Sequences the events chronologically.

Data Handling

• Records data using tally marks.
• Collects data and represents in terms of pictograph choosing appropriate scale and unit for display through pictographs.
• Draws conclusions from the data by discussing with the teacher.

Patterns

• Identifies simple symmetrical shapes and patterns.
• Makes patterns and designs from straight lines and other geometrical shapes.
• Identifies patterns in the numerals for odd and even numbers and in adding odd and even numbers.
• Partitions a number in different ways.
• Identifies patterns in his surroundings
• Identifies patterns in multiplication with, and dividing by 10s.

Number System

(i) Knowing our Numbers:
Consolidating the sense of numberness up to 5 digits, Size, estimation of numbers, identifying smaller, larger, etc. Place value (recapitulation and extension), connectives: use of symbols =, <, > and use of brackets, word problems on number operations involving large numbers up to a maximum of 5 digits in the answer after all
operations. This would include conversions of units of length& mass (from the larger to the smaller
units), estimation of outcome of number operations. Introduction to a sense of the largeness of, and initial
familiarity with, large numbers up to 8 digits and approximation of large numbers)

(ii) Playing with Numbers:
Simplification of brackets, Multiples and factors, divisibility rule of 2, 3, 4, 5, 6, 8, 9, 10, 11. (All these through observing patterns. Children would be helped in deducing some and then asked to derive some that are a
combination of the basic patterns of divisibility.) Even/odd and prime/composite numbers, Co-prime numbers, prime factorisation, every number can be written as products of prime factors. HCF and LCM, prime factorization and division method for HCF and LCM, the property LCM × HCF = product of two numbers. All this is to be embedded in contexts that bring out the significance and provide motivation to the child for learning these ideas.

(iii) Whole numbers
Natural numbers, whole numbers, properties of numbers (commutative, associative, distributive, additive identity, multiplicative identity), number line. Seeing patterns, identifying and formulating rules to be done by children. (As familiarity with algebra grows, the child can express the generic pattern.)

(iv) Negative Numbers and Integers
How negative numbers arise, models of negative numbers, connection to daily life, ordering of negative numbers, representation of negative numbers on number line. Children to see patterns, identify and formulate rules. What are integers, identification of integers on the number line, operation of addition and subtraction of integers, showing the operations on the number line (addition of negative integer reduces the value of the number) comparison of integers, ordering of integers.
(v) Fractions:
Revision of what a fraction is, Fraction as a part of whole, Representation of fractions (pictorially and on number line), fraction as a division, proper, improper & mixed fractions, equivalent fractions, comparison of fractions, addition and subtraction of fractions (Avoid large and complicated unnecessary tasks). (Moving towards abstraction infractions) Review of the idea of a decimal fraction, place value in the context of decimal fraction, inter conversion of fractions and decimal fractions (avoid recurring decimals at this stage), word problems involving addition and subtraction of decimals (two operations together on money, mass, length and temperature)

Algebra

INTRODUCTION TO ALGEBRA
• Introduction to variable through patterns and through appropriate word problems and generalisations (example 5 × 1 = 5 etc.)
• Generate such patterns with more examples.
• Introduction to unknowns through examples with simple contexts (single operations)

Ratio and Proportion

• Concept of Ratio
• Proportion as equality of two ratios
• Unitary method (with only direct variation implied)
• Word problems

Geometry

(i) Basic geometrical ideas (2 -D):
Introduction to geometry. Its linkage with and reflection in everyday experience.
• Line, line segment, ray.
• Open and closed figures.
• Interior and exterior of closed figures.
• Curvilinear and linear boundaries
• Angle — Vertex, arm, interior and exterior,
• Triangle — vertices, sides, angles, interior and exterior, altitude and median
• Quadrilateral — Sides, vertices, angles, diagonals, adjacent sides and opposite sides (only convex quadrilateral are to be discussed), interior and exterior of a quadrilateral.
• Circle — Centre, radius, diameter, arc, sector, chord, segment, semicircle, circumference, interior and exterior.

(ii) Understanding Elementary
Shapes (2-D and 3-D):
• Measure of Line segment
• Measure of angles
• Pair of lines
– Intersecting and perpendicular lines
– Parallel lines
• Types of angles- acute, obtuse, right, straight, reflex, complete and zero angle
• Classification of triangles (on thebasis of sides, and of angles)
• Types of quadrilaterals –Trapezium, parallelogram, rectangle, square, rhombus.
• Simple polygons (introduction)(Upto octagons regulars as well as non regular).
• Identification of 3-D shapes: Cubes, Cuboids, cylinder, sphere, cone, prism (triangular), pyramid (triangular and square) Identification and locating in the surroundings
• Elements of 3-D figures. (Faces, Edges and vertices)
• Nets for cube, cuboids, cylinders, cones and tetrahedrons.

(iii) Symmetry: (reflection)
• Observation and identification of 2-D symmetrical objects for reflection symmetry
• Operation of reflection (taking mirror images) of simple 2-D objects
• Recognising reflection symmetry (identifying axes)

(iv) Constructions (using Straight edge Scale, protractor, compasses)
• Drawing of a line segment
• Construction of circle
• Perpendicular bisector
• Construction of angles (using protractor)
• Angle 60°, 120° (Using Compasses)
• Angle bisector- making angles of 30°, 45°, 90° etc. (using compasses)
• Angle equal to a given angle (using compass)
• Drawing a line perpendicular to a given line from a point a) on the line b) outside the line.

Mensuration

CONCEPT OF PERIMETER AND INTRODUCTION TO AREA
Introduction and general understanding of perimeter using many shapes. Shapes of different kinds with the same perimeter. Concept of area, Area of a rectangle and a square Counter examples to different misconcepts related to perimeter and area. Perimeter of a rectangle – and its special case – a square. Deducing the formula of the perimeter for a rectangle and then a square through pattern and generalisation. Data handling (10 hrs)

(i) What is data - choosing data to examine a hypothesis?

(ii) Collection and organisation of data - examples of organising it in tally bars and a table.

(iii) Pictograph- Need for scaling in pictographs interpretation & construction.

(iv) Making bar graphs for given data interpreting bar graphs+.

Number System

(i) Knowing our Numbers:

Integers
• Multiplication and division of integers (through patterns). Division by zero is meaningless
• Properties of integers (including identities for addition & multiplication, commutative, associative, distributive) (through patterns). These would include examples from whole numbers as well. Involve expressing commutative and associative properties in a general form. Construction of counterexamples, including some by children. ounter examples like subtraction is not commutative.
• Word problems including integers (all operations)

(ii) Fractions and rational

numbers:
• Multiplication of fractions
• Fraction as an operator
• Reciprocal of a fraction
• Division of fractions
• Word problems involving mixed fractions
• Introduction to rational numbers (with representation on number line)
• Operations on rational numbers (all operations)
• Representation of rational number as a decimal.
• Word problems on rational numbers (all operations)
• Multiplication and division of decimal fractions
• Conversion of units (length & mass)
• Word problems (including all operations)

(iii) Powers:
• Exponents only natural numbers.
• Laws of exponents (through observing patterns to arrive at generalisation.)
(i) am- an =am+n

(ii) (am)n = amn

(iii) am / an = am - n where m− n ∈ Ν

(iv) am - am = abm

Algebra

ALGEBRAIC EXPRESSIONS
• Generate algebraic expressions (simple) involving one or two variables
• Identifying constants, coefficient, powers
• Like and unlike terms, degree of expressions e.g., x2 y etc. (exponent≤ 3, number of variables )
• Addition, subtraction of algebraic expressions (coefficients should be integers).
• Simple linear equations in one variable (in contextual problems) with two operations (avoid complicated coefficients)

Ratio and Proportion

• Ratio and proportion (revision)
• Unitary method continued, consolidation, general expression.
• Percentage- an introduction.
• Understanding percentage as a fraction with denominator 100
• Converting fractions and decimals into percentage and vice-versa.
• Application to profit and loss (single transaction only)
• Application to simple interest (time period in complete years).

Geometry

(i) Understanding shapes:
• Pairs of angles (linear, supplementary, complementary, adjacent, vertically opposite) (verification and simple proof of vertically opposite angles)
• Properties of parallel lines with transversal (alternate, corresponding, interior, exterior angles)

(ii) Properties of triangles:
• Angle sum property (with notions of proof & verification through paper folding, proofs using property of parallel lines, difference between proof and verification.)
• Exterior angle property
• Sum of two sides of a it’sthird side
• Pythagoras Theorem(Verification only)

(iii) Symmetry
• Recalling reflection symmetry
• Idea of rotational symmetry, observations of rotational symmetry of 2-D objects. (900, 1200, 1800)
• Operation of rotation through 900 and 1800 of simple figures.
• Examples of figures with both rotation and reflection symmetry (both operations)
• Examples of figures that have reflection and rotation symmetry and vice-versa

(iv) Representing 3-D in 2-D:
• Drawing 3-D figures in 2-D showing hidden faces.
• Identification and counting of vertices, edges, faces, nets (for cubes cuboids, and cylinders, cones).
• Matching pictures with objects (Identifying names)
• Mapping the space around approximately through visual estimation.

(v) Congruence
• Congruence through superposition (examplesblades, stamps, etc.)
• Extend congruence to simple geometrical shapes e.g. triangles, circles.
• Criteria of congruence (by verification) SSS, SAS, ASA, RHS

(vi) Construction (Using scale, protractor, compass)
• Construction of a line parallel to a given line from a point outside it.(Simple proof as remark with the reasoning of alternate angles)
• Construction of simple triangles. Like given three sides, given a side and two angles on it, given two sides and the angle between them.

Mensuration

• Revision of perimeter, Idea of, Circumference of Circle Area Concept of measurement using a basic unit area of a square, rectangle, triangle, parallelogram and circle, area between two rectangles and two concentric circles.

Data handling

(i) Collection and organisation of data – choosing the data to collect for a hypothesis testing.

(ii) Mean, median and mode of ungrouped data – understanding what they represent.

(iii) Constructing bargraphs

(iv) Feel of probability using data through experiments. Notion of chance in events like tossing coins, dice etc. Tabulating and counting occurrences of 1 through 6 in a number of throws. Comparing the observation with that for a coin.Observing strings of throws, notion of randomness.

Number System
(i) Rational Numbers:
• Properties of rational numbers. (including identities). Using general form of expression to describe properties
• Consolidation of operations on rational numbers.
• Representation of rational numbers on the number line
• Between any two rational numbers there lies another rational number (Making children see that if we take two
rational numbers then unlike for whole numbers, in this case you can keep finding more and more numbers that lie between them.)
• Word problem (higher logic, two operations, including ideas like area)

(ii) Powers
• Integers as exponents.
• Laws of exponents with integral powers

(iii) Squares, Square roots, Cubes, Cube roots.
• Square and Square roots
• Square roots using factor method and division method for numbers containing (a) no more than total 4 digits and (b) no more than 2 decimal places
• Cubes and cubes roots (only factor method for numbers containing at most 3 digits)
• Estimating square roots and cube roots. Learning the process of moving nearer to the required number.

(iv) Playing with numbers
• Writing and understanding a 2 and 3 digit number in generalized form (100a + 10b + c , where a, b, c can be only digit 0-9) and engaging with various puzzles concerning this. (Like finding the missing numerals represented by alphabets in sums involving any of the four operations.) Children to solve and create problems and puzzles.
• Number puzzles and games
• Deducing the divisibility test rules of 2, 3, 5, 9, 10 for a two or three-digit number expressed in the general form.

Algebra
(i) Algebraic Expressions
• Multiplication and division of algebraic exp.(Coefficient should be integers)
• Some common errors (e.g. 2 + x ≠ 2x, 7x + y ≠ 7xy )
• Identities (a ± b)2 = a2 ± 2ab + b2,
a2 – b2 = (a – b) (a + b)
Factorisation (simple cases only)
as examples the following types
a(x + y), (x ± y)2, a2 – b2,
(x + a).(x + b)
• Solving linear equations in one variable in contextual problems involving multiplication and division (word problems) (avoid complex coefficient in the equations)

Ratio and Proportion

• Slightly advanced problems involving applications on percentages, profit & loss, overhead expenses, Discount,
tax.
• Difference between simple and compound interest (compounded yearly up to 3 years or half-yearly up to 3 steps only), Arriving at the formula for compound interest through patterns and using it for simple problems.
• Direct variation – Simple and direct word problems
• Inverse variation – Simple and direct word problems
• Time & work problems– Simple and direct word problems

Geometry

(i) Understanding shapes:
• Properties of quadrilaterals – Sum of angles of a quadrilateral is equal to 3600 (By verification)
• Properties of parallelogram (By verification)

(i) Opposite sides of a parallelogram are equal,

(ii) Opposite angles of a parallelogram are equal,

(iii) Diagonals of a parallelogram bisect each other. [Why (iv), (v) and (vi) follow from (ii)]

(iv) Diagonals of a rectangle are equal and bisect each other.

(v) Diagonals of a rhombus bisect each other at right angles.

(vi) Diagonals of a square are equal and bisect each other at right angles.

(ii) Representing 3-D in 2-D
• Identify and Match pictures with objects [more complicated e.g. nested, joint 2-D and 3-D shapes (not more than 2)].
• Drawing 2-D representation of 3-D objects (Continued and extended)
• Counting vertices, edges & faces & verifying Euler’s relation for
3-D figures with flat faces (cubes, cuboids, tetrahedrons, prisms and pyramids)

(iii) Construction:
Construction of Quadrilaterals:
• Given four sides and one diagonal
• Three sides and two diagonals
• Three sides and two included angles
• Two adjacent sides and three angles

Mensuration

(i) Area of a trapezium and a polygon.

(ii) Concept of volume, measurement of volume using a basic unit, volume of a cube, cuboid and cylinder

(iii) Volume and capacity (measurement of capacity)

(iv) Surface area of a cube, cuboid, cylinder.

Data handling

(i) Reading bar-graphs, ungrouped data, arranging it into groups, representation of grouped data through
bar-graphs, constructing and interpreting bar-graphs.

(ii) Simple Pie charts with reasonable data numbers

(iii) Consolidating and generalising the notion of chance in events like tossing coins, dice etc. Relating it to chance in life events. Visual representation of frequency outcomes of repeated throws of the same kind of coins or dice. Throwing a large number of identical dice/coins together and aggregating the result of the throws to get large
number of individual events. Observing the aggregating numbers over a large number of repeated events. Comparing with the data for a coin. Observing strings of throws, notion of randomness

Introduction to graphs

PRELIMINARIES:
(i) Axes (Same units), Cartesian Plane

(ii) Plotting points for different kind of situations (perimeter vs length for squares, area as a function of side of a square, plotting of multiples of different numbers, simple interest vs number of years etc.)

(iii) Reading off from the graphs
• Reading of linear graphs
• Reading of distance vs timegraph.

Number Systems

Real Numbers
Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating/non-terminating recurring decimals, on the number line through successive magnification. Rational numbers as recurring/terminating decimals.
Examples of nonrecurring/non terminating decimals such as root 2, root 3, root 5 etc. Existence of non-rational numbers (irrational numbers) such as root 2, root 3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line, and conversely, every point on the number line represents a unique real number. Existence of root x for a given positive real number x (visual proof to be emphasized). Definition of nth root of a real number. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws). Rationalisation (with precise meaning) of real numbers of the type (and their combinations)
1 / a + b root x and 1 / root x + root y where x and y are natural numbers and a, b are integers.

Algebra

Polynomials
Definition of a polynomial in one variable, its coefficients, with examples and counter examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros/roots of a polynomial/equation. State and motivate the Remainder Theorem with examples and analogy to integers. Statement and proof of the Factor Theorem. Factorisation of ax2 + bx + c, a ≠ 0 where a, b, c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Further identities of the type:
(x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx, (x ± y )3 = x 3 ± y3 ± 3xy (x ± y ),
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx) and their use in factorization of
polynomials. Simple expressions reducible to these polynomials.

Linear Equations in Two Variables
Recall of linear equations in one variable. Introduction to the equation in two variables. Prove that a linear equation in two variables has infinitely many solutions, and justify their being written as ordered pairs of real numbers, plotting them and showing that they seem to lie on a line. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.

Coordinate Geometry

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate
plane, notations, plotting points in the plane, graph of linear equations as examples; focus on
linear equations of the type ax + by + c = 0 by writing it as y =mx + c and linking with the chapter
on linear equations in two variables.

Geometry

1. Introduction to Euclid’s Geometry
History – Euclid and geometry in India. Euclid’s method of formalizing observed phenomenon into rigorous mathematics with definitions, common/obvious notions, axioms/postulates, and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem.
1. Given two distinct points, there exists one and only one line through them.
2. (Prove) Two distinct lines cannot have more than one point in common.

2. Lines and Angles
1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse.
2. (Prove) If two lines intersect, the vertically opposite angles are equal.
3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines.
4. (Motivate) Lines, which are parallel to a given line, are parallel.
5. (Prove) The sum of the angles of a triangle is 180°.
6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

3. Triangles
1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence).
2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence).
3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).
4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle.
5. (Prove) The angles opposite to equal sides of a triangle are equal.
6. (Motivate) The sides opposite to equal angles of a triangle are equal.
7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’; inequalities in a triangle.

4. Quadrilaterals
1. (Prove) The diagonal divides a parallelogram into two congruent triangles.
2. (Motivate) In a parallelogram opposite sides are equal and conversely.
3. (Motivate) In a parallelogram opposite angles are equal and conversely.
4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely.
6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and (motivate) its converse.

5. Area
Review concept of area, recall area of a rectangle.
1. (Prove) Parallelograms on the same base and between the same parallels have the same area.
2. (Motivate) Triangles on the same base and between the same parallels are equal in area and its converse.

6. Circles (Periods 15)
Through examples, arrive at definitions of circle related concepts, radius, circumference, diameter, chord, arc, subtended angle.
1. (Prove) Equal chords of a circle subtend equal angles at the centre and (motivate) its converse.
2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
3. (Motivate) There is one and only one circle passing through three given non-collinear points.
4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre(s) and conversely.
5. (Prove) The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
6. (Motivate) Angles in the same segment of a circle are equal.
7. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle.
8. (Motivate) The sum of the either pair of the opposite angles of a cyclic quadrilateral is 1800 and its converse.

7. Constructions (Periods 10)
1. Construction of bisectors of a line segment and angle, 60°, 90°, 45° angles etc, equilateral triangles.
2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle.
3. Construction of a triangle of given perimeter and base angles.

Mensuration

1. Areas (Periods 4)
Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral.

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

Statistics and Probability

1. Statistics
Introduction to Statistics: Collection of data, presentation of data – tabular form, ungrouped/ grouped, bar graphs, histograms (with varying base lengths), frequency polygons, qualitative analysis of data to choose the correct form of presentation for the collected data. Mean, median, mode of ungrouped data.

2. Probability
History, Repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real-life situations, and from examples used in the chapter on statistics).

Appendix

1. Proofs in Mathematics
What a statement is; when is a statement mathematically valid. Explanation of axiom/ postulate through familiar examples. Difference between axiom, conjecture and theorem. The concept and nature of a ‘proof ’ (emphasize deductive nature of the proof, the assumptions, the hypothesis, the logical argument) and writing a proof. Illustrate deductive proof with complete arguments using simple results from arithmetic, algebra and geometry (e.g., product of two odd numbers is odd etc.). Particular stress on verification not being proof. Illustrate with a few examples of verifications leading to wrong conclusions –
include statements like “every odd number greater than 1 is a prime number”. What disproving means, use of counter examples.

2. Introduction to Mathematical Modelling
The concept of mathematical modelling, review of work done in earlier classes while looking at situational problems, aims of mathematical modelling, discussing the broad stages of modelling – real-life situations, setting up of hypothesis, determining an appropriate model, solving the mathematical problem equivalent, analyzing the conclusions and their real-life interpretation, validating the model. Examples to be drawn from ratio, proportion,
percentages, etc.

Number Systems

Real Numbers
Euclid’s division lemma, Fundamental Theorem of Arithmetic – statements after reviewing work done earlier and after illustrating and motivating through examples. Proofs of results – irrationality of root 2, root 3, root 5 , decimal expansions of rational numbers in terms of terminating/non-terminating recurring decimals.

Algebra

1. Polynomials
Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.

2. Pair of Linear Equations in Two Variables
Pair of linear equations in two variables. Geometric representation of different possibilities of solutions/inconsistency.
Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication. Simple situational problems must be included. Simple problems on equations reducible to linear equations may be included.

3. Quadratic Equations
Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solution of quadratic equations (only real roots) by factorization and by completing the square, i.e., by using quadratic formula. Relationship between discriminant and nature of roots. Problems related to day-to-day activities to be incorporated.

4. Arithmetic Progressions (AP)
Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.

Trigonometry

1. Introduction to Trigonometry
Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° and 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° and 60°. Relationships between the ratios.
Trigonometric Identities: Proof and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.

2. Heights and Distances
Simple and believable problems on heights and distances. Problems should not involve more
than two right triangles. Angles of elevation/depression should be only 300, 450, 600.

Coordinate Geometry

Lines (In two-dimensions)
Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of a triangle.

Geometry

1. Triangles
Definitions, examples, counterexamples of similar triangles.
1. (Prove) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
2. (Motivate) If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
3. (Motivate) If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
4. (Motivate) If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
5. (Motivate) If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
6. (Motivate) If a perpendicular is drawn from the vertex of the right angle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
7. (Prove) The ratio of the areas of two similar triangles is equal to the ratio of the squares on their corresponding sides.
8. (Prove) In a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
9. (Prove) In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angles opposite to the first side is a right triangle.

2. Circles
Tangents to a circle motivated by chords drawn from points coming closer and closer to the point.
1. (Prove) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
2. (Prove) The lengths of tangents drawn from an external point to a circle are equal.

3. Constructions (Periods 8)
1. Division of a line segment in a given ratio (internally).
2. Tangent to a circle from a point outside it.
3. Construction of a triangle similar to a given triangle.

Mensuration

1. Areas Related to Circles
Motivate the area of a circle; area of sectors and segments of a circle. Problems based on areas and perimeter/circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° and 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken.)

2. Surface Areas and Volumes
1. Problems on finding surface areas and volumes of combinations of any two of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of a cone.

2. Problems involving converting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken.)

Statistics and Probability

1. Statistics
Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.

2. Probability
Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation.

Appendix

1. Proofs in Mathematics
Further discussion on concept of ‘statement’, ‘proof ’ and ‘argument’. Further illustrations of deductive proof with complete arguments using simple results from arithmetic, algebra and geometry. Simple theorems of the “Given ……… and assuming… prove ……..”. Training of using only the given facts (irrespective of their truths) to arrive at the required conclusion. Explanation of ‘converse’, ‘negation’, constructing converses and negations of given results/statements.

2. Mathematical Modelling
Reinforcing the concept of mathematical modelling, using simple examples of models where some constraints are ignored. Estimating probability of occurrence of certain events and estimating averages may be considered. Modelling fair instalments payments, using only simple interest and future value (use of AP).

Sets, Relation and function, Mathematical induction, Logarithms, Complex number, Linear inequations, Differentiation, Sequence and series (A.P. & G.P,H.P, Misc.), Trigonometric functions, Cartesian system of rectangular coordinates, Straight line and family of straight lines, Circle, Conic section, Trigonometry, Permutation and combinations, Binomial theorem, Statistics, Mathematical logic, Limits.

SETS AND FUNCTIONS

1. Sets :

Sets and their representations. Empty set. Finite & Infinite sets. Equal sets.Subsets. Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set.

2. Relations & Functions:

Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the reals with itself (upto R x R x R). Definition of relation, pictorial diagrams, domain. codomain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain & range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs. Sum, difference, product and quotients of functions.

3. Trigonometric Functions:

Positive and negative angles. Measuring angles in radians & in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2x + cos2x=1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x+y) and cos (x+y) in terms of sinx, siny, cosx & cosy. Deducing the identities like the following: Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sin= sin , cos= cos and tan= tan .

ALGEBRA

1. Principle of Mathematical Induction:

Processes of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.

2. Complex Numbers and Quadratic Equations:

Need for complex numbers, especially , to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.

3. Linear Inequalities:

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables- graphically.

4. Permutations & Combinations:

Fundamental principle of counting. Factorial n. (n!)Permutations and combinations, derivation of formulae and their connections, simple applications.

5. Binomial Theorem:

History, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications.

6. Sequence and Series:

Sequence and Series. Arithmetic progression (A. P.). arithmetic mean (A.M.) Geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., geometric mean (G.M.), relation between A.M. and G.M. Sum to n terms of the special series n, n2 and n3.

COORDINATE GEOMETRY

1. Straight Lines:

Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axes, point-slope form, slope-intercept form, twopoint form, intercepts form and normal form. General equation of a line. Distance of a point from a line.

2. Conic Sections:

Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three -dimensional Geometry :

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula.

CALCULUS

1. Limits and Derivatives:

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

MATHEMATICAL REASONING

1. Mathematical Reasoning:

Mathematically acceptable statements. Connecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting wordsdifference between contradiction, converse and contrapositive.

STATISTICS & PROBABILITY

1. Statistics: Measure of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

2. Probability: Random experiments: outcomes, sample spaces (set representation). Events: occurrence of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of 'not', 'and' & 'or' events.