# NCERT Math syllabus: Grade 6 to 8

(ncert.nic.in)

## Number System (60 hrs)

(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 of5 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 ofnumber 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 primefactors. HCF and LCM, prime factorization and division methodf or 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 motivationto 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 subtractionof 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 thisstage), word problems involving addition and subtraction of decimals (two operations together on money, mass, length and temperature)

## Algebra (15 hrs)

INTRODUCTIONTO 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 (15 hrs)

• Concept of Ratio

• Proportion as equality of two ratios

• Unitary method (with only direct variation implied)

• Word problems

## Geometry (65 hrs)

(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, interiorand exterior,

• Triangle — vertices, sides, angles,interior and exterior, altitude andmedian

• Circle — Centre, radius,diameter, arc, sector, chord,segment, semicircle, circumference,interior and exterior.

(ii)Understanding ElementaryShapes (2-D and 3-D):

• Measure of Line segment

• Measure of angles

• Pair of lines

– Intersecting and perpendi-cular lines

– Parallel lines

• Types of angles- acute, obtuse,right, straight, reflex, completeand 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 wellas non regular).

•Identification of 3-D shapes: Cubes,Cuboids, cylinder, sphere, cone, prism (triangular), pyramid(triangular and square)Identification and locating in thesurroundings

• Elements of 3-D figures. (Faces,Edges and vertices)

• Nets for cube, cuboids, cylinders,cones and tetrahedrons.

(iii) Symmetry: (reflection)

• Observation and identificationof 2-D symmetrical objects forreflection symmetry

• Operation of reflection (takingmirror images) of simple 2-Dobjects

• 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 (15 hrs)

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 ofdata - 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 (50 hrs)

(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 counter-examples, including some by children. Counter examples like subtraction is not commutative.

• Word problems including integers (all operations)

(ii) Fractions and rationalnumbers:

• Multiplication of fractions

• Fraction as an operator

• Reciprocal of a fraction

• Division of fractions

• Word problems involving mixedfractions

• 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 naturalnumbers.

• Laws of exponents (through observing patterns to arrive at generalisation.)

(i) a^m.a^n = a^(m+n)

(ii) (a^m)^n = a^(m.n)

(iii) a^m / a^n = a^(m-n) , where m - n ∈ N

## Algebra (20 hrs)

ALGEBRAIC EXPRESSIONS

• Generate algebraic expressions(simple) involving one or twovariables

• Identifying constants, coefficient,powers

• Like and unlike terms, degree ofexpressions e.g., x^2.y etc. (exponent≤ 3, number of variables )

• Addition, subtraction of algebraic expressions (coefficients shouldbe integers).

• Simple linear equations in onevariable (in contextual problems)with two operations (avoidcomplicated coefficients)

## Ratio and Proportion (20 hrs)

• Ratio and proportion (revision)

• Unitary method continued, consolidation, generalexpression.

• Percentage- an introduction.

• Understanding percentage as a fraction with denominator 100

• Converting fractions and decimals into percentage andvice-versa.

• Application to profit and loss(single transaction only)

• Application to simple interest(time period in complete years).

## Geometry (60 hrs)

(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, exteriorangles)

(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. (90 degree, 120 degree, 180 degree)

• Operation of rotation through 90 degree and 180 degree 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 (examples-blades, 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 (15 hrs)

• 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 (15 hrs)

(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 datathrough experiments. Notion of chance in events like tossing coins, dice etc. Tabulating andc ounting 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 (50 hrs)

(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 integralpowers

(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 (20 hrs)

(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 = a^2 ± 2ab + b^2, a^2 – b^2 = (a – b).(a + b)

Factorisation (simple cases only) as examples the following types a(x + y), (x ± y)^2, a^2 – b^2, (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 (25 hrs)

• 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(40 hrs)

(i) Understanding shapes:

• Properties of quadrilaterals – Sum of angles of a quadrilateralis equal to 360 degrees (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 for3-D figures with flat faces(cubes, cuboids, tetrahedrons, prisms and pyramids)

(iii) Construction:

• Given four sides and onediagonal

• Three sides and two diagonals

• Three sides and two included angles

• Two adjacent sides and threeangles

## Mensuration (15 hrs)

(i) Area of a trapezium and apolygon.

(ii)Concept of volume,measurement of volumeusing 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 (15 hrs)

(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 (15 hrs)

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.)