1. Number System

• Rational Numbers:
  • Meaning of rational numbers, Properties of rational numbers – addition and multiplication – using the general form of expression to describe the properties –closure, commutative, associative, distributive, the existence of identity element and inverse element – consolidation of operations on rational numbers;
  • Representation of rational numbers on the number line – to reinforce the above properties with simple problems;
  • Between any two rational numbers there lies another rational number unlike for whole numbers (Making children see that if we take two rational numbers then we can keep finding more and more rational numbers that lie between them, unlike for two consecutive whole numbers);
  • Verbal problems (higher logic, any two operations, including ideas like area…..)
• Squares, Square roots, Cubes, Cube roots.
  • Meaning of square and square roots; Finding square roots using factor method;
  • Meaning of Cube and Cube root; Finding Cube root by factor method (limiting to 6 digits, whole number);
  • Estimating square roots and cube roots, learning the process of moving nearer to the required number.
• Playing with numbers
  • Writing and understanding a 2, 3 and 4 digit number in generalized form (e.g. 100a + 10b + c, where a, b and c can be digit 0 – 9) and engaging with various puzzles concerning this, (Like finding the missing numerals represented by alphabet in sums involving any of the four operations);
  • Children to create and solve problems and puzzles;
  • Number puzzles, games, magic squares (3x3 and 5x5 only);
  • Deducing the divisibility test rules of 2, 3, 5, 9, 10, and 11 for a 2, 3 or 4 digit number expressed in the general form.
• Commercial arithmetic
  • Slightly advanced problems involving applications on percentages, profit and loss, discount, commission and simple problems on overhead expenses during commercial transactions and tax;
  • Simple interest and advanced problems on simple interest using formula – completed years and fraction of years.
• Statistics
  • Preparation of frequency distribution table;
  • Representation of grouped data through bar graphs – construction and interpretation;
  • Calculation of mean, median, and mode for grouped data.

2. Algebra

• Algebraic Expressions
  • Meaning and types of polynomials;
  • Revision of addition and subtraction of polynomials;
  • Multiplication of Polynomials – monomials by monomials; binomial by monomial (a+b+c) x; Binomial by binomial (x+a) (x+b), (a+b)², (a-b)² and (a+b) (a-b) types (co-efficients should be integers);
• Factorisation
  • Revision of identities
  • (x+a) (x+b)= x²+(a+b)x+ab;
  • (a ± b)² = a² ± 2ab+b² , a²-b² = (a+b) (a-b);
  • Factorisation of the type – a(x+y), (x ± y)², (x+a) (x+b), a² – b²
• Linear equations
  • Linear equation – meaning and general form, Solving linear equations in one variable in contextual problems involving multiplication and division – word problems (Avoid complicated coefficients in the equations)
• Exponents
  • Integers as exponents;
  • Laws of exponents with integral powers
• Introduction to graphs
  • Preliminaries – Axes (same units), Cartesian plane, plotting points for different kinds of situations (perimeter vs length for square, plotting of multiples of different numbers, simple interest vs the number of years, distance vs time, etc);
  • Reading off from the graphs – graphs obtained for the above situations;
  • Plotting a linear graph; reading of linear graphs.

3. Geometry

• Axioms, Postulates, and Theorems
  • Meaning of axioms, postulates, and enunciations, theorems and statements of these;
  • Verification of the statements-
    • Wherever a ray meets a straight line at a point, the sum of the two adjacent angles formed is equal to two right angles;
    • If two lines intersect the vertically opposite angles are equal
    • Lines that are parallel to the same line are parallel to each other.
    • The angles opposite to equal sides of a triangle are equal – converse statement.
  • Theorem 1 – If a transversal cuts two parallel lines then, a) alternate angles are equal b) the interior angles on the same side of the transversal are supplementary;
  • Problems (numerical) and simple riders based on the theorem.
• Theorem on triangles
  • Theorem 2 – Sum of the three angles of a triangle is equal to two right angles; Exterior angles of a triangle – meaning;
  • Theorem 3 – If one side of a triangle is produced, the exterior angle so formed is equal to the sum of the interior opposite angles.
• Congruency of triangles
  • Meaning of congruency – congruency of plane figures, congruency of triangles;
  • Postulates on congruency of triangles – SAS, SSS, ASA, and RHS (Verification by practical method) – problems.
  • Theorem 4: In an isosceles triangle, the angles opposite to equal sides are equal. (Logical proof based on the different postulates of congruency of triangles)
  • The converse of the theorem, problems, and riders based on the theorem.
  • Theorem 5 -Two right-angled triangles are congruent, if the hypotenuse and a side of one triangle are equal to the hypotenuse and a side of the other triangle, correspondingly.
  • Simple riders based on the theorem.
• Construction of triangles
  • Construction of all types of triangles based on angles and sides; -based on all criteria of data – SAS, SSS, ASA, and RHS;
  • Construction of a triangle given the base and sum/difference of the other two sides;
  • Construction of a triangle given perimeter and base angles.
• Quadrilaterals
  • Definition of quadrilaterals – sides and angles (adjacent & opposite), diagonals;
  • Property of quadrilaterals – the sum of angles of a quadrilateral is equal to 360 (by practical method);
  • Types of quadrilaterals – Parallelogram- Rhombus, rectangle, square; Trapezium and isosceles trapezium;
  • Properties of the parallelogram (by practical method)

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, vi follow from the above)

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

v) Diagonals of a rhombus are equal and bisect each other at right angles

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

• Problems and riders based on the above properties.

4. Mensuration

• The surface area of a cube and cuboid; (both LSA and TSA)
• Volume and capacity – Measurement of capacity - the basic unit of volume;
• The volume of a cube and cuboid.