- 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 non-recurring / non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as √2, √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 √x for a given positive real number x (visual proof to be emphasized).
- Definition of nth root of a real number.
- Rationalization (with precise meaning) of real numbers of the type 1/(a+b√x) and 1/(√x+√y) (and their combinations) where x and y are natural number and a and b are integers.
- 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.)
Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax² + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem.
Recall of algebraic expressions and identities. Verification of identities:
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
(x ± y)³ = x³ ± y³ ± 3xy (x ± y)
x³ ± y³ = (x ± y) (x² ± xy + y²)
x³ + y³ + z³ - 3xyz = (x + y + z) (x² + y² + z² - xy - yz - zx) and their use in factorization of polynomials.
Recall of linear equations in one variable. Introduction to the equation in two variables.
Focus on linear equations of the type ax+by+c=0. 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 lie on a line. Graph of linear equations in two variables. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.
The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.
History - Geometry in India and Euclid's geometry. 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, for example:
- (Axiom) Given two distinct points, there exists one and only one line through them.
- (Theorem) (Prove) Two distinct lines cannot have more than one point in common.