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. Recall of laws of exponents with integral powers.
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 and cubic polynomials; monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem.
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 equationin 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. Graph of linear equations in two variables.
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.
If a ray stands on a line, then the sum of the two adjacent angles so formed is 180 degree and the converse. If two lines intersect, vertically opposite angles are equal. Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines. Lines which are parallel to a given line are parallel. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
Two triangles are congruent if any two sides/angles and the included angle/sides of one triangle is equal to any two sides/angles and the included angle/sides of the other triangle. Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle. Two right triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and a side of the other triangle. The angles/sides opposite to equal sides/angles of a triangle are equal. Triangle inequalities and relation between 'angle and facing side' inequalities in triangles.
The diagonal divides a parallelogram into two congruent triangles. In a parallelogram opposite sides are equal, and conversely. In a parallelogram opposite angles are equal, and conversely. A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. In a parallelogram, the diagonals bisect each other and conversely. In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.
Equal chords of a circle subtend equal angles at the center and its converse. The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord. There is one and only one circle passing through three given non-collinear points. Equal chords of a circle are equidistant from the center and conversely. Angles in the same segment of a circle are equal.
Construction of bisectors of line segments and angles of measure 60, 90, 45 etc. degrees, equilateral triangles. Construction of a triangle given its base, sum/difference of the other two sides and one base angle. Construction of a triangle ofgiven perimeter and base angles.
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).