## Topics with syllabus and resources

1. 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. Operations on real numbers.

2. Examples of non-recurring/non-terminating decimals. Existence of nonrational 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, viz. every point on the number line represents a unique real number.

3. Definition of nth root of a real number.

4. Existence of x for a given positive real number x and its representation on the number line with geometric proof.

5. Rationalization (with precise meaning) of real numbers of the type

`1/(a+bsqrtx) " and " 1/(sqrt(x)+sqrt(y))`

(and their combinations) where x and y are natural number and a and b are integers.

6. 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.)

- Concepts :
- Introduction of Real Number question (58)
- Irrational Numbers question (55)
- Real Numbers and Their Decimal Expansions question (14)
- Representing Real Numbers on the Number Line question (15)
- Operations on Real Numbers question (82)
- Laws of Exponents for Real Numbers question (174)

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 ax^{2} + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem.

x^{3}+y^{3}+z^{3}-3xyz=(x+y+z) (x^{2}+y^{2}+z^{2}-xy-yz-zx) and their use in factorization of polynomials.

- Concepts :
- Remainder Theorem videos (5) question (42)
- Introduction of Polynomials question (21)
- Polynomials in One Variable question (10)
- Zeroes of a Polynomial question (28)
- Factorisation of Polynomials question (138)

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.

- Concepts :
- Introduction of Linear Equations question (9)
- Solution of a Linear Equation question (44)
- Graph of a Linear Equation in Two Variables question (30)
- Equations of Lines Parallel to the X-axis and Y-axis question (22)
- Concept of Linear Equations in Two Variables question (3)

Recall of algebraic expressions

- Concepts :
- Concept of Algebraic Expressions question (115)

Recall of algebraic identities. Verification of identities:-

(x+y+z)^{2} = x^{2}+y^{2}+z^{2}+2xy+2yz+2zx

(x±y)^{3} = x^{3}±y^{3}±3xy (x±y)

x^{3}±y^{3} = (x±y) (x^{2}±xy+y^{2})

- Concepts :
- Algebraic Identities question (198)

The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane.

- Concepts :
- Introduction of Coordinate Geometry question (20)
- Cartesian System question (2)
- Plotting a Point in the Plane If Its Coordinates Are Given question (13)

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) 1. Given two distinct points, there exists one and only one line through them.

(Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common.

- Concepts :
- Concept for Euclid’S Geometry question (26)
- Euclid’S Definitions, Axioms and Postulates question (23)
- Equivalent Versions of Euclid’S Fifth Postulate question (2)

1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180O and the converse.

2. (Prove) If two lines intersect, 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 180O.

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.

- Concepts :
- Concept to Lines and Angles question (30)
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines question (22)
- Pairs of Angles question (61)
- Parallel Lines and a Transversal question (29)
- Lines Parallel to the Same Line question (14)
- Angle Sum Property of a Triangle question (6)

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. (RHS Congruence)

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 triangles.

- Concepts :
- Concept of Triangles question (69)
- Congruence of Triangles question (27)
- Criteria for Congruence of Triangles question (27)
- Properties of a Triangle question (99)
- Some More Criteria for Congruence of Triangles question (7)
- Inequalities in a Triangle question (10)

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 in half of it and (motivate) its converse.

- Concepts :
- Angle Sum Property of a Quadrilateral question (25)
- Types of Quadrilaterals question (1)
- Properties of a Parallelogram question (19)
- Another Condition for a Quadrilateral to Be a Parallelogram question (41)
- The Mid-point Theorem question (31)
- Concept of Quadrilaterals question (17)

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 (or equal base) base and between the same parallels are equal in area.

- Concepts :
- Introduction to Areas of Parallelograms and Triangles question (50)
- Figures on the Same Base and Between the Same Parallels question (7)
- Parallelograms on the Same Base and Between the Same Parallels question (12)
- Triangles on the Same Base and Between the Same Parallels question (25)

Through examples, arrive at definition of circle and related concepts-radius, circumference, diameter, chord, arc, secant, sector, segment, subtended angle.

1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse.

2. (Motivate) 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.

3. (Motivate) There is one and only one circle passing through three given noncollinear points.

4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely.

5. (Prove) The angle subtended by an arc at the center 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 either of the pair of the opposite angles of a cyclic quadrilateral is 180o and its converse.

- Concepts :
- Circles and Its Related Terms question (54)
- Angle Subtended by a Chord at a Point question (9)
- Perpendicular from the Centre to a Chord question (4)
- Circle Through Three Points question (7)
- Equal Chords and Their Distances from the Centre question (18)
- Angle Subtended by an Arc of a Circle question (40)
- Cyclic Quadrilaterals question (42)

1. Construction of bisectors of line segments and angles of measure 60°, 90°, 45° 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.

- Concepts :
- Introduction of Constructions question (1)
- Basic Constructions question (33)
- Some Constructions of Triangles question (15)

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

- Concepts :
- Introduction to Area of a Triangle question (6)
- Area of a Triangle by Heron’S Formula question (20)
- Application of Heron’S Formula in Finding Areas of Quadrilaterals question (39)

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

- Concepts :
- Surface Area of a Cuboid and a Cube question (57)
- Surface Area of a Right Circular Cylinder question (86)
- Surface Area of a Right Circular Cone question (47)
- Surface Area of a Sphere question (55)
- Volume of a Cuboid question (35)
- Volume of a Cylinder question (39)
- Volume of a Right Circular Cone question (39)
- Volume of a Sphere question (52)

Introduction to Statistics:- Collection of data, presentation of data — tabular form, ungrouped / grouped, bar graphs, histograms (with varying base lengths), frequency polygons. Mean, median and mode of ungrouped data.

- Concepts :
- Introduction of Statistics videos (1) question (14)
- Collection of Data question (11)
- Presentation of Data question (37)
- Graphical Representation of Data question (63)
- Measures of Central Tendency question (86)

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).

- Concepts :
- Probability - an Experimental Approach question (43)

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Books For All Subjects- Mathematics Class 9
- Mathematics for Class 9 by R D Sharma (2018-19 Session)