# Class 9 Syllabus

Course StructureMaths Syllabus for CLASS 9

Three Hours
First Term Units Max. Marks -90

 Unit Marks I Number System 17 II Algebra 25 III Geometry 37 IV Co-ordinate Geometry 6 V Mensuration 5 Total 90

Second Term Units

 Unit Marks II Algebra (contd.) 16 III Geometry (contd.) 38 V Mensuration (contd.) 18 VI Statistics 10 VII Probability 8 Total 90

### First Term Units

#### 1. REAL NUMBERS

• Representation of natural numbers, rational numbers on the number line, integers. Representation of terminating and non-terminating recurring decimals, on number line through successive magnification.
• Examples of non-recurring and non-terminating decimals. Non-rational numbers and their representation on the number line.
• Explaining ideal numbers by a unique point on the number line and conversly, each point on the number line shows a unique real number.
• Existence of √x for a specified positive real number x.
• Nth root definition of a real number.
• Laws of exponents with integral powers. Rational exponents with positive real bases.
• Rationalization of real numbers of the type 1/(a+b √x) and 1/(√x + √y) where x and y are natural numbers and a and b are integers.

#### 1. POLYNOMIALS

• Definition of a polynomial in one variable with examples. Coefficients of a polynomial and 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. Further verification of identities of the type (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.

#### 1. INTRODUCTION TO EUCLID’S GEOMETRY

• History – Geometry in India and Euclid’s geometry.
• The five postulates of Euclid. Equivalent versions of the fifth postulate.
• Euclid’s method of formalizing observed phenomenon into rigorous mathematics with definitions, axioms/postulates, common/obvious notions, and theorems.
• Showing the relationship between axiom and theorem.

#### 2. LINES AND ANGLES

Proves

1. If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse.
2. If two lines intersect, vertically opposite angles are equal.
3. Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines.
4. Lines which are parallel to a given line are parallel.
5. The sum of the angles of a triangle is 180°.
6. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.

#### 3. TRIANGLES

Proves

1. 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. 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. Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence).
4. 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.
5. The angles opposite to equal sides of a triangle are equal.
6. The sides opposite to equal angles of a triangle are equal.
7. Triangle inequalities and relation between ‘angle and facing side’ inequalities in triangles.

#### 1. COORDINATE GEOMETRY

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

#### 1. AREAS

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

### Second Term Syllabus

#### 2. LINEAR EQUATIONS IN TWO VARIABLES

• 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 seem to lie on a line.
• Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously.

#### UNIT III: GEOMETRY (Contd.)

Prove

1. The diagonal divides a parallelogram into two congruent triangles.
2. In a parallelogram opposite sides are equal, and conversely.
3. In a parallelogram opposite angles are equal, and conversely.
4. A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal.
5. In a parallelogram, the diagonals bisect each other and conversely.
6. In a triangle, the line segment joining the midpoints of any two sides is parallel to the third side and its converse.

#### 5. AREA

Prove

1. Parallelograms on the same base and between the same parallels have the same area.
2. Triangles on the same (or equal base) base and between the same parallels are equal in area.

#### 6. CIRCLES

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

1. Equal chords of a circle subtend equal angles at the center and (motivate) its converse.
2. 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. There is one and only one circle passing through three given non-collinear points.
4. Equal chords of a circle (or of congruent circles) are equidistant from the center and conversely.
5. 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. Angles in the same segment of a circle are equal.
7. 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.
The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse.

#### 7. CONSTRUCTIONS

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.

#### 2. SURFACE AREAS AND VOLUMES

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

#### UNIT VI: STATISTICS

• Introduction to Statistics: Collection of data, presentation of data – tabular form, bar graphs, histograms (with varying base lengths), frequency polygons.
• Qualitative analysis of data to choose the correct form of presentation for the collected data.
• Mean, median, mode of ungrouped data.

#### UNIT VII: PROBABILITY

• 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).

#### Practise This Question

Which of the following is the multiplicative inverse of (52)3?