# Mathematics (M 907) - Grade 9

Unit Area Covered Marks
Unit 1 Number System and Number Sense 05 Read more
Unit 2 Algebra (core) 05 Read more
Unit 3 Coordinate Geometry (core) 10 Read more
Unit 4 Lines, Angles and Triangles (core) 10 Read more
Unit 5 Quadrilaterals and Area of Parallelograms & Triangles (core) 20 Read more
Unit 6 Circle (core) 05 Read more
Unit 7 Area of Plane Figures and Solid Shapes (core) 10 Read more
Unit 8 Volume of Solids (core) 05 Read more
Unit 9 Introduction to Trigonometry (core) 10 Read more
Unit 10 Introduction to Statistics and Probability (core) 20 Read more
Total Marks 100

# Unit 1

Number System and Number Sense

• Natural numbers, whole numbers, integers and their representation of number line. Symbols to represent them as a system N, W, I respectively.
• Laws of exponents.

Introduction to rational numbers and irrational numbers

• Definition of rational numbers as numbers in the form p/q where p and q are integers and q≠0.
• Difference between rational numbers and fractions.
• representation of rational numbers on number line.
• Symbols Q to represent rational number system.
• Irrational numbers as numbers which are not rational.
• Symbols IR to represent irrational number system.
• Recognition of rational numbers as terminating decimal or non- terminating recurring decimal.
• Recognition of irrational numbers as non- terminating and non- recurring decimal.

Introduction to real numbers

• Real numbers as a system containing both rational as well as irrational numbers.
• Symbols R to represent real number system.
• Representation of real numbers on real line.
• Infiniteness of rational and irrational numbers.

Algebra of real numbers

• Sum, difference, product, quotient of rational numbers. Sum and difference of irrational numbers, Product of two irrational numbers.
• Quotient of two irrational numbers.
• Properties of rational numbers w. r. t. addition and multiplication.
• Properties of real numbers w. r. t. additional and multiplication.

# Unit 2

Review & Recall the terms

• Variable, constant, algebraic expression, equation.

Introduction to rational numbers and irrational numbers

• Definition of rational numbers as numbers in the form p/q where p and q are integers and q≠0.
• Difference between rational numbers and fractions, representation of rational numbers on number line.
• Symbols Q to represent rational number system. Irrational numbers as numbers which are not rational.
• Symbols IR to represent irrational number system.
• Recognition of rational numbers as terminating decimal or nonterminating recurring decimal.
• Recognition of irrational numbers as non- terminating and nonrecurring decimal.

Introduction to real numbers

• Real numbers as a system containing both rational as well as irrational numbers.
• Symbols R to represent real number system.
• Representation of real numbers on real line.
• Infiniteness of rational and irrational numbers.

Algebra of real numbers

• Sum, difference, product and division of polynomial.
• Division Algorithm.
• Factor theorem, Remainder theorem and their applications.

Introduction to algebraic identities

• Difference between polynomial, equation and identity, proving identities using manipulative or otherwise:(a+ b + c)2, a4,– b4 , a3,– b3 , a3 + b3

Factorization of polynomials

• Factorizing polynomial using common factors.
• Factorizing polynomial by splitting middle terms.
• Factorizing using algebraic identities: a2 – b2, (a+ b + c)2, a4,– b4 , a3,– b3 , a3 + b3.

Linear Equations and Linear inequalities

• Linear equation in one variable, solution of linear equations and its representation on number line, expressing word statements into linear equations.
• Linear inequalities in one variable and their representation on number line.

Quadratic equations

• Quadratic equation in one variable, to verify the solution of given quadratic equation.

Patterns

• Number patterns and geometric patterns.

# Unit 3

Cartesian

• Introduction of terms like axes, quadrants, origin, abscissa, ordinate, ordered pair, Cartesian coordinates, Cartesian independent and dependent variables plane.

Point in a plane

• Representation of a given point in Cartesian plane in the form of ordered pair. Plotting of a given point in the plane.

Linear graph

• Recognition and Graph of equations corresponding to straight lines.
• x = constant, y = constant, y = mx + c where m is gradient and c is the y-intercept.
• Gradient of a line as a ratio of 'rise over run'. Relation between.
• gradients of parallel lines.

Interpretation of graphical representation of simultaneous linear equations

• Point of intersection of simultaneous straight line equations drawn in same plane and that the point of intersection represents the solution of two equations.

Interpretation of graphs

• Real life situation graphs including travel graphs and conversion graphs; graphs of quantities that vary against time.

Scatter diagram

• Plot of two independent variables (scatter diagram) and examination by eye for positive or negative correlation.

# Unit 4

Basic Geometrical terms

• Point, Line, line segment, collinear points, non collinear points; Angle: right angle, acute angle, obtuse angle, straight angle, reflex angle, supplementary angles, complementary angles; Parallel lines, perpendicular lines, transversal; Triangle: scalene, isosceles, equilateral, acute angled, obtuse angled, right angled; Median, altitude, bisector of an angle, perpendicular bisector of a line segment.

Lines and Angles

• Pair of angles: adjacent angles, linear pair, vertically opposite angles; Linear pair axiom; Parallel lines and transversal: exterior angles, interior angles, corresponding angles, alternate interior angles, interior angles on the same side of transversal; corresponding angle axiom and converse, if a transversal intersects two parallel lines then each pair of alternate interior angles are equal and converse, if a transversal intersects two parallel lines then each pair of interior angles on the same side of the transversal is supplementary and converse; Proof: sum of interior angles of a triangle is 180 degrees, exterior angle property of triangle.

Triangles

• Congruence criteria: SSS, SAS, ASA, RHS; Properties: angles opposite to equal sides of an isosceles triangle are equal and converse; all angles of an equilateral triangle are 60 degrees.

# Unit 5

Introduction of basic terms

• Polygon, convex and concave polygons, quadrilateral, vertices, diagonal, adjacent sides, adjacent angles, opposite sides, opposite angles, types of quadrilaterals: square, rectangle, parallelogram, rhombus, trapezium, isosceles trapezium, and kite. Base and altitude of parallelogram.

Properties of parallelogram

• Exploration of following properties of parallelogram:
• In a parallelogram, pair of opposite sides is of equal length and the converse.
• A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and is of equal length.
• The diagonals of a parallelogram bisect each other and the converse.
• A parallelogram is a rectangle if its diagonal has equal length and the converse.
• A parallelogram is a rhombus if its diagonals are.
• perpendicular to each other and the converse.
• A parallelogram is a square if its diagonals are equal and are at right angles and the converse.
• Problems based on above properties.

Mid-point theorem

• The line segment joining the midpoint of any two sides.
• triangle is parallel to the third side and equal to half of it. Corollary: The line segment drawn through the mid- point.
• one side of a triangle, parallel to another side, bisects the third side.
• Problems based on above results.

Area of parallelogram

• Investigation into following results:
• A diagonal of a parallelogram divides it into two triangles of equal area.
• Parallelograms on the same base and between the same parallels are equal in area.
• A rectangle and a parallelogram on the same base and lying between the same parallels are equal in area.
• The area of a parallelogram is equal to the product of its base and the corresponding altitude.
• Problems based on above results.

Area of triangle

• Investigation into following results:
• Triangles on the same base and between the same parallels are equal in area.
• The area of a triangle is equal to half the product of one of its base and the corresponding altitude.
• If a triangle and a parallelogram are on the same base and between same parallels then area of triangle is equal to half of the area of the parallelogram.
• Problems based on above results.

# Unit 6

Review and recall basic terms

• Definition of circle, centre, radius, diameter, Interior of circle, circular region, exterior of circle ,arc, chord, minor segment, major segment, minor arc, major arc, sector of circle, minor sector, major sector, semicircular region, circumference of circle, angle subtended by the chord at a point on the circle, angle subtended by the arc at the centre of circle, concentric circles, intersecting circles, congruent circles, concyclic points.

Angle subtended by the chord of the circle at the centre

• Equal chords of a circle (or of congruent circles) subtend equal angles at the centre; If the angles subtended by the chords of a circle (or of congruent circles) at the centre are equal, then the chords are equal

Perpendicular from the centre of a circle to a chord

• The perpendicular from the centre of a circle to a chord bisects the chord; The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord

# Unit 7

Basic terms

• Perimeter & area of plane figures, curved surface area and total surface area of solids

Area of plane

• Area of right triangle, isosceles triangle, equilateral triangle, rectangle, square, parallelogram, rhombus, trapezium.

Heron's formula

• Area of triangle using Heron's formula and its application in finding area of quadrilateral.

Surface Area of solid shapes

• Surface Area of cube, cuboids, curved surface area and total surface area of cylinder, cone, sphere and hemisphere.

Applications in daily life

• Applications in finding the area of field, land etc.

# Unit 8

Introduction to volume

• Volume as product of area of base and height.

Volume of cubes and cuboids

• Formulae for finding volume of cube and cuboid of given dimension.

Volume of right circular cylinder and right circular cone

• Volume of a hollow right circular cylinder. Volume of metal required to cast a solid right circular cylinder. volume of cylindrical pipe of given thickness. volume of a right circular cone, relation between volume of right circular cylinder and right circular cone of given radius and given height.

Volume of sphere

• Volume of sphere and hemisphere of given radius.

# Unit 9

Introduction to Trigonometry

• Trigonometry as study of right angle triangle using relation between its sides and angles.

Defining trigonometric ratios in a right angle triangle

• Right angle triangle: hypotenuse, side containing angles of observation and right angle as adjacent side, side opposite to bearing as perpendicular side.
• Define sine, cosine and tangent of angle as ratio of sides of right triangle.
• Values of T-ratios for 30⁰, 45⁰, 60⁰

Angle of elevation and angle of depression

• Describing angle of elevation and angle of depression for a given point.
• Drawing of figure for given problems involving one right angle triangle.

# Unit 10

Introduction to statistics

• Significance of conducting survey, collecting data, interpreting data etc.
• Meaning and definition of statistics.

Types of data w.r.t. source

• Primary data and secondary data

Classification of data

• Ungrouped data and grouped data
• class interval, class-marks, upper limit, lower limit, frequency, range, cumulative frequency, class-size, discrete data and continuous data

Analysis of Data

• Measure of central tendency: Mean, median, mode of ungrouped data
• Interpretation of Bar graph, histogram of uniform width and frequency polygon for a given data.

Introduction to Probability

• Probability as chance of occurrence of an event
• Basic terms: random experiment, sample space, event, favourable and unfavourable events, sure event and impossible event.

Classical definition of probability

• Probability of an event E is
P(E)= n/N, where n is number of favourable events and N is total number of events in a sample space