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Skills available for India class IX maths curriculum

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National Council of Education Research and Training Syllabus: CISCE Class IX National Council of Education Research and Training Syllabus: CISCE Class IX
National Council of Education Research and Training Syllabus: Class IX National Council of Education Research and Training Syllabus: Class IX

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9.1 Pure Arithmetic

9.1.1 Irrational Numbers
- 9.1.1.a Rational, irrational numbers as real numbers, their place in the number system. Surds and rationalization of surds.
- 9.1.1.b Irrational numbers as non-repeating, nonterminating decimals.
- 9.1.1.c Classical definition of a rational number p/q, p, q ∈ Z, q ≠ 0. Hence, define irrational numbers as what cannot be expressed as above.
  - Identify rational and irrational numbers (X-A.3)
- 9.1.1.d Simplifying an expression by rationalising the denominator.
  - Simplify radical expressions using conjugates (IX-CC.11)

9.2 Commercial Mathematics

9.2.i Profit and Loss
- 9.2.i.1 The meaning of Marked price, selling price and discount, thus giving an idea of profit and loss on day to day dealings. Simple problems related to Profit and Loss and Discount, including inverse working.
9.2.ii Compound Interest
- 9.2.ii.1 Compound Interest as a repeated Simple Interest computation with a growing Principal. Use of formula - A = P ((1 + r)/100)ⁿ. Finding CI from the relation CI = A-P. Simple direct problems based on above formulae.
  - Simple interest (X-C.1)
  - Compound interest (X-C.2)

9.3 Algebra

9.3.i Expansions (a ± b)², (a ± b)³, (x ± a)(x ± b)
9.3.ii Factorisation a² – b², a³ ± b³, ax² + bx + c, by splitting the middle term.
- Find the roots of factorised polynomials (IX-Z.12)
- Factorise quadratics: special cases (IX-AA.5)
9.3.iii Changing the subject of a formula.
- 9.3.iii.1 Concept that each formula is a perfect equation with variables.
- 9.3.iii.2 Concept of expressing one variable in terms, of another various operators on terms transposing the terms squaring or taking square root etc.
9.3.iv Linear Equations and Simultaneous (linear) Equations
- 9.3.iv.1 Solving algebraically (by elimination as well as substitution) and graphically.
- 9.3.iv.2 Solving simple problems based on these by framing appropriate formulae.
9.3.v Indices/ Exponents
- 9.3.v.1 Handling positive, fractional, negative and "zero" indices. Simplification of expressions involving various exponents a to the m power x aⁿ = a to the (m + ⁿ) power, am ÷ aⁿ = a to the (m - ⁿ) power, (a to the m power)ⁿ = a to the mn power, etc. use of laws of exponents. etc use of laws of exponents.
9.3.vi Logarithms
- 9.3.vi.a Logarithmic form vis-à-vis exponential form: interchanging.
  - Convert between exponential and logarithmic form: rational bases (IX-W.1)
- 9.3.vi.b Laws of Logarithms and its use Expansion of expression with the help of laws of logarithm eg. y = a⁴xb²/c³ log y = 4 log a + 3 log b – 3 log c etc.

9.4 Geometry

9.4.i Triangles, Relation between sides and angles of triangles. Types of triangles, Congruent triangles.
- 9.4.i.a Congruency: four cases: SSS, SAS, AAS, RHS. Illustration through cutouts. Simple applications.
- 9.4.i.b Problems based on:
  - 9.4.i.b.1 Angles opposite equal sides are equal and converse.
    - Congruency in isosceles and equilateral triangles (IX-G.10)
  - 9.4.i.b.2 If two sides of a triangle are unequal, then the greater angle is opposite the greater side and converse.
    - Angle-side relationships in triangles (IX-G.5)
  - 9.4.i.b.3 Sum of any two sides of a triangle is greater than the third side.
    - Triangle Inequality Theorem (IX-G.6)
  - 9.4.i.b.4 Of all straight lines that can be drawn to a given line from a point outside it, the perpendicular is the shortest.
  - 9.4.i.b.5 Proofs not required.
9.4.ii Constructions (using ruler and compasses)
- 9.4.ii.1 Constructions of triangles involving 30°, 45°, 60°, 75°, 90°, 120°, 135° angles.
9.4.iii Mid Point Theorem and its converse, equal intercept theorem
- 9.4.iii.a Proof and simple applications of mid point theorem and its converse.
  - Midpoint formula (X-L.7)
- 9.4.iii.b Equal intercept theorem: proof and simple application.
9.4.iv Similarity, conditions of similar triangles.
- 9.4.iv.a As a size transformation.
- 9.4.iv.b Comparison with congruency, keyword being proportionality.
  - Solve proportions (IX-C.5)
  - Solve proportions: word problems (IX-C.6)
- 9.4.iv.c Three conditions: SSS, SAS, AA. Simple applications (proof not included).
  - SSS, SAS and ASA Theorems (IX-G.8)
- 9.4.iv.d Applications of Basic Proportionality Theorem.
9.4.v Pythagoras Theorem
- 9.4.v.1 Proof and Simple applications of Pythagoras Theorem and its converse.
  - Pythagoras' Theorem (X-Q.1)
  - Converse of Pythagoras' theorem (X-Q.2)
9.4.vi Rectilinear Figures
- 9.4.vi.1 Rectilinear figures or polygons, Different kinds of polygons and its names interior and exterior angles and their relations. Types of regular polygons parallelograms, conditions of parallelograms, Rhombus, Rectangles. Proof and use of theorems on parallelogram.
  - 9.4.vi.1.a Sum of interior angles of a polygon.
    - Interior angles of polygons (IX-I.2)
  - 9.4.vi.1.b Sum of exterior angles of a polygon.
    - Exterior angles of polygons (IX-I.3)
  - 9.4.vi.1.c Regular polygons.
    - Polygon vocabulary (IX-I.1)
  - 9.4.vi.1.d Parallelogram:
    - 9.4.vi.1.d.1 Both pairs of opposite sides equal (without proof).
    - 9.4.vi.1.d.2 Both pairs of opposite angles equal.
    - 9.4.vi.1.d.3 One pair of opposite sides equal and parallel (without proof).
    - 9.4.vi.1.d.4 Diagonals bisect each other and bisect the parallelogram.
    - 9.4.vi.1.d.5 Rhombus as a special parallelogram whose diagonals meet at right angles.
      - Properties of rhombuses (IX-H.5)
      - Review: properties of quadrilaterals (IX-H.9)
    - 9.4.vi.1.d.6 In a rectangle, diagonals are equal, in a square they are equal and meet at right angles.
      - Properties of squares and rectangles (IX-H.6)
      - Review: properties of quadrilaterals (IX-H.9)
  - 9.4.vi.1.e Quadrilaterals
    - 9.4.vi.1.e.1 Construction of quadrilaterals (including parallelograms and rhombus) and regular hexagon using ruler and a pair of compasses only.
      - Construct an equilateral triangle or regular hexagon (IX-M.4)
  - 9.4.vi.1.f Proof and use of area theorems on parallelograms:
    - 9.4.vi.1.f.1 Parallelograms on the same base and between the same parallels are equal in area.
      - Proving a quadrilateral is a parallelogram (IX-H.4)
      - Area of parallelograms and triangles (IX-J.3)
    - 9.4.vi.1.f.2 The area of a triangle is half that of a parallelogram on the same base and between the same parallels.
    - 9.4.vi.1.f.3 Triangles between the same base and between the same parallels are equal in area (without proof).
    - 9.4.vi.1.f.4 Triangles with equal areas on the same bases have equal corresponding altitudes.

9.5 Statistics

9.5.1 Introduction, collection of data, presentation of data, Graphical representation of data, Mean, Median of ungrouped data.
- 9.5.1.i Understanding and recognition of raw, arrayed and grouped data.
- 9.5.1.ii Tabulation of raw data using tally-marks.
- 9.5.1.iii Understanding and recognition of discrete and continuous variables.
- 9.5.1.iv Mean, median of ungrouped data
  - Mean, median, mode and range (IX-FF.1)
- 9.5.1.v Class intervals, class boundaries and limits, frequency, frequency table, class size for grouped data.
- 9.5.1.vi Grouped frequency distributions: the need to and how to convert discontinuous intervals to continuous intervals.
- 9.5.1.vii Drawing a histogram and frequency polygon.
  - Create bar graphs, line graphs and histograms (IX-O.2)
- 9.5.1.viii Understanding of how a histogram differs from a bar chart.

9.6 Mensuration

9.6.1 Area and perimeter of a triangle and a quadrilateral. Area and circumference of a circle. Surface area and volume of Cube, Cuboids and Cylinder.
- 9.6.1.a Area and perimeter of triangle (including Heron's formula), square, rhombus, rectangle, parallelogram and trapezium.
- 9.6.1.b (i) Circle: Area and circumference (ii) Simple direct problems involving inner and outer dimensions and cost.
  - Area and circumference of circles (IX-J.5)
- 9.6.1.c Surface area and volume of 3-D solids: cube, cuboid and cylinder including problems of type involving:
  - 9.6.1.c.1 Different internal and external dimensions of the solid.
  - 9.6.1.c.2 Cost.
  - 9.6.1.c.3 Concept of volume being equal to area of cross-section x height.
    - Volume of prisms and cylinders (IX-K.4)
  - 9.6.1.c.4 Open/closed cubes/cuboids/cylinders.

9.7 Trigonometry

9.7.a Trigonometric Ratios: sine, cosine, tangent of an angle and their reciprocals.
9.7.b Trigonometric ratios of standard angles - 0, 30, 45, 60, 90 degrees. Evaluation of an expression involving these ratios.
- Find trigonometric functions of special angles (X-S.4)
9.7.c Simple 2-D problems involving one right-angled triangle.
9.7.d Concept of sine and cosine being complementary with simple, direct application.
- Trigonometric functions of complementary angles (X-S.3)

9.8 Co-ordinate Geometry

9.8.1 Cartesian System, Plotting a point in the plane for given coordinates.
- 9.8.1.a Dependent and independent variables.
  - Identify independent and dependent variables (XI-C.3)
- 9.8.1.b Ordered pairs, co-ordinates of points and plotting them in the Cartesian Plane.
  - Graph quadrilaterals (IX-H.2)
- 9.8.1.c Graphs of x=0, y=0, x=a, y=a, x=y, y= mx+c including identification and conceptual understanding of slope and y-intercept.
- 9.8.1.d Recognition of graphs based on the above.

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9.1 Pure Arithmetic

9.1.1 Irrational Numbers

9.1.1.a Rational, irrational numbers as real numbers, their place in the number system. Surds and rationalization of surds.

9.1.1.b Irrational numbers as non-repeating, nonterminating decimals.

9.1.1.c Classical definition of a rational number p/q, p, q ∈ Z, q ≠ 0. Hence, define irrational numbers as what cannot be expressed as above.

9.1.1.d Simplifying an expression by rationalising the denominator.

9.2 Commercial Mathematics

9.2.i Profit and Loss

9.2.i.1 The meaning of Marked price, selling price and discount, thus giving an idea of profit and loss on day to day dealings. Simple problems related to Profit and Loss and Discount, including inverse working.

9.2.ii Compound Interest

9.2.ii.1 Compound Interest as a repeated Simple Interest computation with a growing Principal. Use of formula - A = P ((1 + r)/100)n. Finding CI from the relation CI = A-P. Simple direct problems based on above formulae.

9.3 Algebra

9.3.i Expansions (a ± b)², (a ± b)³, (x ± a)(x ± b)

9.3.ii Factorisation a² – b², a³ ± b³, ax² + bx + c, by splitting the middle term.

9.3.iii Changing the subject of a formula.

9.3.iii.1 Concept that each formula is a perfect equation with variables.

9.3.iii.2 Concept of expressing one variable in terms, of another various operators on terms transposing the terms squaring or taking square root etc.

9.3.iv Linear Equations and Simultaneous (linear) Equations

9.3.iv.1 Solving algebraically (by elimination as well as substitution) and graphically.

9.3.iv.2 Solving simple problems based on these by framing appropriate formulae.

9.3.v Indices/ Exponents

9.3.vi Logarithms

9.3.vi.a Logarithmic form vis-à-vis exponential form: interchanging.

9.3.vi.b Laws of Logarithms and its use Expansion of expression with the help of laws of logarithm eg. y = a4xb2/c3 log y = 4 log a + 3 log b – 3 log c etc.

9.4 Geometry

9.4.i Triangles, Relation between sides and angles of triangles. Types of triangles, Congruent triangles.

9.4.i.a Congruency: four cases: SSS, SAS, AAS, RHS. Illustration through cutouts. Simple applications.

9.4.i.b Problems based on:

9.4.i.b.1 Angles opposite equal sides are equal and converse.

9.4.i.b.2 If two sides of a triangle are unequal, then the greater angle is opposite the greater side and converse.

9.4.i.b.3 Sum of any two sides of a triangle is greater than the third side.

9.4.i.b.4 Of all straight lines that can be drawn to a given line from a point outside it, the perpendicular is the shortest.

9.4.i.b.5 Proofs not required.

9.4.ii Constructions (using ruler and compasses)

9.4.ii.1 Constructions of triangles involving 30°, 45°, 60°, 75°, 90°, 120°, 135° angles.

9.4.iii Mid Point Theorem and its converse, equal intercept theorem

9.4.iii.a Proof and simple applications of mid point theorem and its converse.

9.4.iii.b Equal intercept theorem: proof and simple application.

9.4.iv Similarity, conditions of similar triangles.

9.4.iv.a As a size transformation.

9.4.iv.b Comparison with congruency, keyword being proportionality.

9.4.iv.c Three conditions: SSS, SAS, AA. Simple applications (proof not included).

9.4.iv.d Applications of Basic Proportionality Theorem.

9.4.v Pythagoras Theorem

9.4.v.1 Proof and Simple applications of Pythagoras Theorem and its converse.

9.4.vi Rectilinear Figures

9.4.vi.1 Rectilinear figures or polygons, Different kinds of polygons and its names interior and exterior angles and their relations. Types of regular polygons parallelograms, conditions of parallelograms, Rhombus, Rectangles. Proof and use of theorems on parallelogram.

9.4.vi.1.a Sum of interior angles of a polygon.

9.4.vi.1.b Sum of exterior angles of a polygon.

9.4.vi.1.c Regular polygons.

9.4.vi.1.d Parallelogram:

9.4.vi.1.d.1 Both pairs of opposite sides equal (without proof).

9.4.vi.1.d.2 Both pairs of opposite angles equal.

9.4.vi.1.d.3 One pair of opposite sides equal and parallel (without proof).

9.4.vi.1.d.4 Diagonals bisect each other and bisect the parallelogram.

9.4.vi.1.d.5 Rhombus as a special parallelogram whose diagonals meet at right angles.

9.4.vi.1.d.6 In a rectangle, diagonals are equal, in a square they are equal and meet at right angles.

9.4.vi.1.e Quadrilaterals

9.4.vi.1.e.1 Construction of quadrilaterals (including parallelograms and rhombus) and regular hexagon using ruler and a pair of compasses only.

9.4.vi.1.f Proof and use of area theorems on parallelograms:

9.4.vi.1.f.1 Parallelograms on the same base and between the same parallels are equal in area.

9.4.vi.1.f.2 The area of a triangle is half that of a parallelogram on the same base and between the same parallels.

9.4.vi.1.f.3 Triangles between the same base and between the same parallels are equal in area (without proof).

9.4.vi.1.f.4 Triangles with equal areas on the same bases have equal corresponding altitudes.

9.5 Statistics

9.5.1 Introduction, collection of data, presentation of data, Graphical representation of data, Mean, Median of ungrouped data.

9.5.1.i Understanding and recognition of raw, arrayed and grouped data.

9.5.1.ii Tabulation of raw data using tally-marks.

9.5.1.iii Understanding and recognition of discrete and continuous variables.

9.5.1.iv Mean, median of ungrouped data

9.5.1.v Class intervals, class boundaries and limits, frequency, frequency table, class size for grouped data.

9.5.1.vi Grouped frequency distributions: the need to and how to convert discontinuous intervals to continuous intervals.

9.5.1.vii Drawing a histogram and frequency polygon.

9.5.1.viii Understanding of how a histogram differs from a bar chart.

9.6 Mensuration

9.6.1 Area and perimeter of a triangle and a quadrilateral. Area and circumference of a circle. Surface area and volume of Cube, Cuboids and Cylinder.

9.6.1.a Area and perimeter of triangle (including Heron's formula), square, rhombus, rectangle, parallelogram and trapezium.

9.2.ii.1 Compound Interest as a repeated Simple Interest computation with a growing Principal. Use of formula - A = P ((1 + r)/100)ⁿ. Finding CI from the relation CI = A-P. Simple direct problems based on above formulae.

9.3.vi.b Laws of Logarithms and its use Expansion of expression with the help of laws of logarithm eg. y = a⁴xb²/c³ log y = 4 log a + 3 log b – 3 log c etc.