ICSE Mathematics Class 9 Syllabus

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MATHEMATICS

Aims:

  1. To acquire knowledge and understanding of the terms, symbols, concepts, principles, processes, proofs, etc. of mathematics.
  2. To develop an understanding of mathematical concepts and their application to further studies in mathematics and science.
  3. To develop skills to apply mathematical knowledge to solve real life problems.
  4. To develop the necessary skills to work with modern technological devices such as calculators and computers in real life situations.
  5. To develop drawing skills and the ability to read tables, charts, and graphs.
  6. To develop an interest in mathematics.

CLASS IX

There will be one paper of two and a half hours duration carrying 80 marks along with an Internal Assessment of 20 marks. Certain questions may require the use of mathematical tables (logarithmic and trigonometric tables). The solution of a question may require the knowledge of more than one branch of the syllabus.

1. Pure Arithmetic

Rational and Irrational Numbers: These are treated as real numbers with a specific place in the number system. Topics include the study of surds, the rationalization of surds to simplify expressions, and the representation of both rational and irrational numbers on the number line. Proofs of irrationality are also discussed.

2. Commercial Mathematics

Compound Interest:

  • (a) Compound interest is approached as a repeated application of simple interest with a growing principal, used to compute the accumulated amount over periods such as 2 or 3 years.
  • (b) The formula is used to determine compound interest:
    \[ \text{CI} = A - P \] where \( A \) is the final amount and \( P \) is the principal. Interest compounded half-yearly is included. Problems involve finding one of the quantities \( A \), \( P \), \( r \), or \( n \) given the others for both compound interest (CI) and simple interest (SI), and understanding the differences between the two. (Note: Repayments in equal installments with a given rate of interest and installment amount are excluded.)

3. Algebra

(i) Expansions: Review of earlier concepts includes expanding:

  • \((a \pm b)^2\)
  • \((a \pm b)^3\)
  • \((x \pm a)(x \pm b)\)
  • \((a \pm b \pm c)^2\)

(ii) Factorisation:

  • \(a^2 - b^2\)
  • \(a^3 \pm b^3\)
  • \(ax^2 + bx + c\) (by splitting the middle term)

(iii) Simultaneous Linear Equations in Two Variables: These equations (with numerical coefficients) are solved using methods such as elimination, substitution, and cross multiplication. Students also solve problems by framing appropriate equations.

(iv) Indices/Exponents: Topics include handling positive, fractional, negative, and zero indices. Examples of laws include:
\[ a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}. \] Simplification of expressions involving various exponents is emphasized.

(v) Logarithms: Focus is on the logarithmic form, its conversion with exponential form, and the use of laws of logarithms. For example, an expression may be expanded as:
\[ \log y = 4\log a + 2\log b - 3\log c. \]

4. Geometry

(i) Triangles:

  • Congruency: Four cases are studied — SSS, SAS, AAS, and RHS. Visual cutouts and simple applications help illustrate the concepts.
  • Problems involve properties such as the fact that angles opposite equal sides are equal, and if two sides are unequal then the greater angle is opposite the greater side, along with the triangle inequality theorem (the sum of any two sides is greater than the third). The perpendicular drawn from a point to a line represents the shortest distance.
  • Mid-Point Theorem and Equal Intercept Theorem: Students prove and apply the midpoint theorem (and its converse) as well as the equal intercept theorem.
  • Pythagoras’ Theorem: An area-based proof is provided along with applications and the converse of the theorem.

(ii) Rectilinear Figures:

  • Theorems on Parallelograms: Topics include:
    • Both pairs of opposite sides are equal (without proof).
    • Both pairs of opposite angles are equal.
    • One pair of opposite sides can be equal and parallel (without proof).
    • The diagonals bisect each other.
    • A rhombus is a special parallelogram whose diagonals intersect at right angles.
    • In a rectangle the diagonals are equal; in a square, the diagonals are both equal and perpendicular.
  • Constructions of Polygons: Construction of various quadrilaterals (including parallelograms and rhombus) and a regular hexagon using only a ruler and compasses.
  • Area Theorems: Includes:
    • Parallelograms on the same base and between the same parallels have equal areas.
    • The area of a triangle is one-half that of a parallelogram on the same base and between the same parallels.
    • Triangles on the same base and between the same parallels are equal in area (without proof).
    • Triangles with equal areas on the same base have equal corresponding altitudes.

(iii) Circle:

  • Chord Properties:
    • A line drawn from the centre to bisect a chord (that is not a diameter) is perpendicular to the chord.
    • The perpendicular from the centre bisects the chord (without proof).
    • Equal chords are equidistant from the centre.
    • Chords equidistant from the centre are equal (without proof).
    • Exactly one circle passes through three non-collinear points.
  • Arc and Chord Properties:
    • If two arcs subtend equal angles at the centre, they are equal (and vice versa).
    • If two chords are equal, they intercept equal arcs (and conversely, without proof).

Note: The proofs for these theorems are to be taught unless specified otherwise.

5. Statistics

This section covers the collection, organization, and presentation of data. It includes the graphical representation of data as well as the calculation of the mean and median for ungrouped data.

  • Understanding and classification of raw, arrayed, and grouped data.
  • Tabulating raw data using tally marks.
  • Differentiating between discrete and continuous variables.
  • Calculating mean and median for ungrouped data.
  • Determining class intervals, boundaries, and frequencies for grouped data.
  • Converting discontinuous intervals into continuous intervals for grouped frequency distributions.
  • Drawing frequency polygons.

6. Mensuration

Topics include the measurement of area, perimeter, surface area, and volume for various shapes.

  • Triangles and Quadrilaterals: Calculation of area and perimeter—including the use of Heron’s formula.
  • Circle: Determination of area and circumference with applications that involve inner and outer area calculations. (Note: Areas of sectors other than quarter-circles and semicircles are not included.)
  • 3-D Solids: Calculation of surface area and volume for cubes and cuboids. Problems involve considerations such as different internal and external dimensions, cost estimation, and understanding that volume equals the area of the cross-section multiplied by the height. Both open and closed cubes/cuboids are discussed.

7. Trigonometry

Trigonometric Ratios: This section covers the sine, cosine, tangent and their reciprocal functions. It includes an analysis of the ratios for standard angles \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\) with example evaluations.

Complementary Angles: The following identities apply:

\[ \sin A = \cos (90^\circ - A), \quad \cos A = \sin (90^\circ - A) \] \[ \tan A = \cot (90^\circ - A), \quad \cot A = \tan (90^\circ - A) \] \[ \sec A = \csc (90^\circ - A), \quad \csc A = \sec (90^\circ - A) \]

8. Coordinate Geometry

This unit introduces the Cartesian coordinate system, the plotting of points, and methods for solving simultaneous linear equations graphically. The distance between two points is determined using the distance formula.

  • Understanding dependent and independent variables.
  • Working with ordered pairs and plotting them in the Cartesian plane.
  • Graphically solving simultaneous linear equations.
  • Distance Formula:
    \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]

Internal Assessment

A minimum of two assignments is to be completed during the year as prescribed by the teacher. Suggested assignments include:

  • Conduct a survey of a group of students (e.g., measuring height, weight, number of family members, pocket money) and then represent the data graphically.
  • Plan delivery routes for a postman or milkman.
  • Operate a tuck shop or canteen project.
  • Investigate ways to raise a loan for purchasing a car or house (e.g., by exploring bank loans or hire purchase options for household items such as refrigerators or televisions).
  • Cut a circle into equal sections of a small central angle to find its area using the formula:
    \[ A = \pi r^2. \]
  • Use flat cutouts to form cubes, cuboids, and pyramids to derive formulae for volume and total surface area.
  • Draw a circle of radius \( r \) on both ½ cm graph paper and 2 mm graph paper. Estimate the area enclosed in each case by counting the squares, compare with the theoretical value:
    \[ \text{area} = \pi r^2. \]
  • Apply any modifications or additional experiments as suggested by the teacher.
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