O Level | O Level Math | O Level Math Tuition | Athena Education — Athena Education

Athena is more than a O level maths tuition centre. It is a way of learning.

Math O Level

Ace the GCE O Levels with Athena

Ace O Level Math Exams the Athena way

No more brute force mathematics and endless, mindless drills. Athena’s SEAB O Level tuition fosters our mentee’s love for learning by advocating for conceptual understanding: we believe fostering an intuitive understanding of key concepts will ensure a student never forgets what is taught. Interactive, kinaesthetic learning using real-life examples are commonplace in our lessons. Our full-time tutors have taught O level maths tuition for at least 8 years, and were themselves top of their cohort in their Math exams (many years ago!)

Many O Level maths tuition centres, especially in Singapore teach in large groups because it helps their bottom line. Athena teaches exclusively one-on-one sessions, in order to ensure each mentee receives full attention, classes can be set at their pace, and mentors can really get to know their mentees. At Athena, an environment is fostered where mentees are comfortable to voice any questions or doubts they have. Mentors also craft personalised learning plans for students after learning their chief stumbling blocks.

Keeping in mind that the O Levels are meant to train students up for the A Level Math exam, we train our mentees to be equipped with the toolkit needed to excel at the E Math O Level or Additional Maths O Level papers. This is done by giving mentees a more complete picture of the content they are studying at O Levels, and to develop their love for and interest in Mathematics. Ultimately, we aim for Athena mentees to score an A1 or A2 in the O Levels and have the best possible head start in life. We’ve helped many students who approached us with C5s and C6s achieve an A in the O Level Exam.

Athena’s tutors are well versed in the O Level A Maths Syllabus and O Level E Maths Syllabus, and have a minimum of 8 years of teaching experience.

Algebra

Equations and inequalities

Conditions for a quadratic equation to have:

(i) two real roots

(ii) two equal roots

(iii) no real roots

and related conditions for a given line to:

(i) intersect a given curve

(ii) be a tangent to a given curve

(iii) not intersect a given curve

Conditions for ax2 + bx + c to be always positive (or always negative)

Solving simultaneous equations in two variables with at least one linear equation, by substitution

Relationships between the roots and coefficients of a quadratic equation

Solving quadratic inequalities, and representing the solution on the number line

Indices and surds

Four operations on indices and surds, including rationalising the denominator

Solving equations involving indices and surds

Polynomials and partial fractions

Multiplication and division of polynomials

Use of remainder and factor theorems

Factorisation of polynomials

Use of:

a^3 + b3 = (a + b)(a^2 – ab + b^2)

a^3 – b^3 = (a – b)(a^2 + ab + b^2)

Solving cubic equations

Partial fractions with cases where the denominator is no more complicated than: – (ax + b)(cx + d) – (ax + b)(cx + d)^2 – (ax + b)(x^2 + c^2)

Binomial expansions

• Use of the Binomial Theorem for positive integer n

Use of the notations n! and (n r)

Use of the general term (n r) a^n-r b^r , 0 < r ⩽ n (knowledge of the greatest term and properties of the coefficients is not required)

Power, exponential, logarithmic and modulus functions

Power functions y = ax^n where n is a simple rational number, and their graphs

Exponential and logarithmic functions a^x , e^x , log ax, ln x and their graphs, including:

– laws of logarithms

– equivalence of y = ax and x = log ay

– change of base of logarithms

Modulus functions |x| and |f(x)| where f(x) is linear, quadratic or trigonometric, and their graphs

Solving simple equations involving exponential, logarithmic and modulus functions

Geometry and Trigonometry

Trigonometric functions, identities and equations

Coordinate geometry in two dimensions

Condition for two lines to be parallel or perpendicular

Midpoint of line segment

Area of rectilinear figure

Graphs of parabolas with equations in the form y2 = kx

Coordinate geometry of circles in the form: – (x – a)^2 + (y – b)^2 = r^2 – x^2 + y^2 + 2gx + 2fy + c = 0 (excluding 2 circles questions)

Transformation of given relationships, including y = axn and y = kbx, to linear form to determine the unknown constants from a straight line graph

Proofs in plane geometry

Use of:

– properties of parallel lines cut by a transversal, perpendicular and angle bisectors, triangles, special quadrilaterals and circles

– congruent and similar triangles

– midpoint theorem

– tangent-chord theorem (alternate segment theorem)

Calculus

Differentiation and integration

Derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point

Derivatives as rate of change

Derivatives of x^n, for any rational n, sin x, cos x, tan x, ex , and ln x, together with constant multiples, sums and differences

Derivatives of products and quotients of functions

Derivatives of composite functions

Increasing and decreasing functions

Stationary points (maximum and minimum turning points and stationary points of inflexion)

Use of second derivative test to discriminate between maxima and minima

Applying differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems

Integration as the reverse of differentiation

Integration of xn, for any rational n, sin x, cos x, sec2 x and ex , together with constant multiples, sums and differences

Integration of (ax + b) n, for any rational n, sin(ax + b), cos(ax + b), and eax+b

Definite integral as area under a curve

Evaluation of definite integrals

Finding the area of a region bounded by a curve and line(s) (excluding area of region between two curves)

Finding areas of regions below the x-axis

Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line

Derivative of f(x) as the gradient of the tangent to the graph of y = f(x) at a point

Derivative as rate of change

Derivatives of x^n, for any rational n, sin x, cos x, tan x, ex , and ln x, together with constant multiples, sums and differences

Derivatives of products and quotients of functions

Derivatives of composite functions

Increasing and decreasing functions

Stationary points (maximum and minimum turning points and stationary points of inflexion)

Use of second derivative test to discriminate between maxima and minima

Applying differentiation to gradients, tangents and normals, connected rates of change and maxima and minima problems

Integration as the reverse of differentiation

Integration of xn, for any rational n, sin x, cos x, sec2 x and ex , together with constant multiples, sums and differences

Integration of (ax + b)^n, for any rational n, sin(ax + b), cos(ax + b), and e^ax+b

Definite integral as area under a curve

Evaluation of definite integrals

Finding the area of a region bounded by a curve and line(s) (excluding area of region between two curves)

Finding areas of regions below the x-axis

Application of differentiation and integration to problems involving displacement, velocity and acceleration of a particle moving in a straight line

Numbers and algebra

Numbers and their operations

Primes and prime factorisation

Finding highest common factor (HCF) and lowest common multiple (LCM), squares, cubes, square roots and cube roots by prime factorisation

Negative numbers, integers, rational numbers, real numbers, and their four operations

Calculations with calculator

Representation and ordering of numbers on the number line

Use of the symbols <, >, ⩽, ⩾

Approximation and estimation (including rounding off numbers to a required number of decimal places or significant figures and estimating the results of computation)

Use of standard form A × 10^n, where n is an integer, and 1 ⩽ A < 10

Positive, negative, zero and fractional indices

Laws of indices

Indices and surds

Ratio and proportion

Ratios involving rational numbers

Writing a ratio in its simplest form

Map scales (distance and area)

Direct and inverse proportion

Percentage

Expressing one quantity as a percentage of another

Comparing two quantities by percentage

Percentages greater than 100%

Increasing/decreasing a quantity by a given percentage

Reverse percentages

Rate and speed

Average rate and average speed

Conversion of units (e.g. km/h to m/s)

Algebraic expressions and formulae

Functions and graphs

Cartesian coordinates in two dimensions

Graph of a set of ordered pairs as a representation of a relationship between two variables

Linear functions (y = ax + b) and quadratic functions (y = ax2 + bx + c)

Graphs of linear functions

The gradient of a linear graph as the ratio of the vertical change to the horizontal change (positive and negative gradients)

Graphs of quadratic functions and their properties:

Positive or negative coefficient of x2

Maximum and minimum points ∗ symmetry

Sketching the graphs of quadratic functions given in the form:

y = – (x − p)^2 + q

y = − (x − p)^2 + q

y = – (x − a)(x − b)

y = − (x − a)(x − b)

Graphs of power functions of the form y = ax^n, where n = −2, −1, 0, 1, 2, 3, and simple sums of not more than three of these

Graphs of exponential functions y = ka^x , where a is a positive integer

Estimation of the gradient of a curve by drawing a tangent

Equations and inequalities

Set language and notation

Matrices

Display of information in the form of a matrix of any order

Interpreting the data in a given matrix

Product of a scalar quantity and a matrix

Problems involving the calculation of the sum and product (where appropriate) of two matrices

Problems in real-world contexts

Solving problems based on real-world contexts:

In everyday life (including travel plans, transport schedules, sports and games, recipes, etc.)

Involving personal and household finance (including simple and compound interest, taxation, instalments, utilities bills, money exchange, etc.)

Interpreting and analysing data from tables and graphs, including distance–time and speed–time graphs

Interpreting the solution in the context of the problem

Geometry and Measurement

Angles, triangles and polygons

Right, acute, obtuse and reflex angles

Vertically opposite angles, angles on a straight line and angles at a point

Angles formed by two parallel lines and a transversal: corresponding angles, alternate angles, interior angles

Properties of triangles, special quadrilaterals and regular polygons (pentagon, hexagon, octagon and decagon), including symmetry properties

Classifying special quadrilaterals on the basis of their properties

Angle sum of interior and exterior angles of any convex polygon

Properties of perpendicular bisectors of line segments and angle bisectors

Construction of simple geometrical figures from given data (including perpendicular bisectors and angle bisectors) using compasses, ruler, set squares and protractors, where appropriate

Congruence and similarity

Congruent figures and similar figures

Properties of similar triangles and polygons:

Corresponding angles are equal

Corresponding sides are proportional

Enlargement and reduction of a plane figure

Scale drawings

Determining whether two triangles are

Congruent

Similar

Ratio of areas of similar plane figures

Ratio of volumes of similar solids

Solving simple problems involving similarity and congruence

Properties of circles

Symmetry properties of circles:

Equal chords are equidistant from the centre

The perpendicular bisector of a chord passes through the centre

Tangents from an external point are equal in length

The line joining an external point to the centre of the circle bisects the angle between the tangents

Angle properties of circles:

Angle in a semicircle is a right angle

Angle between tangent and radius of a circle is a right angle

Angle at the centre is twice the angle at the circumference

Angles in the same segment are equal

Angles in opposite segments are supplementary

Pythagoras’ theorem and trigonometry

Use of Pythagoras’ theorem

Determining whether a triangle is right-angled given the lengths of three sides

Use of trigonometric ratios (sine, cosine and tangent) of acute angles to calculate unknown sides and angles in right-angled triangles

Extending sine and cosine to obtuse angles

Use of the formula 0.5 ab sin C for the area of a triangle

Use of sine rule and cosine rule for any triangle

Problems in two and three dimensions including those involving angles of elevation and depression and bearings

Mensuration

Area of parallelogram and trapezium

Problems involving perimeter and area of composite plane figures

Volume and surface area of cube, cuboid, prism, cylinder, pyramid, cone and sphere

Conversion between cm^2 and m^2 , and between cm^3 and m^3

Problems involving volume and surface area of composite solids

Arc length, sector area and area of a segment of a circle

Use of radian measure of angle (including conversion between radians and degrees)

Coordinate geometry

Finding the gradient of a straight line given the coordinates of two points on it

Finding the length of a line segment given the coordinates of its end points

Interpreting and finding the equation of a straight line graph in the form y = mx + c

Geometric problems involving the use of coordinates

Vectors in two dimensions

Problems in real-world contexts

Statistics and probability

Data Analysis

Analysis and interpretation of:

Tables

Bar graphs

Pictograms

Line graphs

Pie charts

Dot diagrams

Histograms with equal class intervals

Stem-and-leaf diagrams

Cumulative frequency diagrams

Box-and-whisker plots

Purposes and uses, advantages and disadvantages of the different forms of statistical representations

Explaining why a given statistical diagram leads to misinterpretation of data

Mean, mode and median as measures of central tendency for a set of data

Purposes and use of mean, mode and median

Calculation of the mean for grouped data

Quartiles and percentiles

Range, interquartile range and standard deviation as measures of spread for a set of data

Calculation of the standard deviation for a set of data (grouped and ungrouped)

Using the mean and standard deviation to compare two sets of data

Probability

Probability as a measure of chance

Probability of single events (including listing all the possible outcomes in a simple chance situation to calculate the probability)

Probability of simple combined events (including using possibility diagrams and tree diagrams, where appropriate)

Addition and multiplication of probabilities (mutually exclusive events and independent events)

We have helped over 100 O Level A Math and E Math candidates achieve their dream of an A in the GCE O Levels. Join our growing Athena network.