Maths Bridging Course - Algebra, Trigonometry and Calculus.

Master the order of operations by applying brackets, powers (indexes), and left-to-right rules to evaluate expressions like 12 − 8 − 2 and 5 − 10 − 8 + 3^2.

Explore fractions from basics to operations: convert between mixed numbers and improper fractions, add and subtract with common denominators, and multiply or divide fractions with canceling techniques.

Explore rational and irrational numbers, identifying rationals as ratios of two integers and irrationals as not expressible by such ratios. Note pi and e as irrational, non-terminating decimals.

Explore strategies to simplify surds by reducing radicals in the numerator and denominator, applying the order of operations, and canceling like terms in fractions.

Students explore index laws through three examples, expand brackets, apply power rules, and combine exponents by factoring and cancelling to produce simplified expressions with positive powers.

Learn the change of base theorem to evaluate logarithms with any base, using calculator methods to convert to base 10 or natural logs, with log_4(5) approximately 1.161.

define expressions and terms in algebra, identify constants, coefficients, and variables, and illustrate evaluating expressions like six plus three x minus seven x squared with x equals two.

Identify like terms, where variables and powers match, and learn to add or subtract their coefficients; explore multiplication and division of terms and simplify expressions.

Learn to multiply and divide algebraic fractions by applying basic fraction rules, factoring quadratics, canceling common factors, and reducing to lowest terms.

Learn how to add and subtract algebraic fractions by obtaining a common denominator using the lowest common multiple, then combine into a single fraction and simplify.

Learn how to solve linear equations by isolating x, balancing left and right sides, and checking solutions via substitution, using linear expressions where the highest power of x is one.

Master the cross multiply technique to solve equations with fractions in algebra and trigonometry, turning sums into single fractions and verifying solutions.

Learn to solve quadratic equations by factoring, including difference of squares, common factor extraction, and factoring three-term expressions, then apply the zero factor theorem to find solutions and verify.

Explore solving logarithmic equations by using the log definition and ensuring the base is positive, apply logarithm laws to combine terms, solve for x, and verify results.

Recognize indicial equations and convert to a common base, apply exponent laws to equate exponents, and solve using factoring and back-substitution to find x.

Explore triangles, their vertices and opposite sides, and how largest angles correspond to largest sides. Learn that triangle angles sum to 180 degrees and classify by angles and sides.

Explore the core trig ratios for right triangles—sine, cosine, and tangent—defined as opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent, using Sokoto to memorize.

Learn to use the cosine rule to calculate a side from two sides and the included angle, and relate it to Pythagoras for right triangles.

Learn how to convert decimal degrees to degrees, minutes, and seconds and back, using the 60 minutes per degree and 60 seconds per minute, with calculator buttons DMS or deg/min/sec.

Learn to convert between radians and degrees using the formulas 180/π and π/180, with examples and key values like 30, 45, 60, 90, 180, 270, 360.

Two special triangles—30-60-90 and 45-45-90—use Pythagoras to derive sine, cosine, and tangent values in degrees and radians, and memorize their key ratios.

Explore how sine, cosine, and tangent relate to the unit circle across all quadrants, using first quadrant values to map signs in 180 minus theta and 360 minus theta.

Study phase shift by applying horizontal shifts to cosine graphs in degrees, showing how adding inside the function moves the graph left within a standard 360-degree period.

Learn how vertical shift moves the equilibrium line and how amplitude, period, phase shift, and vertical displacement transform sine and cosine graphs.

Transform sine and cosine using the standard forms a cos(bx + c) + d and a sin(bx + c) + d, mastering amplitude, period, phase shift, and vertical shift.

Solve trigonometric equations in radians within zero to two pi by using the unit circle to locate quadrants for tan’s sign, then apply inverse tangent with a calculator.

Explore the concept of functions through a practical pay example, showing inputs mapped to outputs, with domain and range and the distinction between discrete and continuous functions.

Learn to sketch linear functions by converting standard form to gradient-intercept form, identify x- and y-intercepts, plot two points, and draw the line.

Examine special linear equations, y = c and x = a, and how their graphs are parallel to the axes, with examples y = 3 and x = -2.

Derive a line's equation from a point and gradient using the point-slope form, then verify with substitution; when given two points, compute the gradient and form the equation.

Master simultaneous equations with two unknowns using substitution and elimination, demonstrated on a two-equation linear system and verified by a check.

Explore quadratic functions forming parabolas from y = x^2 + Bx + C, find zeros with the quadratic formula, and identify turning point, axis of symmetry, domain, and range.

Understand how the basic quadratic transforms with coefficient a, opening up or down and shifting vertically or horizontally via y = x^2 + k and y = (x ± h)^2.

Explore inverse functions by learning how to swap variables, apply the horizontal line test, and consider domain restrictions using templates across linear and non-linear examples.

Examine the graphs of exponential and logarithmic functions, focusing on y = e^x and y = ln x, their intercepts, and how inverses reflect across y = x.

Explore composite functions, where two mappings combine into a single function by applying the outer function after the inner, using g∘h notation and noting g∘h is not equal to h∘g.

Learn the derivative as the instantaneous rate of change with respect to x and memorize the basic derivative rules, including constants, ax, x^n, sin, cos, e^x, and ln x.

Present a shortcut for the chain rule, called the dido method, using the inner function and power rule to differentiate quickly.

Learn the product rule for differentiating the product of two functions, using u and v with dy/dx = u dv/dx + v du/dx, through practical examples.

Explains how to compute area via definite integrals, using ∫ from 0 to 2 of x^2−1 dx, applying antiderivatives, and splitting at x=1 to ensure positive area.

Compute the integral from 0 to π using sin x and cos x, substitution, and unit circle values to find the area under the curve with symmetry and absolute values.