Mathematics
SPM · 49 topics available
Form 4
Chapter 1 · Quadratic Functions and Equations in One Variable
Roots of a Quadratic Equation
Find the roots of a quadratic equation by factorisation and the formula, and use the discriminant to decide how many real roots exist.
OpenGraphs of Quadratic Functions
Explore the shape, axis of symmetry, vertex and intercepts of the parabola $y=ax^2+bx+c$, and see how the sign of $a$ affects the graph.
OpenQuadratic Functions and Equations in One Variable
Solving quadratic equations by factorisation and the quadratic formula for SPM Mathematics Form 4 — with a worked example and an interactive exercise.
OpenChapter 2 · Number Bases
Converting Between Number Bases
Convert numbers between base 2, base 5, base 8 and base 10 using place values, and change base 10 numbers back by repeated division.
OpenNumber Bases
Place value in different bases and converting between binary and base 10 for SPM Mathematics Form 4 — with a worked example and an interactive exercise.
OpenChapter 3 · Logical Reasoning
Negation and Compound Statements
Learn to negate statements and to form compound statements using 'and' and 'or', then determine their truth values in logical reasoning.
OpenStatements
Learn what makes a sentence a statement, its truth value, how to negate it, and the quantifiers all and some.
OpenArguments
Distinguish premises from conclusions, deductive from inductive arguments, and learn the syllogism and validity.
OpenChapter 4 · Operations on Sets
Intersection of Sets
Find the common elements of two sets using the intersection symbol, Venn diagrams and n(A ∩ B).
OpenUnion of Sets
Combine all elements of two sets using the union symbol, Venn diagrams and n(A ∪ B).
OpenCombined Operations on Sets
Combine complement, intersection and union, and count with n(A ∪ B) = n(A) + n(B) − n(A ∩ B).
OpenChapter 5 · Network in Graph Theory
Vertices, Edges and Weighted Graphs
Understand the basic parts of a graph — vertices, edges, degree and weights — and tell directed graphs from undirected ones in graph theory.
OpenNetwork in Graph Theory
Vertices, edges and degree, and the handshake rule (sum of degrees = 2 × edges) for SPM Mathematics Form 4 — with a worked example and an interactive exercise.
OpenChapter 6 · Linear Inequalities in Two Variables
Applications of Linear Inequalities
Turn word problems into linear inequalities in two variables, shade the feasible region and read off integer solutions.
OpenLinear Inequalities in Two Variables
Represent a linear inequality in two variables as a shaded region, using dashed or solid boundary lines.
OpenSystems of Linear Inequalities in Two Variables
Find the feasible region of a system of inequalities and check which points satisfy every constraint.
OpenChapter 7 · Graphs of Motion
Solving Problems with Graphs of Motion
Read speed from a distance-time gradient, acceleration from a speed-time gradient, and distance from the area under a speed-time graph.
OpenDistance-Time Graphs
Read distance-time graphs, find speed as the gradient, and recognise stationary and return motion.
OpenSpeed-Time Graphs
Read speed-time graphs, find acceleration as the gradient and distance as the area under the graph.
OpenChapter 8 · Measures of Dispersion for Ungrouped Data
Interpreting Measures of Dispersion
Compute range, interquartile range, variance and standard deviation for ungrouped data, and predict the effect of adding or multiplying each value.
OpenDispersion
Understand what dispersion means and read stem-and-leaf plots and dot plots to compare the spread of ungrouped data.
OpenMeasures of Dispersion for Ungrouped Data
Compute range, interquartile range, variance and standard deviation for ungrouped data using their formulas.
OpenChapter 9 · Probability of Combined Events
Combined Events
List the sample space of combined events using tables and tree diagrams, and find their probabilities.
OpenDependent and Independent Events
Distinguish independent from dependent events and apply the multiplication rule of probability.
OpenMutually Exclusive and Non-Mutually Exclusive Events
Apply the addition rule of probability and recognise when events are mutually exclusive.
OpenChapter 10 · Consumer Mathematics: Financial Management
Savings, Investment and Financial Planning
Plan savings, compare simple and compound interest, and weigh investments such as fixed deposits, shares and loans, all measured in RM.
OpenConsumer Mathematics: Financial Management
Simple and compound interest, and personal financial management for SPM Mathematics Form 4 — with a worked example and an interactive exercise.
OpenForm 5
Chapter 1 · Variation
Direct Variation
Model relationships where one quantity varies directly with another, find the constant, and use $y = kx$.
OpenInverse Variation
Model relationships where one quantity varies inversely with another using $y = \frac{k}{x}$.
OpenJoint Variation
Model quantities that vary with two or more variables at once, combining direct and inverse variation.
OpenChapter 2 · Matrices
Inverse Matrix and Simultaneous Equations
Find the inverse of a 2x2 matrix using its determinant, then use it to solve simultaneous linear equations.
OpenMatrices
A matrix is a rectangular array of numbers arranged in rows and columns. This section covers order, equal matrices and the identity matrix.
OpenBasic Operations on Matrices
Matrices can be added, subtracted, multiplied by a scalar and multiplied together. The inverse of a 2×2 matrix lets us solve simultaneous linear equations.
OpenChapter 3 · Consumer Mathematics: Insurance
Types of Insurance and Premiums
Learn life, medical, motor and fire insurance, how premiums are found, and how the average clause reduces a claim when property is under-insured.
OpenConsumer Mathematics: Insurance
How insurance works — premiums, sum insured and types of policy — for SPM Mathematics Form 5, with a worked example and an interactive exercise.
OpenChapter 4 · Consumer Mathematics: Taxation
Income Tax Computation
Compute chargeable income = gross income − reliefs, apply the tax bands, and subtract a rebate to find the tax payable.
OpenConsumer Mathematics: Taxation
Taxable income, tax reliefs and computing income tax for SPM Mathematics Form 5 — with a worked example and an interactive exercise.
OpenChapter 5 · Congruency, Enlargement and Combined Transformations
Congruency
Congruent figures have exactly the same shape and size. This section covers the tests for congruent triangles and how congruency differs from similarity.
OpenEnlargement
An enlargement changes the size of a figure using a centre and a scale factor. Lengths multiply by the scale factor and areas by its square.
OpenCombined Transformation
A combined transformation applies two or more transformations one after another. The order in which they are applied usually matters.
OpenTessellation
A tessellation covers a flat surface with shapes leaving no gaps and no overlaps. Whether a regular polygon tessellates depends on its interior angle.
OpenChapter 6 · Ratios and Graphs of Trigonometric Functions
Solving Trigonometric Equations
Solve sine, cosine and tangent equations for 0 to 360 degrees using the reference angle and the correct quadrants (CAST).
OpenValues of Sine, Cosine and Tangent for 0° ≤ θ ≤ 360°
The sign of a trigonometric ratio depends on the quadrant. A reference angle lets us find any value from the special acute angles.
OpenGraphs of Sine, Cosine and Tangent Functions
The graphs of sine, cosine and tangent are periodic. Each has a characteristic shape, period and range that we can read off directly.
OpenChapter 7 · Measures of Dispersion for Grouped Data
Comparing Grouped Data Dispersion
Find the mean, variance and standard deviation from a frequency table, and compare the spread of two data sets.
OpenDispersion for Grouped Data
When data are grouped into class intervals we use frequency tables, histograms, cumulative frequency and the ogive to display and read the distribution.
OpenMeasures of Dispersion for Grouped Data
For grouped data we estimate the mean using midpoints, compute variance and standard deviation, and read the interquartile range from an ogive.
OpenChapter 8 · Mathematical Modeling
Applying Mathematical Modeling
Follow the modeling steps — identify, form, solve, interpret, evaluate — and apply them to a real situation such as a taxi fare.
OpenMathematical Modeling
Using equations, functions and graphs to represent and solve real-world problems for SPM Mathematics Form 5 — with an interactive exercise.
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