Additional Mathematics
G3N tutors you through the full CTVET Additional Mathematics syllabus offline — from The Concept of Binary Operations, Properties of Binary Operations, Identity and Inverse Elements and more — with adaptive lessons, instant quizzes and exam-ready summaries.
Syllabus
What you’ll cover in Additional Mathematics.
The complete topic outline G3N teaches, mapped to the CTVET curriculum.
Year 1
25 topicsThe Concept of Binary Operations
- Recognise binary operations and apply the knowledge in solving related problems
- Define binary operations
- Identify elements in binary operations
- Evaluate binary operations using given rules and symbols
- Write mathematical rules from word descriptions of operations
- Solve problems involving binary operations
Properties of Binary Operations
- Demonstrate knowledge of the closure property of binary operations
- Define the closure property
- Verify whether a set is closed under a given operation using a table
- Verify closure using algebraic manipulation
- Identify operations that are closed under specific number sets
- Demonstrate knowledge of the commutative property of binary operations
- Define the commutative property
- Show whether an operation is commutative using numerical examples
- Show whether an operation is commutative using algebraic manipulation
- Identify commutative and non-commutative operations
- Demonstrate knowledge of the associative property of binary operations
- Define the associative property
- Show whether an operation is associative using specific values
- Show whether an operation is associative using variables
- Identify associative and non-associative operations
- Demonstrate knowledge of the distributive property of binary operations
- Define the distributive property
- Show whether one operation distributes over another
- Evaluate expressions involving two binary operations
- Determine distributivity using algebraic manipulation
Identity and Inverse Elements
- Determine the identity element for a given binary operation
- Define the identity/neutral element
- Check for commutativity before finding the identity element
- Find the identity element algebraically
- Identify the identity element from a binary table
- Use the identity element to find the inverse of a given element
- Define the inverse of an element under a binary operation
- Find the inverse element algebraically
- Determine specific inverse values
- Identify values for which an inverse does not exist
Sets, Properties of Sets and Operations on Sets
- Demonstrate knowledge of sets and methods of representation
- Define a set
- Represent sets in statement form (word description)
- Represent sets in tabular or roster form (listing)
- Represent sets in rule or set builder notation form
- Identify types of sets: null/empty, unit/singleton, finite, infinite
- Find the cardinality of a set
- Perform operations on three sets
- Find the intersection of three sets
- Find the union of three sets
- Find the complement of a set
- Illustrate three-set problems on Venn diagrams
- Identify and describe regions in a three-set Venn diagram
- Solve word problems involving three sets using Venn diagrams
- Establish properties of operations on sets and apply them to solve problems
- Apply the commutative property of union and intersection
- Apply the associative property of union and intersection
- Apply the distributive property of intersection over union
- Apply the distributive property of union over intersection
- Verify set algebra properties with given sets
Binomial Expansion and Pascal's Triangle
- Expand binomial expressions for positive integer indices using Pascal's triangle
- Identify binomial expressions
- Identify coefficients of terms in algebraic expressions
- Construct Pascal's triangle
- State the characteristics of Pascal's triangle
- Use Pascal's triangle to determine coefficients in binomial expansions
- Expand binomial expressions in descending powers using Pascal's triangle
- Apply binomial expansion to evaluate numerical expressions
The Combination Approach to Binomial Expansion
- Use the combination approach to determine coefficients and exponents in binomial expansions
- Define and evaluate factorial notation n!
- Define and compute combinations nCr
- State the binomial theorem for positive integer indices
- Apply the combination formula to expand binomial expressions
- Write out the first n terms of a binomial expansion
- Find a specific term in a binomial expansion
- Find the coefficient of a specific term in a binomial expansion
- Use binomial expansion to evaluate numerical expressions to given decimal places
Surds
- Investigate the properties of surds and perform basic arithmetic operations on surds
- Define surds and distinguish them from non-surds
- Identify binomial surds of the form a ± √b or √a ± √b
- Apply multiplication rules of surds: √a × √a = a and √a × √b = √(ab)
- Apply division rules of surds: √a ÷ √b = √(a/b)
- Simplify surds by extracting perfect square factors
- Add and subtract like surds by grouping like terms
- Multiply surds using the distributive property
- Identify and find conjugate surds
- Rationalise surds with binomial denominators
- Rationalise the denominator of a fraction with a monomial surd denominator
- Rationalise the denominator of a fraction with a binomial surd denominator
- Simplify expressions after rationalisation
- Find the square root of a surd expression
- Express the square root of a surd in the form √a ± √b
- Set up and solve equations to find a and b
- Verify results by squaring back
Indices
- Recollect the initial laws of indices and establish other laws for negative powers and roots
- Define base, index/exponent, and power
- Write numbers as exponents in simplest form
- Apply the multiplication rule: aᵐ × aⁿ = aᵐ⁺ⁿ
- Apply the division rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- Apply the power of a power rule: (aᵐ)ⁿ = aᵐⁿ
- Apply the power of a fraction rule: (a/b)ᵐ = aᵐ/bᵐ
- Apply the product rule: (ab)ᵐ = aᵐbᵐ
- Evaluate expressions with zero, negative, and fractional indices
- Recognise the relationship between surds and indices and apply laws of indices to simplify expressions
- Express surds in index notation and vice versa
- Simplify expressions involving both surds and indices
- Solve indicial problems that combine surd and index forms
- Pose and solve simple equations involving indices
- Solve indicial equations by equating bases
- Solve simultaneous equations involving indices
- Apply exponential growth formulas to real-world problems
Logarithms
- Establish the relationship between indices and logarithms and use the properties of logarithms to solve related problems in one base
- Define logarithm and state the relationship: N = aˣ ↔ logₐN = x
- Convert between index form and logarithm form
- Evaluate logarithms by converting to index equations
- Solve basic logarithmic equations
- Apply the 1st Law of Logarithm: logₐ(xy) = logₐx + logₐy
- Apply the 2nd Law of Logarithm: logₐ(x/y) = logₐx − logₐy
- Apply the 3rd Law of Logarithm: logₐ(xⁿ) = n logₐx
- Express a sum or difference of logarithms as a single logarithm
Sequences
- Recognise sequences and classify them as linear (arithmetic) or exponential (geometric)
- Define a sequence and identify its terms
- Distinguish between finite and infinite sequences
- Identify patterns in sequences and describe the rule
- Generate terms of a sequence from a given recursive rule
- Classify a sequence as arithmetic, geometric, or neither
- Find the nth term of linear and exponential sequences
- Define Arithmetic Progression (AP) and identify the first term and common difference
- Apply the nth term formula for AP: Uₙ = a + (n − 1)d
- Find a specific term of an AP given the first term and common difference
- Find the first term and common difference given two terms of an AP
- Insert arithmetic means between two terms
- Define Geometric Progression (GP) and identify the first term and common ratio
- Apply the nth term formula for GP: Uₙ = arⁿ⁻¹
- Find a specific term of a GP given the first term and common ratio
Functions
- Identify the difference between a relation and a function and write a function that describes a relationship between two quantities
- Define a relation and a function
- Distinguish between one-to-one and many-to-one functions
- Represent functions using mapping diagrams, tables, and equations
- Use function notation f(x) to describe a function
- Use function notation to evaluate functions for inputs in their domain and outputs in the co-domain
- Evaluate a function for given input values
- Determine the domain and co-domain of a function
- Find the range and zeros (roots) of a function
- Identify values excluded from the domain
- Establish, describe and determine bijective functions
- Define injective (one-to-one) functions
- Define surjective (onto) functions
- Define bijective functions (one-to-one and onto)
- Determine whether a given function is injective, surjective, or bijective
- Find the inverse of simple functions and solve f(x) = c
- Define the inverse of a function
- Find the inverse of a one-to-one function algebraically
- Write an expression for the inverse function f⁻¹(x)
- Solve equations of the form f(x) = c using the inverse
- Determine the composite of two given functions
- Define composite functions fg(x) and gf(x)
- Evaluate composite functions for given input values
- Find the expression for a composite function
- Distinguish fg(x) from gf(x)
Linear Functions and Equations
- Recognise and construct linear functions and draw their graphs
- Define a linear function in the form y = mx + c
- Identify the slope (gradient) and y-intercept from the equation
- Draw the graph of a linear function by hand and with technology (GeoGebra, Desmos)
- Find the x-intercept and y-intercept of a linear function
- Determine the slope between two points
- Determine the area enclosed by linear graphs
- Find the intersection point of two linear graphs
- Determine the area of a region enclosed by two or more lines
- Model and solve linear equations and simultaneous systems
- Formulate linear equations from word problems
- Solve a system of two linear equations simultaneously (substitution and elimination)
- Solve up to three systems of linear equations simultaneously by algebraic manipulation
- Interpret the graphical solution of simultaneous linear equations
- Solve linear inequalities graphically
- Solve and graph linear inequalities in one variable
- Solve and graph linear inequalities in two variables
- Solve systems of linear inequalities graphically
- Apply linear programming to optimisation problems
Quadratic (parabolic) Functions
- Recognise and construct parabolic (quadratic) functions and their graphs
- Define a quadratic function in the form y = ax² + bx + c
- Identify maximum and minimum turning points
- Draw the graph of a quadratic function by hand and with technology
- Find the x-intercepts (roots) and y-intercept of a quadratic function
- Identify the axis of symmetry and vertex of a parabola
- Use completing the square to solve quadratic equations and derive the quadratic formula
- Transform a quadratic equation into the form (x − p)² = q by completing the square
- Solve quadratic equations using completing the square
- Derive the quadratic formula from completing the square
- Solve quadratic equations using the quadratic formula
- Determine the nature of roots using the discriminant
Polynomial Functions
- Describe polynomial functions and perform basic arithmetic operations on them
- Define a polynomial function and identify degree, leading term, and leading coefficient
- Add and subtract polynomial functions
- Multiply polynomial functions
- Divide polynomial functions using long division
- Use the remainder and factor theorems to find factors and remainders
- State and apply the Remainder Theorem
- State and apply the Factor Theorem
- Find the factors of a polynomial of degree not greater than 4
- Determine whether a given expression is a factor of a polynomial
- Draw and interpret graphs of polynomial functions up to degree 3
- Sketch the graph of a polynomial function (degree ≤ 3) by hand and with technology
- Identify x-intercepts, y-intercept, and turning points from the graph
- Describe the end behaviour of polynomial functions
Rational Functions
- Recognise rational functions and determine their domain, range and zeros
- Define a rational function as a ratio of two polynomials R(x) = f(x)/g(x), g(x) ≠ 0
- Determine the domain of a rational function (exclude values where denominator = 0)
- Find the zeros (roots) of a rational function
- Determine the range of a rational function
- Carry out basic arithmetic operations on rational functions
- Add and subtract rational functions
- Multiply and divide rational functions
- Simplify rational expressions
- Apply partial fraction decomposition
- Decompose rational functions into partial fractions with linear factors
- Decompose rational functions with repeated linear factors
- Decompose rational functions with irreducible quadratic factors
- Apply partial fraction decomposition to factors with exponents
Limits and Differentiation
- Describe and interpret the meaning of limit of a function through graphical and algebraic approaches
- Define the limit of a function: lim f(x) as x→c = L
- Evaluate limits numerically by constructing tables of values near a point
- Evaluate limits graphically by reading the height of a graph near a given point
- Evaluate limits algebraically by direct substitution for polynomial and rational functions
- Evaluate limits at infinity for polynomial and rational functions
- Distinguish between determinate and indeterminate forms (0/0, ∞/∞)
- Resolve indeterminate forms by factoring, simplifying, or rationalising before substitution
- Classify left-hand and right-hand limits algebraically and determine whether a limit exists
- Define the left-hand limit: lim x→c⁻ f(x) as x approaches c from below
- Define the right-hand limit: lim x→c⁺ f(x) as x approaches c from above
- State the condition for a limit to exist: left-hand limit must equal the right-hand limit
- Determine whether a limit exists at a given point using numerical tables or graphs
- Apply properties of limits and distinguish between continuous and discontinuous functions
- Apply limit properties: scalar multiple, sum, difference, product, quotient, and power rules
- State the three conditions for continuity: f(a) defined, lim f(x) exists, lim f(x) = f(a)
- Determine whether a function is continuous or discontinuous at a given point
- Identify intervals of continuity and points of discontinuity from a graph or equation
- Use limits of a function to find its derivative from first principles
- Define the derivative: dy/dx = lim(h→0) [f(x+h) − f(x)] / h
- Apply the four steps of first principles: find f(x+h), subtract f(x), divide by h, take the limit as h→0
- Differentiate linear, quadratic, cubic, and rational functions from first principles
- Interpret the derivative as the gradient of the tangent to the curve at any point
- Differentiate polynomial functions using the power rule and analyse the behaviour of curves
- State the power rule: if y = xⁿ, then dy/dx = nxⁿ⁻¹
- Apply the power rule to differentiate polynomial functions term by term
- Apply the power rule to functions with negative and fractional indices
- State that the derivative of a constant is zero
- Determine whether a function is increasing (f'(x) > 0), decreasing (f'(x) < 0), or stationary (f'(x) = 0)
- Find stationary (turning) points by solving f'(x) = 0 and substituting back into f(x)
- Identify intervals on which a function is strictly increasing or strictly decreasing
- Use differentiation to find equations of tangents and normals to curves at a given point
- Define a tangent to a curve as the line that touches it at a point with the same gradient
- Find the gradient of a tangent by evaluating dy/dx at the given x-coordinate
- Find the equation of a tangent using y − y₁ = m(x − x₁)
- Define a normal to a curve as the line perpendicular to the tangent at the same point
- Find the gradient of the normal: m_normal = −1 / m_tangent
- Find the equation of a normal to a curve at a given point
- Find the point on a curve where the tangent is parallel or perpendicular to a given line
- Apply differentiation to find the rate of change of a function
- Interpret the derivative as the instantaneous rate of change of y with respect to x
- Calculate the average rate of change: Δy/Δx = [f(x₂) − f(x₁)] / (x₂ − x₁)
- Calculate the instantaneous rate of change by evaluating f'(x) at a specific point
- Apply differentiation to real-world problems involving velocity, area, and population growth
Trigonometric Functions and Their Applications
- Recall basic trigonometric ratios and use the knowledge to solve problems relating to triangles
- Name the six trigonometric functions: sine, cosine, tangent, cosecant, secant, cotangent
- Identify the sides of a right-angled triangle relative to a reference angle: opposite, adjacent, hypotenuse
- Apply primary ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
- Apply reciprocal ratios: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
- Find unknown sides using Pythagoras' theorem together with trigonometric ratios
- Find unknown angles using inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹)
- Determine the signs of trigonometric ratios in each quadrant using the CAST rule
- Find all six trigonometric functions of an angle given a point on its terminal side
- Use special triangles and the unit circle to determine the geometrical and functional values of trigonometric ratios including special angles
- Derive exact trig ratios for 45° from an isosceles right triangle of side 1
- Derive exact trig ratios for 30° and 60° from a bisected equilateral triangle of side 2
- State exact values of sin, cos, and tan for 30°, 45°, and 60°
- State exact values of csc, sec, and cot for 30°, 45°, and 60°
- Apply special angle values to evaluate trigonometric expressions without a calculator
- Construct a unit circle (radius = 1, centre at origin) and relate coordinates to (cos θ, sin θ)
- Use reference angles and quadrant signs to evaluate trig functions for angles beyond 90°
- Determine radian measure and apply the knowledge to solve practical arc length problems
- Define one radian as the angle subtended by an arc equal in length to the radius
- State the relationship: 360° = 2π radians and 180° = π radians
- Convert degree measures to radians by multiplying by π/180
- Convert radian measures to degrees by multiplying by 180/π
- Apply the arc length formula: l = θr where θ is in radians
- Calculate the perimeter of a sector: arc length + two radii
- Solve real-world problems involving arc length, sector perimeter, and angle in radians
- Identify the coordinates of the quadrantal angles in a unit circle and use them to find trigonometric values of quadrantal angles
- Define quadrantal angles as angles terminating on the x or y axis: 0°, 90°, 180°, 270°, 360°
- Use the unit circle to determine the coordinates (cos θ, sin θ) at each quadrantal angle
- Evaluate sin, cos, and tan at each quadrantal angle
- Evaluate csc, sec, and cot at each quadrantal angle, identifying undefined values
- Solve trigonometric problems by combining quadrant signs with quadrantal angle values
Vectors
- Recognise and explain various forms of vectors and apply the knowledge to find unit vectors
- Define vector quantities and distinguish them from scalar quantities
- Identify types of vectors: collinear, co-initial, parallel, equal, position, free, negative, and unit vectors
- Represent vectors graphically using directed line segments (e.g. AB with arrow)
- Write vectors in column/component form (x, y)
- Write vectors in algebraic form xi + yj
- Write vectors in magnitude-direction form (r, θ)
- Identify parallel vectors as scalar multiples of each other
- Calculate the magnitude of a vector using |v| = √(x² + y²)
- Perform algebraic and graphical operations (addition, subtraction, scalar multiplication) and their geometrical interpretation
- Add two vectors algebraically by summing corresponding components: a + b = (x₁+x₂, y₁+y₂)
- Subtract two vectors algebraically: a − b = (x₁−x₂, y₁−y₂)
- Multiply a vector by a scalar: ka = (kx, ky)
- Evaluate expressions combining vector addition, subtraction, and scalar multiplication
- Interpret vector addition and subtraction geometrically on a coordinate plane
- Express a free vector in terms of position vectors: BC = OC − OB
- Determine the resultant of vectors using triangle and parallelogram laws of addition
- State the triangular law of vector addition: AC = AB + BC
- Apply the triangular law to find the resultant of two vectors graphically and algebraically
- State the parallelogram law of vector addition: OC = OB + OD for co-initial vectors
- Apply the parallelogram law to find the resultant of two co-initial vectors
- Find missing coordinates of parallelogram vertices using vector relationships
- Calculate the magnitude and direction of a resultant vector
Matrices — Definition, Order and Types
- Recognise a matrix, state its order and identify types of matrices
- Define a matrix and identify its elements (entries)
- Describe a matrix by its order (m × n): number of rows and columns
- Identify and reference elements using row and column indices (e.g. a₂₃)
- Identify a square matrix (equal rows and columns)
- Identify a rectangular matrix (unequal rows and columns)
- Identify a zero (null) matrix
- Identify a diagonal matrix
- Identify an identity matrix
Matrices — Determinants
- Find the determinant of a (2 × 2) matrix
- State the determinant formula for a 2×2 matrix: det(A) = ad − bc
- Evaluate the determinant of a 2×2 matrix with numerical entries
- Evaluate the determinant of a 2×2 matrix with algebraic entries
- Solve equations where the determinant equals a given value
- Interpret a zero determinant (non-invertible matrix)
- Apply determinants to real-world contexts (area, systems of equations)
Straight Lines
- Describe the properties of lines including parallel, perpendicular and midpoints
- Define and identify types of straight lines: horizontal, vertical, slanted
- Define parallel lines and identify real-world examples (railway tracks, ruler edges)
- Define perpendicular lines and identify real-world examples (T-junctions, corners)
- State properties of a straight line: no curves, one-dimensional, shortest distance between two points
- Calculate the distance between two points using the formula d = √((x₂−x₁)² + (y₂−y₁)²)
- Work out the midpoint of a line segment given two points and find the generalisation of the midpoint
- Define the midpoint as the point that divides a line segment into two equal parts
- Apply the midpoint formula: M = (½(x₁+x₂), ½(y₁+y₂))
- Find the midpoint given the coordinates of two endpoints
- Find a missing endpoint given the midpoint and one endpoint
- Apply the knowledge of ratio to divide a line segment in a given ratio either internally or externally
- Explain internal division: placing a point between two endpoints in ratio m:n
- Apply the internal division formula: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
- Explain external division: placing a point outside the line segment in ratio m:n
- Apply the external division formula: Q = ((mx₂−nx₁)/(m−n), (my₂−ny₁)/(m−n))
- Solve problems involving both internal and external division of line segments
- Recall the formula for finding the gradient of a line and apply it to find the equation of a straight line in various forms
- Recall the gradient formula: m = (y₂−y₁)/(x₂−x₁)
- Derive the point-slope form of a line equation: y − y₁ = m(x − x₁)
- Derive the slope-intercept form: y = mx + c
- Write the equation of horizontal lines (y = c) and vertical lines (x = c)
- Find the equation of a line passing through two given points
- Express the equation of a line in the form ax + by + c = 0
- Use standard algebraic manipulations to find the equation of parallel and perpendicular lines including the perpendicular bisector
- State that parallel lines have equal gradients: m₁ = m₂
- State that perpendicular lines have gradients whose product is −1: m₁ × m₂ = −1
- Find the gradient of a line perpendicular to a given line: m₂ = −1/m₁
- Find the equation of a line parallel to a given line passing through a given point
- Find the equation of a line perpendicular to a given line passing through a given point
- Find the equation of the perpendicular bisector of a line segment
- Deduce the shortest distance between a point and a line and find the perpendicular distance from an external point to a line
- State that the shortest distance from an external point to a line is perpendicular
- Apply the perpendicular distance formula: D = |ax₁ + by₁ + c| / √(a² + b²)
- Rewrite a line equation in the form ax + by + c = 0 before applying the formula
- Solve real-world problems involving perpendicular distance from a point to a line
- Determine the acute angles between two intersecting lines with the aid of technological tools
- Find the point of intersection of two straight lines by solving simultaneous equations
- Apply the acute angle formula: tan θ = |(m₁ − m₂) / (1 + m₁m₂)|
- Calculate the acute angle between two lines given their gradients
- Use GeoGebra to construct intersecting lines and measure the angle between them
- Verify calculated angles using dynamic geometry software
Matrices — Arithmetic Operations
- Add and subtract (2 × 2) matrices
- State the condition for matrix addition/subtraction (same order)
- Add two 2×2 matrices by adding corresponding elements
- Verify that matrix addition is commutative: A + B = B + A
- Verify that matrix addition is associative: A + (B + C) = (A + B) + C
- Subtract two 2×2 matrices by subtracting corresponding elements
- Verify that matrix subtraction is not commutative: A − B ≠ B − A
- Apply matrix addition and subtraction to real-world problems
- Multiply a matrix by a scalar and multiply two (2 × 2) matrices
- Multiply a matrix by a scalar (scale factor k): kA
- Find the negative of a matrix: −A
- Evaluate expressions combining scalar multiples (e.g. 2A − 3B + 4C)
- State the rule for multiplying two 2×2 matrices (row-by-column dot product)
- Multiply two 2×2 matrices
- Verify that matrix multiplication is not commutative: AB ≠ BA in general
- Multiply a matrix by a column or row vector
- Solve for unknown elements given a matrix product equation
Statistics — Data Collection and Representation
- Identify and present appropriate ways of collecting and representing data, and describe sampling techniques
- Define data and distinguish between primary and secondary data
- Collect data through interviews, questionnaires, and observations
- Define sampling and explain why it is used
- Describe simple random sampling: each member has equal chance of selection
- Describe systematic sampling: select every kth member from an ordered list
- Describe stratified sampling: divide population into strata and sample proportionally from each
- Describe cluster sampling: randomly select entire clusters from the population
- Describe convenience, purposive, snowball, and quota sampling as non-probability methods
- Categorise data and determine which scale of measurement describes the data
- Distinguish between qualitative (categorical) and quantitative (numerical) data
- Distinguish between discrete data (countable, specific values) and continuous data (any value in a range)
- Identify the nominal scale: classifies data into categories with no ranking
- Identify the ordinal scale: classifies and ranks data but with unequal intervals
- Identify the interval scale: ordered data with equal intervals but no true zero
- Identify the ratio scale: ordered data with equal intervals and a true zero
- Classify given variables according to their level of measurement
- Give real-world examples of data at each level of measurement
- Organise data into appropriate frequency distribution tables manually and with technology
- Construct ungrouped frequency distribution tables using tally marks for raw data
- Define class interval, lower class limit, and upper class limit
- Calculate class boundaries by adjusting class limits by 0.5 (or half the gap between classes)
- Calculate class midpoints as the mean of upper and lower class limits
- Construct grouped frequency distribution tables using appropriate class intervals
- Use Microsoft Excel to organise and display frequency distribution tables
- Present categorical data using appropriate graphs and justify the choice of representation
- Construct single bar charts for categorical data
- Construct double (comparative) bar charts to compare two data sets
- Calculate sector angles using (frequency/total) × 360° and construct pie charts
- Interpret values and draw conclusions from bar charts and pie charts
- Justify the choice between a bar chart and a pie chart based on data type and purpose
- Present continuous and grouped data using appropriate graphs and justify the choice of representation
- Construct simple line graphs for time-series data and identify trends
- Construct multiple-line graphs to compare two or more data sets over time
- Construct histograms for equal-width class intervals with frequency on the vertical axis
- Calculate frequency density (frequency ÷ class width) for unequal class widths
- Construct histograms with frequency density on the vertical axis for unequal class widths
- Build cumulative frequency tables by accumulating frequencies up to each upper class boundary
- Plot cumulative frequency curves (ogives) by graphing cumulative frequency against upper class boundaries
- Read cumulative frequency curves to estimate the median, quartiles, and percentiles
Statistics — Measures of Central Tendency and Dispersion
- Calculate measures of central tendency (mode, median, mean) for ungrouped and grouped data and determine which measure is most appropriate
- Identify the mode of ungrouped data as the most frequently occurring value
- Estimate the mode of grouped data using the formula: Mode = L + [Δ₁/(Δ₁ + Δ₂)] × c
- Find the median of ungrouped data by ordering values and locating the middle value(s)
- Estimate the median of grouped data using the formula: Median = L₁ + [(N/2 − cfl)/fm] × c
- Calculate the arithmetic mean of ungrouped data: x̄ = Σx/n
- Calculate the mean from a frequency table: x̄ = Σfx/Σf
- Use the assumed mean method: Mean = A + Σd/n or Mean = A + Σfd/Σf, where d = x − A
- Estimate the mean of grouped data by replacing class values with class midpoints
- Calculate and interpret measures of dispersion (range, interquartile range, standard deviation, and variance) for raw and grouped data
- Calculate the range of ungrouped data: Range = largest value − smallest value
- Estimate the range of grouped data using class limits or midpoints
- Calculate the interquartile range: IQR = Q₃ − Q₁
- Explain why the IQR is preferred over the range when outliers are present
- Calculate standard deviation of ungrouped data: SD = √[Σ(x − x̄)²/n]
- Calculate standard deviation using the equivalent formula: SD = √[Σx²/n − (Σx/n)²]
- Calculate standard deviation using the assumed mean method: SD = √[Σd²/n − (Σd/n)²]
- Calculate standard deviation of grouped data: SD = √[Σfx²/Σf − (Σfx/Σf)²]
Combinations, Permutations and Probability
- Use the fundamental counting principle to identify and determine the number of ways an event can occur
- State the multiplication rule: if event A can occur in m ways and event B in n ways, both can occur together in m × n ways
- Apply the multiplication rule to two or more sequential independent choices
- Construct possibility tables or tree diagrams to enumerate all outcomes
- Distinguish between ordered arrangements (permutations) and unordered selections (combinations)
- Calculate permutations and solve related problems
- Define factorial notation: n! = n × (n−1) × (n−2) × … × 1, and 0! = 1
- Evaluate factorial expressions and simplify ratios of factorials
- State the permutation formula: nPr = n!/(n − r)!
- Apply nPr to count ordered arrangements of r objects chosen from n distinct objects
- Evaluate nPn = n! as the total arrangements of all n distinct objects
- Find the number of permutations when objects are identical: n!/(r₁! × r₂! × … × rk!)
- Calculate cyclic (circular) permutations of n objects: (n − 1)!
- Solve permutation problems with restrictions (e.g., specific letters must be adjacent or separated)
- Calculate combinations, establish their relationship with permutations, and solve related problems
- State the combination formula: nCr = n!/[r!(n − r)!]
- Evaluate combination expressions and verify using a calculator (nCr button)
- Apply combinations to selection problems where order does not matter
- Solve multi-group selection problems by multiplying combinations (e.g., choosing from boys and girls separately)
- State and apply the relationship between permutations and combinations: r! × nCr = nPr
- Solve equations involving nCr and nPr using their algebraic relationship
- Simplify and evaluate mixed expressions involving factorials, permutations, and combinations
- Recall basic probability terminology and identify types of events
- Define experiment, trial, outcome, sample space (S), and event (E)
- List the sample space for common experiments (tossing a coin, rolling a die, drawing a card)
- Define mutually exclusive events: A and B cannot occur simultaneously
- Define independent events: the occurrence of A does not affect the probability of B
- Identify whether pairs of events are mutually exclusive, independent, or neither
- Give real-world examples of each type of event
- Calculate and interpret the probability of events using relative frequency, theoretical probability, and subjective probability
- Define relative frequency: Relative Frequency = frequency of event / total number of trials
- Calculate relative frequency after each trial and observe how it changes
- Observe that relative frequency approaches theoretical probability as the number of trials increases
- State the theoretical probability formula: P(E) = n(E)/n(S)
- State that probability is always between 0 and 1 inclusive: 0 ≤ P(E) ≤ 1
- Apply theoretical probability to experiments with coins, dice, playing cards, and spinners
- Calculate probabilities from frequency distribution tables
- Describe subjective probability as an estimate based on personal judgement or experience
Year 2
14 topicsSets and Binomial Expansions
- Establish and apply De Morgan's Laws of set theory
- State De Morgan's Laws for union and intersection of sets
- Verify De Morgan's Laws using Venn diagrams and set elements
- Apply De Morgan's Laws to simplify set expressions with complements
- Use properties of complements (union, intersection, double complement, empty set, universal set)
- Rewrite complex set expressions using De Morgan's Laws
- Apply the Binomial Theorem to expand binomial expressions
- State the Binomial Theorem for non-negative integer powers
- Use Pascal's Triangle to find binomial coefficients
- Use the formula nCr to find binomial coefficients
- Expand binomial expressions of the form (a + b)^n
- Find specific terms in a binomial expansion
- Apply binomial expansion to approximate exponential and other values
Sequences and Inequalities
- Find the sum of sequences and determine convergence or divergence of series
- Use sigma notation to represent sums of sequences
- Find the sum of arithmetic progressions (AP) using the sum formula
- Find the sum of geometric progressions (GP) using the sum formula
- Evaluate sums using sigma notation
- Determine whether a series is convergent or divergent
- Find the sum to infinity of a convergent geometric series
- Analyse recursive sequences using the method of undetermined coefficients
- Find arithmetic and geometric means of sequences
- Define the arithmetic mean between two terms
- Insert arithmetic means between two given values
- Define the geometric mean between two terms
- Insert geometric means between two given values
- Solve quadratic inequalities and related real-life problems
- Solve linear inequalities and represent solutions on a number line
- Solve systems of linear inequalities and determine maximum/minimum values
- Solve quadratic inequalities using graphical methods
- Solve quadratic inequalities algebraically
- Solve systems of quadratic inequalities
- Apply quadratic inequalities to model and solve real-life problems
Polynomial Functions
- Find factors and zeros of polynomial functions
- Factorise quadratic expressions using various methods
- Apply the Factor Theorem to determine factors of polynomials
- Apply the Remainder Theorem to find remainders
- Use the Zero-Product Property to find roots of polynomials
- Find zeros of cubic, quartic and quintic polynomial functions
- Sketch polynomial functions with degrees higher than 2
- Identify end behaviour of polynomial functions by degree and leading coefficient
- Find x-intercepts (zeros) and y-intercepts of polynomial functions
- Determine multiplicity of zeros and their effect on the graph
- Sketch graphs of polynomial functions with degrees higher than 2
- Apply advanced theorems to analyse polynomial functions
- Apply Descartes' Rule of Signs to determine possible number of positive and negative roots
- State and apply the Fundamental Theorem of Algebra
- Apply the Complex Conjugates Theorem to identify complex roots
- Apply the Linear and Quadratic Factor Theorems to factorise polynomials completely
Circles and Loci
- Explore properties of circles and derive their equations
- Identify and name parts of a circle (radius, diameter, chord, arc, sector, segment, tangent)
- Derive the standard equation of a circle with centre (h, k) and radius r
- Derive the general equation of a circle x² + y² + 2gx + 2fy + c = 0
- Convert between standard and general forms of the circle equation
- Determine the centre and radius from a given circle equation
- Derive and apply equations of circles from given conditions
- Find the equation of a circle given its centre and radius
- Find the equation of a circle given three points on its circumference
- Find the equation of a circle given endpoints of a diameter
- Find equations of tangents and normals to a circle at a given point
- Determine the point of intersection of a line and a circle
- Deduce the relation of various loci under given conditions
- Define a locus as the path traced by a point satisfying given conditions
- Find the equation of a locus given geometric conditions
- Identify loci as circles, lines or other curves
- Solve problems involving loci in coordinate geometry
Vectors
- Perform operations on vectors including transposing and dividing in a ratio
- Transpose column vectors to row vectors and vice versa
- Add and subtract vectors using column and unit vector notation
- Divide a vector internally in a given ratio
- Find the position vector of a point dividing a line segment in a given ratio
- Use unit vectors (i, j) to express and manipulate vectors
- Find and apply the dot product of vectors and related angle calculations
- Define and compute the scalar (dot) product of two vectors
- Find the angle between two vectors using the dot product formula
- Determine if two vectors are perpendicular using the dot product
- Apply the dot product in two and three dimensions
- Apply the sine and cosine rules and find vector projections
- State and derive the sine rule for triangles
- Apply the sine rule to find unknown sides and angles
- State and derive the cosine rule for triangles
- Apply the cosine rule to find unknown sides and angles
- Find the projection of one vector onto another vector
- Solve real-life problems involving vectors, sine and cosine rules
Matrices
- Perform matrix operations and identify types of matrices
- Identify types of matrices (square, rectangular, zero, diagonal, identity, triangular)
- Perform matrix addition, subtraction and scalar multiplication
- Perform matrix multiplication including with the identity matrix
- Transpose a matrix by switching rows and columns
- Verify matrix properties (commutativity, associativity, distributivity)
- Find the determinant and inverse of matrices
- Find the determinant of a 2×2 matrix
- Find minors and cofactors of elements in a 3×3 matrix
- Find the determinant of a 3×3 matrix using cofactor expansion
- Find the inverse of a 2×2 matrix using the adjugate method
- Verify that a matrix multiplied by its inverse gives the identity matrix
- Determine when a matrix is singular (non-invertible)
- Use matrices to solve systems of linear equations and model real-life problems
- Write a system of linear equations in matrix form AX = B
- Solve systems of linear equations using matrix inverses
- Solve systems of linear equations using Cramer's Rule
- Use matrices to model and solve real-life problems in economics, science and engineering
Correlation
- Distinguish between univariate and bivariate data and understand correlation
- Define univariate data and give examples
- Define bivariate data and give examples
- Describe univariate data using measures of central tendency and dispersion
- Define correlation as the measure of relationship between two variables
- Distinguish between positive, negative and zero correlation
- Describe the strength of correlation (weak, moderate, strong, perfect)
- Construct and interpret scatter plots for bivariate data
- Construct a scatter plot for a given set of bivariate data
- Describe the direction and strength of a relationship from a scatter plot
- Draw and interpret the line of best fit on a scatter plot
- Use the line of best fit to make predictions (interpolation and extrapolation)
- Identify outliers in scatter plot data
- Calculate and interpret Spearman's rank correlation coefficient
- Define Spearman's rank correlation coefficient
- Rank data for both variables and handle tied ranks
- Apply the Spearman's rank formula to compute the correlation coefficient
- Interpret the value of Spearman's rank correlation coefficient
- Compare Spearman's rank correlation with Pearson's product-moment correlation
Indices and Logarithms
- Apply laws of indices and logarithms to solve equations
- Review and apply laws of indices (multiplication, division, power of a power)
- Review and apply laws of logarithms (product, quotient, power rules)
- Solve equations involving logarithms by expressing both sides as a single logarithm
- Solve simultaneous equations involving logarithmic and exponential functions
- Change the base of logarithms using the change of base formula
- Model real-life phenomena using exponential and logarithmic functions
- Model exponential growth using A = P(1 + r)^t
- Model exponential decay using A = P(1 − r)^t
- Apply exponential models to compound interest, population growth and radioactive decay
- Draw linear graphs for Y = ab^X by plotting log Y against X
- Reduce exponential functions of the form Y = aX^n to linear form by plotting log Y against log X
- Determine constants from linearised exponential models
Trigonometric Identities
- Derive and apply fundamental trigonometric identities
- State and apply reciprocal identities (csc, sec, cot)
- State and apply quotient identities (tan θ = sin θ/cos θ)
- Derive and apply the three Pythagorean identities
- Derive compound angle (addition and subtraction) formulas for sin, cos and tan
- Derive and apply double angle formulas for sin 2θ, cos 2θ and tan 2θ
- Simplify trigonometric expressions using identities
- Prove trigonometric identities
- Solve trigonometric equations
- Solve basic trigonometric equations for angles in specified ranges
- Solve trigonometric equations involving multiple angles
- Solve trigonometric equations using identities to simplify
- Find the general solution of trigonometric equations
- Solve real-life problems using trigonometric equations
Differentiation
- Identify and apply rules of differentiation
- Differentiate constant functions and linear functions
- Apply the power rule to differentiate polynomial functions
- Apply the product rule to differentiate products of two functions
- Apply the quotient rule to differentiate quotients of two functions
- Apply the chain rule to differentiate composite functions
- Differentiate implicit and transcendental functions
- Differentiate implicit functions using implicit differentiation
- Differentiate exponential functions (e^x and e^f(x))
- Differentiate natural logarithmic functions (ln x and log_a x)
- Differentiate trigonometric functions (sin x, cos x, tan x)
- Differentiate combinations of transcendental and polynomial functions
Integration
- Understand the concept of integration as antidifferentiation
- Define partitioning of an interval and identify subintervals
- Calculate the width of subintervals given an interval and step size
- Approximate the area under a curve by summing rectangular areas
- Understand the connection between limits, area and integration
- Define an antiderivative as the reverse process of differentiation
- Find indefinite and definite integrals and apply the Fundamental Theorem of Calculus
- Find indefinite integrals using the power rule for integration
- Find indefinite integrals of standard functions (e^x, sin x, cos x, 1/x)
- Apply integration rules (sum, constant multiple)
- Find definite integrals over a given interval
- Apply the Fundamental Theorem of Calculus to evaluate definite integrals
- Find the area enclosed between a curve and the x-axis
- Find the area enclosed between two curves
Applications of Differentiation
- Determine the nature of gradients and classify stationary points
- Interpret the first derivative as the gradient function
- Determine intervals where a function is increasing or decreasing
- Find stationary (turning) points by setting the first derivative to zero
- Use the second derivative test to classify stationary points as maximum, minimum or saddle points
- Identify points of inflection where concavity changes
- Sketch polynomial functions and solve real-life optimisation problems
- Sketch polynomial functions using derivatives, intercepts and stationary points
- Find the maximum and minimum values of functions over a closed interval
- Set up and solve real-life optimisation problems using differentiation
- Apply differentiation to solve problems involving rates of change in science and economics
- Interpret the meaning of the second derivative in context
Probability
- Apply addition and multiplication laws of probability
- State and apply the addition law for mutually exclusive events: P(A ∪ B) = P(A) + P(B)
- State and apply the addition law for non-mutually exclusive events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- State and apply the multiplication law for independent events: P(A ∩ B) = P(A) × P(B)
- State and apply the multiplication law for dependent events using conditional probability: P(A ∩ B) = P(A) × P(B|A)
- Use Venn diagrams to illustrate probability laws
- Solve probability problems using addition and multiplication laws
- Investigate the axioms of probability and solve related problems
- State the three axioms of probability
- Verify probability axioms for given sample spaces and events
- Apply the complement rule: P(A') = 1 − P(A)
- Use De Morgan's Laws in probability contexts
- Solve real-life problems using probability laws and axioms
Combinations and Permutations
- Apply fundamental counting rules to count arrangements and selections
- State and apply the fundamental counting principle (multiplication rule)
- Use tree diagrams and sample space tables to list outcomes
- Distinguish between situations where order matters (permutations) and where it does not (combinations)
- Understand and compute factorial notation n!
- Solve problems involving permutations and combinations
- Define permutations as ordered arrangements
- Apply the permutation formula nPr = n! / (n − r)!
- Solve permutation problems with restrictions and repeated elements
- Define combinations as unordered selections
- Apply the combination formula nCr = n! / ((n − r)! × r!)
- Relate nCr to binomial coefficients in Pascal's Triangle
- Solve real-life problems involving permutations and combinations
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