Mathematics Study Guides

For weekly lessons, explore our Sec 1 Maths tuition programme for teachers, fees and class details.

• Mathematics01

  Primes, factors and multiples

  Build prime factorisations, choose HCF or LCM for a situation, and use prime powers to understand squares, cubes and roots.

  5 worked examples · 15 min read

  Jump to a section

  • Factors divide; multiples are made by multiplying
  • A quick divisibility check
  • Prime factorisation and index notation
  More sections (9)

  • Why HCF uses smaller powers and LCM uses larger powers
  • Explore the prime powers
  • Choose HCF or LCM from the situation
  • Squares, cubes and the missing prime factors
  • Example: factorise 360
  • Example: HCF and LCM of 72 and 120
  • Example: equal ribbon pieces
  • Example: events repeating together
  • Example: make 540 into a square or cube
  Read guide →: Primes, factors and multiples
• Mathematics02

  Integers and rational numbers

  Use a number line to reason about signs, distinguish types of numbers, and calculate accurately with fractions, powers and roots.

  5 worked examples · 14 min read

  Jump to a section

  • Know which kind of number you are using
  • Use position to compare signed numbers
  • Add and subtract by thinking about a change
  More sections (8)

  • Explore a signed move
  • Multiply, divide and keep fractions exact
  • Brackets, powers and calculator entry
  • Example: a root sign does not always mean irrational
  • Example: compare fractions and decimals
  • Example: a temperature change
  • Example: a fraction calculation
  • Example: combine roots and signed powers
  Read guide →: Integers and rational numbers
• Mathematics03

  Approximation and estimation

  Round with purpose, distinguish decimal places from significant figures, estimate calculations and interpret rounded measurements.

  5 worked examples · 13 min read

  Jump to a section

  • An approximate answer should say how precise it is
  • Decimal places and significant figures answer different questions
  • Round to decimal places and ordinary place values
  More sections (9)

  • Count significant figures from the first non-zero digit
  • Explore the rounding boundary
  • Estimate before trusting a calculator
  • What values could have rounded to this measurement?
  • Example: a carry crosses the decimal point
  • Example: small numbers and significant zeros
  • Example: estimate a quotient
  • Example: retain accuracy until the final line
  • Example: a measurement interval and whole-number endpoints
  Read guide →: Approximation and estimation
• Mathematics04

  Basic Algebra

  Learn algebraic notation, substitution, like terms, brackets, fractional coefficients and common-factor factorisation through explained examples and an area explorer.

  5 worked examples · 16 min read

  Jump to a section

  • Algebra is a language for numbers
  • Read each symbol carefully
  • Collect like terms
  More sections (8)

  • Expand by multiplying every term
  • Explore the distributive law
  • Factorise by reversing expansion
  • Example: translate, then substitute
  • Example: keep each sign with its term
  • Example: a negative multiplier
  • Example: fractional coefficients
  • Example: take out the highest common factor
  Read guide →: Basic Algebra
• Mathematics05

  Linear equations in one unknown

  Understand why equation steps work, handle brackets and fractions accurately, and turn word problems into equations you can check.

  6 worked examples · 15 min read

  Jump to a section

  • What does it mean to solve an equation?
  • The balance principle: do the same to both sides
  • Explore one equation step at a time
  More sections (14)

  • Two operations undo two operations
  • Collect the unknown terms first
  • A negative coefficient is still a multiplier
  • Brackets and negative signs
  • Fractions: multiply every term by the LCM
  • Turn a word problem into an equation
  • When the unknown is in a denominator
  • Going further: linear inequalities
  • Example: unknowns on both sides
  • Example: a negative multiplier
  • Example: two different denominators
  • Example: two related prices
  • Example: clear a variable denominator
  • Example: an inequality with a negative coefficient
  Read guide →: Linear equations in one unknown
• Mathematics06

  Number Patterns

  Connect term numbers, tables and diagrams to rules for number patterns, including constant differences, decreasing sequences and simple square patterns.

  5 worked examples · 14 min read

  Jump to a section

  • A term has a position and a value
  • Constant differences
  • Explore the starting value and the change
  More sections (8)

  • Decreasing sequences and membership
  • Use the structure of a diagram
  • Patterns that multiply instead of add
  • Example: find a distant term
  • Example: can a number occur?
  • Example: a row of joined squares
  • Example: a pattern without a constant difference
  • Example: repeated doubling
  Read guide →: Number Patterns
• Mathematics07

  Ratio, rate and proportion

  Compare quantities in the same units, share totals, join ratios and track changes using clear bar models.

  5 worked examples · 18 min read

  Jump to a section

  • A ratio compares equal-sized parts
  • Share a total, or use a difference
  • Explore a fixed ratio
  More sections (7)

  • When a ratio changes, track what stays the same
  • A unit rate compares different units
  • Example: units and fractional ratios
  • Example: divide a total in three parts
  • Example: join two ratios using the shared quantity
  • Example: transfer between two containers
  • Example: choose the lower unit price
  Read guide →: Ratio, rate and proportion
• Mathematics08

  Percentages

  Identify the quantity that represents 100%, calculate percentage changes and reverse percentages, and combine successive changes accurately.

  7 worked examples · 16 min read

  Jump to a section

  • Start by identifying what counts as 100%
  • Compare amounts and calculate percentage change
  • Use multipliers for forward and reverse percentages
  More sections (11)

  • Successive changes act on the latest amount
  • Explore two successive changes
  • Profit, prices and other percentage contexts
  • Percentage points compare two percentage rates
  • Example: a part of a whole and a value above 100%
  • Example: the denominator depends on the comparison
  • Example: recover the price before a discount
  • Example: two changes in a bill
  • Example: profit after a discount
  • Example: simple and repeated interest
  • Example: percentage points or percentage increase?
  Read guide →: Percentages
• Mathematics09

  Rate, speed and time

  Connect distance, time and speed, convert units, combine rates and explain average speeds with journey graphs.

  6 worked examples · 20 min read

  Jump to a section

  • A rate measures change per unit
  • Average speed uses totals
  • Explore the effect of a stop
  More sections (9)

  • Combine rates, then find the time
  • Use equal distances to form an equation
  • Read a distance–time graph
  • Example: speed and flow-rate conversions
  • Example: a journey with a rest
  • Example: equal distances do not mean equal times
  • Example: two taps fill a tank
  • Example: an earlier arrival
  • Example: interpret each graph section
  Read guide →: Rate, speed and time
• Mathematics10

  Linear Graphs

  Plot ordered pairs, build straight-line graphs, interpret gradient and intercepts, and explore how changing a linear rule changes its graph.

  4 worked examples · 15 min read

  Jump to a section

  • Coordinates connect numbers to a position
  • A linear rule makes a straight line
  • From rule to graph
  More sections (7)

  • Read a value from the plotted segment
  • Explore gradient and intercept
  • Measure gradient using coordinate changes
  • Example: plot and describe a point
  • Example: make and check a table
  • Example: a falling line
  • Example: interpret a graph in context
  Read guide →: Linear Graphs
• Mathematics11

  Angles and Parallel Lines

  Recognise angle relationships, give geometric reasons and solve unknown angles with a parallel-line explorer.

  5 worked examples · 14 min read

  Jump to a section

  • Read the angle before choosing a rule
  • Three facts that do not need parallel lines
  • A transversal crosses two parallel lines
  More sections (6)

  • Explore which pairs stay equal
  • Example: distinguish the smaller and reflex angles
  • Example: turn a straight-line fact into an equation
  • Example: work around a point
  • Example: chain two angle reasons
  • Example: parallel-line angles with algebra
  Read guide →: Angles and Parallel Lines
• Mathematics12

  Triangles, Quadrilaterals and Polygons

  Understand triangle and quadrilateral properties, interior and exterior angles, symmetry and ruler-and-compass constructions.

  6 worked examples · 20 min read

  Jump to a section

  • Use triangle properties, not the appearance of the sketch
  • Classify quadrilaterals by their properties
  • Interior sums and exterior turns
  More sections (9)

  • Count the triangles inside a polygon
  • Construct accurately from given measurements
  • Bisect an angle: equal angles, not equal side lengths
  • Example: an isosceles triangle and an exterior angle
  • Example: angles in a parallelogram
  • Example: recover the number of sides
  • Example: an irregular polygon
  • Example: construct a triangle from three sides
  • Example: construct a perpendicular bisector
  Read guide →: Triangles, Quadrilaterals and Polygons
• Mathematics13

  Perimeter, Area and Volume

  Choose perpendicular heights, separate perimeter from area, and work with prisms, cylinders and composite solids.

  7 worked examples · 22 min read

  Jump to a section

  • Decide what is being measured
  • Perpendicular height gives the area
  • Split the area; trace the boundary
  More sections (12)

  • Prisms: repeat the same cross-section
  • Cylinders: circles and an unrolled rectangle
  • Build a cylinder from its net
  • Composite solids: joins disappear from the outside
  • Square and cube the conversion factor
  • Example: the sloping side has a different job
  • Example: the average of two parallel sides
  • Example: a rectangular corner is removed
  • Example: a right triangular prism
  • Example: closed cylinder or open container?
  • Example: a cylinder fixed to a cuboid
  • Example: area, volume and capacity conversions
  Read guide →: Perimeter, Area and Volume
• Mathematics14

  Data Handling and Statistics

  Collect and organise data, choose and interpret charts, recognise misleading displays and calculate simple averages.

  6 worked examples · 20 min read

  Jump to a section

  • Start with a question and reliable data
  • Choose a display that answers the question
  • Pie charts convert a fraction into an angle
  More sections (10)

  • Turn counts into a pie chart
  • Read a line graph using both axes
  • Ask whether the display supports the claim
  • Dot diagrams, grouped tables and averages
  • Example: a frequency table, bar graph and pictogram
  • Example: find an angle and recover a count
  • Example: change over time, with a limit to the conclusion
  • Example: a bar can look three times as tall
  • Example: place boundary values in the right class
  • Example: averages from a frequency table
  Read guide →: Data Handling and Statistics

For weekly lessons, explore our Sec 2 Maths tuition programme for teachers, fees and class details.

• Mathematics01

  Simultaneous Linear Equations

  Connect the intersection of two straight lines to elimination and substitution, then form equations from practical conditions and recognise all three solution cases.

  4 worked examples · 9 min read

  Jump to a section

  • One equation gives a line; two equations give a shared condition
  • Graphical method: read the intersection
  • Elimination: make one unknown cancel
  More sections (8)

  • Substitution: replace an unknown with an equal expression
  • Word problems: translate each condition separately
  • One solution, no solution or infinitely many?
  • Explore the number of solutions
  • Example: two lines, one solution
  • Example: multiply both equations
  • Example: keep the brackets
  • Example: tickets and total revenue
  Read guide →: Simultaneous Linear Equations
• Mathematics02

  Linear Inequalities

  Connect inequality rules to number-line solution sets, handle negative divisors and translate practical limits into valid whole-number answers.

  4 worked examples · 12 min read

  Jump to a section

  • An inequality describes a set of possible values
  • Read the answer on a number line
  • A negative divisor reverses the direction
  More sections (8)

  • Fractions and brackets: keep every term
  • Translate a limit into an inequality
  • Explore the effect of a negative or zero coefficient
  • Solve ax+b compared with c
  • Example: an included boundary
  • Example: a strict negative boundary
  • Example: clear fractions, then select special values
  • Example: the largest affordable whole number of hours
  Read guide →: Linear Inequalities
• Mathematics03

  Expansion and Factorisation

  Connect distribution to factorisation, check every cross-product and use common factors, two-variable quadratics and grouping with confidence.

  4 worked examples · 12 min read

  Jump to a section

  • Expansion and factorisation describe the same expression
  • Expand first, then collect like terms
  • Factorise by checking the four products
  More sections (7)

  • Grouping: create the same bracket twice
  • Explore how the middle term is made
  • A multiplication frame for (x+a)(x+b)
  • Example: four products and a subtraction
  • Example: common factor, then two brackets
  • Example: keep the second letter in every factor
  • Example: grouping and a related expression
  Read guide →: Expansion and Factorisation
• Mathematics04

  Special Algebraic Identities

  Understand the three special identities through area, expansion and factorisation, then apply them to numerical shortcuts and algebraic reasoning.

  4 worked examples · 14 min read

  Jump to a section

  • Recognise a structure before multiplying
  • Why the middle term is there
  • Opposite middle terms cancel
  More sections (6)

  • Use identities without finding each variable
  • Explore the missing rectangles
  • Example: a square with a negative middle term
  • Example: calculate efficiently without a calculator
  • Example: factorise completely
  • Example: reason from a sum and a product
  Read guide →: Special Algebraic Identities
• Mathematics05

  Quadratic Equations

  Make one side zero, use factorisation to find every root and decide which solutions make sense in a word problem.

  4 worked examples · 12 min read

  Jump to a section

  • A quadratic equation can have more than one answer
  • Keep the possibility that the unknown is zero
  • An equation may need expanding before factorising
  More sections (8)

  • A given root is a value that makes the equation true
  • Form an equation and interpret its roots
  • What a repeated root means
  • Explore roots as zero factors and x-intercepts
  • Example: do not divide away a solution
  • Example: the displayed brackets are not yet zero factors
  • Example: find a coefficient, then the other root
  • Example: a rectangular garden
  Read guide →: Quadratic Equations
• Mathematics06

  Linear and Quadratic Graphs

  Connect gradient, intercepts, quadratic tables and turning points to graphical solutions and real situations, with four worked examples and an interactive graph.

  4 worked examples · 18 min read

  Jump to a section

  • Read a graph as a relationship
  • Gradient and intercepts
  • Plot a quadratic with a smooth curve
  More sections (8)

  • Use intersections to solve equations
  • Interpret the part of the graph that makes sense
  • Explore two, one or no intersections
  • Move the line
  • Example: find a falling line
  • Example: build the table and locate the minimum
  • Example: choose the right straight line
  • Example: height, maximum and duration
  Read guide →: Linear and Quadratic Graphs
• Mathematics07

  Algebraic Fractions and Formulae

  Use factorisation to handle algebraic fractions, retain excluded inputs and rearrange formulae by collecting and isolating the required letter.

  4 worked examples · 13 min read

  Jump to a section

  • Fractions still follow the same rules
  • Simplify, multiply and divide using factors
  • Add and subtract with a common denominator
  More sections (8)

  • Fractional equations: clear denominators without losing restrictions
  • Make a different letter the subject
  • Explore why cancellation does not fill a missing value
  • Same values, with one excluded input
  • Example: keep the original restrictions
  • Example: a reversed bracket changes the sign
  • Example: a fractional equation becomes quadratic
  • Example: the new subject appears twice
  Read guide →: Algebraic Fractions and Formulae
• Mathematics08

  Direct and Inverse Proportion

  Choose a proportional model, find its constant and handle powers, roots, work rates and percentage changes through four worked examples and a graph explorer.

  4 worked examples · 16 min read

  Jump to a section

  • First identify what stays constant
  • Direct proportion and the origin
  • Compare a constant ratio with a constant product
  More sections (9)

  • Match the whole expression, including powers and roots
  • Inverse proportion in a fixed job
  • Inverse powers and transformed quantities
  • Explore the effect of changing x
  • Same starting pair, different models
  • Example: find the unit cost
  • Example: a square law and a percentage increase
  • Example: workers and the remaining work
  • Example: an inverse square with a changed input
  Read guide →: Direct and Inverse Proportion
• Mathematics09

  Map Scale

  Convert map lengths, perimeters and areas, recover an unknown scale and compare maps with four worked examples and an interactive scale drawing.

  4 worked examples · 16 min read

  Jump to a section

  • A scale compares corresponding lengths
  • Convert the scale before using it
  • Find an unknown scale
  More sections (8)

  • Area uses the square of the length scale
  • Work backwards from area to a length scale
  • Explore a rectangular region
  • Change the map scale or the map dimensions
  • Example: a road length and a reverse calculation
  • Example: turn a measured pair into 1 : n
  • Example: convert map area and actual area
  • Example: compare two maps of the same lake
  Read guide →: Map Scale
• Mathematics10

  Congruence and Similarity

  Match corresponding vertices, use equal angles and scale factors, and solve nested-triangle and algebraic-length problems with four worked examples and an explorer.

  4 worked examples · 16 min read

  Jump to a section

  • Same shape, same size, or both?
  • Name corresponding sides and angles
  • Use equal angles before doing arithmetic
  More sections (9)

  • Use one consistent scale factor
  • Turn a side ratio into an equation
  • Explore enlargement, reduction and reflection
  • Keep the correspondence as the size changes
  • Looking ahead: tests and higher-dimensional scale factors
  • Example: read a congruence statement
  • Example: congruence and a straight line
  • Example: nested similar triangles and a missing length
  • Example: similarity with algebraic lengths
  Read guide →: Congruence and Similarity
• Mathematics11

  Pythagoras’ Theorem

  Understand the relationship between the three sides of a right-angled triangle, use its converse carefully, and choose the right triangle in a longer problem.

  4 worked examples · 14 min read

  Jump to a section

  • Three sides, one right angle
  • Find the hypotenuse: add the squares
  • Find a shorter side: subtract before taking the root
  More sections (9)

  • The converse: can you prove a right angle?
  • A longer problem: choose one triangle at a time
  • Explore how the sides are connected
  • Build a right-angled triangle
  • Choose the next step
  • Example: the diagonal of a rectangle
  • Example: a cable and a vertical pole
  • Example: compare both sides of the test
  • Example: a diagonal and the area of a kite
  Read guide →: Pythagoras’ Theorem
• Mathematics12

  Trigonometric Ratios

  Label sides relative to an acute angle, choose a ratio from the available information, and connect right triangles without rounding too early.

  4 worked examples · 14 min read

  Jump to a section

  • Ratios connect an angle to side lengths
  • Label sides from the chosen angle
  • Finding a side: choose the ratio before substituting
  More sections (8)

  • Finding an angle: use the inverse function
  • Connect two right triangles through one shared length
  • Explore why angle determines a ratio
  • Change the angle and size independently
  • Example: the same triangle, two viewpoints
  • Example: vertical and horizontal components
  • Example: choose tangent when no hypotenuse is given
  • Example: two positions on level ground
  Read guide →: Trigonometric Ratios
• Mathematics13

  Mensuration: Pyramids, Cones and Spheres

  Separate perpendicular height from slant height, choose volume or surface area, and count only the exposed surfaces of a composite solid.

  4 worked examples · 14 min read

  Jump to a section

  • Decide what is being measured
  • A pyramid has a vertical height and face heights
  • A cone: distinguish its curved surface from its base
  More sections (9)

  • Spheres and hemispheres: half the volume, but which area?
  • Composite solids: join volumes, trace exposed surfaces
  • A truncated cone: use subtraction and similarity
  • Explore a joined solid’s volume and exposed area
  • Cone and hemisphere explorer
  • Example: a square-based pyramid
  • Example: a cone with its base included
  • Example: recasting a sphere as hemispheres
  • Example: a cone joined to a hemisphere
  Read guide →: Mensuration: Pyramids, Cones and Spheres
• Mathematics14

  Probability

  Build a clear sample space before counting, distinguish outcomes from event labels, and use probabilities to reason about changing groups.

  4 worked examples · 14 min read

  Jump to a section

  • Probability starts with a clear set of outcomes
  • List outcomes without omissions or repetitions
  • Combine mutually exclusive events carefully
  More sections (8)

  • Distinguish a model from results of trials
  • When the group changes, change the denominator too
  • Explore a bag’s sample space
  • Change the counts and the event
  • Example: a prime number on a fair die
  • Example: one draw from a bag
  • Example: unequal colour regions and experimental probability
  • Example: adding balls to reach a target probability
  Read guide →: Probability
• Mathematics15

  Statistical Diagrams

  Read dot diagrams, stem-and-leaf diagrams and histograms, choose useful representations and spot misleading claims.

  4 worked examples · 14 min read

  Jump to a section

  • Choose a diagram that answers the question
  • Dot diagrams: one dot for each observation
  • Stem-and-leaf diagrams: retain the actual values
  More sections (7)

  • Histograms: continuous intervals share boundaries
  • Read the scale before trusting the impression
  • Explore what grouping hides
  • Example: count carefully and interpret a condition
  • Example: build and read a stem-and-leaf diagram
  • Example: interpret grouped data without inventing exact values
  • Example: evaluate a misleading claim
  Read guide →: Statistical Diagrams
• Mathematics16

  Statistics: Mean, Median and Mode

  Connect an average to the question being asked: use totals for the mean, positions for the median and frequencies for the mode.

  4 worked examples · 14 min read

  Jump to a section

  • An average answers a particular question
  • Raw data: totals, positions and frequencies
  • Frequency tables: weight each value by its count
  More sections (9)

  • Grouped data: use midpoints and state the limitation
  • Recover totals before combining averages
  • Explore an extreme value’s effect
  • Move one observation
  • Choose a measure and explain it in context
  • Example: three averages from the same data
  • Example: goals scored in twenty matches
  • Example: an estimated mean travel time
  • Example: unequal groups and a missing observation
  Read guide →: Statistics: Mean, Median and Mode

For weekly lessons, explore our A Maths tuition programme for teachers, fees and class details.

• Additional Maths01

  Quadratic functions, equations and inequalities

  Connect the turning point, real roots and sign of a quadratic. Learn when to complete the square, when to use the discriminant, and how to state a complete solution.

  9 worked examples · 24 min read

  Jump to a section

  • One quadratic, three useful forms
  • Completing the square: find the turning point
  • The discriminant: count real roots
  More sections (13)

  • Explore: move the curve through the x-axis
  • Quadratic inequalities: find intervals, not just roots
  • Line–curve intersections: form a new quadratic
  • Equations, substitutions and modelling
  • Example: a minimum with a non-unit coefficient
  • Example: find a parameter for strict positivity
  • Example: a non-positive quadratic
  • Example: find the tangent in a family of lines
  • Example: exact roots from the quadratic formula
  • Example: a quadratic after substitution
  • Example: a linear and a nonlinear equation
  • Example: a maximum with a restricted domain
  • Example: when a parameter removes the quadratic term
  Read guide →: Quadratic functions, equations and inequalities
• Additional Maths02

  Surds, indices, exponentials and logarithms

  Simplify surds, solve exponential and logarithmic equations, sketch inverse graphs and interpret growth and decay, with exact worked methods and topic practice.

  11 worked examples · 32 min read

  Jump to a section

  • Keep exact values and valid inputs
  • Surds: simplify, rationalise and solve
  • Indices: powers, roots and equations
  More sections (16)

  • Logarithms: the index is the answer
  • Exponential equations and substitutions
  • Graphs: growth, decay and inverse pairs
  • Explore an exponential and its logarithm
  • Models and logarithmic scales
  • Example: simplifying and rationalising exactly
  • Example: a radical equation and an unwanted root
  • Example: fractional indices and a common base
  • Example: laws and change of base
  • Example: combine logarithms, then check the domain
  • Example: logarithms with different bases
  • Example: an exponential equation without a common base
  • Example: a quadratic in an exponential
  • Example: a shifted logarithm and a graphical equation
  • Example: fit a decay model and find a time
  • Example: a logarithmic scale measures ratios
  Read guide →: Surds, indices, exponentials and logarithms
• Additional Maths03

  Polynomials, cubic equations and partial fractions

  Use division, remainder and factor theorems to solve cubic equations, then decompose proper and improper fractions with all denominator restrictions.

  9 worked examples · 29 min read

  Jump to a section

  • One division identity connects the chapter
  • Division, remainders and unknown coefficients
  • Cubic factors and repeated factors
  More sections (11)

  • Explore: the remainder is a graph height
  • Partial fractions: choose the complete form
  • Example: a non-monic divisor
  • Example: use two conditions, then solve a cubic
  • Example: a quadratic divisor needs a linear remainder
  • Example: a cubic need not have three real roots
  • Example: a repeated factor needs more than one root check
  • Example: two distinct linear factors
  • Example: a repeated linear factor
  • Example: a quadratic factor that stays whole
  • Example: divide an improper fraction first
  Read guide →: Polynomials, cubic equations and partial fractions
• Additional Maths04

  Finite binomial expansions and coefficients

  Understand binomial coefficients, target terms, products, unknown powers and numerical estimates through worked examples and a coefficient explorer.

  7 worked examples · 25 min read

  Jump to a section

  • Why the binomial coefficients appear
  • Expand in the requested order
  • Find a target term without expanding everything
  More sections (10)

  • Explore coefficients, signs and term position
  • Products and unknown constants
  • Substitution, estimation and remainders
  • Example: a complete expansion with a minus sign
  • Example: a constant term and an absent term
  • Example: one middle term or two?
  • Example: coefficient of a product
  • Example: recover an unknown power and coefficient
  • Example: estimate a power and quantify the error
  • Example: expand around a divisor
  Read guide →: Finite binomial expansions and coefficients
• Additional Maths05

  Coordinate Geometry: Lines and Areas

  Connect geometric conditions to coordinate equations, then check lengths, vertices and areas.

  6 worked examples · 20 min read

  Jump to a section

  • Turn a geometrical condition into an equation
  • Perpendicular bisectors and unknown points
  • Divide a segment in a given ratio
  More sections (9)

  • Explore: the same fraction in both coordinates
  • Use properties of a shape to find a vertex
  • Areas: use a perpendicular height or the shoelace rule
  • Example: length, midpoint and a line
  • Example: equidistant from two points
  • Example: a point two fifths along a segment
  • Example: complete and verify a rectangle
  • Example: the area of a quadrilateral
  • Example: an area condition can have two answers
  Read guide →: Coordinate Geometry: Lines and Areas
• Additional Maths06

  Coordinate Geometry of Circles

  Use the distance definition of a circle to connect its equation, centre, radius, intersections and tangents.

  6 worked examples · 20 min read

  Jump to a section

  • A circle is a fixed distance from its centre
  • Build the equation from geometric information
  • Intersect a circle and a line
  More sections (8)

  • Explore: a secant becomes a tangent
  • Tangents are perpendicular to radii
  • Example: recover the centre and radius
  • Example: a diameter determines the circle
  • Example: a circle through three points
  • Example: find both intersection coordinates
  • Example: a tangent at a known point
  • Example: recover a circle from its centre and tangent
  Read guide →: Coordinate Geometry of Circles
• Additional Maths07

  Linear law

  Choose what to plot, interpret gradient and intercept, and return to the original variables. Connect algebraic, power and exponential models to straight-line graphs.

  8 worked examples · 23 min read

  Jump to a section

  • A straight line in new variables
  • Algebraic rearrangements and restrictions
  • Power and exponential models
  More sections (11)

  • Explore the meaning of a logarithmic gradient
  • From a table to a useful line of best fit
  • Solve a new equation on the same axes
  • Example: dividing by a variable
  • Example: reciprocal axes
  • Example: a power law
  • Example: exponential decay
  • Example: subtract the background before taking logs
  • Example: fit data and undo the logarithm
  • Example: recognise an anomalous reading
  • Example: translate the second condition too
  Read guide →: Linear law
• Additional Maths08

  Trigonometric Identities and R-Formula

  Build identities from the unit circle, preserve their domains and use R-formula to combine sine and cosine.

  8 worked examples · 28 min read

  Jump to a section

  • An identity is true wherever both sides are defined
  • Addition and subtraction formulae
  • Double angles: choose a form that reduces the work
  More sections (10)

  • R-formula: combine two waves of the same frequency
  • Explore: coefficients, amplitude and phase
  • Example: simplify and preserve the domain
  • Example: prove an identity from one side
  • Example: exact values without a calculator
  • Example: use quadrants before expanding
  • Example: signs of double-angle values
  • Example: a proof using double angles
  • Example: maximum on a full cycle and on a restricted interval
  • Example: a reciprocal reverses the order
  Read guide →: Trigonometric Identities and R-Formula
• Additional Maths09

  Trigonometric Functions, Graphs and Equations

  Read periodic graphs and find every permitted solution, keeping angle units, transformed intervals and excluded values precise.

  11 worked examples · 32 min read

  Jump to a section

  • The angle unit and interval are part of the question
  • Inverse functions return principal values
  • Read amplitude, period, midline and asymptotes
  More sections (15)

  • Explore: a horizontal level meets a sine graph
  • Solve within the interval, including its boundaries
  • Use algebra and identities without losing solutions
  • Build and interpret a periodic model
  • Example: principal values are not unrestricted cancellation
  • Example: sketch from the input angle
  • Example: tangent branches and a half-turn period
  • Example: negative angles and tangent's shorter period
  • Example: a multiple angle in radians
  • Example: shift the whole interval before solving
  • Example: a quadratic in cosine
  • Example: a zero factor must be retained
  • Example: a reciprocal equation with an impossible branch
  • Example: solve after using the R-formula
  • Example: form a model and interpret a threshold
  Read guide →: Trigonometric Functions, Graphs and Equations
• Additional Maths10

  Proofs in Plane Geometry

  Turn given angle and length facts into complete geometric proofs with named reasons and correct triangle correspondence.

  6 worked examples · 24 min read

  Jump to a section

  • A proof is a chain of justified statements
  • Similar triangles turn angle facts into length ratios
  • Midpoint and intercept theorems
  More sections (8)

  • Explore: equal side fractions make a parallel segment
  • Circle theorems and their converses
  • Example: use congruence to prove bisection
  • Example: prove a product using an altitude
  • Example: prove and use the midpoint conclusion
  • Example: use a converse to prove parallel lines
  • Example: transfer an angle from a tangent
  • Example: prove concyclicity before using it
  Read guide →: Proofs in Plane Geometry
• Additional Maths11

  Differentiation Techniques

  Connect tangent gradients to rates of change, choose a differentiation rule and keep every domain restriction.

  8 worked examples · 20 min read

  Jump to a section

  • A derivative is a local rate of change
  • Standard derivatives and their conditions
  • Chain rule: account for the inner change
  More sections (12)

  • Product and quotient rules
  • Logarithms: simplify only within the domain
  • Second derivatives and derivative relations
  • Explore a moving tangent
  • Example: rewrite roots and reciprocals as powers
  • Example: a square root and its endpoint
  • Example: a chain with two inner layers
  • Example: an exponential times a trigonometric function
  • Example: a quotient with a forbidden input
  • Example: use log laws without enlarging the domain
  • Example: check a relation between a function and its derivatives
  • Example: simplify first, but keep the hole
  Read guide →: Differentiation Techniques
• Additional Maths12

  Applications of Differentiation

  Turn gradients into equations of lines, classify stationary points, optimise models and connect rates of change.

  10 worked examples · 25 min read

  Jump to a section

  • Translate the question into a derivative condition
  • Tangents and normals
  • Increasing, decreasing and stationary points
  More sections (14)

  • Optimisation: build one feasible-variable model
  • Related rates: retain the time variable and the sign
  • Small increments and extreme gradients
  • Explore when turning points merge
  • Example: find both lines and check the point
  • Example: a horizontal tangent has a vertical normal
  • Example: classify two stationary points and state the intervals
  • Example: zero second derivative: two different outcomes
  • Example: maximum area with a fixed fence length
  • Example: minimum material for a closed cylinder
  • Example: a changing sphere
  • Example: a falling level with net outflow
  • Example: estimate a small change with units and an error check
  • Example: minimum gradient is not minimum height
  Read guide →: Applications of Differentiation
• Additional Maths13

  Integration and Area

  Reverse differentiation, prepare difficult integrands and distinguish signed accumulation from geometric area.

  13 worked examples · 30 min read

  Jump to a section

  • Integration reverses differentiation
  • Standard integrals and the reverse chain factor
  • Prepare the integrand before integrating
  More sections (17)

  • Recover the function from its derivatives
  • Definite integrals give signed accumulation
  • Regions bounded by a curve and straight lines
  • Explore cancellation across the x-axis
  • Example: powers, a root and a reciprocal
  • Example: a linear inner expression with a negative power
  • Example: exponential and trigonometric standard forms
  • Example: a logarithm with a negative inner coefficient
  • Example: expand a square first
  • Example: split a rational function into partial fractions
  • Example: use an identity for a squared trigonometric function
  • Example: use a given derivative to integrate a product
  • Example: one point fixes one constant
  • Example: integrate twice and use two independent conditions
  • Example: signed integral versus total area
  • Example: a line and a parabola
  • Example: horizontal strips avoid integrating a logarithm
  Read guide →: Integration and Area
• Additional Maths14

  Kinematics in a Straight Line

  Use calculus to describe straight-line motion, find reversals and calculate the whole distance travelled.

  7 worked examples · 22 min read

  Jump to a section

  • One straight line, three connected quantities
  • Read motion from the displacement function
  • Total distance: split at direction changes
  More sections (11)

  • Recover motion from acceleration or velocity
  • Trigonometric motion: keep the time interval and radians
  • Returning and approaching a limiting position
  • Explore speed, direction and distance together
  • Example: rest times and changes of direction
  • Example: distance, displacement and two different averages
  • Example: two integrations with two initial conditions
  • Example: a moment of rest without a reversal
  • Example: two reversals in a stated interval
  • Example: return to the origin is not a rest condition
  • Example: a limit need not be attained
  Read guide →: Kinematics in a Straight Line

For weekly lessons, explore our JC H2 Maths tuition programme for teachers, fees and class details.

• H2 Mathematics01

  Functions

  Work from the stated domain to inverse branches, inverse graphs and composite ranges. Follow the method through complete question-based applications.

  Study guide · 14 min read

  Jump to a section

  • What to look for in a Functions question
  • Restrict the domain, then find the inverse
  • Sketch the inverse and the identity composites
  More sections (6)

  • Check whether the composite exists, then find its range
  • Other question types
  • Before you finish
  • Which square-root branch should we use?
  • Use the inner range as the domain of the outer graph
  • Follow an input through the same composite
  Read guide →: Functions
• H2 Mathematics02

  Arithmetic and Geometric Progressions

  Use term positions, selected sums and contribution timing to set up AP/GP questions, with complete aligned working and checks.

  Study guide · 15 min read

  Jump to a section

  • What is the question asking you to find?
  • Given the sum, find the term and prove AP
  • AP terms form consecutive terms of a GP
  More sections (10)

  • Even-numbered, odd-numbered and remaining terms
  • How many terms give the required accuracy?
  • Contextual problems: deposits and interest
  • Why does January have power 12?
  • Before you finish
  • Recover the sequence from its sum
  • Use the stated term positions
  • Sum the first five odd-numbered GP terms
  • Within 0.1% of the sum to infinity
  • Compare two ways of saving over one year
  Read guide →: Arithmetic and Geometric Progressions
• H2 Mathematics03

  Inequalities

  Solve rational, modulus and hence inequalities, carrying critical points, ranges and excluded inputs through the working.

  Study guide · 7 min read

  Jump to a section

  • Put everything on one side first
  • Rational inequalities: zeros and excluded points
  • Test the sign without crossing a denominator zero
  More sections (13)

  • Remove a factor that is always positive
  • Open up modulus only after considering its sign
  • Hence questions: solve for the replacement
  • A trigonometric replacement also has a restricted range
  • When the replacement is a modulus
  • Read a graphical inequality correctly
  • Before you finish
  • A repeated factor gives an isolated solution
  • Complete the square before drawing signs
  • Compare two moduli by squaring
  • A reciprocal substitution can lose a valid input
  • Use the cosine graph on the stated interval
  • Discard a negative replacement interval
  Read guide →: Inequalities
• H2 Mathematics04

  Systems of Linear Equations

  Form and solve systems from purchases, stationary points and recurrence data; handle substitutions and exceptional parameter cases completely.

  Study guide · 7 min read

  Jump to a section

  • Define the unknowns before forming equations
  • Prices, comparisons and discounts
  • A stationary point supplies two conditions
  More sections (10)

  • Equations that become linear after substitution
  • Decide when the solution is unique
  • Choose different members of the same solution family
  • Find unknown recurrence constants from consecutive terms
  • Before you finish
  • Three pieces of information give three equations
  • Find a quadratic from a point and its minimum
  • Solve for the replacements, then return to the original unknowns
  • Keep the exceptional parameter value before dividing
  • Use the correct index in every equation
  Read guide →: Systems of Linear Equations
• H2 Mathematics05

  Graphing Techniques and Transformations

  Build complete rational and conic sketches, use them to solve range and intersection tasks, and work through transformations, derived graphs and parametric limits.

  Study guide · 10 min read

  Jump to a section

  • A sketch needs features, not just a calculator picture
  • Long division reveals the asymptotes
  • Prove which y-values are possible
  More sections (19)

  • A parameter question: do not count an excluded point
  • Complete the square and select the required half
  • Count intersections without relying on the scale of a sketch
  • For parametric asymptotes, follow both coordinates
  • Track points to keep the order of transformations clear
  • Modulus and reciprocal: change the right feature
  • A derivative graph records gradients
  • Recover a graph from its modulus and gradient signs
  • Before you finish
  • Sketch a rational curve completely
  • Use a horizontal line and the discriminant
  • Use the given asymptotes to recover coefficients
  • A conic written in expanded form
  • Changing an ellipse can change four intersections into two
  • Different parameter limits can give different vertical lines
  • An input change and an output change together
  • Three derived graphs from the same quadratic
  • Use stationary points and increasing or decreasing intervals
  • Choose the sign on each part
  Read guide →: Graphing Techniques and Transformations
• H2 Mathematics06

  Recurrence Relations

  Use data to recover a recurrence, work backwards, obtain a shifted GP, and solve limits, integer thresholds, periodic sums and nonlinear restrictions.

  Study guide · 7 min read

  Jump to a section

  • Generate terms using the stated starting index
  • Use consecutive readings to find the rule
  • Subtract the limit to obtain a GP
  More sections (12)

  • Follow the reading hour by hour
  • The multiplier decides the behaviour
  • A repeated term can start a complete cycle
  • Check that a nonlinear recurrence stays defined
  • Prove a direction of change on a stated interval
  • Before you finish
  • Find constants, then work backwards
  • Find a formula and the first reading below a threshold
  • Compare starting values in an unstable model
  • Count complete cycles and leftover terms
  • A forbidden starting value and a constant sequence
  • A local sign conclusion is not a global claim
  Read guide →: Recurrence Relations
• H2 Mathematics07

  Summation and Series

  Work through mixed sums, changed bounds, hence replacements, telescoping tails, selected terms and convergence tests with full index working.

  Study guide · 7 min read

  Jump to a section

  • Read the general term and both limits
  • Split a mixed geometric and polynomial sum
  • Change the starting point by subtracting earlier terms
  More sections (14)

  • Hence: make the term and both limits match
  • Method of differences: show what cancels
  • See the partial sum and the remaining tail
  • Select odd terms by removing the even terms
  • Convergence concerns the sum, not only its terms
  • Use an unfamiliar test exactly as supplied
  • Before you finish
  • The index in an exponent gives a GP
  • Both limits depend on n
  • Shift the index, then remove the unwanted beginning
  • A finite sum, an infinite sum and an accuracy requirement
  • Find a sum of odd cubes
  • Combine two convergent series from their correct starting terms
  • Apply a given root test
  Read guide →: Summation and Series
• H2 Mathematics08

  Binomial Series

  Expand recognisable binomial tasks: numerical estimates, products, general coefficients, finite terms and replacements containing more than one power.

  Study guide · 7 min read

  Jump to a section

  • Put the constant term into the form 1
  • Normalise, expand, and state validity
  • How far from zero is the substitution?
  More sections (10)

  • Track every contribution to the required power
  • Use partial fractions for a general coefficient
  • A finite binomial: find the required term before expanding
  • When the replacement itself contains several powers
  • Before you finish
  • A square root whose constant is not 1
  • A product of two binomial expansions
  • An alternating contribution and a repeated denominator
  • The term independent of x
  • Keep the quadratic contribution from the square
  Read guide →: Binomial Series
• H2 Mathematics09

  Maclaurin Series

  Derive and use Maclaurin expansions through differential relations, implicit equations, standard-series composition, hence operations, integrals and small-angle geometry.

  Study guide · 7 min read

  Jump to a section

  • Choose derivatives at zero or standard series
  • Use a differential relation to find coefficients
  • An implicit relation: find the starting value first
  More sections (11)

  • Expand the inner expression far enough
  • If the inner expression does not vanish at zero
  • Track degrees through multiplication and integration
  • Small-angle geometry: set up the exact length first
  • Before you finish
  • Find the first two non-zero terms
  • Keep the product rule when differentiating
  • Compose sine and logarithm, then deduce cosine
  • Factor out the constant before using the logarithmic series
  • Approximate an integral and measure the error
  • Use the sine rule before expanding
  Read guide →: Maclaurin Series
• H2 Mathematics10

  Differentiation Techniques

  Work through derivatives, implicit stationary points, logarithmic differentiation, parametric tangents, a midpoint locus and a further normal intersection.

  Study guide · 10 min read

  Jump to a section

  • Choose the rule before differentiating
  • Product, quotient and chain rules
  • Implicit and logarithmic differentiation
  More sections (12)

  • Parametric differentiation and a midpoint locus
  • Watch the tangent and its intercepts
  • A normal can meet the curve again
  • Horizontal and vertical tangents
  • Before you finish
  • Differentiate a product, then simplify
  • Differentiate a quotient containing a logarithm
  • An inverse trigonometric function is a composite
  • Find stationary points on an implicit curve
  • Both the base and the power vary
  • Find a tangent, its intercepts and the locus
  • Substitute the line back into the parametric equations
  Read guide →: Differentiation Techniques
• H2 Mathematics11

  Applications of Differentiation

  Work from derivative conditions to stationary-point tests, first-derivative graph reading, related rates and complete geometric optimisation models.

  Study guide · 10 min read

  Jump to a section

  • Turn the question into a derivative condition
  • Find the point and determine its nature
  • Read the graph of the first derivative
  More sections (11)

  • Rate of Change: WANT, GIVEN, NEED
  • Maxima and Minima: form one-variable models
  • Tangent conditions in applications
  • Before you finish
  • A parameter and a restricted domain
  • When the second derivative test is inconclusive
  • Height gives increasing; slope gives concavity
  • Water rises in a trough
  • An angle is decreasing
  • Choose a box size from a sheet
  • Optimise a trigonometric model
  Read guide →: Applications of Differentiation
• H2 Mathematics12

  Integration Techniques

  Choose and carry out reverse-chain integration, rational and quadratic methods, trigonometric identities, substitutions, integration by parts, hence integration and modulus splitting.

  Study guide · 10 min read

  Jump to a section

  • Choose the method from the integrand
  • Polynomials, exponential and logarithm: match the derivative
  • Factorisable quadratic denominators
  More sections (17)

  • Numerator not constant: split off the derivative
  • Trigonometry: rewrite before integrating
  • By substitution: change the integrand, differential and limits
  • By parts and “hence integrate”
  • Modulus: split where the expression changes sign
  • Before you finish
  • The same inside function gives three different integrals
  • Make the fraction proper, then use partial fractions
  • Long division followed by completing the square
  • A square-root quadratic denominator
  • Odd powers and even powers use different identities
  • A trigonometric substitution with exact limits
  • A substitution that removes two different roots
  • Differentiate inverse tangent; integrate the polynomial
  • Integration by parts twice returns the original integral
  • Differentiate, rearrange, hence integrate
  • Use one antiderivative on both sides of the split
  Read guide →: Integration Techniques
• H2 Mathematics13

  Applications of Integration

  Set up split and parametric areas, volumes about coordinate and shifted axes, disc–washer changes, symmetric rotations, Riemann sums and rigorous rectangle bounds.

  Study guide · 10 min read

  Jump to a section

  • Draw the strip before writing the integral
  • Area: a boundary changes inside the region
  • Parametric area: replace dx as well as y
  More sections (13)

  • Volume of revolution: find the radii
  • Cut the region where a disc becomes a washer
  • Integration as the limit of a sum
  • Before you finish
  • One region, two valid strip directions
  • A signed integral is not the total area
  • Subtract the tangent area with parameter limits
  • Rotate about the y-axis: use x as a function of y
  • Rotate about a shifted horizontal axis
  • A region crosses the axis of rotation
  • A half-turn can already fill a symmetric solid
  • Identify width, height and endpoints
  • Use upper and lower rectangles to prove a bound
  Read guide →: Applications of Integration
• H2 Mathematics14

  Differential Equations

  Form and solve exponential, input–output and nonlinear models; handle changing volume, initial conditions, equilibria, limiting values and three forms of given substitution.

  Study guide · 10 min read

  Jump to a section

  • Form the rate equation, then solve it
  • Separation of variables and initial conditions
  • Forming the differential equation: input minus output
  More sections (12)

  • Initial value and limiting value
  • A nonlinear model and its equilibrium solutions
  • Given substitutions: differentiate the substitution first
  • Interpret the solution within its model
  • Calibrate exponential growth from two observations
  • A time-dependent growth rate can give a finite limit
  • An extraction system and its steady-state
  • A changing tank volume: distinguish amount and concentration
  • An initial value above the limiting value
  • Recognise a product derivative
  • For u = y/x, remember the product rule
  • A substitution can reduce the order
  Read guide →: Differential Equations
• H2 Mathematics15

  Vectors

  Work through ratios, vector algebra, equations and relationships of lines and planes, angles, shortest distances, perpendicular feet, reflections and geometrical loci.

  Study guide · 10 min read

  Jump to a section

  • Identify points, directions and normals
  • Ratio, scalar product and vector product
  • Form equations of lines and planes
  More sections (20)

  • Intersections, angles and distances
  • Foot of perpendicular and reflection
  • Move the point and keep both reflection conditions
  • Distances between parallel planes
  • Geometrical meanings: translate the vector condition
  • Before you finish
  • Ratios lead to an intersection and an area
  • A vector equation determines a scalar parameter
  • Rearrange a vector-product identity
  • Use two directions in a plane
  • Non-parallel lines may be skew
  • Two intersecting planes and a line–plane angle
  • A line parallel to a plane may lie inside it
  • Three planes: one point, a line, or no common point
  • From a point to a line: locate the foot first
  • From a point to a plane: move along the normal
  • Use the same normal before comparing constants
  • Reflect enough points to determine the image
  • A dot-product locus is a sphere
  • Collinearity and a greatest projection
  Read guide →: Vectors
• H2 Mathematics16

  Complex Numbers

  Solve complex equations and polynomial replacements, then use modulus, argument and Argand vectors to recover exact values and square vertices.

  Study guide · 7 min read

  Jump to a section

  • Choose Cartesian form, factors or geometry
  • Solve a quadratic with complex coefficients
  • A non-real root of a real polynomial gives a factor
  More sections (15)

  • Hence roots: identify what replaces the original variable
  • Eliminate the variable appearing without a conjugate
  • If a quadratic remains, solve it as a quadratic
  • A modulus is a non-negative real number
  • Use the quadrant when finding an argument
  • Opposite vertices of a square: rotate about the midpoint
  • Before you finish
  • Find a complex square root by comparing parts
  • Find missing coefficients and all remaining roots
  • Reversed coefficients suggest a reciprocal
  • Only then write the remaining variable as x + iy
  • Use the modulus condition to choose between two assignments
  • Keep the sign condition before squaring
  • A quotient gives an exact trigonometric value
  • Find the other vertices in the stated order
  Read guide →: Complex Numbers
• H2 Mathematics17

  Permutations & combinations

  Choose and arrange under restrictions, with complete working for repeated objects, distributions, groups, blocks, gaps and rotational symmetry.

  Study guide · 10 min read

  Jump to a section

  • What is being counted?
  • Choose first, then arrange if needed
  • Repeated letters and repeated digits
  More sections (17)

  • Are the groups labelled?
  • Distribute objects: identify what can be distinguished
  • Blocks, gaps and “exactly”
  • A selection of any non-zero size
  • Around a circle: arrange the blocks, then use the gaps
  • Before you finish
  • Select four objects with a colour restriction
  • Arrange repeated letters, then form shorter words
  • A six-digit number with exactly three even digits
  • Three teams of equal size
  • Five distinct gifts, at least one each
  • “Either … or … or both”: subtract the overlap
  • Exactly three of four friends sit together
  • Separate two different pairs of repeated letters
  • Each person is chosen or not chosen
  • Four pairs and two separated hosts
  • Choose the hosts’ gaps
  Read guide →: Permutations & combinations
• H2 Mathematics18

  Probability

  Solve conditional, independent-event, region-bound, tree, counting and repeated-game probability questions with complete event setups.

  Study guide · 10 min read

  Jump to a section

  • Translate the wording into events
  • Use the overlap to recover missing probabilities
  • Independence needs a calculation
  More sections (13)

  • Find bounds by making every region non-negative
  • Multiply along paths, add the required paths
  • Conditional counting without replacement
  • A repeated game can give a geometric series
  • Before you finish
  • Two conditional probabilities and “neither”
  • Mutually exclusive and independent are different tests
  • An unknown overlap
  • Move the overlap while keeping both totals fixed
  • Reverse a conditional probability
  • When the next result depends on the previous result
  • Restrict the denominator to the given selections
  • Alternating turns until the first success
  Read guide →: Probability
• H2 Mathematics19

  Discrete random variables

  Construct probability tables from stopping rules and outcomes, recover moments, handle independent observations and evaluate net winnings.

  Study guide · 10 min read

  Jump to a section

  • Define the score before finding its distribution
  • Stop at the first success: distinguish the last two cases
  • Use total probability, mean and second moment
  More sections (12)

  • Build a distribution from equally likely outcomes
  • Independent observations: list the ordered pairs
  • Winnings and a mean over many games
  • Before you finish
  • At most three boxes are opened
  • Recover three missing probabilities
  • A squared expression is not a variance
  • The smaller of two die scores
  • An absolute difference uses both directions
  • Which ordered pairs belong in the event?
  • Condition on the sum of several scores
  • Translate the score into money before judging the game
  Read guide →: Discrete random variables
• H2 Mathematics20

  Binomial distribution

  Choose a binomial model, translate tails and conditions, solve unknown parameters and sample sizes, and count packets, inspections and modes.

  Study guide · 10 min read

  Jump to a section

  • Count successes in a fixed number of trials
  • State the assumptions in context
  • Translate the integer boundaries before calculating
  More sections (17)

  • Intersect the condition with the requested event
  • Unknown parameters: form equations from what is given
  • Count packets or count individual chocolates?
  • Split off a specified trial or inspection stage
  • Most probable means the largest probability
  • A total over many independent samples
  • Count flavoured chocolates in one packet
  • At least three; more than four but not more than eight
  • Fewer mint than plain, given at least three mint
  • Recover n and p from the mean and variance
  • An unknown probability inside a mixed pack
  • Find the least sample size
  • Ten packets: every packet versus the total
  • The eighth selection is the sixth blue
  • An inspection may require a second sample
  • Find a range of p for a unique mode
  • Watch the boundary cases produce a tie
  Read guide →: Binomial distribution
• H2 Mathematics21

  Normal distribution

  Translate totals, weighted observations, absolute differences, means and counts into the correct normal model, with complete probability calculations.

  Study guide · 10 min read

  Jump to a section

  • Define the quantity whose probability is required
  • Use a percentile to find an unknown parameter
  • Different observations are not multiples of one observation
  More sections (15)

  • “Differs by” means an absolute difference
  • A sample mean has a smaller variance
  • Heaviest and lightest: translate into “all”
  • A normal probability becomes a binomial success probability
  • Percentage changes and price conversions
  • Before you finish
  • Recover both mean and standard deviation
  • Use symmetry before standardising
  • A combined packet mass
  • Two visits compared with one visit
  • Move the “differs by at least” threshold
  • Choose enough observations for a one-sided requirement
  • The heaviest of eight components
  • Count acceptable bolt–hole pairs
  • Wrapping increases every fruit’s mass by eight percent
  Read guide →: Normal distribution
• H2 Mathematics22

  Sampling & Central Limit Theorem

  Work with sampling distributions, CLT probabilities, unbiased estimates, coded and frequency data, combined samples and required sample sizes.

  Study guide · 10 min read

  Jump to a section

  • Is the question about one observation, a total or a mean?
  • The normal population case is exact
  • Use the CLT when individual observations are not normal
  More sections (14)

  • The same method applies to money
  • Unbiased estimates from coded data
  • A frequency is a count of observations
  • Combine raw totals, not sample variances
  • Choose n for a required precision
  • Before you finish
  • Four independent packet masses
  • Average heads over fifty games
  • A mean net gain in repeated games
  • Data are given as deviations from fifty
  • Use all ten observations, not five row labels
  • Two groups with different means
  • Within two seconds with probability at least 0.95
  • How n changes the same two-second requirement
  Read guide →: Sampling & Central Limit Theorem
• H2 Mathematics23

  Hypothesis testing

  Carry out and interpret one- and two-sided mean tests, estimate variance, find critical regions and solve reverse-test inequalities.

  Study guide · 10 min read

  Jump to a section

  • Choose the alternative from the claim
  • Test a directional claim
  • A two-sided test counts equally extreme results in both directions
  More sections (14)

  • Explain p-value and significance level in context
  • Unknown variance: estimate it before testing
  • Find a critical region in sample-mean units
  • Work backwards from rejection or non-rejection
  • Recover the significance level from a rejection rule
  • Before you finish
  • Has a new method reduced the filling time?
  • Has the mean changed?
  • The same sample, two different pre-specified questions
  • Coded sample summaries
  • Which sample means reject a claimed minimum?
  • Find the possible population standard deviations
  • Find null means that are not rejected
  • A fixed cutoff and a different null mean
  Read guide →: Hypothesis testing
• H2 Mathematics24

  Correlation & regression

  Choose and fit regression models, use the mean point and residuals, transform variables, assess predictions and apply a fitted model.

  Study guide · 10 min read

  Jump to a section

  • Read the scatter diagram before trusting r
  • A regression line passes through the mean point
  • Fit the response on the predictor
  More sections (12)

  • Residuals explain what least squares minimises
  • A curved relationship needs the right transformed predictor
  • Use the fitted model for a further calculation
  • Before you finish
  • Find a missing assessment score
  • Estimate in each direction
  • Calculate the minimum sum of squared residuals
  • Change the slope while retaining the mean point
  • Growth that slows down
  • Changing units in a transformed model
  • Predict renovation time, then minimise estimated wages
  • An exponential model transforms the response
  Read guide →: Correlation & regression