Chapter 3 · Functions
Chapter 3 · Functions
From mappings and graphs to justified choices
MATHEMATICS · EXPERIMENT · EXPLAIN
One mathematics learning space
School lessons, inquiry laboratories, Python workshops and research models in one workspace. Open an activity here, keep its original records and carry your reasoning between lessons.
No sign-in. Records stay in this browser. Export a backup before clearing browser data.
My prediction, evidence and explanation
These notes belong to this activity. They complement its original notebook and do not change question results.
YOUR NEXT TASK
Choose by level, topic or purpose
Each card opens the working activity inside this workspace. Original pages and saved question IDs remain available.
442 activities and resources
Chapter 3 · Functions
Recognising functions through mappings
Check that every allowed input has exactly one output; distinguish many-to-one from one-to-many.
Chapter 3 · Functions
Reading function notation and the three elements
Read f(input) as an output, substitute the complete input, and identify domain, rule and range.
Chapter 3 · Functions
Writing allowed inputs with intervals
Translate real-input restrictions into inequalities, intervals and open or closed endpoints.
Chapter 3 · Functions
Combining domain restrictions and finding ranges
Intersect simultaneous input restrictions, preserve excluded inputs after cancellation, and find attained outputs on a specified domain.
Chapter 3 · Functions
Moving between tables, graphs and formulas
Connect inputs and outputs across representations while preserving the domain and stating any linear-model assumption.
Chapter 3 · Functions
Piecewise functions and boundary points
Choose the branch containing an input and check coverage and output consistency at boundaries.
Chapter 3 · Functions
Composition as two successive steps
Track the inner and outer steps, compare their order, and check whether the intermediate output is allowed.
Chapter 3 · Functions
Reading increase, decrease and extrema from graphs
Describe increasing and decreasing intervals, distinguish input locations from extreme values, and check whether a bound is attained.
Chapter 3 · Functions
Explaining monotonicity with its definition
Use arbitrary ordered inputs to prove monotonicity on a stated interval, and use one counterexample to refute an overbroad claim.
Chapter 3 · Functions
Understanding odd and even functions through symmetry
Check domain symmetry before comparing f(−x) with f(x) and −f(x); include functions that are both or neither.
Chapter 3 · Functions
Recognising five basic power functions
Compare x, x², x³, √x and 1/x by their real domains, ranges, symmetry and monotonic intervals.
Chapter 3 · Functions
Using power-function properties to compare values
Choose a valid monotonic interval for equal-exponent comparisons and prove square–cube comparisons by factoring their difference.
Chapter 3 · Functions
Composition rules and simple substitution
Substitute an entire expression into a rule, recover a rule by a reversible change of variable, and compare the order of two discounts.
Chapter 3 · Functions
Using functions to make justified choices
Model and compare two costs, locate every valid crossover, and make a recommendation consistent with quantities, units and budget.
Sets
An element or a subset?
Does changing 3 into {3} change the question?
Sets
Containment, equality and the empty set
When is a subset proper?
Sets
Operations select different elements
Can union, intersection and difference use the same rule?
Sets
Can these class counts coexist?
How many pupils choose neither option?
Mathematical logic
A counterexample to an implication
When does p⇒q fail?
Mathematical logic
Necessary or sufficient?
Which interval is contained in which?
Mathematical logic
All or at least one?
Can a statement be existentially true but universally false?
Mathematical logic
Negating a quantified statement
Does negating “all are greater” mean “all are smaller”?
Quadratic inequalities
Does the middle always solve it?
What changes when the leading coefficient changes sign?
Quadratic inequalities
How many real zeros?
Can a quadratic be positive without crossing the axis?
Quadratic inequalities
Do the endpoints belong?
What is the difference between < and ≤ at a root?
Quadratic inequalities
A profitable value may be impossible
Which non-loss solutions are feasible in the context?
Polynomial division and remainders
What does division preserve?
How can the quotient and remainder reconstruct f?
Polynomial division and remainders
Where did the missing terms go?
Can missing coefficients be omitted in synthetic division?
Polynomial division and remainders
Read a remainder from one point
Why is the remainder f(a)?
Polynomial division and remainders
Design a factor by changing one coefficient
What constant makes x−a a factor?
Functions and their properties
Repair one mapping arrow
Can two inputs share an output? Which single input breaks the function rule?
Functions and their properties
Same formula: the same function?
Does x = 0.5 distinguish a finite domain from the interval [−2,2]?
Functions and their properties
Which endpoint belongs?
Why is 3 allowed in √(x−3), while 2 is forbidden in 1/(x−2)?
Functions and their properties
Keep every restriction after simplification
Does cancelling x−1 restore the missing input x=1?
Functions and their properties
Should the points be joined?
The same cost formula y=2x+4 describes time or item count. Which graph permits x=2.5?
Functions and their properties
A missing boundary or two outputs?
If both pieces include the boundary, is the relation always invalid?
Functions and their properties
Swap two function machines
Does doubling then adding 3 give the same result as adding 3 then doubling?
Functions and their properties
Can the largest value be reached?
Does y=x on (0,2) have a maximum just because the graph approaches 2?
Functions and their properties
Decreasing on each piece or the whole domain?
Is 1/x decreasing on all nonzero real numbers?
Functions and their properties
Symmetry needs the matching input
Is x² still even when its domain is restricted to [0,3]?
Functions and their properties
Five powers, different forbidden inputs
Which of x, x², x³, √x and 1/x are defined at −1 and at 0?
Functions and their properties
When does the larger exponent give less?
For positive x, does x³ always exceed x²?
Functions and their properties
Undo the input rule in the right order
If f(2x+1)=6x+5, how do you obtain f(u) rather than f((u−1)/2)?
Functions and their properties
Cheapest at this quantity or most within budget?
Can the best plan change when the decision question changes?
Functions and their properties
Squaring can create a false root
Why does squaring √(x+2)=x produce two candidates but only one solution?
Exponential and logarithmic functions
A negative exponent is a reciprocal
Does 2⁻³ mean a negative number or a reciprocal?
Exponential and logarithmic functions
A fractional exponent names a root
What does 16^(3/4) ask you to do?
Exponential and logarithmic functions
Equal additions or equal ratios?
Does adding 20 each round match growing by 20% each round?
Exponential and logarithmic functions
Crossing the base 1 reverses growth
What changes in aˣ when a passes through 1?
Exponential and logarithmic functions
A logarithm asks for an exponent
What number does log₂8 ask for, and why is log₂0 impossible?
Exponential and logarithmic functions
Swap coordinates for inverse graphs
If (2,4) lies on y=2ˣ, which point lies on y=log₂x?
Exponential and logarithmic functions
Products become sums of logarithms
Is logₐ(u+v)=logₐu+logₐv?
Exponential and logarithmic functions
The threshold between two integer rounds
When does 100rⁿ first reach a target, and why might rounding a logarithm fail?
Exponential and logarithmic functions
A decreasing base reverses the inequality
Does logₐx<logₐb always imply x<b?
Trigonometric functions
What does one radian measure?
Does doubling a circle radius double its angle or its arc?
Trigonometric functions
One circle, two coordinates
Which coordinate is sine, and which signs depend on the quadrant?
Trigonometric functions
Reflect an angle without recalculating
Why do θ and180°−θ share a sine but not always a cosine?
Trigonometric functions
A square loses a sign
Does sin²θ+cos²θ=1 determine the sign of cosine?
Trigonometric functions
Unroll the circle into a wave
How does a moving circle point produce a sine graph?
Trigonometric functions
Where does tangent break?
Can a tangent graph pass through90°?
Trigonometric functions
Can sine distribute over addition?
Is sin(α+β)=sinα+sinβ?
Trigonometric functions
Double an angle, not its sine
How does sin2θ relate to sinθ?
Trigonometric functions
Amplitude and a reflected wave
Does a negative coefficient mean a negative amplitude?
Trigonometric functions
Frequency squeezes the horizontal axis
Why is the period2π/|ω| rather than2πω?
Trigonometric functions
Phase is divided by frequency
Does adding φ shift every sine graph by−φ?
Trigonometric functions
A wheel needs a starting position
Can amplitude and period alone determine a height model?
Plane vectors
Move the arrow without changing the vector
Must equal vectors start at the same point?
Plane vectors
A detour and its final displacement
Does changing the order of two displacements change the destination?
Plane vectors
A negative scalar reverses direction
What does multiplication by−2 change?
Plane vectors
Two directions need to be independent
Can any pair of vectors locate every point uniquely?
Plane vectors
What does the sign of a dot product tell us?
Does a dot product of0 always mean a right angle?
Plane vectors
Translate a midpoint and a distance
What changes when both endpoints receive the same translation?
Solving triangles
A triangle and three equal ratios
If one side changes scale, what happens to a/sinA?
Solving triangles
Pythagoras with a correction term
What changes when the included angle is not90°?
Solving triangles
The same SSA data can make two triangles
How can two sides and a nonincluded angle give0,1 or2 solutions?
Solving triangles
The same two sides, different areas
Which included angle gives the greatest area for fixed sides?
Solving triangles
Measure a distance without crossing the river
What can a baseline and two sighting angles determine?
Solid geometry
Classify by structure, not by viewpoint
Does rotating a solid change its type?
Solid geometry
A drawing does not preserve every length
Can equal perpendicular sides look unequal in an oblique drawing?
Solid geometry
A cone height is not its slant height
Which length belongs in the lateral-area formula?
Solid geometry
The same base and vertical height
Which height determines a prism or cone volume?
Solid geometry
A frustum is a difference of cones
Can the volume use the average of the two end areas?
Solid geometry
Double a sphere: four or eight times?
Do surface area and volume scale at the same rate?
Variation
Which power controls the multiplier?
Does doubling x always double y?
Variation
A fixed product instead of a fixed ratio
What remains fixed in inverse variation?
Variation
Change one factor or both?
In joint variation, do two doublings still give a doubling?
Variation
The fixed part changes the ratio
Does a straight graph always mean direct proportion?
Polynomial factors and multiples
Which factor copies are shared?
Why does a greatest common divisor use the lower exponent?
Polynomial factors and multiples
The smallest shared multiple
Why is multiplying A and B usually excessive?
Polynomial factors and multiples
Why do remainders preserve the GCD?
Can replacing A by a remainder lose a common factor?
Partial fractions
Two denominators, one identity
How can substituting a forbidden value still find a coefficient?
Partial fractions
Why include every repeated power?
Can one term represent every numerator over (x−a)²?
Partial fractions
Divide before decomposing
Where does the polynomial part come from?
Partial fractions
Does cancellation repair a hole?
Are equal values on a shared domain the same as equal domains?
Parallelism and perpendicularity in space
Not intersecting does not imply parallel
Can two space lines neither intersect nor run parallel?
Parallelism and perpendicularity in space
A parallel line must lie outside the plane
Can the outside-plane condition be omitted?
Parallelism and perpendicularity in space
Two test lines must intersect
Do two parallel test lines prove that two planes are parallel?
Parallelism and perpendicularity in space
Perpendicular to two parallel lines is not enough
Why must the two lines in the perpendicularity test intersect?
Parallelism and perpendicularity in space
Perpendicular planes do not make every line normal
If two planes are perpendicular, is every line of one perpendicular to the other?
Geometry · congruent triangles
Properties of an angle bisector
A 15–20 minute classroom inquiry: predict, measure perpendicular distances, explain and test the converse.
School curriculum
Rational numbers
Chapter concepts, worked examples and tiered practice.
School curriculum
Operations with rational numbers
Chapter concepts, worked examples and tiered practice.
School curriculum
Algebraic expressions
Chapter concepts, worked examples and tiered practice.
School curriculum
Adding and subtracting polynomials
Chapter concepts, worked examples and tiered practice.
School curriculum
Linear equations
Chapter concepts, worked examples and tiered practice.
School curriculum
Geometric figures and angles
Chapter concepts, worked examples and tiered practice.
School curriculum
Intersecting and parallel lines
Chapter concepts, worked examples and tiered practice.
School curriculum
Real numbers
Chapter concepts, worked examples and tiered practice.
School curriculum
Cartesian coordinates
Chapter concepts, worked examples and tiered practice.
School curriculum
Systems of linear equations
Chapter concepts, worked examples and tiered practice.
School curriculum
Inequalities and systems of inequalities
Chapter concepts, worked examples and tiered practice.
School curriculum
Collecting and describing data
Chapter concepts, worked examples and tiered practice.
School curriculum
Triangles
Chapter concepts, worked examples and tiered practice.
School curriculum
Congruent triangles
Chapter concepts, worked examples and tiered practice.
School curriculum
Reflection symmetry
Chapter concepts, worked examples and tiered practice.
School curriculum
Multiplying polynomials
Chapter concepts, worked examples and tiered practice.
School curriculum
Factorisation
Chapter concepts, worked examples and tiered practice.
School curriculum
Algebraic fractions
Chapter concepts, worked examples and tiered practice.
School curriculum
Square-root expressions
Chapter concepts, worked examples and tiered practice.
School curriculum
Pythagorean theorem
Chapter concepts, worked examples and tiered practice.
School curriculum
Quadrilaterals
Chapter concepts, worked examples and tiered practice.
School curriculum
Functions
Chapter concepts, worked examples and tiered practice.
School curriculum
Linear functions
Chapter concepts, worked examples and tiered practice.
School curriculum
Analysing data
Chapter concepts, worked examples and tiered practice.
School curriculum
Sets
Chapter concepts, worked examples and tiered practice.
School curriculum
Mathematical logic
Chapter concepts, worked examples and tiered practice.
School curriculum
Quadratic inequalities
Chapter concepts, worked examples and tiered practice.
School curriculum
Polynomial division and remainders
Chapter concepts, worked examples and tiered practice.
School curriculum
Functions and their properties
Chapter concepts, worked examples and tiered practice.
School curriculum
Exponential and logarithmic functions
Chapter concepts, worked examples and tiered practice.
School curriculum
Trigonometric functions
Chapter concepts, worked examples and tiered practice.
School curriculum
Plane vectors
Chapter concepts, worked examples and tiered practice.
School curriculum
Solving triangles
Chapter concepts, worked examples and tiered practice.
School curriculum
Solid geometry
Chapter concepts, worked examples and tiered practice.
School curriculum
Direct and inverse variation
Chapter concepts, worked examples and tiered practice.
School curriculum
Polynomial factors and multiples
Chapter concepts, worked examples and tiered practice.
School curriculum
Partial fractions
Chapter concepts, worked examples and tiered practice.
School curriculum
Parallel and perpendicular lines in space
Chapter concepts, worked examples and tiered practice.
School inquiry
A meeting point on a number line
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Powers and order of operations
Predict, change one condition, save evidence and revise your explanation.
School inquiry
A growing matchstick pattern
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Explain a number trick
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Which transformations preserve solutions?
Predict, change one condition, save evidence and revise your explanation.
School inquiry
What changes an angle?
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Angles and parallel conditions
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Bounds for a square root
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Coordinates, displacement and route length
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Two kinds of tickets
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Compare pricing plans
Predict, change one condition, save evidence and revise your explanation.
School inquiry
What can a sample tell us?
Predict, change one condition, save evidence and revise your explanation.
School inquiry
A line and two parameters
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Investigate the Pythagorean theorem
Predict, change one condition, save evidence and revise your explanation.
School inquiry
When can three lengths form a triangle?
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Area explains algebra
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Mean, median and an outlier
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Growth and decay
Predict, change one condition, save evidence and revise your explanation.
School inquiry
The unit circle
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Sets and overlapping groups
Predict, change one condition, save evidence and revise your explanation.
School inquiry
Signs of a quadratic
Predict, change one condition, save evidence and revise your explanation.
Textbook lessons and reviews
Concept of a set
Build understanding of concept of a set through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basic relations between sets
Build understanding of basic relations between sets through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basic operations on sets
Build understanding of basic operations on sets through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Sufficient and necessary conditions
Build understanding of sufficient and necessary conditions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Universal and existential quantifiers
Build understanding of universal and existential quantifiers through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Properties of equalities and inequalities
Build understanding of properties of equalities and inequalities through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basic inequality: AM–GM
Build understanding of basic inequality: am–gm through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Quadratic functions, equations and inequalities
Build understanding of quadratic functions, equations and inequalities through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Function concept and representations
Build understanding of function concept and representations through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basic properties of functions
Build understanding of basic properties of functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Power functions
Build understanding of power functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Applications of functions I
Build understanding of applications of functions i through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Exponents
Build understanding of exponents through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Exponential functions
Build understanding of exponential functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Logarithms
Build understanding of logarithms through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Logarithmic functions
Build understanding of logarithmic functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Applications of functions II
Build understanding of applications of functions ii through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
General angles and radian measure
Build understanding of general angles and radian measure through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Concept of trigonometric functions
Build understanding of concept of trigonometric functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Reduction formulas
Build understanding of reduction formulas through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Graphs and properties of trigonometric functions
Build understanding of graphs and properties of trigonometric functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Trigonometric identities and transformations
Build understanding of trigonometric identities and transformations through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
The function y=A sin(ωx+φ)
Build understanding of the function y=a sin(ωx+φ) through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Applications of trigonometric functions
Build understanding of applications of trigonometric functions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Concept of plane vectors
Build understanding of concept of plane vectors through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Operations on plane vectors
Build understanding of operations on plane vectors through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basis theorem and coordinate representation
Build understanding of basis theorem and coordinate representation through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Applications of plane vectors
Build understanding of applications of plane vectors through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Concept of complex numbers
Build understanding of concept of complex numbers through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Arithmetic of complex numbers
Build understanding of arithmetic of complex numbers through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Trigonometric representation of complex numbers (starred)
Build understanding of trigonometric representation of complex numbers (starred) through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Basic solid figures
Build understanding of basic solid figures through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Oblique drawings of solids
Build understanding of oblique drawings of solids through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Surface areas and volumes of simple solids
Build understanding of surface areas and volumes of simple solids through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Positions of points, lines and planes in space
Build understanding of positions of points, lines and planes in space through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Parallel lines and planes in space
Build understanding of parallel lines and planes in space through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Perpendicular lines and planes in space
Build understanding of perpendicular lines and planes in space through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Random sampling
Build understanding of random sampling through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Estimating a population from a sample
Build understanding of estimating a population from a sample through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Statistical case study
Build understanding of statistical case study through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Random events and probability
Build understanding of random events and probability through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Independence of events
Build understanding of independence of events through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Frequency and probability
Build understanding of frequency and probability through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Space vectors and their operations
Add, scale and multiply space vectors; distinguish scalar and vector results.
Textbook lessons and reviews
The fundamental theorem of space vectors
Recognise a basis and use unique vector coordinates and affine combinations.
Textbook lessons and reviews
Coordinate representation of space vectors
Calculate spatial lengths, angles, projections and perpendicular directions from coordinates.
Textbook lessons and reviews
Applications of space vectors
Use direction and normal vectors to establish angles, distances and spatial relations.
Textbook lessons and reviews
Inclination and slope of a line
Connect slope, inclination and parallel/perpendicular directions, including vertical lines.
Textbook lessons and reviews
Equations of a line
Choose point-slope, two-point, intercept or general form without losing exceptional lines.
Textbook lessons and reviews
Intersections and distance formulas
Solve intersections and calculate point-line and parallel-line distances with geometric checks.
Textbook lessons and reviews
Equations of a circle
Construct circle equations from centers, radii, diameters and point constraints.
Textbook lessons and reviews
Relations between lines and circles
Classify intersection and tangency using distances, discriminants and common chords.
Textbook lessons and reviews
Ellipse
Use the focal definition, standard equation and geometric properties of an ellipse.
Textbook lessons and reviews
Hyperbola
Use the distance-difference definition, asymptotes and branch restrictions.
Textbook lessons and reviews
Parabola
Relate focus, directrix and vertex, and retain conditions in line-parabola intersections.
Textbook lessons and reviews
Concept of a sequence
Connect general terms, recurrence relations and partial sums.
Textbook lessons and reviews
Arithmetic sequences
Derive terms and sums and solve indexed arithmetic models.
Textbook lessons and reviews
Geometric sequences
Connect ratios, indexed powers and finite geometric sums.
Textbook lessons and reviews
Mathematical induction
Build proofs with a verified base, an explicit induction hypothesis and a valid step.
Textbook lessons and reviews
Derivative: concept and meaning
Distinguish average and instantaneous rates and derive tangents from a limit.
Textbook lessons and reviews
Derivative operations
Apply power, product, quotient and chain rules with elementary derivatives.
Textbook lessons and reviews
Applications of derivatives
Use derivative signs, boundary values and models to justify extrema and optimization.
Textbook lessons and reviews
Addition and multiplication counting principles
Count disjoint alternatives and sequential choices without overlap or omission.
Textbook lessons and reviews
Permutations and combinations
Distinguish ordered selections, unordered sets and symmetry in arrangements.
Textbook lessons and reviews
Binomial theorem
Use the general term to locate coefficients, constants and coefficient sums.
Textbook lessons and reviews
Conditional and total probability
Use conditional denominators, disjoint partitions and Bayes inversion.
Textbook lessons and reviews
Discrete random variables and distributions
Construct a distribution and compute event probabilities from its support.
Textbook lessons and reviews
Numerical characteristics of random variables
Compute expectation and variance and distinguish transformation from random independence.
Textbook lessons and reviews
Binomial and hypergeometric distributions
Choose the correct model for repeated independent trials or sampling without replacement.
Textbook lessons and reviews
Normal distribution
Standardise a normal variable and interpret supplied table probabilities and symmetry.
Textbook lessons and reviews
Correlation of paired data
Interpret scatter patterns and compute a correlation from centered paired summaries.
Textbook lessons and reviews
Linear regression and applications
Fit a least-squares line and distinguish prediction, residual and extrapolation.
Textbook lessons and reviews
Contingency tables and independence tests
Compute expected counts and a chi-square statistic and state a bounded conclusion.
Textbook lessons and reviews
Polynomial division and partial fractions
Divide polynomials and choose decomposition numerators matched to each denominator factor.
Textbook lessons and reviews
Variation and financial models
Identify direct, inverse, joint and partial variation and apply stated interest or depreciation models.
Textbook lessons and reviews
Euclidean figures and circle theorems
Use similarity, angle and circle theorems with labeled points and compatible arcs.
Textbook lessons and reviews
Linear programming and discrete feasibility
Draw a feasible region and optimise a linear objective, then check integer restrictions when required.
Textbook lessons and reviews
Matrices, determinants and linear systems
Add and multiply matrices, use determinants and solve systems without assuming invertibility.
Textbook lessons and reviews
Polar coordinates
Convert representations and eliminate polar variables without losing the geometric range.
Textbook lessons and reviews
Integration, area and school extensions
Find antiderivatives, evaluate definite integrals and distinguish signed accumulation from geometric area.
Textbook lessons and reviews
Spatial lines, planes, spheres and cross products
Construct spatial equations and use normals, cross products and direction tests.
Textbook lessons and reviews
Parametric curves and preserved domains
Eliminate parameters while retaining ranges, missing points and the geometry of the locus.
Textbook lessons and reviews
Infinite geometric series and error control
Check convergence before summing and interpret recurring decimals and truncation errors.
Textbook lessons and reviews
Radicals, rational equations and simultaneous equations
Build understanding of radicals, rational equations and simultaneous equations through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Absolute values and solution sets
Build understanding of absolute values and solution sets through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Inverse functions and domain restrictions
Build understanding of inverse functions and domain restrictions through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Exponential and logarithmic equations
Build understanding of exponential and logarithmic equations through definitions, contrasting cases and justified applications.
Textbook lessons and reviews
Sets and commonly used logic: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Quadratic functions, equations and inequalities: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Functions: concept and properties: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Exponential and logarithmic functions: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Trigonometric functions: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Plane vectors and applications: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Complex numbers: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Introduction to solid geometry: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Statistics: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Probability: mixed assessment
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Space vectors and solid geometry: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Equations of lines and circles: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Conic sections: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Sequences: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Derivatives and their applications: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Counting principles: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Random variables and distributions: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Textbook lessons and reviews
Statistical analysis of paired data: mixed review
A mixed assessment: identify the method, justify it and revise your reasoning.
Skill lesson and interactive model
Vertex form and completing the square
Explain turning points, symmetry and constrained optima through a square.
Skill lesson and interactive model
Roots and completeness of solutions
Choose a root method, then check reality, location and the original domain.
Skill lesson and interactive model
Quadratic signs and solution sets
Use roots to partition a line, then retain endpoint and domain conditions.
Skill lesson and interactive model
Quadratic extrema on restricted intervals
Compare admissible vertex and endpoints, including open and moving intervals.
Skill lesson and interactive model
Unit-circle coordinates and signs
Read sine and cosine as coordinates, retaining angle direction and quadrant information.
Skill lesson and interactive model
Identity choice and domain preservation
Transform expressions with a purpose, explain excluded angles and test tempting false rules.
Skill lesson and interactive model
Amplitude, period and phase in models
Read parameters as observable changes and reconstruct models without hiding ambiguity.
Skill lesson and interactive model
Complete trigonometric solution sets
Solve a transformed angle, recover every branch and intersect the original interval.
Skill lesson and interactive model
Vectors as displacements
Translate, add and scale arrows; distinguish a journey from its net displacement.
Skill lesson and interactive model
Coordinates, bases and division
Convert arrows to coordinates, recover unknown endpoints and test when a basis actually determines coordinates.
Skill lesson and interactive model
Dot products and constrained projection
Connect signed projection, angle and squared distance; distinguish a line projection from a nearest point on a segment.
Skill lesson and interactive model
Vector proofs in triangles
Turn geometric hypotheses into vector equations; prove midpoint, centroid and orthocenter properties with their assumptions visible.
Skill lesson and interactive model
Ellipse: definition, axes and constraints
Build an ellipse from a distance sum, identify its major direction and distinguish a genuine ellipse from boundary loci.
Skill lesson and interactive model
Hyperbola: branches and asymptotes
Keep signed and absolute focal differences separate; inspect degree loss when a line is parallel to an asymptote.
Skill lesson and interactive model
Parabola: focus and directrix
Derive a parabola from equal distances, track orientation under translation and use the definition in a distance optimization.
Skill lesson and interactive model
Conic intersections, chords and inverse problems
Solve line–conic systems with degree and domain checks; use Vieta for midpoint loci and reconstruct a chord from its midpoint.
Skill lesson and interactive model
Rates and tangent meaning
Derive tangent slope from shrinking secants, then connect slope, point and local approximation.
Skill lesson and interactive model
Derivative rules and domains
Choose power, product, quotient or chain rules from the expression structure, retaining domains.
Skill lesson and interactive model
Monotonicity and constrained extrema
Use derivative sign changes, endpoints and feasible domains to distinguish local extrema from global optima.
Skill lesson and interactive model
Curvature, inflections and derivative graphs
Verify a concavity change at a point on the curve, distinguishing inflections from stationary points.
Skill lesson and interactive model
Conditions, partitions and posterior probability
Identify the conditioning group, verify partitions, and update priors with observed evidence.
Skill lesson and interactive model
Distribution means, variances and dependence
Compute weighted moments and apply shifts/scales with correct units and dependence conditions.
Skill lesson and interactive model
Choosing binomial, hypergeometric and normal models
Choose a model from its assumptions and support before computing its probability or moments.
Skill lesson and interactive model
Paired data, fitted lines and responsible prediction
Calculate a least-squares line from paired centered data and interpret residuals and units.
Past papers
2026 JM01
Official source links, independently written explanations and local practice records.
Past papers
2026 JM02
Official source links, independently written explanations and local practice records.
Past papers
2025 JM01
Official source links, independently written explanations and local practice records.
Past papers
2025 JM02
Official source links, independently written explanations and local practice records.
Past papers
2024 JM01
Official source links, independently written explanations and local practice records.
Past papers
2024 JM02
Official source links, independently written explanations and local practice records.
Past papers
2023 JM01
Official source links, independently written explanations and local practice records.
Past papers
2023 JM02
Official source links, independently written explanations and local practice records.
Past papers
2022 JM01
Official source links, independently written explanations and local practice records.
Past papers
2022 JM02
Official source links, independently written explanations and local practice records.
Past papers
2021 JM01
Official source links, independently written explanations and local practice records.
Past papers
2021 JM02
Official source links, independently written explanations and local practice records.
JAE pilot
Quadratics
The original JAE pilot, with the same question IDs and saved work.
JAE pilot
Trigonometry
The original JAE pilot, with the same question IDs and saved work.
Math Inquiry Lab
Quadratic functions
Core, deeper and challenge tasks with an optional local notebook.
Math Inquiry Lab
Probability experiments
Core, deeper and challenge tasks with an optional local notebook.
Math Inquiry Lab
Regression and influential points
Core, deeper and challenge tasks with an optional local notebook.
Modeling and research
Logistic growth
Change a model assumption, inspect the response and state its limits.
Modeling and research
SIR intervention
Change a model assumption, inspect the response and state its limits.
Modeling and research
Stopping distance
Change a model assumption, inspect the response and state its limits.
Modeling and research
Critical transitions
Change a model assumption, inspect the response and state its limits.
Modeling and research
Equation discovery
Change a model assumption, inspect the response and state its limits.
Modeling and research
Collective motion
Change a model assumption, inspect the response and state its limits.
Modeling and research
Higher-order contagion
Change a model assumption, inspect the response and state its limits.
Modeling and research
Operator learning
Change a model assumption, inspect the response and state its limits.
Modeling and research
Persistent homology
Change a model assumption, inspect the response and state its limits.
Modeling and research
Melt-pond topology
Change a model assumption, inspect the response and state its limits.
Modeling and research
Quantum repeater scheduling
Change a model assumption, inspect the response and state its limits.
Modeling and research
Parameter identifiability
Change a model assumption, inspect the response and state its limits.
Modeling and research
Network interference
Change a model assumption, inspect the response and state its limits.
Modeling and research
Robust decisions
Change a model assumption, inspect the response and state its limits.
Browser Python · model and evidence
A tank, a boundary, a balance
Build a mass balance, uncover its dimensionless structure, and verify your first numerical solver.
Browser Python · model and evidence
A good fit, an uncertain future
Fit competing models, withhold future data and inspect the shape of parameter uncertainty.
Browser Python · model and evidence
Mechanisms become trajectories
Derive compartment flows and compare Euler, RK4 and an adaptive reference.
Browser Python · model and evidence
Local choices, collective patterns
Build an original Schelling-type experiment and compare paired stochastic replications.
Browser Python · model and evidence
The assumptions behind an optimum
Formulate a linear program, verify it geometrically and challenge its decision assumptions.
Browser Python · model and evidence
When a smooth picture lies
Derive a diffusion equation and test explicit and implicit schemes against an exact mode.
One-class activity · 45 min
Sampling: does more mean better?
Compare who is asked, how many are asked and what changes when you draw again.
One-class activity · 45 min
Sampling: does more mean better? · Teacher reference
Compare who is asked, how many are asked and what changes when you draw again.
One-class activity · 50 min
Mixing tank: can a smooth curve be trusted?
Derive a balance, complete two short functions and separate numerical checks from a physical assumption.
One-class activity · 50 min
Mixing tank: can a smooth curve be trusted? · Teacher reference
Derive a balance, complete two short functions and separate numerical checks from a physical assumption.
Guided paper practice · 40 min
Linear equations: collect like terms
Collect variable terms on the same side, handle negative coefficients and check each solution. A 40-minute projection and paper lesson.
Guided paper practice · 40 min
Linear equations: collect like terms · Teacher reference
Collect variable terms on the same side, handle negative coefficients and check each solution. A 40-minute projection and paper lesson.
Guided paper practice · 40 min
Linear equations: transpose and verify
Solve equations with variable terms on both sides, explain each sign change and verify in the original equation. A 40-minute paper-first lesson.
Guided paper practice · 40 min
Linear equations: transpose and verify · Teacher reference
Solve equations with variable terms on both sides, explain each sign change and verify in the original equation. A 40-minute paper-first lesson.
Guided paper practice · 40 min
Angle bisectors: perpendicular distances and the forward proof
Distinguish distance from an oblique segment, practise equal lengths, and fade an AAS proof scaffold.
Guided paper practice · 40 min
Angle bisectors: perpendicular distances and the forward proof · Teacher reference
Distinguish distance from an oblique segment, practise equal lengths, and fade an AAS proof scaffold.
Guided paper practice · 40 min
Angle bisectors: the converse and applications
Separate the two directions, construct an HL proof, and use equal distances in angle and perimeter problems.
Guided paper practice · 40 min
Angle bisectors: the converse and applications · Teacher reference
Separate the two directions, construct an HL proof, and use equal distances in angle and perimeter problems.
Guided paper practice · 40 min
Recognising reflection symmetry and counting axes
Use the fold test, distinguish a figure’s own symmetry from a pair of reflected figures, and count complete axes.
Guided paper practice · 40 min
Recognising reflection symmetry and counting axes · Teacher reference
Use the fold test, distinguish a figure’s own symmetry from a pair of reflected figures, and count complete axes.
Guided paper practice · 40 min
Perpendicular bisectors: properties and the SSS converse
Separate perpendicularity from midpoint information, prove equal endpoint distances, and use SSS for the converse.
Guided paper practice · 40 min
Perpendicular bisectors: properties and the SSS converse · Teacher reference
Separate perpendicularity from midpoint information, prove equal endpoint distances, and use SSS for the converse.
Guided paper practice · 40 min
Constructing symmetry axes with ruler and compass
Construct perpendicular bisectors, check one common axis against all supplied pairs, and draw complete axes of familiar figures.
Guided paper practice · 40 min
Constructing symmetry axes with ruler and compass · Teacher reference
Construct perpendicular bisectors, check one common axis against all supplied pairs, and draw complete axes of familiar figures.
Guided paper practice · 40 min
Rational numbers: position, opposites and absolute value
Use signed quantities, number-line order, opposite numbers and distance from zero in a fixed core-practice lesson.
Guided paper practice · 40 min
Rational numbers: position, opposites and absolute value · Teacher reference
Use signed quantities, number-line order, opposite numbers and distance from zero in a fixed core-practice lesson.
Guided paper practice · 40 min
Rational numbers: operations, powers and large numbers
Select core signed arithmetic, fractions, powers, scientific notation and rounding after the relevant textbook periods.
Guided paper practice · 40 min
Rational numbers: operations, powers and large numbers · Teacher reference
Select core signed arithmetic, fractions, powers, scientific notation and rounding after the relevant textbook periods.
Guided paper practice · 40 min
Expressions: terms, coefficients and values
Read the structure of a monomial or polynomial, distinguish coefficient from degree, and substitute signed values.
Guided paper practice · 40 min
Expressions: terms, coefficients and values · Teacher reference
Read the structure of a monomial or polynomial, distinguish coefficient from degree, and substitute signed values.
Guided paper practice · 40 min
Polynomials: collect terms and remove brackets
Keep signed coefficients, distinguish unlike powers, apply the distributive law and check by substitution.
Guided paper practice · 40 min
Polynomials: collect terms and remove brackets · Teacher reference
Keep signed coefficients, distinguish unlike powers, apply the distributive law and check by substitution.
Guided paper practice · 40 min
Equations: meaning, solutions and equal operations
Distinguish an expression from an equation in an unknown, test a solution and explain operations on both sides.
Guided paper practice · 40 min
Equations: meaning, solutions and equal operations · Teacher reference
Distinguish an expression from an equation in an unknown, test a solution and explain operations on both sides.
Guided paper practice · 40 min
Equations: brackets and constant denominators
Distribute to every term, clear constant denominators on the whole equality and verify in the original equation.
Guided paper practice · 40 min
Equations: brackets and constant denominators · Teacher reference
Distribute to every term, clear constant denominators on the whole equality and verify in the original equation.
Guided paper practice · 40 min
Equations in context: quantities, rates and conditions
Choose representative cost, travel, work-rate, percentage, ratio and tiered-price models; verify the solution in context.
Guided paper practice · 40 min
Equations in context: quantities, rates and conditions · Teacher reference
Choose representative cost, travel, work-rate, percentage, ratio and tiered-price models; verify the solution in context.
Guided paper practice · 40 min
Geometric objects, segments and midpoints
Distinguish lines, rays and segments; count cube elements and use point order and midpoint conditions.
Guided paper practice · 40 min
Geometric objects, segments and midpoints · Teacher reference
Distinguish lines, rays and segments; count cube elements and use point order and midpoint conditions.
Guided paper practice · 40 min
Angles: addition, bisectors, complements and supplements
Use ray order, degree-minute conversion, internal bisectors and the correct total for complementary or supplementary angles.
Guided paper practice · 40 min
Angles: addition, bisectors, complements and supplements · Teacher reference
Use ray order, degree-minute conversion, internal bisectors and the correct total for complementary or supplementary angles.
Guided paper practice · 40 min
Triangle sides, medians and altitudes
Consolidate strict triangle inequalities, perimeter and the definitions of supporting segments.
Guided paper practice · 40 min
Triangle sides, medians and altitudes · Teacher reference
Consolidate strict triangle inequalities, perimeter and the definitions of supporting segments.
Guided paper practice · 40 min
Triangle and polygon angles
Choose the correct angle sum, distinguish adjacent and remote angles, and count polygon sides.
Guided paper practice · 40 min
Triangle and polygon angles · Teacher reference
Choose the correct angle sum, distinguish adjacent and remote angles, and count polygon sides.
Guided paper practice · 40 min
Congruence: correspondence, SSS and SAS
Read the vertex order, check included angles and build short justified proofs.
Guided paper practice · 40 min
Congruence: correspondence, SSS and SAS · Teacher reference
Read the vertex order, check included angles and build short justified proofs.
Guided paper practice · 40 min
Congruence: ASA, AAS and right-triangle HL
Select a criterion from the actual givens and complete a short proof without circular assumptions.
Guided paper practice · 40 min
Congruence: ASA, AAS and right-triangle HL · Teacher reference
Select a criterion from the actual givens and complete a short proof without circular assumptions.
Guided paper practice · 40 min
Reflecting whole figures and coordinates
Use a given axis to reflect every vertex, then connect the construction with x/y-coordinate rules.
Guided paper practice · 40 min
Reflecting whole figures and coordinates · Teacher reference
Use a given axis to reflect every vertex, then connect the construction with x/y-coordinate rules.
Guided paper practice · 40 min
Isosceles, equilateral and special-triangle practice
Apply properties and converses with conditions; use 30° right triangles and an optional reflection shortest-path extension.
Guided paper practice · 40 min
Isosceles, equilateral and special-triangle practice · Teacher reference
Apply properties and converses with conditions; use 30° right triangles and an optional reflection shortest-path extension.
Guided paper practice · 40 min
Powers and polynomial multiplication
Consolidate the three power laws and distribute monomial or polynomial products accurately.
Guided paper practice · 40 min
Powers and polynomial multiplication · Teacher reference
Consolidate the three power laws and distribute monomial or polynomial products accurately.
Guided paper practice · 40 min
Special products and polynomial division
Practise difference of squares, complete squares and every-term division while retaining nonzero divisors.
Guided paper practice · 40 min
Special products and polynomial division · Teacher reference
Practise difference of squares, complete squares and every-term division while retaining nonzero divisors.
Guided paper practice · 40 min
Factorisation: common factors and square patterns
Choose a method, factor completely over integers and check by expanding.
Guided paper practice · 40 min
Factorisation: common factors and square patterns · Teacher reference
Choose a method, factor completely over integers and check by expanding.
Guided paper practice · 40 min
Rational expressions: meaning, domain and cancellation
Separate zero numerators from zero denominators and retain restrictions after cancellation.
Guided paper practice · 40 min
Rational expressions: meaning, domain and cancellation · Teacher reference
Separate zero numerators from zero denominators and retain restrictions after cancellation.
Guided paper practice · 40 min
Rational operations, integer powers and scientific notation
Consolidate common denominators, multiplication/division and negative powers, with short scientific-notation checks.
Guided paper practice · 40 min
Rational operations, integer powers and scientific notation · Teacher reference
Consolidate common denominators, multiplication/division and negative powers, with short scientific-notation checks.
Guided paper practice · 40 min
Rational equations: checking and applications
Clear denominators completely, check extraneous candidates, and model work-rate and travel problems.
Guided paper practice · 40 min
Rational equations: checking and applications · Teacher reference
Clear denominators completely, check extraneous candidates, and model work-rate and travel problems.
Guided paper practice · 40 min
Intersections, perpendiculars and distance
Distinguish angle position from angle size, and perpendicular distance from an oblique segment.
Guided paper practice · 40 min
Intersections, perpendiculars and distance · Teacher reference
Distinguish angle position from angle size, and perpendicular distance from an oblique segment.
Guided paper practice · 40 min
Parallel lines: conditions and reasoning
Use angle relations in the correct direction and distinguish a hypothesis from a conclusion.
Guided paper practice · 40 min
Parallel lines: conditions and reasoning · Teacher reference
Use angle relations in the correct direction and distinguish a hypothesis from a conclusion.
Guided paper practice · 40 min
Translation: matching points and invariant lengths
Move a whole figure in one direction by one distance; distinguish translation from rotation or reflection.
Guided paper practice · 40 min
Translation: matching points and invariant lengths · Teacher reference
Move a whole figure in one direction by one distance; distinguish translation from rotation or reflection.
Guided paper practice · 40 min
Square roots and cube roots: meanings and signs
Distinguish two square roots from one principal square root, and allow negative cube roots.
Guided paper practice · 40 min
Square roots and cube roots: meanings and signs · Teacher reference
Distinguish two square roots from one principal square root, and allow negative cube roots.
Guided paper practice · 40 min
Real numbers: classification, bounds and approximation
Use exact definitions and square comparisons; keep an approximation distinct from an equality.
Guided paper practice · 40 min
Real numbers: classification, bounds and approximation · Teacher reference
Use exact definitions and square comparisons; keep an approximation distinct from an equality.
Guided paper practice · 40 min
Coordinates: location and translation
Use ordered pairs, quadrants, map scales and one common coordinate change.
Guided paper practice · 40 min
Coordinates: location and translation · Teacher reference
Use ordered pairs, quadrants, map scales and one common coordinate change.
Guided paper practice · 40 min
Two-variable systems: substitution and elimination
Reduce two unknowns to one, then recover the complete solution and check both equations.
Guided paper practice · 40 min
Two-variable systems: substitution and elimination · Teacher reference
Reduce two unknowns to one, then recover the complete solution and check both equations.
Guided paper practice · 40 min
Two-variable systems: quantities and models
Name unknowns, keep units consistent, build independent conditions and check contextual feasibility.
Guided paper practice · 40 min
Two-variable systems: quantities and models · Teacher reference
Name unknowns, keep units consistent, build independent conditions and check contextual feasibility.
Guided paper practice · 40 min
Three-variable systems: eliminate, solve and check
Eliminate the same unknown to obtain a two-variable system, then back-substitute all three values.
Guided paper practice · 40 min
Three-variable systems: eliminate, solve and check · Teacher reference
Eliminate the same unknown to obtain a two-variable system, then back-substitute all three values.
Guided paper practice · 40 min
Linear inequalities: direction and endpoints
Keep equivalent conditions, reverse for negative division and show the complete solution set.
Guided paper practice · 40 min
Linear inequalities: direction and endpoints · Teacher reference
Keep equivalent conditions, reverse for negative division and show the complete solution set.
Guided paper practice · 40 min
Inequality systems: overlap and feasible choices
Translate every restriction, take the overlap and then apply meaningful integer conditions.
Guided paper practice · 40 min
Inequality systems: overlap and feasible choices · Teacher reference
Translate every restriction, take the overlap and then apply meaningful integer conditions.
Guided paper practice · 40 min
Statistical surveys: samples, proportions and claims
Choose a defensible survey method, read proportions and state the limits of sample-based estimates.
Guided paper practice · 40 min
Statistical surveys: samples, proportions and claims · Teacher reference
Choose a defensible survey method, read proportions and state the limits of sample-based estimates.
Guided paper practice · 40 min
Grouped data and histograms: frequency and area
Count boundary values once, label the vertical measure, and use frequency per class width for unequal classes.
Guided paper practice · 40 min
Grouped data and histograms: frequency and area · Teacher reference
Count boundary values once, label the vertical measure, and use frequency per class width for unequal classes.
Guided paper practice · 40 min
Square roots: meaning, simplification and domain
Distinguish a square root from the solutions of a squared equation, and retain every real-domain condition.
Guided paper practice · 40 min
Square roots: meaning, simplification and domain · Teacher reference
Distinguish a square root from the solutions of a squared equation, and retain every real-domain condition.
Guided paper practice · 40 min
Square-root operations: factor, multiply and combine
Simplify square factors, apply real-domain product and quotient laws, then combine like radicals.
Guided paper practice · 40 min
Square-root operations: factor, multiply and combine · Teacher reference
Simplify square factors, apply real-domain product and quotient laws, then combine like radicals.
Guided paper practice · 40 min
Pythagoras: identify the sides and calculate lengths
Model right triangles, choose addition or subtraction of squares, and retain positive exact lengths.
Guided paper practice · 40 min
Pythagoras: identify the sides and calculate lengths · Teacher reference
Model right triangles, choose addition or subtraction of squares, and retain positive exact lengths.
Guided paper practice · 40 min
Pythagoras converse: conditions and combined problems
Test actual side data and combine the converse with length, midpoint and perpendicularity reasoning.
Guided paper practice · 40 min
Pythagoras converse: conditions and combined problems · Teacher reference
Test actual side data and combine the converse with length, midpoint and perpendicularity reasoning.
Guided paper practice · 40 min
Parallelograms: properties, criteria and midsegments
Separate given facts from conclusions, use diagonals carefully, and prove the triangle midsegment theorem.
Guided paper practice · 40 min
Parallelograms: properties, criteria and midsegments · Teacher reference
Separate given facts from conclusions, use diagonals carefully, and prove the triangle midsegment theorem.
Guided paper practice · 40 min
Rectangles and rhombi: diagonal conditions and converses
Keep the parallelogram premise, distinguish equal from perpendicular diagonals, and use the right-triangle median theorem.
Guided paper practice · 40 min
Rectangles and rhombi: diagonal conditions and converses · Teacher reference
Keep the parallelogram premise, distinguish equal from perpendicular diagonals, and use the right-triangle median theorem.
Guided paper practice · 40 min
Squares and inclusive quadrilateral classification
Combine rectangle and rhombus conditions, compare diagonals and reject unsupported converse statements.
Guided paper practice · 40 min
Squares and inclusive quadrilateral classification · Teacher reference
Combine rectangle and rhombus conditions, compare diagonals and reject unsupported converse statements.
Guided paper practice · 40 min
Functions: inputs, domains, tables and graphs
Decide whether a relation is a function, use three representations and check mathematical and contextual domains.
Guided paper practice · 40 min
Functions: inputs, domains, tables and graphs · Teacher reference
Decide whether a relation is a function, use three representations and check mathematical and contextual domains.
Guided paper practice · 40 min
Linear functions: coefficients, graphs and determination
Relate slope and intercept to the graph, distinguish constant functions, and determine a line from two distinct inputs.
Guided paper practice · 40 min
Linear functions: coefficients, graphs and determination · Teacher reference
Relate slope and intercept to the graph, distinguish constant functions, and determine a line from two distinct inputs.
Guided paper practice · 40 min
Linear models: equations, inequalities and intersections
Translate contextual conditions and connect graphical solutions with exact algebra.
Guided paper practice · 40 min
Linear models: equations, inequalities and intersections · Teacher reference
Translate contextual conditions and connect graphical solutions with exact algebra.
Guided paper practice · 40 min
Data centres: weights, ordering and choosing a summary
Calculate weighted means, medians and all tied modes; distinguish grouped estimates from raw-data answers.
Guided paper practice · 40 min
Data centres: weights, ordering and choosing a summary · Teacher reference
Calculate weighted means, medians and all tied modes; distinguish grouped estimates from raw-data answers.
Guided paper practice · 40 min
Variance: calculation, stability and transformations
Use the textbook’s n-denominator definition, compare data with the same centre and reason about shifts and scales.
Guided paper practice · 40 min
Variance: calculation, stability and transformations · Teacher reference
Use the textbook’s n-denominator definition, compare data with the same centre and reason about shifts and scales.
Resources and tools
Question bank
Learning resources and tools from the existing collection.
Resources and tools
Senior classroom list
Learning resources and tools from the existing collection.
Resources and tools
Senior teacher resources
Learning resources and tools from the existing collection.
Resources and tools
Macau competition archive
Learning resources and tools from the existing collection.
Resources and tools
Python laboratory guide
Learning resources and tools from the existing collection.
Resources and tools
Mathematical modeling lecture programme
Learning resources and tools from the existing collection.
A classroom sequence across all activities
Build a sequence of up to 30 activities. The share link contains activity IDs and a title; it contains no answers or notes.
My learning across the collection
Saved drafts, checked answers, self-assessments, experiment notes and practice exposure are different kinds of evidence. They are shown separately; there is no combined mastery score.
Inspect retained raw recovery data
These bytes are preserved for recovery. Download the complete backup before replacing an unreadable store.
Saved activity notebooks
Original answer history and corrections
Open detailed answers, retries and unknown records here
Complete backup
Includes the original mathematics record stores, inquiry notes, optional lab notes, shared activity notes, classroom sequence and IndexedDB Python drafts. A backup can contain your written work and code; keep it privately.
Review before restoring
Existing records are kept unless you select an overwrite. A copy of the current records is downloaded before applying. Workshop imports keep local conflicts by default.