4 LESSONS
Build confidence with numbers and algebra
Connect fractions, ratios, powers and equations through examples from everyday life.
Follow this series →MAKE SENSE OF NUMBERS
Free mathematics: understand numbers, patterns and data
Learn the reasoning behind the calculation. Work through fractions, equations, measurement and data with solved examples and practice you can check.
A guided place to begin
4 LESSONS
Understand graphs, measurement and data
Read relationships, measure space and reason about averages and chance.
Follow this series →5 LESSONS
Functions, models and change
Move from domains and quadratic models through growth and trigonometry to derivatives and optimisation.
Follow this series →3 LESSONS
Probability and statistical inference
Track conditional probabilities, quantify sampling uncertainty and interpret a regression without confusing association with cause.
Follow this series →Explore every topic
Guides 1–16 of 16
Mathematics
BeginnerAlgebra: turn a situation into an expression and an equation
Algebra gives a name to an unknown quantity and expresses relationships that stay true.
Learning outcome: Build and simplify expressions, solve equations with one unknown, explain each operation and recognise impossible or unrestricted equations.
Read the guide →Mathematics
AdvancedConditional probability and independence
“How often is an item defective when an alarm sounds?” differs from “How often does the alarm sound when an item is defective?” Conditioning changes the reference group.
Learning outcome: Choose the correct denominator, distinguish independence from mutual exclusion, calculate a posterior probability, and explain why association does not establish causation.
Read the guide →Mathematics
AdvancedCorrelation, regression and confounding
A pattern between two variables can help summarise data or make cautious predictions.
Learning outcome: Calculate and interpret a fitted line, inspect residuals, distinguish prediction from causation, and show numerically how changing group composition can reverse an overall comparison.
Read the guide →Mathematics
BeginnerData: describe a typical value and the variation around it
An average can hide both an unusually large value and a wide gap between observations.
Learning outcome: Calculate and interpret mean, median, mode, range and interquartile range, compare data sets with the same mean, and avoid misleading averages of unequal groups.
Read the guide →Mathematics
AdvancedDerivatives and optimisation
An average rate describes change across an interval.
Learning outcome: Derive a simple derivative from first principles, differentiate polynomials, interpret rates with units, and solve a constrained optimisation problem with a justified maximum.
Read the guide →Mathematics
IntermediateExponents, roots and scientific notation: reason across scales
A tiny thickness and a huge count are easier to compare when powers of ten organise the numbers.
Learning outcome: Use exponent rules with their conditions, distinguish a square root from solutions of a squared equation, and calculate with large and small quantities in scientific notation.
Read the guide →Mathematics
BeginnerFractions, decimals and percentages: one quantity, three forms
A half-filled bottle, 0.5 litre and 50% can describe related ideas, but only when the whole is clear.
Learning outcome: Convert among the three forms, combine fractional amounts, calculate a percentage change, and spot misleading comparisons that use different starting quantities.
Read the guide →Mathematics
IntermediateFunctions, domains and composition
A formula becomes useful when you know which inputs it accepts, what its outputs mean, and how it connects to another rule.
Learning outcome: Find restrictions before simplifying, calculate compositions in the correct order, and explain when an inverse function exists.
Read the guide →Mathematics
BeginnerGeometry: measure boundaries, surfaces and space
Fencing a garden, covering a floor and filling a container ask three different measurement questions.
Learning outcome: Calculate rectangular, triangular and circular measures, split a compound shape into simpler pieces, and distinguish container capacity from the material needed to build it.
Read the guide →Mathematics
IntermediateLinear equations and graphs: read a constant rate of change
A graph turns pairs of numbers into a visible relationship.
Learning outcome: Plot ordered pairs, calculate slope, recover a line's equation from two points and evaluate predictions against the assumptions of the model.
Read the guide →Mathematics
IntermediateProbability and risk: reason about uncertain outcomes
Uncertainty does not mean every outcome is equally likely.
Learning outcome: Construct a sample space, calculate complements and combined events, compare sampling with and without replacement, and interpret expected counts without treating them as promises.
Read the guide →Mathematics
IntermediateQuadratic equations and models
Products of changing lengths and models of accelerating motion lead to squared terms.
Learning outcome: Choose a solution method, explain the discriminant, find an extreme value, and reject a root only for a stated mathematical or contextual reason.
Read the guide →Mathematics
BeginnerRatios, proportion and unit rates: compare fairly
A larger packet may cost more but offer a lower price per item.
Learning outcome: Interpret ratios, scale a mixture, compare unit prices and test whether a situation supports proportional reasoning.
Read the guide →Mathematics
AdvancedSampling variation and confidence intervals
Different random samples produce different estimates even when the population is unchanged.
Learning outcome: Distinguish standard deviation from standard error, calculate mean and proportion intervals, interpret confidence correctly, and identify uncertainty that the formulas omit.
Read the guide →Mathematics
IntermediateSequences and compound growth
Adding a fixed amount and multiplying by a fixed factor produce very different patterns.
Learning outcome: Build recursive and explicit rules, sum finite sequences, compare linear and compound change, and check the assumptions behind a projection.
Read the guide →Mathematics
IntermediateTrigonometry and measurement
A distance and an angle can reveal a height that cannot be measured directly.
Learning outcome: Choose a trigonometric ratio, solve one- and two-position height problems, convert degrees and radians, and explain how measurement errors affect a result.
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