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Upper secondary · Final-exam preparation

Following a solution is not the same as being able to solve the problem.

The difference only shows on a problem that does not look like the one you practised. These materials train the step that is missing then: recognising which method a problem calls for.

View problemsHow it is built
xy
f(x)=x3−3x2+2xf(x) = x^3 - 3x^2 + 2xThe area between a graph and the x-axis — the class of problem that reveals whether the method is understood or only the arithmetic.

§01The problem

The usual route trains calculation, not decision.

  1. 01

    Read the theory

    The statement makes sense as long as it is in front of you.

  2. 02

    Work a problem

    The chapter has already prescribed the method. The actual decision has been made for you.

  3. 03

    Compare with the solution

    The route is easy to follow. That feels like competence, but it is recognition.

  4. 04

    The exam

    No chapter heading above the problem. The decision that was never practised is the first one required.

This is why the problems here are not sorted by method but mixed, graded, and labelled with the competency they require.

§02Structure of a practice block

Hints instead of the solution — in the order you actually need them.

Someone who is stuck does not need the full solution but the smallest next step. Every problem is therefore staged.

  1. 1Problem

    Stated with its difficulty level and the competency it trains.

  2. 2Hint 1

    Points out what to look at, without naming the method.

  3. 3Hint 2

    Names the method, not how to carry it out.

  4. 4Starting point

    The first step, from which you can continue on your own.

  5. 5Full solution

    Complete, including the rearrangements that are usually skipped.

  6. 6Common mistake

    The error this problem tends to provoke — and why it happens.

§03Difficulty

Six levels, kept separate from competency.

The level says how far a problem sits from the basic form. It does not say how much arithmetic is involved. A short calculation can be conceptually demanding — and the reverse.

Both values appear on every problem, so practice can be targeted instead of sequential.

1/6
Foundations
The basic form of the method, with no detour.
2/6
Standard
The form that appears in written tests as routine work.
3/6
Application
The method is embedded in a context and has to be uncovered first.
4/6
Exam level
Matches the level and scope of a final-exam part-question.
5/6
Transfer
Requires combining two methods or handling an unfamiliar representation.
6/6
Beyond exam level
Above the usual examination standard. For advanced courses and competition preparation.

§04Subject areas

Three areas, broken down by topic.

Calculus

The behaviour of functions: how they change, where they reach extremes, and what area or volume lies beneath them.

  • Functions and their properties
  • Limits and continuity
  • DerivativesRules of differentiation, higher derivatives, implicit differentiation
  • Curve analysisRoots, monotonicity, extrema, concavity, inflection points, asymptotes
  • IntegrationAntiderivatives, definite integrals, substitution, integration by parts
  • Areas and volumes of revolution
  • Applied problemsOptimisation, growth and decay, quantity and rate of change
  • Families of functions
  • Differential equationsPart of some syllabuses only. Each product states its curriculum reference.

Analytic geometry

Lines, planes and solids in space described by vectors — and how they are positioned relative to one another.

  • Vectors and vector arithmetic
  • Lines in space
  • PlanesParametric, normal and coordinate form, and converting between them
  • Relative positionPoint, line, plane — parallel, intersecting, skew, coincident
  • Systems of linear equations
  • Dot productOrthogonality, angles between vectors
  • Cross productPart of some syllabuses only. Each product states its curriculum reference.
  • DistancesPoint–line, point–plane, line–line, line–plane
  • Angles
  • Areas and volumes of solids
  • SpheresPart of some syllabuses only. Each product states its curriculum reference.

Probability and statistics

How likely an outcome is, how outcomes are distributed, and when an observation counts as evidence against an assumption.

  • ProbabilityEqually likely outcomes, tree diagrams, two-way tables, conditional probability
  • Counting methods
  • Random variables, expectation, variance
  • Binomial distribution
  • Normal distributionPart of some syllabuses only. Each product states its curriculum reference.
  • Hypothesis testingOne- and two-tailed tests, type I and type II errors
  • Confidence intervalsPart of some syllabuses only. Each product states its curriculum reference.
View all topics

§05Curriculum

German states set different syllabuses. That is not glossed over here.

Where a topic is recorded for a particular state and year group, it says so on the product. Where the information is absent, it means exactly that: not recorded. It does not mean the topic is examinable everywhere.

These entries are added gradually rather than claimed wholesale.