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How it is built

How it is built

The ordering of the problems is the actual content. Two values appear on every problem: its difficulty level and the competency it requires.

§01Difficulty

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.

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.

§02Difficulty

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

xy
∫0212x2 dx\int_0^2 \tfrac{1}{2}x^2\,\mathrm{d}x1/6Foundations
xy
∫02(x3−3x2+2x)dx\int_0^2 \left(x^3-3x^2+2x\right)\mathrm{d}x3/6Application

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

§03Competencies trained

Calculation
Rearranging and computing accurately and without slips.
Applying a method
Carrying out a known procedure correctly.
Identifying the method
Deciding which procedure the problem calls for.
Seeing connections
Relating concepts and representations to one another.
Interpreting
Reading a result back into its context.
Modelling
Translating a situation into a mathematical model.
Reasoning
Deriving or refuting a statement so others can follow it.
Transfer
Carrying known material over to an unfamiliar form of problem.
Problem solving
Finding your own route with no procedure given.

§04Structure 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.

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