Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Mathematics – grade 12
Qualification phase, third and fourth half-year (Q3, Q4; at the Gymnasium with Abitur after grade 12: grade 12). Q4 also deepens and links all three areas. LK = advanced course only. The numbers are our own.
Unit Q3Third half-year: analytic geometry
Coordinates, vectors, the dot product, lines and planes with relative positions, angles and distances.
Q3.1Vectors and the dot product
Describing geometry with coordinates, calculating with vectors, testing for collinearity and using the dot product for lengths and angles.
Adding and scaling vectors (3D)
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
Dot product and cross product
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
Q3.2Lines and planes
Lines and planes in parametric, coordinate and normal form, relative positions and distances; LK: also families and distances point–line and line–line.
Lines and planes in space
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Relative positions and distances in space
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Unit Q4Fourth half-year: calculus, probability and tasks from all areas
The binomial distribution with its parameters, inferring from the sample to the population, hypothesis tests (LK) and the normal distribution (LK), and further calculus.
Q4.1Calculus (further study)
Deepening and linking differential and integral calculus; LK: solids of revolution and improper integrals.
Optimisation problems
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Integral function and fundamental theorem
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
Q4.2Probability (binomial distribution, inferential statistics)
The binomial distribution with expected value and standard deviation, k-σ intervals and significance; LK: hypothesis tests with type I and type II errors, the bell shape of the normal distribution.
Galton board
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Normal distribution
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
Q4.3Complex tasks from all three areas
Tasks that link calculus, analytic geometry and probability.
No separate simulation is planned for this.
Unit ZAdditional courses (offered by the school)
Possible additional courses: non-Euclidean geometry, logic, number theory, numerical mathematics, differential equations, sequences and series with Taylor series, Markov chains, complex analysis, conic sections, further calculus, reasoning and proof.
Taylor polynomials
Term by term a polynomial hugs a function such as sin x ever more closely. That is pretty much how a calculator does it.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Rahmenlehrplan, Gymnasium, Mathematics, grade 12