Home Mathematics Saxony curriculum · Gymnasium Grade 11/12 Mathematics Saxony curriculum · Gymnasium
Mathematics – grade 11/12 Grades 11/12: the curriculum describes the basic and the advanced course for both years together, so they are shown as one stage 11/12. The GK/LK prefixes are ours; the numbers after them are the curriculum’s; W marks optional areas.
Unit GK-1 Differential calculus (basic course)Limits and behaviour at infinity, the derivative as slope of secant and tangent, rules of differentiation, curve sketching with poles and asymptotes, function equations from conditions and optimisation problems.
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
Grade 10–11 Baden-Württemberg: grade 11 Bavaria: grade 11 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grades 11, 12 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 11 Berlin: grade 11 Brandenburg: grade 11 Bremen: grades 10, 11/12 Hamburg: grades 11, 12 Hesse: grades 11, 12 Lower Saxony: grade 11 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 11, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 11, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
Grade 10–13 Baden-Württemberg: grade 12/13 Bavaria: grades 11, 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 10 Hamburg: grade 11 Hesse: grade 12 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12
What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.
Grade 10–12 Baden-Württemberg: grades 9/10, 12/13 Bavaria: grade 11 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 10 Hamburg: grade 11 Hesse: grades 11, 12 Lower Saxony: grades 9/10, 11, 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grades 10, 11/12 Saxony-Anhalt: grades 10, 11/12 Schleswig-Holstein: grades 10, 11
Does the graph cross the x-axis or only touch it? The multiplicity of the zero decides, the highest power decides what happens far out.
Grade 10–11 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grades 10, 11/12 Schleswig-Holstein: grade 11
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
Drag three points and the matching parabola always passes through all of them. The equation is worked out step by step.
Grade 9–11 Baden-Württemberg: grade 9/10 Bavaria: grade 9 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 7–9 Hamburg: grade 9/10 Lower Saxony: grade 9/10 North Rhine-Westphalia: grade 9/10 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grade 9 Saxony: grade 11/12 Saxony-Anhalt: grade 9 Schleswig-Holstein: grades 7–9, 11
Unit GK-2 Matrices (basic course)Linear systems in matrix notation, multiplying matrices and a further application.
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Grade 12–13 · University Baden-Württemberg: grade 12/13 Hesse: grade 12 Lower Saxony: grade 12/13 Rhineland-Palatinate: grade 11–13 Saxony: grade 11/12
Unit GK-3 Vectors, lines and planes (basic course)The 3D coordinate system, calculating with vectors, lines and planes in parametric and coordinate form and their relative positions.
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 11
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 11
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit GK-4 Binomially distributed random variables (basic course)Multi-stage experiments with tree diagrams, two-way tables and simulation, independence, Bernoulli chains and the measures of the binomial distribution.
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Grade 8–13 Baden-Württemberg: grade 9/10 Bavaria: grades 9, 11 Berlin: grades 9, 11 Brandenburg: grades 9, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grades 9/10, 12/13 North Rhine-Westphalia: grades 9/10, 12/13 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grades 7/8, 11/12 Schleswig-Holstein: grade 11
The histogram of the binomial distribution: choose a range and read off the probability directly.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 13 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 12, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Grade 9–12 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 9/10 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grades 10, 11/12 Saxony-Anhalt: grades 9, 11/12 Schleswig-Holstein: grade 12
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Grade 8–13 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grade 7/8 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grades 8, 11/12 Saxony-Anhalt: grade 7/8 Schleswig-Holstein: grades 7–9, 11
Unit GK-5 Integral calculus (basic course)Antiderivatives, the definite integral as reconstructed total and area, the fundamental theorem and calculating areas.
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
Grade 11–13 · University Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
Unit GK-6 Inferential statistics (basic course)Estimating parameters and one- and two-sided significance tests for the binomial distribution.
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Grade 12–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 13 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Schleswig-Holstein: grade 13
Unit GK-7 Distances and angles (basic course)Dot and cross product, normal vectors, angles of intersection and distances between points and planes.
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
Grade 10–12 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 12
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit GK-8 Further applications (basic course)Extremal distances, reflection in planes, areas, and families of functions and lines.
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grades 12, 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grades 12, 13 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 12, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 13
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit GK-W1 Conic sections (optional, basic course)Circle, ellipse, parabola and hyperbola as conic sections; constructing the ellipse.
No separate simulation is planned for this.
Unit GK-W2 Numerical methods for solving equations (optional, basic course)Graphical methods, bisection and Newton’s method.
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Grade 10–12 Baden-Württemberg: grade 12/13 Bavaria: grade 11 Bremen: grade 10 Hamburg: grade 12 Hesse: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
Unit GK-W3 Dynamical systems and fractals (optional, basic course)Predator–prey systems and fractal structures.
No separate simulation is planned for this.
Unit GK-W4 Conditional probability (optional, basic course)Conditional probability, Bayes’ theorem and stochastic independence.
A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.
Grade 9–13 Baden-Württemberg: grade 9/10 Bavaria: grade 11 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grades 9/10, 12/13 North Rhine-Westphalia: grades 9/10, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Grade 8–13 Baden-Württemberg: grade 9/10 Bavaria: grades 9, 11 Berlin: grades 9, 11 Brandenburg: grades 9, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grades 9/10, 12/13 North Rhine-Westphalia: grades 9/10, 12/13 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grades 7/8, 11/12 Schleswig-Holstein: grade 11
Play yourself or simulate thousands of games: is it worth switching doors?
Grade 10–12 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 9/10 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
Unit GK-W5 Numerical integration (optional, basic course)Rectangle and trapezium rules.
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
Unit GK-W6 Differential equations (optional, basic course)Simple differential equations for growth and motion.
No separate simulation is planned for this.
Unit LK-1 Differential calculus (advanced course)As in the basic course, in more depth: limits and continuity, rules of differentiation, curve sketching with asymptotes, function equations from conditions and optimisation problems.
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
Grade 10–11 Baden-Württemberg: grade 11 Bavaria: grade 11 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grades 11, 12 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 11 Berlin: grade 11 Brandenburg: grade 11 Bremen: grades 10, 11/12 Hamburg: grades 11, 12 Hesse: grades 11, 12 Lower Saxony: grade 11 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 11, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12
Which box holds the most, which can uses the least metal? Five everyday problems show how to find the best solution with the derivative.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 11, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
Grade 10–13 Baden-Württemberg: grade 12/13 Bavaria: grades 11, 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 10 Hamburg: grade 11 Hesse: grade 12 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12
What happens to a graph when x gets very large? A point runs outwards and a magnifier shows how close it gets to the asymptote.
Grade 10–12 Baden-Württemberg: grades 9/10, 12/13 Bavaria: grade 11 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 10 Hamburg: grade 11 Hesse: grades 11, 12 Lower Saxony: grades 9/10, 11, 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grades 10, 11/12 Saxony-Anhalt: grades 10, 11/12 Schleswig-Holstein: grades 10, 11
Does the graph cross the x-axis or only touch it? The multiplicity of the zero decides, the highest power decides what happens far out.
Grade 10–11 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grades 10, 11 Brandenburg: grades 10, 11 Bremen: grade 10 Hamburg: grades 9/10, 11 Hesse: grade 11 Lower Saxony: grade 11 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grades 10, 11/12 Schleswig-Holstein: grade 11
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
Drag three points and the matching parabola always passes through all of them. The equation is worked out step by step.
Grade 9–11 Baden-Württemberg: grade 9/10 Bavaria: grade 9 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 7–9 Hamburg: grade 9/10 Lower Saxony: grade 9/10 North Rhine-Westphalia: grade 9/10 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grade 9 Saxony: grade 11/12 Saxony-Anhalt: grade 9 Schleswig-Holstein: grades 7–9, 11
Unit LK-2 Matrices (advanced course)Linear systems in matrix notation, multiplying matrices and a further application.
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Grade 12–13 · University Baden-Württemberg: grade 12/13 Hesse: grade 12 Lower Saxony: grade 12/13 Rhineland-Palatinate: grade 11–13 Saxony: grade 11/12
Unit LK-3 Vectors, lines and planes (advanced course)As in the basic course, plus linear dependence of vectors and lines of intersection of planes.
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 11
Drag vector arrows in space, add them and multiply them by numbers. The calculation is shown alongside in column form.
Grade 10–12 Baden-Württemberg: grade 11 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 11
A point travels along a line through space, two vectors span a plane. Parametric, normal and coordinate forms all describe the same thing.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit LK-4 Binomially distributed random variables (advanced course)As in the basic course, plus approaches to probability, counting methods, conditional probability and Bayes’ theorem.
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Grade 8–13 Baden-Württemberg: grade 9/10 Bavaria: grades 9, 11 Berlin: grades 9, 11 Brandenburg: grades 9, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grades 9/10, 12/13 North Rhine-Westphalia: grades 9/10, 12/13 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grades 7/8, 11/12 Schleswig-Holstein: grade 11
A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.
Grade 9–13 Baden-Württemberg: grade 9/10 Bavaria: grade 11 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grades 9/10, 12/13 North Rhine-Westphalia: grades 9/10, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
Play yourself or simulate thousands of games: is it worth switching doors?
Grade 10–12 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 9/10 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
The histogram of the binomial distribution: choose a range and read off the probability directly.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 13 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 12, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
Left or right at every pin, purely by chance. Yet every time the balls pile up into the same bell shape.
Grade 9–12 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 9/10 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grades 10, 11/12 Saxony-Anhalt: grades 9, 11/12 Schleswig-Holstein: grade 12
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Grade 8–13 Baden-Württemberg: grade 9/10 Bavaria: grade 10 Berlin: grades 9, 10, 11 Brandenburg: grades 9, 10, 11 Bremen: grade 11/12 Hamburg: grades 9/10, 11 Hesse: grade 13 Lower Saxony: grade 7/8 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grades 9/10, 11–13 Saarland: grades 10, 12 Saxony: grades 8, 11/12 Saxony-Anhalt: grade 7/8 Schleswig-Holstein: grades 7–9, 11
Unit LK-5 Integral calculus (advanced course)As in the basic course, plus the integral function, linear substitution and volumes of solids of revolution.
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
The area under a graph grows as you move along. Its rate of change is exactly the function itself: this is the fundamental theorem.
Grade 11–13 · University Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
Unit LK-6 Normally distributed random variables (advanced course)Density and distribution function, measures and the link to the binomial distribution.
How many bottles are underfilled? Under the bell curve every probability is an area, with the sigma rules as a rule of thumb.
Grade 12–13 · University Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 13 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 12, 13
Unit LK-7 Inferential statistics (advanced course)Estimating parameters and one- and two-sided significance tests for the binomial distribution.
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Grade 12–13 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 13 Rhineland-Palatinate: grade 11–13 Saarland: grade 13 Saxony: grade 11/12 Schleswig-Holstein: grade 13
Unit LK-8 Distances and angles (advanced course)Dot and cross product, Hesse normal form, angles of intersection and all distances between points, lines and planes.
The dot product tells you the angle between two vectors; the cross product gives a perpendicular vector and an area.
Grade 10–12 Baden-Württemberg: grade 12/13 Bavaria: grade 12 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 10 Schleswig-Holstein: grade 12
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit LK-9 Further applications (advanced course)Extremal distances and angles, reflection, bounded and unbounded areas, and families.
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grades 12, 13 Berlin: grades 11, 12 Brandenburg: grades 11, 12 Bremen: grade 11/12 Hamburg: grades 11, 12 Hesse: grades 12, 13 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grades 12, 13 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 13
Do the lines meet, run parallel or pass each other by? In a 3D space you can turn, intersections, angles and distances become visible.
Grade 11–13 Baden-Württemberg: grades 11, 12/13 Bavaria: grade 13 Berlin: grade 12 Brandenburg: grade 12 Bremen: grade 11/12 Hamburg: grade 12 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grades 11, 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grades 11, 12, 13
Unit LK-W1 Conic sections (optional, advanced course)Conic sections, the ellipse with construction and tangents.
No separate simulation is planned for this.
Unit LK-W2 Numerical methods for solving equations (optional, advanced course)Bisection, Newton’s method and general iteration.
Approximate a zero with tangent lines. Each step gets you closer, but sometimes it goes wrong.
Grade 10–12 Baden-Württemberg: grade 12/13 Bavaria: grade 11 Bremen: grade 10 Hamburg: grade 12 Hesse: grade 11 Rhineland-Palatinate: grade 11–13 Saarland: grade 11 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 11
Unit LK-W3 Dynamical systems and fractals (optional, advanced course)Predator–prey systems, fractals, the Feigenbaum diagram and chaos.
No separate simulation is planned for this.
Unit LK-W4 Functions of two variables (optional, advanced course)Graphs in space, cross-sections and partial derivatives.
No separate simulation is planned for this.
Unit LK-W5 Numerical integration (optional, advanced course)Rectangle, trapezium and Simpson’s rules.
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Grade 11–13 Baden-Württemberg: grade 12/13 Bavaria: grade 13 Berlin: grade 11 Brandenburg: grade 11 Bremen: grade 11/12 Hamburg: grade 11 Hesse: grade 12 Lower Saxony: grade 12/13 North Rhine-Westphalia: grade 12/13 Rhineland-Palatinate: grade 11–13 Saarland: grade 12 Saxony: grade 11/12 Saxony-Anhalt: grade 11/12 Schleswig-Holstein: grade 12
Unit LK-W6 Differential equations (optional, advanced course)Differential equations for growth, motion and the harmonic oscillator, slope fields.
No separate simulation is planned for this.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Lehrplan, Gymnasium, Mathematics, grade 11/12
