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.
Mathematics – grade 11
In the nine-year Gymnasium, grade 11 is a separate stage before the course phase.
Unit 3.4.1Guiding idea: number – variable – operation
Calculating with tuples (vectors) and first rules of differentiation including the derivatives of sine and cosine.
3.4.1 (1)Working with vectors as tuples
Adding and scaling tuples, writing them as linear combinations and interpreting the operations geometrically.
3.4.1 (2)–(3)Differentiating function terms
Applying the constant-factor, power and sum rules and stating the derivatives of sine and cosine.
No separate simulation is planned for this.
Unit 3.4.2Guiding idea: measuring
Distances between points in space and the magnitude of a vector as a length.
3.4.2 (1)–(2)Calculating lengths in Cartesian coordinate systems
Finding the distance between two points and calculating the magnitude of a vector as a length.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
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.
Unit 3.4.3Guiding idea: space and shape
Points, vectors and lines in a three-dimensional coordinate system; investigating how lines lie relative to each other.
3.4.3 (1)–(7)Working with geometric objects in Cartesian coordinate systems
Plotting points in an oblique view, calculating midpoints, checking vectors for collinearity, describing lines with parametric equations, investigating their relative position and illustrating them with trace points.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
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 3.4.4Guiding idea: functional relationships
Introduction to differential calculus: from average to instantaneous rate of change, tangents and the derivative function; using derivatives to investigate monotonicity, extrema and points of inflection.
3.4.4 (1)–(2)Working with characteristic properties of functions
Defining monotonicity and distinguishing local from global extrema.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
3.4.4 (3)–(14)Understanding the basic idea of differential calculus and working with derivatives
Difference quotient as secant gradient, derivative as limit and tangent gradient, tangent and normal, derivative function and monotonicity theorem; investigating graphs with higher derivatives and reasoning graphically between f and f′.
From secant to tangent
Let a point Q move towards P: the secant becomes the tangent, and the average slope becomes the derivative.
The derivative function graphically
A tangent slides along the graph and draws its slope as a new graph: the derivative function.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Unit 3.4.5Guiding idea: data and chance
Bernoulli chains and the binomial distribution with its key figures; effect of the parameters n and p in graphs.
3.4.5 (1)–(7)Working with binomial distributions
Recognising Bernoulli experiments, explaining Bernoulli’s formula and binomial coefficients, calculating probabilities, plotting and interpreting distributions, finding missing parameters and calculating expected value and standard deviation.
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.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Bildungsplan, Gymnasium, Mathematics, grade 11