The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Mathematics – grade 12
Qualification phase Q1 “Calculus” and Q2 “Analytic geometry, linear algebra and further calculus”. Q1.1–Q1.3 and Q2.1–Q2.3 are mandatory. Assigning them to grade 12 is an assumption (G9). “Advanced course only” marks the higher level.
Unit Q1.1Introduction to integral calculus
Mandatory topic area. The integral as reconstructed total and as signed area, rectangle sums, fundamental theorem and first integration rules.
Q1.1.1Meaning of the integral as accumulated quantity and as signed area
Reconstructing a total from rate of change and initial value; integral as a sum of products.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Q1.1.2Areas under a function graph
Approximating areas with upper and lower sums and passing to the definite integral via a limit.
The integral as signed area (rectangle sums)
Fill the area under a graph with rectangles. The more rectangles, the more precisely you get the value of the integral.
Q1.1.3Fundamental theorem of calculus
Justifying the link between differentiation and integration intuitively; function and antiderivative graphically.
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.
From a graph to its antiderivative
The height of one graph becomes the slope of another. If you understand bank balances, the antiderivative comes quicker than you think.
Q1.1.4Developing integration rules from differentiation rules
Antiderivatives of powers, eˣ, sin x and cos x; constant-factor and sum rule.
No separate simulation is planned for this.
Q1.1.5Integrating e^(ax+b), sin[b·(x − c)], cos[b·(x − c)]
Finding antiderivatives of functions with a linear inner function.
No separate simulation is planned for this.
Q1.1.6Integrating (ax + b)^r with r ∈ ℚ \ {−1}, integrating g(x) = f′(x)·e^f(x) (advanced course only)
Further antiderivatives by reversing the chain rule.
No separate simulation is planned for this.
Unit Q1.2Applications of integral calculus
Mandatory topic area. Areas between graphs, totals and mean values in context; in the advanced course also solids of revolution and improper integrals.
Q1.2.1Calculating areas
Calculating areas between graphs and lines parallel to the axes, also backwards (unknown bound or parameter).
No separate simulation is planned for this.
Q1.2.2Applying the integral to calculations in context
Reconstructed total, mean value and average rate of change.
The integral as total change: from velocity to distance
From the speed of a car you get the distance travelled: the area under the curve shows the total change.
Q1.2.3Reasoning and justifying with integral expressions
Interpreting integral expressions in context and justifying statements.
No separate simulation is planned for this.
Q1.2.4Solids of revolution (advanced course only)
Justifying the volume formula and calculating volumes of solids of revolution about the x-axis.
Solids of revolution
The area under a graph rotates about the x-axis and becomes a solid. Its volume is calculated with thin discs.
Q1.2.5Improper integrals (advanced course only)
Examining infinitely extended regions for finite area.
No separate simulation is planned for this.
Unit Q1.3Deepening differential and integral calculus
Mandatory topic area. Confident use of the introductory-phase content, combined exponential functions and growth models; in the advanced course functions with parameters and logistic growth.
Q1.3.1Competent use of the content developed in the introductory phase
Applying functions, derivatives and differentiation rules in an integrated way.
Monotonicity, extrema and inflection points
Analyse a graph step by step: zeros, maximum and minimum points, inflection points and where it rises or falls.
Q1.3.2Investigating exponential functions combined with polynomial functions
Investigating sums, products and compositions of exponential functions and polynomials, also in context.
The natural exponential function
Why is the number e so special? For the exponential function, the graph and its derivative are exactly the same.
Q1.3.3Growth and decay processes
Modelling exponential and bounded growth with measured data; solving exponential equations.
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
Q1.3.4Investigating functions with parameters combining exponential and polynomial functions, as well as catenaries and bell curves (advanced course only)
Investigating the influence of parameters on combined exponential functions.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Q1.3.5Logistic growth (advanced course only)
Applying logistic growth models (without deriving them from the differential equation).
No separate simulation is planned for this.
Unit Q1.4Integration techniques
Optional topic area. Further methods for finding antiderivatives.
Q1.4.1Finding an antiderivative by an ansatz with comparison of coefficients
Determining an antiderivative with a suitable ansatz and comparison of coefficients.
No separate simulation is planned for this.
Q1.4.2Integration by parts (advanced course only)
Solving integrals of products by integration by parts.
No separate simulation is planned for this.
Q1.4.3Integration by substitution (advanced course only)
Simplifying integrals by substitution.
No separate simulation is planned for this.
Unit Q1.5Ordinary differential equations
Optional topic area. Describing growth models with differential equations and solving first-order equations.
Q1.5.1Describing and developing models for growth and decay with differential equations
Translating growth assumptions into differential equations.
No separate simulation is planned for this.
Q1.5.2Solution methods for first-order differential equations
Slope fields, separation of variables and other elementary methods.
No separate simulation is planned for this.
Q1.5.3Investigating second-order differential equations (advanced course only)
Second-order differential equations, e.g. for periodic processes.
No separate simulation is planned for this.
Unit Q2.1Deepening of calculus
Mandatory topic area. Further function classes (simple rational and root functions), graph transformations and inverse functions; in the advanced course the natural logarithm and compositions.
Q2.1.1Simple rational functions and simple root functions
Properties and derivatives of f(x) = 1/xⁿ and f(x) = √x.
Poles and asymptotes of rational functions
A zero or a pole, a hole or a leap to infinity? Drag the terms of the fraction and the graph shows the difference.
Limits and asymptotes
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.
Q2.1.2Shifting and stretching
Shifting, stretching and reflecting graphs; interpreting g(x) = a·f(b·(x − c)) + d.
No separate simulation is planned for this.
Q2.1.3Inverse function
Finding the inverse and interpreting it as reflection in y = x; domain and range.
Inverse functions: reflecting in y = x
Reflect the graph in y = x and the inverse function appears. It only works if no function value occurs twice.
Q2.1.4The natural logarithm function f(x) = ln(x) (advanced course only)
Describing ln as the inverse of the exponential function and transforming it.
Inverse functions: reflecting in y = x
Reflect the graph in y = x and the inverse function appears. It only works if no function value occurs twice.
Logarithms as the inverse of exponentiation
The logarithm answers the question: to what power do I raise the base to get this number? With graphs, rules and exponential equations.
Q2.1.5The natural logarithm as antiderivative of 1/x (advanced course only)
Using ln(x) as the antiderivative of 1/x.
No separate simulation is planned for this.
Q2.1.6Combining and composing functions (advanced course only)
Justifying properties of sums, products and compositions from the graphs.
Function plotter with custom terms
Up to three of your own functions at once, with a table of values, zeros and intersections. The tool for anything that can be a graph.
Unit Q2.2Orientation and movement in space
Mandatory topic area. Spatial coordinate system, vectors, dot product and angles, and investigation of simple solids and figures in space.
Q2.2.1Spatial coordinate systems
Representing points and solids in a 3D coordinate system, also with software.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Q2.2.2Vectors
Translations as vectors, vector arithmetic, collinearity, magnitude and distance.
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.
Q2.2.3Angles
Dot product, orthogonality and angles between vectors.
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.
Q2.2.4Simple geometric solids and figures in space
Investigating and calculating cubes, prisms, pyramids and plane figures in space.
Points and solids in 3D coordinates
Place points in 3D coordinates, rotate the view and calculate distances with Pythagoras' theorem.
Unit Q2.3Lines and planes in space
Mandatory topic area. Systems of linear equations, lines and planes in different forms, relative positions, angles, distances and applications such as shadows.
Q2.3.1Systematic solution of systems of linear equations (including over- and underdetermined)
Solving systems algorithmically and interpreting solution sets geometrically.
No separate simulation is planned for this.
Q2.3.2Parametric representations
Lines and segments with parametric equations, point test and trace points.
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.
Q2.3.3Relative position of lines
Intersecting, parallel, identical or skew; intersection point and angle.
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.
Q2.3.4Application to linear motion
Describing motion with lines; direction vector as velocity.
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.
Q2.3.5Planes in different forms
Parametric and coordinate form, normal vector.
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.
Q2.3.6Representing planes and bounded regions (especially triangle, parallelogram) in space with a parametric equation
Point test and trace points of planes.
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.
Q2.3.7Converting between the forms of a plane
Converting forms; determining the position of a line relative to a plane and intersection points.
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.
Q2.3.8Calculating the angle between line and plane and between two planes
Intersection angles using direction and normal vectors.
No separate simulation is planned for this.
Q2.3.9Investigating angles (especially orthogonality and parallelism) and the relative positions mentioned, each depending on an (additional) parameter
Investigating positions and angles depending on a parameter.
No separate simulation is planned for this.
Q2.3.10Reflection and orthogonal projection
Reflecting a point in a plane; foot-of-perpendicular method for point–plane distance.
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.
Q2.3.11Application in context
Shadows, projections onto coordinate planes and area calculations.
No separate simulation is planned for this.
Q2.3.12Normal form of a plane (advanced course only)
Describing planes via normal vector and position vector.
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.
Q2.3.13Further relative positions and distance calculations (advanced course only)
Position of two planes, lines of intersection, distances between points, lines and planes, general reflections.
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.
Q2.3.14Cross product (advanced course only)
Using the cross product for normal vectors and areas.
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.
Q2.3.15Families of lines (advanced course only)
Investigating lines with a parameter.
No separate simulation is planned for this.
Q2.3.16Families of planes, especially in coordinate form (advanced course only)
Investigating planes with a parameter.
No separate simulation is planned for this.
Q2.3.17Representing parabolic trajectories using parameters (advanced course only)
Describing trajectories (e.g. of a ball) parametrically.
No separate simulation is planned for this.
Unit Q2.4Matrices for describing transition processes
Optional topic area. Modelling transition processes with graphs, state vectors and transition matrices; matrix arithmetic and Markov chains.
Q2.4.1Describing transition processes using state/transition graphs, state vectors and transition matrices
Modelling e.g. populations, voter movements or customer flows.
No separate simulation is planned for this.
Q2.4.2Calculating with matrices
Scalar, matrix–vector and matrix multiplication, addition, simple inverses.
No separate simulation is planned for this.
Q2.4.3Markov chains
Calculating states step by step (also backwards) and finding stable states via fixed vectors.
No separate simulation is planned for this.
Q2.4.4Determining and interpreting entries of matrix powers
Interpreting entries of M² etc. in context.
No separate simulation is planned for this.
Q2.4.5Long-term development of transition processes (advanced course only)
Interpreting matrix powers, limiting processes and limit matrices.
No separate simulation is planned for this.
Unit Q2.5Matrices for representing linear geometric transformations
Optional topic area. Representing and composing geometric transformations such as reflections, projections and dilations with matrices.
Q2.5.1Describing geometric transformations using matrices
Describing e.g. shadows as a mapping given by a matrix.
Matrices as linear maps
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Q2.5.2Calculating with matrices
Adding and multiplying matrices and finding simple inverses.
No separate simulation is planned for this.
Q2.5.3Representing linear geometric transformations with matrices
Transformation matrices for reflections, parallel projections, central dilations and their compositions.
Matrices as linear maps
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
Q2.5.4Representing linear geometric transformations in three-dimensional Euclidean space ℝ³ with matrices (advanced course only)
Rotations about coordinate axes, projections onto arbitrary planes through the origin, fixed points.
Matrices as linear maps
Two arrows decide everything: wherever they point, the matrix rotates, stretches or reflects the whole grid.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 12