Two-way tables and Venn diagrams
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Mathematics – grade 13
Qualification phase Q3 “Probability and statistics” and Q4 “Families of functions, problem solving and reasoning”. Q3.1–Q3.2 (in the advanced course also Q3.3) and Q4.1 are mandatory. Assigning them to grade 13 is an assumption (G9). “Advanced course only” marks the higher level.
Unit Q3.1Basic concepts and methods of probability
Mandatory topic area. Concept of probability, multi-stage experiments, events and sets, conditional probability, independence and combinatorics.
Q3.1.1Statistical concept of probability
Absolute and relative frequency and the empirical law of large numbers.
No separate simulation is planned for this.
Q3.1.2Foundations of probability theory
Outcome, event, Laplace probability and comparison with experimental results.
No separate simulation is planned for this.
Q3.1.3Determining probabilities in multi-stage random experiments
Using tree diagrams and path rules.
No separate simulation is planned for this.
Q3.1.4Combining events
And, or and complementary events, set notation, also with a two-way table.
Q3.1.5Conditional probabilities
Conditional probabilities in tree diagrams and contingency tables; stochastic independence.
Conditional probability with medical tests
A medical test is positive, but how likely is it that you are really ill? 1000 people show why the answer is often surprising.
Two-way tables and Venn diagrams
A two-way table, a Venn diagram and a tree diagram show the same data and are linked. Change one and the others follow.
Q3.1.6Solving combinatorial counting problems
Product rule, samples with/without replacement, factorial, binomial coefficient.
No separate simulation is planned for this.
Q3.1.7Determining Laplace probabilities with counting methods
Comparing urn models with and without replacement (e.g. lottery).
Simulating the birthday problem
How likely is it that two people in a class share a birthday? From just 23 people on, it is more likely than not.
Unit Q3.2Probability distributions
Mandatory topic area. Random variables with expected value and spread, Bernoulli chains and binomial distribution; in the advanced course also the normal distribution.
Q3.2.1Developing basic concepts
Random variable, distribution as histogram, expected value, variance, standard deviation; fair games.
No separate simulation is planned for this.
Q3.2.2Bernoulli chains
Justifying Bernoulli's formula and applying it in context; limits of the model.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
Q3.2.3Binomially distributed random variables
Parameters, histograms and cumulative probabilities, including inverse problems.
Binomial distribution
The histogram of the binomial distribution: choose a range and read off the probability directly.
Q3.2.4Normally distributed random variables, probability density of the normal distribution (advanced course only)
Density function, bell shape, μ and σ, maximum and inflection points.
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.
Q3.2.5Probabilities of normally distributed random variables (advanced course only)
Cumulative distribution function, sigma rules and inverse problems.
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.
Unit Q3.3Hypothesis tests
[mandatory in the advanced course only (optional in the basic course)] Significance tests for binomially distributed random variables with decision rule and type I and II errors; in the advanced course also the operating characteristic.
Q3.3.1Hypothesis tests (right-, left- and two-tailed significance tests) for binomially distributed random variables
Setting hypotheses, finding the rejection region, calculating type I and II errors.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Q3.3.2Graphical representation of the probabilities of keeping or rejecting the null hypothesis depending on the true value of the success probability p (advanced course only)
Plotting and interpreting the operating characteristic or power function.
No separate simulation is planned for this.
Unit Q3.4Prediction and confidence intervals
Optional topic area. Using sigma rules to infer from population to sample and vice versa.
Q3.4.1Prediction intervals for relative frequencies
Determining expected ranges for sample results.
No separate simulation is planned for this.
Q3.4.2Confidence intervals for probabilities
Estimating a range for the unknown probability from a sample.
No separate simulation is planned for this.
Q3.4.3Confidence diagrams (confidence ellipse) (advanced course only)
Representing confidence regions graphically.
No separate simulation is planned for this.
Q3.4.4Sample sizes (advanced course only)
Calculating the sample size needed for a confidence level.
No separate simulation is planned for this.
Unit Q3.5Statistics and further probability distributions
Optional topic area. Own data collection and simulations, empirical statistics, Poisson distribution; in the advanced course approximations and further continuous distributions.
Q3.5.1Statistical concept of probability
Carrying out own random experiments and simulating them with software.
No separate simulation is planned for this.
Q3.5.2Working with data
Assessing own data with mean, empirical variance and standard deviation.
No separate simulation is planned for this.
Q3.5.3Poisson distribution
Calculating rare events with the Poisson distribution.
No separate simulation is planned for this.
Q3.5.4Normal distribution as approximation of the binomial distribution (advanced course only)
Approximating binomial histograms with large spread by the bell curve.
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.
Q3.5.5Poisson distribution as approximation of the binomial distribution with constant expected value (advanced course only)
Proving the limit as n → ∞.
No separate simulation is planned for this.
Q3.5.6Further continuous distributions (advanced course only)
E.g. the exponential distribution.
No separate simulation is planned for this.
Unit Q4.1Families of functions
Mandatory topic area. Investigating and integrating functions with a parameter, especially polynomial families and families combined with exponential functions; in the advanced course loci.
Q4.1.1Investigating and integrating families of polynomial functions, meaning of the parameter for the graph
Influence of the parameter on zeros, extrema and areas.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Q4.1.2Investigating families combining exponential and polynomial functions
Investigating and integrating families built from exponential functions and polynomials.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Q4.1.3Investigating families of all other known function classes if the parameter only causes a stretch or shift
Interpreting the parameter as a shift or stretch of the graph.
Families of functions
One parameter, infinitely many graphs. Where do the maxima go as k grows? Along a curve of their own, the locus.
Q4.1.4Loci (advanced course only)
Determining the loci of extreme and inflection points.
No separate simulation is planned for this.
Unit Q4.2Problem solving and reasoning
Optional topic area. Process-oriented topic area: recognising problems, using heuristic strategies and developing arguments, applied to freely chosen content from all areas.
Q4.2.1Recognising and formulating mathematical problems
Framing questions mathematically.
No separate simulation is planned for this.
Q4.2.2Applying heuristic strategies and principles to solve mathematical problems
E.g. working forwards/backwards, systematic trial, extremal and symmetry principles.
No separate simulation is planned for this.
Q4.2.3Assessing given statements or commenting on them
Checking statements and evaluating them with reasons.
No separate simulation is planned for this.
Q4.2.4Following, explaining and developing arguments and logical conclusions
Understanding chains of reasoning and building them oneself.
No separate simulation is planned for this.
Q4.2.5Application to topic areas chosen by the teacher as examples
Examples from calculus, analytic geometry, linear algebra and probability.
No separate simulation is planned for this.
Q4.2.6Probability and statistics (advanced course only)
Strategies for hypotheses and critical values; influence of n on the operating characteristic.
One-sided significance test: type I and II errors
A test decides between two hypotheses. Coloured areas show how often it can be wrong.
Unit Q4.3Complex numbers
Optional topic area. Calculating with complex numbers in algebraic and polar form and solving equations in ℂ.
Q4.3.1Algebraic form
Real and imaginary part, basic operations.
No separate simulation is planned for this.
Q4.3.2Polar form
Representation in the complex plane and geometric interpretation of the operations.
No separate simulation is planned for this.
Q4.3.3Solving equations in the set ℂ of complex numbers
Solving equations that have no solution in ℝ.
No separate simulation is planned for this.
Q4.3.4Representing complex numbers with 2×2 matrices (advanced course only)
Complex numbers as rotation-dilation matrices.
No separate simulation is planned for this.
The summaries describe the curriculum in our own words. The original text is authoritative. Original text (German): Kerncurriculum, Gymnasium, Mathematics, grade 13