Comprehensive Guide to the Mathematics Basisfach Oral Abitur Examination Baden-Württemberg (from 2023)

General Educational Framework for the Mathematics Basisfach Oral Abitur

According to the 2016 Education Plan in Baden-Württemberg, the basic subject (Basisfach) in mathematics is designed to help students acquire and expand competencies that enable them to recognize and explain mathematical relationships while acting mathematically with understanding. The lessons are structured more strictly than the intensive course, often utilizing heuristic observations and descriptive arguments. A primary goal is to foster an understanding of mathematics as a practical tool for addressing real-world problems outside the classroom through a reality-based approach. These guidelines and task examples apply to oral examinations starting from the year 2023.

Examination Structure and Procedures

Every oral examination in the mathematics Basisfach consists of two distinct parts: a presentation (Vortrag) and an examination discussion (Prüfungsgespräch). Each part typically lasts approximately $10$ minutes, resulting in a total exam duration of roughly $20$ minutes per student. The exam covers two subject areas: Analysis is mandatory and can be included in either the first or second part of the exam. The second subject area is a choice between either Analytic Geometry or Stochastics.

The structure of the task involves a written assignment for the first part (the presentation), which includes fully formulated subtasks provided with specific mathematical operators. This is the only task the student receives in writing before the preparation period begins. The second part of the exam, the discussion, is based on a short written input that the student receives only at the start of the second part. The second part is designed to cover a range of aspects across all three requirement levels (Anforderungsbereiche), which must be clearly identified.

Requirements and Task Design

Tasks for the oral exam must be designed so that they allow an easy entry point for the student while remaining complex enough to allow for any grade to be achieved across the three requirement levels (I, II, and III). There are no "easy" or "difficult" tasks; instead, every task must contain sub-tasks that differentiate between sufficient and excellent performance. The tasks are derived from the curriculum taught at the gymnasiale Oberstufe level and must balance breadth and depth of content. They should cover a wide range of competencies without becoming too superficial, while also focusing on specific areas for in-depth understanding.

The tasks differ from those in the written examination in that they should avoid extensive calculations or time-consuming geometric constructions. The emphasis is on the student's ability to present mathematical facts freely and respond to questions. Tasks are particularly suitable if they require the explanation of a solution path without performing all individual calculations, or if they provide sketches and results for the student to explain the underlying logic. Open-ended tasks that gradually expand the problem offer students the best chance to demonstrate the depth of their mathematical understanding.

Evaluation Principles and Grading Criteria

A mathematical performance is graded as "good" ($11$ points) if the student delivers performances in all three requirement levels ($I$, $II$, and $III$). A grade of "sufficient" ($05$ points) requires that the student provides performances in requirement level $I$ and at least one other level. The fundamental criteria for evaluation include the quantity and quality of demonstrated competencies, the logical structure and clarity of the presentation, the mastery of technical language, and the appropriate use of presentation media.

Further criteria include the student's insight into mathematical problems and their ability to recognize connections, judge facts, respond to questions or objections, and utilize hints provided by the examiners. Additionally, the examination evaluates the student's creativity, ability to reflect, and independence throughout the process. All the specific requirement levels for sub-tasks must be indicated in a written expectation horizon (Erwartungshorizont), though specific point values are not assigned to individual sub-tasks.

Organizational Guidelines and Permitted Aids

The preparation time for the oral exam is $20$ minutes. This time is intended to allow the student to complete the task for the first part and prepare their presentation. During preparation, students may be allowed to use specific aids such as a scientific calculator (WTR) or a formula sheet (Merkhilfe), but the specific task must dictate whether these are permitted. During the actual examination itself, no aids are allowed. Students should have the opportunity to visualize the notes they took during preparation, but they should avoid wasting time writing down things they have already prepared.

Teachers must submit at least two more tasks than are required for the group of students to allow the head of the examination committee to make a selection. Tasks can be used for up to three consecutive examinations and across parallel courses happening simultaneously. The subject areas must be represented in sufficient numbers across all submitted tasks. These assignments must be submitted in writing by the course teacher at least one week before the examination date.

Analysis Task Examples and Solution Logic

In Task Example 1, students are asked to match three types of functions—polynomial $f$, trigonometric $g$, and exponential $h$—to their respective graphs based on characteristic properties. They might derive a function term such as $f(x) = -\frac{1}{6} \times (x - 2) \times (x - 3) \times (x + 4)$ or $h(x) = 2 \times e^{-x}$. Analysis tasks also include determining the area between a graph and the $x$-axis using the procedure of finding zeros, calculating sub-areas with integrals, and summing them: $A = -\textstyle\int_{-4}^{2} f(x) dx + \textstyle\int_{2}^{3} f(x) dx$. In Requirement Level III, students might find values for parameters to satisfy an integral equation, such as finding $a = -\ln(3)$ given $\textstyle\int_{0}^{a} (e^{-x} - 1) dx = -2$.

Task Example 2 focuses on a real-world scenario involving a car's velocity given by $f(x) = 24 + 24 \times e^{-0.08 \times x}$, where $x$ is in seconds and $f(x)$ is in $\text{m/s}$. Students must calculate and interpret values like $f(0) = 48$ (initial velocity) and the initial acceleration $f'(0) = -1.92 \, \text{m/s}^2$. The total distance over $60$ seconds is calculated via the integral $\textstyle\int_{0}^{60} f(x) dx \approx 1737.53 \, \text{m}$. Advanced questions involve determining when a motorcycle (velocity $g(x)$) overtakes the car by solving the equation for the distance: $\textstyle\int_{0}^{x_0} f(x) dx = \textstyle\int_{0}^{x_0} g(x) dx$.

Task Example 4 involves analyzing the graph of a function and its antiderivative $F$. Students determine slopes graphically (e.g., $f'(0) \approx -1$) and use "box counting" to approximate integrals: $\textstyle\int_{0}^{2} f(x) dx \approx -4.25$. They must relate the properties of $f$ (zeros and extrema) to the features of $F$ (extrema and inflection points). For instance, a zero with a sign change from negative to positive in $f$ at $x_1 = -2$ indicates a local minimum for $F$. Students also match function terms like $f_2(x) = (x - 2) \times e^x$ to the graph based on specific points and zeros.

Analytic Geometry Task Examples and Aspects

Analytic Geometry tasks often utilize inputs involving geometric bodies or planes. In Task Example 1, Part 2, a square pyramid with height $4$ is examined. Discussion aspects in requirement level I include identifying vertex coordinates like $S(1.5|1.5|4)$, finding volumes through formulas, and stating plane equations for side faces. Requirement level II covers proving that a side triangle is isosceles, describing processes for finding the total surface area, and determining the formula for the angle between the base and a side face. Requirement level III asks students to find planes that bisect the pyramid's volume or analyze shadow projections from a light source at point $L$.

Task Example 2, Part 2, uses coordinate equations of planes, such as $E: 2x_1 - x_2 + 3x_3 = 6$ and $F: 2x_1 - x_2 = 6$. Students interpret plane positions through trace points (Spurpunkte) and normal vectors $\mathbf{n} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$. They must explain why plane $F$ has a special position (parallel to the $x_3$-axis) and calculate distances to points or other objects. Advanced discussion points involve reflecting a line across the plane or explaining subsets of a plane in parameter form when parameters are restricted to intervals like $[0, 1]$.

Task Example 7, Part 1, requires students to work with lines through points. For points $A(2|1|0)$ and $B(4|0|2)$, the line equation is $g: \mathbf{x} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + t \times \begin{pmatrix} 2 \\ -1 \\ 2 \end{pmatrix}$. Students check if a point like $C(0|2|-2)$ lies on the line and prove that the line lies within a specific plane $E: x_1 + 2x_2 = 4$ by verifying that both points $A$ and $B$ lie on $E$. Requirement level III tasks include describing a procedure to find a point $P$ that forms an equilateral triangle with $A$ and $B$ by finding the midpoint $M$, a perpendicular vector $\mathbf{v}$, and setting the distance $|\text{AP}| = |\text{AB}|$

Stochastics Task Examples and Statistical Logic

Stochastics tasks cover probability distributions, expected values, and binomial/normal distributions. Task Example 9 involves an urn with six blue and four white balls. If three balls are drawn with replacement, the probability for event A (all blue) is $P(A) = 0.6^3 = 0.216$. If drawn $15$ times, the number of blue balls $Y$ is binomial-distributed because the trials are independent and the probability remains constant ($p = 0.6$). The expected value is $E(Y) = n \times p = 15 \times 0.6 = 9$, meaning that in the long run, an average of $9$ out of $15$ balls will be blue. Probabilities for ranges like $5 \le Y \le 10$ are found via $P(Y \le 10) - P(Y \le 4) \approx 0.773$.

Task Example 10 focuses on the normal distribution of lengths $L$ for plastic parts with $\mu = 100$ and $\sigma = 3.4$. Students find probabilities such as $P(L < 95) \approx 0.071$ and identify symmetric events like $P(L > 105)$. They must interpret the bell curve's maximum point as $\mu$ and the inflection points as $\mu \pm \sigma$. In Requirement level III, students evaluate combined probabilities, such as the term $\binom{200}{200} 0.14^{200} + \binom{200}{199} 0.14^{199} \times 0.86^1 + \binom{200}{198} 0.14^{198} \times 0.86^2$, which represents the probability that at least $198$ out of $200$ parts are defective (where the defect rate is $14\%$).

Task Example 11 utilizes four-field tables to explore conditional probability and stochastic independence. Given that $37\%$ of men have an ideal height and $22\%$ both have the ideal height and interest in a photoshoot, the conditional probability that a man with ideal height has interest is $P_B(A) = \frac{22}{37} \approx 0.595$. Independence is checked by comparing $P(A) \times P(B)$ to $P(A \cap B)$; here, $0.41 \times 0.37 = 0.1517 \neq 0.22$, so the events are dependent. Students must also explain why a binomial distribution is inappropriate for small groups drawn without replacement, typically requiring a hypergeometric approach for exactness even if the binomial is used as an approximation in larger populations.