Teaching series
Math make visible
Interactive courses on functions, geometry, trigonometry, vectors, and statistics with visual experiments.
Didactic concept
The series connects mathematical content with a consistently visual and interactive working method: concepts are not only described, but implemented as manipulable models. The competency development targets the secure application of coordinate and function models, the structured implementation in program flows, and the interpretation of mathematical representations in dynamic visualizations. The progression is thematically grouped and leads from geometric basics over trigonometric and vectorial representations to chance experiments and statistical visualizations. Methodically, the connection between subject idea and implemented representation is in the foreground, so that theory and application are combined in a common working process.
For teachers, the series offers a clearly structured, modular structure with thematically closed modules that can be used individually or as a connected sequence. The respective contents are transparently described and support planned competency development along central mathematical fields of study.
Competency Development
- Modeling mathematical facts through interactive visualizations (e.g., functions, geometric objects, vectors, chance experiments)
- Working with coordinate systems, scaling, and graphical representation as a basis for mathematical interpretation
- Structured implementation of calculations and representations in program flows with event processing and interaction
- Analysis and comparison of function courses as well as parameter effects through zooming, shifting, and mode switching
- Transfer between mathematical model, numerical calculation, and visual representation in different thematic contexts.
Structure of the series.
Fundamentals of plane geometry.
This course establishes geometric basic models as interactive objects by visually constructing and computationally evaluating circles, triangles, points, and distances. Thus, coordinate reference, scaling, and fundamental measurement and calculation quantities are secured as a common basis for the further modules of the series.
Trigonometry & Unit Circle
This module uses the unit circle as a visual reference model to make angle representations and the connections between sin/cos immediately observable. The coupling with a sinusoidal time series extends the understanding of cyclic trends by connecting geometric situations with functional representation.
Vectors & Directions
The course introduces vectorial representations as graphical action objects and connects drawing, decomposition, and addition with immediate feedback through live evaluations. Simulating movement with acceleration creates an application-oriented transition from static vectors to dynamic state quantities.
Probabilities & Statistics – Visualizing Randomness
This module makes experiments with randomness visible by repeatedly executing and immediately graphically aggregating them, for example through distributions and histograms. Scatter plots and a simple regression extend the view from discrete frequencies to interpreting data patterns in a point-based representation.
Visually Explore Functions
Interactive courses on functions: Sine, Parabolas, Roots, Exponential Functions, and more.
This course bundles function theory as a visual comparison and analysis field by investigating different function types (linear, quadratic, cubic, absolute value, root, hyperbola, exponential functions) through uniform plotter interaction. The recurring control patterns (zoom, shift, mode switch) support a systematic progression from individual observations to justified comparisons of properties.
Contents at a Glance
Basics of Plane Geometry
| Content | Focus | Duration |
|---|---|---|
| Circle – Radius, Circumference, Area | Interactive Circle Sandbox: Create a circle with the mouse, live calculate geometric sizes, switch scale and units. | 0 min |
| Triangle – Geometry Sandbox | Interactive sandbox for exploring triangles: Set points, draw, live calculate and visualize geometric characteristics. | 0 min |
| Coordinate System – Sandbox | Sandbox for experimenting with coordinate systems in pygame: Origin, axes, points, and simple interactions. | 0 min |
| Distance between two points | Sandbox to explore distances between two points. Interactive visualization with two points, distance measurement, scaling, and visualization without fixed tasks. | 0 min |
Trigonometry & Unit Circle
| Content | Focus | Duration |
|---|---|---|
| Sandbox: Unit Circle Experiment | Sandbox to try out a simple unit circle canvas. Move the radius endpoint with the mouse or keyboard, switch the angle display between rad and deg, and observe cos/sin values. The content is intentionally reduced fragments that can be gradually expanded. | 0 min |
| Unit Circle → Sinus Time Series | Sandbox to experiment with the unit circle and sinus time series. Includes a graphical visualization, interactive controls, and basic event handling. | 0 min |
Vectors & Directions
| Content | Focus | Duration |
|---|---|---|
| Vectors draw and add | Interactive Sandbox for drawing, breaking down, and adding vectors with live stats. | 0 minutes |
| Vector Decomposition – Experimentation Sandbox | Sandbox for experimenting with 2D vector decomposition with graphical representation and mouse/keyboard interaction. | 0 minutes |
| Vector movement with acceleration | Simulation of position, speed, and acceleration with interactive control. | 0 minutes |
Probabilities & Statistics – Visualizing Randomness
| Content | Focus | Duration |
|---|---|---|
| Visualize random distributions | Interactive Sandbox approach to visualizing random distributions (coin vs dice) with live statistics and experimentation possibilities. | 0 minutes |
| Histograms for Dice | Sandbox guide: operation, didactic hints, and tips for experimenting with the die histogram. Understand how random rolls are converted into visual bars. | 0 minutes |
| Scatter Plots & Simple Regression | Simulation of position, speed, and acceleration with interactive control. | 0 min |
Explore functions visually
| Content | Focus | Duration |
|---|---|---|
| Explore sine functions | Function plotter sandbox with sine curve: zoom, move by keyboard, mouse drag, and additional on-screen buttons. | 10 min |
| Lines and slopes | Interactive plotter for linear functions f(x)=x, 2x, -x, x+b. Observe: slope, y-intercept, effects of zoom and axis shift. Control: arrows, +/- , 1-4, mouse drag. New: on-screen buttons for shifting and zooming (parallel to mouse-drag usage). | 20 min |
| Investigate parabolas | Interactive analysis of quadratic functions: observe opening, width, symmetry, and vertex. Plotter with three parabola variants, zoom, axis shift, mouse interaction, and on-screen buttons. | 20 min |
| Cubic functions – S-Form | Interactive visualization of cubic functions (S-Form). Switch between f(x)=x^3 and stretched variant, zoom, shift, observe inflection point and edge behavior. | 20 min |
| Knock and reflection | Interactive investigation of the absolute function | Observe knock and reflection | Switch between |x| and |x|-1 | Compare with f(x)=x | Zoom and axis shift | 15 min |
| Observe square root functions | Interactive visualization of f(x)=sqrt(x) and comparison with f(x)=x^2. Observe: only x>=0, faster start, later flattening. Control: mouse drag, +/- zoom, 1-4 choose mode. | 20 min |
| Sinus: Amplitude and Period | Interactive plotter for investigating f(x)=sin(x) and f(x)=2*sin(x): Zoom, Shift, Switch, new: on-screen buttons for navigation; Observe amplitude and period; short input of observation via command prompt. | 15 min |
| Observe initial value & phase position | Interactive Plotter shows sin, cos and shifted sin functions. Control over mode, phase (A/D), zoom (+/-), arrow keys and additional on-screen buttons. Mouse click centers, drag pans the view. Observe: Initial value f(0) and how phase shift moves the curve horizontally. | 15 minutes |
| Hyperbola: Approximation without contact | Interactive Plotter for hyperbolas (f(x)=1/x variants). Observe two branches, approach to axes/asymptotes and behavior near x=0. Control: 1-3 switch, +/- zoom, arrow keys move, click+drag axis, i input. Additional on-screen buttons for shifting and zooming (click has priority over drag). | 20 minutes |
| Compare exponential functions | Interactive Plotter for exponential growth (2**x) and decay (0.5**x) with additional on-screen buttons for left/right/up/down/zoom. Control via buttons, keyboard, and mouse drag possible. | 20 minutes |
| Compare functions | Interactive Plotter for visual comparison of two functions; Control: 1-4 switch, +/- zoom, arrow keys move, click+drag; Tasks: Find intersection points, compare growth, justify asymptotic behavior. | 25 minutes |
For teachers, a connectable lesson sequence with clear progression and modular course modules is created, which can be combined flexibly.
Request a demo access to try out the series in your own teaching context and adapt it as needed.