# Laboratory on Mathematics and Mathematics Education/DGS

About Bathtubes and Rabbits (DGS)

## Task 1: Exploring Dynamic Geometry Systems! edit

- Download GeoGebra. If you are not allowed to install software on your system, I will give you GeoGebra on a flash.
- Try to construct the perpendicular bisector of two points on at least three different ways.
- Try to create the construction which "proofs" Thales' Theorem.
- Show that the sum of a triangle's angles is always 180°.
- Show that in a circle the angle at the center is double of the angle at the circumference.
- Given a triangle, show that its perpendicular bisectors always intersect in one single point, the circumcenter. Do this also for its median lines (centroid) and for the heights of its sides (orthocenter). Afterwards, show that these three points always are on one line (Euler line)
- Construct a regular hexagon.
- "Proof" some other geometric theorems. If you can't remember one, select on from the following list.

## Task 2: Dynamic aspects of functions edit

- Invent a bathtube story for a given a "bathtube graph".
- Create filling graphs for a given jar (and vice versa).
- How could the EIS principle be realized in this context? Collect ideas!

## Task 3: Making functions dynamic! edit

- Show how the unit circle is connected to the sinus curve in GeoGebra.
- Do the same for the graphs of cosine and tangent.
- Show the effects of functional parameters on graphs of different functions (for example, quadratic functions in different representations, (co)sine, exponential functions, ...). Create dynamic worksheets using sliders!
- Show that the loci of points which have the same distance from a given point and line are on a parabola. Are all parabolas graphics of a function?

## Task 4: Using spreadsheet calculation edit

- Create a boxplot from data given in the spreadsheet view. What happens if you change the data? Observe!

## Task 5: The Rabbit Problem edit

You have fence material with the length of 18 m, and you want to build a rectangular enclosure for your rabbit. How do you have to choose the side lengths of the rectangle that the area for your rabbit is maximal?

- Use a spreadsheet calculation program to try out some different solutions.
- Solve the problem formally (on a sheet of paper)!
- Create a dynamic sheet in GeoGebra showing the graph and the enclosure in order to visualize the solution. Be creative!

## Links edit

- Introduction to DGS
- Introduction to Action Learning (Principle of operation)
- Filling Graphs
- Combination of dynamic geometry, algebra and calculus in the software system GeoGebra (by Hohenwarter & Fuchs)
- GeoGebra help
- Geogebra: Dynamic mathematics made easy