Have you ever heard about a thing called "Spirograph" ?
Actually it is a toy for kids, but it has a nice mathematical background.
How it works: You take a pencil, put it through a small hole in the inner circle, while you make the inner circle roll around the inside of the outer circle continuously.
What is going to happen then? You will find it out by doint this worksheet.
Important (!) before you start playing:
-To delete the trace in the application, simply press "Strg + F".
If you are working with a Mac, press "Command + F".
Now play with the construction below and work through the points at the bottom of the page.

1. Start the application as the text above the geogebra-file tells you to do.
2. Check what happens, if you move the sliders "Radius" and "Distance of Drawingpoint"
3. Activate the Checkbox and try to make the Spiralo paint the three pictures shown.
Write down the settings of the sliders you have to set, to draw each picutre.
4. Let's name the radius of the inner circle "a" and the sliderparameter,
(here its set: min: 0, max: 50) which has to run to animate the application, "b".
Write down the angle, which the drawingpoint is rotated by, if the inner circle mad a half turn.
Can you write down a general formula, which gives you the rotationangle of the drawingpoint (dependent of the sliderparameter and the radius of the inner circle)?
(You can compare your result with the one you get by activating the checkbox at the bottom right of the construction above).
5. If the radius of the outer circle is 2, how big does a have to be, to draw the picture of challange three, which of on top of all?
(If you think, you have the answer, activate the checkbox "Answer to point 5:"
6. Which important attribute does the ratio of the radius of the outer circle and the radius of the inner circle need, so that the spirograph can finish drawing one picture?
(After you have a conclusion, compare it with the solution given by activating the checkbox at the bottom of the application.)
Vocabulary:
ratio - Verhältnis (hier: von 2 Zahlen)