Lessons taught; Lessons learnt

As mentioned in my last post, I frequently use Geogebra to create interesting little visualisations which can be used as starters, or just to fill in a transition gap. Several of them are animated using javascript, but I can’t yet figure out how to get javascript into Wordpress in a consistent manner, so no animations yet!

I like playing with these without initially setting any questions, and wait to see if any interesting questions arise from the students. If not, I let them know some of the questions which have crossed my mind. I didn’t pursue any of these questions last year, but may well next year.

So, in that spirit, have a play with these:

Geogebra applet will be here if you enable Java.

Geogebra applet will be here if you enable Java.

(Source: rotating-triangles1.ggb and rotating-squares1.ggb.)

Can you see how these were generated?

Pretty Pictures

The first thing I’d do with something like this is to play around with the slider, and see what happens, to try and understand what’s going on. You soon find that changing the number alters the picture, and some of the numbers produce ‘nicer’ pictures in some sense than others.

Here are some interesting configurations for the triangles:

And for the squares:

The Mechanics

Each file is created by starting with a shape (either a triangle or square) at the centre, and one parameter which you can change: an angle. The angle represents the amount by which each triangle (or square) is rotated anti-clockwise to create the shapes at the next ‘level’. Each level is given a different colour.

If you want to explore in more detail how the files were created, then (as with all my Geogebra worksheets) you just have to open them in Geogebra, and play — unhiding everything and looking at the definitions at points is always a good start.

Questions

• What other angles produce ‘interesting’ patterns?
• What angle would I need to have the shapes surround a pentagon, and what would the pattern look like?
• Can we calculate which angles result in exactly overlapping triangles/squares?
• What would a similar pattern look like using pentagons, or non-regular polygons?
• What would it look like if we added another ‘level’ of shapes?
• What about if we alternate clockwise and anticlockwise rotation?

I’d be interested to hear any answers to these questions… and (more importantly) any further related questions.

Related Posts (automatically generated)

1. Visualisation: More rotating squares
2. Visualisation: Yet more rotating squares!
3. Visualisation: Six moving lines