If you are not careful, hours can disappear playing with problems like this. Once the first animation is understood then it is a natural development to play with the idea and see where it goes. As such this is a very rich activity. The kaleidoscopic animation is visually appealing and students do want to make it for themselves, so they become interested in figuring out how to do it! There are lots of great questions and discussion points here that are outlined below and help both to understand and develop the idea behind the problem. Of course, the associated learning objective is rotation, and whilst this is a great tool for teaching rotation there is a lot more mathematics going on!
The following is some practical advice about how the activity might be run.
Dynamic Geometry Software
This activity rather depends on teachers and students being reasonably familiar with dynamic geometry software. This animation was made using Cabri Geometre and the screencast below shows how. It is equally possible if you are more familiar with Geogebra or Geometer's Sketchpad. I suggest that teachers just make sure they know how it was constructed so they can answer questions from students during the activity. Sharing the screencast below with students would rather defeat the object of the exercise!
Making the animation
The following screencast shows how the Kaleidoscope animation was put together using dynamic geometry software (cabri geometre).
Students will need access to computers and dynamic geometry software for this activity. Obviously computers can be shared, but one to a computer is probably preferable in this case.
As ever, this is variable and probably driven by how much time is available! It could be a quick starter to a lesson or an in depth project lasting a couple of lessons. This depends how the teacher wants to present the problem.
Starting and Finishing
- My favourite way to start this activity is to ask the open question 'Can you recreate this animation?' and let students start making conjectures as soon as possible!
- Students could be given the list of questions below (also ready on the printable worksheet) as prompts for investigating the scenario. This should bring out the key points discussed in the screencast below.
- Teachers should help indivduals as they see fit, but hopefully the class will finish with students making their own animations.
- It is possible to set up more objects around the same centre and create an even more kaleidoscopic looking image.
The following screencast outlines some of the points to consider whilst looking at the animation. A summary is written below.
If time allows, then it is fantastic if students can create their own animations based on the principles in this activity. These can be kept by students or posted on Youtube or the school website, for example, and students will show them off proudly to others.
This is one of a series of related activities for transformations.
Dr Who - A great puzzle based on enlargement