John was invited to give the opening address (jointly with Kath Cross) at the Association of Teachers of Mathematics (UK) 2006 Easter Conference. This note develops from experiences at the conference. It is based around a pencil and paper investigation which, if you set it up outside or in a multi-purpose room, can also physically involve students. Maths300 Lesson 53, Spirolaterals, is based on the same concept and offers software support for a related investigation.

At this year's ATM Easter Conference I went to a workshop run by an old friend and maths enthusiast, David Cain. David now lives and works in Poland where he has had an old ATM classic Points of Departures 1 translated into Polish for use in his workshops for students and teachers. Using the English version he had the group that I joined working on the conditions about forming a triangle when a stick is broken into three bits.

In the opening address at the same conference I made a comment about an investigation in the same book called Worms. Here I quote from Page 62.

Worms leave tracks in layers of mud.
A worm forms a piece of track, turns through 90 degrees, forms another piece, turns, forms another piece, turns ,...

The drawing shows a 1, 2, 3 worm; it makes one unit of track, turns, makes two units of track, turns, makes three units of track, turns, makes one unit of track,... Investigate worms.

This page from my notebook includes a worm formed by using my 'phone number. Kids enjoy this activity and in one school pupils made badges using their 'phone numbers to wear and amaze their friends!   (Discover more in Maths300 Lesson 53, Spirolaterals.)

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