Tiling and induction

Matthias Kawski and Michael Hall took the lead in developing this circle session on Tiling and induction for the 2011 Circle on the Road workshop.

Apprentices: Linda Green, Martin Montgomery, Nina Otterson

Target audience: Advanced middle school and high school students (teachers welcome)

Given a set of bricks of whatever shape and a box of whatever
shape, every child knows what to do: Start packing the bricks
into the box. Sometimes it does not work without leaving empty
spaces, sometimes an elaborate strategy is needed.
This session focuses on planar problems: Use a set of flat
tiles of rectangular, triangular and other shapes to tile
the floor of a more or less odd shaped bathroom (without
cutting any tiles, without any gaps or overlays).

Mathematically, sometimes impossibility can be established using
some kind of parity argument parity. As the rooms get larger (or
the tiles get smaller) it becomes natural to use solutions of
smaller problems. This naturally suggests and leads to formal
induction arguments.

A lesson plan is attached below. In addition templates for five different
triangular grids are attached, a copy of the student hand out, and a proof are attached.

Triangle 1.pdf
Triangle 2.pdf
Triangle 3.pdf
Triangle 4.pdf
Triangle 5.pdf