Introduction
Covering the plane with an unlimited supply of identical tiles is called tiling the plane, or tessellation. You can use one or more initial shapes, and you must have a convincing argument that your pattern can extend infinitely in all directions.
Tiling provides many opportunities in the classroom, as it illustrates basic geometry concepts, and offers opportunities for mathematical exploration and discussion. Student engagement is nearly guaranteed, the opportunities for creativity abound, and student work can be displayed on bulletin boards or online. I share some arguments for why tiling belongs in the classroom in this blog post and the following one. You might read those first.
An alternate starting place is Geometric Puzzles for Prospective Math Teachers, where I describe and explain a coherent approach to the topic, essentially a unit on tiling: start with increasingly challenging tiling puzzles, and use those to introduce basic ideas of high school geometry, followed by basic definitions and theorems of transformational geometry.
On this page, I share links to tiling materials on this site.
Geometry Basics
In Geometry Labs:
- Section 7 is all about tessellation: Tiling with Polyominoes, Tiling with Pattern Blocks, Tiling with Triangles and Quadrilaterals, Tiling with Regular Polygons. The last two are very "curricular", in that they address topics central to secondary school geometry.
- My Geometry Labs template is ideal for exploring tiling with triangles and quadrilaterals (Lab 7.3), and regular polygons (Lab 7.4). Check it out here. Purchase it here.
- Section 10 includes material on tiling finite figures. Here is an enhanced version of Lab 10.4: Rep-Tiles.
- Section 11 includes the final lab in the book, which uses a pattern block tiling in a lesson about "π for regular polygons".
After Lab 7.3, you can show or share these GeoGebra files: Tiling Triangles | Tiling Quadrilaterals
Transformations and Symmetry
Tessellations offer an excellent context to introduce and explore geometric transformations and symmetry.
Wallpaper Starters is a challenging puzzle-like activity which combines symmetry and tiling. The idea is to extend the given pattern, using the given mirror lines as the only guide.
If you create more pattern block wallpaper starters, I will post them here and credit you.
Wallpapers Catalog, using pattern block tilings
And More...
In Polyomino Lessons, the section on tiling includes specific challenges on grid paper, or perhaps Virtual Grid Paper.
Zome Geometry, the book I co-authored with George Hart, has a section on tiling, including activities about dual tessellations, spiral tilings, and non-periodic tilings.
If you use the Notebook software for the SmartBoard, you can explore tiling with this document. (Includes clonable pentominoes, pattern blocks, triangles, quadrilaterals, and regular polygons.)