Covering the plane with an unlimited supply of identical tiles is called tiling the plane, or tessellation. Over the years, I’ve developed a number of classroom activities about tiling. You can find links to those on the Tiling home page on my website.
In a conversation with a teacher a few months ago, I realized that not everyone appreciated the pedagogical value of tiling as a curricular springboard. In this post, I’ll try to argue that tiling activities and the subsequent discussions can play a useful role in grades 6-10. When possible, I link to relevant activities.
Students can create tessellations using polyominoes (grid paper polygons). [See Geometry Labs 7.1] Here is an example:
How do we know this tiling extends infinitely in all directions? One argument is that the middle two rows (gold-blue, and green-pink) can be extended to the left and right, following the same pattern. The result is a horizontal “stripe” with straight-line edges. Such stripes can be placed next to each other, thus covering the whole plane. (The example above shows one such arrangement, but there are other ways to line up the stripes.)
Students can create pattern block tilings. Here is an example:
The same question can be asked about those: how do we know it extends infinitely? There are no straight lines between stripes here, so a more sophisticated argument is needed. Another interesting question is to find a notation to describe pattern block tilings. (One is suggested in Geometry Labs 7.2.)
Admittedly, few teachers feel they have time for activities that “merely” help develop mathematical habits of mind, even if (like the above) they are aesthetically pleasing and guaranteed to generate student interest and engagement. Read on for explicit curricular connections.
Students can create tessellations using triangles or quadrilaterals. [See Geometry Labs 7.3, though in that lab I failed to point out the wealth of curricular connections this offers.] One way to do it is with the help of my Geometry Labs drawing template, which includes eight types of triangles (equilateral, acute isosceles, right isosceles, and so on) and ten types of quadrilaterals (square, rhombus, rectangle, etc.) Be sure to let the students color their creations!
Another way is to do it online, using this set of Google slides.
Of course, there are many ways to make the tilings. Here is one possibility, using a scalene triangle:
(I chose this one because it has a lot of useful math embedded in it.) Ask students to label the angles, using color to distinguish the three angles, or perhaps numbers, as I did here:
In this figure, we can readily see the sum of the angles in a triangle. If you look at three consecutive angles around the vertex I marked, it is clear that they add to 180°. Other geometry facts lurk in this tiling: the exterior angle theorem; parallels and transversals; decomposing a larger triangle into four smaller similar figures by joining the midpoints, and perhaps more. All this is a lot more interesting to students if these questions are explored on tilings they created. Encountering these ideas in this context makes for a smoother approach when they arise as axioms or theorems later on.
Tiling with combinations of regular polygons offers an opportunity to explore the angles in regular n-gons and how they could be combined to go around a point. This provides motivation to figure out the measurements of those angles, with values of n from 3 to 12. [See Geometry Labs 7.4.] Again, making the actual tilings can be done with my Geometry Labs template, which includes all the regular polygons you need, or by using this online tool.
Here is a tessellation made from regular octagons and squares:
After some unsuccessful attempts, many students are surprised that a general asymmetric quadrilateral can be used to tile the plane, even if it is not convex. Here is an example:
The secret is the same one as in the above triangle tiling: (spoiler alert!) place adjacent tiles so that they are 180° rotations of each other around the midpoints of sides.
Tilings offer many other opportunities to discuss isometries (rigid motions). Students can explore the translations, rotations, and reflections in the tilings they create or in the ones they find. For example, in Islamic art:
or in Escher prints:
[See my Pattern Blocks Wallpaper Catalog for more on tilings and transformations.]
Finally, here’s an idea I have not tested with students: using GeoGebra to make tessellations using transformations and the Sequence tool. The quadrilateral tiling above involved using expressions like this one:
list1=Sequence(Translate( q1, u*i), i, -5, 5)
- q1 is the original quadrilateral,
- u is the translation vector,
- i is an integer ranging from -5 to 5, used to scale the vector
This would create 11 copies of the original quadrilateral, evenly spaced. We would need another original, rotated 180° around the midpoint of a side, and the corresponding sequence, to make a stripe. Also, we would need to figure out how to create the needed translation vector. Once we have the stripe, we can again use the Sequence tool and a different translation vector to create the tessellation.
I’m guessing that some lessons could be devised to make this accessible to high schoolers, and that students would find the possibility of creating repeating designs motivating. Experience with this would complement work on sequences and series, and help preview sigma notation. If you create such lessons, let me know how it goes! I may publish them on my Tiling page — acknowledging you, of course.
Well, that’s all for now. I hope I convinced you to find a place for tiling in your math classes!