# Symmetry

Henri Picciotto

Symmetry is an important topic in mathematics, but it is not explored much in school. This is unfortunate, because it is of interest to many students, and provides many "real world" connections to art and culture.

Still, there are many opportunities to introduce symmetry at all levels from 1st to 12th grade, both in the classroom and in math circles. I have developed a fair amount of curriculum to do just that. Read on for links to relevant pages on this site.

## A Unit

I was asked to develop some material on symmetry, to be used in New York's Museum of Math summer program for kids. The unit I created involves much work with concrete materials, and combines puzzles and creative opportunities. It consists of ten lessons, but you don't have to do all ten: specific activities can be extracted from it.

The lesson plans are intended for grades 1-6, but I suspect many of the ideas and most of the handouts can be used with older kids. I link to those in my online unit for middle school and high school. (See below.)

## An Online Unit

Symmetry Labs is an enrichment unit for middle school and high school which consists of a dozen activities that can be carried out remotely. Some of it should work in elementary school.

Presumably the unit can be used in the classroom as well. For that reason, I also link to many in-person activities.

## Geometry Labs

Geometry Labs includes a number of activities about symmetry, intended for middle school and high school.

Lab 2.4: Symmetric tangram polygon puzzles

Lab 4.2: Polyominoes and Symmetry

The eight labs in Section 5 are about symmetry.

Slight edit of Lab 5.6, an attempt to make the table headings easier to understand.

Online applet for Lab 5.6

Some Lab 5.6 photos for discussion. (Only show these after your students have worked through the lab, and found their own examples! Better yet, use photos of your students' work.)

Extension of Lab 5.6: Cover the Big Dodecagon with pattern blocks.

## A Set of Puzzles

Using paper and scissors: Cutting Out Hearts, by Katherine Paur

## Wallpapers Catalog

A pattern blocks catalog of the 17 wallpaper symmetry groups.

Two blog posts to complement and (somewhat) explain the catalog: Part 1 | Part 2

Wallpaper Starters are pattern block puzzles where a whole design is generated by a few mirror lines.

Vacation photos: Symmetry in Spain, especially at the Alhambra. (Related blog post.)

## Math Connections

Symmetry is related to geometric transformations. I have a lot of material on that topic on my Transformational Geometry page, some of it classroom-ready, and some intended for teachers and curriculum developers. Most relevant to symmetry:
Symmetric Polygons
Glide Reflection

Symmetry is also related to group theory. I share some pre-college lessons on that topic on my Abstract Algebra page. Most relevant are Lessons 4 and 5 in the Lessons packet.

One way to create fascinating symmetric designs is with my Spirograph simulation. (There's a math connection to cycloids and perhaps other curves, but unfortunately, I don't know anything about that.)

## A Course

Along with some other advanced geometry topics, I taught much of the material I link to on this page in my Space course, a one-semester math elective for students who have had Algebra 2. Such courses may become more prevalent if the NCTM's call to diversify the last two years of high school math catches on.

Some symmetric designs by Space students: finite | friezes.

## Elsewhere

Pages 59-64 of Working with Pentominoes and Unit 5 of Zome Geometry are about symmetry.