# 6.B.11. symmetry (& Group Theory) v.3

#### 1. TEACHER INTERVIEW

Bronna Silver teaches grade 2/3. Here are some of her reflections after exploring symmetry + coding with her class.

#### 2. TRY IT WITH YOUR STUDENTS

Symmetry story

The digital version of the SYMMETRY story used in this activity is freely available at researchideas.ca/sym/story.html.

Here is the lesson plan.

To receive a class set of the story for your students to take home (as long as our printer will last), please email George Gadanidis (ggadanid@uwo.ca) your name, school address, grade, number of students. If possible, please email back a brief summary of what you did and what you learned.

The last page of the story has room for parents to record for the teacher what their child shared with them and what they themselves learned.

Explore symmetry with code

The coding environment used to explore symmetry is available at researchideas.ca/sym.

There are also links in the digital version of the story to this coding environment.

#### 3. WHAT IS SYMMETRY?

We often think of symmetry as something quite simple and obvious.

Something has symmetry or it does not.

For example, some letters of the alphabet have lines of symmetry.

Because symmetry appears to be so simple and so obvious, mathematicians ignored it for a long time.

It wasn't until about three centuries ago that mathematicians realized that symmetry is actually quite complex and interesting.

#### 4. THE MORE IT CHANGES THE MORE IT STAYS THE SAME

Here is a puzzle to help you see symmetry in a new way.

• I hold a square in my hands.
• I ask you to close your eyes and I change the square in some way.
• Then you open your eyes and tell me how I changed it.
• Here is the before and after picture of the square.
• How did I change the square?

The idea for the symmetry puzzle in this story comes from Marcus du Sautoy's book, Symmetry - A Journey into the Patterns of Nature. See also his TED-Ed talk on Symmetry, reality's riddle.

#### 5. HOW MATHEMATICIANS SEE SYMMETRY

Mathematicians define a symmetry as a change (a transformation) that leaves a shape or object looking unchanged.

So, in the above puzzle, you could turn the square 90 degrees or 180 degrees and it would still look the same.

• A turn of 90 degrees is a symmetry of the square.
• A turn of 180 degrees is another symmetry of the square.
• How many different symmetries does a square have?

One way to keep track of how the square changes yet seems unchanged is to label its vertices.

So, with the above square, where the vertices are numbered 1234 starting with the upper left-hand corner, a turn of 90 degrees would result in 4123.

Each different arrangement of the numbers 1,2,3 and 4 is called a permutation.

Some questions to explore:

• How many different permutations are possible with the numbers 1,2,3 and 4?
• How many of these permutations are symmetries of the square?

#### 6. GROUP THEORY

These symmetry ideas explored by the grade 2/3 students in Bronna Silver's class are part of the foundation of Group Theory.

Group Theory is a part of Abstract Algebra, which is typically studied in first or second year university mathematics.

Isn't it interesting that something as everyday and as beautiful as symmetry can lead to the development of one of the most abstract branches of mathematics?

Isn't it also interesting that if we put our own school expefriences and prejudices aside, we can see that what is abstract, what is simple and everyday, and what is beautiful are all one thing, and that young children thirst for such experiences?

As one of the grade 2 students in Bronna Silver's class said, "Math is so cool!"

It really is.

Pass it on.