Re-Discovering Symmetry

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SWBAT describe the transformations that carry a polygon or a tiling onto itself. Students will understand the meaning of congruence in terms of rigid motions.

Big Idea

Students describe symmetry in the art of M. C. Escher and others. In the process, they learn to describe transformations that bring corresponding parts of a figure together.

Lesson Open

4 minutes

I begin the lesson by displaying the warm-up slide and reciting the first stanza of a poem by William Blake in a loud, clear voice:  "Tyger! Tyger! burning bright."  Blake, after all, knew his geometry.

I ask students to define symmetry in their own words.  The warm-up follows our Team Warm-up routine.  I choose students at random to write the team's answer on the board. 

We do not review the teams' answers yet.  I display the agenda and learning targets for the lesson.  I tell the class that today we will be learning about symmetry.


What is Symmetry?

15 minutes

To engage students in this part of the lesson, I display the Who Would You Hire? slide and ask, If you were hiring someone to look after your dog, would you hire the man on the left?  Or right?  

I'll also make sure to share that research in the social sciences has consistently shown that, when two job candidates are in all ways equally qualified, managers of both sexes tend to hire the one who has the most attractive physical features.  

From here I'll pose the following question: What might make the man on the left more attractive than the man on the right?  Since my students know that I am always on topic, someone will surely suggest that it is because his facial features are more symmetrical.

All this leads to the discussion I want to engage the students in: So, what is symmetry?

Since we have been studying congruence, students will not be surprised to hear that symmetry is related.  I tell the class: an object or pattern has symmetry when it can be rotated, reflected, or translated onto itself so that all parts perfectly coincide.  

Displaying the What is Symmetry? slide, I ask the class to help me describe the symmetry in this work of the Dutch artist M. C. Escher.  I press the class to reason abstractly first, then use the animations in the slides to help them see what it actually looks like.  (You must download the slide-show to see the animations.  Different transformations are represented in different slides, so that I can navigate in the order of students' suggestions.)

Taking student suggestions, I ask the whole class to help us refine the good ideas that are offered (MP1):

  • What point would we rotate the pattern around?  What angle would we rotate it by (MP6)? 
  • Do students see that the original pattern reappears 3 times as we rotate it in a full circle? Then, what is the smallest angle of rotation that will carry the pattern onto itself?  What other angles will work (MP2)?
  • What line would we reflect the pattern across (MP6)? Do students see that the line of reflection is equidistant from parts of the pattern that look alike (MP7)?
  • What vector would we translate the pattern along (MP6)?  Do students see that the vector must join two matching parts (MP7)?

Although I take student suggestions, I refine their language as I use an interactive whiteboard to write descriptions of the reflections, rotations, and translations that carry the pattern onto itself .  I am modeling the way I want students to describe these transformations in the activity that follows, as well as on the quiz at the end of the unit.  [Upload pdf file of mimeo board]

Describing Symmetry

20 minutes

I display the Describing Symmetry slide as I ask students to work together to practice describing symmetry in a few more of M. C. Escher's works.

The first problem is scaffolded so that students will know what information they must include to make their descriptions of rigid motions complete and precise.  As students progress through the problems, I check to see whether they are meeting this standard.  

It helps to have some markers on hand for students to use.  It is too hard to see points, lines, and vectors drawn on the pictures in pencil.

This activity  uses the Team Jigsaw format.  Ideally, each team will get through at least 3 of the 4 problems.

Before class:  Reproduce the handout (2 per team of 3-4 students) and cut into half-sheets.

Summarizing Symmetry

10 minutes

We summarize symmetry and regular polygons with the help of the Guided Notes for the lesson.  

I emphasize that students should follow the examples in the notes on symmetry when describing the specific rotations, reflections, and translations that carry a symmetric figure onto itself (MP6).  

For now, students will be drawing lines of reflection, centers of rotation, and vectors of translation right in the drawings.  Later, they will learn to use the properties of rigid motions to reason what those points, lines and segments must be.

More on how I use Guided Notes can be found in my Strategies folder.




Lesson Close and Homework

5 minutes

The lesson close follows our individual size-up routine.  The prompt allows me to check whether students are able to determine the angle of rotational symmetry of a pattern or polygon.  They will need this skill in the next lesson.

Recognizing Good Work

While the class is completing the lesson close activity, I invite a student from each team to assign his or her team a score for the lesson.  Student scorekeepers write the score in a spot on the front board, and I write the scores I assign to each team next to them. 

You can read more about how I assign Team Points for cooperative learning.


For  homework, I assign problems #9-11 of Homework Set 1 for this unit.  Problem #9 provides additional practice in describing the symmetry of a tesselation.  Problem #10 asks students to tell which quilt patterns have rotation or reflection symmetry, then design a quilting pattern themselves and describe its symmetry.  Problem #11 provides additional practice in writing congruence statements to identify congruent corresponding parts of two figures.