These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop

Circle Symmetry

A kinesthetic approach to rotational symmetry

This is a clear, self-contained lesson on rotational symmetry. It is geared toward elementary age students, but it is also a good introduction for any age (older students might want to skip the hexagon-folding part of the lesson). As we say in one of our shows, "It’s much easier to see symmetry than to actually do it." Mirror, or reflection, symmetry is so natural because we’ve been trained by looking into mirrors: we raise our left hands, the people in the mirror raise their right hands; we raise our right legs, the people in the mirror raise their left legs. As you’ll discover, rotational symmetry is more circular, and with a bit of practise it can start to feel natural. Examples of rotational symmetry are everywhere (flowers, logos, games), and the study of symmetry serves as a wonderful bridge between the sciences and the arts: the same ideas that appear in geometry, graphing and chemistry also appear in architecture, art and dance. Moreover, rotational symmetry is a building block for other symmetries.

(Insert Illustration)

Grades: K-6

Time: 20-40 minutes

Topics: Symmetry, visual thinking

Groups: of 2 or 3

Space: Requires clear floor

Related Activities: Dance classes

Materials: Handout (page XXX) for each student

Pencil with eraser for each student

(optional) Tape deck and music



7-1. Facing A Partner (5-10 minutes)

Mirroring. Everyone get a partner. Standing a few feet apart, face your partner. K-3 students might want to start sitting cross-legged rather than standing. The teacher could also demonstrate in the center with a student. In each group, one person be the leader and the other person be the follower. Leaders, make a simple shape. Followers, copy the shape in mirror image, as if looking into a mirror. For instance if the leader holds up the right hand, the follower holds up the left hand. Leaders, start moving slowly and followers follow in mirror image. Take turns leading and following. Move slowly so your partner can follow!

Rotational Symmetry. Now try rotational symmetry. One person leads and one person follows, but this time you do the exact same thing as your partner. For example, if your partner raises her right arm, you raise your right arm. If she leans to her left, you lean to your left. Take turns leading and following. Instead of imitating your partner in mirror image, you will find yourself leaning in opposite directions. Dancers have to do this often and anyone can learn to do it. Many people find this more difficult than mirroring.


7-2. Folding the Hexagon (10-20 minutes)

Fold your hexagon handout (diagram needed) (Note: older students may want to skip this section and continue on to Section 3)

Hand out a copy of the handout shown on page XXX to each student. Fold the paper down along the edges of the hexagon in the order listed. In other words, fold down along the line numbered "1" first, then "2," until all six sides are folded. Fold tab "7" down and insert into the slot. You will end up with a shape that has six sides, called a hexagon.

Now, get in groups of three. Place your hexagons on the floor and move them together so the pointed fingers touch in the center and the hexagons fit together. Now make that pointed-finger shape with your trio. Do the same with the foot shape and the crossed arms shape.

Math: The trios have just made shapes that have rotational symmetry. One way of thinking of rotational symmetry is that each person is doing the exact same shape and facing toward the center of the circle.

Can the trio make all three shapes on the hexagon at the same time?


7-3. Inventing Trio Shapes (5-15 minutes)

Make up new shapes. Using the hexagon as a guide, you’ve made three shapes that have three-fold rotational symmetry. Now, using your whole body, each trio needs to invent 3 to 6 shapes that have three-fold rotational shapes (Note: the younger the students, the fewer the shapes). You can be standing up, lying down, upside down, whatever you want. Be sure that all three "petals" of the flower are all the same. For instance, if one person crosses right arm over left, all three people must cross the same way. Or, another way to look at it is: If you turn the shape around once, it will coincide with itself three times. Try it!

Put your shapes in a sequence. Practice your shapes until you can remember them exactly. Then put your shapes in an order and learn how to move from one shape to the next. Practice your shape sequence until you can move smoothly through all the shapes without talking.


7-4. Performing the 3-fold Rotational Shapes (5-20 minutes, depending on the size of the class)

Go around the room and have each group perform their shapes. You can have each group come to the center of the room to perform, or just stay where they are. You may want to have the groups count off an order before you start. Then you can go uninterrupted through all the trios. Try playing music while trios perform. Music encourages students to treat this exercise more like a performance, and pay attention to how they move.

Dance Suggestions:

• Vary the movement

• You can change the type of music and see how they look. You can change the speeds (for example: you move into shapes 1, 3 and 5 quickly and into shapes 2, 4 and 6 slowly).

• You can break the symmetry in between the shapes. You can make one shape that is not symmetrical and see if other groups can tell which is which. These are some suggestions, but of course there are tons of other things you can do.

• How do you choose which variation to do? You can discuss it as a group, or each person in a trio can come up with an idea. Do what you think would be best or most interesting or most enjoyable.

• EXTRA CREDIT: Make one of your shapes asymmetrical, that is, not symmetrical. There are many ways to do this.


7-5. Further Ideas

Draw your shapes. Back to your pieces of paper. The paper has three shapes pre-drawn on it: a finger-pointing shape, a foot shape, and a arm-crossing shape. Can you draw in the other three spaces of the hexagon the shapes that you made up? This is difficult but do-able. Try to line up your drawings the way you did at the beginning of the class. Do the shapes work? If they don’t, you can erase them and try drawing them again. Save your hexagons.

Color your hexagons. You may want to save this activity for a separate session. Color your hexagon any way you want. When everyone is done, fit all the hexagons together on the floor. The resulting mosaic might have some fun patterns. Look for points where there is three-fold rotational symmetry. Do you think the symmetry is broken if the colors are different? Why? You can also decide as a group which areas should be colored in which ways and create another mosaic. Experiment.

• You just did rotational symmetry in groups of three. Would it work for groups of 4 or 5 or 6? How about for groups of 20?

• Look through picture books and try to find the two types of symmetry we’ve be working with: Mirror (sometimes called Reflection) and Rotation.

• How are Mirror symmetry and Rotational symmetry different? How are they similar?

• Can a shape have Mirror symmetry and Rotational Symmetry at the same time?

• In the pictures you saw, did you find any types of symmetry that are not one of these two kinds?

• Can you list some of the ways to make a symmetrical shape asymmetrical?

• Which symmetry dances did you like? Why?

• What makes the dances interesting?


7-6. Assessment

• Did the clarity of the movement increase with practise?

• Did the dance phrase include the ideas we were working with?

• Did they commit to creating and working on the dances?

• Did the students perform or were they hesitant or afraid?

• For this particular exercise, was the symmetry correct?

• Was each dancer in a trio doing the same shape/movement?

• Were the transitions between shapes clear and quietly done (unless they added sound as part of the dance)?


7-7. Resources



7-8. Handout blackline master

These and other activities included in our book, Math Dance with Dr. Schaffer and Mr. Stern, now available from our Book Shop