Measuring Symmetries
Ralf Youtz
Master’s program, Department of Mathematics, San Francisco State University ryoutz@sfsu.edu
Summary
Lesson plan
• Engage students with concept of symmetry through guided discovery.
Goal
• Introduce students to “mathematics without numbers”
Students discover the symmetries associated with different geometric objects can be essentially
the same.
• Show students mathematics is not only arithmetic, geometry, and basic algebra.
Handout
Process
Background
Measuring Symmetry
1. (private think time) Rank these seven figures in order from least to
most symmetric.
Students rank geometric objects by symmetry, and find a method to measure the symmetry of a
given object.
Think about how you would measure—that is, assign a number to—
the symmetry of a figure.
(There is not one correct way to do this. Use your instinct, intuition,
and artistic sensibility.)
Themes
Symmetry abounds in nature and design:
Name:
Date:
• Justification — Groups justify their quantification scheme so that it aligns with their intuitive
understanding of symmetry.
B.
A.
C.
D.
• Equivalence — Which symmetries do students consider equivalent? Why?
• Student thinking — There is no right answer here; students should be encouraged to listen
carefully to each other’s thinking during the activity.
E.
3. (group brainstorm) Come to consensus on one ranking of the seven
figures from least to most symmetric. Your group should agree on a
method for measuring symmetry, that is, a way to assign a number
any figure that tells how symmetric the figure is.
1. Reflection
• promote a safe, supportive, challenging classroom environment;
2. Rotation
• highlight the role of student thinking in teaching and learning;
3. Translation
• develop students’ understandings of symmetry and equivalence; and
Not all patterns are reflected, rotated, or translated the same way.
• make connections between algebra and geometry.
Which symmetry types are exhibited by the snowflake, leaf, Taj Mahal, and corporate logo above?
Which are exhibited by the M. C. Escher design below?
G.
2. (go-around) Share your ranking and measurement system with your
group.
Rewards
Types of symmetry:
F.
4. Create a poster that shows your group’s ranking and describes your
method of measuring symmetry. Use the seven given figures to illustrate your method.
5. Poster Q & A and gallery walk. Visit the other groups and compare
their ideas to those of your group.
Suggested questions for students
• When are two symmetries the same?
• Regarding Figure F, is a 90 degree rotation more like a 450 degree rotation or more like a 180
degree rotation?
(1) & (2) : Students share their thinking and build on their individual understanding.
• Should no rotation at all count as a symmetry? What about a 360 degree rotation?
(3) : Discussion and thinking about symmetry and equivalence.
• What criteria should be used to determine if two symmetries are the same? Is this a general
enough criteria to apply to any figure?
(4) & (5) : Synthesize student learning; question and challenge each other’s thinking.
• Why did you rank object G as less symmetric than Figures E and F?
Reflection
• Why is E more (or less—depending on the group) symmetric than F?”
I led this activity in two of Taica Hsu’s ninth grade algebra classes at Mission High School.
Students had taken their final the previous class. This was presented as a fun pre-break activity.
Of these examples, only the snowflake and Escher’s design have rotational symmetry (can you
see Escher’s point of rotation?). Escher’s print cannot be reflected across any line, unlike our
other examples.
A warm-up activity asked students if objects were symmetric or not and why. This got students
thinking about reflection symmetries. I hinted that rotations could be considered symmetries,
but left discovery of this symmetry type—which differentiates Figures E, F, and G—to student
groups.
The symmetries of different objects and patterns are formalized mathematically as the symmetry
groups of abstract algebra.
The basic notion of abstract algebra is that of a group, which is a set together with an operation on
the set satisfying three axioms. The operation is associative, the set contains an identity element,
and each element of set has an inverse element.
Another set-up issue is to provide a reason for ranking these figures. Why do we care if one figure
is more or less symmetric than another? (This is a good question to ask a mathematician!)
Familiar examples of groups include the integers with the operation addition and the rational
numbers with the operation multiplication. (Quick quiz: Are the integers still a group with the
operation multiplication? Why not?)
References
[1] John R. Durbin, Modern algebra: an introduction. John Wiley & Sons, Hoboken, NJ 2005.
A symmetry group is a set of symmetries (called isometries in formal mathematics) which leave
a figure in the plane unchanged (viewed as functions) with the operation of function composition.
[2] Sean Larsen, personal correspondence.
This lesson engages students in classifying geometric patterns by symmetry type, based on their
own classification schemes.
2
Supported by San Francisco State University’s (CM) program, funded by the National Scienc Foundation GK-12 Program (grant DGE-0841164)