We encourage you to adapt this lesson to fit your needs. We anticipate it will take two 50-minute class periods.
Begin with the wallpaper patterns included in this document. You could have your students view the patterns on the computer. However, we recommend printing enough copies on a color printer for groups of four and handing them out so students can view all the patterns at the same time. Remember, you can easily make more handouts by printing one large picture and cutting it into smaller pieces. If a color printer is not available, ask a wallpaper store in your area for samples to use in your classroom. Wrapping paper may also work well.
We have provided more pictures than you may want to use. Some of the patterns are more complicated than others. You can eliminate all or some of the more difficult ones from introductory discussions. Glide reflections are generally the most difficult to see.
Give each group the same set of approximately 12 pictures and ask them to classify the patterns in any manner they wish. You might suggest they classify them by color, size, shapes - whatever they want. It is important at this point to accept ALL systems. The idea is that the students learn there are many possible classification systems; some are more useful than others. There is not one right answer. BE CREATIVE!
Students should develop a system that classifies all the patterns. We estimate 5-10 minutes to work on this activity. Then bring all the groups back together to share their classification systems with each other. We recommend providing supplies that will allow students to present their classification system on a poster board or butcher paper. (See our supply list for more details.)
Once the class has presented its results, you have two options. 1.) You could ask students to reclassify their patterns based on what they've heard from their classmates, or 2.) You could introduce the mathematical classification system.
For some classes, revisiting their original classification systems after having heard everyone else's may be helpful. The class discussion may have showed areas where a group's system had weaknesses or given the group additional ideas for classifying. If your students do revisit their own system, you may or may not need to have a class discussion afterwards. See how they are doing and use your own judgment.
For more advanced classes or classes who may not need to revisit their classifications, move on to a discussion of specific types of symmetry. The student lesson* introduces four types of symmetry.** We have provided a question early in our tutorial that you may want to discuss as a class.
Once your students have been introduced to the various types of planar symmetry, have them reclassify their pictures according to this structure. In our student page we have asked them to make a chart or Venn diagram to classify their patterns. A sample chart is provided. You may want to discuss their chart with them before they begin their reclassification process. It is important that they understand that some patterns have more than one type of symmetry and thus will have more than one checkmark in their row. We have provided a Hint Book for students to help them determine the type of symmetry a figure has. We have recommended that they use rulers, mirrors, or Mira, if available, to determine lines of reflection.
This is where your original selection of patterns will be critical. Patterns that have glide-reflection symmetry can be difficult to classify. While any pattern of symmetry that is created by reflection will also have a glide reflection, we have omitted this discussion from the students' pages. It is our hope that through the process of classifying, students will come to this realization and at that point you should initiate a class discussion. Therefore, a Venn diagram for classifying symmetries would look like the following:
An added note: All wallpaper patterns contain translational symmetry. For more advanced classes you may want to have them classify by the types of rotational symmetry that a pattern exhibits. For the purposes of our lesson, we have not specified types of rotation (e.g.,30,60,90,120, etc.), but you may want to. If you are using this as a review session for a geometry class, for example, you will want to include the more difficult patterns and topics.
After the students have classified their original group of pictures according to the patterns of planar symmetry, they can surf the Web and look at some patterns that we have selected to be classified. CAUTION: because we have linked to sites outside of our own, you will want to warn your students that they only need to look at the pictures at those sites - they do not need to read the documents attached to them. Instead, they should hit the BACK button on your browsing software to return to our document.
Once students have completed the above assignment, you may want to have a discussion with them about the number of planar symmetry patterns that exist. There are actually 17 types of symmetry patterns possible on a plane if you include degrees for rotations. Thus your students will have found at most 6 of these symmetries if you don't differentiate rotations. The reason there are only 17 types of planar patterns possible is an entirely different story for another time and another place.
Other discussion questions might be:
We have provided a diagram for symmetry classification for your use. We suggest, however, that you first have the students try to create a diagram on their own. We have also provided answers for the classification of the symmetry patterns we have referenced.
We hope this has been helpful and interesting to you. Please send comments to us.
Thank you!
*Other related lessons available on the Web are listed on the Teachers' Resource page.
**We have chosen not to mention that the translation vector in a glide reflection is parallel to the line of reflection. We feel that this would only confuse the issue and individual teachers should deal with it on an as needed basis.
