Wednesday, April 4, 2007


Looking back on our topic some questions still remain in our minds. Throughout our research it was very difficult to determine the origin of symmetry. There have been examples of symmetry in nature and around the world since the beginning of time, however we were unable to find an exact date or mathematician relating to its discovery. We feel that this information would be interesting to note, as there is much information regarding the origin of geometry in general. Therefore, a question we would like to have answered is:
-Who first discovered symmetry and when was it discovered?

We found a lot of information on the topic of symmetry, especially with regards to the various types (i.e. line symmetry, point symmetry, rotational symmetry, etc). With so many aspects of symmetry, teachers may question the necessity of which topics to focus on in their lessons.
-How do we, as teachers, determine what aspects of symmetry to emphasize in the teaching of symmetry?

In reflecting on the topic of symmetry, we noted that there are a vast amount of resources available for use in the implication of a unit, such as this, in mathematics. Helpful to us, as future teachers in a technology-based world, online activities are valuable resources in gaining the attention of our students. A helpful online resource that we found is listed at the bottom of the main page of this blog.

Interesting to us, and possibly our future students, is the concept of the Golden Ratio. As we learned, in creating this blog, children have an innate sense to look for symmetry in the human body. For this reason, we feel that this topic would be of interest to our students - an explanation for why we see beauty in different people.

The issues we addressed in our blog were those that we felt would be most important when implicating a mathematics unit on symmetry. Though there are many other issues that can be addressed, as primary/elementary teachers, we feel that we cannot be experts in this field, yet we certainly feel more capable of teaching symmetry in our classroom thanks to our researched information.

Tuesday, March 27, 2007

List of Referenced Materials

Reference List

1) Clements, Douglas H. and Julie Sarama.(2000) The Earliest Geometry. Teaching Children Mathematics, 7(2), 82-86. Retrieved March 25, 2007, from Wilson Web.

2) Coffin, Tom. (n.d.). The Symbol of Beauty. Retrieved March 22, 2007 from

3) Johnson, Iris DeLoach and Sarah KatherineBomholt (2000). Picture This: Second Graders “See” Symmetry and Reflection. Teaching Children Mathematics, 7(4), 208-209. Retrieved March 23, 2007, from Wilson Web.

4) Knuchel, Christy. (n.d.). Teaching Symmetry in the Elementary Curriculum. The Montana Mathematics Enthusiast, 1(1), 3-8. Retrieved March 24, 2007, from Wilson Web.

5) Liebeck, Helen and Elaine Pollard. (1995). The Oxford English Dictionary. Fourth Edition. New York: Clarendon Press.

6) Palermo, Patricia. (2003). Symmetry? Could This be the Answer to the Age Old Question; “What is Beauty?”. Retrieved on March 23, 2007 from

7) Thompson, Mark. (2006). Why is Symmetry Important. Retrieved on March 24, 2007 from

8) Van de Walle, John and Sandra Folk (2004). Elementary and Middle School Mathematics: Teaching Developmentally. Canadian Edition. Ontario: Pearson Education Canada Inc.

9) (2007). Symmetry. Retrieved March 24, 2007 from

Monday, March 26, 2007

Useful Student Links

The links below present websites that offer helpful and interactive activities for students.
- Fun game for students
- Make a symmetrical pattern online
- Symmetry WebQuest
- Create reflections of images
- Variety of activities on symmetry and patterns

Helpful Lesson Plans

The following are a selection of lesson plans that may assist in the teaching of the topic of symmetry.

As symmetry can be taught in several subject areas, we have provided lesson plans from the areas of Math, Science and Art.

- Provides several activities on symmetry
- Lesson teaching the concepts of flip, rotate and slide

- Teaches the concept of symmetry through exploration of butterflies

- Investigates facial symmetry with the use of digital art
- Students create symmetrical illustrations with the use of crayons and an iron

Suggestions for Teaching Symmetry in the Primary/Elementary Grades

As previously mentioned, symmetry lends itself to many subjects, especially Mathematics, Science and Art. The internet provides many great suggestions for how to teach symmetry. This post will offer some suggestions on what to consider when implementing a unit on symmetry.

When teaching symmetry, it is important to note the various terms associated with it. The terms presented in the first blog posting will aid in the teaching of symmetry.

Materials are an important aspect to the implementation of lessons on symmetry. Students begin learning about symmetry with the concept of reflective symmetry in grade three. To best teach this concept, reflective materials, such as a mira, can be very useful in identifying reflective symmetry of any object.

When learning the concepts of flipping, rotating and sliding, manipulatives are great as students can use a variety of shapes and physically move the objects, such as pattern blocks. Geoboards are a useful tool as well. Students can be placed in pairs, with a geoboard: one student creates a shape, and the other shows its flip, rotation or slide. Again, the mira can be used to check and see if the flip, rotation or slide is correct.

Tracing paper is another useful tool, as students can trace an image onto the tracing paper, then rotate, flip or slide it to learn about its symmetrical properties.

The authors of the article The Earliest Geometry note that children begin forming concepts about shape before they enter school, and they are able to relate shapes to real-life objects. An example of this is the relationship made between a rectangle and a door. In this case, it is important, as teachers, to remember to make such connections when teaching symmetry. Allow students to relate new information to what they already know.

With regards to Science, symmetry can be taught with reference to a variety of natural organisms. Children are familiar with the shape of butterflies, leaves and shells, and this allows the concept of symmetry to easily be related.

As previously discussed, many art forms are created with the use of symmetry. Symmetry is appealing to the eye, therefore many artists base their work around symmetrical shapes. Encouraging students to create their own shapes and designs with the concepts of symmetry in mind, allows them to form their own understandings of symmetry and what constitutes as being symmetrical.

There are many other suggestions for the instruction of symmetry listed in the curriculum documents.

Interesting Facts About Symmetry

Among the most interesting facts found in the study of symmetry involve nature, the creation artwork, and even the human body. When each of these items has perfect symmetry, they appear more beautiful.

Coffin states on his website, The Symbol of Beauty, that nature has a strong tendency towards symmetry. Many animals and plants are created in perfect symmetry. Butterflies, star fish, leaves and flowers can each be divided into at least two identical parts. He goes on to say that symmetry in nature has influenced art and architecture.

A unifying trend throughout art history is the use of symmetry. In almost every piece of art one can find symmetrical patterns that make them unique. Leonardo DaVinci's famous drawing of a man's physical proportions is a wonderful example of symmetry in artwork.

Mark Thompson wrote an article entitled "Why is Symmetry Important?" This article states symmetry is "essential for a body to function correctly and avoid injury." This symmetry aids in the everyday functions of the body, and when out of alignment poses great problems, such as limited mobility.

Of all the facts presented about symmetry, the most interesting is that of the tendency of humans to be more attracted to people who are perfectly symmetrical. Known as "The Golden Ratio", the face must be of particular proportions to be considered symmetrical. Mathematically speaking, the width of an ideal face would be two-thirds its length, while a nose would be no longer than the distance between the eyes. Patricia Palermo states that even babies are drawn to symmetrical faces over asymmetrical ones. Animals, too, are more attracted to the most symmetrical of their species. This is very interesting, as she considers symmetry to be an answer to "What is beauty?"

Why Teach Symmetry?

Teachers often find themselves defending their reasons for teaching a topic, thus it is important to be knowledgeable about the benefits of certain areas of study. Symmetry is one such topic. It seems to be such a small aspect of the study of Geometry, however it is an integral component connecting Mathematics to the real world. There is much information on why teachers should teach symmetry, and the following are the points we feel are the most important.

Christy Knuchel states that there are a number of benefits to the students in the study of symmetry. Symmetry can be found in everyday items, however the connections to Mathematics are rarely noted. Teaching Mathematics involves more than simply adding, subtracting, multiplying and dividing. The study of symmetry and its properties instills an awareness that mathematics is truly used throughout our lives (Knuchel). Symmetry, in the real world, is expressed in many pieces of art, for example, quilts are highly mathematical in their creation, and depict how symmetry and mathematics are linked to real-life uses.

A number of reasons for teaching symmetry are outlined in the article Picture This: Second Graders "See" Symmetry and Reflection. The first point states that children have an innate sense of symmetry, in that they look for balance and order in the real world naturally. As teachers, it is important to build on this inner ability, as it is appealing to students. In addition, students are better able to learn a concept when they can relate to it, therefore teaching symmetry gives all students a chance for success.

Another point notes that learning about symmetry aids students in learning how to "classify objects according to the arrangement of their constituent parts." Ordering and classification are skills that are used throughout many daily tasks, and the ability to notice patterns or similarities will make these tasks much easier to carry out.

The study of symmetry in schools looks beyond geometric forms to organic shapes, meaning animals, plants, everyday items, etc. Johnson and Bomholt comment that children have a natural curiosity about the world around them, and learning about symmetry encourages this interest.

Lastly, children learn concepts about geometric shapes at a very early age. They learn, first, about a shape as a whole, but, with the help of symmetry, children learn how to focus on the characteristics and parts of an object.

These points may assist teachers in their reasoning for why they should teach symmetry in their classroom. Furthermore, the teaching of symmetry holds great importance in the development of mathematical minds of students as it gives students a different perspective of the world around them.

What is Symmetry?

Symmetry can be a very abstract concept for both teachers and students. To begin, there are many definitions explaining what symmetry truly is. Numerous responses to the question "What is symmetry?" can be found, however there are certain commonalities amongst them. Symmetry is a topic found within the field of geometry. Geometry deals with the spatial relations of lines, angles, surfaces and solids, while symmetry involves the parts of a figure corresponding in size, shape and position on either side of a dividing line, or round a center (Liebeck and Pollard, 1995). Simply put, a figure has symmetry when it can be divided equally in a minimum of two parts, with both parts having identical characteristics.

The website identifies three types of symmetry: reflection, rotational and point. A brief overview of each type is listed below.

Reflection Symmetry- Often called line or mirror symmetry, it is when a figure is divided into two parts and one half is the reflection of the other half. The line separating both sides of the figure is referred to as the Line of Symmetry.

Rotational Symmetry- Occurs when a figure can be rotated around a central point, at least two times, and still looks the same. The number of matches the figure makes as it is rotated once around is called the Order.

Point Symmetry- A figure that looks the same upside down or from opposite direction and has all parts matching has point, or origin symmetry. The origin is the central point around which the figure is symmetrical.

Symmetry can be found in mathematics, science, nature, art and the body, just to name a few. This blog will attempt to discuss symmetry with a focus on mathematics instruction.