I-MATH
Symmetry and Transformations
I-MATH
Symmetry and Transformations
The National Council of Teachers of Mathematics Standards for School Mathematics in geometry says:
"Instructional programs from pre kindergarten through grade 12 should enable all students to apply transformations and use symmetry to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems."
We see examples of symmetry in almost everything around us - from the structure of a fern to the pattern of a patchwork quilt.
The fern has reflection symmetry, as it can be folded across the spine and one side will match the other. This quilt has 8 lines of reflection symmetry, because it can be folded across both sets of angle bisectors, and across both sets of "midlines" (lines connecting midpoints of opposite sides). The quilt also has rotation symmetry of 45 degrees, because a copy of it will match the original if rotated 45 degrees about the center.
When a figure is reflected across a line, wherever that line may be, a "transformation" has occurred. In the diagram below, triangle ABC has been reflected, and the reflection is labeled A' B'C':
Rather than reflecting triangle ABC, we can rotate it. The diagram below illustrates a rotation of 90 degrees about point P:
Translation is a transformation that "slides" a figure in any straight-line direction. The example below shows a translation to the right of a distance equal to the length of segment PQ. In GSP, and in transformational geometry, the directed length is called a vector.
Our final transformation is dilation. Dilation is a transformation that "shrinks" or "makes is bigger". As you can, the lengths of the sides of triangle ABC have been dilated by a factor of 2. The center of dilation is point P:
There is more information about symmetry and geometry for both you and your students to explore, at the following web pages called "Creative Geometry", on the Math Fourm website. The Creative Geometry pages are intended for use by teachers and by students interested in learning about creative applications of geometry. The following page is an introduction to symmetry. You will find links throughout I-MATH to other topics in the "Creative Geometry" pages.
Symmetry is an important aspect of geometry, and you might find that your students benefit from an exploration of symmetry, and transformations.
My students find transformations, especially on Sketchpad, to be very interesting - and a lot of fun! They enjoy doing geometric artwork, using triangles and transformations, as well as angle bisectors, perpendicular bisectors, and other geometric concepts. The following example was constructed by a student on GSP using transformations, then copied and pasted into a drawing program and colored there.
You can see some of my students work at the following link. These beautiful graphics have been published as a poster series by the National Council of teachers of Mathematics, and my students were thrilled with the idea of being published! I believe that it is a very valuable experience for students to have the opportunity to use mathematics creatively. Some of these constructions involve topics from other chapters of geometry (and chapters of I-MATH, and you will find more information about geometric graphics constructions in later I-MATH chapters.
Another geometric topic that is based on transformations, and connected to polygons and parallel lines, is the creation of Tessellations. There are some interesting programs that will create tessellations for students, but the value of using GSP to create tessellations is that the students learn and apply the concepts of transformations in creating them. Additionally this provides opportunities for students to learn the properties of polygons - which polygons will "tessellate" and which will not? Why or why not? What transformations can be used to form tessellations? What types of tessellations are there, and how are they created? All of these are valuable and intriguing questions to pursue with your students. You will find information on this topic at the following link:
The Math Forum is one of the most interesting and content-rich mathematics sites on the web. I hope you will explore some of the many hundreds of pages of information and ideas for teachers and for students that are available on the Math Forum's website.
Many people are intrigued by the tessellations of M.C. Escher. Escher was a Dutch artist who created fascinating artwork, much of it based on mathematics. There are many wonderful websites on Escher and his work on the internet. Some of my favorite sites are linked below, beginning with the "official Escher website". You may find them interesting and inspiring to your students.
http://www.nga.gov/collection/gallery/ggescher/ggescher-main1.html
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