Introduction
Many students have difficulties when attempting to solve geometry problems. Many reasons are suggested or put forward for the students' lack of success in this area. These reasons include students' lack of exposure to life outside of their neighborhoods, minimal visual skills, and difficulty in understanding basic geometric concepts. I do not dispute these reasons for students' failure, but I propose that there is a way to ensure that geometric concepts, especially transformations and symmetry, permeate the mathematics classroom while, at the same time, maintaining student interest. Wallpaper is an example of a real world item that is seen everyday in most homes across the nation. It is also one of the most important when it comes to the topic of geometry. Some people look at wallpaper and go "wow! That's an incredible pattern" but mathematicians see wallpapers and go "hmm! What's the fundamental domain? What type of symmetry can be found here?" among others. I have tried to develop a series of lessons that will help teachers develop various strategies to teach geometry, with the help of symmetry, in their classrooms. It is my hope that implementing this curriculum unit will help teachers to teach geometry in a way that will excite students, assist their connection and application of "real world" scenarios to the concepts, aid their use of various strategies, and extend students' abilities to solve math problems in other contexts.
I teach in a school district with approximately 23,000 students. They are 86% African American, and 75% are eligible for free or reduced lunch. I have taught in under-resourced, urban areas and my students come with various academic deficiencies. Some of these deficiencies can be positively affected in the classroom, some cannot. Since NCLB has come into existence, for student outcomes, there is not much that matters except for their standardized test scores at the end of each school year. If students meet standards according to state test, schools are viewed as successful and the stress in "passing the test" is put off for another year. Sometimes, students can pass the test while doing poorly on specific domains. One of the domains that my students have consistently performed poorly on is Geometry and Measurement.
Each year, students come into my classroom who are supposed to possess skills that are prerequisites for the math activities that I teach. Usually, most of them don't. Most of the time, there is a great difference between what the students need to know to "get started" and what they actually know. Of course, I have to begin my instruction "where they are." This means that I will not have the advantage of merely working on the concepts and strategies. I will have to teach my students the fundamental parts of geometry, nature and shapes. If they master these skills, I will need to teach them how to approach geometry in an investigative manner using such techniques as collaborative learning; exploration and problem solving to formulate, test, and locally prove or disprove conjectures; and written and oral assignments to develop effective communication skills; and such tools as physical manipulatives, models, and software.
Symmetry is a fundamental part of geometry, nature, and shapes. It creates patterns that help us organize our world conceptually. We see symmetry everyday but often don't realize it. People use concepts of symmetry, including translations, rotations, reflections, and their geometric figures and patterns as part of their careers. Examples of people whose careers that incorporate these ideas are artists, craftspeople, musicians, choreographers, not to mention, mathematicians. It is important for students to grasp the concepts of geometry and symmetry as a means of exposing them to things they see everyday that aren't obviously related to mathematics but have a strong foundation in it. According to the National Council of Teachers of Mathematics grades 6-8 should be able to apply transformations and use symmetry to analyze mathematical situations. This includes predicting and describing the results of translating, reflecting and rotating (aka sliding, flipping, and turning) two-dimensional shapes. They should also be able to describe a motion or a series of motions that will show that two shapes are congruent, and identify and describe line and rotational symmetry in 2 and 3- dimensional shapes and designs.
This unit, "Geometry and the real world" is designed for sixth and seventh grade mathematics classes. The unit could be used in eighth grade classes as well. It will be taught over approximately 2 weeks for 90 minutes each day. The unit will cover basic concepts of geometry beginning with the core assumptions about points, lines, and planes. These are the undefined terms that will provide a starting place for basic mathematical applications used in the real world. We will also examine geometry that exists around us in the real world, both the obvious and not so obvious. Geometry deals with extensive visual reasoning and the ability to picture how certain shapes will look after being transformed into different shapes. This unit will not only bridge the gap but will also help them see how these ideas can be easily related to the environment in which they live. Once they have gotten the basic understanding of all the geometric shapes, the students will investigate isometries. The four basic isometries they will look at are; translation, rotation, reflection, and glide reflections.
All geometric diagrams are comprised of the same basic components: points, lines (and rays or line segments), planar regions. Man-made objects that are made of these geometric structures would be almost everything. If a person looks closely, they would see many geometric shapes in structures. Buildings, cars, airplanes, ships, textbooks, television sets, dishes, pictures, computers, cups all have geometric structures to name a few. Some of these (dishes and cups) are curved rather than being made of flat pieces. However, the curved ones often exhibit circular symmetry. But keep in mind that not only man-made objects are geometric. Nature has its own geometric structures. The world is a big sphere, so is the moon and the other 8 planets in the solar system. The entire world can be thought of as a geometric structure. Measurements on maps are geometric which proves that nature has geometry and that geometry exists even in things humans cannot see but we just know it's there.
Points, lines, and planes are the undefined terms that provide the starting place for geometry. When we define words, we normally use simple words; and these simple words are in turn defined by simpler words. The process must eventually terminate; some definition must use a word whose meaning is accepted as immediately clear. Because that meaning is accepted without definition, we refer to those terms by italicizing or underlining. These undefined terms will be used in defining other terms. Although the terms are not formally defined, a brief discussion is needed.
The unit will begin by providing the students an intuitive history about geometry and its relevance in our environment. We will also look briefly at Euclid and other famous mathematicians' contributions to geometry. In doing so, we will look at real world objects like ships and bridges and figure out the geometric shapes involved and how they are used. We will then explore the differences between basic types of angles and shapes in geometry. With this knowledge, regarding different types of angles and their representations, they will be introduced to transformational geometry. This is where they will learn how to reflect, rotate, and translate, manually, most of the geometric shapes. Transformations have been created way back in ancient civilization- both oriental and occidental. This refers to the art of ornament, called the "oldest aspect of higher mathematics expressed in an implicit form" according to the famous twentieth century mathematician Hermann Weyl.
Computers have become a great resource in our educational system these days. Very soon most of the state standardized tests will be taken online. It is very crucial that as we make strides in preparing our students for these hurdles. I am a crusader in teaching students with technology. Most of my students get very focused anytime an activity is being done using the smart board and computers. In this regard, as I teach this unit, I intend to show them how to use computer programs to achieve the same results they have manually achieved. They will be shown how reach that goal using a computer aided program like the Geometer's Sketchpad.
One of the main concepts of geometry, especially advanced geometry, is the notion of sound logic and proof. In an attempt to show the students how to relate what we have learned in the classroom to our everyday lives and environments, we will look at a very simple everyday item like wallpaper. Creating wallpaper designs is guided by formal geometrical principles. It starts with identifying a fundamental design and then repeating it over and over again. The fundamental design could be a simple square, rectangle, or even a parallelogram. Translating this basic design over and over again in one direction (and its opposite) will produce what is called a frieze pattern. Translating repeatedly in two independent directions produces a wallpaper pattern. Repeating the fundamental designs many times in a specific direction can be achieved by the combination or composition of a translation with itself many times. These ideas will help them realize that mathematics is not always paper and pencil.
Most wallpaper motifs have several designs: some very complicated, some very simple and some in between. We will begin this discussion by looking at the design and trying to figure out what is the basic design element (also known as fundamental domain). In many attractive wallpaper designs, besides translations, there are reflections or rotations or other symmetries of the overall design. The students will eventually come up with their own postulates on how to figure out various fundamental domains of wallpaper. With all this knowledge, the students will each design their own wallpaper and discuss how they came up with their designs.
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