Introduction
This curriculum unit is on symmetry and fractals with an emphasis on fractal structures in the lung. I urge that there is a way to ensure that geometry concepts diffuse the mathematics environment while still engaging to students. I have tried to come up with lessons that will help teachers realize strategies to teach geometry to their students. It is my hope that completing this curriculum unit will aid teachers to teach fractals in a way that will motivate students, assist their connection and approach of the "real world" situations to the concepts, aid their use of different strategies, and extend students' skills to solve math problems in other situations.
I teach in a school district with approximately 24,000 students. They are 88% African American, and 80% are eligible for free or reduced lunch. I teach mathematics to 7 th and Pre-Algebra students. The unit can be used, with some modifications to the activities, by higher level grades as well. In my situation, the unit will be taught over approximately 2 weeks for 90 minutes each day. I teach in under-resourced, urban areas and my students come with various educational needs. Some of these needs can be easily changed in the classroom, some cannot. Since NCLB 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.
The students I receive each year must have some prerequisite skills in describing and implementing symmetry. Usually, they 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 do not have the advantage of merely working on the concepts and strategies. I have to start my students off with the elemental parts of geometry, nature and shapes. If they master these skills, I need to teach them how to approach geometry in an investigative manner using such techniques as collaborative learning; analysis and problem solving to express, test, and locally confirm or disconfirm conclusions; written and oral assignments to establish useful communication skills; and tools such as representations, and application software.
The unit will scope concepts of geometry starting with types of symmetry. That will give a starting point for mathematical relevance in the real world. In our case, we will consider the lungs. We will then examine fractals and, in particular, the fractals that exist in the lungs. If time permits, we will consider a final project. With this project, the students will design and build a prototype of an artificial lung. The reason for this project is for them to apply the knowledge they already know in ratio and proportion, symmetry, fractals, scale values and area.
Figure 1 Coastline of Europe
Fractals have really interesting ideas connected to them. Imagine that the image in Figure 1 is a representation of the coastline of Europe. Should you trail it with a mile-long ruler you will get a satisfied measurement. Now, let's say you come back the next day to measure it with a meter-long ruler, should you get another satisfied measurement? Which of the two measurements would give you a larger measurement? Since the coastline is jagged, you could get into corners and crevices better with the meter-long ruler, so it would yield a much satisfied measurement. What if someone decides to use an inch-long ruler to measure it? They could really get into the smallest and insignificant of fissures there. The analysis will get bulky since we know that the coastline is broken at a scale smaller than an inch. What if it were broken at every point on the coastline? You could analyze it with smaller and smaller tools, and the analysis would get longer and longer. That's an example of a fractal design. Therefore, the length trait of a fractal design depends on the scale at which you analyze it.
Most mathematics we teach in school today is old knowledge. For example, the study of the geometric shapes was established around 300 B.C. by Euclid. On the other hand, fractal geometry is newer and that research is still underway to prove some aspect of it. In the meantime, there is a lot we can learn and understand about fractals. A lot of objects around us are not made with your basic geometric shapes like squares or triangle, but are made with much more complex ones. For example things like ferns and snowflakes have fractal attributes that people use to solve real- world problems. Engineers are using the knowledge in developing and building fractals to solve engineering problems.
One of the main concepts of geometry, especially advanced geometry, is the notion of sound logic and proof. In an attempt to show students how to relate what we learn in the classroom to our everyday lives and environments, we will look at a very simple everyday item like the human lung. We will look at the development of the lungs, paying particular attention to the fractals that exists within it. We will also be looking at the lungs because it is the one thing that the students do study in their science classes. Studying them in a math class will strengthen their understanding, and will help them realize that the two subjects, math and science, truly do move hand in hand. We will look at the mathematics in the branching of the lungs, the type of branching in them, and calculate their fractal dimensions. We will accomplish this in two approaches. One, by manually calculating the dimensions and two, by using computer generated software. This will enable the students to see that mathematics does not always need to be done with paper and pencil.
Computers have become a great resource in our educational system these days. Now, most of the state standardized tests are taken online. It is crucial that we make strides in preparing our students for the hurdles introduced by new technology. I am a crusader in teaching students with technology. Research shows K to 12 students spend about 75% of their time on some type of computer-aided program: playing computer games, video chatting on the computer, downloading and uploading podcasts, etc. So it is imperative that, as educators, we use these avenues to our advantage as well. Also, most of my students become excited and focused anytime an activity or lesson is being done using computer aided software. In this regard, as I teach this unit, I intend to show them how to use computer programs to achieve the same results they achieved manually or with the aid of a calculator. They will be shown how to calculate fractal dimension using a computer-aided program like Winfeed.
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