E-Mail Address: c_davis28@hotmail.com
School: Niles West High School: Skokie, IL
School Home Page URL: http://www.niles-hs.k12.il.us
| Author: Colleen Davis E-Mail Address: c_davis28@hotmail.com School: Niles West High School: Skokie, IL School Home Page URL: http://www.niles-hs.k12.il.us |
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THEME: Fractals
BROAD CONCEPT: Students will learn how fractals relate to everyday life.
GRADES: 7-12
INTEGRATED SUBJECTS: Reading, Writing, Mathematics, Science, Art, and Computers
UNIT GOALS AND PURPOSE: Students will learn how fractals, which are a mathematics and science concept, can relate to real-world situations and cultures. Students will be given various assessments throughout the unit to determine their understanding. However the major assessment will be in the form of a portfolio at the end of the unit comprised of activities completed throughout the unit, and a few new activities to summarize what the students have learned about fractals. All of the activities in the unit will be technology based, and students should become more proficient with the use of software, the Internet, and word processing skills. Since fractals are a new topic within mathematics, the teacher should take the role of a guide or co-investigator throughout the unit.
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TIME FRAME: Approximately 12 days OBJECTIVES: In this lesson, students will:
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MATERIALS:
X Brainstorming (Inspiration) X Word Processing (MS Word)
X Multimedia (Power Point or Kid Pix) X Other software pertaining to unit
(Version 20.0 Fractint-Fractal Software)
STATE STANDARDS ADDRESSED BY THIS UNIT:
Reading and Writing
1.B.3a Preview reading materials, make predictions and relate reading to information from other sources.
1.B.3d Read age-appropriate material with fluency and accuracy.
1.C.3a Use information to form, explain and support questions and predictions.
1.C.3d Summarize and make generalizations from content and relate them to the purpose of the material.
1.B.4b Analyze, interpret and compare a variety of texts for purpose, structure, content, detail and
effect.
1.C.4a Use questions and predictions to guide reading.
1.C.4b Explain and justify an interpretation of a text.
1.C.4c Interpret, evaluate and apply information from a variety of sources to other situations (e.g., academic, vocational, technical, personal).
1.C.4d Summarize and make generalizations from content and relate them to the purpose of the material.
3.A.4 Use standard English to edit documents for clarity, subject/verb agreement, adverb and adjective agreement and verb tense; proofread for spelling, capitalization and punctuation; and ensure that documents are formatted in final form for submission and/or publication.
3.C.4a Write for real or potentially real situations in academic, professional and civic contexts (e.g., college applications, job applications, business letters, petitions).
3.C.4b Using available technology, produce compositions and multimedia works for specified audiences.
5.A.4b Design and present a project (e.g., research report, scientific study, career/higher education opportunities) using various formats from multiple sources.
5.C.4a Plan, compose, edit and revise information (e.g., brochures, formal reports, proposals, research summaries, analyses, editorials, articles, overheads, multimedia displays) for presentation to an audience.
Mathematics
8.A.4b Represent mathematical patterns and describe their properties using variables and mathematical symbols.
9.A.4a Construct a model of a three-dimensional figure from a two-dimensional pattern.
9.A.4b Make perspective drawings, tessellations and scale drawings, with and without the use of technology.
Science
11.B.3c Select the most appropriate design and build a prototype or simulation.
STUDENT ACTIVITIES WHICH WILL PROMOTE ENGAGED LEARNING:
Day 1 & 2: The teacher should introduce the unit and topic of fractals. Students will research the topic on the Internet from the website http://mathforum.org/library/topics/fractals/. The website contains many different links to other fractal websites. The students should discover through research what a fractal is, how fractals are formed, examples of fractals, who discovered fractals, why fractals are significant to the real world, how fractals are related to mathematics, what is the Mandelbrot Set, and how Chaos Theory relates to fractals. Students should take notes on the information they find either on paper or by using cut and paste in a Microsoft Word document.
Day 3: Begin class by breaking into groups of four. Students should discuss information that they found about fractals from their research. After about 10 minutes, use the Inspiration software to create an outline or attribute web about fractals. Encourage the students to include all of the different information they found about fractals in the outline. Students should save their outline or attribute web to be used during a later class and in their final portfolio. As a class, discuss the information the groups included in their outlines about fractals. Encourage groups to write down any missing information that other groups included.
Day 4: The teacher should lead students in making a fractal by hand. From the students’ research, students should have encountered the Koch snowflake. Each student should draw Koch’s snowflake by hand through five iterations. The students should see that it is difficult to draw a fractal by hand. Students should access the website http://spanky.triumf.ca/www/fractint/fractint.html. First, have students access the “deep zooming” option to see that fractals consist of iterations. Next, have the students download the Version 20.0 Fractint software. The students have the ability to create high iteration fractals, interactive fractals, and colorful fractals. Allow students to create a detailed fractal to be printed out by the end of class.
Day 5: Student should get into the same group of four as Day 3. Students should go to the website http://www.manteno.k12.il.us/webquest/middle/Math/Fractals/fractalquest.htm. Each student in the group should take a role as the mathematician, historian, application expert, or artist. The website gives detailed instructions as to what the students are expected to do and how they will be graded. The students should have the class period to research their role in the project on the websites provided within the project guidelines.
Day 6: Students will continue to work on the project listed at the website http://www.manteno.k12.il.us/webquest/middle/Math/Fractals/fractalquest.htm. The group members should bring the information found together and create a Power Point presentation. The presentations should be no more than 5 minutes and will be presented the next day. The presentations should include each student’s role as described in the project.
Day 7: Begin class with the Power Point presentations. Give each group roughly 5 minutes to present their information about the math, history, application, and art of fractals. After the presentations, have students go back to their outlines from Day 3 using the Inspiration software, and add any additional information they have learned about fractals in the past few days.
Day 8: The focus of today’s class is finding real-world examples of fractals and how fractals have affected different cultures. Give the students 10 minutes to search the website from Day 1, http://mathforum.org/library/topics/fractals/. This site will allow the students to quickly find a real-world example of fractals through other sites. The teacher should make a list on the board of the different real-world examples students find. Next, the students should access the website http://www.rpi.edu/~eglash.dir/afractal.htm. This site, titled African Fractals, provides students with information on how fractals are found in the African culture. The teacher could use an Epson projector so that the class could explore the site together. Be sure to access the links to fractals in African development and newspaper and journal articles.
Day 9: During the first half of class, watch the video “Math and Science through the Eyes of Culture.” The video can be purchased from the African Fractals website from Day 8. As a class, discuss how fractals relate to the real world. The teacher should try to get students to connect the real world to fractals, fractals to mathematics and science, and finally the real world to mathematics and science. Try to show the students through examples that the real world relates to math and science. At the end of class, the students should write a one-page journal on Microsoft Word about how fractals relate to the real world.
Day10-12: For the last few days of the unit, students will be working on a final project in the form on a portfolio. The students must include the following:
1) From Day 3 & 7, the outline or attribute web using the Inspiration software
2) From Day 5, the individual research found from the group project
3) From Day 9, the journal written on Microsoft Word
AND
4a) Create a fractal using the Fractint software and enter the fractal into the contest listed on the website http://spanky.triumf.ca/www/fractint/fractint.html The fractal must be detailed and the “deep zooming” option should reveal the iterations.
OR
4b) Create a 12-month fractal calendar using Fractint to make your own fractals, or download free fractals from the Internet. Use Kid Pix to create the calendar with the fractal pictures.
5a) Write a 3-5 page paper using Microsoft Word explaining what are fractals, who discovered fractals, how are fractals formed, how do fractals connect with mathematics and science, and how do fractals relate to the real-world.
OR
5b) Write a 3-page interview between you and the person who discovered fractals. Include the questions you would ask and the replies to those questions. You are basically creating a dialogue between yourself and the founder of fractals.
SHARE YOUR IDEAS:
Throughout the lesson, students will exhibit their knowledge to the teacher and each other. Students discuss in groups what they learned from research, and then bring the information to class through discussion and a Power Point presentation. At the end of the unit, students will create a portfolio which will demonstrate what each student has learned about fractals. The following rubric outlines how the students will be assessed on the portfolio.
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Total |
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Outline |
Complete information is present such as who created fractals, what are fractals, significance, Mandelbrot Set, Chaos Theory, how created. All items listed in class are present. |
Some information from the previous column is missing. Information from class discussion is not present. |
Significant information is missing. Did not find the research or information to answer the guidelines set in Day 1 and 2. |
Missing |
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Individual research from group project |
All questions from the project description have been answered in detail. |
One question not answered, or answers to questions are not complete. |
Two or more questions are not answered and answers are missing detail. |
Missing |
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Journal |
Thorough explanation and connection between fractals and real world have been made. Complete sentences, grammar is correct, no typo’s. |
Minor typo’s or connections between real world and fractals are not clear. |
Doe not meet length requirement and no connections have been made between fractals and real world. |
Missing |
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Fractal Creation |
Proof of submission to contest and picture is detailed, elaborate, and colorful. Fractal must have many iterations that can use the “deep zooming” option to clearly see iterations. |
Proof of submission into the fractal contest is missing. |
Proof of submission into the fractal contest is missing, and the picture has few iterations with little detail. |
Missing |
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Calendar |
Detailed and elaborate pictures displaying multiple iterations. Months and days are clear with an organized layout. |
One to two months missing or fractals are not complex or do not use multiple iterations. |
Days and months are missing, and pictures are not complex or detailed. |
Missing |
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Paper |
Thorough explanation of required guidelines. Complete sentences, grammar is correct, no typo’s. |
Minor typo’s are present or one-two requirements have not been detailed. |
Does not meet length requirement and many typo’s are present. Paper does not give detail to the assigned requirements. |
Missing |
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Interview |
Complete dialogue is created with appropriate questions and answers. Grammar is correct with no typo’s. |
Minor typo’s or dialogue is not complete. Some questions and answers may not be thorough. |
Does not meet length requirement, and questions may not be appropriate to an interview on fractals. Multiple typo’s are present. |
Missing |
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STUDENT AND TEACHER ROLES:
Throughout this unit, the teacher will take the role as guide and/or co-learner. All lessons have been planned in advance, however the teacher should allow the students to discover new information on their own. Since fractals are a newer topic in mathematics, few teachers will be experts on the topic. The teacher and students should be learning together from the research on the Internet.
The student role throughout the unit would be that of an explorer and producers of knowledge. Students are required to explore Internet sites in order to gather information about fractals. After each activity the students gathered information for, they are required to produce an outline, journal, presentation, etc…to display their knowledge. At the end of the unit, the students produce a portfolio which is a collection of the student’s knowledge of fractals.
ADDITIONAL REFERENCE PRINT RESOURCES STUDENTS MAY USE:
The following books could be used as additional resources for a fee. I have not previewed any of these books, so please be careful that the books are age appropriate for your students. All of these books can be purchased on http://www.amazon.com.
1. Fractals by John Briggs $15.40 new, $6.99 used.
2. Fractals in Music: Introductory Mathematics for Musical Analysis by Charles Madden $39.95 new, $9.95 used
3. African Fractals: Modern computing and Indigenous Design by Ron Eglash $25.00 new, $9.95 used
4. Chaos and Fractals: New Frontiers of Science by Heinz-Otto Peitgen $69.95 new, $45.00 used
FOR ADDITIONAL ONLINE INFORMATION AND MEDIA PERTAINING TO THIS UNIT, TEACHERS MAY WANT TO USE THE FOLLOWING INTERNET ARTICLES:
Title1: The Math Forum at Drexel
URL1: http://mathforum.org/library/topics/fractals/
Description1: This website lists hundreds of other sites teachers or students can visit to gain information about fractals. The site connects teachers to lesson plans, software, and interactive activities pertaining to fractals. This is a great site to start research on fractals.
Title2: Fractals
URL2: http://math.rice.edu/~lanius/frac/
Description2: This site offers lesson plans for teachers to use within their classroom. The lesson plans include access to projects that students can work on to learn more about fractals. The site also provides background information about fractals to allow teachers and students to research the topic further.
Title3: The Wonder-World of Mandelbrot
URL3: http://mathforum.org/library/view/3459.html
Description3: The site provides lesson plans for teachers in grades 6-8 and 9-12. The lesson plans implement a video, bibliographies, frequently asked questions, graphics, and a listing of links.
Title4: 10 Lessons in Fractals, Complex Patterns, and Chaos
URL4: http://mathforum.org/te/exchange/hosted/lee/
Description4: The site contains lesson plans for middle and high school teachers on fractals. The lesson plans concentrate on how fractals relate to complex patterns and chaos theory. The lessons are available on MS Word, PC-compatible, and flash version.
Title5: The Topic: Fractals
URL5: http://www.42explore.com/fractal.htm
Description5: Great site with links to five different fractal projects for students. The students can access the projects online, or print the projects out and answer the questions.
Title6: African Fractals
URL6: http://www.rpi.edu/~eglash.dir/afractal.htm
Description6: The site reveals how fractals are seen in different cultures. The site offers order information for the purchase of videos and publications on African fractals. There is also software available, FractaSketch, that allows students to create their own African fractals.
Title7: Fractals
URL7: http://staff.harrisonburg.k12.va.us/~dflick/fractal/fractals.htm
Description7: The site contains a project where students have to research and create their own fractals.
Title8: Fractals: Math or Art?
URL8: http://www.dcet.k12.de.us/teach/quest/shari.htm
Description8: The site contains a project where students are instructed to write a one-page paper on fractals, create a fractal, and create a calendar using the desktop publishing program.
Title9: Fractal or Not to Fractal: That is the Question
URL9: http://www.manteno.k12.il.us/webquest/middle/Math/Fractals/fractalquest.html
Description9: The site contains a final project for students to work together to create a portfolio on fractals. Each student takes a different role, as assigned in the project, and students bring their individual information together in a portfolio. Students are to present their information in a PowerPoint presentation to the class.
RELEVANT INFORMATION, STUDENTS WILL VISIT THE FOLLOWING WEB SITES:
Title1: The Fractal Microscope
URL1: http://archive.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html
Purpose of Use1: Students will use this site for research.
Description1: This site provides an interactive approach to exploring different parts of fractals. With the fractal microscope, students can explore the patterns within the Mandelbrot set and other fractals. This site addresses a variety of topics and how they relate to fractals including scientific notation, coordinate systems and graphing, number systems, convergence, and similarity.
Title2: The Geometry Junkyard
URL2: http://www.ics.uci.edu/~eppstein/junkyard/fractal.html
Purpose of Use2: Students will use this site for research
Description2: This site provides an extensive list of links to material on fractals.
Title3: Fractint
URL3: http://spanky.triumf.ca/www/fractint/fractint.html
Purpose of Use3: Download software to construct fractals
Description3: Free software is available to download that allows students to generate their own fractals. This site provides fractal tutorials, wonderful pictures of fractal art, animated fractal art, and a contest where students can submit their own fractal art.
Title4: Welcome to the Spanky Fractal Database
URL4: http://spanky.triumf.ca/
Purpose of Use4: Students can use this site for research and interactive explorers.
Description4: The site offers students with many links to fractal information. The site also provides students with a link to interactive fractal explorers. Students can magnify previously made fractals, make their own fractals, and fully examine the Mandelbrot and Julia Sets.
Title5: Studying Mandelbrot Fractals
URL5: http://mathforum.org/alejandre/applet.mandelbrot.html
Purpose of Use5: Students can use this site for research and interactive explorers.
Description5: The site defines and explorers the Mandelbrot set by continually enlarging the screen. The students can magnify the Mandelbrot set to see how the images compare to each other. The site also links to other sites with fractal information.
Title6: World of Fractals
URL6: http://www.angelfire.com/art2/fractals/
Purpose of Use6: Students can research fractals and download software
Description6: The site describes different types of fractal animations, explanations, and free program downloads to create fractals.
Title7: African Fractals
URL7: http://www.rpi.edu/~eglash/eglash.dir/afractal.htm
Purpose of Use7: Students can see how fractals are relevant to different cultures.
Description7: The site allows students to research information about fractals and the African culture. Students can read reviews, publications, newspaper articles, and journals about African Fractals.
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