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Fractals, An Inquiry: Day 2
Grade Level(s): 6–8
This unit is designed as a UDL approach to supporting student understanding of scientific inquiry. Students experience first-hand each step in the inquiry process and how the steps systematically build toward understanding. A simple art activity that involves creating fractals with paint serves as the context for the inquiry. By situating an introduction to inquiry in art, students who might feel reluctant or incompetent in science have an alternative environment for engaging in the inquiry process. Exploring scientific inquiry through art is also a way to enable students to experience the inquiry process as a natural, sometimes spontaneous process that is intrinsic to many learning experiences.
Fractals, intriguing in their complexity and beauty, have been charted mathematically, and serve as the connection for the inquiry. By experimenting with the effects of various art media on the formation of fractals, such as the thickness of paint or types of paper used to make them, students' initial encounter with the inquiry process occurs in a non-threatening, intuitive way, so they will arrive at an understanding of the inquiry process inductively.
Lesson Description for Day
In today's lesson, students will engage in both an informal exploration and a formal overview of the meaning of fractals. They will observe several examples, manipulate a digital tool that creates them, and dialogue with partners about their understanding of fractals.
Students will then use paint media in order to create fractals of their own in preparation for a scientific inquiry.
Florida Sunshine State Standards:
The student uses the scientific processes and habits of mind to solve problems. (SC.H.1.3)
The student understands that most natural events occur in comprehensible, consistent patterns. (SC.H.2.3)
1. Recognizes that patterns exist within and across systems
California State Standards:
Investigation and Experimentation
Scientific progress is made by asking meaningful questions and conducting careful investigations. As a basis for understanding this concept and addressing the content in the other three strands, students should develop their own questions and perform investigations.
- Students will gain an understanding of the process of scientific inquiry.
- Students will gain independence in conducting and reporting on their own inquiries.
- Students will become aware that scientific inquiries can happen anywhere, and are a natural part of exploring the world.
Time Frame: 10 Minutes
Distribute copies of the following goals to students. After sharing them, invite students to write down other goals they think are pertinent to the lesson. Discuss them with the class and vote whether to adopt any of them. Students' suggestions can remain anonymous during this process.
- Students will gain an understanding of the meaning of fractals.
- Students will understand that fractals can be found in many places.
- Students' powers of observation will be increased as they become more motivated to look closely for fractals in their environment.
Time frame: 5 minutes
Ask the class if they have ever spent time standing under an oak or maple tree looking up into its branches. Ask for one or two volunteers to describe what they see when they do this. (Ideally, they'll describe how the smaller branches extend outward and seem to 'copy' the shape of the tree itself). Explain to students that today's lesson will focus on many such patterns in nature, and that they can be called 'fractals.'
Build Background on Fractals as the Context for the Inquiry
Time Frame: 15 minutes
A. Explore Fractals Informally:
In order to provide a context and connection for the scientific inquiry, it is necessary to first build background on the concept explored, in this case, fractals. Share some visual examples of fractals with students, without defining them explicitly at this point. Discuss the images briefly, then ask students to share their observations of what the images might have in common. The following websites have examples you can project or print for viewing:
Students can manipulate this fractal making tool for a hands-on experience in understanding fractals:
(Use the above site's fractal-making tool by dragging your cursor to the border of the image until a square forms. An enlargement appears. Repeat the process for each magnified image that appears in order to demonstrate self-similarity of fractals into infinity.)
B. Define Fractals:After discussing possible commonalities of the images, provide a simple definition of fractals:
- Fractals: (In simplest language): when a section of an image or object resembles the whole object's shape, or another smaller part of the object, it can be called a fractal.
- From Merriam-Webster Unabridged Dictionary: Any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.
C. Share a section of a descriptive narrative that defines fractals:
Fractals in nature and applications
Fractals are not just complex shapes and pretty pictures generated by computers. Anything that appears random and irregular can be a fractal. Fractals permeate our lives, appearing in places as tiny as the membrane of a cell and as majestic as the solar system. Fractals are the unique, irregular patterns left behind by the unpredictable movements of the chaotic world at work.In theory, one can argue that everything existent on this world is a fractal:
- the branching of tracheal tubes,
- the leaves in trees,
- the veins in a hand,
- water swirling and twisting out of a tap,
- a puffy cumulus cloud,
- tiny oxygene molecule, or the DNA molecule,
- the stock market
The basic concept of fractals is that they contain a large degree of self similarity. This means that they usually contain little copies of themselves buried deep within the original. And the also have infinite detail. Like the costal problem, the more you zoom in on a fractal, the more detail (coastline) you get. And this keeps going on forever and ever, so you could make a pretty movie of a fractal zooming in. Or two. So far I've made a Mandelbrot Zoom (1.1 meg) and a Julia Set Zoom (784 k).
Generate and Refine Meanings:
Time Frame: 10 minutes
Share some of the same visuals again with students, and encourage them to identify similarities between parts of the image and its general shape or pattern.
During the discussion, help clear up any misperceptions, as you guide students toward a clearer understanding of fractals. Have students discuss their observations with partners and if time allows, to sketch, audio record, or share examples of their observations orally.
Create fractal patterns:
Time Frame: 25 minutes (Allow time for clean up)
Have students work in small groups to create their own fractals using finger or oil paints, a variety of brushes, and large sheets of paper.
Step-by-step directions can be found at:
Alternatively, if students prefer a more controlled approach, they can use colored pencils, markers, and other media to draw fractals. Or if they have drawing software available, they might wish to generate fractals on the computer.
Time Frame: 10 minutes
Provide each small group with the opportunity to display the fractals they made. Ask them to point out the traits that make their images fractals.
As students engage in discussions of fractals and while they are making their own, circulate among them. Prompt as many students as possible for their insights and understandings of fractals. Jot down notes about the students' understanding after each informal 'mini-conference'.
None in this lesson.
- finger or oil paints
- containers for holding paint and diluting with water if necessary
- enough paint brushes for every student
- blank sheets of paper (all of one texture and weight)—enough so every student has several
- sponges and toweling for clean-up