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Fractals, An Inquiry: Day 2


Lesson Overview | Unit Description | Lesson Description for Day | State Standards
Goals | Methods | Wrap-Up | Assessment | Materials

Lesson Overview

Title: Fractals, An Inquiry: Day 2
Author: CAST
Subject: Science
Grade Level(s): 6–8
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Unit Description

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.

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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.

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State Standards

Florida Sunshine State Standards:
Standard 1:
The student uses the scientific processes and habits of mind to solve problems. (SC.H.1.3)
Standard 2:
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.

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Goals

Unit Goals:
  1. Students will gain an understanding of the process of scientific inquiry.
  2. Students will gain independence in conducting and reporting on their own inquiries.
  3. Students will become aware that scientific inquiries can happen anywhere, and are a natural part of exploring the world.

Lesson Goals:

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.

  1. Students will gain an understanding of the meaning of fractals.
  2. Students will understand that fractals can be found in many places.
  3. Students' powers of observation will be increased as they become more motivated to look closely for fractals in their environment.
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Methods

Anticipatory Set:

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.'


Introduce and Model New Knowledge:

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:

http://en.wikipedia.org/wiki/Frond

http://classes.yale.edu/fractals/Panorama/Nature/MountainsReal/Mountains1.gif

Mandelbrot Zoom (1.1 meg)

http://en.wikipedia.org/wiki/Blood_vessels

Students can manipulate this fractal making tool for a hands-on experience in understanding fractals:

http://www.coolmath.com/fractals/fractalgenerators/generator1/index.html

(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

Reference:
http://kluge.in-chemnitz.de/documents/fractal/node2.html
and:
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).

Reference:
http://www.jracademy.com/~jtucek/math/fractals.html


Provide Guided Practice:

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.


Provide Independent Practice:

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:
http://classes.yale.edu/Fractals/Panorama/Art/Decalcomania/Decalcomania.html

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.

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Wrap-Up

Group Share:

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.

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Assessment

Formative/Ongoing Assessment:

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'.


Summative/End Of Lesson Assessment:

None in this lesson.

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Materials

Other Resources
  • 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
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Print

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Represent information in multiple media and formats.

Listen

UDL Reflection:

By situating scientific inquiry in a different learning context, namely art, I'm hoping to capture a greater variety of learners. Since the art component to this lesson doesn't require generating a skilled artistic product, students with less background knowledge of the scientific method will hopefully have a greater comfort level when learning about it for the first time.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Represent information in multiple media and formats.

Listen

UDL Reflection:

By presenting several different examples of fractals, a straightforward definition, a narrative description, a manipulable tool for generating fractals, and finally the opportunity to create their own with art media, I'm hoping students are provided with well-rounded options that support their recognition networks.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Affective Network – Provide adjustable levels of challenge.

Listen

UDL Reflection:

By ensuring that the outcomes of learning rather than the means of achieving them are expressed as goals, diverse learners can be included in all accountabilities for the learning process. Also, involving students in creating their own goals for learning, rather than having them always imposed from above enhances motivation, because they are now stakeholders in the process. When goals are perceived as being relevant to students' needs, their motivation to achieve them increases.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Provide supports for limited background knowledge, and provide a context for learning.

Strategic Network – Provide flexible opportunities for demonstrating skill, (e.g. written, oral, or visual presentation, explanations, word process).

Listen

UDL Reflection:

By connecting to students' prior knowledge and experiences, I'm hoping to offer a platform to which they can add and modify new information, skills, and knowledge. The variety of presentation options offered later in the lesson will augment and support this initial anticipatory set.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Provide multiple examples, show the range of examples, provide examples and counter-examples.

Listen

UDL Reflection:

It is my goal that in providing three different linguistic formats for defining fractals (informal definition, formal, and descriptive narrative) that I'll be supporting a greater range of students than if I had just kept to the traditional dictionary definition.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Provide multiple examples, show the range of examples, provide examples and counter-examples.

Listen

UDL Reflection:

This section offers multiple ways for students to explore fractals, so that ideally, I've provided a broad umbrella of options in the hope that all students' individual learning needs are met.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Strategic Network – Provide novel problems to solve (e.g., unique problems outside the initial instructional set to promote generalization and transfer).

Listen

UDL Reflection:

By not defining fractals explicitly at first, I'm attempting to give students opportunities to construct their own knowledge by connecting their prior experiences to this exploration. Since I reminded them in the anticipatory set of a fundamental experience they most likely will have had, I'm hoping they can develop new connections to it through this open exploration.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Recognition Network – Provide multiple examples, show the range of examples, provide examples and counter-examples.

Listen

UDL Reflection:

By revisiting the original examples again, students have another opportunity to refine the knowledge they're constructing, this time by applying new knowledge acquired through discussing the definitions. Also, revisiting the original examples while providing guidance and feedback, allows me to observe how well learners can connect new information to their prior knowledge.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Strategic Network – Provide flexible opportunities for demonstrating skill, (e.g. written, oral, or visual presentation, explanations, word process).

Listen

UDL Reflection:

It's difficult to always provide options for expressing one's knowledge—time, technology, and other constraints can often hinder this. But at the least, offering a few simple options for expression that cover a range of media, such as paper, recorder, camera, and computer, increases the likelihood that more students will be able to accurately express what they have learned, and additionally, it provides teachers with more accurate windows through which to view student understanding.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Affective Network – Offer choices of content and tools (e.g., choice of books to study literature).

Listen

UDL Reflection:

By letting go of my need to channel student engagement into a narrow range of options, due to fear of losing control of the classroom, I've discovered that actually, a greater order descends. Options tend to facilitate student ownership in the learning process. Also, the classroom becomes a 'laboratory' where students are engaging in meeting objectives through a variety of channels better suited to their needs.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.

UDL Connections
UDL Connection
UDL Feature:

Strategic Network – Provide ongoing, relevant feedback (e.g., questions and answers in classroom).

Listen

UDL Reflection:

Ongoing assessment provides me with a rich set of data for making decisions about my students' needs. I can't imagine a day without my clipboard and rotation around the room to jot down my observations. This information really helps me make decisions about the very next day—what the students I observe need in terms of support, and in terms of extra challenges. It also allows me to provide on the spot feedback to students, so I can encourage them and/or clear up any misunderstandings. Of course I don't have time to observe every student every day, but by making sure that I rotate and observe sets of students over three days, at least I've reached everyone within a short time frame. This in turn means that I can 'turn on a dime' with my teaching methods.

Listen

Note: If you do not get sound when you click on the audio Listen button within a UDL connection, simply close and reopen the UDL Connection, and click on the audio Listen button again.