by Scott Nelson

Remote sensing is an important way to acquire data indirectly via satellites and airplanes. Its use is critical to our understanding of our planet and the processes that are continually changing it. There are many methods satellites can remotely sense features about our planet. One type of sensor on a satellite, classified as active instrumentation, is known as a non-imaging sensor. An example of a non-imaging sensor is called an altimeter, which uses radar pulses or lasers to determine the "height" of the land. Aerial photographs taken via aircraft in sequence with an approximate overlap of 60% creates a stereogram, which viewed using stereoscopic glasses, creates a three-dimensional image.

To determine whether remote sensing satellites are measuring accurately, scientists use ground measurements to double check and/or add data. These actions are called ground-truth studies and insure data is accurate. One type of ground measurement is called a topographic, or contour, map which consists of a series of contour lines. These are imaginary lines on the earths surface that connect points of equal elevation.

Teachers of science:

* plan an inquiry based science program for their students

* guide and facilitate learning

* design and manage learning environments that provide students with the time, space, and resources needed for learning science

Students will be able to:

* compare and contrast different types of data collection

* create a contour map

* build a 3-D model using a contour map

* evaluate the effectiveness of remote sensing

* explain the importance of remote sensing technology

* justify the importance of ground-truthing data to remotely sensed data

6-12 - each activity can be modified to fit the appropriate achievement level

To conduct a ground-truth study on remote sensing imagery using contour maps, models, and stereograms.

pencil scissors pipe cleaners stereoscopic glasses

tracing paper ruler poster board clear plastic shoebox/lid

glue Styrofoam masking tape modeling clay

needle/pin transparency film

Part A: creating a contour map

1. Place one strip of masking tape vertically on one side of the clear plastic shoebox. Create a scale on the tape using 1cm intervals starting from the bottom of the box and moving upwards. (start with 0cm at the bottom)

2. Use the modeling clay to create a land feature (volcano, island chain, mountainous terrain) inside the shoebox. The land feature should not be taller than the top of the shoebox.

3. Pour enough water into the shoebox so that the level is even with the 1cm mark. Place the clear lid on the box.

4. Place a piece of transparency film over the lid and secure it with a small piece of tape. With a marking pen, trace the outline of the land feature that appears above the water level. Label this line "1cm".

5. Repeat steps 3 and 4 filling the shoebox with water for each cm marking until no land feature appears above the water level.

6. Remove the transparency from the lid and return all materials to their proper place.

7. Convert your vertical scale in centimeters to real measurements of elevation using the ratio you have established (example: 1cm = 100 ft). Label each line with the proper elevation in feet. Your contour map is now complete.

8. Compare the contour map you created with the clay model. Notice how various features such as cliffs or gentle slopes are represented on map. Write down at least 5 observations based on your comparisons.

1. Study the contour map in Figure 1.

2. Place a piece of tracing paper over the contour map. Trace an outline of the perimeter of the contour map, the river, and the labels x,y, and z. Label this 4800 and place to one side.

3. Place a piece of tracing paper over the contour map. Trace the 4900 foot contour line. Cut along this line and discard all elevations less than 4900 .Label this 4900 feet and place to one side.

4. Repeat step 3 for each contour line in your contour map, cutting out and/or discarding all elevations less than the line you are cutting. Be sure to label the "x" , "y", and "z" in all your cuttings. You will notice there is another 5000 ft. contour line closer to the center of the contour map. Do not forget to trace this contour line, cut it out, label it, and place it to the side.

5. Glue each cutout onto a piece of poster board and cut to shape.

6. Arrange all of your cuttings, one on top of the other, so that they appear in their proper location on your contour map(all of the xs, ys and zs should line up one on top of the other). Use a pin to poke a small hole through each of the letters.

7. Obtain at least three pipe cleaners and a 4x5 piece of Styrofoam. Place your cuttings from step 6 on the Styrofoam and notice the position of the "x", "y", and "z". Place one of the pipe cleaners in the Styrofoam where the "x" would be located, one where the "y" would be located, and one where the "z" would be located (after removing your cuttings).

8. The cuttings will be placed over the pipe cleaners to create a three dimensional image of the land feature depicted in your contour map. Before doing this, you must create a vertical scale. An example of a vertical scale could be: for every 1/2 inch up the pipe cleaner, 100 feet of elevation are represented, or, for every 1cm up the pipe cleaner, 100 feet of elevation are represented. Once you choose your vertical scale, mark off each interval on all three pipe cleaners with a pencil or marker.

9. Glue your 4800 foot cutting (the one with the river) to the Styrofoam. Then push each successive cutting through the pipe cleaners to the proper elevation (determined by your vertical scale). Your three dimensional model is now complete.

10. Compare your three dimensional model to the contour map. Write down at least 5 observations about your comparison of these two images.

1. Review the use of stereograms and stereoscopes with your teacher.

2. Compare the stereogram in Figure 2 with the model you created in Part B and the contour map in Figure 1. Compose a final statement that determines whether the remotely sensed data is accurate using the ground truthing data from Part B.

3. Create a chart that compares and contrasts all three types of imagery.

1. What type of land feature is represented by the contour map from Part B,the model, and the stereogram?

2. Did the ground-truthing data coincide with the remote sensing imagery?

Explain.

3. How many feet are represented by 1 vertical inch on your model?

4. The gentlest slopes of the land feature from Part B can be described using which of the following directions? (circle the correct choice)

N NE E SE S SW W NW

5. The highest elevation is represented by which of the following letters from your contour map in Part B? (circle the correct choice)

x y z

6. Describe other possible uses of remote sensing imagery besides elevation.

Resources that can be obtained from a NASA Teacher Resource Center

1. Mission Earthbound: Remote Sensing Activities, p.5

2. Earths Mysterious Atmosphere: A Remote Possibility, p. 48.

3. Handout: Binary numbers

4. Handout: Paint by the numbers

5. Handout: Magic Wand.

6. Handout: Persistence of Vision

7. Handout: Computer Simulation

8. Atmospheric Detectives: Nothing but the truth, p.15.

9. NASA Activities in Planetary Geol.: Planets in Stereo, p.124.

Texts:

1. Laboratory Investigations in Earth Science. Silver Burdett Co., 1970.

2. Exercises in Physical Geology. Macmillan Publishing Co., 1986.

3. Stereogram Book of Contours. Hubbard Scientific, 1995.

4. The Complete Encyclopedia of Space Satellites. Portland House, 1986.

Websites:

http://geo.arc.nasa.gov/esdstaff/landsat/15USCch82.html

http://www.amproductions.com/rs.html

http://mollisol.agry.purdue.edu/~helt/syllabus.html

http://www.ilri.nl/lswlinks.html

http://rsd.gsfc.nasa.gov/rsd/RemoteSensing.html

http://ceps.nasm.edu:2020/RPIF/EARTH/earthcraft.html

http://acorn.educ.nottingham.ac.uk/ShellCent/maps/relief.html

http://www.colorado.com/trails/talk/dir/topo.html

Are images more than what we see? Are images sources of data waiting to be calculated into information? Images are all that and more. We receive images through remote sensing, i.e., things that we actually see, feel, hear, smell and taste are all forms of remote sensing. But speaking from a technological point of view, we are receiving images that we would have never imagined. With the use of remote sensing instruments such as telescopes, satellites, and microscopes, we are receiving images of very complex forms and processes called chaotic systems. Chaotic systems are determinististic, i.e., they have some determining equation ruling their behavior. Chaotic systems are very sensitive to their initial conditions. A very slight change in their starting point can lead to enormously different outcomes. This makes the system fairly unpredictable. In order to analyze and better predict these spatial form images, they are modeled using fractal geometry. The link between chaos and fractals is strong. Fractal geometry is the geometry which models-describes the chaotic system we find in nature. Fractal geometry is the study of shapes that contain an infinite amount of fine detail. Thus, no matter how much a shape is enlarged, there is still more detail to be revealed by enlarging it further.

Extending beyond the typical perception of Mathematics as a body of sterile formulas, fractal geometry mixes art with mathematics to demonstrate that equations are more then just a collection of numbers. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines, mountains, weather formation, growth( cell, population) and soil erosion. But , beyond potential applications of describing complex natural patterns, with their visual beauty fractals can alter the public belief that mathematics is dry and inaccessible and may help motivate mathematical discovery.

Goal:

The goal of this classroom activity is to create a type of scenario such that students discover chaotic systems using hands-on experience, technology and geometric visualization, to explore fractal geometry and its curriculum interrelationship to mathematics concepts. That is to say, students will be directly involved in constructing, counting, computing, visualizing, and measuring. Students will understand how fractal geometry mixes art with mathematics to demonstrate that equations are more than just a collection of numbers, but a visual model of what we witness in nature through various forms of remote sensing.

.

Concepts/Skills:

Mathematics -- Fractal geometry and its interrelated concepts: Number patterns, Geometric pattern, Geometric sequence, Data plotting, Similarity vs. Congruency, Visualization , Family of functions, Modeling, Technology usage, Area, Perimeter;

Science -- Chaotic systems

• be able to model chaotic systems using fractal geometry
• see the power and beauty of mathematics when they view it as an integrated whole
• recognize equivalent representations of the concept
• use and value the connections between mathematics and other disciplines.
• be able to enhance their understanding of topics and interrelations among disciplines by the use of technology as a tool .
• acquire confidence in using mathematics meaningfully.
• express mathematical ideas orally and in writing.
• develop an appreciation of geometry as a means of describing the world.

Grade Level: 7^th - 12^th Grade -NOTE: Activities are extended for upper levels.

Desired Number of Participants: 24 Students ( 6 cooperative groups of 4 students)

Project Timeline: 2-3 weeks

Prerequisites: Students should be able to measure length; count; and compute fractions.

Content Accommodations: NCTM Standards: Grade (6 - 8) #1- 10, 12, 13;

Grade (9 - 12) #1- 8, 10;

State Standards: Note- since all state standards are based

on NCTM standards, no state standards

are documented for these lessons.

Materials for all Activities: For each group:

Broccoli or Cauliflower; Plastic butter knives(optional); Dot paper;

Paper/pencil Magnifying glasses; Xerox paper; graphing calculator(optional)

This activity is designed for students to discover, observe, and understand how fractal geometry and the chaos theory are interconnected in their world. With the aid of technology and hands-on activities, students will observe and discover fractal geometry as an art and a complex system that is chaotic in nature.

Objectives: The students will be able to:

• make an observation of broccoli or cauliflower.
• estimate quantities.
• predict and draw conclusions

Materials : For each group:

Broccoli or Cauliflower; Plastic butter knives(optional); Paper/pencil

Magnifying glasses; Xerox paper;

1. Place the students in groups of four(one above-average student, two average students, and one below-average student) and assign roles: a reporter, a data recorder, a scientist, and a mathematician. Discuss roles/responsibilities with students.

• Reporter - ask/answer questions and orally summarize group results; will also

• Data Recorder - write and represent data and results in various form; will also

assist the mathematician.

• Scientist - systematically studies, observes and examines specimen
• Mathematician - studies the science of numbers, of shapes and performs all

Note: All members have primary roles but should work cooperatively to

summarize/ conclude the results.

2. Distribute the materials to each group and a worksheet on activity #1 to each student.
3. Instruct students to examine the plant(s) and begin working on the worksheet.
3. Discuss key questions from the worksheet after students have finished.
5. Instruct each group to come up with some previous learned concepts that they think could be used in collecting data from plants and images modeling plants to better understand the topic "Images of Chaos". Allow 10 - 15 minutes for group discussion.

F. Instruct each group to come up with some procedures they would follow to collect

data from plants like broccoli in order to describe growth patterns. Allow 10 - 15

minutes for a group discussion.

F. Check for understanding: Teacher/Students class discussion on procedure E and F.

Question/Answer session

G. Closure:

The following activity is a construction which uses a recursive, scaling, and substitution process to produce a fractal image.

Objectives: The students will be able to:

• use their experience and their concept of fractal geometry to aid in constructing a model of a chaotic system.
• select and use computation techniques appropriate to constructing self-similar fractals and determine whether the results are reasonable
• apply mathematical thinking and model chaotic systems to solve problems that arise in other disciplines, such as art, and science (geology, geography, biology,)
• interpret and draw dimensional objects
• classify figures in terms of congruence and similarity and apply these relationships.

Materials: For each group

. Ruler; pencil; colored pencils; dot paper; Xerox paper(optional);

1. Place the students in groups of four(one above-average student, two average students, and one below-average student) and assign roles: a reporter, a data recorder, a scientist, and a mathematician.

B. Distribute the material and Worksheet #2.

3. Instruct the groups to complete the following activity.

1. Using dot paper and a straight edge, construct an equilateral triangle with a base of

27 units. (Optional: Using xerox paper and a straight edge, construct an

equilateral triangle with a base of 18 centimeters.)

3. Repeat the iterative process starting from the line segments generated in the

previous procedure. Continuously apply this construction process three successive

times for all three sides of the equilateral triangle. Use different colored pencils

following the chart below for each generation.

Generation Colored Pencil

0 Regular lead pencil

1 Red

2 Blue

3 Green

4 Orange

5 Yellow

4. Students will do a journal writing summarizing activity #1 and #2.

D. Closure: Class discussion on the word "chaos" and its interrelations.

Objective: The student will be able to:

• compute the perimeter and/or area of a geometric model
• represent problem situations with geometric models and apply properties of figures
• use a coordinate system to plot data and categorize it as a particular function.
• express generalizations discovered through investigation

Materials: For each group

. Ruler, pencil, colored pencils, dot paper and graph paper and previous construction

from activity #2.

Place the students in groups of four(one above-average student, two average students, and one below-average student) and assign roles: a reporter, a data recorder, a scientist, and a mathematician.

1. Distribute the material and Worksheet #3 (Level 1 or Level 2)
2. Instructed student to complete investigation in Part I.
3. After discussing investigation #1, have student to proceed with the investigation in Part II and Part III(optional).

Books-

Peitgen, Heinz-Otto, et. al. Fractals for the Classroom: Part I, Springer-Verlag , 1992.

Peitgen, Heinz-Otto, et. al. Fractals for the Classroom: Part II, Springer-Verlag , 1992.

On-Line-

http://www.lib.rmit.edu.au/fractals/exploring.html

http://tqd.advanced.org/3703/frame.html

http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html

http://www-tep.ucsd.edu/SpaceMath2/SpaceMath/FracDimension.doc

http://www.gypsymoth.ento.vt.edu/~sharov/PopEcol/lec3/fracdim.html

In an effort to strengthen cross- curriculum ties, teachers are introducing subjects of contemporary interest like remote sensing in todays classroom. Remote sensing and its interrelated topics -chaos and fractals-are being presented via technology and hands-on activities. Consequently, topics such as these are more appreciated in this regard. Many ideas in this interrelated math-science topic was conceived during the students lifetime. Students must realize that if fractals and chaos are applicable to nature, then it must be applicable in fields as diverse as medicine, business, art, and music. When fractals beauty burst open in the early 1970s, it captivated many layman and professionals.

Therefore, these activities are intended as part of an introductory lesson on Chaotic system interrelationship to fractal geometry and its correlation of mathematics concepts. Students will develop a feel for how fractal geometry describe natural phenomena. Students shall be asked to solve problems by gathering information, making inferences about the data and transforming the data into other forms. Students will be encouraged to use the scientific method.

The National Council of teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989, p. 11) suggests that, "the curriculum should include deliberate attempts though specific instructional activities, to connect ideas and procedures both among different mathematical topics and with other content areas." NCTM (1989, p.7) further states, "the curriculum for all students must provide opportunities to develop and understanding of mathematical models, structures and simulations applicable to many disciplines."

The NCTM Standard broadly emphasizes integrating school mathematics and science within a problem solving framework. This approach is consistent with the recommendations of the American Association for the Advancement of Science as explained in Science for All Americans: Project 2061 (1989, p. 19).

1. On-Line:

 

http://spacelink.msfc.nasa.gov

http://math.bu.edu/DYSYS/chaos-game/chaos-game.html

http://math.bu.edu/DYSYS/chaos-game/node6.html

http://www.nctm.org

http://polymer.bu.edu/~trunfio/java/coastline/coast1.html

http://polymer.bu.edu/~knelson/CHS_Fractals.html

http://www.woodrow.org/programs/teachers/math/institutes/1993/22fall.html

http://members.australis.net.au/~engineer/publications/ew/wefractaldim12.html

http://www.glenbrook.k12.il.us/gbsmat/fractals/iter.html

http://www.lib.rmit.edu.au/fractals/exploring.html

 

2. Off-Line:

Bannon, Thomas J. 03/1991 Algorithm Demo Fractal: Program 1- Dragon On The AppleIIe, Program 2- Produce The Koch Curve On The AppleIIe, Program 3-Sierpinski Triangle On The AppleIIe. The Mathematics Teacher, pp. 182-184

Camp, Dane R. 04/1991 The Fractal Excursion. The Mathematics Teacher, pp. 265-275

Coes III, Loring. 1993 Building Fractal Models with Manipulatives. The Mathematics Teacher, 88(8): 646-651

Dale Seymour Publication Co.;1995 Calculator Program Explores A Star Fractal Pattern. Graphing Power High School Activities for the TI-81 and TI-82. pp190

Jurgens, Hartmut; Peitgen, Heinze-Otto; Saupe, Dietmar; 08/1990. The Language of Fractals. The Mathematics Teacher, pp. 60- 67

Marks, Tim K. 1992 Fractal. The Science Teacher,pp. 22-27

NCTM Addenda Series/Grades 5-8

Peitgen, Heinz-Otto, et. al. Fractals for the Classroom: Part I, Springer-Verlag , 1992.

Peitgen, Heinz-Otto, et. al. Fractals for the Classroom: Part II, Springer-Verlag , 1992.

Reinstein, David; Sally, Paul; and Camp, Dane R. 1997 Generating Fractals through Self-Replication. The Mathematics Teacher, 90(1):34-43

Atmospheric Detectives: Atlas II

Earth Mysterious Atmosphere : Atlas I

http://www-tep.ucsd.edu/SpaceMath2/SpaceMath/FracDimension.doc

Earth Mysterious Atmosphere : Atlas I

Science and Math Experience Manual: Light, Color, and Their Uses

Seeing in a New Light

Spacelink

Lam, N. S-N. and Quattrochi, Dale A; 2/92 On the Issue of Scale, Resolution, and Fractal Analysis in the Mapping Sciences. Professional Geographer, 44(1) :88-98

Extend - Topics in Mathematics in a Fractal Class:

Congruency, Ratios, Numerical patterns, Geometric patterns, Geometric sequences,

Pascal triangle, Probability, Similarity, Area, Visualization, Modular arithmetic,

Coordinate System, Logic and Truth Table, Power functions, Linear function,

Exponential functions, Logarithms, Data plotting, Slope, Limit concept, Box and

Whisker plots, Curve fitting, Coordinate Systems, Negative Numbers, Scientific

Notation, Multiplication, Distributive Law, and the Slope concept.

1. History
2. Art
3. Communications
4. Software
5. Programming
6. Science

Student Evaluation Method:

Student will write up a detailed lab report relating graphs and charts to functions.

NAME DATE PERIOD GROUP

ACTIVITY #1 : MAGNIFYING BROCCOLI

OBJECTIVES: The students will be able to:

• make an observation of the broccoli.
• estimate quantities.
• predict and learn methods for testing them in later activity

Materials: For each group:

Magnifying glasses; Broccoli; Plastic butter knives(optional); Pencil and

paper; xerox paper

Directions: Answer the following questions to be discussed.

1. The broccoli is classified as a __ __ __ n __ .

2. The broccoli grows from s __ __ __ s.

3. The broccoli goes through the process of __ __ p __ __ __ __ __ t __ __ __ to

create other generations like it.

4. Do you know if you have a whole broccoli? Explain.

5. What do you see when you observe the top of the broccoli with a magnifying glass ?

6. Does a living broccoli plant represent a system/process? Explain

7. Observe any similar repetitive fractional features in the broccoli and be prepared to

show and tell what you think are the self-similar features .

8. How many similar repetitive fractional features(self-similar fractals) make-up your

broccoli?

9. Briefly observe fractal growth in the broccoli and roughly sketch a picture graph

representing the growth.

NAME DATE PERIOD GROUP

OBJECTIVES: The students will be able to:

• use their experience and their concept of fractal geometry to aid in constructing an illustration of a chaotic system.
• select and use computation techniques appropriate to constructing self-similar fractals and determine whether the results are reasonable
• apply mathematical thinking and model chaotic systems to solve problems that arise in other disciplines, such as art and science( geology, geography, biology,)
• interpret and draw dimensional objects
• classify figures in terms of congruence and similarity and apply these relationships.

Materials: For each group

. Ruler, pencil, colored pencils, dot paper and graph paper

Directions: Complete the following construction and answer all questions.

1. Using dot paper and a straight edge, construct an equilateral triangle with a base of

27 units. (Optional: Using lineless paper and a straight edge, construct an

equilateral triangle with a base of 18 centimeters.)

3. Repeat the iterative process starting from the line segments generated in the

previous procedure. Continuously apply this construction process three successive

times for all three sides of the equilateral triangle. Use different colored pencils

following the chart below for each generation.

Generation Colored Pencil

0 Regular lead pencil

1 Red

2 Blue

3 Green

4 Orange

5 Yellow

4. Students will do a journal writing summarizing activity #1 and #2.

NAME DATE PERIOD GROUP

ACTIVITY #3 (Level 1) : COLLECTING DATA: COUNTING, GRAPHING,

OBJECTIVES: The students will be able to:

• represent situations with geometric models and apply properties of figures
• use a coordinate system to plot data and summarize it as a particular function.
• compute perimeter and/or area using geometric models

Ruler, pencil, colored colors, dot paper and graph paper

Directions: Complete this exercise using the model constructed from Activity # 2.

1. Count the number of segments in generations(stages) 1, 2, and 3. Discover and extend

the number pattern through the first five stages of growth. Use the chart below to input

the data.

2. Generalize to find the number of segments for stage n.

3. Graph the data from the chart above on graph paper in the

4. Imagine repeating the process without end. Visualize and describe how the figure

Set-up your own investigation on perimeter using the snowflake as a chaotic system. Starting from stage zero, let the equilateral triangle base equal one unit.

NAME DATE PERIOD GROUP

OBJECTIVES: The students will be able to:

• represent problem situations with geometric models and apply properties of figures
• use a coordinate system to plot data and summarize it as a particular function.
• compute perimeter and/or area using geometric models

Ruler, pencil, colored pencils, dot paper and graph paper

A. Set-up your own investigation on perimeter and area using the snowflake as a chaotic

system. Starting from stage zero, let the equilateral triangle base equal one unit.

B. Elaborate on the area and perimeter for the snowflake using the terms finite and

infinite.