Standard Benchmarks and ValuesCommon Core Standards for Mathematics:High School Geometry - Modeling

• Make geometric constructions• Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). • Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

National Council of Teachers of Mathematics, Geometry grades 6-8, 9-12

Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, architecture, and everyday life.

National Standard for Art Education, Grade 9-12, Visual Art

Standard 1 - Understanding and applying media, techniques, and processes.

Standard 2 - Using knowledge of structures and functions.

Standard 3 - Choosing and evaluating a range of subject matter, symbols, and ideas.

Standard 4 - Understanding the visual arts in relation to history and cultures.

Standard 5 - Reflecting upon and assessing the characteristics and merits of their work and the work of others.

Standard 6 - Making connections between visual arts and other disciplines.

Photo by Jitze Couperus

Ethnomathematics 2

FRACTAL LANDSCAPESPauline Sawai

9th - 12th Grade2 - 3 Week Periods

What are fractals and how are they applied in society?Enduring Understandings

• Fractals are patterns found in nature and part of everyday life.

• Knowledge of fractals is applicable to nature, science and entertainment.

• Elements and Principles of Art can create meaning in artwork.

• Culture has influence on the artist or the artwork.

Critical Skills and ConceptsStudents will:1. Create a 4-5 generation fractal.2. Create or imitate a landscape for digital or

media use.3. Apply organizational principles (i.e., repetition,

balance, contrast, etc.) in original works.4. Self assess their work.5. Compare the use of creativity and imagination

in artistic pursuits to the use of creativity and imagination in the field of science.

Cultural Extension:6. Assess fractals in Hawaii using native or

endemic plants on land or in the ocean through observation and comparison to fractal examples. Take students on a field trip to collect images (through drawing or photographing) to use in their study of fractals.

Fractal Landscapes3

LESS

ONAuthentic Performance Task• Students will research and find

6 or more examples of types of fractals: Sierpinski, Mandelbrot, and Koch.

• Students will create a landscape or artwork based on authentic natural inspirations from photographs.

Students will apply Enduring Understanding: • Understanding the elements and

principles of design provide the tools for visual expression.

• Art provides a framework for viewing the world.

1. What is a fractal? View PBS clip on Fractals: The Hidden Dimension. See resources page.

2. In groups of 2-3 students, distribute the worksheet Fractal Problem, and have students work together to answer the questions.

3. Distribute the reference sheet 3 Examples of Fractals

4. Discuss and review fractals using the reference sheet. Show presentation of different types of fractals and system that it may be a part of. (Fractals: Natural or Man Made? resources page)

5. Compare or point out examples of fractals in landscapepictures or pictures of nature. (See Fractals in My Backyard ppt. or pdf.)

6. Have students reflect on the following as an exit pass: What are fractals and how are fractals important in everyday life and for the future?

Authentic Audience

Learning Plan

Other Evidence

Teacher

Student

Viewers of movies, natural images, and fantastical images.

Students provide a written reflection of the work completed:a. What were the influences for this artwork? b. How did the elements and principles of art help create

this image?c. What are the parts of the image that stand out and why?

What parts should be changed? Explain.d. What are some examples of fractals that exist naturally

and that are man made? How did the knowledge of fractals help to create your image?

Ethnomathematics 4

LESSON

• Use SHAPE TUTORIAL to create fractal. Post requirements from previous instruction on board for reference.

• Print final creations and complete reflection from Other Evidence section.

• Display and discuss several images from nature.

• Homework: Bring in images from the environment of fractals that occur in the naturally.

• Display and discuss images and how fractals are applied to them.

• Using Illustrator or Photoshop computer program to create a fractal landscape or image based on the inspiration of the environment.

• Critique and discuss the final images and review the inspiration or the genesis for the images.

9. Create a computer generated image of a landscape based on their knowledge of fractals.

7. Demonstrate fractals on Photoshop. (see Shape Tutorial)

8. Create a fractal on the computer that replicates at least 20 times with the requirements and rubric for distribution. Think about the different images and how they influenced your project as you create it.

Project Requirementsa. Use 3-4 different shapes, 3-4 different layer styles for the main replicating

shape. Include a background. Resulting image should reflect understanding of space, shape, form, color, repetition, unity, variety and emphasis.

b. Set format to 8 1 ⁄2”X11”, 300 ppi.c. Using the shape tool, combine 3 to 4 different shapes using the layer styles

to create the desired shapes. Use the final combined shapes to fill the space with those shapes to create a final fractal pattern.

(Write following directions on overhead or board and keep up until student has recorded or the project is finished for reference.)

Teacher

Student

Method

Cultural Connection and Project Extension

Procedure

Fractal Landscapes5

RESO

URCE

S Fractals in my Backyard. Presentation reference for fractalsaround us. Ppt and pdf format.

Movie to Review with class: Hunting the Hidden Dimensionhttp://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html

3 Examples of Fractals.pdf Wikipedia source

Review with students “Fractals: Natural or Man Made?”Show a powerpoint presentation of various images and have them guess are they natural or man made? (All are natural.)

Fractal Problem. Pdf

Shape Tutorial.pdf

Fractal Rubric and assessment questions

Image Gallery: Fractals.pdf Students samples as well as nature samples to print and hang in your room.

Movie to review about Fractals: History and comparison withAfrican Culturehttp://www.ted.com/talks/ron_eglash_on_african_fractals

Another tutorial on the web: How to simulate fractals in Photoshophttp://design.tutsplus.com/tutorials/how-to-simulate-fractals-in-photoshop--psd-340

Fractals resources:What are Fractals? http://fractalfoundation.org/resources/what-are-fractals/

What is a fractal?http://fractalfoundation.org/fractivities/WhatIsaFractal-1pager.pdf

Cool Math Fractal Galleryhttp://www.coolmath.com/fractals/gallery.htm

Earth’s Most Stunning Natural Fractal Patternshttp://www.wired.com/2010/09/fractal-patterns-in-nature/?pid=170 - slideid-170

Resources

Other Resources and Background Information

Ethnomathematics 6

Repeating Shapes TutorialProgram: Adobe Photo-Format: 8 ½ × 11”, Resolution 300 ppi, landscape or portrait.

Save as: p1_lastname_shapes

Create a new layer, use the shape tool to draw a shape, 2” wide.Go to Layer > Layer Style > Blending options. Choose 4 - 5 different effects to change the shape and make it interesting. (EX: Add drop shadow, different stroke, color, pattern overlay, etc.)

Group 3 -4 shapes with the desired effects and resize them about 5” width.

Results: your finished shapes

Follow this sequence to make repeated shape layers:• Command + E > Combine layers.• Select combined shapes to make your repeated pattern.• Command + ALT + SHIFT + T > repeat commands.• Resize (use transform corners)• Rotate (use transform corners)• Move reference point out of the selection.• Return (to store commands)

Reference point moved

Fractal Landscapes7

Fractal Landscapes

The Sierpiński arrowhead curve is a fractal curve similar in appearance and identical in limit to the Sierpiński triangle.

Evolution of Sierpiński arrowhead curve The Sierpinski arrowhead curve draws an equilateral triangle with triangular holes at equal intervals. It can be described with two substituting production rules: (A →B-A-B) and (B →A+B+A). A and B recur and at the bottom do the same thing — draw a line. Plus and minus (+ and -) mean turn 60 degrees either left or right. The terminating point of the Sierpinski arrowhead curve is always the same provided you recur an even number of times and you halve the length of the line at each recursion. If you recur to an odd depth (order is odd) then you end up turned 60 degrees, at a different point in the triangle.

The Koch snowflake (also known as the Koch star and Koch island) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” (original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémenatire) by the Swedish mathematician Helge von Koch.The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:1. Divide the line segment into three segments of equal length.2. Draw an equilateral triangle that has the middle segment from step 1 as its base and

point outward.3. Remove the line segment that is the base of the triangle from step 2.After one iteration of this process, the resulting shape is the outline of a hexagram.The Koch snowflake is the limit approached as the above steps are followed over and over again. The Koch curve originally described by Koch is constructed with only one of three sides of the original triangle. In other words, three Koch curves make a Koch snowflake.

Koch Snowflake

Sierpiński Arrowhead CurveInformation From Wikipedia

Ethnomathematics 8

The Mandelbrot set is the set of complex numbers ‘c’ for which the sequence ( c, c² + c, (c²+c)² + c, ((c²+c)²+c)² + c, (((c²+c)²+c)²+c)² + c, ...) does not approach infinity. The set is closely related to Julia sets (which include similarly complex shapes) and is named after the mathematician Benoit Mandelbrot, who studied and popularized it. Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.More precisely, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial

z_{n+1}=z_n^2+cremains bounded.[1] That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0,..., which is bounded, and so −1 belongs to the Mandelbrot set.Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The “style” of this repeating detail depends on the region of the set being examined. The set’s boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules, and is one of the best-known examples of mathematical visualization.

Fractal Landscapes9

Create a fractal by beginning with an equilateral triangle (Step 1).Using the midpoints of the sides of the original triangle, draw a smaller equilateral triangle. Now there are four triangles. We remove the center triangle, leaving behind three shaded triangles (Step 2).Repeat Step 2 on each of the 3 shaded triangles. Now there are 13 triangles, 9 of which are shaded (Step 3).(Note the features of the fractal; constructed shapes are “self-similar” and the process repeats.

Extend the drawing above for at least one more step.

Answer the following:a. At Step 1 there was on equilateral triangle, at Step 2 there were 4 seperate and non-

overlapping triangles, and at Step 3 there were 13 such triangles. How many triangles would there be at Step 4? Step 10? Step N?

b. If the area of the shaded portion at Step 1 was 1 unit, the area of the shaded portion at Step 2 is ¾ of a unit. What is the area at Step 3? Step 6? Step N? (Is the shaded area increasing or decreasing?) The original area was 1 unit and this does not change throughout this problem.

c. If the perimeter of the shaded portion at Step 1 is 3 units and the perimeter at Step 2 is 4 ½ units or ⁹/₂ units. What is the perimeter of the shaded portions at Step 3? Step 6? Step N? (Is the perimeter increasing or decreasing?)

For each part a, b, and c above, are the measures increasing or decreasing at each step? How?

The Fractal Problem

Ethnomathematics 10

(From the Fractal Foundation)