Lesson Plan

Lesson title: Patterns to Fractals

Step 1—Desired Results
Established goals:
·  Elementary (E): Strand 3, Concept 1: Identify him/herself in relation to his/her primary, immediate, and extended clan family through the concept of the Navajo clan system. Also, students should be able to recognize and create patterns.
·  Secondary (S): 6.RP: Understanding ratios between two quantities. Strand 3, Concept 1: Identify him/herself in relation to his/her primary, immediate, and extended clan family through the concept of the Navajo clan system. Also, students should be able to find perimeters and Areas and define, give examples, and create fractals. Extension: Use areas and perimeters to come up with infinite sums.
Understandings:
Students will understand…
·  E: how to count triangles, how to color patterns
·  S: ratios of areas and perimeters of triangles, what a fractal is, when it is used in real-life, how to infinitely sum. / Essential questions:
·  E: What pattern do you see? How many triangles are there?
·  S: What are the areas of the triangles? What are the ratios? What is a fractal? Where are they used?
Students will know:
·  E: About patterns
·  S: what a fractal is, etc… (see performance tasks) / Students will be able to:
·  See performance tasks
Step 2—Assessment Evidence
Performance tasks:
·  E: Students will count triangles and create patterns with colors. Students will use pattern blocks to form their own triangle Students could even draw the triangle or their own patterns.
·  S: Students will complete the Sierpinski Triangle problem and the Koch Snowflake problem (see below). Students will explain what a fractal is and give a real-life example of one. Students will come up with their own fractal. / Other evidence:
·  E/S: The teacher will talk with the students, circulating the classroom as they work, looking for a clear understanding from all students.
Step 3—Learning Plan
Learning activities:
W = E: Students should know colors and shapes, as well as how to count before this lesson.
S: Students should know how to find area and perimeter. Explain the objectives of the lesson to the students and have them posted on the board.
H = E/S: Hold class discussion about clan chart and family tree.
E = E: Explain that there is a pattern with the clan charts and family trees.
S: What is the pattern in the clan charts and family trees? Does it end?
R = E: Students will create/color the Sierpinski triangle. Students will count the number of triangles. Students will also use pattern blocks to form their own patterns.
S: Students will do the Sierpinski Triangle problem and the Koch Snowflake problem (edited depending on level of class and knowledge of students). During a discussion, lead the class to come up with the definition of fractals. Assign homework: have them come up with their own fractal and/or give a real-life example of fractals.
E = E/S: Students will share their work with their classmates, comparing patterns and answers. Look at and discuss patterns in Navajo rugs.
T = E/S: Give hints as needed. Encourage extensions for gifted students. Group students if needed.
O = E/S: Have materials ready and on hand.
Step 4—Reflection
What happened during my lesson? What did my students learn? How do I know?
What did I learn? How will I improve my lesson next time?

Extension: For students in higher classes the teacher can have them come up with series and sequences and discuss limits of areas and perimeters for these fractals.

fractals

  1. Consider an equilateral triangle with sides of length 1. This triangle is considered to be stage number 0 of the Sierpinski triangle. Then the central triangle obtained by joining the midpoints of each side is removed. This is considered to be stage number 1 of the Sierpinski triangle. Stage number 2 is obtained when from each of the three remaining triangles the central triangle is removed, as was done on the initial triangle when going from stage number 0 to stage number 1 (see figure below). Appropriately shade the last triangle to examine stage number 3. Then complete the table below.

Stage number / 0 / 1 / 2 / 3 / n
Number of shaded triangles
Total perimeter of shaded triangles
Total area of shaded triangles

(a)  Find U1, U2, U3, P1, P2 and P3.

(b)  Write a recursive description of Un in terms of Un-1. Find an explicit formula for Un.

(c)  Write a recursive description of Pn in terms of P. Find an explicit formula for P.

(d)  Use your work to evaluate the sum 14+316+964+…+3994100+31004101.

  1. In 1904 Helge von Koch invented his snowflake, which is probably the first published example of a fractal. It is the result of an endless sequence of stages: Stage 0 (the initial configuration) consists of an equilateral triangle, whose sides are 1 unit long. Stage 1 is obtained from stage 0 by replacing the middle third of each edge by a pair of segments, arranged so that a small equilateral triangle protrudes from that edge. In general, each stage is a polygon that is obtained by applying the middle-third construction to every edge of the preceding stage.

(a)  Stages 0, 1, and 3 are shown above. Make your own sketch of stage 2.

(b)  Stage 0 has three edges, and stage 1 has twelve. How many edges do stages 2 and 3 have? How many edges does stage n have?

(c)  Stage 1 has twelve vertices. How many vertices does stage n have?

(d)  How long is each edge of stage 1? of stage 2? of stage n?

(e)  What is the perimeter of stage 1? of stage 2? of stage n?

(f)  Does the snowflake have finite perimeter? Explain.

(g)  Is the area enclosed by the snowflake finite? Explain.