LESSON PLAN

Name: Reem Kattura

Title of lesson: Getting to Know Fractals Using Sierpinski's Triangle

Date of lesson: Fall 2006

Length of lesson: 50 Minutes

Description of the class:

                     Name of course: Geometry

                     Grade level: 9^th/10^th

                     Honors or regular: Regular

Source of the lesson:

            PBI Fall 2003: Fractals - Lesson 2 “Serpienski Triangle” by Derek McDaniel

TEKS addressed:

§111.34. Geometry
(G.5)  Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems.
(A)  use numeric and geometric patterns to develop algebraic expressions representing geometric properties.

(G.9)  Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures.
(D)  analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.

I. Overview

Self-symmetry, recursion and infiniteness are properties of fractals. An exercise using Sierpinski Triangle (an example of a fractal) will give students an opportunity to identify and use these properties.

II. Performance or learner outcomes

Students will be able to:
1. List the properties of fractals.
2. Write n-th term explict and recursive formulas.

III. Resources, materials and supplies needed

Each student will need 4 pieces of paper each with an equilateral triangle drawn on it covering the entire page. Each page will be identical except one will be labeled stage 0, stage 1, etc.

IV. Supplementary materials, handouts.

Shaded vs. Unshaded worksheet

Five-E Organization

Teacher Does                    Probing Questions                    Student Does      

Engage: Learning Experience(s) Students will begin to think about fractals and the properties they exhibit. Teacher shows students a few different pictures of famous fractals and asked to write down one property they see that each of the pictures has in common. (Color may not be one of the properties)

Critical questions 1. What do each of these pictures have in common? What about size? Shape? What are the dimensions of each picture?

Expected Student Responses/Misconceptions 1. Each picture is a smaller version of itself. Each picture is continuous, no beginning or end.

Evaluate
Teacher will call on different students randomly to share and explain their ideas with the class. A class discussion will then take place to tell everyone that the pictures they were just looking at are called fractals..

Explore: Learning Experience(s) The students will physically reproduce the Sierpinski Triangle. The students will have to measure the lengths of the sides and find the midpoints of all the triangles. They will then have to shade in the appropriate triangles. The four pages with the triangle will be passed out and the students will follow as the teacher shows how to recreate the triangle on the overhead. The Teacher demonstrates how to recreate the first stage of the triangle. The students will then have to recreate the last three stages on their own individually.

Critical questions 1. How should we recreate the triangle? How many triangles will we need to do? How will we measure the midpoint of each side? 2. How many shaded triangles are there for stage 0? What about stage 1?

Expected Student Responses/Misconceptions 1. Connect the midpoints of each one of the sides of the triangles then shade in the middle triangle. 2. Stage 0 has 1 triangle, stage 1 has 3 triangles

Evaluate
Teacher calls on students to share their triangles at each stage and explain to the class how they continued on past stage 1.

Explain: Learning Experience(s) The students fill the chart given to them. After the students have completed their charts they will have to find a way to represent the information in the charts for the n-th stage.

Critical questions 1. Do you notice a pattern between stage number and number of shaded triangles? What about stage number and number of unshaded triangles? 2. What formula did you come up with for shaded triangles? 3. What formula do you come up with for unshaded triangles?

Expected Student Responses/Misconceptions The students will complete their charts. 1. Yes (for both) 2. an = 3^n 3. bn+1 = 3^n + bn

Evaluate
Teacher picks on a couple of students to give the formulas they come up with and explain the process they used to find them.  

Extend / Elaborate: Learning Experience(s) Students use information from the shaded and and shaded triangles to find a relationship between the two that gives the next term for unshaded triangles.

Critical questions There is a relationship that relates unshaded and shaded triangles from one stage to give the unshaded triangle in the next stage (e.g. Unshaded stage 1 relates to shaded and unshaded stage 0). What is it?

Expected Student Responses/Misconceptions If shaded is an and unshaded is bn, then bn+1 = an + bn.

Evaluate
Students have the right relationship and can explain how they got it.   

Wrap Up: Lesson Objective(s) Teacher explains that today’s activity used the properties of fractals. Teacher writes out these properties. “Your activity today demonstrates 3 properties that fractals have: self-symmetry, infiniteness, and recursion.”

Critical questions How did we see each of these properties?

Expected Student Responses/Misconceptions Self-symmetry was when each time we redrew the triangles, they had the same characteristics. Infinite was coming up with the n-th term formula. Recursion was when we did the same thing over and over to create Sierpinski's Triangle.

Name: Date:

Partner:

Shaded Vs. Unshaded

Stage	# of Shaded Triangles	Number of Unshaded Triangles
0
1
2
3
4

1. Give n-th term formula for shaded triangles. Show your thought process.

2. Give n-th term formula for unshaded triangles. Show your thought process

3. Relate shaded and unshaded. Show your thought process.