This activity is designed to further the work of the
Infinity, Self-Similarity, and Recursion lesson by showing students other classical fractals, the Sierpinski Triangle and Carpet, this
time involving iterating with a plane figure.

Objectives

Upon completion of this lesson, students will:

have seen the classic geometric fractals

have reinforced their sense of infinity, self-similarity and recursion

have practiced their fraction, pattern recognition, perimeter and area skills

Standards Addressed:

Grade 10

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 6

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

Geometry

The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 9

Geometry

The student demonstrates an understanding of geometric relationships.

The student demonstrates a conceptual understanding of geometric drawings or constructions.

Fifth Grade

Operations and Algebraic Thinking

Analyze patterns and relationships.

Geometry

Similarity, Right Triangles, and Trigonometry

Prove theorems involving similarity

Grades 6-8

Geometry

Use visualization, spatial reasoning, and geometric modeling to solve problems

Grades 9-12

Geometry

Use visualization, spatial reasoning, and geometric modeling to solve problems

Geometry

Geometry and Measurement

Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Grade 8

Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Integrated Mathematics III

Geometry and Measurement

Competency Goal 2: The learner will use properties of geometric figures to solve problems.

Introductory Mathematics

Data Analysis and Probability

COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Geometry and Measurement

COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts to solve problems.

Technical Mathematics I

Geometry and Measurement

Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

Geometry and Measurement

Competency Goal 1: The learner will use properties of geometric figures to solve problems.

5th Grade

Patterns, Functions, and Algebra

5.20 The student will analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship, using words, tables, graphs, or a mathematical sentence. Concrete materials and calculators will be used.

6th Grade

Geometry

6.15 The student will determine congruence of segments, angles, and polygons by direct comparison, given their attributes. Examples of noncongruent and congruent figures will be included.

Student Prerequisites

Geometric: Students must be able to:

recognize and sketch objects such as lines, rectangles, triangles, squares

understand the concepts of and use formulas for area and perimeter

Arithmetic: Student must be able to:

build fractions from ratios of sizes

manipulate fractions in sums and products

Technological: Students must be able to:

perform basic mouse manipulations such as point, click and drag

use a browser for experimenting with the activities

Teacher Preparation

Access to a browser

Pencil and Graph Paper

Copies of supplemental materials for the activities:

The bent line-segment or figure that replaces the initiator at each iteration of a fractal

infinity

Greater than any fixed counting number, or extending forever. No matter how large a number one thinks of, infinity is larger than it. Infinity has no limits

initiator

A line-segment or figure that begins as the beginning geometric shape for a fractal. The initiator is then replaced by the generator for the fractal

iteration

Repeating a set of rules or steps over and over. One step is called an iterate

recursion

Given some starting information and a rule for how to use it to get new information, the rule is then repeated using the new information

self-similarity

Two or more objects having the same characteristics. In fractals, the shapes of lines at different iterations look like smaller versions of the earlier shapes

Lesson Outline

Focus and Review

Does anyone remember what infinity means?

Can someone explain to the class what an iteration is?

Who knows what self-similarity is?

Objectives

Today, class, we are going to learn more about fractals, the idea of self-similarity, and
recognizing patterns within fractals.

We are going to use the computers to learn more about fractals, the idea of self-similarity,
and recognizing patterns with in fractals, but please do not turn your computers on until I
ask you to. I want to show you a little about this activity first.

Teacher Input

Walk students through several steps of the
Sierpinski Triangle. The students should look at the patterns made by the areas of the individual triangles and
the total area. It may take drawing two or three iterations before the number pattern becomes
obvious.

Discuss the number of triangles present in each iteration see if any of your students can
recognize the pattern.

Have the students discuss what they believe will happen to the area of Sierpinki's Triangle as
the number of iterations go beyond the computers computational capability. Will the area of
the triangle ever reach zero?

Guided Practice

Have the students repeat the previous exercise with
Sierpinski Carpet .

If you choose to hand out the worksheets that accompany these applets you can have the
students work on them.

An alternative is to have the students calculate the area Sierpinski's carpet and triangle at
several different iterations.

Closure

You may wish to bring the class back together for a discussion of the findings. Once the
students have been allowed to share what they found, summarize the results of the lesson.

Leave out the concept discussion and focus on pattern recognition and fractions.

Suggested Follow-Up

After these discussions and activities, the students will have seen a few of the classic plane
figure fractals to compare with those from the
Infinity, Self-Similarity, and Recursion lesson. The next lesson,
Fractals and the Chaos Game, continues the student's exploration of fractals by showing how other, seemingly different, ideas
can generate the same kinds of fractals.