· Activity Name: Fractals: Sierpinski Gaskets

· Objectives:

This activity focuses on the students’ ability to explain what is happening when a recursion takes place. It asks the students to do calculations with perimeter and area, and to determine the formula for a geometric sequence. Students will discover that it is possible for a shape to have its perimeter going to infinity while its area is going to zero.

· EALR/Standards:

1.3 understand and apply concepts and procedures from geometric sense.

1.5 understand and apply concepts and procedures from algebraic sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

· Materials:

Sierpinski generating stamp – one per group

Stamp pad – one per group

Large sheet of paper (butcher paper at least 3 feet long) – one per group

Data collection sheet – one per group or student

· Teacher Notes

o Prerequisites for the learner:

Able to work with geometric sequences.

o Teacher hints for the activity:

This activity works on almost any schedule. Working in groups of four is best for this activity. Tables or hard floors are needed for the paper to sit on while the students are stamping. Some groups may need rulers if they want to line up their stamps perfectly.

o Introductory questions:

What do you know about fractals?

Do you think a shape can have an infinite perimeter and an area of zero?

o Wrap-up questions:

What will happen if you change the base shape to a square?

o Solutions:

Introductory questions: Many students will have heard about fractals, but

they may not know many details.

Some will think it is possible, but most will say no.

Wrap-up questions: To do the same thing with a square, the square must be split into 9

little squares, and then the four corners and center are colored black, OR the

four side squares can be colored black. The answers will very according to how it is colored.

o Assessment suggestions:

Collect the worksheets to verify participation and understanding.

· The Activity:

Begin by handing out the worksheets stamps, stamp pads, and large pieces of paper. Explain that the object for the day is to create a level 4 Sierpinski gasket, and to recognize the patterns in the gasket. The rule for recursion is: Take an equilateral triangle, connect the midpoints so that there are now four equilateral triangles inside the original, and then remove the center triangle. This triangle will be the only one that is “upside down.” If the original triangle has an area of 1, and a perimeter of 3, the next level will have a perimeter of 9/2 (nine triangle edges, each ½ the length of the original), and an area of ¾ (3 of the small triangles are still counted and one has been removed).

Because these gaskets are created with stamps, and therefore the middle triangles cannot be erased, the gaskets will grow in size each layer. It will therefore be necessary to scale the gaskets down to do the mathematics.

Using the stamp pads, stamps and paper, have each group create gaskets of levels 1, 2, 3 and 4. (Level zero is simply one triangle.)

When the stamping is finished, have students work through the worksheet in their groups, and then as a class.

· Extensions:

Create square gaskets and have the students repeat the process with the squares.

Sierpinski Gaskets

You are to create a separate stamping of each of the first four generations of a Sierpinski gasket. Generation zero and one are below, but you are to repeat generation 1 and then create 2, 3, and 4.

When done stamping, gather the necessary data for the chart below.

Generation 0 Generation 1

Generation 0 1 2 3 4 n

Segment

Length 1

Perimeter 3

Number of

colored triangles 1

Density *** 1/1

*** Density is equal to the number of colored triangles divided by the total number of triangles.

For the chart below, assume that the gaskets were shrinking each generation.

Perimeter 3

Area 1 or Ö3/4

Conclusions: _________________________________________________________________________

Here are the perimeter tracings of the 0 and 1 generations. Try to draw a second-generation gasket and a third generation gasket.

Generation 0 Generation 1

Generation 2

Generation 3

Sierpinski Gaskets

You are to create a separate stamping of each of the first four generations of a Sierpinski gasket. Generation zero and one are below, but you are to repeat generation 1 and then create 2, 3, and 4.

When done stamping, gather the necessary data for the chart below.

Generation 0 Generation 1