· Activity Name: Area vs. Perimeter: A Lesson in Fractals

· Objectives:

The students will discover that the perimeter of a figure can go to infinity while the area is held constant. They will also work with the Pythagorean theorem and discover the formula for the perimeter of their fractal pattern.

· EALR/Standards:

1.2 understand and apply concepts and procedures from measurement.

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:

Pattern Blocks (Do not use the Squares or Tan Rhombi)

Stamp Pads (one of each color of the blocks per group is best)

Legal size Paper

Colored Pencils

· Teacher Notes

o Prerequisites for the learner:

Pythagorean theorem

Able to find the formula for a geometric sequence (can be walked though this part)

o Teacher hints for the activity:

This activity works on almost any schedule. Allow the students a few extra minutes at the beginning of the lesson to explore the pattern blocks especially if they have not used them in class before.

Working on tables in groups of four is best for this activity. When a student finds a shape that he/she believes satisfies the conditions, the group members should check it before stamping it onto the paper. Each group member should have a design that is different from the other group members.

o Introductory questions:

How is area related to perimeter?

What do you think of when you hear the word fractal?

o Wrap-up questions:

What is the formula for the perimeter given the layer?

What would happen if all the triangles were on the inside?

o Solutions:

Introductory questions: Listen to what the students think, but do not give an answer.

Wrap-up questions: X = 4 x 2^x((Ö5)^x/4^x-1).

The area would be decreasing.

o Assessment suggestions:

Verify that students participated and understand the concepts by collecting their stamp papers. For homework, ask the students to do the square activity again, only this time place all of the triangles on the outside of the square. Redo the calculations to show the changes in perimeter and area.

· The Activity:

Begin by handing out a pile of pattern blocks to each group, and a sheet of legal size paper to each student. (Allow a few minutes of experimenting with the blocks if this is the first time they have been used in the class.)

To start the lesson, have the students place the blocks in order from smallest to largest. The students may ask, “Do you mean area wise or perimeter?” If this comes up you will want to ask them if it matters for these blocks. Once they all agree that the order is triangle, rhombus, trapezoid, hexagon, have them make a table that gives each shape’s area and perimeter. You will want to specify that the green triangles have an area of one and a perimeter of three. Have the students hold the legal paper in the landscape position and then divide it into three columns. This table should be written fairly small in the upper left corner of the legal size paper.

After building the table, ask the students to use the pattern blocks to build a polygon that has an area of 20 and perimeter of 12. Each person in a group should come up with a different design. Have them discuss how each of them could move the pieces of his/her design to get a second design that would also work.

There should be a discussion about what classifies as a design. Can there be holes in the middle? Can the blocks be stacked up? Can two shapes share only a vertex and no sides with any other shape? Can shapes not match perfectly edge to edge? These should all be answered in the negative for this activity.

Next, have them build a design with an area of 20 and perimeter of 14, and a design with area 20 perimeter 16. These designs, after being checked by group members for correctness, should be stamped onto the first and second columns of the paper for use later.

For the last column, ask the students to build a shape with area 20 and the largest perimeter they believe possible. This design should also be stamped onto their papers.

When all students in a group have their three stamped designs, the following questions should be asked:

1) Can you determine the perimeter given the area of a shape or vice versa?

In other words, is there a constant relationship between the perimeter and the area in a shape?

***No

2) Is there a minimum perimeter for an area of 20 using these pattern blocks?

***Yes,

3) How about a maximum?

***Yes,

4) How can you show/prove to a skeptic that you have found a minimum or maximum

perimeter?

***Use only green triangles, and place them either as tightly together as possible or as loosely as possible. This is not a proof, but it does show what is happening.

5) If you were not limited to the pattern blocks, could the perimeter become larger than

your maximum or smaller than your minimum values?

***Yes, see below

When the groups have discussed these questions and come to some conclusions, the whole class should discuss what a shape would look like as its area stayed constant, but its perimeter minimized or maximized.

(Minimized perimeter = circle)

(Maximum perimeter = infinite--Fractal)

To show the maximum perimeter, have the students turn their papers over and draw a square with an area of sixteen square inches (this is approximate, rulers are not necessary, but it is important for them to have a fairly big square with room on all sides for growth). While they are drawing this square, hand out colored pencils to each group.

In order to make the fractal, it is easiest to think of the original square on a coordinate system with the origin in the middle and the four vertices at

(2, 2), (2, -2), (-2, -2), and (-2, 2).

Using a second color, the students should now draw segments joining the following points:

(-2, 2) and (0, 3), (0, 3) and (2, 2)

forming an isosceles triangle on the top of the square.

(2, 2) and (1, 0), (1, 0) and (2, -2)

forming an isosceles triangle inside the right side of the square.

(2, -2) and (0, -3), (0, -3) and (-2, -2)

>

forming an isosceles triangle below the square.

(-2, -2) and (-1, 0), (-1, 0) and (-2, 2)

forming an isosceles triangle inside the left side of the square.

It should be noted that by starting at the upper left corner, you have drawn a triangle on the outside of the original shape on the first side, the same triangle on the inside along the second side, outside for the third, and inside for the fourth. These triangles have a height of one-fourth the original side length. Therefore, our algorithm for this fractal is to take every line segment, starting at (-2, 2), and to draw an isosceles triangle onto that segment with a height of ¼ the original segment, using the segment as the base. The triangles are added alternating outside the shape, then inside the shape.

Using a third color, the students should again use this algorithm to draw a third shape around the square.

With each of these, it should be discussed that the area is staying the same even though the perimeter is growing and a table should be built.

***Area is constant because each triangle that is on the outside has a partner that is on the inside; thus, you have “bumps and bites.”

Continuing to use different colors each time, the students should build four or five layers and complete the table.

Layer	Area	Perimeter (long hand)	Perimeter (decimal expansion)
1 (Black)	16 sq. in	(4 sides)(4-inch ea.)	16
2 (Blue)	16 sq. in	(8 sides)(SQRT(5) ea.)	17.88854
3 (Pink)	16 sq. in	(16 sides)(5/4 ea.)	20
4	16 sq. in	(32 sides)(5*SQRT(5)/16 ea.)	22.36068
5	16 sq. in	(64 sides)(25/64 ea.)	25
6	16 sq. in	(128 sides)(25*SQRT(5)/256 ea.)	27.95085
7	16 sq. in	(256 sides)(125/1024 ea.)	31.25

· Extensions:

Have students explore different fractals such as the Koch snowflake or Sierpinski triangle.