Today I introduced a new Math-Art integration project to the 8th grade classes at Evergreen. I showed the students a really cool video that illustrates the depth of mysterious information that can be experienced by looking at and learning about fractals.

After watching the video, I gave the students a bit of background information about Sierpinski who was a Polish Mathematician whom in 1915 first explained, in a mathematical language, the Sierpinski Triangle, the Sierpinski Carpet and more.

I explained as well as I could what a fractal is. This is what Google says:

I explained as well as I could what a fractal is. This is what Google says:

I also explained and demonstrated to the students how the Sierpinski triangle is created.

You begin with an equilateral triangle (all sides are the same length, all angles are the same measurement) pointing upward and you measure each side, making a mark at the center. Then you draw a line connecting each dot. When you do this, it creates four new equilateral triangles all of the same size within the original triangle. Three of these triangles are similar to the original in that they are all pointing upward, the forth is the opposite, the point faces downward. Next, in each upward facing triangle, you do the same thing: measure each side and mark the center, then connect the marks with a line. Each time that you do this step, it is called an Iteration.

Next, you do the same thing, but just within the newly created upward facing triangles. In theory, you can do this infinity times.

You begin with an equilateral triangle (all sides are the same length, all angles are the same measurement) pointing upward and you measure each side, making a mark at the center. Then you draw a line connecting each dot. When you do this, it creates four new equilateral triangles all of the same size within the original triangle. Three of these triangles are similar to the original in that they are all pointing upward, the forth is the opposite, the point faces downward. Next, in each upward facing triangle, you do the same thing: measure each side and mark the center, then connect the marks with a line. Each time that you do this step, it is called an Iteration.

Next, you do the same thing, but just within the newly created upward facing triangles. In theory, you can do this infinity times.

The result is a really cool looking triangle design.

I gave the students all a handout with just the outline of an equilateral triangle. I told them that each side was 32 units. Their assignment was to do at least 5 iterations to the original triangle. At each iteration, they must fill in the new downward facing triangle with a new color with colored pencil and they must find the perimeter of the upward facing non-colored in triangles.

This is sort of difficult to explain in words (especially because Art is my specialty much more than math) but they had to set up a chart as such:

1 32 * 3 = 96 (Each side of original is 32 units)

2 (16*3) 3 = 144

3

4

5

I gave the students all a handout with just the outline of an equilateral triangle. I told them that each side was 32 units. Their assignment was to do at least 5 iterations to the original triangle. At each iteration, they must fill in the new downward facing triangle with a new color with colored pencil and they must find the perimeter of the upward facing non-colored in triangles.

This is sort of difficult to explain in words (especially because Art is my specialty much more than math) but they had to set up a chart as such:

__Iteration #____Perimeter of (upward pointing) Triangles__1 32 * 3 = 96 (Each side of original is 32 units)

2 (16*3) 3 = 144

3

4

5

It was a really good experience for me to teach this lesson and to see how the students reacted. I was also worried about the difficulty level because I haven't much experience with 8th grade math, but my cooperating teacher said that it went well. I look forward to seeing how they turn out! I'll post some pictures when they get them completed!

A close up of my example with the beginning of the 6th iteration:

**Overview:**

8th grade

Fractal study and creation of fractal patterns.

Integration of Math and Art.

**Learning Objectives:**

The learner will be able to successfully use math tools and concepts to create fractal designs.

The learner will incorporate Elements and Principles of Art into their fractal designs.

The learner will recognize interdisciplinary concepts of education.

**NC Essential Standards:**

8.V.1.2 Apply the Elements of Art and Principles of Design in the planning and creation of personal art.

8.V.2.1 Create art that uses the best solutions to identified problems.

8.CX.2.2 Analyze skills and information needed from visual arts to solve problems in art and other disciplines.

8.CX.2.4 Exemplify the use of visual images from media sources and technological products to communicate in artistic contexts.

**Vocabulary:**

- Fractals-Equilateral triangle, algorithm, infinity, SIerpinski's Triangle

- Elements of Art-line, shape

- Principles of Design-pattern, rhythm